# Regular singularities

*Written by GPT-6.1 Sol (OpenAI), October 2026. Public domain (CC0).*

A meromorphic change of basis can alter the poles in a connection matrix. Regularity therefore concerns the connection, rather than the appearance of one matrix. A useful basis has only a logarithmic pole; analytically, its horizontal sections grow at most like a power. On an algebraic variety the test includes the points at infinity.

We use left D-modules and the conventions of The de Rham functor. In particular, if $\partial_t e=a(t)e$, a horizontal section $g(t)e$ satisfies $g'+ag=0$. A scalar solution of the defining operator $\partial_t-a(t)$ satisfies $u'=au$ instead. These two functions have opposite exponential signs.

The formal lattice argument works over a characteristic-zero field. Statements involving monodromy, growth or analytic spaces use $\mathbb C$. The formal classification below also uses $\mathbb C$ explicitly. We assume finite rank throughout. We use the existence of holomorphic fundamental matrices on simply connected nonsingular domains from D-modules, flat connections and local systems.

## 1. A lattice that controls all iterated derivatives

Put $R=k[[t]]$, $K=k((t))$. A connection on the punctured formal disc is a finite-dimensional $K$-vector space $V$ with a $k$-linear map $\nabla_{\partial_t}$ satisfying the Leibniz rule. Write
\[
\Theta=t\nabla_{\partial_t},\qquad
\Theta(fv)=t f'v+f\Theta(v).                                      \tag{1.1}
\]
A lattice is a free $R$-submodule $L\subset V$ of rank $\dim_K V$ with $K L=V$.

Here is a definition that makes the lattice criterion substantive. Call the connection **regular singular** if, for some lattice $L_0$, all its iterated logarithmic derivatives remain within a fixed pole bound:
\[
\sum_{m\geq0}R\Theta^m L_0\subset t^{-N}L_0
\quad\text{for some integer }N\geq0.                             \tag{1.2}
\]
This is a boundedness condition, not a bound on the ordinary derivatives $\nabla_{\partial_t}^m$.

**Lemma 1.0 (lattice algebra).** In either $R=k[[t]]$ or $R=\mathbb C\{t\}$, every submodule of a finite free module is finite free. A finite free $R$-submodule of $K^r$ whose $K$-span is $K^r$ is a lattice. Any two lattices in the same $K$-vector space contain one another after multiplication by sufficiently large powers of $t$.

**Proof.** Every nonzero series is $t^m u$ with $u$ a unit: the inverse of a series with nonzero constant term is obtained recursively in the formal case, and by a convergent geometric series in the analytic case. In a nonzero ideal choose an element $a$ of least valuation. Every other element divided by $a$ belongs to $R$, so the ideal is $Ra$.

For a submodule $N\subset R^r$, project to the first coordinate. If the image is $Ra\ne0$, choose $w\in N$ projecting to $a$. Subtracting a unique multiple of $w$ from any element gives
$N=Rw\oplus\ker(N\to R)$, and the kernel is a submodule of $R^{r-1}$. If the image is zero, $N$ itself lies in $R^{r-1}$. Induction on $r$ proves the assertion, including finite generation. Its free rank is the dimension of its $K$-span, since the resulting basis stays independent after allowing denominators. Finally, express a basis of one lattice in a basis of the other. There are finitely many coefficients in $K$, and their valuations have a common lower bound. Clearing their poles gives one inclusion; exchanging the bases gives the other. $\square$


**Theorem 1.1 (lattice criterion).** Condition (1.2) holds if and only if $V$ has a lattice $L$ with $\Theta L\subset L$. It then holds for every initial lattice $L_0$.

**Proof.** The sum $H=\sum_{m\geq0}R\Theta^m L_0$ contains $L_0$. Under (1.2) it is an $R$-submodule of the finite free module $t^{-N}L_0$, so it is finitely generated. It is torsion free over the discrete valuation ring $R$, hence free; its rank is $\dim_K V$. Formula (1.1) shows that $\Theta H\subset H$, because $tf'\in R$ for $f\in R$.

Conversely, suppose $L$ is stable. Every lattice $L_0$ is contained in $t^{-a}L$ for some $a\geq0$. Formula
\[
\Theta(t^{-a}v)=t^{-a}(\Theta-a)v                               \tag{1.3}
\]
makes $t^{-a}L$ stable too. Thus the entire hull of $L_0$ stays there. Commensurability gives $t^{-a}L\subset t^{-N}L_0$ for a sufficiently large $N$. This also proves independence of $L_0$. $\square$

In a basis of a stable lattice, write the basis as a row $e$ and
\[
\Theta(e v)=e\bigl(t v'+A(t)v\bigr),\qquad A(t)\in M_r(R).        \tag{1.4}
\]
The connection matrix for $\nabla_{\partial_t}$ is $A(t)/t$. Thus the theorem gives precisely a logarithmic basis. For convergent meromorphic germs the same proof uses $R=\mathbb C\{t\}$ and $K=R[t^{-1}]$, also a discrete valuation ring.

For rank one every lattice is $t^mR e$. If $\nabla_{\partial_t}e=a(t)e$, its logarithmic coefficient in this lattice is $t a(t)+m$. Consequently
\[
V\text{ is regular singular}\quad\Longleftrightarrow\quad
t a(t)\in R.                                                   \tag{1.5}
\]
An integral shift of the residue cannot remove a term $t^{-q}$ with $q\geq2$. Regular connections have regular subconnections and quotients: intersect a stable lattice with the subconnection, or take its image in the quotient. Each is a finite torsion-free lattice, spans the required vector space and remains $\Theta$-stable.

## 2. Constant models and the formal classification

From now through the analytic discussion, $k=\mathbb C$. Choose the strip
\[
\Sigma=\{\alpha\in\mathbb C:0\leq\operatorname{Re}\alpha<1\}.     \tag{2.1}
\]
It supplies one representative of each class in $\mathbb C/\mathbb Z$.

**Lemma 2.1 (normalizing the residue).** Every regular singular formal connection has a stable lattice whose residue eigenvalues lie in $\Sigma$.

**Proof.** Start with a stable lattice and its residue $A(0)$ on $L/tL$. Split this finite-dimensional space into a chosen generalized eigenspace $W$ and the sum $C$ of the others. Lift bases of $W,C$ to $L$. In the resulting block matrix the off-diagonal blocks are divisible by $t$.

Replace the lifts of $W$ by $t$ times those lifts, leaving $C$ unchanged. The new lattice is stable: one off-diagonal block is multiplied by $t$, the other divided by $t$, and the latter remains holomorphic. Its residue is block triangular, with the eigenvalues on $W$ increased by $1$ and those on $C$ unchanged. Replacing the lifts by $t^{-1}$ times themselves gives the analogous downward shift. These are modifications of a whole generalized eigenspace, not a claim that a meromorphic invertible matrix is a scalar power times a holomorphic invertible matrix.

Move each eigenvalue by integral steps toward its representative in $\Sigma$. The sum, with multiplicities, of the absolute integral distances to those representatives decreases at each step. If eigenvalues collide, their representatives coincide, so the new generalized eigenspace can be moved together. The process terminates. $\square$

**Lemma 2.2 (removing the higher terms).** For a lattice as in Lemma 2.1, the connection is formally isomorphic to one with constant matrix $A_0=A(0)$. If the initial matrix is convergent, the isomorphism is convergent meromorphic.

**Proof.** Seek $G(t)=I+\sum_{n\geq1}G_n t^n$ so that the new basis $eG$ has logarithmic matrix $A_0$. This asks for
\[
tG'+A(t)G=GA_0,\qquad
(nI+\operatorname{ad}A_0)G_n
 =-\sum_{i=1}^n A_iG_{n-i},                                    \tag{2.2}
\]
where $\operatorname{ad}A_0(H)=A_0H-HA_0$.
The eigenvalues of this operator are $n+\alpha-\beta$. No difference of two members of $\Sigma$ is a nonzero integer, so they are nonzero for every $n\geq1$. This proves existence and uniqueness recursively, and $G(0)=I$ makes $G$ formally invertible.

For convergence choose a submultiplicative matrix norm. There are constants $C,B>0$ and a radius $\rho>0$ such that
\[
\|(nI+\operatorname{ad}A_0)^{-1}\|\leq C/n,\qquad
\|A_i\|\leq B\rho^{-i}.                                        \tag{2.3}
\]
The first bound follows for large $n$ by a Neumann series and for the remaining finitely many $n$ by increasing $C$. If $g_0\geq\|I\|$ and
$n g_n=CB\sum_{i=1}^n\rho^{-i}g_{n-i}$, then induction gives $\|G_n\|\leq g_n$. The generating series is
$g_0(1-t/\rho)^{-CB}$, by differentiating it. Hence $G$ converges near zero; its determinant is nonzero there. Lemma 2.1 used only finitely many meromorphic basis changes, so the whole change is convergent meromorphic. $\square$

**Theorem 2.3 (formal regular monodromy).** The category of regular singular connections over $\mathbb C((t))$ is equivalent to the category of finite-dimensional complex vector spaces with an invertible automorphism. For a normalized constant model with matrix $A$ the automorphism is
\[
T=\exp(-2\pi i A).                                             \tag{2.4}
\]
Isomorphism classes correspond to conjugacy classes of such $T$.

**Proof.** The lemmas put every object in constant form with spectrum in $\Sigma$. A morphism from the constant model $A$ to $B$ is a meromorphic matrix $H=\sum_{n\geq N}H_n t^n$ satisfying
\[
tH'+BH-HA=0,\qquad (nI+B(\,\cdot\,)-(\,\cdot\,)A)H_n=0.        \tag{2.5}
\]
For $n\neq0$ the operator has eigenvalues $n+\beta-\alpha$, all nonzero. Therefore $H=H_0$ and $BH_0=H_0A$. Thus normalized constant models have exactly the constant intertwining maps.

Given an invertible $T$, decompose its vector space into generalized eigenspaces for eigenvalues $\tau$. Choose the unique $\alpha\in\Sigma$ with $\tau=\exp(-2\pi i\alpha)$. On that block put $U=\tau^{-1}T-I$, a nilpotent operator, and set
\[
A=\alpha I-\frac1{2\pi i}\log(I+U),\qquad
\log(I+U)=\sum_{j\geq1}\frac{(-1)^{j+1}}{j}U^j.                \tag{2.6}
\]
This is a finite sum; its exponential is $T$. A map intertwining two $T$'s preserves the corresponding generalized eigenspaces and intertwines these polynomial logarithms. It therefore intertwines the corresponding $A$'s. Conversely an $A$-intertwiner intertwines its exponential. This proves essential surjectivity and full faithfulness. $\square$

The word monodromy here is a name for the automorphism obtained by the constant model. The formal disc has no analytic loop. Its constant model does have an analytic realization: a horizontal fundamental matrix is $t^{-A}=\exp(-A\log t)$, with actual monodromy (2.4). Formal gauges need not converge for a general formal input.

For example, $A=\alpha I+N$, $N^2=0$, gives
\[
t^{-A}=t^{-\alpha}(I-N\log t),\qquad
T=e^{-2\pi i\alpha}(I-2\pi iN).                                \tag{2.7}
\]
Logarithmic growth and nontrivial unipotent monodromy belong to regular singularities.

## 3. Moderate growth and Fuchs's criterion

A multivalued horizontal section has **moderate growth** if its coefficients in a meromorphic frame satisfy a bound $C|t|^{-N}$ on every closed angular interval of a sufficiently small sector on the universal cover of the punctured disc. The constants may depend on the interval and the section. Meromorphic changes of frame preserve this property, because their entries have finite pole order.

**Theorem 3.1.** A convergent meromorphic connection is regular singular if and only if all its horizontal sections have moderate growth.

**Proof.** In a logarithmic frame, horizontal columns solve $tv'=-A(t)v$. On a fixed radial ray, $\|dv/dr\|\leq C\|v\|/r$ for a uniform bound on $A$. The integral form of this inequality, or Grönwall's inequality with variable $-\log r$, gives
$\|v(r)\|\leq\|v(r_0)\|(r_0/r)^C$. On a compact angular interval the initial values at $r_0$ are bounded, proving moderate growth.

Conversely let $S$ be a horizontal fundamental matrix, with continuation $S\mapsto ST$. Choose a logarithm $A$ with $e^{-2\pi iA}=T$, as in (2.6). The matrix
\[
G(t)=S(t)t^A                                                   \tag{3.1}
\]
is single valued: $T$ commutes with $t^A$, and the two continuation factors cancel. Entries of $t^A$ are powers of $t$ times polynomials in $\log t$, hence moderate. By hypothesis the entries of $S$ are moderate as well. A finite collection of angular intervals covers one turn, giving a uniform polynomial bound for each entry of $G$. A single-valued holomorphic function with this bound becomes bounded after multiplication by a sufficiently large power of $t$, so the removable-singularity theorem makes it meromorphic at zero.

The determinant of $G$ is a nonzero meromorphic germ. The adjugate formula makes $G^{-1}$ meromorphic too. If the original logarithmic matrix was $B(t)$, horizontality of $S$ gives $tG'+BG=GA$. The basis change $G$ therefore produces the constant logarithmic matrix $A$. Its lattice is stable. $\square$

**Corollary 3.2 (completion does not change regularity).** A convergent meromorphic connection is regular singular if and only if its formal completion is regular singular.

**Proof.** Completing a convergent stable lattice gives a formal one. Conversely write the convergent logarithmic matrix in an arbitrary meromorphic frame as $B$, allowing poles. A formal stable lattice has basis matrix $g\in\operatorname{GL}_r(\mathbb C((t)))$ with
$tg'+Bg=gA$, where $A$ is a formal power-series matrix. Choose nonnegative integers $b,c$ so every entry of $B$ has valuation at least $-b$ and every entry of $g^{-1}$ at least $-c$. Truncate $g$ to a Laurent polynomial matrix $G$ with $\Delta=g-G\in t^M M_r(\mathbb C[[t]])$, where $M>b+c$.

Then $g^{-1}\Delta$ has strictly positive valuation, so $G=g(I-g^{-1}\Delta)$ is formally invertible and $G^{-1}$ has entries of valuation at least $-c$. Its convergent determinant is nonzero, hence $G^{-1}$ is convergent meromorphic. The new logarithmic matrix is
\[
G^{-1}(tG'+BG)
=A+G^{-1}\Delta A-G^{-1}(t\Delta'+B\Delta).                    \tag{3.2}
\]
The error terms have valuation at least $M-b-c>0$. Thus the left side, a convergent meromorphic matrix, has no negative Laurent coefficients and is holomorphic. It gives the required convergent lattice. $\square$

### 3.1. Fuchs's criterion in every order

**Theorem 3.3 (Fuchs's criterion).** Let
\[
P=\partial_t^n+a_1(t)\partial_t^{n-1}+\cdots+a_n(t),             \tag{3.3}
\]
with convergent meromorphic coefficients. Its connection is regular singular at zero if and only if
\[
t^j a_j(t)\in\mathbb C\{t\}\quad(1\leq j\leq n).                \tag{3.4}
\]
The same assertion holds for formal coefficients over any characteristic-zero field $k$, with $\mathbb C\{t\}$ in (3.4) replaced by $k[[t]]$. The proof below treats both rings. In particular, neither an analytic solution basis nor a restriction on exponents is needed for the formal statement.

**Proof.** Write $R=\mathbb C\{t\}$ or $k[[t]]$, and let $K$ be its fraction field. Put $D=\nabla_{\partial_t}$ and $\Theta=tD$. The cyclic connection defined by $P$ has generator $v$ and $K$-basis $v,Dv,\ldots,D^{n-1}v$. To see this directly, divide operators by the monic $P$ on the left: subtracting the appropriate left multiple cancels the highest derivative and decreases the order. A nonzero left multiple of $P$ has order at least $n$, since its highest coefficient cannot cancel. Thus the remainder of order below $n$ is unique.

Set $F_0(z)=1$ and $F_m(z)=z(z-1)\cdots(z-m+1)$. The Leibniz rule gives
\[
(\Theta-m)t^mD^m=t^{m+1}D^{m+1},\qquad
t^mD^m=F_m(\Theta).                                           \tag{3.4a}
\]
The second identity follows by induction from the first. Consequently $v,\Theta v,\ldots,\Theta^{n-1}v$ are a $K$-basis too: their transition from the ordinary derivatives is triangular with diagonal $1,t,\ldots,t^{n-1}$. With $\alpha_i=t^i a_i$, multiplication of the defining relation by $t^n$ gives
\[
t^nP=F_n(\Theta)+\sum_{i=1}^n\alpha_i F_{n-i}(\Theta),\qquad
(t^nP)v=0.                                                    \tag{3.4b}
\]
All coefficients here are placed on the left. If every $\alpha_i$ belongs to $R$, this monic relation expresses $\Theta^n v$ in the $R$-span of the preceding powers. The Leibniz rule and $tf'\in R$ then make
$Rv+R\Theta v+\cdots+R\Theta^{n-1}v$ a stable lattice. Theorem 1.1 proves sufficiency.

Conversely, let the connection be regular singular. Theorem 1.1 supplies a stable lattice $L$. Choose $q\geq0$ with $v\in t^{-q}L$; formula (1.3) shows that this enlarged lattice remains stable. Consider the algebraic span
\[
H=\sum_{m\geq0}R\Theta^m v\ \subset\ t^{-q}L.                 \tag{3.4c}
\]
Every element of the sum has finitely many terms. Lemma 1.0 makes $H$ finite free, even though its displayed generating family is infinite. It contains a $K$-basis of the connection, so its rank is $n$. The formula
$\Theta(f\Theta^m v)=tf'\Theta^m v+f\Theta^{m+1}v$ proves that it is stable. Also $\Theta(th)=t(\Theta h+h)$, and $tf'\in tR$. Therefore $\Theta$ induces a $k$-linear operator $\overline\Theta$ on $H/tH$, whose dimension is $n$; in the analytic case the residue field is $\mathbb C$.

The image $\overline v$ generates $H/tH$ under $\overline\Theta$. Its first $n$ iterates form a basis. Indeed, if the first dependence occurred at degree $d<n$, it would express the $d$-th iterate in the preceding ones and make their span invariant. All later iterates would remain in that space of dimension at most $d$, contradicting cyclic generation of the $n$-dimensional quotient. This also rules out $\overline v=0$.

In any $R$-basis of $H$, the coordinate matrix of $v,\Theta v,\ldots,\Theta^{n-1}v$ thus has invertible reduction modulo $t$. Its determinant is a unit, so the adjugate formula gives an inverse over $R$. These vectors are an $R$-basis of $H$ itself. In particular there are $b_i\in R$ such that
\[
Qv=0,\qquad
Q=\Theta^n+b_1\Theta^{n-1}+\cdots+b_n.                        \tag{3.4d}
\]
Both $Q$ and the expression (3.4b) are monic of degree $n$ in $\Theta$ and annihilate $v$. Their difference has degree below $n$; independence of the first $n$ iterates over $K$ makes every coefficient of that difference zero.

It follows that $Q-F_n(\Theta)=\sum_i\alpha_iF_{n-i}(\Theta)$ has coefficients in $R$. Its coefficient of $\Theta^{n-1}$ is $\alpha_1$, so $\alpha_1\in R$. After subtracting $\alpha_1F_{n-1}(\Theta)$, its coefficient of $\Theta^{n-2}$ is $\alpha_2$ when $n\geq2$. Repeating the subtraction and coefficient comparison proves $\alpha_i\in R$ for every $i$, including zero coefficients. This uses monic polynomials with integer coefficients and left coefficient placement, so it introduces no derivatives of the $\alpha_i$. These are exactly the bounds (3.4). $\square$

For convergent meromorphic coefficients, formal regularity is therefore equivalent to the same bounds: a convergent Laurent series with no negative coefficients is holomorphic. The stable lattice in the sufficiency proof is then convergent. This gives the completion assertion for scalar equations in every order, consistent with Corollary 3.2. Schnell's free lecture notes, Lecture 20, are supplementary reading for Fuchs's theorem.

### 3.2. Low-order solution calculations

The preceding proof covers every order. The following analytic and formal computations give additional descriptions in orders one and two.

For sufficiency, multiply $P$ by $t^n$. The identities
$t^m\partial_t^m=\Theta(\Theta-1)\cdots(\Theta-m+1)$ rewrite it as a monic polynomial in $\Theta=t\partial_t$, with holomorphic coefficients when (3.4) holds. In its rank-$n$ cyclic differential module, the lattice generated by
$v,\Theta v,\ldots,\Theta^{n-1}v$ is therefore stable. These vectors are a $K$-basis, since the transition from ordinary derivatives is triangular with nonzero diagonal $1,t,\ldots,t^{n-1}$.

For necessity in order one, take a nonzero solution $u$ of $u'+a_1u=0$. It is moderate, by Theorem 3.1 applied to the dual connection, which is regular whenever the original one is: a stable lattice has a stable dual lattice. Its scalar monodromy is some $\tau\neq0$. Choose $\sigma$ with $e^{2\pi i\sigma}=\tau$. Then $u=t^\sigma h$ with single-valued, moderate $h$, hence $h$ is a meromorphic germ. Write $h=t^m h_0$, $h_0$ holomorphic and invertible. It follows that
\[
\frac{u'}u=\frac{\sigma+m}{t}+\frac{h_0'}{h_0}.                 \tag{3.5}
\]
Since $a_1=-u'/u$, its pole is at most simple.

For necessity in order two, choose independent solutions $u,v$ and form
$W=uv'-u'v$. The fundamental companion matrix is invertible, so $W\neq0$. Derivatives of moderate holomorphic functions are moderate on smaller sectors: Cauchy's formula on a circle of radius $c|t|$ loses at most one further power of $|t|$. Thus $W$ is moderate. It has scalar monodromy $\det T_{\rm sol}$, and the same argument as (3.5) shows that $W'/W$ has at most a simple pole. Direct differentiation gives Abel's identity
\[
W'=-a_1W,                                                     \tag{3.6}
\]
so $ta_1$ is holomorphic.

The two-dimensional scalar-solution monodromy has an eigenvector. The corresponding nonzero solution $w$ has scalar monodromy, and hence $w=t^\sigma t^m h_0$ as above. Therefore $w'/w$ has pole order at most one and $w''/w$ at most two. The equation gives
\[
a_2=-\frac{w''}{w}-a_1\frac{w'}w,                              \tag{3.7}
\]
which has pole order at most two. This proves necessity. In order two sufficiency can also be seen directly from
\[
\Theta^2u+(ta_1-1)\Theta u+t^2a_2u=0.                         \tag{3.8}
\]
The scalar system on $(u,\Theta u)^t$ has holomorphic logarithmic matrix
$\left(\begin{smallmatrix}0&1\\-t^2a_2&1-ta_1\end{smallmatrix}\right)$.

The same order-one and order-two criterion holds for formal coefficients over $\mathbb C$. Here is the formal replacement for the growth step, to avoid appealing to analytic functions that may not exist. Theorem 2.3 puts a regular connection and its dual in constant form. Its scalar solutions can consequently be written in a formal differential extension as finite sums of $t^\alpha$ times polynomials in $\ell=\log t$ with coefficients in $\mathbb C((t))$. Differentiate by $\partial_t(t^\alpha)=\alpha t^{\alpha-1}$ and $\partial_t\ell=t^{-1}$. Formal continuation sends $t^\alpha$ to $e^{2\pi i\alpha}t^\alpha$ and $\ell$ to $\ell+2\pi i$.

A scalar eigenvector for continuation has exactly one exponent class modulo integers and no logarithmic powers: a polynomial invariant under a nonzero translation is constant, and distinct exponent classes have distinct continuation eigenvalues. Such a nonzero solution is $t^\sigma h$, $h\in\mathbb C((t))^\times$. For a fundamental order-two companion matrix the determinant $W$ is nonzero and is an eigenvector with eigenvalue the determinant of continuation. Formulas (3.5)–(3.7) apply as formal Laurent identities; $h=t^m h_0$ with $h_0\in\mathbb C[[t]]^\times$. This proves the two necessity bounds formally. Sufficiency already used a formal lattice. In particular, for convergent scalar coefficients of order at most two, formal regularity implies the convergent logarithmic lattice constructed in the sufficiency proof.

## 4. Rank one on the multiplicative line

Let $x$ be the coordinate on $\mathbb G_m$. Its coordinate ring $\mathbb C[x,x^{-1}]$ is a principal ideal domain, so an algebraic line bundle is trivial. Write a connection as
\[
\partial_x e=a(x)e,\qquad a(x)\in\mathbb C[x,x^{-1}].            \tag{4.1}
\]
At zero, (1.5) permits only powers $x^j$ with $j\geq-1$. At infinity put $y=x^{-1}$. Since $\partial_x=-y^2\partial_y$, the coefficient becomes
\[
\partial_y e=-y^{-2}a(y^{-1})e.                                \tag{4.2}
\]
Regularity there permits only $j\leq-1$. Both conditions hold exactly when $a(x)=\lambda/x$.

**Theorem 4.1.** The rank-one algebraic connections on $\mathbb G_m$ regular at both ends are
\[
E_\lambda=\mathcal O_{\mathbb G_m}e_\lambda,\qquad
\partial_xe_\lambda=\frac{\lambda}{x}e_\lambda.                 \tag{4.3}
\]
Two are isomorphic exactly when $\lambda-\mu\in\mathbb Z$.

**Proof.** The classification of coefficients was just proved. An isomorphism $e_\lambda\mapsto g e_\mu$ uses a unit $g=cx^m$ of the Laurent polynomial ring. Connection compatibility says $xg'/g=\lambda-\mu$. Thus $\lambda-\mu=m$; conversely this formula constructs the isomorphism. $\square$

The notation $\mathcal O.x^\lambda$ denotes $E_\lambda$, even when $x^\lambda$ is not an algebraic function. Its horizontal coefficients are $x^{-\lambda}$, so its de Rham monodromy is $e^{-2\pi i\lambda}$. On the punctured line one may write the cyclic presentation $\mathcal D/\mathcal D(x\partial_x-\lambda)$. At the origin, different coherent extensions can share that restriction; Preservation of holonomicity and minimal extensions distinguishes them.

Here are the assigned pole calculations, with connection rather than scalar-solution signs:

| Operator on the punctured line | Coefficient $a(x)$ | Coefficient at infinity | At zero | At infinity |
| --- | --- | --- | --- | --- |
| $x^2\partial_x-x$ | $x^{-1}$ | $-y^{-1}$ | regular | regular |
| $x^3\partial_x-1$ | $x^{-3}$ | $-y$ | irregular | regular |
| $\partial_x-1$ | $1$ | $-y^{-2}$ | nonsingular | irregular |

For the second example a horizontal coefficient is $\exp(1/(2x^2))$, whereas a scalar solution of the operator is $\exp(-1/(2x^2))$. For the third it is $\exp(-x)=\exp(-1/y)$, growing faster than every pole on some sectors at infinity.

If $\mathcal O.e^{1/x}$ means the connection obtained by formally differentiating the symbol $e^{1/x}$, its coefficient is $-x^{-2}$ and it is irregular at zero. Its horizontal coefficient is $e^{-1/x}$. Changing the sign of the exponential changes the growth sectors but not regularity.

**Hypergeometric example.** For parameters $a,b,c\in\mathbb C$, the equation
\[
x(1-x)u''+[c-(a+b+1)x]u'-ab\,u=0                              \tag{4.4}
\]
has regular singularities among $0,1,\infty$. Indeed its monic coefficients are
$a_1=[c-(a+b+1)x]/[x(1-x)]$ and $a_2=-ab/[x(1-x)]$. At $0$ and $1$ the first has at most a simple pole and the second at most a double pole. At infinity the transformed monic coefficients are
\[
\widetilde a_1(y)=\frac2y-\frac{a_1(1/y)}{y^2},\qquad
\widetilde a_2(y)=\frac{a_2(1/y)}{y^4}.                         \tag{4.5}
\]
The estimates $a_1(x)=O(x^{-1})$, $a_2(x)=O(x^{-2})$ give the same Fuchs bounds there. Degenerate parameters may make a singularity removable; the assertion is an upper bound on singularity type.

## 5. Global regularity and regular holonomic modules

For a connection on a smooth algebraic curve $C$, regularity means the formal or convergent logarithmic test at **every** point of its smooth complete compactification $\overline C$. The points of $\overline C\setminus C$ are part of the definition. A flat bundle on a smooth algebraic $X$ is regular if its pullback to every smooth algebraic curve $C\to X$ is regular in this sense. Testing only points already in $X$ would admit $\mathcal O.e^x$ on $\mathbb A^1$ and would give the wrong category.

### Canonical logarithmic extensions

The flat-connection/local-system equivalence used here is proved in D-modules, flat connections and local systems, Lemma 4.1 and Theorem 4.2. The parameter Cauchy and Taylor facts are proved in Cauchy's theorem for cycles, Lemma 0.1 and Theorems 2.3, 3.2 and 4.2 and Holomorphic functions of several variables, Theorems 1.2, 2.1 and 2.3. The Laurent, growth, finite-dimensional logarithm and category-gluing arguments needed below are proved explicitly.

#### Normalization and the local theorem

Let \(\overline U\) be a complex manifold, \(D\) a reduced divisor locally given by a product of distinct coordinate functions, and \(j:U=\overline U\setminus D\hookrightarrow\overline U\). This includes a simple normal-crossing boundary; no global ordering of its components is needed. Put
\[
\Sigma=\{\alpha\in\mathbb C:0\leq\operatorname{Re}\alpha<1\}.
\tag{5.0a}
\]
We use the connection convention \(\nabla=d+\Gamma\); for a constant residue \(A\), horizontal coefficients are \(z^{-A}\) and positive-loop monodromy is \(\exp(-2\pi iA)\).

A logarithmic lattice is a locally free finite-rank \(\mathcal O_{\overline U}\)-module \(E\) in a meromorphic flat bundle, spanning it after allowing poles along \(D\), with
\[
\nabla E\subset E\otimes\Omega^1_{\overline U}(\log D).
\]
Its residue along a local component \(D_i=(z_i=0)\) is the coefficient of \(dz_i/z_i\) restricted to \(D_i\). Call it normalized if the eigenvalues of every residue lie in \(\Sigma\).

**Theorem 5.0 (canonical logarithmic extension).** Every finite-rank complex local system \(L\) on \(U\) has a normalized logarithmic extension \((E_\Sigma(L),\nabla)\) on \(\overline U\), with restriction identified with \((\mathcal O_U\otimes_\mathbb C L,d\otimes1)\). This extension is unique up to the unique horizontal isomorphism inducing that identification. It is functorial and exact.

Every horizontal map on \(U\) between logarithmic bundles extends uniquely meromorphically across \(D\). Between normalized bundles it extends holomorphically. Consequently restriction identifies the category of regular meromorphic flat bundles on \((\overline U,D)\), defined locally by logarithmic lattices, with finite-rank local systems on \(U\). The local logarithmic criterion is equivalent to regular singularity on every holomorphic disc meeting the boundary and not contained in it; testing transverse coordinate-disc families already suffices.

The construction includes every Jordan block. The normalization functor is generally not tensor compatible. The meromorphic correspondence does preserve the usual tensor constructions.

**Proof.**

#### Topology of the punctured polydisc

An adapted local chart is
\[
V=\Delta^{k}\times\Delta^{m},\quad
D\cap V=(z_1\cdots z_k=0),\quad
V^*=(\Delta^*)^k\times\Delta^m.
\tag{5.0b}
\]
Radial contraction of each punctured disc to a circle and contraction of the other discs to their centers reduce its fundamental group to that of a product of \(k\) circles. More explicitly, the logarithmic covering has domain
\[
\{\operatorname{Re}w_i<\log\rho\}_{1\leq i\leq k}\times\Delta^m,
\qquad z_i=e^{w_i}.
\]
This domain is convex and simply connected, and its deck transformations are \(w_i\mapsto w_i+2\pi in_i\), \(n_i\in\mathbb Z\). A loop lifts with endpoint differing by one such integer tuple; homotopies preserve it. Conversely equal endpoint tuples give homotopic lifts by contraction in the convex domain. Thus \(\pi_1(V^*)=\mathbb Z^k\), with the positive coordinate loops as generators.

A local system is trivial on this cover. To verify the usual continuation assertion, subdivide a path into finitely many intervals lying in trivializing opens and compose their constant transition maps. Refinements give the same map. Subdivide a homotopy square into rectangles lying in trivializing opens; transition cancellations on each rectangle show homotopy invariance. A simply connected cover then identifies every fibre with a fixed one by path continuation. Deck continuation gives a representation of \(\mathbb Z^k\), and conversely its constant deck gluing gives a local system. Maps are precisely fibre maps commuting with the representation.

Hence \(L|_{V^*}\) is specified by a finite vector space \(W\) and commuting invertible maps \(T_1,\ldots,T_k\). Its rank is \(\dim W\), including rank zero.

The holomorphic flat-bundle part has an exact earlier proof, rather than an assumed analytic existence theorem. In lesson 02's proof the iterated integrals for \(\partial_zH=-A(z,w)H\), \(H(0,w)=I\), are bounded by \(C^n|z|^n/n!\); they converge with parameters and uniqueness follows by iterating the same estimate. The determinant satisfies a scalar equation and never vanishes. Killing the first matrix makes the other matrices independent of that coordinate by zero curvature, so induction gives a horizontal frame. Constant transitions then give the equivalence with local systems. We use that proved equivalence on \(V^*\) and on overlaps.

#### Commuting logarithms with all nilpotent parts

For an invertible \(T\) on a finite complex vector space, decompose into its generalized eigenspaces \(W_\tau\), \(\tau\ne0\). The decomposition and its projectors are polynomial in \(T\): the relatively prime factors \((t-\tau)^{r_\tau}\) of an annihilating polynomial admit Bézout projectors by the Euclidean algorithm. Evaluating those projectors at \(T\) gives their direct-sum images and the nilpotent restrictions.

For each \(\tau\), choose the unique \(\alpha_\tau\in\Sigma\) satisfying
\(\exp(-2\pi i\alpha_\tau)=\tau\). The kernel of the scalar exponential here is \(\mathbb Z\): its modulus first forces the imaginary part to be zero, then its argument forces an integer. This proves existence and uniqueness after shifting the real part by an integer.

On \(W_\tau\), put
\[
N_\tau=\tau^{-1}T-I,\qquad
A(T)=\alpha_\tau I-\frac1{2\pi i}
\sum_{a=1}^{r_\tau-1}\frac{(-1)^{a+1}}aN_\tau^a.
\tag{5.0c}
\]
The sum is finite. The formal identity \(\exp(\log(1+t))=1+t\), modulo \(t^{r_\tau}\), proves \(\exp(-2\pi iA(T))=T\). One elementary verification of that identity is to differentiate the formal exponential: it satisfies \(F'=F/(1+t)\), \(F(0)=1\); coefficient recursion gives \(F=1+t\). Truncation is legitimate because \(N_\tau\) is nilpotent. Its residue eigenvalues are the \(\alpha_\tau\).

Using the polynomial projectors, (5.0c) is a polynomial in \(T\) on the whole space. If a map \(C:W\to W'\) intertwines \(T,T'\), it preserves matching generalized eigenspaces and intertwines each finite nilpotent logarithm. Therefore
\[
A(T')C=CA(T).
\tag{5.0d}
\]
This assertion remains true across changes of rank and along exact sequences.

Set \(A_i=A(T_i)\). Since these are polynomials in pairwise commuting \(T_i\),
\[
[A_i,A_j]=0,\qquad \exp(-2\pi iA_i)=T_i.
\tag{5.0e}
\]
No diagonalizability assumption occurs. If \(T_i\) is unipotent, all its \(\alpha\)'s are zero and \(A_i\) is nilpotent. On a simultaneous generalized eigenblock, the horizontal coefficients are scalar powers of the \(z_i\) times polynomials in their logarithms. Each individual nilpotent exponential terminates, and their commuting product is still a finite polynomial in all the logarithms.

#### The local model and intrinsic residues

On \(V\), take
\[
E=\mathcal O_V\otimes_\mathbb CW,\qquad
\nabla=d+\sum_{i=1}^k A_i\,\frac{dz_i}{z_i}.
\tag{5.0f}
\]
Its curvature is zero: the logarithmic forms are closed and (5.0e) kills their matrix commutators. Its residues are the \(A_i\) and have spectrum in \(\Sigma\). A horizontal fundamental matrix on the logarithmic cover is
\[
S(z)=\exp\!\left(-\sum_{i=1}^kA_i\log z_i\right).
\tag{5.0g}
\]
Continuation around the \(i\)-th positive loop sends \(S\) to \(ST_i\). Thus its horizontal local system is the given one. A branch-dependent constant normalization at the chosen base point identifies the fibres; changing that identification conjugates the matrices and gives the same identified flat bundle.

A monodromy intertwiner \(C\) is a constant horizontal map between these local models by (5.0d). Short exact sequences of representations give short exact sequences of the bundles (5.0f), since their underlying bundles are \(\mathcal O_V\otimes W\). Polynomial logarithms restrict and descend to subrepresentations and quotients, so their connections are the induced ones. This proves local functoriality and exactness.

For an arbitrary logarithmic connection in a local frame write
\[
\Gamma=\sum_{i=1}^k B_i(z,w)\frac{dz_i}{z_i}
+\sum_{a=1}^m C_a(z,w)\,dw_a,
\tag{5.0h}
\]
where all matrix coefficients are holomorphic. Zero curvature gives
\[
z_i\partial_iB_j-z_j\partial_jB_i+[B_i,B_j]=0.
\]
Restricting to \(z_i=z_j=0\) shows that its residue endomorphisms commute on every intersection of those components. The residues are intrinsic: replacing \(z_i\) by \(u_i z_i\), with \(u_i\) a holomorphic unit, changes \(d\log z_i\) by the holomorphic form \(d\log u_i\). Frame changes conjugate their restrictions. Thus residue normalization is unchanged by adapted-coordinate choices and by permuting local components.

#### Finite poles from growth and transverse families

**Lemma 5.0.1 (all boundary directions).** A horizontal map \(F:E_1|_{V^*}\to E_2|_{V^*}\) between two logarithmic bundles has, in their local holomorphic frames, a bound
\[
\|F(z,w)\|\leq C\prod_{i=1}^k|z_i|^{-N}
\tag{5.0i}
\]
on a smaller polydisc, uniformly for all arguments of the single-valued \(F\) and for \(w\) in a smaller closed polydisc.

**Proof.** The Hom connection has logarithmic coefficient
\(H\mapsto B_iH-HA_i\), whose norm is uniformly bounded on the closed polydisc. Horizontality on a fixed radial line gives a linear equation with coefficient norm at most \(c_i/r_i\). With \(s=\log(r_0/r_i)\), the integral inequality is \(g(s)\leq g(0)+c_i\int_0^s g(u)\,du\). Iterating the integral inequality gives \(g(s)\leq g(0)e^{c_i s}\): the \(n\)-fold integrals are bounded by \(s^n/n!\), and the remainder tends to zero on each finite interval. Thus varying the \(i\)-th radius multiplies the bound by at most \((r_0/r_i)^{c_i}\).

Start on the compact product torus \(|z_i|=r_0\), with the smaller parameter polydisc. The holomorphic map \(F\) is bounded there. Vary the radii successively, keeping all other coordinates fixed. The coefficient bounds include the simultaneous intersections, so the same estimates hold at every step. Choose one integer \(N\geq\max c_i\) and absorb the fixed \(r_0\)'s into \(C\). This proves (5.0i). No relation between the rates at which different \(z_i\) approach zero was imposed. \(\square\)

**Lemma 5.0.2 (growth means an actual finite pole).** A single-valued holomorphic function on \(V^*\) satisfying (5.0i) belongs to
\(\mathcal O_V[(z_1\cdots z_k)^{-1}]\).

**Proof.** The Laurent formula being used follows directly from the proved Cauchy cycle formula: on a closed annulus, subtract the inner-circle Cauchy integral from the outer-circle integral. For a point strictly between them, expand the outer kernel as \(\sum_{n\ge0}z^n\zeta^{-n-1}\) and the inner kernel as \(-\sum_{n<0}z^n\zeta^{-n-1}\). Both geometric series converge uniformly on compact subannuli; termwise integration gives the Laurent series. Applying the cycle formula to the intervening annulus makes each coefficient independent of its contour radius. The parameter-integration and polydisc Cauchy proofs make the coefficients holomorphic in the other variables and justify iterating in the \(k\) punctured coordinates.

Multiply by \((z_1\cdots z_k)^N\). The result is bounded. Iterated Cauchy integrals on coordinate tori give its Laurent coefficients, holomorphic in the ordinary parameters. For a coefficient with negative exponent in \(z_i\), move that contour radius to zero while holding the others fixed. Its estimate contains a positive power of that radius, so the coefficient is zero. All remaining exponents are nonnegative. On a smaller polydisc their Cauchy bounds give a normally convergent power series, whose restriction is the original Laurent expansion. Hence it is holomorphic across all components and their intersections. This also proves the bounded-removal assertion jointly in all variables. \(\square\)

We will also use the following uniform finite-pole test, which requires no a priori growth estimate.

**Lemma 5.0.3 (transverse-family test).** Let \(f(t,w)\) be holomorphic on \(\Delta^*\times W\), with \(W\) a connected product of punctured and ordinary discs. Suppose that for every \(w\) in some nonempty open subset of \(W\), \(f(\,\cdot\,,w)\) is meromorphic at zero. Then there is one \(N\) such that its Laurent coefficients below degree \(-N\) vanish identically on \(W\).

**Proof.** The Laurent coefficients \(a_q(w)\), computed on one fixed \(t\)-circle, are holomorphic. Let
\[
F_N=\{w:a_q(w)=0\text{ for all }q<-N\}.
\]
These are closed sets. Their union contains the specified open subset. The elementary Baire argument gives one \(F_N\) with interior in it. Here that argument needs only Euclidean completeness: if all the closed sets had empty interior, choose nested closed balls of radii tending to zero, the \(N\)-th avoiding \(F_N\), each in the interior of the preceding ball and in the initial open set. Their centers form a Cauchy sequence; its limiting point lies in every ball and outside every \(F_N\), a contradiction.

Each \(a_q\), \(q<-N\), now vanishes on an open subset, hence on connected \(W\) by the holomorphic identity theorem. That theorem can be verified by Taylor expansion: the locus where a holomorphic function vanishes on a neighbourhood is open, and is closed along connected coordinate domains because all Taylor derivatives vanish at a limit of such points. Thus those coefficients vanish identically. \(\square\)

Apply Lemma 5.0.3 separately to each punctured coordinate of a function on \(V^*\). If its restrictions in every coordinate are meromorphic for a nonempty open parameter family, one obtains bounds \(N_i\) on all negative Laurent exponents. Multiplication by \(\prod z_i^{N_i}\) leaves only nonnegative multi-exponents. Fixed outer-torus Cauchy bounds make that power series converge jointly on a smaller polydisc, including the intersections. This proves meromorphic extension there.

#### Meromorphic maps and normalized uniqueness

Lemma 5.0.1 and Lemma 5.0.2 prove that every horizontal map between logarithmic bundles extends meromorphically. Its connection equation remains true after extension: it is a meromorphic identity already true on the dense open complement of \(D\). Such an extension is unique, since a meromorphic function zero on that open set is zero.

Now suppose both bundles are normalized. If an extended matrix \(F\) had a pole of positive order \(q\) along \(z_i=0\), let \(F_{-q}\) be its nonzero leading Laurent coefficient. Work first away from the other components, where this is an ordinary divisor Laurent expansion with holomorphic parameter coefficients. The horizontal equation is
\[
z_i\partial_iF+B_iF-FA_i=0.
\]
Its leading coefficient gives
\[
(-qI+B_i|_{D_i}(\,\cdot\,)-(\,\cdot\,)A_i|_{D_i})F_{-q}=0.
\tag{5.0j}
\]
The displayed Sylvester operator has eigenvalues \(-q+\beta-\alpha\), where \(\alpha,\beta\in\Sigma\) are residue eigenvalues. None is zero: the real part of \(\beta-\alpha\) lies strictly between \(-1\) and \(1\), whereas \(q\) is a positive integer.

This eigenvalue assertion includes all nilpotent parts. Decompose the source and target into generalized eigenspaces. On the corresponding Hom block the operator is a nonzero scalar plus the difference of left and right nilpotent multiplication. Those nilpotent operators commute, and their sum is nilpotent by the binomial formula. A finite geometric series inverts the operator. Thus (5.0j) forces \(F_{-q}=0\), a contradiction. This eliminates poles along every component.

An entry of \(F\) already has the form \(h/\prod z_i^{N_i}\), with \(h\) holomorphic. Absence of a pole along \(z_i\), away from the other components, makes the first \(N_i\) Taylor coefficients in that coordinate vanish on an open set of the remaining variables. The identity theorem makes them vanish everywhere, so \(z_i^{N_i}\) divides \(h\). Repeating for the different coordinates shows that the full product divides \(h\), by its convergent power-series expansion. Thus \(F\) is holomorphic at all intersections too.

A horizontal isomorphism on \(V^*\) extends together with its inverse, so its normalized extension is a holomorphic isomorphism. In particular any normalized logarithmic bundle with the given local system is uniquely isomorphic to the model (5.0f). This proves local uniqueness and, as a consequence rather than an assumed normal-form theorem, every normalized logarithmic bundle is locally isomorphic to the constant commuting-residue model.

#### The local analytic disc criterion

Let \(M\) be a locally free finite-rank meromorphic bundle over
\(\mathcal O_V[(z_1\cdots z_k)^{-1}]\), with integrable connection and no poles on \(V^*\).

If it has a logarithmic lattice, pull back to any holomorphic disc
\(\gamma:\Delta\to V\) not contained in \(D\). Each boundary coordinate has
\[
z_i\circ\gamma=t^{m_i}u_i(t),\qquad
m_i\geq0,\quad u_i(0)\ne0,
\]
after shrinking. Hence
\[
\gamma^*\frac{dz_i}{z_i}
=m_i\frac{dt}{t}+\frac{du_i}{u_i}.
\tag{5.0k}
\]
The pulled-back lattice is free of the same rank and its connection has only a logarithmic pole. Lesson 13's proved lattice criterion makes it regular singular. Formula (5.0k) includes tangent, ramified and simultaneous approaches to any number of boundary components.

Conversely, it suffices to assume regular singularity along the transverse coordinate-disc slices, for a nonempty open family of parameters for each \(z_i\). The restriction of \(M\) to \(V^*\) has a local system, by the earlier flat-connection theorem. Form its normalized model \(E_\Sigma\) from (5.0f). That same theorem and the monodromy description give a horizontal isomorphism
\[
G:E_\Sigma|_{V^*}\longrightarrow M|_{V^*}.
\tag{5.0l}
\]
Both \(G\) and \(G^{-1}\) are single-valued holomorphic matrices on \(V^*\) in chosen meromorphic frames.

On each regular transverse slice, \(M\) has a convergent logarithmic lattice by the proved one-variable criterion. The model \(E_\Sigma\) is logarithmic there as well. The one-variable case of the meromorphic-map proof therefore makes \(G\) and its inverse meromorphic in that slice. Lemma 5.0.3 gives a uniform finite pole bound in that coordinate; repeat for every boundary coordinate. The joint Laurent argument after that lemma makes \(G,G^{-1}\) meromorphic on \(V\). Transporting \(E_\Sigma\) by (5.0l) now gives a free logarithmic lattice in \(M\).

Thus the following local conditions are equivalent:

1. A logarithmic lattice exists.
2. Every holomorphic-disc pullback not contained in \(D\) is regular singular at its boundary intersections.
3. Regularity holds on nonempty open families of transverse coordinate-disc slices in each boundary direction.

This is the local analytic part of the global curve criterion. The global proof below shows how the lesson's tests on smooth algebraic curves, including their compactification points, supply its local hypothesis. No assumption that analytic and algebraic curve tests are automatically interchangeable appears here.

For the constant model, the residue on a disc approaching several components is \(\sum_i m_iA_i\). These matrices commute, and their nilpotent parts are retained. Its eigenvalues need not lie in \(\Sigma\); regularity survives, but the pulled-back logarithmic lattice need not itself be the normalized lattice.

#### Gluing across every boundary intersection

Cover \(\overline U\) by adapted charts, using ordinary flat frames on charts disjoint from \(D\). On each adapted chart construct (5.0f) with its identified restriction to the given \(L\). On an overlap, those identifications give a horizontal isomorphism between the restrictions on \(U\). Locally near every boundary point of the overlap, both extensions are normalized logarithmic bundles for the same reduced divisor, irrespective of the chosen component order or equations. the meromorphic-map proof extends the isomorphism and its inverse holomorphically.

These local extensions agree on smaller overlaps because they agree on the dense boundary complement and are unique. The cocycle identity on a triple overlap holds there and hence everywhere as a holomorphic identity. Thus the bundles and logarithmic connections glue on \(\overline U\), with rank unchanged. This is ordinary sheaf gluing: identify sections by the transition maps; the cocycle relation makes the equivalence relation consistent, and each chart gives a local free frame.

Maps of local systems extend by the same argument and commute with composition by uniqueness. Exactness can be checked on each adapted chart, where it is the exact \(\mathcal O\)-extension of a finite-dimensional representation sequence from the local model construction. The construction is therefore globally functorial and exact. Any other identified normalized extension is glued to this one by its unique local normalized isomorphisms.

Allowing meromorphic coefficients gives \(E_\Sigma(L)(*D)\). the meromorphic-map proof proves that every horizontal map of the restrictions extends uniquely between regular meromorphic bundles, after choosing local logarithmic lattices. the local analytic disc criterion proves that every locally regular meromorphic bundle is identified with this construction. Hence the regular meromorphic correspondence is fully faithful and essentially surjective; the construction is independent of the local lattices.

Regularity is preserved by a holomorphic map of normal-crossing pairs carrying the source open complement into the target open complement, for which the inverse-image boundary is supported in the source boundary. Locally, a nonzero pulled-back boundary equation is a unit times a product of source boundary coordinates with nonnegative integer multiplicities.

Here is the factor assertion without an extra divisor theorem. If a holomorphic \(f\) has no zero off a coordinate boundary, \(1/f\) is holomorphic on its complement. On every transverse coordinate slice it is meromorphic, since the nonzero holomorphic one-variable \(f\) has a zero of finite order. Lemma 5.0.3 and its joint Laurent consequence give \(1/f=h/\prod w_j^{N_j}\). Thus \(fh=\prod w_j^{N_j}\). Factor from \(f\) and \(h\) their maximal coordinate powers using convergent Taylor expansions. Coordinate valuations are additive: their leading coefficient product is nonzero in the convergent ring in the other variables, which is a domain by the identity theorem. After cancelling those powers the remaining factors multiply to \(1\). Therefore the remaining factor of \(f\) is a unit, proving the assertion.

The logarithmic differential is the corresponding sum of logarithmic differentials plus a holomorphic unit differential, by (5.0k)'s multivariable version. A pulled-back logarithmic lattice remains logarithmic and preserves rank. This proves the local pullback statement, including boundary multiplicities. The global proof below supplies proper algebraization and compactification independence.

#### Tensor products and normalization

Normalized extension is not generally tensor compatible. In rank one in one boundary coordinate, two canonical residues \(3/4\) have tensor residue \(3/2\), while the normalized extension of their tensor local system has residue \(1/2\). Multiplication by an integral coordinate power identifies the two meromorphic bundles but changes their holomorphic lattices. The same issue affects canonical pullback when boundary multiplicities change. Exactness and horizontal-map functoriality above require no tensor-compatibility claim.

The meromorphic tensor and dual of logarithmic bundles remain logarithmic, by the usual connection Leibniz rules; their residues are sums and negatives. Hence these operations remain regular, even when renormalization changes the chosen lattice.


This proves Theorem 5.0 and its local criterion. $\square$

### Global correspondence and compactification

**Theorem 5.1 (Deligne correspondence and compactification criterion).** If $X$ is a smooth separated complex algebraic variety, horizontal sections after analytification give an equivalence
\[
\operatorname{Conn}_{\rm reg}(X)
\ \simeq\ \operatorname{Loc}_{\rm fd}(X^{\rm an}).               \tag{5.1}
\]
Morphisms on the left are algebraic connection maps. On the right are finite-rank complex local systems. For every smooth proper compactification \(P\) with simple normal-crossing boundary \(D\), an algebraic connection on \(X=P\setminus D\) is curve regular if and only if it has an algebraic logarithmic extension on \(P\); equivalently it is regular at the generic points of every boundary component by the transverse discrete-valuation-ring test. The criterion is independent of compactification.


**Proof.** The global prerequisites have earlier programme proofs. AG-QC, Appendix N, Theorem N.E.1 proves Nagata compactification for separated finite-type Noetherian schemes. After taking the reduced closure, Principalization and resolution, Theorem 4.8 and Corollary 3.3 resolve while preserving the smooth open and principalize its complement. This is the proof of that lesson's Corollary 5.2, with its Nagata step supplied by the preceding programme theorem. It gives a smooth proper SNC compactification of the entire \(X\), including non-quasi-projective \(X\).

Proper GAGA is AG-QC, Theorem 6.11, with full faithfulness in Proposition 6.9 and analytic compactness in Lemma 6.4; it includes proper nonprojective schemes. AG-QC, Theorem 4.1 and Proposition 5.1 prove faithfully flat analytic stalk comparison and exact coherent analytification. The coherent cokernel argument uses Oka coherence, Theorem 2.1 and the analytic Nullstellensatz, Theorem 5.1. The coherent extension, finite poles, jet splitting and descent needed in the rest of the proof are established below.

The finite-type closed-point test follows from the weak and strong Nullstellensatz, Theorems 2.1 and 2.2. Extending curve maps uses Proper morphisms and valuative criteria, Theorems 1.2 and 3.1, with uniqueness from Valuation rings and separatedness, Theorem 4.2. The local rings of smooth curves are regular by the smooth local-ring criterion, Theorem 3.2, hence are DVRs by Regular local rings, Solution 7.2.

#### Algebraizing the bundle and its logarithmic connection

Let \(P\) be smooth and proper over \(\mathbf C\), let \(D\) be a reduced SNC divisor, and put \(U=P\setminus D\). Suppose \((L,\nabla_{\mathrm{an}})\) is an analytic locally free bundle with an integrable logarithmic connection on \(P^{\mathrm{an}}\).

Proper GAGA gives a coherent algebraic \(F\) with an identification \(F^{\mathrm{an}}\cong L\). This \(F\) is locally free. At a closed algebraic point, lift a basis of its residue fibre to a map \(\mathcal O_{P,x}^r\to F_x\). Nakayama makes it surjective. After faithfully flat analytic base change it is a map between rank-\(r\) free modules with invertible residue determinant, hence an isomorphism. Faithfulness kills the algebraic kernel. The free-stalk presentation extends to a neighbourhood. The complement of the locally free locus is closed and has no closed point, hence is empty on a finite-type complex scheme.

The logarithmic one-forms have local basis \(dt_i/t_i\) in boundary directions and the other coordinate differentials. Changing a boundary equation by a unit adds a regular form, so they glue and their analytification is the analytic logarithmic sheaf.

Construct an algebraic logarithmic first-jet extension
\[
0\longrightarrow\Omega_P^1(\log D)\otimes F
\longrightarrow J^1_{\log D}(F)\xrightarrow{p}F\longrightarrow0.
\tag{5.1a}
\]
In a local frame the middle module is \(F\oplus(\Omega_P^1(\log D)\otimes F)\). If coefficient columns satisfy \(e_\beta=g_{\beta\alpha}e_\alpha\), glue pairs by
\[
(e_\alpha,v_\alpha)\longmapsto
(g_{\beta\alpha}e_\alpha,(dg_{\beta\alpha})e_\alpha+g_{\beta\alpha}v_\alpha).
\tag{5.1b}
\]
These are invertible \(\mathcal O_P\)-linear block matrices. The derivative product rule proves their cocycle identity, so they define a finite locally free algebraic bundle. Projection to the first entry is \(p\), with the indicated tensor kernel. The \(\mathbf C\)-linear map \(j(e)=(e,de)\) is well defined by the same transition rule and satisfies \(j(fe)=f j(e)+df\otimes e\). Splittings and logarithmic connections therefore correspond by
\[
\nabla=j-\sigma,\qquad \sigma=j-\nabla,\qquad p\sigma=1.
\tag{5.1c}
\]
The Leibniz terms cancel in \(j-\nabla\), making it \(\mathcal O_P\)-linear. These formulas commute with analytification.

The given analytic connection supplies an analytic splitting. Proper GAGA full faithfulness lifts it to a unique algebraic \(\sigma\); faithfulness detects \(p\sigma=1\). Hence \(\nabla=j-\sigma\) algebraizes the connection itself. The logarithmic exterior algebra is closed under \(d\): its coordinate generators are closed and derivatives of regular coefficients are regular. Extend \(\nabla\) by the graded Leibniz rule. Its square is \(\mathcal O_P\)-linear because the terms involving \(df\) cancel and \(d^2f=0\). This curvature map \(F\to\Omega_P^2(\log D)\otimes F\) has zero analytification and therefore vanishes. The connection is integrable. The same GAGA argument algebraizes horizontal maps: their defect of horizontality is \(\mathcal O_P\)-linear and vanishes after analytification. This applies to proper nonprojective \(P\).

The transverse-curve and coherent-comparison arguments below prove that this construction recovers every curve-regular algebraic connection.


#### From algebraic curves to finite poles at crossings

Let \(E\) be a curve-regular algebraic flat bundle on \(U\). First fix a point \(p\) on the smooth part of a component of \(D\), away from the others. On an affine algebraic neighbourhood \(W\), choose a local boundary equation \(t\) and functions completing its nonzero differential to a cotangent basis at \(p\). The prescribed-coordinate form of Smooth morphisms, Theorem 4.1 makes their map étale after shrinking:
\[
(t,y_2,\ldots,y_d):W\longrightarrow\mathbf A^d,
\qquad D\cap W=(t=0).
\tag{5.1d}
\]
The fibres \(y_i=a_i\) are smooth algebraic curves. Each small analytic \(t\)-slice in this étale chart is an open piece of such a curve. Its punctured piece maps into \(U\), and the missing point \(t=0\) occurs in its smooth complete model. Thus the assumed algebraic curve test applies there. Corollary 3.2 turns a formal test into a convergent logarithmic lattice.

Along any such curve, choose a basis of the generic fibre of its pulled-back algebraic bundle. These are rational sections. Every finite algebraic generator, and every coefficient of an algebraic map into a free ambient bundle, has finite Laurent poles relative to that basis near the missing point: each rational function is a quotient of convergent regular functions, and a nonzero one-variable denominator has finite order. The curve local ring is a DVR by Regular local rings, Solution 7.2. One can instead pull back a coherent extension to the curve and remove its DVR torsion. A generic basis and common denominators embed that finite torsion-free module in a finite free module. The coordinate-projection induction of Lemma 1.0 applies to this DVR too and proves it free of the full generic rank. The frames may be chosen separately for each curve. No injectivity of a pulled-back coherent ambient map is needed for these scalar calculations.

Compare with the normalized analytic logarithmic bundle \(L\) of the same local system. In the curve's convergent logarithmic lattice, the horizontal comparison and its inverse are meromorphic by the one-variable meromorphic-map proof above. Changing to rational frames preserves this property. Thus the forward scalar columns obtained by an algebraic ambient injection, and the inverse coordinates of finite algebraic generators, are meromorphic on every slice of this algebraic family. The same assertion holds for every horizontal analytic map between two curve-regular algebraic bundles.

Lemma 5.0.3 in these étale coordinates proves actual meromorphy on an analytic neighbourhood of \(p\). This is independent of coordinates: another equation of the same reduced smooth boundary component differs from \(t\) by a holomorphic unit.

Now fix any analytic SNC chart \(V\) at a boundary intersection. For each local component \((z_i=0)\), choose a smooth point \(p\) in \(V\), away from the other components. It has a product neighbourhood in the fixed \(z\)-coordinates on which the comparison entries are already meromorphic. This supplies a nonempty open subfamily of the fixed \(z_i\)-slices. The other normal coordinates are punctured and their product parameter domain is connected. The fixed-chart Laurent argument below propagates the finite pole bound to that entire domain; repeating in every \(z_i\) extends across the intersection. This argument does not require arbitrary analytic coordinate slices themselves to be algebraic curves.

Here is the uniformity argument, including crossings. For a holomorphic function \(f\) on \(\Delta^*\times V\), its Laurent coefficients \(a_j(y)\) are holomorphic on the connected parameter domain \(V\), by parameter Cauchy integrals. Slice meromorphy on a nonempty open \(V_0\) means
\[
V_0\subset\bigcup_{N\ge0}
\{y:a_j(y)=0\text{ for all }j<-N\}.
\]
The displayed sets are closed. If each had empty relative interior in \(V_0\), nested closed balls, with the \(N\)-th ball avoiding the \(N\)-th set and radius at most \(2^{-N}\), would have a common point avoiding their union, a contradiction. Thus one has interior; the identity theorem makes all its forbidden Laurent coefficients vanish on all of \(V\). Cauchy bounds give a holomorphic extension of \(t^Nf\).

Apply this in each normal coordinate, with the other normal coordinates punctured and all remaining coordinates as parameters. The joint Laurent expansion has a fixed lower bound in every normal exponent. Multiplication by a finite product \(\prod t_i^{N_i}\) leaves a convergent holomorphic power series on the full polydisc. Thus finite poles extend across every SNC crossing. No meromorphic Hartogs theorem or a generic uniform pole-order assertion is imported. 

#### Coherent extensions and algebraic full faithfulness

**A quasi-compact open direct image is quasi-coherent.**

Let \(V=\operatorname{Spec}A\), \(W\subset V\) be quasi-compact open, and \(j:W\hookrightarrow V\). For a quasi-coherent sheaf \(Q\) on \(W\), choose a finite principal cover \(W=\bigcup_iD(s_i)\). The sheaf equalizer gives
\[
\Gamma(W,Q)=
\ker\left(
\prod_i\Gamma(D(s_i),Q)
\longrightarrow
\prod_{i,j}\Gamma(D(s_is_j),Q)
\right),
\tag{5.1e1}
\]
where the arrow is the difference of restrictions.

For \(a\in A\), localization at \(a\) is exact and commutes with these finite products. One can verify exactness directly with fractions: a fraction whose image is zero has some denominator power killing that image, making its numerator a representative from the original kernel. On the principal affines, quasi-coherence identifies localization with restriction to \(D(a)\). Thus localizing (5.1e1) gives the same equalizer on \(W\cap D(a)\):
\[
\Gamma(W,Q)_a=\Gamma(W\cap D(a),Q).
\tag{5.1e2}
\]
This is the affine quasi-coherence criterion. Consequently \(j_*Q\) is quasi-coherent. Applied on every affine chart, it proves the same statement for a quasi-compact open of a Noetherian scheme. Kernels and quotients of quasi-coherent sheaves are quasi-coherent, since on each affine chart they are the associated sheaves of module kernels and quotients, and exact localization identifies their restrictions.

No finite generation of \(j_*Q\) is asserted.

**Finite coherent submodules extend across an open.**

Let \(X\) be Noetherian separated, \(Q\) quasi-coherent on \(X\), and \(G\subset Q|_W\) a finite-type quasi-coherent submodule on a quasi-compact open \(W\).

First take \(X=\operatorname{Spec}A\). The quotient \(Q|_W/G\) is quasi-coherent. Define
\[
H=\ker\bigl(Q\longrightarrow j_*(Q|_W/G)\bigr).
\tag{5.1e3}
\]
The quasi-coherent direct-image argument makes \(H\) quasi-coherent, and restriction to \(W\) makes it exactly \(G\). Cover \(W\) by finitely many \(D(s_i)\). Choose finite module generators of \(G\) on each \(D(s_i)\); this is possible because the coefficient rings are Noetherian and \(G\) is of finite type. By quasi-coherence of \(H\), every chosen generator is a fraction of an element of \(\Gamma(X,H)\), with a power of \(s_i\) as denominator. Let \(N\) be the finite \(A\)-submodule generated by all those numerators.

Its associated sheaf lies in \(H\), and on each \(D(s_i)\) contains all the chosen generators, since \(s_i\) is invertible there. Hence
\[
\widetilde N|_W=G.
\tag{5.1e4}
\]
The finite \(A\)-module \(N\) is finitely presented because \(A\) is Noetherian: the kernel of a finite free surjection is a finite submodule of a finite free module. Thus \(\widetilde N\) is coherent.

For general \(X\), choose a finite affine cover. Start with \(G\) on \(W\), and add these affine opens one at a time. If \(W'\) is the already treated open and \(V\) the next affine, \(W'\cap V\) is quasi-compact because \(X\) is Noetherian. The affine construction extends \(G|_{W'\cap V}\) to a coherent submodule of \(Q|_V\), agreeing exactly with it on the intersection. Glue the two submodules inside \(Q\). The glued sheaf is quasi-coherent and coherent because these properties are local and it has coherent restrictions on the old open and on \(V\). After finitely many steps it is defined on all of \(X\).

This proves the full finite-submodule extension assertion. It uses no ample line bundle, projectivity, extension of a locally free sheaf as locally free, or affine assumption on the original \(j\).

**Coherent extension of a bundle on a quasi-compact open.**

Let \(j:U\hookrightarrow X\) be quasi-compact open, and \(E\) a coherent sheaf on \(U\), in particular a finite-rank vector bundle. The quasi-coherent direct-image argument gives the quasi-coherent sheaf
\[
Q=j_*E
\]
on \(X\), with \(Q|_U=E\). Apply the finite-submodule extension argument to the finite-type submodule \(G=E\subset Q|_U\). We obtain a coherent subsheaf
\[
F\subset j_*E,\qquad F|_U=E.
\tag{5.1e5}
\]
Thus the required coherent extension exists in the entire Noetherian separated scope. It is unnecessary to assert a pre-existing coherent extension theorem as an additional leaf.

For the smooth pair \(X=P\), \(D=P\setminus U\), every integral component meets \(U\). The chosen \(F\subset j_*E\) is already torsion free on that component. A nonzero coefficient cannot kill a section of \(E\) on its dense open domain: in local vector-bundle coordinates over a domain it kills no nonzero coordinate. Equivalently, \(j_*E\) embeds into its generic-fibre vector space. If one starts instead with any coherent extension, quotienting its finite torsion submodule is legitimate over a Noetherian domain and leaves its restriction to the bundle \(E\) unchanged.

For an effective Cartier boundary \(D\), the open immersion is affine. On an affine chart \(V=\operatorname{Spec}A\) trivializing its equation \(q\), one has \(U\cap V=D(q)=\operatorname{Spec}A_q\). If \(F|_V=\widetilde N\), (5.1e5) gives
\[
N_q=\Gamma(D(q),E).
\tag{5.1e6}
\]
The module on the right is finite projective over \(A_q\); this follows from the vector-bundle affine module description. We use (5.1e6), not a claim that \(N\) is projective over \(A\).


On an affine neighbourhood, a rational basis and the finitely many denominators of generators give an injection
\[
F\hookrightarrow\mathcal O_P^r.
\tag{5.1f}
\]
Exact analytification preserves that injection. This does not assert that \(F\) is locally free at the boundary. The original algebraic connection already defines a meromorphic connection on \(F(*D)\): on an affine \(W=\operatorname{Spec}A\) with \(D=(q)\), if \(F\) corresponds to \(N\), the equality \(F|_{W\setminus D}=E\) says \(N[1/q]\) is precisely the module of \(E\) there. Its connection is defined on that localization. Images of finitely many generators have finite algebraic \(q\)-denominators, so analytification gives a meromorphic connection on \(F^{\mathrm{an}}[1/q]\). No free meromorphic frame is needed for this step.

Suppose first that \(L\) is the normalized analytic logarithmic bundle for \(E^{\mathrm{an}}\)'s local system. The analytic horizontal comparison \(v:L|_U\to E^{\mathrm{an}}\) is an isomorphism. In a local frame of \(L\), compose its columns with (5.1f). They are scalar holomorphic functions on the punctured SNC product. the transverse-curve and Laurent argument above makes their restrictions meromorphic on algebraic transverse slices and hence gives a product pole bound across the whole polydisc.

Their extensions belong to \(F^{\mathrm{an}}(*D)\). To see this precisely, let \(q=\prod t_i\). After multiplying a column by \(q^N\), it is a holomorphic vector in \(\mathcal O^r\). Its class \(c\) in the coherent cokernel of (5.1f) vanishes off \(D\). The cyclic coherent module generated by \(c\) has support in \(D\). Analytic Nullstellensatz therefore puts \(q\) in the radical of its annihilator, so \(q^Mc=0\) for some \(M\), on a smaller neighbourhood. The column multiplied by \(q^{N+M}\) lies in \(F^{\mathrm{an}}\).

For the inverse comparison, take finitely many generators of \(F\). Their images under \(v^{-1}\) have scalar coordinates in a frame of \(L\). The same curve and Laurent argument extends those coordinates meromorphically. Every finite relation is preserved, since its meromorphic image is zero on the dense complement. Thus
\[
F^{\mathrm{an}}(*D)\simeq L(*D)
\tag{5.1g}
\]
as meromorphic flat bundles. The inverse identities can be checked using the localized injection \(F^{\mathrm{an}}[1/q]\hookrightarrow\mathcal O^r[1/q]\): a difference becomes scalar meromorphic functions zero off \(D\), hence zero. This avoids treating a meromorphic localization itself as an \(\mathcal O\)-coherent module. The connection identities hold by the same localized argument.

The same argument works for an arbitrary horizontal analytic map between two curve-regular algebraic bundles \(E_1,E_2\). Take torsion-free coherent extensions \(F_1,F_2\), embed \(F_2\) in a finite free sheaf as in (5.1f), and apply the scalar argument to the images of finitely many generators of \(F_1\). The cokernel argument puts them in \(F_2^{\mathrm{an}}(*D)\). Relations give a map \(F_1^{\mathrm{an}}\to F_2^{\mathrm{an}}(*D)\).

On proper \(P\), compactness provides one finite global pole bound \(n\). The local maps therefore glue to
\[
F_1^{\mathrm{an}}\longrightarrow F_2^{\mathrm{an}}(nD).
\tag{5.1h}
\]
After increasing smaller local bounds to their maximum, the maps agree on overlaps: they agree off \(D\), and the torsion-free target makes a section with that dense-open restriction unique. Full faithful proper GAGA algebraizes (5.1h). Its restriction to \(U\) is the desired algebraic map. The difference \(\nabla_2\phi-(1\otimes\phi)\nabla_1\) is \(\mathcal O_U\)-linear, since its two Leibniz terms cancel. It has zero analytification, so faithfully flat analytic stalks make it zero algebraically. Algebraizing the inverse in (5.1g) similarly gives an algebraic horizontal isomorphism on \(U\).

This proves algebraic full faithfulness for curve-regular connections without requiring a free coherent extension at the boundary. 

#### The curve and divisorial criteria

**Logarithmic extension implies curve regularity.** Let \(F\) be an algebraic logarithmic flat bundle on the proper pair \((P,D)\), and let \(h:C\to U\) be any map from a smooth algebraic curve. Its map from the function field extends to the smooth complete curve \(\overline C\) by properness of \(P\), applied at the discrete valuation rings of its missing points. The valuative criterion gives a map on each missing-point discrete valuation ring. Choose an affine target neighbourhood of its closed image: the entire spectrum of that local ring maps into it. Finitely many coordinate images are rational functions regular at the point, so extend to a Zariski neighbourhood on the curve; finitely many equations and inverted coordinates remain valid after shrinking. Thus the local-ring maps spread to actual curve maps. Separatedness makes the extensions agree on overlaps because they agree at the dense generic point, and they glue. Locally at such a point, each pulled-back boundary equation is
\[
t_i\circ\overline h=u_i s^{m_i},\qquad
\overline h^*(dt_i/t_i)=m_i\,ds/s+du_i/u_i,
\quad m_i\ge0.
\tag{5.1i}
\]
Hence \(\overline h^*F\) has a logarithmic connection at every missing point. This proves regularity for every curve map, including curves approaching intersections with nontrivial multiplicities.

**Curve regularity implies logarithmic extension on this pair.** Let \(E\) be curve regular on \(U\). Build \(L\) from its local system, using the proved local SNC theorem. The logarithmic jet algebraization above algebraizes \(L\) and its logarithmic connection to \(F_\Sigma\) on \(P\). By (5.1i), \(F_\Sigma|_U\) is curve regular. The analytic horizontal identification \(E^{\mathrm{an}}\cong F_\Sigma^{\mathrm{an}}|_U\) algebraizes by the coherent comparison and full-faithfulness proof above. Therefore \(F_\Sigma\) is an algebraic logarithmic extension of \(E\) on this very pair.

This proves the criterion for every given proper smooth SNC compactification, including nonprojective ones. Existence of at least one such pair is supplied separately by the compactification proof chain above.

The generic divisorial formulation follows as well. At the generic point of each boundary component, require a lattice over its discrete valuation ring \(R\) preserved by \(t\nabla_v\), for an algebraic vector field \(v\) regular and transverse there.

If the test was first expressed after completion, it gives the same criterion over \(R\). Put \(K=\operatorname{Frac}R\). A basis of the completed stable lattice is a matrix \(G\in\operatorname{GL}_r(\widehat K)\). Approximate its finitely many entries by entries of \(K\) to sufficiently high valuation, obtaining \(\widetilde G\) with \(G^{-1}\widetilde G=H=1+\) a positive-valuation matrix. Such approximations exist by the definition of completion, after multiplying entries by fixed powers of \(t\). The derivation \(tv\) preserves valuation: \(v(R)\subset R\) and \(v(t)\) is a unit, so \(tv(t^na)\in t^nR\), for every integer \(n\) and \(a\in R\). It therefore extends continuously to the completion with the same property. Write \(t\nabla_v=tv+B_0\) in the initial rational frame. In the completed lattice frame its coefficient matrix is \(A=G^{-1}B_0G+G^{-1}tv(G)\in\operatorname{Mat}_r(\widehat R)\). The coefficient matrix in the \(\widetilde G\) frame is \(H^{-1}AH+H^{-1}tv(H)\), still integral over \(\widehat R\). It also lies in \(K\), because \(\widetilde G\) and the original connection do. The valuation test gives \(K\cap\widehat R=R\). Thus \(\widetilde G\) supplies an actual \(R\)-lattice preserved by \(t\nabla_v\). This proves the descent of the formal test rather than assuming it.

A basis of this lattice consists of rational sections. Its finitely many connection coefficients, and the finitely many denominators of that basis and its inverse, extend to an algebraic open meeting the component, with \(t\nabla_v\) regular on the chosen basis.

Near a complex point of this open, straighten \(v\) holomorphically using its local flow from the divisor. Picard iteration on a smaller polydisc is a contraction, uniformly in the divisor parameters, so it gives a holomorphic flow. Its derivative is invertible at the initial point because \(v\) is transverse; the proved holomorphic inverse-function argument gives coordinates \((s,y)\) with \(v=\partial_s\) and \(t=s\) times a unit. In the rational basis just chosen, the connection along every \(s\)-slice therefore has at most a simple pole. The one-variable criterion and the same parameter argument give meromorphy of the comparison with \(L\) on a nonempty open part of the component.

Every local boundary component meets that good open in a nonempty analytic open subset. Indeed if the proper algebraic exceptional closed set contained an analytic neighbourhood on the irreducible smooth divisor, its finite defining ideal would have zero analytification at that point by the identity theorem. Faithful local flatness makes the algebraic ideal zero there; finite generation then makes it zero on a Zariski neighbourhood, contradicting properness of the closed subset. At a good smooth point, the flow-chart boundary equation and the fixed analytic \(z_i\) differ by a holomorphic unit. A small product box in this meromorphic neighbourhood supplies a nonempty open family of the fixed \(z_i\)-slices. The connected punctured-product Laurent argument above now propagates the finite pole bound and handles all crossings. The coherent comparison and full-faithfulness proof above and proper GAGA then give the logarithmic extension. Conversely a logarithmic lattice is preserved by \(t\nabla_v\) for every such regular transverse \(v\). Thus the criterion is independent of the selected transverse vector field. In this implication the analytic flow is a consequence of the divisorial lattice itself; it is not substituted for an algebraic curve test.

#### Correspondence on every smooth separated variety

For a proper SNC pair, a finite-rank local system on \(U^{\mathrm{an}}\) first gives its analytic logarithmic bundle, then the logarithmic jet algebraization above algebraizes it to an algebraic logarithmic flat bundle. the curve and divisorial criterion above makes its restriction curve regular. The coherent comparison and full-faithfulness proof above proves full faithfulness for all curve-regular algebraic connections. This establishes the correspondence on \(U\), with no projectivity assumption on the pair.

There is also a complete descent proof preserving arbitrary smooth separated \(X\). Choose a finite affine cover \(X=\bigcup U_i\). Each \(U_i\) has an explicit projective closure; resolution and principalization give a projective smooth SNC compactification of \(U_i\), preserving its entire smooth open. Thus the already proved pair argument establishes the correspondence on every \(U_i\). Separatedness makes \(U_i\cap U_j\) affine: its diagonal description is a closed subscheme of the affine \(U_i\times U_j\).

Regularity defined by all curves is Zariski local. To prove this, take a curve \(h:C\to X\) and one irreducible component of \(C\). Its generic point maps into some \(U_i\), so \(C_0=h^{-1}U_i\) is a dense open curve. The restriction of \(h^*E\) to \(C_0\) is regular at every point of its smooth complete model, by regularity on \(U_i\). That complete model is also the smooth complete model of \(C\): a rational map between two smooth proper models extends at every discrete valuation ring by properness, and the two extensions are inverse by separatedness. The restriction therefore proves regularity at infinity for \(h^*E\). At the remaining points of \(C\) the given algebraic connection is already nonsingular. This proves the local assertion. Restriction of a curve-regular connection to an open is immediate by composition of curve maps.

Given a local system \(A\) on \(X^{\mathrm{an}}\), algebraize it on each \(U_i\) to \(E_i\), with the specified identification of its horizontal local system. On \(U_i\cap U_j\), full faithfulness gives the unique algebraic horizontal isomorphism inducing those identifications. On triple overlaps their compositions coincide analytically and hence algebraically, proving the cocycle identity. Ordinary open gluing produces a vector bundle and a connection on all of \(X\); integrability is local. The preceding curve argument makes it globally regular. Its horizontal local system is \(A\).

Likewise a morphism of local systems algebraizes on each \(U_i\); uniqueness on overlaps glues the algebraic connection maps. Faithfulness is local. Thus
\[
\operatorname{Conn}_{\mathrm{reg}}(X)
\simeq\operatorname{Loc}_{\mathrm{fd}}(X^{\mathrm{an}})
\tag{5.1j}
\]
holds for every smooth separated finite-type complex \(X\). The proof covers every smooth separated finite-type complex \(X\).

#### Independence of compactification

By the curve and divisorial criterion above, existence of a logarithmic extension on one proper smooth SNC compactification is equivalent to the intrinsic curve property. That property supplies a logarithmic extension on every such compactification. Together with the actual existence theorem, this proves the asserted compactification criterion and its independence.

A common proper SNC model can also be constructed explicitly. For two compactifications \(P,Q\), take the reduced closure of the graph of \(X\) in \(P\times Q\). Its inverse image of \(X\) under either projection is exactly that graph: the graph is already closed over this open, by separatedness. Resolve the closure while preserving the smooth open \(X\), and principalize its complement. This gives a smooth proper SNC \(W\) and proper maps \(W\to P,Q\) restricting to the identity on \(X\).

Pullback preserves logarithmic connections. Each pulled-back boundary equation factors locally as a unit times a monomial in the boundary equations of \(W\); its logarithmic differential is a sum of their logarithmic differentials plus a holomorphic unit differential, exactly as in (5.1i). The two pulled-back extensions of the same connection are uniquely identified after allowing boundary poles: their horizontal identity on \(X\) extends meromorphically by the proved local logarithmic morphism lemma.

One must not claim equality of the normalized holomorphic lattices under this pullback. New boundary residues are sums with integral multiplicities and may leave the chosen strip; re-normalization can change the lattice by integral boundary powers. The meromorphic identification and the intrinsic curve property are independent of that choice. This distinction preserves all multiplicities and Jordan blocks without adding an invalid tensor or pullback compatibility claim.


The correspondence and all compactification assertions have now been proved. The freely accessible [IAS-hosted Deligne text](https://publications.ias.edu/sites/default/files/Number9.pdf), Chapter II, §§4–5, is the human reading source for these constructions. The arguments above supply the proofs. $\square$

Now use finite length and the description of simple holonomic modules from the preceding lesson. A holonomic module is **regular holonomic** if every composition factor can be presented as the minimal extension $j_{!*}E$ of an irreducible regular connection on a smooth locally closed subvariety with affine inclusion. The zero module is regular. This is the definition in Bernstein's freely accessible lecture “Holonomic D-modules with regular singularities.” Theorem 5.2 below proves that, for a module that is itself a connection, it agrees with the preceding all-curve definition, including infinity.

### Connections and the regular-holonomic definition

Theorem 5.1 was proved using the all-curve definition of regular connections, independently of the regular-holonomic identification below.

#### Connection subquotients and finite length

We work first on a connected smooth variety \(X\). Smooth local rings are regular by the smooth local-ring criterion, Theorem 2.1. They are domains by Regular sequences and depth, Theorem 6.1. Its proof uses the polynomial associated graded ring, Krull separation and multiplication of initial forms. The finitely many irreducible components of a Noetherian scheme cannot meet when all local rings are domains: at an intersection the corresponding local ring would have two minimal primes. They are therefore disjoint closed and open components. Smoothness also makes \(X\) reduced, so connected \(X\) is integral. The disconnected case is the finite direct sum over these components.

Let \(E\) be a finite-rank vector bundle with integrable connection. Any quasi-coherent differential-operator submodule \(N\) of \(E\) is coherent over the structure sheaf: on a Noetherian affine open, its module of sections is a submodule of a finite module, hence finite. Its quotient \(E/N\) is coherent as well. Both inherit integrable connections, so the coherent-connection theorem, Theorem 3.1 makes \(N\) and \(E/N\) finite-rank vector bundles. The same argument applies successively to all subquotients.

The constant filtration \(F_iE=0\) for \(i<0\) and \(F_iE=E\) for \(i\geq0\) is a good order filtration. Its associated graded module is \(E\) in degree zero, and positive-degree vector symbols act by zero. Its characteristic support is the zero section over the support of \(E\). Hence a nonzero \(E\) on this connected \(X\) is holonomic and its support is all of \(X\).

A nonzero bundle on connected \(X\) has positive constant rank. Thus a proper inclusion of connection subobjects strictly increases rank: its nonzero quotient is again a bundle. Every increasing or decreasing chain of connection subobjects therefore has at most \(\operatorname{rank}E\) strict steps. A composition series exists without any further finiteness theorem: among nonzero subobjects choose one of least positive rank; it is simple, and repeat in the lower-rank quotient. All simple factors are themselves connections of full support \(X\). A simple connection is exactly an irreducible connection because every differential-operator subobject is a subbundle with bundle quotient.

This also excludes nonzero boundary-supported subobjects or quotients of a connection. Such an object would be a bundle of generic rank zero, hence zero.

#### Extensions of regular singular differential modules

Let \(R\) be either \(\mathbf C[[t]]\) or \(\mathbf C\{t\}\), let \(K\) be its fraction field, and set \(\theta=t\,d/dt\). A differential module \(V\) over \(K\) has an operator \(\Theta\) satisfying
\[
\Theta(fv)=\theta(f)v+f\Theta(v).
\tag{5.2a}
\]
Regular means that \(V\) has a full \(R\)-lattice \(L\) preserved by \(\Theta\). The following argument works over each ring separately, including arbitrary extension classes.

**Lemma 5.2.1.** In an exact sequence of differential \(K\)-modules
\[
0\longrightarrow V_1\longrightarrow V\longrightarrow V_2\longrightarrow0,
\tag{5.2b}
\]
\(V\) is regular if and only if both \(V_1\) and \(V_2\) are regular.

**Proof for subobjects and quotients.** Suppose \(L\) is a stable lattice in \(V\). Then \(L_1=L\cap V_1\) is preserved by \(\Theta\). It is finite free by Lemma 1.0. It spans \(V_1\): for any \(K\)-basis of \(V_1\), multiply the finitely many basis vectors by sufficiently large powers of t to put them in \(L\). The image \(L_2\) of \(L\) in \(V_2\) is finite, torsion free and spans \(V_2\); it is a lattice by the same lattice algebra. It is preserved by the induced operator. Thus both ends are regular.

**Proof for extensions.** Choose stable lattices \(L_1\) and \(L_2\) and a \(K\)-linear splitting \(s:V_2\to V\) of (5.2b). The defect
\[
B=\Theta s-s\Theta_2:V_2\longrightarrow V_1
\tag{5.2c}
\]
is \(K\)-linear: the two \(\theta(f)\) terms in (5.2a) cancel. There are finitely many coefficients of \(B\) on an \(R\)-basis of \(L_2\). Therefore some integer \(N\geq0\) satisfies
\[
B(L_2)\subset t^{-N}L_1.
\]
Put
\[
L=t^{-N}L_1\oplus s(L_2)\subset V.
\tag{5.2d}
\]
This is a full finite free lattice. Moreover
\[
\Theta(t^{-N}a)=t^{-N}(\Theta_1a-Na),\qquad
\Theta(s(b))=s(\Theta_2b)+B(b).
\tag{5.2e}
\]
Both terms lie in \(L\) whenever a belongs to \(L_1\) and b belongs to \(L_2\). Hence \(L\) is stable. The minus sign in -Na is essential and follows from \(\theta(t^{-N})=-Nt^{-N}\). No splitting of the differential module is assumed. This proves the lemma over formal and convergent rings. The zero-rank cases use the zero lattice. \(\square\)

The middle lattice may require an integral shift of the submodule lattice; no assertion is made that a normalized extension lattice is the direct sum of two normalized end lattices.

#### Curve-regular connections form a Serre category

Let
\[
0\longrightarrow E_1\longrightarrow E\longrightarrow E_2\longrightarrow0
\tag{5.2f}
\]
be a short exact sequence of finite-rank algebraic integrable connections on \(X\). This sequence is locally split as a sequence of structure-sheaf modules, since its quotient \(E_2\) is locally free. Its pullback by every algebraic map \(h:C\to X\) is therefore an exact sequence of flat bundles. No flatness assumption on h is needed.

Fix an irreducible component of the smooth curve \(C\) and any point p of its smooth complete model, including every point absent from \(C\). The sequence on \(C\) is exact at the function field. Choose rational frames for these generic fibres. Tensoring with the formal or convergent meromorphic field at p is exact, since this is a field extension of the rational function field; the rational connection coefficients define the resulting differential modules. Thus (5.2f) gives (5.2b) at p. Lemma 5.2.1 shows that the middle pullback is regular at p if and only if its two ends are regular there.

Doing this for every h and p proves:

**Proposition 5.2.2.** Curve-regular connections on \(X\) are closed under connection subobjects, quotients and extensions. In particular, a connection is curve regular if and only if all the factors of any one composition series of that connection are curve regular.

For the forward implication apply the subquotient part of Lemma 5.2.1 to every factor. For the reverse implication reconstruct the composition series by successive applications of its extension part. The proof retains every compactification point and every curve map. It does not merely test the nonsingular points already in \(C\).


#### Extending a horizontal map from a dense open

The next lemma concerns nonsingular connections on all of \(X\). It makes no assertion about arbitrary singular holonomic modules.

**Lemma 5.2.3.** Let \(E\) and \(F\) be finite-rank algebraic integrable connections on a smooth integral \(X\), and let \(V\) be a nonempty open subset. Every algebraic horizontal map \(\phi:E|_V\to F|_V\) extends to a unique algebraic horizontal map on \(X\). If \(\phi\) is an isomorphism on \(V\), its extension is an isomorphism.

**Proof.** Write \(K=\mathbf C(X)\), and let \(G\) be the graph of the generic \(K\)-linear map \(\phi_K\) inside \(E_K\oplus F_K\). It is stable under all derivations, because \(\phi\) is horizontal on \(V\). Derivations and connections extend to \(K\) by the quotient rule.

Inside \(E\oplus F\) define the graph closure \(\Gamma\) by intersection with \(G\). This is a coherent quasi-coherent submodule, a point worth checking explicitly. On an affine open \(\operatorname{Spec}A\), let M be the finite module of sections of \(E\oplus F\), viewed inside \(M\otimes_A K\); set \(\Gamma_A=M\cap G\). It is finite because A is Noetherian. For every nonzero f in A,
\[
(\Gamma_A)_f=M_f\cap G.
\tag{5.2g}
\]
Indeed, a vector \(m/f^a\) belongs to \(G\) precisely when m does, since \(G\) is a \(K\)-vector subspace. Consequently these finite modules sheafify compatibly and glue. On \(V\) the intersection recovers exactly the ordinary graph of \(\phi\).

For any local vector field \(\xi\), both the lattice of regular sections \(E\oplus F\) and its rational subspace \(G\) are preserved by the connection. Their intersection \(\Gamma\) is preserved as well. It is therefore a coherent module with integrable connection. Projection
\[
\pi:\Gamma\longrightarrow E
\tag{5.2h}
\]
is horizontal and is an isomorphism at the generic point. Its kernel and cokernel are structure-sheaf coherent modules with induced integrable connections. Theorem 3.1 of the preceding coherent-connection proof makes both bundles, and each has generic rank zero, so both are zero. Thus \(\pi\) is an isomorphism. The required extension is \(\operatorname{pr}_F\circ\pi^{-1}\).

Two extensions differ by an algebraic bundle map zero on a dense open. In local free frames its entries are regular functions in an integral ring, and hence zero. This proves uniqueness. If \(\phi\) is invertible on \(V\), apply the proved extension statement to \(\phi^{-1}\); the two compositions are the identity by uniqueness. \(\square\)

The graph construction does not assume that the original connection or any graph lattice is regular at infinity. Its only freeness input is the earlier local theorem for coherent nonsingular connections. It proves that restriction to a dense open is fully faithful for nonsingular algebraic connections, but essential surjectivity of that restriction is not asserted.

#### Dense-open regularity for a nonsingular connection

**Lemma 5.2.4.** Suppose \(Q\) is a finite-rank algebraic integrable connection on all of smooth separated \(X\). If \(Q|_V\) is curve regular for a dense open \(V\), then \(Q\) is curve regular on \(X\).

**Proof.** Let A be the horizontal local system of the full analytic connection \(Q^{\mathrm{an}}\) on \(X^{\mathrm{an}}\). Apply the already proved Theorem 5.1 on \(X\): it gives a curve-regular algebraic connection \(R\) and a horizontal analytic isomorphism
\[
u:R^{\mathrm{an}}\xrightarrow{\sim}Q^{\mathrm{an}}.
\tag{5.2i}
\]
It realizes the same full local system A, without any semisimplicity restriction.

On \(V\), both \(R|_V\) and \(Q|_V\) are curve regular. Theorem 5.1's full faithfulness, applied on this smooth separated open variety \(V\) to the restriction of u, gives an algebraic horizontal isomorphism
\[
u_V:R|_V\xrightarrow{\sim}Q|_V.
\tag{5.2j}
\]
For clarity, full faithfulness algebraizes u and its inverse on \(V\); their compositions are the identity by faithfulness.

Apply Lemma 5.2.3 on each connected component of \(X\) to extend (5.2j) to an algebraic horizontal isomorphism \(R\simeq Q\) on all of \(X\). Since \(R\) is curve regular by construction, \(Q\) is curve regular. In particular, for a curve \(h:C\to X\) whose image is contained in \(X\) minus \(V\), the conclusion follows by pulling back this global algebraic isomorphism. Such curves are not discarded, moved into \(V\) or implicitly treated as restrictions of curves meeting \(V\). \(\square\)

This lemma is the needed dense-open argument. Curve regularity of \(Q|_V\) alone says nothing directly about curve maps contained in the complement. The full-local-system realization and the nonsingular graph-closure theorem fill that gap.

#### The connection and regular-holonomic definitions agree

Use the definition given above: a holonomic module is regular holonomic when every simple composition factor has a presentation \(j_{!*}H\), where \(H\) is an irreducible curve-regular connection on a smooth locally closed subvariety and its inclusion j is affine.

**Theorem 5.2.** A finite-rank algebraic integrable connection \(E\) on a smooth separated complex algebraic variety \(X\) is regular holonomic under this definition if and only if \(E\) is curve regular at every compactification point of every smooth algebraic curve mapping to \(X\).

**Proof.** Reduce to connected components, which are open and closed; their inclusions are affine. A module and its subobjects split by these open-and-closed components, so every nonzero simple factor belongs to one component. A connected smooth curve maps into one such component; disconnected curves are tested componentwise. Minimal extension from a component is just its exact open-and-closed direct image. Thus both conditions are componentwise. The zero connection satisfies both conditions.

Suppose \(E\) is curve regular. By the connection subquotient argument above it is holonomic and has a finite composition series whose factors \(Q\) are simple connections. Proposition 5.2.2 makes every \(Q\) curve regular. Present \(Q\) as the minimal extension of itself under the identity inclusion \(X\to X\). This inclusion is affine even when \(X\) is not affine. The identity has no boundary, so its minimal extension is \(Q\) itself, either directly from the image definition or by the minimal-extension Theorem 4.1. Thus every simple factor meets the regular-holonomic definition.

Conversely suppose \(E\) is regular holonomic and choose a composition series. Each simple factor \(Q\) is a connection on all of this connected \(X\), of positive rank and support \(X\), by the connection subquotient argument above. The definition supplies some presentation
\[
Q\simeq j_{!*}H,\qquad j:S\hookrightarrow X,
\tag{5.2k}
\]
with \(H\) irreducible and curve regular, \(S\) smooth locally closed and j affine. If necessary remove components of \(S\) on which \(H\) has rank zero: irreducibility permits only one component of positive rank, and the inclusion of that component is open and closed, hence affine. We can therefore take \(H\) of positive rank on an integral \(S\) without changing its minimal extension or the affineness condition.

Minimal-extension uniqueness and the simple-factor classification, Theorems 4.1 and 5.1 identify the support of this presentation with the closure of \(S\) and identify its restriction to \(S\) with \(H\). Since \(Q\) has support \(X\), the closure of \(S\) is \(X\). A dense locally closed subset is open: writing \(S=O\cap Z\) with O open and Z closed, density forces \(Z=X\). Hence \(S\) is a nonempty dense smooth open, and
\[
Q|_S\simeq H.
\tag{5.2l}
\]
Lemma 5.2.4 now proves that \(Q\) is curve regular on all of \(X\). This is the step that handles a presentation made only on a dense open. It neither replaces \(S\) by \(X\) by assertion nor infers all-curve regularity from (5.2l) without proof.

All simple factors of \(E\) are therefore curve regular. Proposition 5.2.2's extension assertion, applied successively to the composition series, makes \(E\) curve regular. This proves the converse and the theorem. \(\square\)

The affineness condition is retained: it is supplied by the defining presentation in the converse and satisfied by the identity presentation in the forward direction. The proof imposes no affine hypothesis on \(X\) itself, no properness hypothesis on \(X\), and no rank-one or semisimple restriction.


### Analytic regularity of SNC meromorphic bundles

**Theorem 5.3.** Let \((Z,D)\) be a complex-manifold simple normal-crossing pair, allowing a locally finite set of globally smooth boundary components. Let \(E_\Sigma\) be the normalized logarithmic extension of any finite-rank local system supplied by Theorem 5.0. Then \(E_\Sigma(*D)\) is a coherent analytic regular holonomic left \(\mathcal D_Z\)-module. On a chart with \(k\) boundary coordinates, its characteristic variety is the union of all coordinate-intersection conormals, and each component has characteristic-cycle coefficient equal to the rank. Formula (5.3n) is a global good filtration annihilated, in its analytic associated graded, by the reduced characteristic ideal.

We use the analytic good-filtration definition in [Kashiwara's freely accessible paper, Definition 5.2](https://www.kurims.kyoto-u.ac.jp/~kenkyubu/kashiwara/RiemannHilbert.pdf), printed pp. 351–352. The proof below establishes the required property directly. The normalized local model and Laurent estimates were proved in Theorem 5.0 and Lemma 5.0.2.

#### Analytic regularity and the normalized local model

As usual, the zero module is included among regular holonomic modules. For an analytic holonomic operator module \(M\), call its good-filtration condition regular when some good filtration \(F\) has analytic associated graded annihilated by the reduced ideal of its actual characteristic variety. We will prove this condition explicitly for the module below. This concerns local analytic regularity; it does not identify analytic regularity on a nonproper algebraic variety with algebraic regularity at infinity.

Let
\[
V=\Delta_z^k\times\Delta_w^m,\quad d=k+m,\quad
D=(q=0),\quad q=z_1\cdots z_k,
\qquad H=\mathcal O_V.
\tag{5.3a}
\]
Let \(W\) be a complex vector space of dimension \(r\), and let \(A_i\) commute with all eigenvalues in the half-open strip \(0\leq\operatorname{Re}\alpha<1\). Write
\[
M=H[1/q]\otimes W,\qquad
\nabla=d+\sum_{i=1}^kA_i\,d\log z_i.
\tag{5.3b}
\]
The positive-loop monodromy is \(e^{-2\pi iA_i}\). All nilpotent parts remain. Put \(v(u)=q^{-1}\otimes u\), for \(u\in W\). The module is zero for \(r=0\), in which case all the asserted regularity properties are immediate. Below assume \(r>0\).

For each multi-index \(a\in\mathbf N^k\), define
\[
C_a=\prod_{i=1}^k\prod_{\ell=1}^{a_i}(A_i-\ell I).
\tag{5.3c}
\]
Every factor is invertible: a positive integer cannot be a residue eigenvalue in the chosen strip. On a generalized eigenspace its inverse is the terminating geometric series in the nilpotent part. All factors commute. Differentiating gives exactly
\[
\partial_z^a v(u)=q^{-1}z^{-a}\otimes C_a u,\qquad
\partial_{w_j}v(u)=0.
\tag{5.3d}
\]
In particular the signs are \(A_i-\ell I\), not \(A_i+\ell I\).

#### The generated order filtration and Laurent deficits

Let \(\mathcal D\) have its ordinary analytic order filtration, and set \(F_pM=0\) for \(p<0\). Formula(5.3d) and the coordinate normal ordering of differential operators prove
\[
F_pM=\mathcal D_{\leq p}v(W)
=\sum_{\substack{a\in\mathbf N^k\\|a|\leq p}}
H\,q^{-1}z^{-a}\otimes W.
\tag{5.3e}
\]
Normal ordering is the PBW coordinate calculation already proved for differential operators; with holomorphic coefficients the same finite Leibniz argument applies. Tangential derivatives of the chosen generators vanish. Invertibility of \(C_a\) gives every displayed pole term, not only an upper bound.

The filtration is exhaustive, since each finitely bounded Laurent pole can be placed in one of these terms after increasing \(|a|\). Every \(F_p\) is a finite sum of coherent \(H\)-submodules of the common coherent lattice \(q^{-(p+1)}H\otimes W\): for \(|a|\leq p\), the factor \(q^p z^{-a}\) is holomorphic. Compatibility with the operator filtration follows from its definition by generated order. In fact
\(\mathcal D_{\leq b}F_p=F_{p+b}\) for \(p,b\geq0\): each normally ordered derivative of total order at most \(p+b\) factors into derivatives of orders at most \(b\) and \(p\), with its coefficient on the left.

Here is the exact scalar description that will prove goodness and its annihilator. For a Laurent exponent \(n\in\mathbf Z^k\), put
\[
\nu(n)=\sum_{i=1}^k\max(-n_i-1,0).
\tag{5.3f}
\]
A convergent Laurent germ belongs to the scalar filtration in (5.3e) if and only if all its nonzero terms have \(\nu(n)\leq p\). The forward implication follows by expanding the holomorphic coefficients of each pole term.

For the converse, group the Laurent terms by
\(a_i=\max(-n_i-1,0)\). Only finitely many \(a\) have \(|a|\leq p\). If \(a_i>0\), the exponent in that coordinate is fixed at \(-a_i-1\); if \(a_i=0\), its exponent is at least \(-1\). Multiplication by \(qz^a\) leaves a convergent power series in the remaining coordinates. The parameter Cauchy bounds from the Laurent proof in Lemma 5.0.2 give convergence of every grouped series on a smaller polydisc. Thus the finite grouping has exactly the form in (5.3e). This argument is entrywise for \(W\).

Consequently, in degree \(p\), the terms with \(\nu(n)<p\) disappear and distinct top-degree \(a\)'s are independent:
\[
\operatorname{gr}_p^F M
\cong
\bigoplus_{\substack{a\in\mathbf N^k\\|a|=p}}
\bigl(H/(z_i:a_i>0)\bigr)\otimes W.
\tag{5.3g}
\]
The summand is represented by \(q^{-1}z^{-a}\). Multiplication by a \(z_i\) with \(a_i>0\) lowers the deficit by one, giving zero in this graded degree. The converse independence is the uniqueness of the grouped Laurent exponents. The description is an \(H\)-module description of the graded piece; no direct-sum decomposition of the ungraded filtration is claimed.


![Laurent exponents in the two-component pole filtration](assets/snc-laurent-pole-filtration.png)

*The points represent Laurent exponents for two boundary coordinates, in the displayed finite window. Color records the exact deficit (5.3f), so \(F_p\) consists of the convergent terms with deficit at most \(p\). Differentiating the generator at \((-1,-1)\) gives the coefficient \(A_1-I\) in (5.3d); multiplication by \(z_1\) lowers the resulting filtration degree. Thus \(z_1\xi_1\) annihilates the graded module, as proved in (5.3i). The admitted regions continue rightward and upward. This is an exponent-lattice figure. Reproducible figure source.*

#### Associated graded module and the full operator presentation

Write
\[
S=H[\xi_1,\ldots,\xi_k,\eta_1,\ldots,\eta_m],
\qquad J=(z_1\xi_1,\ldots,z_k\xi_k,\eta_1,\ldots,\eta_m).
\tag{5.3h}
\]
In degree \(p\), \(S/J\) has precisely the direct summands of (5.3g), with basis monomials \(\xi^a\) and coefficient ring \(H/(z_i:a_i>0)\). The graded map
\[
(S/J)\otimes W\longrightarrow\operatorname{gr}_F M,\qquad
f\xi^a\otimes u\longmapsto
\bigl[f\,\partial_z^a v(u)\bigr]
\tag{5.3i}
\]
is an isomorphism. It respects the symbol action by construction. On each summand (5.3g) it multiplies \(W\) by \(C_a\), which is invertible; hence it has no kernel and is surjective in every degree. This proves a good filtration and the exact equality \(\operatorname{Ann}_S(\operatorname{gr}_F M)=J\).

There is also a concrete finite operator presentation. Start with the free left module \(\mathcal D\otimes W\) mapping to \(M\) by \(P\otimes u\mapsto Pv(u)\). Its relations are generated by
\[
z_i\partial_i\otimes u-1\otimes(A_i-I)u,
\qquad \partial_{w_j}\otimes u,
\tag{5.3j}
\]
for a finite basis of \(W\). These displayed relations map to zero by(5.3d).

To see they generate the entire kernel, the map is strict for the generated filtration in(5.3e). Formula(5.3i) makes its graded kernel exactly \(J\otimes W\), generated by the first-order symbols of(5.3j). Subtract lifts of a finite principal-symbol expression from any kernel element; its order strictly decreases. Repeating terminates because order is bounded below. In order zero there is no relation: \(q^{-1}\otimes W\) is an \(H\)-basis of its lattice. This proves the presentation. Analytic operator coherence was proved in Holonomic D-modules and duality, Theorem 3.0c; in particular this finite presentation defines a coherent analytic operator module.

#### The radical characteristic ideal

For each \(I\subset\{1,\ldots,k\}\), let
\[
P_I=(z_i:i\in I,\ \xi_j:j\notin I,\ \eta_1,\ldots,\eta_m).
\tag{5.3k}
\]
At a stalk on the corresponding coordinate component, these ideals are prime: their quotient is a polynomial ring over the convergent ring in the remaining coordinates, a domain by its power-series expansion. At a stalk where a defining coordinate is a unit, the corresponding ideal is the unit ideal and contributes no component. Moreover
\[
J=\bigcap_I P_I.
\tag{5.3l}
\]
Here is a direct membership proof. Modulo all \(\eta\)'s, expand in the polynomial \(\xi\)'s and convergent \(z\)'s. A monomial not divisible by any \(z_i\xi_i\) has no index at which both exponents are positive. Choose \(I\) to contain every index at which its \(\xi_i\)-exponent is positive and to omit every index at which its \(z_i\)-exponent is positive. That monomial survives in \(S/P_I\). In a power series, quotienting by these coordinate ideals only removes forbidden monomials; distinct surviving monomials do not cancel. Thus a series in every \(P_I\) has every monomial divisible by some generator \(z_i\xi_i\). Assign each such monomial to the first generator dividing it and divide by that fixed generator. There are finitely many generators, and Cauchy bounds preserve convergence after a fixed division on a smaller polydisc. This gives membership in \(J\).

The same argument proves the equality in the analytic cotangent local rings. At a point where a \(z_i\) or \(\xi_i\) is nonzero it is a unit, so that paired generator reduces to the other coordinate; if both are zero it remains their squarefree product. A nonzero \(\eta_j\) makes the localized module zero. Otherwise the convergent Taylor argument in the remaining zero coordinates is exactly the one just given. Therefore the analytic extension of \(J\) is radical.

The actual characteristic variety, including the fibre analytic structure, is
\[
\operatorname{Ch}(M)=
\bigcup_{I\subset\{1,\ldots,k\}}
\{z_i=0\ (i\in I),\ \xi_j=0\ (j\notin I),\ \eta=0\}.
\tag{5.3m}
\]
Each component has complex dimension \(d\), with \(k+m=d\) independent coordinate equations in a \(2d\)-dimensional cotangent chart. It is the conormal to the coordinate intersection \((z_i=0:i\in I)\). Formula(5.3i) gives \(r\) copies of the reduced quotient on every generic component, so its generic length and characteristic-cycle coefficient are exactly \(r\).

Thus \(M\) is holonomic and its analytic associated graded is annihilated by the reduced ideal of its actual characteristic variety. This proves the analytic regularity condition from the analytic good-filtration definition above. 

#### A global good filtration on the SNC pair

Let \(E_\Sigma\) be the normalized analytic logarithmic extension from Theorem5.0 on a simple normal-crossing pair \((Z,D)\), and put \(M=E_\Sigma(*D)\). Label the globally smooth components \(D_i\). The formula
\[
F_pM=\sum_{\substack{a_i\geq0,\ \sum a_i\leq p\\ a\text{ has finite support}}}
E_\Sigma\bigl(D+\textstyle\sum_i a_iD_i\bigr),\qquad p\geq0
\tag{5.3n}
\]
defines a coherent good filtration. Only finitely many components meet any sufficiently small chart, and all terms from components disjoint from that chart have the same trivial twist there. Hence the sum is locally the finite sum in(5.3e), even if \(D\) has infinitely many locally finite components. Effective locally finite divisor twists are defined by their local boundary equations; unit changes give the same invertible sheaf.

The local normalized isomorphism with(5.3b) is holomorphic and has holomorphic inverse, so it preserves these lattice twists and their filtration. Every preceding conclusion therefore holds on every chart. In particular(5.3n) supplies one global good filtration meeting the analytic regularity definition; no unproved locality theorem for that definition is needed.


### The natural SNC de Rham comparison

**Theorem 5.4.** Let \(j:U=Z\setminus D\hookrightarrow Z\) be the complement of an analytic simple normal-crossing divisor on a complex manifold of dimension \(d\). For any finite-rank local system \(L\) on \(U\), let \(E_\Sigma\) be its normalized logarithmic extension from Theorem 5.0. The natural maps
\[
\Omega_Z^\bullet(\log D)\otimes E_\Sigma
\longrightarrow \Omega_Z^\bullet(*D)\otimes E_\Sigma
\longrightarrow Rj_*L
\]
are quasi-isomorphisms. These are unshifted de Rham complexes. With the course's left-module convention the result is
\(\operatorname{DR}_Z(E_\Sigma(*D))\simeq Rj_*L[d]\).
All residue Jordan blocks and all boundary crossings are included.

It suffices to prove the assertion on adapted polydiscs. Theorem 5.0 supplies the model
\[
V=\Delta_z^k\times\Delta_w^m,\quad U=V\setminus D,\quad
E=\mathcal O_V\otimes W,\quad
\nabla=d+\sum_{i=1}^k A_i\,d\log z_i,\quad
T_i=\exp(-2\pi iA_i).
\tag{5.4a}
\]
The matrices \(A_i\) commute, and their spectra lie in
\(0\leq\operatorname{Re}\alpha<1\). We use the proved
analytic flat-frame equivalence, Lemma 4.1 and Theorem 4.2.
The parameter-integration and Cauchy estimates are the earlier proofs linked before Theorem 5.0; the convergent Laurent expansion is proved in Lemma 5.0.2. The smooth-resolution and finite-cover comparison arguments needed below are also given explicitly.

#### The convergent Laurent homotopy

Order the normal exterior generators \(e_i=d\log z_i\) increasingly. If normal degree is p and tangential degree is a, order all normal generators before the tangential ones. The differential is \(d_N+(-1)^p d_w\). If \(H_w\) is the ordinary tangential radial homotopy, take \(H=(-1)^pH_w\). The normal matrices commute with \(H_w\); the two normal terms in \(dH+Hd\) cancel because \(d_N\) increases the normal degree \(p\), reversing the factor \((-1)^p\) in \(H\). The remaining identity is
\[
dH+Hd=1-\operatorname{ev}_{w=0}.
\tag{5.4b}
\]
The contraction is integral in the tangential variables and never changes any z-exponent or its finite lower bound. Normal convergence and parameter differentiation follow from the explicitly proved parameter-integral estimate.

After this retraction, write a form as a normally convergent Laurent sum \(\sum_n z^n v_n\), with exterior-valued coefficients. In degree n the normal differential is
\[
d_n=\sum_j (n_jI+A_j)e_j\wedge.
\]
For n different from zero choose the least i with \(n_i\ne0\) and define
\[
h_n=(n_iI+A_i)^{-1}\iota_i .
\tag{5.4c}
\]
Here \(\iota_i(e_{j_1}\wedge\cdots\wedge e_{j_p})\) is zero unless i occurs, and otherwise is \((-1)^{a-1}\) times the exterior term with i omitted, where i is in position a. Thus
\(\iota_i\epsilon_j+\epsilon_j\iota_i=\delta_{ij}\).
Commutativity of the matrices gives \(d_nh_n+h_nd_n=1\), with this exact exterior sign.

The inverse exists because its only possible zero eigenvalue would require an eigenvalue of \(A_i\) equal to the nonzero integer -n_i, excluded by the strip. On every generalized eigenspace it is a terminating geometric series in the nilpotent part.

For \(|n_i|>2\|A_i\|\), the Neumann series gives inverse norm at most \(2/|n_i|\). Only finitely many nonzero integers remain for each of the finitely many i, and all their inverses exist. Hence a single constant bounds every \(h_n\). Absolute normally convergent coefficient sums on a smaller polydisc remain convergent after applying this bounded multiplier. No exponent changes, including negative exponents. The logarithmic complex retains only nonnegative normal exponents; the meromorphic complex retains its original finite lower Laurent bounds.

Set \(p_w=\operatorname{ev}_{w=0}\), and let \(i_w\) insert w-independent forms. If \(h_N\) is the coefficient homotopy with h_0=0, the combined homotopy is \(h=H+i_wh_Np_w\). Substitution using the commutation of \(i_w\) and \(p_w\) with the normal differential gives
\[
dh+hd=1-i_Ap_A
\tag{5.4d}
\]
at the boundary stalk, where \(p_A\) extracts the zero normal coefficient after (5.4b), and \(i_A\) inserts the constant-coefficient normal forms. The residual complex is \(K^\bullet(A_i;W)\). The logarithmic-to-meromorphic inclusion is the identity on that residual complex. This proves the Laurent homotopy, including convergence and arbitrary residue Jordan blocks.

#### Fine resolutions and actual derived sections

Here all manifolds on which we require fine acyclicity are open subsets of real Euclidean space: \(U\), its open subsets, and the sector products below. A direct partition construction suffices.

Let \(O\) be such an open set with any open cover. For n positive set
\[
K_n=\{x\in O:|x|\leq n,\ 
\operatorname{dist}(x,\mathbb R^N\setminus O)\geq 1/n\},
\]
interpreting the distance as infinite if the complement is empty, and put \(K_0\) and \(K_{-1}\) empty. These sets are compact, exhaust \(O\), and satisfy \(K_n\subset\operatorname{int}K_{n+1}\). The compact band \(K_n\setminus\operatorname{int}K_{n-1}\) can be covered by finitely many smaller balls whose concentric larger closed balls lie in assigned cover members, in \(\operatorname{int}K_{n+1}\), and outside K_{n-2}. Such balls exist since the band misses K_{n-2}. All the smaller balls together cover \(O\). The larger balls are locally finite: near any point choose a neighbourhood inside some \(\operatorname{int}K_N\); all bands with n at least N+2 miss that neighbourhood, and there are only finitely many balls in the remaining bands.

On each larger ball use a smooth positive bump, zero outside its closed ball. For example \(e^{-1/s^2}\) for \(s=r^2-|x-a|^2>0\), extended by zero for s nonpositive, is smooth: successive derivatives are a polynomial in 1/s times the exponential, and the exponential-series bound dominates every such polynomial and its difference quotient at zero. The locally finite sum of these bumps is positive. Divide each by that sum. This proves the required smooth subordinate partition, with compact supports, including the boundary smoothness and local finiteness. The zero-dimensional and empty open sets are immediate.

Let \(\mathcal S=\mathcal C^\infty_O\) denote the sheaf of complex-valued smooth functions. Global sections are exact on \(\mathcal S\)-modules. Given a local lift of a section of a quotient on each cover member, multiply by the corresponding partition functions and extend by zero; the support is closed and contained in that member, so gluing with zero is valid. Their locally finite sum is a global lift. Kernels give left exactness.

Sheaves of modules and their derived categories, Theorem 5.4, supplies enough injectives for \(\mathcal S\)-modules. Forgetting to complex-vector-space sheaves preserves injectives: its left adjoint \(\mathcal S\otimes_\mathbb C-\) is exact on stalks. An injective \(\mathcal S\)-module resolution is consequently an injective complex-sheaf resolution, and exactness of global sections makes every \(\mathcal S\)-module acyclic for sections. This applies on every open subset of \(O\).

For completeness, the homotopy identity used here can be checked without a separate topological theorem. For a smooth homotopy \(F:O\times[0,1]\to O'\), write
\(F^*\alpha=\beta_t+dt\wedge\gamma_t\) and define \(H\alpha=\int_0^1\gamma_t\,dt\). The dt component of \(d(F^*\alpha)\) is \(\partial_t\beta_t-d_O\gamma_t\). Integration and the fundamental theorem give
\(d_OH+Hd=F_1^*-F_0^*\).
For the radial homotopy from a point to \(x\), this integrates the contraction with the radial vector field; in degree zero it gives the function minus its value at that point. Parameter differentiation justifies the calculation, and the same argument for holomorphic coefficients proves the tangential homotopy used in the Laurent homotopy argument above. Thus smooth de Rham forms on a convex lifted sector have only constant cohomology in degree zero.

Let \(L\) be the horizontal local system. The smooth flat de Rham complex \(\mathcal A^\bullet(L)\) resolves \(L\). In a local flat frame the preceding contraction proves exactness. Each term is an \(\mathcal S\)-module and hence acyclic. The bounded-below acyclic-term comparison, Theorem 6.1(4), therefore identifies
\[
R\Gamma(O,L)=\Gamma(O,\mathcal A^\bullet(L))
\tag{5.4e}
\]
in the derived category, compatibly with its augmentation from \(L\).

The same resolution computes \(Rj_*L\) by \(j_*\mathcal A^\bullet(L)\). To check this assertion directly, take a complex-sheaf injective resolution. Restrictions to an open remain injective by Corollary 5.5 of the same sheaf lesson. Thus at x the stalk of \(R^aj_*F\) is the filtered colimit of \(H^a(B\cap U,F)\) over neighbourhoods \(B\): stalks are such colimits, and filtered colimits of vector spaces are exact and commute with complex cohomology. For a smooth-form term F those groups vanish for a positive by the proved open-set acyclicity. The bounded acyclic-term comparison for j_* now applies.

Consequently the actual natural map of complexes representing the de Rham comparison is
\[
\Omega_V^\bullet(*D)\otimes E
\longrightarrow j_*\mathcal A^\bullet(L).
\tag{5.4f}
\]
The map restricts to \(U\) and regards each holomorphic flat form as a smooth flat form.
On \(U\) the two inclusions from \(L\) into holomorphic and smooth flat de Rham forms commute, and both are local resolutions. This verifies that (5.4f) represents the adjunction de Rham-to-\(Rj_*L\) map with its ordinary identification on \(U\); it is not a separately chosen derived isomorphism.

#### Two arcs and the finite Čech resolution

Fix positive radii \(r_i\) inside the polydisc and w=0. Use angle \(\theta_i\) increasing from zero to \(2\pi\); set \(u_i=i\theta_i\) along the chosen torus. Backwards transport from a point of a lifted path to the base point identifies coefficients with \(W\) by multiplication by
\[
\exp\!\left(\sum_i u_iA_i\right).
\]
Forward parallel transport has matrix \(T_i\), whereas these coefficients satisfy the deck relation with
\[
R_i=T_i^{-1}=\exp(2\pi iA_i).
\tag{5.4g}
\]
There is no replacement of the actual positive-loop monodromy by its inverse; the inverse occurs in the backwards coefficient trivialization.

For one angle take \(0<\varepsilon<\pi/4\) and the arcs represented by
\[
I_0=(-\varepsilon,\pi+\varepsilon),\qquad
I_1=(\pi-\varepsilon,2\pi+\varepsilon).
\tag{5.4h}
\]
Their intersection has two components. Call the component at \(\pi\) the middle seam and the one at zero the end seam. At the middle seam the chosen lifts and backwards frames agree. At the end seam use the \(I_1\) lift near \(2\pi\); the coefficient from the \(I_0\) lift near zero is multiplied by \(R\). Thus, with Čech differential section 1 minus section 0, the local-system Čech complex is
\[
C^0=W\oplus W,\quad C^1=W\oplus W,\quad
\delta(a,b)=(b-a,\ b-Ra).
\tag{5.4i}
\]
The first entry of \(C^1\) uses the middle frame; the second uses the end frame.

Define a projection to the oriented-circle complex \(K^0=W,\ K^1=W,\ d_K=R-I\) by
\[
p^0(a,b)=a,\qquad p^1(c,d)=c-d.
\tag{5.4j}
\]
A section and contraction are
\[
\begin{aligned}
i^0(v)&=(v,v),& i^1(v)&=(0,-v),&
h(c,d)&=(0,c).
\end{aligned}
\tag{5.4k}
\]
Direct substitution gives \(pi=1\), \(\delta i=id_K\), and
\(\delta h+h\delta=1-ip\). Thus this is a proved chain retraction, with the sign selecting the positive orientation from zero through \(\pi\) to \(2\pi\).

For k angles use the \(2^k\) sector-product cover of \(U\), with the corresponding radial variables and w variables unrestricted. It is more convenient to totalize its k successive two-cover Čech resolutions than to keep the redundant ordinary Čech complex of the \(2^k\) sets. This is a genuine finite open-cover resolution: in each coordinate the augmented two-cover complex is exact, and its contraction locally chooses a cover member containing the point. Iterating these finite exact resolutions gives the augmentation for the product cover. Its terms are finite direct sums over the connected components of sector intersections.

Every such connected component has convex logarithmic coordinates: log-radius in an open interval, angle in one of the intervals in (5.4h) or an overlap interval, and real coordinates of w in a convex polydisc. The local system is trivial there. The real radial homotopy in a flat frame and (5.4e) give its higher-cohomology vanishing.

Here is also a direct resolution comparison, avoiding any assumption that open direct image is exact. Let \(B\) be the iterated Čech complex of local-system sections and let \(C\) be the total iterated Čech complex of smooth flat forms. Inclusion
\[
b:B\longrightarrow C
\]
is a quasi-isomorphism: in every sector component the smooth complex has cohomology only the constant sections in degree zero. Finite filtration by Čech degree, and induction on the short exact sequences of that filtration, proves the total assertion. The augmentation
\[
a:\Gamma(U,\mathcal A^\bullet(L))\longrightarrow C
\tag{5.4l}
\]
is also a quasi-isomorphism. Indeed, for each two-cover the Čech sequence of each smooth-form sheaf is exact on sections. Surjectivity can be written explicitly: given c on the overlap, choose a subordinate pair \(\chi_0+\chi_1=1\), take \(-\chi_1c\) on the first open and \(\chi_0c\) on the second, extending by zero; their difference is c. Apply this in every coordinate. Filter the augmentation cone by form degree: its associated graded complexes are these exact Čech rows, so finite induction makes the cone acyclic. For b use instead the filtration by Čech degree and the already proved sector-form quasi-isomorphisms. All Čech degrees and all smooth form degrees are bounded here. Therefore the finite complex \(B\) computes actual derived sections through the concrete zigzag \(B\xrightarrow{b}C\xleftarrow{a}\Gamma(U,\mathcal A^\bullet(L))\).

#### Čech reduction equals backwards-transport integration

First retract the logarithmic radial coordinates to their chosen \(r_i\) and the tangential coordinates to zero, without changing angles. On each lifted sector this is a real convex homotopy with flat coefficients, and it agrees on overlaps because every transition is a constant \(R_i\). Restriction to the torus therefore gives a chain homotopy equivalence of smooth-form Čech totals. Local-system constants are unchanged. Denote the resulting angular Čech total by \(C_{\mathrm{ang}}\).

For one angle the angular de Rham degree a is zero or one, and the Čech degree p is zero or one. Totalize as \(D=\delta+(-1)^p d_\theta\). In total degree zero an element is a pair of functions \((f_0,f_1)\). In degree one it is
\[
(g_0\,d\theta,\ g_1\,d\theta;\ c,\ d),
\]
with the last pair functions on the middle and end seams in their chosen frames. Define
\[
\begin{aligned}
F^0(f_0,f_1)&=f_0(0),\\
F^1(g_0,g_1;c,d)
&=\int_0^\pi g_0(\theta)\,d\theta
 +\int_\pi^{2\pi}g_1(\theta)\,d\theta
 +c(\pi)-d(2\pi),\\
F^q&=0\quad(q\geq2).
\end{aligned}
\tag{5.4m}
\]
This is a chain map to \(K^\bullet(R-I;W)\). The only nontrivial degree check is on a degree-zero element:
\[
\begin{aligned}
F^1D(f_0,f_1)
={}&f_0(\pi)-f_0(0)+f_1(2\pi)-f_1(\pi)\\
 &+[f_1(\pi)-f_0(\pi)]
   -[f_1(2\pi)-Rf_0(0)]\\
={}&(R-I)f_0(0).
\end{aligned}
\tag{5.4n}
\]
The remaining degrees land in zero target degrees. This proof also works with coefficients in any complex of forms in the other variables: evaluation and integration commute with their differential and with all other commuting transition matrices.

On the inclusion of local-system Čech constants, F is exactly the reduction p in (5.4j). On the augmentation of global angular forms, the seam terms are zero, so F is exactly backwards-transport integration around the positive circle. This is equality of chain maps, not only equality on cohomology classes.

For multiple coordinates totalize the pairs in the interleaved order
\[
(p_1,a_1),\ (p_2,a_2),\ldots,(p_k,a_k).
\]
The differential in the i-th pair is
\[
(-1)^{\sum_{j<i}(p_j+a_j)}
\bigl(\delta_i+(-1)^{p_i}d_{\theta_i}\bigr).
\tag{5.4o}
\]
The standard total order with all Čech degrees first and all form degrees second is converted to this order by multiplication in each multidegree by
\[
(-1)^{\sum_{i<j}a_ip_j}.
\tag{5.4p}
\]
When \(p_i\) increases, the flip exponent changes by \(\sum_{j<i}a_j\), adding exactly the missing earlier form degrees to the standard Čech sign. When \(a_i\) increases, it changes by \(\sum_{j>i}p_j\); adding this to the standard form sign \(\sum_jp_j+\sum_{j<i}a_j\) leaves \(\sum_{j\leq i}p_j+\sum_{j<i}a_j\), as in (5.4o). On global-form augmentation all \(p_i\) are zero, and on constant Čech inclusion all \(a_i\) are zero; therefore the flip has sign one on both sides of the required comparison diagram.

Apply the maps (5.4m) one coordinate at a time, with the order (5.4o). They commute with the remaining operators. The resulting map
\[
\mathcal F:C_{\rm ang}\longrightarrow K^\bullet(R_i-I;W)
\tag{5.4q}
\]
is a chain map. Its composite with the constant Čech inclusion is exactly the iterated projection (5.4j). The latter is a chain homotopy equivalence: apply (5.4k) in each coordinate, multiplying its contraction in the i-th coordinate by \((-1)^{\sum_{j<i}p_j}\) to cancel the other coordinate differentials. After an earlier coordinate has been reduced, its Koszul degree replaces \(p_j\) in this sign. All matrices involved commute. Finite induction proves the retraction.

Since b and the radial restriction are quasi-isomorphisms, and \(\mathcal F b\) is this retraction, \(\mathcal F\) is a quasi-isomorphism too. Most importantly, its composite with the actual smooth-form augmentation and radial restriction is the cubical period map:
\[
\begin{aligned}
\mathcal I^p(\alpha)_I
={}&
\int_{[0,2\pi]^I}
\operatorname{PT}_{\gamma_I}^{-1}(\gamma_I^*\alpha),\\
I={}&\{i_1<\cdots<i_p\}.
\end{aligned}
\tag{5.4r}
\]
Here the coordinates outside I have angle zero, all radii are \(r_i\), w=0, the integration orientation is \(d\theta_{i_1}\wedge\cdots\wedge d\theta_{i_p}\), and the backwards transport uses the lifted paths with those increasing angles. For p zero this is evaluation at the base point.

The following figure records the chain maps computing the actual comparison.


The composite of restriction, smooth-form augmentation, radial restriction and \(\mathcal F\) is exactly \(\mathcal I\). On the sector-constant complex its composite is exactly the proved Čech retraction. The augmentations and sector-constant comparisons used to identify the target were proved quasi-isomorphisms above. This verifies the natural map rather than selecting an abstract isomorphism between its source and target cohomology.

![The two overlaps and the actual chain-map comparison](assets/snc-period-comparison.png)

*The two colored arc portions, together with the purple overlap portions, lie on the same circle; \(\varepsilon=\pi/8\) in this drawing. The upper arc is \(I_0\) and the lower arc is \(I_1\). Positive orientation runs leftward at the top and rightward at the bottom. The end seam uses the \(I_1\) lift near \(2\pi\), forcing \(R\) in (5.4i). The right panel displays the actual comparison whose one-coordinate chain map is (5.4m), and the lower panel is its zero-Laurent restriction (5.4t). Exact proof locators: (5.4i)–(5.4p), (5.4s)–(5.4u). Reproducible figure source. Earlier freely accessible human reading for the normalized logarithmic model: [Deligne, IAS author edition, Chapter II, Section 5](https://publications.ias.edu/sites/default/files/Number9.pdf); all new period and Čech arguments are proved here.*

#### The oriented cube differential

Write the Koszul differential as \(\sum_i e_i\wedge(R_i-I)\). In an increasing index set \(J=\{j_1<\cdots<j_{p+1}\}\), its component is
\[
(d_Kv)_J
=\sum_{a=1}^{p+1}(-1)^{a-1}
(R_{j_a}-I)v_{J\setminus\{j_a\}}.
\tag{5.4s}
\]
This is also the exact integration boundary formula. For the positively oriented J-cube, its boundary is the signed sum
\[
\sum_a(-1)^{a-1}
\bigl(\text{face }\theta_{j_a}=2\pi-
      \text{face }\theta_{j_a}=0\bigr),
\]
with the remaining coordinates in increasing order. On the upper face backwards coefficients are R_{j_a} times those on the lower face. The coefficient matrices commute with transport in all remaining directions. Iterated one-variable integration and the fundamental theorem of calculus therefore give
\(\mathcal I(d_\nabla\alpha)=d_K\mathcal I(\alpha)\).
This proves the needed Stokes identity directly for each exterior component; no independent general Stokes theorem is imported.

The signs in (5.4o)–(5.4p), the seam subtraction in (5.4m), the face signs in (5.4s), and the minus sign in the original forward monodromy \(T_i\) are mutually consistent. Replacing \(R_i\)-I by \(T_i\)-I would require an additional explicit coordinate cochain isomorphism; it is not done here.

#### Invertible periods and the natural comparison

For a zero-Laurent coefficient \(v_I\) in the normalized frame, the associated form is \(v_I\,d\log z_{i_1}\wedge\cdots\wedge d\log z_{i_p}\). On its cube,
\[
\operatorname{PT}^{-1}\gamma_I^*\alpha
=\exp\!\left(\sum_{i\in I}u_iA_i\right)v_I
\,du_{i_1}\wedge\cdots\wedge du_{i_p}.
\]
Use ordered iterated integration of this finite-dimensional continuous matrix-valued function on the compact rectangle. Factor the commuting exponentials and pull the independent factors through each one-variable integral. The period is
\[
\mathcal I^p i_A(v)_I
=\left(\prod_{i\in I}Q_i(A_i)\right)v_I,\qquad
Q_i(A_i)=\int_0^{2\pi i}e^{uA_i}\,du.
\tag{5.4t}
\]
For the empty I the product is the identity. The factors of i from \(du_i=i\,d\theta_i\) are already included in each \(Q_i\); there is no additional phase. Thus these are the actual comparison periods on the proved Laurent retract.

Differentiating the matrix exponential and integrating gives \(Q_i(A_i)A_i=R_i-I\). For a scalar eigenvalue alpha, its value is \(2\pi i\) at zero and \((e^{2\pi i\alpha}-1)/\alpha\) otherwise. Its only possible nonzero zeros are nonzero integers, excluded by the strip. On a generalized eigenspace the value is the finite Taylor polynomial in the nilpotent part, with nonzero scalar leading coefficient; its inverse is another terminating geometric series. Every \(Q_i\) is therefore invertible, including all commuting Jordan blocks, and commutes with every \(A_j\) and Q_j.

The finite map (5.4t) is consequently a chain isomorphism \(K^\bullet(A_i;W)\to K^\bullet(R_i-I;W)\), with the differential (5.4s). To connect it with the full natural source, use the actual homotopy (5.4d):
\[
\mathcal I-\mathcal I i_Ap_A
=d_K(\mathcal I h)+(\mathcal I h)d_\nabla.
\tag{5.4u}
\]
All operations are defined on smaller polydiscs: h preserves normal convergence and finite pole bounds, and \(\mathcal I\) integrates smooth forms along a compact torus disjoint from D. Thus this is an actual chain homotopy, not a formal exchange of divergent Laurent sums. It identifies the full natural comparison with the isomorphism (5.4t) on the proved Laurent retract.

Taking boundary stalks is legitimate. Shrinking polydiscs are cofinal; each meromorphic germ is represented with finite pole bounds on one of them. The fine-resolution model for \(Rj_*L\) applies on every one. On each sufficiently small polydisc choose radii inside it: the exact diagram and (5.4t) prove the quasi-isomorphism for its finite-pole coefficient complex and its target of smooth forms. Those actual maps (5.4f), independent of the chosen radii, commute with restriction. Their filtered colimit therefore gives the stalk comparison. This uses no asserted equality of the radius-dependent period chain models. The bounded coefficient multiplier in (5.4c) preserves convergence on every compact subpolydisc, so it defines the homotopy throughout the original open polydisc as well as at its stalk.

It follows that the actual natural maps
\[
\Omega_V^\bullet(\log D)\otimes E
\longrightarrow
\Omega_V^\bullet(*D)\otimes E
\longrightarrow Rj_*L
\tag{5.4v}
\]
are quasi-isomorphisms in the normalized regular model. At a point on fewer components, repeat the same proof with precisely those boundary coordinates. Off D use the ordinary analytic flat Poincaré lemma. This retains every crossing, arbitrary finite rank and all nilpotent parts.


### Algebraic de Rham comparison for regular connections

**Theorem 5.5.** Let \(X\) be any smooth separated finite-type complex algebraic variety. Let \((E,\nabla)\) be a finite-rank integrable algebraic connection regular along every algebraic curve, including infinity, and let \(L=\ker\nabla^{\mathrm{an}}\) be its horizontal local system. Restriction of algebraic forms to analytic forms, followed by the analytic flat de Rham augmentation, induces a natural quasi-isomorphism
\[
R\Gamma\bigl(X,(\Omega_X^\bullet\otimes E,\nabla)\bigr)
\longrightarrow R\Gamma(X^{\mathrm{an}},L).
\tag{5.5a}
\]
Both de Rham complexes here are unshifted. With the lesson12 left-module convention, on each connected component of dimension \(d\) apply the common shift \([d]\) to both sides. The theorem allows nonproper and non-quasi-projective \(X\), arbitrary rank and residue Jordan blocks.

This is the regular-coefficient comparison discussed in [Deligne's freely accessible IAS-hosted lecture notes, Chapter II, Theorem 6.2](https://publications.ias.edu/sites/default/files/Number9.pdf). We prove it below using the preceding SNC results and the exact earlier cohomology theorems.

The comparison in (5.5a) is defined before choosing a compactification. If \(a:X^{\mathrm{an}}\to X\) is the continuous map to the Zariski space, restriction gives a map of complexes of complex-vector-space sheaves from algebraic forms to \(a_*\) of analytic forms. It commutes with the connection differential because analytic differentiation restricts to algebraic differentiation. Passing to derived sections gives the comparison to analytic de Rham cohomology. The analytic flat-frame proof, Lemma 4.1 and Theorem 4.2, and the holomorphic Poincaré proof, Proposition 1.2, identify the latter complex with \(L\). The augmentation from \(L\) is a natural quasi-isomorphism; its inverse in the derived category introduces no choice of bases or periods.

Theorem 5.1 provides a smooth proper compactification \(P\) of \(X\) with SNC boundary \(D\), including non-quasi-projective \(X\). It also supplies the normalized algebraic logarithmic bundle \((\bar E,\bar\nabla)\) on \(P\), restricting to \((E,\nabla)\). These assertions were proved there using the earlier Nagata, resolution and full proper GAGA proofs. Write \(j:X\hookrightarrow P\).

We first prove the algebraic pole-removal step; it has no analytic convergence assumption.

#### Removing finite algebraic pole layers

**Lemma 5.5.1.** On a smooth algebraic SNC pair over \(\mathbf C\), let \((F,\nabla)\) be an integrable logarithmic bundle all of whose boundary residue eigenvalues lie in \(0\leq\operatorname{Re}\alpha<1\). Then
\[
(\Omega_P^\bullet(\log D)\otimes F,\nabla)
\longrightarrow
(\Omega_P^\bullet(*D)\otimes F,\nabla)
\tag{5.5b}
\]
is a quasi-isomorphism of complexes of complex-vector-space sheaves.

**Proof.** Work locally in étale coordinates \(z_1,\ldots,z_k,w_1,\ldots,w_m\) adapted to the SNC divisor, and choose a frame of \(F\). Such coordinates are the smooth SNC coordinates used in the earlier local construction. Write \(e_i=d\log z_i\). The local logarithmic connection is
\[
\nabla=d+\sum_i B_i e_i+\sum_\ell C_\ell\,dw_\ell,
\tag{5.5c}
\]
where all matrix entries are regular. No constant-matrix algebraic gauge is assumed.

Fix a boundary coordinate \(z_i\). Allow any already chosen other boundary coordinates to be inverted; localization is exact and preserves the connection identities. For \(n\geq0\), the logarithmic complex with coefficients \(z_i^{-n}F\) is stable under the differential: differentiating \(z_i^{-n}\) contributes only \(-n e_i\). There are termwise injective inclusions from the \(n-1\) complex into the \(n\) complex. Their quotient, for \(n\geq1\), is the complex supported on \(D_i=(z_i=0)\) with coefficient bundle
\[
F|_{D_i}\otimes\mathcal O_P(nD_i)|_{D_i}.
\tag{5.5d}
\]
Indeed, multiplication by \(z_i^n\) identifies its local coefficient module with \(F/z_iF\). Logarithmic forms are locally free with basis the \(e_j,dw_\ell\), so tensoring their degree pieces preserves this quotient description. The line twist in (5.5d) records the normal-frame changes; it is not globally replaced by a trivial bundle.

Locally separate the exterior generator \(e_i\) from all remaining generators, putting \(e_i\) first. On this quotient the differential is
\[
d_Q=\delta+e_i\wedge N_i,\qquad
N_i=\operatorname{Res}_{D_i}(\nabla)-nI.
\tag{5.5e}
\]
Here \(\delta\) is the tangential connection differential, with its exterior sign when acting on a term containing \(e_i\). Derivatives \(z_i\partial_i\) of regular coefficients vanish modulo \(z_i\); therefore the normal differential is exactly the displayed residue minus \(n\).

The endomorphism \(N_i\) is invertible. At every complex closed point its eigenvalues are \(\alpha-n\), which cannot be zero in the specified strip. Its determinant is thus a unit locally: any nonempty determinant-zero locus on this finite-type complex scheme would contain a closed point, by the earlier closed-point Nullstellensatz test. The adjugate formula supplies its inverse in the algebraic coefficient ring, and localization in other coordinates preserves it. Nilpotent residue parts cause no exception.

Integrability makes \(N_i\) parallel for \(\delta\). Explicitly, the commutator of \(z_i\partial_i+B_i\) with each tangential connection operator is zero. Modulo \(z_i\), its derivative in the normal direction disappears, leaving
\[
V(B_i|_{D_i})+[B_V|_{D_i},B_i|_{D_i}]=0
\quad\text{for each tangential operator }V+B_V.
\]
Equivalently, the tangential connection commutes with the residue endomorphism. Differentiating \(N_iN_i^{-1}=I\) then gives the same commutation for \(N_i^{-1}\).

Let \(\iota_i\) be contraction with the normal exterior generator. It anticommutes with the tangential exterior differential and satisfies
\(\iota_i(e_i\wedge\,\cdot)+e_i\wedge\iota_i=1\).
Consequently the actual algebraic map
\[
h_Q=N_i^{-1}\iota_i
\quad\hbox{satisfies}\quad
d_Qh_Q+h_Qd_Q=1.
\tag{5.5f}
\]
Thus every finite added-pole quotient is contractible. The short exact sequence of complexes makes every finite inclusion a quasi-isomorphism.

The union over \(n\) gives localization in \(z_i\). Filtered colimits of modules, and of their underlying complex-vector-space sheaves, are exact on stalks; they therefore commute with cohomology of these complexes. This union preserves the quasi-isomorphism. Repeat for each of the finitely many boundary coordinates in the chart. Every meromorphic algebraic section has a finite denominator power in each coordinate. After all localizations, logarithmic and ordinary differential forms have the same basis module because every \(z_i\) is a unit. This gives precisely the right-hand complex in (5.5b).

The inclusions in (5.5b) are intrinsic, so these stalkwise proofs prove the global quasi-isomorphism. The contractions need not agree on overlaps: only acyclicity of the quotient is used for that global conclusion. Off the boundary the inclusion is already the identity. The zero-rank and boundary-free cases are immediate. \(\square\)

#### The affine open and its meromorphic complex

The open immersion \(j\) is affine. Here is the affine-base check even if its boundary ideal on an affine open is not principal. The quasi-coherent algebra
\(\mathcal A=\bigcup_{n\geq0}\mathcal O_P(nD)\) has, on a neighbourhood with boundary equation \(q\), the algebra \(\mathcal O_P[1/q]\). Its relative spectrum is therefore locally the distinguished open \(D(q)\); these identifications agree in their common rational-function algebra and glue to \(X\). On any affine \(V=\operatorname{Spec}R\), the affine sheaf/module equivalence, Theorem 1.2 identifies \(\mathcal A|_V\) with the algebra associated to \(A=\Gamma(V,\mathcal A)\). Its relative spectrum there is \(\operatorname{Spec}A\): on every distinguished open \(D(f)\subset V\), the rings are \(A_f\), by the same localization equivalence. Thus \(j^{-1}V\) is affine for every affine \(V\), proving the assertion. The algebra is quasi-coherent because each divisor twist is invertible and filtered unions preserve quasi-coherence on these local module charts.

Degree by degree, localization gives the intrinsic equality
\[
j_*(\Omega_X^p\otimes E)
=\Omega_P^p(*D)\otimes\bar E.
\tag{5.5g}
\]
The equality is first the elementary module formula on \(D(q)\), where sections are localized at \(q\); it glues because both sides are the subsheaf of rational forms restricting to the given regular form on the open. The connection differentials agree by the same Leibniz calculation.

Cohomology of affine schemes and Serre's criterion, Theorems 2.2 and 3.2, proves that every quasi-coherent sheaf is \(j_*\)-acyclic for this affine morphism. This applies to each \(\Omega_X^p\otimes E\), including its underlying sheaf cohomology. The complex has only degrees zero through \(\dim X\). The bounded acyclic-term comparison, Theorem 6.1(4) therefore identifies its derived open image with its ordinary termwise image (5.5g), even though its differential is \(\mathbf C\)-linear rather than \(\mathcal O\)-linear.

Composing derived sections with this derived image gives
\[
\begin{gathered}
R\Gamma\bigl(X,\Omega_X^\bullet\otimes E\bigr)
\\ \simeq
R\Gamma\bigl(P,\Omega_P^\bullet(*D)\otimes\bar E\bigr)
\\ \simeq
R\Gamma\bigl(P,\Omega_P^\bullet(\log D)\otimes\bar E\bigr).
\end{gathered}
\tag{5.5h}
\]
The last arrow is the inverse of the proved inclusion (5.5b). Composition of derived sections and direct image here is the usual sheaf identity: direct image preserves injectives as right adjoint to exact inverse image of complex-vector-space sheaves. Thus no unproved nonproper coherent or analytic direct-image comparison has entered.

#### Proper GAGA for the complex-linear differential

Each degree \(\Omega_P^p(\log D)\otimes\bar E\) is coherent. Proper cohomological GAGA, Theorem 6.7, proves the natural cohomological comparison for every coherent sheaf on every proper complex scheme, including nonprojective ones.

We apply it to the finite logarithmic complex as a complex of \(\mathbf C\)-sheaves. Its differential is not \(\mathcal O_P\)-linear, so this requires the following explicit naturality check rather than treating it as a complex in the coherent \(\mathcal O_P\)-module category.

Restriction of algebraic logarithmic forms gives a map of \(\mathbf C\)-sheaf complexes into the analytic logarithmic complex under \(a:P^{\mathrm{an}}\to P\). In every frame it commutes with exterior differentiation and with the algebraic connection matrices in (5.5c). On each individual term its map on cohomology is exactly the canonical GAGA map from Theorem 6.7, since both maps extend the same restriction on sections. Derived restriction/adjunction is natural for these \(\mathbf C\)-linear sheaf maps.

Filter both derived-section totals by logarithmic degree. Their first pages are the cohomology of those individual coherent terms. The actual restriction map just constructed is an isomorphism on every first-page group by GAGA and respects all differentials. The filtration has \(\dim X+1\) columns, hence the filtration of each total degree is finite. Kernels and quotients give isomorphisms on each subsequent page, and induction over the finite abutment filtration proves
\[
\begin{gathered}
R\Gamma\bigl(P,\Omega_P^\bullet(\log D)\otimes\bar E\bigr)
\\ \simeq
R\Gamma\bigl(P^{\mathrm{an}},
 \Omega_{P^{\mathrm{an}}}^\bullet(\log D^{\mathrm{an}})
 \otimes\bar E^{\mathrm{an}}\bigr).
\end{gathered}
\tag{5.5i}
\]
Analytification of these logarithmic bundles is the displayed analytic bundle: their local free bases \(d\log z_i,dw_\ell\), algebraic transition matrices and connection formula simply extend to holomorphic coefficients. Exact coherent analytification was proved earlier; no infinite meromorphic union is passed through GAGA.

#### The intrinsic global comparison

Theorem 5.4, now already proved earlier in this lesson, identifies the analytic logarithmic complex in (5.5i) with \(Rj^{\mathrm{an}}_*L\) by the actual natural map. Taking derived sections gives \(R\Gamma(X^{\mathrm{an}},L)\).

Combining (5.5h), (5.5i) and this analytic SNC comparison proves an isomorphism between the two sides of (5.5a). It proves the specified natural map itself. Indeed the restriction map from the algebraic logarithmic complex to the analytic one commutes with restriction to \(X\) and with both inclusions into meromorphic forms. The algebraic localization equality (5.5g) is restriction of the same forms. The analytic map of Theorem 5.4 is restriction into smooth flat forms, commuting with the local-system augmentation. These are exactly the maps used to define (5.5a). Their derived-section diagram therefore commutes, and its other arrows are the proved quasi-isomorphisms. The remaining arrow (5.5a) is consequently a quasi-isomorphism.

This also proves independence of compactification and of the normalized lattice used in the proof: the comparison (5.5a) was defined intrinsically on \(X\). It is natural in every horizontal algebraic bundle map; all restriction, augmentation and inclusion maps in its construction are natural, and Theorem 5.0 extends such maps to the normalized boundary bundles.

With \(E=\mathcal O_X\) and its ordinary differential, this gives the full algebraic de Rham comparison with complex constant-sheaf cohomology. The proof does not assert general singular regular-holonomic RH, any arbitrary nonproper D-module comparison, or analytic proper-image regularity. Those retain their original required scope.


### Gauss–Manin regularity on every smooth base

For a smooth algebraic map with regular coefficient connection, the generic relative de Rham images are finite-rank regular integrable connections. Theorem 5.6 below proves this for every smooth separated finite-type complex base, with no properness or quasi-projectivity assumption on the original map. We first construct its logarithmic lattices over a curve, then prove integrability and base change along every curve on the general base.

Freely accessible reading: [Deligne, Chapter II, Theorem 7.9 and Lemma 7.10](https://publications.ias.edu/sites/default/files/Number9.pdf), printed pp. 113–120. The proof here supplies the lattice, corrected Čech action, finite flat-complex base change and actual transfer calculation; that reading reference replaces none of those arguments.

#### The curve lattice and the completed pair

**Lemma 5.6.1 (the curve lattice).** Let \(f:X\to C\) be a smooth morphism of smooth separated finite-type complex varieties, with \(C\) a smooth connected algebraic curve. Let \((E,\nabla)\) be an integrable algebraic connection on \(X\), regular along every algebraic curve including infinity.

There is a dense open \(C_0\subset C\) such that
\[
H^a=R^a f_*(\Omega_{X/C}^\bullet\otimes E)|_{C_0}
\]
are finite-rank regular connections, with the actual Gauss–Manin action constructed below. Every such generic image has a logarithmic lattice at every point of the smooth complete model \(\bar C\). All ranks and residue Jordan blocks are allowed.

We use the following rank criterion for maps between smooth complex algebraic varieties. Such a map is smooth at a point exactly when its differential is surjective there. Necessity follows from the differential basis in a standard smooth chart. For sufficiency, choose étale coordinates \(y_1,\ldots,y_b\) on the base and complete their pulled-back differentials to a source coframe with differentials of additional local functions. The chosen-coordinate theorem, Smooth morphisms, Theorem 4.1 applied to the source over \(\mathbf C\), makes the resulting source coordinate map étale over affine space. The target, the product of the base with the additional affine coordinates, is étale over the same affine space. A morphism between two such étale objects is étale: in a square-zero lifting problem, first take the unique source lift over affine space; its map to the target agrees with the prescribed target map by the target's own uniqueness. Finite presentation follows from the finite-type affine presentations over the Noetherian target. Composition with the smooth affine projection to the original base proves sufficiency. Thus Jacobian rank failure is precisely the critical locus used below. The same chosen-coordinate argument allows prescribed independent boundary functions to be retained among the relative coordinates.

Take a smooth proper absolute compactification \(Z\) of \(X\) with a morphism \(g:Z\to\bar C\) extending \(f\). Namely, take the closure of the graph in an absolute compactification times \(\bar C\), and resolve/principalize outside the smooth open. Nagata existence, Appendix N, Theorem N.E.1, and resolution and SNC compactification, Theorem 4.8 and Corollary 5.2, provide these operations and preserve \(X\).

Remove from \(C\) the finitely many bad values of \(g\) and of its finitely many closed boundary intersections, obtaining \(C_0\), so that \(g\) and every horizontal boundary intersection are smooth over \(C_0\). Each bad image is closed by properness. Here is the needed rank and image argument. A function field dominating the curve has a transcendence basis containing its nonconstant local base parameter \(t\). The derivation sending \(t\) to one and the remaining basis elements to zero extends through the finite algebraic field extension: differentiating the minimal polynomial of an algebraic element solves for its derivative because its polynomial derivative is nonzero in characteristic zero. Thus \(dt\) is nonzero in that function field. The smooth Jacobian criterion, Theorem 2.1 and Corollary 2.2, together with local standard coordinates, Theorems 4.1 and 5.1, identifies the nonsmooth locus of a map between the smooth source and curve with the vanishing of its differential. No irreducible component of this critical locus can dominate the curve: if it did, the differential's vanishing in \(\Omega_Z^1\) restricted to its function field would imply \(dt=0\) in the differentials of that function field, contradicting the derivation just constructed. Its proper image is therefore a proper closed subset of the curve, hence finite. Apply the same argument to each smooth closed boundary intersection. Nondominating closed intersections also have finitely many images and are removed. This treats the closed intersections, which are proper, before their open strata.

Add fibres over \(T=\bar C\setminus C_0\) to the complement and principalize its reduced ideal on the already smooth \(Z\), preserving \(X_0=f^{-1}(C_0)\). This also preserves a smooth relative SNC pair over \(C_0\), for the following reason. In a relative SNC chart over \(C_0\), the boundary ideal is generated by \(z_1\cdots z_k\), and the chart is étale over a product of these normal affine coordinates, the remaining relative coordinates and a base neighbourhood. Principalization commutes with smooth morphisms, by principalization, Theorem 3.1(4). The sequence on this chart is consequently the pullback of the sequence for the ideal \((z_1\cdots z_k)\) on \(\mathbf A^k\), with the relative and base factors retained. Its resulting ambient scheme is smooth and its divisor SNC by that theorem; their products and étale pullbacks are smooth and SNC relative to the base. These local assertions apply on the charts of the global sequence. Thus the new \(Z\) is smooth proper; its reduced boundary \(D=Z\setminus X_0\) is SNC, contains \(g^{-1}T\), and is relative SNC over \(C_0\). Theorem 5.1 supplies a normalized logarithmic extension \(\bar E\) of \(E|_{X_0}\) on \(Z\). No semistable reduction is asserted: components of \(g^*T\) may have any positive multiplicity.

#### Relative logarithmic forms and explicit lifts

Set
\[
\mathcal A=\Omega_Z^1(\log D),\qquad
\mathcal B=g^*\Omega_{\bar C}^1(\log T),\qquad
\mathcal Q=\mathcal A/\mathcal B.
\tag{5.6a}
\]
The inclusion is a subbundle. Over \(C_0\) this is the relative SNC coordinate calculation. At a point over \(t_0\in T\), choose a base parameter \(t\) and SNC coordinates. The local equation is
\[
g^*t=u\prod_i z_i^{a_i},\qquad
u\text{ a unit},\quad a_i\geq0,\quad
\text{some }a_i>0.
\tag{5.6b}
\]
The exponents are the vertical divisor multiplicities. This algebraic factorization can be checked without a new factoriality theorem: the coordinate-factor proof of Theorem 5.0 gives the corresponding analytic monomial and unit; the faithfully flat analytic stalk comparison, Theorem 4.1 contracts divisibility to the algebraic stalk. The resulting algebraic quotient is a unit because its analytic image is a unit and faithful flatness contracts the unit ideal.

Pullback of \(dt/t\) is
\[
\alpha=\sum_i a_i\,d\log z_i+du/u.
\tag{5.6c}
\]
Choose a vertical index \(i\) through the point. The coefficient
\(b_i=a_i+z_i(\partial_i u)/u\) is a unit locally, since modulo \(z_i\) it equals the nonzero integer \(a_i\). Replacing \(d\log z_i\) by \(\alpha\) proves the subbundle assertion. The logarithmic base field \(\theta=t\partial_t\) has the lift
\[
v_i=b_i^{-1}z_i\partial_i,\qquad v_i(g^*t)=g^*t.
\tag{5.6d}
\]
This field is tangent to every boundary component. It contracts logarithmic forms to regular logarithmic forms. Arbitrary positive \(a_i\) cause no obstruction. Over \(C_0\), relative SNC coordinates similarly lift ordinary base fields.

Since \(d\alpha=0\), the exterior ideal of pulled-back base forms is differential-stable. Quotienting the absolute logarithmic connection gives
\[
K^\bullet=(\Lambda^\bullet\mathcal Q\otimes\bar E,\nabla_{\mathrm{rel}}).
\tag{5.6e}
\]
It is a finite complex of coherent locally free terms; its differential is \(g^{-1}\mathcal O_{\bar C}\)-linear, though not \(\mathcal O_Z\)-linear.

Each individual term has coherent ordinary proper higher images, by ordinary proper coherence, Theorem 4.1. These are also its underlying complex-sheaf higher images: forgetting \(\mathcal O_Z\)-modules preserves injectives because its left adjoint \(\mathcal O_Z\otimes_{\mathbf C}-\) is exact on stalks. The finite-degree spectral sequence of this complex is a spectral sequence of \(\mathcal O_{\bar C}\)-modules. Its pages are kernels and quotients of coherent modules and its abutment has a finite coherent filtration. Thus every \(A^a=\mathcal H^a(Rg_*K^\bullet)\) is coherent. This argument does not treat the relative differential as \(\mathcal O_Z\)-linear and does not invoke D-module proper regularity.

#### The corrected Čech Gauss–Manin connection

Over an affine base neighbourhood choose a finite affine adapted cover \(U_i\) of its inverse image, and logarithmic lifts \(v_i\) of a base logarithmic field \(\theta\). Finite intersections are affine by separatedness. Affine acyclicity, Theorems 2.2 and 3.2, and the finite Čech comparison and basis criterion, Theorems 3.2 and 4.1, make the ordered Čech total compute \(Rg_*K^\bullet\) over that neighbourhood.

On relative forms put
\[
L_i=\nabla\iota_{v_i}+\iota_{v_i}\nabla.
\tag{5.6f}
\]
It descends to the relative quotient: Lie derivative by a lift preserves the ideal of pulled-back base forms. It commutes with the relative differential by integrability and obeys
\[
L_i(g^*a\,s)=g^*(\theta a)s+g^*a\,L_i(s).
\tag{5.6g}
\]
All terms preserve the logarithmic bundle.

For differences of lifts define \(H_{ij}=\iota_{v_j-v_i}\). These are well-defined contractions on relative forms because their vector fields are vertical. Subtraction of (5.6f) and addition of differences give
\[
L_j-L_i=d_KH_{ij}+H_{ij}d_K,\qquad
H_{ij}+H_{jk}=H_{ik}.
\tag{5.6h}
\]

For a cochain \(c\) of Čech/form bidegree \((p,q)\), totalize by
\(D=\delta+(-1)^p d_K\). Define the degree-zero operator with diagonal component
\[
(\mathcal Lc)_{i_0\ldots i_p}
=L_{i_0}c_{i_0\ldots i_p}
\]
and extra component in bidegree \((p+1,q-1)\)
\[
(\mathcal Lc)_{i_0\ldots i_{p+1}}
=(-1)^pH_{i_0i_1}c_{i_1\ldots i_{p+1}}.
\tag{5.6i}
\]
Both components are added.

The diagonal Čech commutator is
\[
(\delta L-L\delta)c_{i_0\ldots i_{p+1}}
=(L_{i_1}-L_{i_0})c_{i_1\ldots i_{p+1}}.
\]
The extra component's internal commutator is its negative by (5.6h): the output internal sign is \((-1)^{p+1}\), the input one is \((-1)^p\), and its own factor is \((-1)^p\). Its Čech commutator vanishes by \(H_{i_0i_2}=H_{i_0i_1}+H_{i_1i_2}\); terms deleting later indices cancel in pairs. Each \(L_i\) commutes with \(d_K\). Consequently
\[
D\mathcal L=\mathcal LD.
\tag{5.6j}
\]

For new lifts \(v_i'\), let \(h_i=\iota_{v_i'-v_i}\). The degree-minus-one diagonal operator \(\mathcal H\) acts on bidegree \((p,q)\) by \((-1)^p h_{i_0}\). The exact identity is
\[
\mathcal L'-\mathcal L=D\mathcal H+\mathcal HD.
\tag{5.6k}
\]
Its diagonal internal part is \(d_Kh_i+h_id_K=L_i'-L_i\); its extra part is \((-1)^p(h_{i_1}-h_{i_0})\), the change in (5.6i). Thus the operator on cohomology is independent of lifts. The intrinsic construction below proves compatibility with arbitrary refinements and independence of the ordered covers.

There is also an intrinsic definition that proves cover compatibility. The absolute logarithmic complex has the short exact sequence of complex-vector-space complexes
\[
0\longrightarrow
g^*\Omega_{\bar C}^1(\log T)\otimes K^{\bullet-1}
\xrightarrow{\ \alpha\wedge\ }
(\Omega_Z^\bullet(\log D)\otimes\bar E,\nabla)
\longrightarrow K^\bullet\longrightarrow0.
\tag{5.6l}
\]
The first differential is \(-d_K\), since \(\nabla(\alpha\wedge s)=-\alpha\wedge\nabla s\). Base dimension one makes every exterior term with two base forms zero, so the kernel is exactly this first complex. Locally the chosen base logarithmic form trivializes its first factor. The connecting morphism on derived images defines
\(A^a\to\Omega_{\bar C}^1(\log T)\otimes A^a\).

To identify its cochain action, lift each relative form to its \(v_i\)-horizontal representative \(\sigma_i\), whose contraction with \(v_i\) is zero. The internal lift error is \(\alpha\wedge L_i\); on the first overlap, \(\sigma_{i_1}-\sigma_{i_0}=-\alpha\wedge H_{i_0i_1}\). In the total, the internal error has factor \((-1)^p\). The identification of the first complex's total with the shifted relative total multiplies Čech degree \(p\) by \((-1)^p\). Hence its connecting map is exactly the diagonal \(L_i\) plus extra component \((-1)^pH_{i_0i_1}\) in (5.6i). The short exact sequence is intrinsic and compatible with every restriction and refinement; thus its connecting map proves the asserted gluing independently of choices of ordered cover.

For a base function \(a\), differentiating the lifted cocycle for \(a\,s\) adds precisely \(da\wedge s\). The intrinsic connecting map therefore has the Leibniz rule \(\nabla(a s)=da\otimes s+a\nabla s\), agreeing with (5.6g). Multiplication of the base field by a base function multiplies the operator by that function. This gives a logarithmic connection. Since a curve has rank-one tangent sheaf, integrability follows directly:
\[
[a\nabla_\theta,b\nabla_\theta]
=(a\theta(b)-b\theta(a))\nabla_\theta.
\tag{5.6m}
\]
Expanding with (5.6g) proves the identity; the scalar second-order terms cancel.

#### A stable coherent lattice at every boundary point

At a boundary point the torsion of \(A^a\) is stable. If \(t^Ns=0\), then
\[
t^N\mathcal L_\theta(s)
=\mathcal L_\theta(t^Ns)-\theta(t^N)s
=-Nt^Ns=0.
\]
At an ordinary point differentiation loses at most one power of its parameter, so increasing the annihilating exponent by one proves stability there too. Hence the torsion-free quotient
\[
\bar H^a=A^a/\operatorname{torsion}
\tag{5.6n}
\]
is stable. Here is the algebraic local freeness argument. At a closed point of the smooth curve, its regular local ring \(R\) has maximal ideal \((t)\). The faithful-flat analytic injection into \(\mathbf C\{t\}\) makes \(\bigcap_n t^nR=0\). Thus every nonzero \(a\in R\) has a largest divisibility exponent \(m\), and \(a=t^m u\) with \(u\notin(t)\), hence \(u\) a unit. Choosing least valuation in an ideal makes every ideal principal. A finite torsion-free \(R\)-module embeds in its generic vector space: the kernel of localization is its torsion. Choose a generic basis and clear the finitely many denominators of module generators to embed it into a finite free \(R\)-module. The coordinate-projection induction of Lemma 1.0, now using the principal ideals just proved, makes that submodule finite free. Freeness at the generic point is already a vector-space assertion. Thus \(\bar H^a\) is a finite-rank bundle with logarithmic connection, equal to \(A^a\) on a smaller dense open.

Over \(C_0\), the horizontal boundary coordinates are relative coordinates. Lemma 5.5.1's finite pole proof applies to the relative complex: the normal operator \(\operatorname{Res}_i-nI\) is invertible and parallel for the remaining relative differential; its inverse times contraction makes the quotient acyclic. Other-coordinate poles remain allowed. Exact filtered localization gives
\[
K^\bullet|_{g^{-1}C_0}
\simeq Rj_{0*}(\Omega_{X_0/C_0}^\bullet\otimes E),
\tag{5.6o}
\]
where \(j_0:X_0\hookrightarrow g^{-1}C_0\) is affine by the proved Cartier-complement argument, and its quasi-coherent form terms are image-acyclic. The actual inclusions commute with logarithmic Lie derivatives and vertical contractions. They therefore identify the connections from the two corrected Čech totals.

Properness of \(g\), (5.6o) and derived-image composition identify \(A^a|_{C_0}\) with the relative de Rham image of \(f\), including its actual connection action. The lattice (5.6n) supplies its logarithmic extension at every point of \(\bar C\). The proved one-variable lattice criterion therefore makes the generic Gauss–Manin connection regular.

This proves the curve-base input with all fibre multiplicities and all coefficient Jordan blocks, for arbitrary smooth \(f\) without a properness assumption. We now prove the general-base reduction; arbitrary singular regular-holonomic images and full analytic proper regularity remain separate.

#### The theorem on every smooth algebraic base

**Theorem 5.6 (generic Gauss–Manin regularity).** Let \(f:X\to B\) be any smooth morphism of smooth separated finite-type complex varieties, and let \(E\) be any finite-rank integrable connection regular on every algebraic curve, including infinity. On every connected component of \(B\), there is a dense open \(U\) on which all relative de Rham images
\[
H^a=\mathcal H^a\bigl(Rf_*(\Omega_{X/B}^\bullet\otimes E)\bigr)|_U
\tag{5.6p}
\]
are finite-rank regular integrable connections. Their connection is the actual Gauss–Manin connection, and formation of these connections commutes with pullback by every map from a smooth curve to \(U\). The statement allows nonproper and non-quasi-projective \(X\) and \(B\). Empty source components give zero images; work componentwise otherwise.

Take a smooth proper SNC compactification \(\bar B\) of \(B\) and an absolute proper compactification of \(X\). Close the graph of \(f\) in their product, take its reduced closure, resolve while preserving \(X\), and principalize its boundary. This gives a smooth proper \(Z\), a proper \(g:Z\to\bar B\), and an SNC divisor \(D=Z\setminus X\). Theorem 5.1 extends \(E\) to a normalized logarithmic bundle \(\bar E\) on this absolute proper pair.

There is a dense \(U\subset B\) over which \(g\) and every nonempty closed boundary intersection are smooth. Here is the higher-dimensional version of the earlier bad-image argument. For a closed intersection dominating an integral component of \(B\), choose a transcendence basis \(t_1,\ldots,t_b\) of its base function field and extend it to a transcendence basis of the intersection's function field. Each derivation \(\partial/\partial t_j\) extends over the remaining finite separable field extension by the minimal-polynomial argument above. Therefore the pulled-back differentials \(dt_1,\ldots,dt_b\) are linearly independent there. An irreducible component of the critical locus cannot dominate \(B\): its ambient differential rank would be less than \(b\), and restriction to its function field could not increase that rank, contradicting this independence. The Jacobian minors cut out a closed critical locus. Its image is closed by properness and is a proper subset. Remove these images for \(Z\) and for the finitely many closed intersections, and remove the proper images of the nondominating intersections. Their finite union is a proper closed subset of each base component.

This makes \(D|_{Z_U}\) relative SNC. At a point of a \(k\)-fold boundary intersection the differential of \(g\) on that intersection is surjective. The base coordinate differentials together with \(dz_1,\ldots,dz_k\) are consequently independent in the ambient cotangent space. Complete them to a basis with remaining relative coordinates. The Jacobian criterion gives an étale relative coordinate chart with those normal coordinates. In particular \(g:Z_U\to U\) is smooth and
\[
\Omega^1_{Z_U}(\log D)/g^*\Omega_U^1
\]
is locally free. Let \(K_U\) be its exterior complex with coefficients \(\bar E|_{Z_U}\). The same ordinary proper-coherence spectral sequence used for (5.6e) proves that every
\(A^a=\mathcal H^a(Rg_*K_U)\) is coherent over \(\mathcal O_U\).

#### Integrability and arbitrary curve base change

Filter the absolute logarithmic complex on \(Z_U\) by the number of pulled-back base one-forms. Integrability of \(\bar E\) makes this a finite differential-stable filtration, with quotients
\[
\operatorname{gr}_F^p
(\Omega_{Z_U}^\bullet(\log D)\otimes\bar E)
\simeq
g^*\Omega_U^p\otimes K_U^{\bullet-p}.
\tag{5.6q}
\]
The internal differential under the wedge identification is \((-1)^p d_{K_U}\). Choose an affine base chart and a finite affine cover of its inverse image; the termwise affine Čech total computes every quotient. The base form modules are finite locally free, so tensoring this total with them commutes with cohomology. Its first filtered page is
\[
E_1^{p,q}=\Omega_U^p\otimes A^q,\qquad
d_1^{0,q}:A^q\longrightarrow\Omega_U^1\otimes A^q.
\tag{5.6r}
\]
The last map defines \(\nabla_{\rm GM}\).

This is a connection, with the same cochain operators as (5.6i). Modulo two base factors, contraction with a lift of any base field gives the horizontal representative and the overlap correction already computed. Moreover \(d(a s)=da\wedge s+a\,ds\), so the connecting map obeys
\[
\nabla_{\rm GM}(a s)=da\otimes s+a\nabla_{\rm GM}s.
\tag{5.6s}
\]
For a base \(p\)-form \(\beta\), the same calculation identifies the first-page differential as
\[
d_1(\beta\otimes s)
=d\beta\otimes s+(-1)^p\beta\wedge\nabla_{\rm GM}s.
\tag{5.6t}
\]
Its square is zero. To see this directly, lift a cocycle of the quotient of filtration degree \(p\) to a cochain whose differential lies in \(F^{p+1}\). That differential represents its \(d_1\) image and is closed, since the absolute differential squares to zero. Its next first-page differential is therefore zero. Changing the lift changes the representative only by a quotient boundary. Applying this to \(p=0\) and using (5.6t) gives zero curvature. Thus \(\nabla_{\rm GM}\) is integrable. The coherent-connection theorem, Theorem 3.1, makes every \(A^a\) locally free of finite rank.

Here is the required base change, without using a proper base-change theorem for D-modules. On an affine \(V=\operatorname{Spec}A\subset U\), choose the finite affine cover of \(Z_V\) just used. Every term of its Čech total for \(K_U\) is flat over \(A\). Indeed \(g\) is smooth and hence flat by smooth flatness, Theorems 5.1 and 6.1, and a locally free sheaf on each affine intersection gives a projective module over its flat coordinate algebra. The total is bounded in both directions. Its cohomology modules are finite projective over \(A\), by the local freeness just proved.

For completeness, a bounded complex of flat modules with flat cohomology commutes with arbitrary scalar extension. Starting in the highest degree, the exact sequence with quotient the top cohomology shows that its boundary module is flat. Descend: the exact sequence
\(0\to Z^n\to C^n\to B^{n+1}\to0\)
makes \(Z^n\) flat; then
\(0\to B^n\to Z^n\to H^n\to0\)
makes \(B^n\) flat. The kernels are flat because the long exact Tor sequence has flat middle and quotient terms. All these sequences remain exact after tensoring, since their quotient modules are flat. Thus kernels, images and cohomology of the tensor complex are the tensor products of the corresponding modules.

Now let \(h:C\to U\) be any map from a smooth algebraic curve. Take an affine \(V=\operatorname{Spec}A\subset U\) and an arbitrary affine open \(W\subset h^{-1}(V)\), with coordinate ring \(A'\). The open \(h^{-1}(V)\) itself need not be affine; a constant map from a proper curve already shows this. Over \(W\), the pulled-back Čech intersections are affine and their vector-bundle sections are the scalar extensions along \(A\to A'\) of the original sections, by the affine sheaf/module equivalence, Theorem 1.2 and Proposition 1.3. Relative logarithmic differentials commute with this base change in the relative SNC coordinates above. Hence the pulled-back total is precisely the total for the pulled-back proper pair and bundle. The preceding flat-complex argument proves, locally and then by gluing,
\[
h^*A^a
\simeq
\mathcal H^a(Rg_{C*}K_C).
\tag{5.6u}
\]
This also respects connections, by the following actual cochain calculation. Shrink \(V\) to have étale coordinates \(y_1,\ldots,y_b\), choose lifts \(v_{ij}\) of \(\partial/\partial y_j\) on its adapted affine cover, and write \(\mathcal L_j\) for the corrected operator (5.6i) and \(H_{ik,j}=\iota_{v_{kj}-v_{ij}}\). For a vector field \(w\) on \(W\), put \(a_j=w(h^*y_j)\). Its lift to a pulled-back affine chart is the derivation
\[
v_i^C(r\otimes b)
=\sum_j a_j(v_{ij}r\otimes b)+r\otimes w(b).
\]
This respects the tensor relations: on a base function \(a\), the chain rule in the étale coordinates says \(\sum_j a_j h^*(\partial_j a)=w(h^*a)\). Differences of these lifts are vertical, with contraction \(H^C_{ik}=\sum_j a_j H_{ik,j}\). The pulled-back relative differential kills every \(a_j\), so the corrected target operator on the Čech total is
\[
\mathcal L_w^C(b\otimes c)
=w(b)\otimes c+b\sum_j a_j(1\otimes\mathcal L_jc).
\tag{5.6v}
\]
Its off-diagonal component is precisely (5.6i) with \(H^C_{ik}\). Thus the natural scalar-extension map intertwines it with the usual pullback connection, proving connection compatibility in (5.6u). Equivalently, pullback of absolute forms as complex-vector-space sheaves respects the base-form filtration and the connecting morphisms; no scalar tensor of the non-\(\mathcal O_U\)-linear absolute differential is being taken. No flatness of \(h\), immersion hypothesis or dominance assumption is required.

Finally, the finite-pole proof of Lemma 5.5.1 applies relative to \(U\), exactly as in (5.6o). It identifies \(A^a\) with the images (5.6p), with their actual connection. It applies after the curve base change too. The morphism
\(X_C=X\times_B C\to C\)
is smooth, and the pulled-back connection is curve-regular, since any curve mapping to \(X_C\) composes to a curve mapping to \(X\). The curve theorem above makes the generic Gauss–Manin connection on \(C\) regular, with logarithmic lattices at every completion point. By (5.6u) it is the restriction of \(h^*A^a\), which is already a nonsingular finite-rank connection everywhere on \(C\). At an omitted point lying inside \(C\), its own bundle gives a stable logarithmic lattice; at points outside \(C\), retain the lattices from the curve theorem. The one-variable criterion therefore proves regularity of the whole connection \(h^*A^a\), including infinity. As \(h\) was arbitrary, \(A^a\) is curve-regular on \(U\).

This proves the full generic Gauss–Manin theorem for arbitrary smooth bases and all regular coefficient connections. Theorem 5.17 supplies arbitrary singular algebraic regular-holonomic map stability. Full analytic proper regularity remains required.

#### The relative Spencer transfer and its shift

For clarity, the same Gauss–Manin action is the operator action in the smooth direct image with arbitrary coefficients. The relative Spencer complex has terms \(\mathcal D_X\otimes_{\mathcal O_X}\Lambda^qT_{X/B}\) in degree \(-q\), with the multiplication and bracket differential of the relative Spencer calculation, Theorem 2.1. It augments to the forward transfer \(\mathcal O_X\otimes_{f^{-1}\mathcal O_B}f^{-1}\mathcal D_B\) by \(P\mapsto P(1\otimes1)\). In commuting adapted étale coordinates, PBW identifies the augmentation quotient with this transfer and the shifted order-graded complex with the Koszul complex on the \(r\) vertical symbols. Those polynomial variables are a regular sequence, so that Koszul complex is exact. Lifting a graded preimage and descending its finite order proves exactness of the augmented Spencer complex. Its terms are locally free over \(\mathcal D_X\), and the bracket formula is intrinsic, so the resolutions glue. This proves the same resolution for every smooth \(f\), without assuming it is a global product.

If \(f\) has relative dimension \(r\), tensoring this resolution with the right side-changed \(E\) gives
\[
f_+E\simeq Rf_*(\Omega_{X/B}^\bullet\otimes E)[r].
\tag{5.6w}
\]
In adapted commuting étale coordinates, its vertical scalar derivatives become the vertical covariant derivatives of \(E\), so its differential is \(\nabla_{\rm rel}\). The right action of a lifted horizontal field on \(\omega_X\otimes E\) is minus the covariant derivative, including the volume Lie derivative; changing sides on the base removes this minus sign. In the coordinate frame, its left action is therefore \(\nabla_{\partial_{y_j}}\). On form factors the bracket terms give ordinary Lie derivative, so intrinsically the action is \(L_v=\nabla\iota_v+\iota_v\nabla\).

This last identity can be checked on a coefficient section, where contraction of its connection gives \(\nabla_v s\), and on a one-form, where it is the ordinary Cartan identity; the product rule extends it to every wedge. Changing a lift by a vertical field \(w\) changes the operator by \(\nabla_{\rm rel}\iota_w+\iota_w\nabla_{\rm rel}\). Thus the transfer action glues by exactly the correction (5.6i), and agrees with the intrinsic connecting map, also for nontrivial \(E\). The shift \([r]\) changes the cohomological index and not this degree-zero action: the bundle \(H^a\) is \(H^{a-r}(f_+E)|_U\).


### Duality of every regular holonomic object

**Theorem 5.7 (regular holonomic duality).** Let \(X\) be a smooth separated finite-type complex algebraic variety. With the left-module duality
\[
\mathbb D_XK=
\operatorname{SC}_X
 R\mathcal Hom_{\mathcal D_X}(K,\mathcal D_X)[d_X],
\qquad
\operatorname{SC}_X(Q)=Q\otimes_{\mathcal O_X}\omega_X^{-1},
\tag{5.7a}
\]
duality is an exact contravariant equivalence of the category of regular holonomic modules, as defined by regular connection presentations of their simple factors. It induces a contravariant equivalence of the full subcategory \(D^b_{\mathrm{rh}}(\mathcal D_X)\) of bounded complexes with regular holonomic cohomology. In particular
\[
H^q(\mathbb D_XK)\simeq
\mathbb D_X H^{-q}(K).
\tag{5.7b}
\]
There is no restriction on the dimension or singularities of the closed supports. There is no properness or quasi-projectivity assumption on \(X\). On different open-and-closed components use their respective dimensions in (5.7a).

Here \(f_*\) denotes the differential-operator direct image, also often written \(f_+\). Thus the canonical map considered below is exactly \(j_!\to j_+\) in the plus notation. It is not an ordinary underived sheaf map.

The following earlier **proved** results suffice:

1. Holonomic duality, Lemma 3.0 and (3.4): finite local operator resolutions, intrinsic derived bidual evaluation. Theorem 3.1 and Corollary 3.2: Ext concentration, exact holonomic duality, preservation of simple objects and finite length. Proposition 5.1: its dual of a flat bundle is its usual dual connection, proved with the density-correct Spencer resolution.
2. Direct images, Propositions 3.1–3.2: exact closed direct image with its normal derivative basis, and derived open direct image. The special closed-then-open composition follows directly from these transfer descriptions, as explained in the canonical-map argument above below.
3. Kashiwara equivalence, Theorem 3.1 and (3.2)–(3.5): exact supported equivalence, intrinsic determinant-twisted inverse, and its actual unit and evaluation maps.
4. Adjunctions, Lemma 1.2, closed duality (4.3), and the open adjunction, Theorem 5.1: closed duality and the two embedding adjunctions with their units and counits.
5. Minimal extensions, Theorem 4.1: the degree-zero image is the unique supported extension with no boundary subobject or quotient. Its full proof, rather than its subsequent one-line duality assertion (4.4), is the input.
6. Lemma 1.0 and Theorem 1.1 above: the formal and convergent lattice algebra and the logarithmic lattice criterion. The regular connection definition tests every smooth algebraic curve map and every point of its complete curve, including infinity.

We prove all the additional steps here. In particular we do not invoke the regular-holonomic stability statement, a curve criterion for singular holonomic objects, or Riemann–Hilbert.

The freely accessible [Bernstein lecture notes, lecture 4, §§2–3 and the beginning of §5](https://www.math.columbia.edu/~khovanov/resources/Bernstein-dmod.pdf), printed pages 31–33, give the regular composition-factor definition and list duality among the stability operations. They are a reading reference; the proof below supplies this operation.

#### Dual logarithmic lattices, including infinity

Let \(R\) be \(\mathbf C[[t]]\) or \(\mathbf C\{t\}\), \(K=\operatorname{Frac}R\), and \(\theta=t\,d/dt\). Suppose \((V,\Theta)\) is a finite-dimensional differential \(K\)-module:
\[
\Theta(fv)=\theta(f)v+f\Theta(v).
\]
Define \(V^\vee=\operatorname{Hom}_K(V,K)\) with
\[
(\Theta^\vee\phi)(v)
=\theta(\phi(v))-\phi(\Theta v).
\tag{5.7c}
\]
The right side is \(K\)-linear in \(v\): applying it to \(fv\) cancels the two terms \(\theta(f)\phi(v)\). It satisfies the differential Leibniz identity in \(\phi\). This proves that (5.7c) is the dual differential module and is precisely the pairing formula of the earlier dual-connection Proposition 5.1.

If \(L\subset V\) is a full stable lattice, its dual lattice is
\[
L^\vee=\{\phi\in V^\vee:\phi(L)\subset R\}
       =\operatorname{Hom}_R(L,R).
\tag{5.7d}
\]
An \(R\)-basis of \(L\) and its dual \(K\)-basis identify (5.7d) with \(R^{\operatorname{rank}V}\). Consequently it is finite free, spans \(V^\vee\), and has the full rank, including the zero-rank case. For \(\phi\in L^\vee\) and \(v\in L\), both terms in (5.7c) lie in \(R\), because \(\theta(R)\subset R\) and \(\Theta L\subset L\). Hence
\[
\Theta^\vee L^\vee\subset L^\vee.
\]
In the dual basis the matrix and residue are
\[
\Theta=\theta+A(t),\qquad
\Theta^\vee=\theta-A(t)^{\mathsf T},\qquad
\operatorname{Res}(L^\vee)=-\operatorname{Res}(L)^{\mathsf T}.
\tag{5.7e}
\]
This is the actual dual lattice, without a semisimplicity assumption. Dualizing twice recovers \(L\), its evaluation pairing, and \(\Theta\). The lattice criterion therefore proves
\[
V\text{ is regular singular}\quad\Longleftrightarrow\quad
V^\vee\text{ is regular singular}
\tag{5.7f}
\]
over each of the two rings separately.

The higher-dimensional logarithmic formula is the same. If \(F\) is a bundle with logarithmic connection on a smooth SNC pair \((P,D)\), define
\[
\langle\nabla^\vee\phi,v\rangle
=d\langle\phi,v\rangle-\langle\phi,\nabla v\rangle.
\tag{5.7g}
\]
The logarithmic forms constitute an \(\mathcal O_P\)-module containing the differentials of regular functions, so (5.7g) has coefficients in
\(\Omega_P^1(\log D)\otimes F^\vee\). In a frame \(\nabla=d+\Gamma\), the dual is \(d-\Gamma^{\mathsf T}\). Evaluating the curvature on two vector fields and a section shows that it is minus the original curvature under the pairing, so the dual is integrable. Every boundary component has the residue in (5.7e). Commuting residues remain commuting after negative transposition. All Jordan blocks and all crossings are retained. A normalized residue strip is not in general retained by taking this lattice dual; regularity requires existence of a logarithmic lattice and imposes no such normalization.

For any algebraic map \(h:C\to U\), the natural bundle isomorphism
\[
h^*(E^\vee)\simeq(h^*E)^\vee
\tag{5.7h}
\]
respects the connections: apply the pairing formula to local sections and the chain rule for \(h\). On every irreducible component of \(C\), and at every point of its complete smooth model, pass from rational frames to the formal or convergent meromorphic field. Formula (5.7h) remains the same dual pairing there. Equations (5.7d)–(5.7f) apply at that point, whether it lies in \(C\) or outside \(C\). Thus an all-curve regular algebraic connection has an all-curve regular dual, and conversely.

This argument includes constant curve maps, arbitrary finite rank and nonsemisimple monodromy. It does not replace the test at infinity by a test on the nonsingular interior.

#### Support and coherent biduality

The duality in (5.7a) commutes with restriction to any Zariski open \(W\subset X\). Indeed, restriction of a finite local operator resolution computes both internal derived Hom complexes, the operator ring restricts to \(\mathcal D_W\), and the canonical density line restricts to \(\omega_W\). These termwise identifications respect the differentials and evaluation maps.

In particular for a holonomic module
\[
\operatorname{Supp}\mathbb D_X M=\operatorname{Supp}M.
\tag{5.7i}
\]
If \(M\) vanishes on \(W\), its dual vanishes there. Applying this implication to the dual and using biduality gives the converse. Taking the complements of all such opens gives (5.7i). The same argument detects vanishing of a bounded holonomic complex on an open.

For clarity, the biduality used here is a coherent involution, not merely an unspecified objectwise isomorphism. For an unshifted bounded finite projective complex \(P\), the evaluation on a homogeneous element \(x\in P^p\) is
\[
e_P(x)(f)=(-1)^p f(x)\quad(f\in(P^\vee)^{-p}).
\]
With \(d_{P^\vee}f=(-1)^{|f|+1}f\,d_P\), this is a cochain map. Applying it twice multiplies the two signs \((-1)^p\) and \((-1)^{-p}\), so
\[
\mathbb D(e_K)\circ e_{\mathbb D K}=\mathrm{id}_{\mathbb D K}.
\tag{5.7j}
\]
The side-changing equivalence and the dimension shifts transport this evaluation; the two shifts cancel when Hom reverses the shift. This is the intrinsic evaluation of the earlier biduality (3.4). The shifted evaluation can be made explicit in a volume trivialization, using the dualizing target \(\mathcal D_X[d_X]\). An element of degree \(p\) pairs with a dual element of degree \(-d_X-p\), so the evaluation sign is \((-1)^{p(p+d_X)}\). On the second evaluation, put \(q=-d_X-p\); then \(q(q+d_X)=p(p+d_X)\), and the two signs cancel. The local finite projective computations prove (5.7j), naturality, and their gluing. Thus dualizing an adjunction really exchanges its unit and counit, with their triangular identities. There is no freely chosen scalar or sign in that exchange.

Closed duality has the same normalization. In the earlier closed-embedding proof, the map is formed by applying direct image to evaluation and then the Kashiwara counit trace. On a finite free source term, tensor–Hom adjunction makes it the identity on the transfer term, with \(d_X-c=d_U\). These are precisely the unit and evaluation in the supported equivalence; applying evaluation twice uses (5.7j). The local comparison is therefore compatible with biduality and with those units and counits. Resolving by finite projectives extends these compatibilities to all coherent bounded objects, and their intrinsic formulas glue. We may consequently use, with that normalization,
\[
\mathbb D_V i_*A\simeq i_*\mathbb D_UA
\tag{5.7k}
\]
for a smooth closed embedding \(i:U\hookrightarrow V\). This is the already proved closed case of the adjunction lesson (4.3), with its map retained.

Explicitly, if \(\alpha_A:\mathbb D_Vi_*A\to i_*\mathbb D_UA\) is this comparison, its bidual compatibility is the equality of maps from \(i_*A\) to \(\mathbb D_V^2i_*A\)
\[
\mathbb D_V(\alpha_A)\,
\alpha_{\mathbb D_UA}^{-1}\,i_*(e_A)=e_{i_*A}.
\]
In the finite free transfer calculation, tensor–Hom adjunction turns the two composites into the same signed evaluation: the normal Koszul evaluation is the unit followed by its counit, whose composite is the identity. The remaining complex evaluation is the one in (5.7j). This verifies the equality locally, with the determinant and dimension shift included. Its naturality extends it to the finite resolutions and their cones. This is the compatibility required below when the closed and open transpositions are combined.

The inverse to \(i_*\) in the supported heart is the actual \(H^0i^!\), including \(\det(\mathcal I/\mathcal I^2)\); replacing it by an untwisted normal kernel would change this compatibility. No extra codimension shift is appended.

#### The canonical extension map and its transpose

We first prove the assertion for an open embedding \(u:V\hookrightarrow X\), allowing an arbitrary bounded holonomic input \(A\) on \(V\). Put
\[
\rho=u^!,\qquad R=u_*,\qquad L=u_!
=\mathbb D_XR\mathbb D_V.
\]
Restriction is \(\rho=u^*\) as well. Let
\(\eta:\mathrm{id}\to R\rho\) and \(c:\rho R\to\mathrm{id}\) be the unit and counit of \(\rho\dashv R\). The latter is an isomorphism for an open embedding. Dualize this adjunction using (5.7j). It gives \(L\dashv\rho\), with unit \(b:\mathrm{id}\to\rho L\) and counit \(\varepsilon:L\rho\to\mathrm{id}\), where
\[
b_A=\mathbb D_V(c_{\mathbb D_VA}),\qquad
\varepsilon_M=\mathbb D_X(\eta_{\mathbb D_XM})
\tag{5.7l}
\]
under the natural restriction and bidual identifications. In particular \(b_A\) is an isomorphism. This is exactly the open-embedding adjunction in the earlier lesson, constructed by dualizing restriction–direct-image adjunction.

The canonical arrow is
\[
\operatorname{can}_{u,A}
=R(b_A^{-1})\,\eta_{LA}
=\varepsilon_{RA}\,L(c_A^{-1}): LA\longrightarrow RA.
\tag{5.7m}
\]
Both displayed composites have mate \(c_A^{-1}:A\to\rho RA\) under \(L\dashv\rho\). For the second this is the defining adjunction formula. For the first, naturality of \(\eta\) and the triangular identity for \(\rho\dashv R\) give
\(c_A\,\rho(\operatorname{can}_{u,A})\,b_A=\mathrm{id}_A\).
The adjunction bijection is injective, so the composites agree. This verifies that (5.7m) is the arrow corresponding to \(\mathrm{id}_A\) under \(\rho RA\simeq A\); it is not merely some extension map.

Dualizing the first composite and using (5.7l) yields
\[
\begin{aligned}
\mathbb D_X(\operatorname{can}_{u,A})
&=\mathbb D_X(\eta_{LA})\,
  \mathbb D_X\!\bigl(R(b_A^{-1})\bigr)\\
&=\varepsilon_{R\mathbb D_VA}\,
  L(c_{\mathbb D_VA}^{-1})
=\operatorname{can}_{u,\mathbb D_VA}.
\end{aligned}
\tag{5.7n}
\]
Here \(\mathbb D_XRA=L\mathbb D_VA\) and
\(\mathbb D_XLA=R\mathbb D_VA\). Also
\(\mathbb D_V(b_A^{-1})=c_{\mathbb D_VA}^{-1}\) by (5.7l) and (5.7j). Equation (5.7n) proves the transpose with the actual adjunction units and counits, including its sign.

Now let \(j:U\hookrightarrow X\) be any smooth locally closed embedding. Choose an open \(V\subset X\) with \(U\) closed in \(V\), and factor \(j=u i\). Write \(Z=\overline U\); then \(Z\cap V=U\). The transfer for the open embedding is the identity bimodule. Therefore the backward transfer of \(j\), on \(U\), is the backward transfer of \(i\) with the ambient operators restricted from \(X\) to \(V\). Closed sheaf direct image and its transfer tensor are exact, while open sheaf direct image is derived. Their tensor and sheaf compositions give
\[
j_*=u_*i_*,\qquad
j_!=u_!i_*
\tag{5.7o}
\]
using (5.7k) in the second identity. This particular composition needs no quasi-projective factorization theorem. All density factors are those in (5.7a) and in the closed transfer, and cancel by the side-changing equivalence.

For modules supported on \(Z\), restriction to \(V\) followed by \(H^0i^!\) is the exact restriction functor to \(U\); for bounded supported complexes it is the corresponding derived supported equivalence. Its compatibility with duality follows from open restriction, (5.7k), and the inverse equivalence. In particular the restriction of the transpose of an identity is the identity on the dual connection.

Under (5.7o), the canonical map for \(j\) is
\(\operatorname{can}_{u,i_*N}\). Indeed its restriction to \(V\) is the identity on \(i_*N\), and application of \(i^!\) gives the identity on \(N\). The adjunction bijection
\[
\operatorname{Hom}_X(j_!N,j_*N)
\simeq\operatorname{Hom}_U(N,j^!j_*N)
\simeq\operatorname{End}_U(N)
\]
uniquely characterizes that map. Using (5.7k), naturality of (5.7m), and (5.7n) now proves
\[
\mathbb D_X(\operatorname{can}_{j,N})
=\operatorname{can}_{j,\mathbb D_UN}
\tag{5.7p}
\]
under the normalized natural identifications
\(\mathbb D_Xj_*N\simeq j_!\mathbb D_UN\) and
\(\mathbb D_Xj_!N\simeq j_*\mathbb D_UN\).
The factorization was used to prove equality of these actual canonical maps, so no choice of \(V\) remains in (5.7p).

#### Minimal extensions and the boundary

For a holonomic module \(N\) on \(U\), \(j_*N\) has cohomology in degrees \(\geq0\), by its closed-exact/open-right-derived construction. Holonomic exact duality gives \(j_!N\) cohomology in degrees \(\leq0\). The minimal extension is
\[
j_{!*}N=
\operatorname{im}\bigl(H^0\operatorname{can}_{j,N}\bigr).
\tag{5.7q}
\]

Both embedding images vanish on \(X\setminus Z\): the direct image there comes from the empty intersection with \(U\), and the exceptional image has the same vanishing by its dual definition and the support and biduality argument above. Thus all the supported restriction arguments above apply to their bounded cohomology objects.

On the holonomic heart an exact anti-equivalence takes the image of a map to the image of its reversed dual map. To check this explicitly, factor \(f:A\to B\) as a surjection \(A\to I\) followed by an injection \(I\to B\). Exact duality makes
\(\mathbb D B\to\mathbb D I\) surjective and
\(\mathbb D I\to\mathbb D A\) injective. Their composite is \(\mathbb D f\), so its image is \(\mathbb D I\).
Duality exchanges cohomological bounds and induces
\(H^0(\mathbb D K)=\mathbb D H^0(K)\) on bounded holonomic complexes; the complete truncation proof is supplied in the bounded-complex argument below. Apply this and (5.7p) to (5.7q):
\[
\mathbb D_X(j_{!*}N)\simeq
j_{!*}(\mathbb D_UN).
\tag{5.7r}
\]
This proves the image identity from the actual transpose, rather than assuming the assertion following the earlier minimal-extension Theorem 4.1.

There is also a direct support check. The object \(j_{!*}N\) is supported on \(Z\), hence its dual has the same support by (5.7i). Its supported restriction is \(\mathbb D_UN\) by (5.7k). If a nonzero subobject \(B\) of this dual were supported on \(Z\setminus U\), exact duality would give a nonzero boundary quotient \(\mathbb D_XB\) of \(j_{!*}N\), contradicting the earlier no-boundary theorem. If it had a nonzero boundary quotient, duality would give a nonzero boundary subobject of \(j_{!*}N\). Thus the dual has both no-boundary properties. The uniqueness clause of that theorem recovers exactly (5.7r). The support argument retains boundary delta modules and all normal derivatives; it uses neither a structure-sheaf fibre nor restriction detection for general derived morphisms.

If \(N=E\) is a finite-rank connection, the earlier dual-connection Proposition 5.1 identifies \(\mathbb D_UE=E^\vee\). Therefore
\[
\mathbb D_X(j_{!*}E)\simeq j_{!*}(E^\vee).
\tag{5.7s}
\]
If \(E\) is irreducible, \(E^\vee\) is irreducible because holonomic duality preserves simple modules. If it is curve regular, the dual-lattice argument above proves curve regularity of \(E^\vee\) at all compactification points. The same embedding \(j\), including its affine-inclusion property when required by the regular-holonomic definition, presents the dual simple factor. No change or shrinking of that regular connection presentation is needed.

#### Every regular simple factor and every support

Let \(M\) be regular holonomic. Choose a finite composition series
\[
0=M_0\subset M_1\subset\cdots\subset M_\ell=M,
\qquad S_a=M_a/M_{a-1}.
\]
Each \(S_a\) has a presentation \(j_{a,!*}E_a\) required by the definition: \(E_a\) is an irreducible all-curve regular connection on a smooth locally closed subvariety, and \(j_a\) is affine. Equation (5.7s) and the dual-lattice argument above prove that \(\mathbb D_XS_a\) has the same sort of presentation with \(E_a^\vee\). Each is simple.

Set
\(F_a=\mathbb D_X(M/M_a)\subset\mathbb D_XM\).
Exactness gives a reversed composition series
\[
0=F_\ell\subset F_{\ell-1}\subset\cdots\subset
F_0=\mathbb D_XM,\qquad
F_{a-1}/F_a\simeq\mathbb D_XS_a.
\tag{5.7t}
\]
Every factor is regular, so \(\mathbb D_XM\) is regular holonomic. Applying this implication to \(\mathbb D_XM\) and using biduality proves the equivalence in both directions and the asserted exact anti-equivalence.

For use with complexes, the regular modules form a Serre subcategory directly from the composition-factor definition. Intersect a composition series of \(M\) with a submodule \(A\); each successive quotient injects into the corresponding simple factor, so it is zero or that simple factor. The quotient filtration has the complementary zero or simple quotients. This proves regularity of submodules and quotients without assuming a curve criterion for singular modules. Conversely concatenate a composition series of a submodule with the inverse images of one in its quotient to obtain a composition series of the extension. Thus extensions are regular as well.

All the arguments are on the ambient holonomic heart, allowing different supports in the same extension. They apply to point supports, nonclosed smooth strata and singular closures. Finite type gives finitely many open-and-closed smooth components, and the construction acts componentwise if their dimensions differ.

#### Bounded complexes and degree reversal

The earlier holonomic dual is a contravariant triangulated equivalence on the ambient \(D_h^b(\mathcal D_X)\), with
\(\mathbb D(A[s])=(\mathbb D A)[-s]\). No identification of this category with the abstract derived category of its heart is needed.

A bounded object is obtained by finitely many standard truncation triangles from \(H^a(K)[-a]\). The dual of this piece is
\((\mathbb D H^a(K))[a]\), whose only cohomology is in degree \(-a\). Finite induction on these triangles proves
\[
\mathbb D(D_h^{[a,b]})\subset D_h^{[-b,-a]}.
\tag{5.7u}
\]
In particular duality exchanges the upper and lower standard cohomological bounds for bounded objects.

Fix \(q\) and put \(a=-q\). The triangle
\(\tau_{\leq a-1}K\to K\to\tau_{\geq a}K\)
dualizes to
\[
\mathbb D\tau_{\geq a}K\longrightarrow
\mathbb DK\longrightarrow
\mathbb D\tau_{\leq a-1}K\longrightarrow.
\]
The last term has cohomology only in degrees \(\geq -a+1\), by (5.7u). Its long cohomology sequence identifies
\(H^{-a}(\mathbb DK)\) with
\(H^{-a}(\mathbb D\tau_{\geq a}K)\).
Next dualize the triangle
\(H^a(K)[-a]\to\tau_{\geq a}K\to\tau_{\geq a+1}K\).
Its first dual term, \(\mathbb D\tau_{\geq a+1}K\), has cohomology only in degrees \(\leq-a-1\). The long sequence therefore identifies
\[
H^{-a}(\mathbb D\tau_{\geq a}K)
\simeq\mathbb D H^a(K).
\tag{5.7v}
\]
The maps are the functorial truncation and evaluation maps, so the resulting isomorphism is natural. This proves (5.7b), including \(q=0\) used in the minimal-extension argument above.

If every \(H^a(K)\) is regular, the composition-series argument above and (5.7b) make every cohomology module of \(\mathbb DK\) regular. Boundedness is retained by (5.7u). Thus \(\mathbb D\) preserves \(D^b_{\mathrm{rh}}\); coherent biduality gives its inverse. Conversely regularity of \(\mathbb DK\) gives regularity of \(K\). This completes the theorem in its full original singular-support and bounded-complex scope.

![The dual lattice, transposed extension map, and reversed composition series](assets/regular-duality-mechanism.png)

**Figure.** The lattice panel records the pairing and negative-transpose residue (5.7c)–(5.7f). The two extension rows are identified by duality in the reversed direction; the equality is the actual canonical-map calculation (5.7l)–(5.7p), and their images are (5.7r). The last panel records (5.7t) and the exact degree reversal (5.7b). All statements allow arbitrary rank, Jordan blocks and closed supports. Source reading: the free Bernstein text linked in Section 1; every displayed mechanism is proved above. The reproducible figure source records the same proof locators.



### Relative analytic generation in every projective dimension

We now prove finite twist generation near a projective fibre for every coherent analytic sheaf. This supplies an analytic input for later direct-image arguments, including sheaves with torsion or singular support and sheaves that are not flat over the base.

Freely accessible reading: [Norguet, Images de faisceaux analytiques cohérents](https://www.numdam.org/item/SL_1957-1958__1__A7_0.pdf), Section II.1–II.2, printed pp. 11-10–11-14, for the projective-line mechanism. The fixed-chart estimates, matrix factorization, product induction and finite split-trace descent are proved below.

**Theorem 5.8 (relative analytic generation).** Let \(D\subset\mathbf C^m\) be a polydisc, let \(y_0\in D\), and let \(F\) be coherent analytic on \(\mathbf P^n(\mathbf C)\times D\). There are a smaller polydisc \(D'\ni y_0\) and \(k_0\ge0\) such that \(F(k_0)\) is generated by finitely many global holomorphic sections on the entire product. For every \(k\ge k_0\), the same \(D'\) works, with a finite generating list for that \(k\).

For \(n=0\), choose finite local generators near \(y_0\) and shrink. The complete projective-line argument proves \(n=1\) by Cartan A/B, Laurent approximation and convergent matrix factorization. We first deduce its particular coherent-image theorem, then induct on products and descend through a finite quotient.


#### The projective-line statement and its local inputs

**Lemma 5.8.1 (projective-line generation).** Let \(D\) be a polydisc, let \(y_0\in D\), and let \(A\) be a coherent analytic sheaf on \(\mathbf P^1\times D\). There are a smaller polydisc \(D_0\ni y_0\) and an integer \(k_0\) such that, for every integer \(k\ge k_0\), finitely many sections of \(A(k)\) on \(\mathbf P^1\times D_0\) generate that sheaf. Equivalently a finite sum of \(\mathcal O(-k)\) surjects onto \(A\) there.

The inputs are Oka coherence, Theorem 2.1, and the coherent-kernel property, Theorem 3.1, Cartan A and B, Theorems 5.1 and 4.1, and Cauchy/Laurent expansion with holomorphic parameters. Every open used for Cartan B is a product of a polydisc with a disc, an exterior disc including infinity, or an annulus, hence is Stein by the explicit exhaustions in the programme. No fibrewise cohomological finiteness enters the argument.

#### Bounded Laurent splitting on fixed charts

Fix \(0<r<R<\infty\). Let \(U_1=D_0\times\{|z|<R\}\), \(U_2=D_0\times\{|z|>r\}\), where infinity belongs to the second factor of \(U_2\), and put \(U_{12}=U_1\cap U_2\). A bounded holomorphic function \(f\) on \(U_{12}\) has its normally convergent Laurent expansion
\[
f(y,z)=\sum_{j\in\mathbf Z}c_j(y)z^j=f_+(y,z)+f_-(y,z),
\]
where \(f_+\) extends to \(U_1\) and \(f_-\), its strictly negative part, extends to \(U_2\) and vanishes at infinity. The coefficients are holomorphic in \(y\) by their circle integrals.

There is a constant \(K_{r,R}\) independent of \(f,D_0\) such that
\[
\|f_+\|_{U_1},\ \|f_-\|_{U_2}\le K_{r,R}\|f\|_{U_{12}}.
\tag{5.8a}
\]
Here and below suprema are taken over the open domains; the estimates hold whenever these suprema are finite. To prove them put \(s=(r+R)/2\), \(M=\|f\|\). Cauchy estimates on radii tending to \(R\) give \(|c_j|\le MR^{-j}\) for \(j\ge0\), uniformly in \(y\). Thus \(|f_+|\le MR/(R-s)\) on \(|z|\le s\). On radii tending to \(r\), \(|c_{-j}|\le Mr^j\) for \(j\ge1\); hence \(|f_-|\le Mr/(s-r)\) for \(|z|\ge s\). On \(s<|z|<R\), use \(f_+=f-f_-\); on \(r<|z|<s\), use \(f_-=f-f_+\). Together these give (5.8a), for example with
\[
K_{r,R}=1+\frac{2R}{R-r}.
\]
This proves a bounded splitting on the *fixed* full chart domains; no successive domain shrink is necessary. The two different annulus boundary circles, separated by \(R-r>0\), are essential to this estimate.

For square matrices use a submultiplicative row-sum norm, with \(\|I\|=1\). Applying the scalar splitting entrywise produces, for a fixed matrix size, a constant \(K\ge1\) with
\[
L=U_1-U_2,
\qquad \|U_i\|\le K\|L\|,
\tag{5.8b}
\]
where \(U_i\) is holomorphic on the corresponding chart. The same estimates are uniform in all base parameters.

#### Convergent multiplicative factorization

For \(K\) from (5.8b), let \(\epsilon<1/(4K)\). Every matrix \(T\), holomorphic on \(U_{12}\), with \(\|T-I\|<\epsilon\), factors
\[
T=G_2^{-1}G_1,
\tag{5.8c}
\]
where \(G_i\) and \(G_i^{-1}\) are holomorphic on \(U_i\).

Set \(T_0=T=I+L_0\). At step \(j\), split \(L_j=U_{1,j}-U_{2,j}\) by (5.8b), and put
\[
T_{j+1}=(I+U_{2,j})T_j(I+U_{1,j})^{-1}.
\]
Because \(\|U_{i,j}\|\le K\|L_j\|<1/2\), both correction matrices are invertible by the convergent Neumann series and their inverses have norm at most 2. The exact residual identity is
\[
L_{j+1}=U_{2,j}L_j(I+U_{1,j})^{-1},
\qquad \|L_{j+1}\|\le2K\|L_j\|^2\le\tfrac12\|L_j\|.
\tag{5.8d}
\]
Thus \(\|L_j\|\le2^{-j}\epsilon\). Define products recursively by
\[
G_{i,0}=I,\qquad
G_{i,j+1}=(I+U_{i,j})G_{i,j}.
\]
They satisfy \(T_j=G_{2,j}T G_{1,j}^{-1}\). The sum of the correction norms is finite. Products and their inverse products therefore converge uniformly on the full fixed domains, by the submultiplicative product estimate and the estimate
\(\|(I+U)^{-1}-I\|\le2\|U\|\).
Their limits are holomorphic, remain mutual inverses, and \(T_j\to I\). Hence \(I=G_2TG_1^{-1}\), proving (5.8c). No compact-operator theorem, sheaf-frame assumption or flatness assumption is involved.

#### Redundant generators and coefficient matrices

Choose nested polydiscs \(D_0\Subset D_1\Subset D_2\Subset D\), all containing \(y_0\), and radii
\[
0<r_2<r_1<r_0<R_0<R_1<R_2<\infty.
\]
The two Stein opens
\(D_2\times\{|z|<R_2\}\) and
\(D_2\times\{|z|>r_2\}\)
cover \(\mathbf P^1\times D_2\).
Cartan A, then a finite subcover of the compact smaller closed chart sets, provides lists \(s_1,s_2\) of sections on these larger opens generating \(A\) on
\[
D_1\times\{|z|<R_1\},\qquad
D_1\times\{|z|>r_1\}.
\]
Pad the shorter list by zeros so that both are columns of the same length \(t\).

On the Stein overlap \(D_1\times\{r_1<|z|<R_1\}\), each list gives a surjection \(\mathcal O^t\to A\) with coherent kernel. Cartan B makes its map on global sections surjective. Componentwise lifting supplies holomorphic \(t\times t\) matrices \(B,C\) there with
\[
s_2=B s_1,\qquad s_1=C s_2.
\tag{5.8e}
\]
The matrices need not be inverses. The sheaf need not be free or flat. Equation (5.8e) is all that will be used.

These matrices are bounded on
\(\overline{D_0}\times\{r_0\le|z|\le R_0\}\).
For any \(\delta>0\), truncate only the far negative Laurent tail of \(B\) and the far positive Laurent tail of \(C\). Normal convergence with parameters gives matrices \(B',C'\) satisfying
\[
\|B-B'\|,\ \|C-C'\|<\delta
\tag{5.8f}
\]
on that compact overlap. The matrix \(B'\) is meromorphic on the inner chart, with a pole only at 0 and of a finite order; its nonnegative Laurent part extends holomorphically to \(|z|<R_1\). Similarly \(C'\) is meromorphic on the outer chart, with a pole only at infinity and of a finite order; its negative part extends holomorphically to \(|z|>r_1\). Let \(b\) dominate both finite pole orders. The holomorphic parameter dependence and the common finite pole bound follow from this one finite truncation, not from an unproved extension of fibre sections.

#### Pole compensation and the exact block matrix

For any \(k\ge2b\), choose a section \(\tau_k\) of \(\mathcal O(k)\) whose zeros are precisely 0 and infinity, with orders \(b\) and \(k-b\), respectively. If \(b=0\), increase it to 1 first. The binary monomial \(X_1^bX_0^{k-b}\), in coordinates where \(z=X_1/X_0\), supplies this section. Multiplication by \(\tau_k\) cancels every pole of \(B's_1\) and \(C's_2\). Put
\[
\begin{array}{ll}
S_1=\binom{\tau_k s_1}{\tau_k B's_1}
&\text{on }U_1=D_0\times\{|z|<R_0\},\\[3pt]
S_2=\binom{\tau_k C's_2}{\tau_k s_2}
&\text{on }U_2=D_0\times\{|z|>r_0\}.
\end{array}
\]
These are columns of \(2t\) genuine holomorphic sections of \(A(k)\). On their overlap, (5.8e) gives the exact equality
\[
S_2=T S_1,
\qquad
T=I_{2t}+
\begin{pmatrix}
(C'-C)B&0\\
B-B'&0
\end{pmatrix}.
\tag{5.8g}
\]
For instance the top difference is
\(\tau_k(C'-C)s_2=\tau_k(C'-C)Bs_1\),
and the bottom difference is \(\tau_k(B-B')s_1\).
Consequently \(\|T-I\|\le\delta(1+\|B\|)\).
Choose \(\delta\) so that this is below the factorization threshold in the factorization argument above. The matrix \(T\) and its smallness estimate are independent of \(k\).

Factor \(T=G_2^{-1}G_1\). Then
\[
G_1S_1=G_2S_2
\]
on the overlap. These lists glue to \(2t\) global holomorphic sections of \(A(k)\) on \(\mathbf P^1\times D_0\). Each \(G_i\) is invertible, so it does not alter the local submodule generated by the corresponding list. The first half of \(S_1\) already generates away from 0; the second half of \(S_2\) already generates away from infinity. Hence the glued sections generate everywhere except possibly at those two endpoints.

Repeat the construction using a projective coordinate whose zero and infinity are two different points, neither of them one of the original endpoints. Shrink the base finitely once more if necessary and raise \(k\) above the maximum of the two bounds. Each construction works for every such \(k\). Their combined finite global list generates at every point, since the two exceptional endpoint sets are disjoint. This proves the theorem.

![The pole compensation, exact block matrix and convergent gluing of redundant generator lists](assets/relative-p1-generation-mechanism.png)

The diagram records the actual chart sections and the exact comparison matrix in (5.8g). The matrix corrections in (5.8b)–(5.8d) are functions on those fixed charts, and the products are invertible there. The union of the two endpoint-complement constructions generates every stalk. The reproducible drawing source is original CC0 explanatory material; the full argument and estimates remain above.


#### Coherent projective-line images from generation

**Lemma 5.8.2 (projective-line coherent images).** For a complex manifold \(Z\), put \(p:\mathbf P^1\times Z\to Z\). We prove that \(R^q p_*H\) is coherent for coherent \(H\), and zero for \(q>1\).

Work near \(z_0\) in a coordinate polydisc \(V\). The two standard projective charts and their intersection, multiplied by any smaller polydisc, are \(\mathbf C\times V\), \(\mathbf C\times V\), and \(\mathbf C^*\times V\). They are Stein: sums of \(|w|^2\), of \(|w|^{-2}\) for punctured coordinates, and of reciprocal distances to polydisc boundaries give strictly plurisubharmonic exhaustions. Cartan B makes this cover acyclic for every coherent sheaf. Its Čech complex has degrees 0 and 1; the open-set description of higher direct-image sheaves gives
\[
R^q p_*K=0\quad(q>1)
\tag{5.8h}
\]
for every coherent \(K\). This bound is independent of proper coherence.

The directly proved parameter Laurent calculation in Serre's comparison theorems and Chow's theorem, Lemma 6.5, gives
\[
R^q p_*\mathcal O(a)
 =H^q(\mathbf P^1_{\mathbf C},\mathcal O(a))
       \otimes_{\mathbf C}\mathcal O_V,\qquad q=0,1.
\tag{5.8i}
\]
Its proof decomposes the finite Čech complex by Laurent negative-coordinate sets. Cauchy estimates justify coefficient projections and the contracting homotopies on normally convergent series with holomorphic parameters. The surviving monomial sets are finite. This calculation invokes neither Grauert nor arbitrary analytic coherent images.

Apply projective-line generation to \(H\), then to two successive coherent kernels. Three finite base shrinkings give an exact sequence on the whole product
\[
0\to K\to E_2\to E_1\to E_0\to H\to0,
\qquad E_i=\mathcal O(-a_i)^{N_i}.
\tag{5.8j}
\]
The kernels are coherent by Oka's coherent-category consequence. Locally, one can see the needed kernel assertion by lifting the finite generators of the source to a target presentation: coherence gives finitely many relations among their images; quotienting these relations by the source relations presents the kernel.

Put \(C=[E_2\to E_1\to E_0]\) in degrees \(-2,-1,0\). The augmentation has
\[
\operatorname{Cone}(C\to H)\simeq K[3].
\tag{5.8k}
\]
Indeed the kernel in degree \(-2\) is the only cohomology besides \(H^0(C)=H\), and the cone shifts it to degree \(-3\). The finite termwise spectral sequence
\[
E_1^{a,b}=R^b p_*C^a,\qquad
-2\le a\le0,\quad0\le b\le1
\]
has coherent entries by (5.8i), six positions, and a finite filtration of its abutment by coherent subquotients. Thus all cohomology sheaves of \(Rp_*C\) are coherent. By (5.8h), \(Rp_*K[3]\) has cohomology only in degrees \(-3,-2\); the triangle (5.8k) identifies
\[
\mathcal H^q(Rp_*C)\xrightarrow{\sim}R^q p_*H
\qquad(q\ge0).
\tag{5.8l}
\]
This proves coherence near \(z_0\), hence on \(Z\). Open restriction identifies these calculations with the intrinsic direct-image sheaves.

#### Generation on products of projective lines

Write
\[
Q_n=(\mathbf P^1)^n,\qquad
L_n(a_1,\ldots,a_n)=\boxtimes_i\mathcal O_{\mathbf P^1}(a_i).
\]
Inductively we prove that any coherent \(A\) on \(Q_n\times D\) has, after shrinking \(D\), finitely many global sections generating
\[
A\otimes L_n(h,\ldots,h)
\tag{5.8m}
\]
for some \(h\ge0\). The cases \(n=0,1\) are established.

Project the last factor,
\[
p:Q_n\times D\longrightarrow Q_{n-1}\times D=:Z.
\]
For each point of the compact fibre \(Q_{n-1}\times\{y_0\}\), apply projective-line generation over a target coordinate polydisc. It gives a smaller target open \(V_j\), a degree \(d_j\ge0\), and actual finite sections generating \(A\otimes L_n(0,\ldots,0,d_j)\) on the whole \(p^{-1}V_j\). Finitely many such opens cover that target fibre.

Projection \(Q_{n-1}\times D\to D\) is closed, by the compactness of \(Q_{n-1}\). The image of the closed complement of \(\bigcup V_j\) avoids \(y_0\). Thus a smaller polydisc \(D_1\ni y_0\) makes these opens cover every point of \(Q_{n-1}\times D_1\). Let \(d=\max_j d_j\). Multiply the \(j\)-th list by the finite monomial generators of \(\mathcal O_{\mathbf P^1}(d-d_j)\). Those lists generate \(A(d):=A\otimes L_n(0,\ldots,0,d)\) on their whole inverse images.

Lemma 5.8.2 gives coherence of \(G=p_*A(d)\). The lists are sections of \(G\) on the \(V_j\); consequently the intrinsic evaluation
\[
p^*G\longrightarrow A(d)
\tag{5.8n}
\]
is surjective everywhere over \(D_1\). No gluing of the local lists is needed to establish this assertion about the evaluation map.

Apply the induction hypothesis to the coherent \(G\) on \(Q_{n-1}\times D_1\). On a smaller \(D_2\), finitely many sections generate \(G\otimes L_{n-1}(b,\ldots,b)\), for some \(b\ge0\). Pull that finite surjection back and compose with (5.8n). Pullback is right exact, which suffices. This gives generation of \(A\otimes L_n(b,\ldots,b,d)\) on the entire \(Q_n\times D_2\). Put \(h=\max(b,d)\) and multiply by the finite monomial generators of
\[
L_n(h-b,\ldots,h-b,h-d).
\]
All its degrees are nonnegative, so this proves (5.8m). Every fixed induction dimension uses only finitely many neighbourhood shrinkings.

#### The finite symmetric-root quotient and split trace

The product of binary linear forms defines
\[
\rho:Q_n\longrightarrow\mathbf P^n,\quad
([a_1:b_1],\ldots,[a_n:b_n])
\longmapsto\left[\prod_i(a_iX+b_iY)\right].
\tag{5.8o}
\]
The coefficient sections have degree one in each pair and no common zero, since over a residue field the product of nonzero linear forms is nonzero. They define the morphism and the exact twist identification
\[
\rho^*\mathcal O_{\mathbf P^n}(1)=L_n(1,\ldots,1).
\tag{5.8p}
\]
On each nonvanishing-coefficient open the same coefficient trivializes both line bundles, and ratios of coefficients give the transition identifications.

Here is the finite quotient calculation. Choose \(n+1\) distinct points \(\xi_j\in\mathbf P^1\). Let \(U_j\) be the open of binary forms not vanishing at \(\xi_j\). These cover: a nonzero form of degree \(n\) has at most \(n\) distinct zeros, by division by its distinct linear zero factors. Move \(\xi_j\) to \([1:0]\). On \(U_j\) write the form as
\[
X^n+e_1X^{n-1}Y+\cdots+e_nY^n.
\]
In the corresponding linear-form coefficient coordinates, the inverse image has every \(a_i\ne0\); put \(x_i=b_i/a_i\). Then \(U_j=\operatorname{Spec}\mathbf C[e_1,\ldots,e_n]\), its inverse image is \(\operatorname{Spec}\mathbf C[x_1,\ldots,x_n]\), and the \(e_i\) pull back to elementary symmetric functions of the \(x_i\). Each \(x_i\) satisfies
\[
T^n-e_1T^{n-1}+\cdots+(-1)^ne_n=0.
\tag{5.8q}
\]
Successive reduction by these monic equations shows that the monomials with all exponents \(<n\) span the root ring over the coefficient ring. Hence \(\rho\) is finite on this finite affine cover.

For completeness, the invariant ring is exactly the coefficient ring. In the lexicographic order \(x_1>\cdots>x_n\), the leading exponent sequence of a symmetric homogeneous polynomial is \(a_1\ge\cdots\ge a_n\); otherwise a permutation gives a larger monomial. The product
\[
e_1^{a_1-a_2}e_2^{a_2-a_3}\cdots e_n^{a_n}
\]
has precisely that leading monomial with coefficient 1. Subtract its multiple. There are finitely many monomials of the fixed total degree, so repetition terminates. Applying this to each homogeneous component expresses every invariant polynomial in the \(e_i\). Distinct monomials in the \(e_i\) have distinct leading exponent sequences, proving algebraic independence as well. Thus
\[
\mathbf C[x_1,\ldots,x_n]^{\mathfrak S_n}
=\mathbf C[e_1,\ldots,e_n].
\tag{5.8r}
\]

Let \(B=\rho_*\mathcal O_{Q_n}\), an algebraic coherent sheaf by the finite module calculation. Permuting factors acts globally on \(B\); (5.8r) identifies \(B^{\mathfrak S_n}\) with the unit copy of \(\mathcal O_{\mathbf P^n}\). Those identifications agree on overlaps. Averaging therefore gives a global \(\mathcal O\)-linear retraction
\[
\tau:B\longrightarrow\mathcal O_{\mathbf P^n},
\qquad \tau=\frac1{n!}\sum_{\sigma\in\mathfrak S_n}\sigma,
\qquad \tau(1)=1.
\tag{5.8s}
\]
The codomain denotes the invariant summand. This formula is defined at branching points.

We identify this fixed algebra on the analytic product by an earlier comparison proved for algebraic sheaves. The algebraic morphism \(\rho\times\operatorname{id}_{\mathbf A^m}\) is projective: the iterated Segre map is a closed embedding \(Q_n\hookrightarrow\mathbf P^{2^n-1}\), its rank-one equations define its image, and its coordinate ratios give the inverse on each chart. The closed graph of \(\rho\), together with that embedding, gives a closed immersion into \(\mathbf P^{2^n-1}_{\mathbf P^n}\); take the product with \(\mathbf A^m\).

Its algebraic direct-image algebra is the pullback of \(B\): on the preceding affine cover the root algebra is tensored with \(\mathbf C[u_1,\ldots,u_m]\). Apply projective comparison for algebraic sheaves, Proposition 6.6 to the algebraic structure sheaf of this fixed morphism. Restricting the natural comparison to \(D_2\subset\mathbf C^m\) gives
\[
B_{D_2}:=(\rho\times\operatorname{id}_{D_2})_*
       \mathcal O_{Q_n\times D_2}
 \cong\operatorname{pr}_1^*B^{\mathrm{an}}.
\tag{5.8t}
\]
The proof of Proposition 6.6 uses algebraic finite twist presentations, the directly proved parameter twist calculation of Lemma 6.5, and finite cohomological bounds. It never assumes generation or coherent proper images of arbitrary analytic coherent sheaves. Its naturality for permutations carries \(\tau\) to an analytic retraction \(\tau_{D_2}\), still with \(\tau_{D_2}(1)=1\). In particular this fixed \(B_{D_2}\) is coherent.

#### Finite exact pushforward descends the generators

Write \(\rho_D=\rho\times\operatorname{id}_{D_2}\). Its continuous map is proper: inverse images of compact sets are closed subsets of \(Q_n\) times their compact base projection. Its fibres are finite; on the preceding affine charts each root coordinate has at most \(n\) possible values by (5.8q).

For any sheaf \(S\) of abelian groups on its source, there is a canonical stalk identification
\[
(\rho_{D*}S)_v=\bigoplus_{w\in\rho_D^{-1}(v)}S_w.
\tag{5.8u}
\]
Indeed representatives of the finitely many fibre germs can be chosen on pairwise disjoint neighbourhoods. The complement of their union is closed; properness gives a target neighbourhood whose inverse image is contained in that union. The representatives glue there. This proves surjectivity. If a section germ is zero at every fibre point, choose smaller neighbourhoods on which it is zero and repeat the same argument, proving injectivity. Properness gives the closed-map assertion used here by restricting over a relatively compact target neighbourhood and using compactness to extract a convergent subsequence.

Stalk exactness and the finite sum show that \(\rho_{D*}\) is exact on sheaves of modules. For any analytic \(\mathcal O\)-module \(T\), the natural projection formula is
\[
\rho_{D*}\rho_D^*T=B_{D_2}\otimes_{\mathcal O}T.
\tag{5.8v}
\]
At \(v\) this is the distributivity isomorphism between
\[
\bigoplus_{w\mid v}\left(\mathcal O_{Q_n\times D_2,w}
          \otimes_{\mathcal O_{\mathbf P^n\times D_2,v}}T_v\right)
\quad\text{and}\quad
\left(\bigoplus_{w\mid v}\mathcal O_{Q_n\times D_2,w}\right)
          \otimes_{\mathcal O_{\mathbf P^n\times D_2,v}}T_v.
\]
It applies to arbitrary modules.

The pullback of the original \(F\) to \(Q_n\times D\) is coherent: pull back a finite local presentation, and use Oka. the product-induction argument above and (5.8p) give a smaller \(D_2\), \(h\ge0\), and an actual global surjection
\[
\mathcal O_{Q_n\times D_2}^{\,N}
\twoheadrightarrow \rho_D^*F(h).
\tag{5.8w}
\]
Apply finite exact pushforward, (5.8v), and the retraction:
\[
B_{D_2}^{\,N}\twoheadrightarrow
B_{D_2}\otimes F(h)
\xrightarrow{\ \tau_{D_2}\otimes1\ }F(h).
\tag{5.8x}
\]
The second map is surjective because \(t\mapsto1\otimes t\) is its right inverse.

Choose a fixed \(e\ge0\) such that \(B(e)\) is generated by finitely many algebraic global sections. The earlier algebraic Serre proof, Lemma 1.1 and Proposition 1.2, applies to this algebraic \(B\). Its construction chooses finite module generators on each standard affine chart. A section on that principal chart extends after multiplication by a power of its coordinate: clear finitely many localized denominators on a finite affine trivializing cover, then kill the finitely many overlap differences by a further common power and glue. Raise these finitely many exponents to one common \(e\). This proves the asserted finite generation without any analytic generation input.

Analytify that finite surjection, pull it across \(D_2\), and use (5.8t):
\[
\mathcal O_{\mathbf P^n\times D_2}^{\,K}
\twoheadrightarrow B_{D_2}(e).
\]
Tensor pullback preserves surjections. Twist (5.8x) by \(\mathcal O(e)\) and compose \(N\) copies of this surjection. The result is
\[
\mathcal O_{\mathbf P^n\times D_2}^{\,KN}
\twoheadrightarrow F(h+e).
\tag{5.8y}
\]
The images of the standard global basis are finitely many global holomorphic sections generating on the entire product. This proves Theorem 5.8 with \(D'=D_2\), \(k_0=h+e\). For \(k\ge k_0\), multiplication by the finite monomial generators of \(\mathcal O(k-k_0)\) proves the assertion on the same base.

![Product induction, finite symmetric-root quotient and split-trace descent of generators](assets/relative-generation-descent-mechanism.png)

The diagram records the actual twist (5.8p) and surjections (5.8w)–(5.8y). Its trace is normalized averaging, defined at every branch point. The reproducible drawing source is original CC0 material.


### Coherence of the actual locally projective analytic operator image

**Theorem 5.9.** Let \(f:X\to Y\) be a locally projective holomorphic map of complex manifolds, \(y_0\in Y\), and \(M\) a coherent analytic operator module. Suppose that on a neighbourhood of the compact fibre a coherent holomorphic submodule \(L\subset M\) generates \(M\) under \(\mathcal D_X\). Then, near \(y_0\), the actual derived direct image \(f_+M\) has bounded coherent operator cohomology. Every closed or singular support is allowed; holonomicity of \(M\) is unnecessary. The statement also holds for bounded complexes whose cohomology modules each have such a generator near the fibre. The proof applies to right modules and transfers to the programme's left-module conventions by the exact density equivalence.

The initial coherent generator is an explicit hypothesis. Section5.90, GG.A–GG.G constructs it near every compact set for every analytic regular holonomic module. We first prove ordinary coherent holomorphic images in every projective dimension, then construct a finite operator resolution and check the actual target action and every output degree.

Freely accessible support reading: [Schapira, An Introduction to Sheaves on Grothendieck Topologies](https://webusers.imj-prg.fr/~pierre.schapira/LectNotes/SHV.pdf), Sections 4.2–4.4 and 5.1. The exact earlier programme proofs are linked below; the operator and finite-cone arguments are proved here.

#### Coherent holomorphic images in every projective dimension

**Lemma 5.9.1 (locally projective coherent holomorphic images).** Let \(p:\mathbf P^n\times D\to D\) be projection, where \(D\) is a coordinate polydisc. Theorem 5.8 gives, for any coherent analytic \(A\) and chosen \(y_0\), a finite epimorphism
\[
\mathcal O(-k)^r\twoheadrightarrow A
\tag{5.9a}
\]
on the entire product over a smaller polydisc. The proof includes torsion, nonflat coefficients and singular support.

The \(n+1\) standard projective charts and every finite nonempty intersection, multiplied by a base polydisc, are Stein. In a fixed affine chart an intersection is a product of affine and punctured affine coordinates; sums of squared coordinate moduli, inverse squared punctured-coordinate moduli and reciprocal boundary distances give its strictly plurisubharmonic exhaustion. Cartan B, Theorem 4.1 therefore makes this finite cover acyclic for each coherent analytic sheaf. Its Čech complex gives
\[
R^b p_*A=0\qquad(b>n).
\tag{5.9b}
\]
This uses no coherent-image theorem. The independently proved parameter Laurent calculation gives
\[
R^b p_*\mathcal O(k)
 =H^b(\mathbf P^n,\mathcal O(k))\otimes_{\mathbf C}\mathcal O_D;
\tag{5.9c}
\]
the surviving Laurent monomial sets are finite. Its exact earlier proof is the parameter Laurent calculation, Lemma 6.5.

Apply (5.9a) to \(A\) and then to \(n+1\) successive coherent kernels. There are only \(n+2\) shrinkings. This gives
\[
0\to K\to E_{n+1}\to\cdots\to E_0\to A\to0,
\qquad E_j=\mathcal O(-k_j)^{r_j}.
\tag{5.9d}
\]
Let \(C=[E_{n+1}\to\cdots\to E_0]\) in degrees \(-n-1,\ldots,0\). Its augmentation has cone \(K[n+2]\). Formula (5.9b) puts \(Rp_*K[n+2]\) in degrees at most \(-2\). Consequently
\[
\mathcal H^q(Rp_*C)\xrightarrow{\sim}R^q p_*A
\qquad(q\ge0).
\tag{5.9e}
\]
The bounded termwise spectral sequence of \(C\) has finitely many entries, coherent by (5.9c). Kernels, images, quotients and finite extensions in the coherent holomorphic category are coherent by the proved Oka/coherent-category argument. This proves coherence of every \(R^q p_*A\). Thus \(Rp_*\) sends \(D^b_{\mathrm{coh}}(\mathcal O)\) to \(D^b_{\mathrm{coh}}(\mathcal O_D)\), with amplitude \([0,n]\) on modules, by the finite Postnikov argument.

For a closed holomorphic embedding of complex manifolds, ordinary pushforward is exact: its stalk at a point of the closed submanifold is the source stalk, and elsewhere is zero. In adapted coordinates the defining ideal is generated by the normal coordinates. Lift the finite matrix entries of a coherent source presentation to the ambient coordinate neighbourhood, and add the normal-coordinate relations on its finite generators. This gives a finite target holomorphic presentation, so pushforward is coherent by Oka. Factoring a locally projective holomorphic map of complex manifolds through this embedding proves the image theorem for every coherent analytic source sheaf. The local projection dimension supplies a sufficient upper bound.

#### A coherent global generator and finite induced resolutions

Work with right operator modules. The exact left/right equivalence, Theorem 5.1 is \(M\mapsto\omega_Z\otimes M\); tensoring a coherent generator by that line bundle preserves coherence. The local volume and transpose proof uses only PBW and the Leibniz identities, which hold for holomorphic operators, so it proves the same analytic equivalence. In a volume frame the right vector-field action is minus the left action and minus the holomorphic divergence. Induction on operator order therefore shows that the density-twisted generator generates the converted module, and conversely. Here \(Z=\mathbf P^n\times D\), \(d=\dim_{\mathbf C}Z\). Suppose a coherent right \(\mathcal D_Z\)-module \(M\) has one coherent \(\mathcal O_Z\)-submodule \(L\) with \(L\mathcal D_Z=M\) on a neighbourhood of the compact fibre.

Properness makes that neighbourhood contain the entire inverse image of a smaller base: the image of its closed complement is closed and avoids \(y_0\). Apply (5.9a) to \(L\). If \(E_0=\mathcal O(-k_0)^{r_0}\), the composed action map is an actual right-operator epimorphism
\[
A_0:=E_0\otimes_{\mathcal O_Z}\mathcal D_Z\twoheadrightarrow M.
\tag{5.9f}
\]
Here \(E_0\) is finite locally free; no connection on \(E_0\) is required. Right multiplication acts on the last factor. The globally defined order pieces
\[
G_jA_0=E_0\otimes_{\mathcal O_Z}\mathcal D_{Z,\le j}
\quad(j\ge0),\qquad G_jA_0=0\quad(j<0)
\tag{5.9g}
\]
are coherent right holomorphic modules: in a frame of \(E_0\), these are finite copies of the finite-order operator pieces, which are finite free for the right holomorphic action by right normal ordering. Changing the frame multiplies by order-zero coefficients and preserves these pieces. PBW shows their symbol module is generated by the finite frame in degree zero. Thus (5.9g) is a global good filtration.

Put \(N_1=\ker(A_0\to M)\). Analytic operator coherence makes \(N_1\) coherent. The global intersections \(G_jN_1=N_1\cap G_jA_0\) are coherent holomorphic and locally good. Indeed locally \(A_0\) is a finite free filtered operator module and \(N_1\) is the image of a finite operator matrix; the uniform image-and-relation proof, Lemma 3.0c.4, proves goodness of the **induced** filtration. It also proves coherence of each of its finite-degree pieces by Oka. Thus no filtration of \(M\) was needed to obtain the first kernel's global pieces.

For completeness the same assertion holds for a coherent submodule of any already good filtered coherent module \(B\). Locally choose a strict finite filtered free epimorphism \(F\to B\), by lifting finite homogeneous symbol generators and lowering order. The inverse image \(H\) of the submodule is coherent and has a good induced filtration in \(F\) by Lemma 3.0c.4. Strictness identifies the desired submodule's degree \(j\) with the image of \(H\cap F_j\) in \(B_j\). Both are coherent holomorphic, so the image is coherent. The graded image is a quotient of the finite graded module of \(H\), hence finite. These identities prove the assertion, rather than merely assuming that coherence of an operator module implies coherence of an arbitrary holomorphic submodule.

The pieces of \(N_1\) exhaust it. At each point of the compact fibre, finitely many local operator generators lie in one degree. Their membership in that sheaf piece holds on a neighbourhood of the point. A finite cover and the maximum of these finitely many degrees give one global coherent piece generating \(N_1\) on a neighbourhood of the fibre. Properness again supplies a smaller entire inverse image. Apply (5.9a) to that piece and obtain
\[
A_1=E_1\otimes\mathcal D_Z\twoheadrightarrow N_1,
\qquad E_1=\mathcal O(-k_1)^{r_1}.
\tag{5.9h}
\]
Its kernel has the induced global good filtration from \(A_1\). Repeat a fixed finite number of times. Each kernel and finite coherent piece has just been constructed; only finitely many base shrinkings occur. This yields, for any fixed integer \(s\ge0\), an actual exact sequence
\[
0\to K\to A_s\to\cdots\to A_0\to M\to0,
\qquad A_j=E_j\otimes\mathcal D_Z,
\quad E_j=\mathcal O(-k_j)^{r_j}.
\tag{5.9i}
\]
Consequently the initial coherent global generator, and not a separately presumed filtration of every syzygy, is the sufficient input. Conversely that generator gives \(L\mathcal D_{\le j}\) as global good pieces: locally place its finitely many generators in a coherent local good piece of \(M\); the finite-order images are coherent inside that piece by Oka, exhaustive, and their symbol module is generated in degree zero. This recovers the generator/good-filtration equivalence near the compact fibre.

#### The induced-term images and actual target action

The right-module direct image is
\[
p_+N=Rp_*\bigl(N\otimes^L_{\mathcal D_Z}\mathcal D_{Z\to D}\bigr),
\qquad
\mathcal D_{Z\to D}
 =\mathcal O_Z\otimes_{p^{-1}\mathcal O_D}p^{-1}\mathcal D_D.
\tag{5.9j}
\]
In a local frame \(E_j\), \(A_j\) is finite free over \(\mathcal D_Z\). Thus its derived transfer tensor is its ordinary tensor, canonically
\[
A_j\otimes^L_{\mathcal D_Z}\mathcal D_{Z\to D}
 =E_j\otimes_{p^{-1}\mathcal O_D}p^{-1}\mathcal D_D.
\tag{5.9k}
\]
There is no assertion that the forward transfer is left flat over \(\mathcal D_Z\); the local freeness of this particular source term is what removes its Tor.

On a coordinate base the left holomorphic PBW basis makes \(\mathcal D_D\) the sheaf coproduct of its normally ordered finite derivative monomials. Proper derived image commutes with those coproducts in the bounded-below situation used here. The exact earlier proofs are compact germs and c-soft coproducts, (C1)–(C4), proper coproducts, (F4), and derived fibres and image acyclicity, (F6). Briefly, coproducts of c-soft sheaves are c-soft because a section on a compact subset uses a single finite set of summands; proper underived image has the same finite-summand property on its compact fibres. Sum injective resolutions in nonnegative degrees. The resulting resolution has termwise c-soft, image-acyclic terms, and computes the image of the coproduct. Hence the comparison is the specific proper projection map, not an assertion about unrestricted sections on a fixed open.

Finite-order projection and this coproduct comparison give
\[
p_+A_j\simeq Rp_*E_j\otimes_{\mathcal O_D}\mathcal D_D.
\tag{5.9l}
\]
This is a right \(\mathcal D_D\)-linear identity. At the sheaf level it takes \(s\otimes P\) to the coefficient section multiplied by its pulled-back target operator. Right multiplication by a coordinate derivative shifts the PBW monomial. Right multiplication by a target holomorphic function uses
\(\partial^\alpha a=\sum_{\beta\le\alpha}\binom\alpha\beta\partial^\beta(a)\partial^{\alpha-\beta}\); on the source those coefficients are exactly their pullbacks. The finite-order comparisons commute with these operations. Proper coproduct passage and derived resolution comparisons are natural for these coefficient maps, so they commute with the full right action. This verifies the action explicitly.

By (5.9c), every cohomology module of (5.9l) is a finite locally free right operator module, and vanishes outside degrees \(0,\ldots,n\). The identity is stronger here than mere coherent-image preservation. It uses only finite-order analytic operators and the proved twist calculation.

#### Finite transfer bounds and removal of the resolution tail

We prove the required transfer bound for any holomorphic map \(f:X\to Y\) of complex manifolds. In target coordinates its forward transfer
\[
\mathcal T=\mathcal D_{X\to Y}
=\mathcal O_X\otimes_{f^{-1}\mathcal O_Y}f^{-1}\mathcal D_Y
\]
is free as a left \(\mathcal O_X\)-module on the target PBW monomials. Its augmented absolute Spencer complex has terms
\[
\mathcal D_X\otimes_{\mathcal O_X}\bigwedge^rT_X
       \otimes_{\mathcal O_X}\mathcal T,\qquad 0\le r\le d_X,
\]
in degrees \(-r\). Its differential is the alternating sum of \(P\xi_i\otimes t-P\otimes\xi_i t\), with the alternating Lie-bracket terms. These formulas are coordinate invariant, glue and commute with the right \(f^{-1}\mathcal D_Y\)-action.

Filter the first operator factor by its order plus the wedge degree. The terms involving \(\xi_i t\) and brackets have lower filtration degree. The graded augmented complex is Koszul on the source cotangent variables with the flat coefficient \(\mathcal T\). Multiplication by a new polynomial variable is injective on a polynomial module with any coefficient module, and its quotient removes that variable. The one-variable two-term exact sequence and induction on the variables prove Koszul exactness. For a Spencer cycle, lift a graded preimage of its leading symbol and subtract its boundary. The order decreases; its lower bound makes this process terminate. Thus the augmented Spencer complex is exact.

Each term is flat as a left \(\mathcal D_X\)-module: its coefficient is \(\mathcal O_X\)-flat, and extension by the right \(\mathcal O_X\)-flat ring \(\mathcal D_X\) preserves that flatness, as tensor associativity tests on a short exact sequence of right operator modules. This finite flat resolution computes the actual transfer tensor on every right operator module, in degrees \([-d_X,0]\). The analytic PBW and coherent-category arguments are proved in the earlier operator lesson.

The earlier uniform real-manifold bound and finite c-soft resolution, (D8)–(D10), give proper sheaf-image amplitude \([0,2d_X]\) for every complex sheaf. Apply these two bounds to the projection, with \(d=\dim_{\mathbf C}Z\). Thus for every operator module \(T\),
\[
p_+T\in D^{[-d,2d]}(\mathcal D_D).
\tag{5.9m}
\]
All transfer tensors, resolution triangles and images above are formed in right operator-module categories. For these bounds, forgetting the \(p^{-1}\mathcal D_D\)-action gives the ordinary sheaf-derived image: forgetting to complex sheaves preserves injectives because its left adjoint \(p^{-1}\mathcal D_D\otimes_{\mathbf C}(-)\) is exact. Hence the underlying sheaf bound applies to the actual operator image. The displayed interval is sufficient; no optimality is claimed.

Take the finite resolution (5.9i) with
\[
s=3d+1,
\qquad C=[A_s\to\cdots\to A_0]
\text{ in degrees }-s,\ldots,0.
\tag{5.9n}
\]
Its augmentation to \(M\) has exactly
\[
\operatorname{Cone}(C\to M)\simeq K[s+1].
\tag{5.9o}
\]
The sole unresolved leftmost syzygy is in degree \(-s\) of \(C\) and degree \(-s-1\) of that cone. Apply the actual triangulated operator functor (5.9j). Formula (5.9m) gives
\[
p_+K[s+1]\in
D^{[-d-s-1,\,2d-s-1]}
 =D^{[-4d-2,\,-d-2]}.
\tag{5.9p}
\]
Both degrees \(q-1\) and \(q\) of this cone vanish for every \(q\ge-d\). The long exact sequence therefore gives the natural right-operator isomorphisms
\[
\mathcal H^q(p_+C)\xrightarrow{\sim}\mathcal H^q(p_+M)
\qquad(q\ge-d).
\tag{5.9q}
\]
The independent bound (5.9m) already kills every degree \(q<-d\) of \(p_+M\). Thus (5.9q) covers **every possibly nonzero degree** of the actual output.

The complex \(C\) has finitely many terms. Successive brutal filtration triangles express \(p_+C\) as finitely many cones of shifts of the \(p_+A_j\). Each latter object has bounded coherent cohomology by (5.9l). Analytic operator coherence makes kernels, images and quotients of coherent operator morphisms coherent, and finite extensions are coherent by the block-presentation argument. The long exact sequences for these finitely many cones therefore show that \(p_+C\) has bounded coherent operator cohomology. Formula (5.9q) proves the same conclusion for \(p_+M\). No convergence or infinite-width totalization is used in this deduction. Equivalently, the augmentation identifies the actual output with the canonical truncation \(\tau_{\ge-d}p_+C\).

![The exact finite augmentation and the independent degree separation](assets/finite-operator-image-coherence.png)

The figure is a schematic of cohomological degree intervals, not a numerical estimate or a claim of sharp bounds. Its maps are (5.9o) after applying \(p_+\); its upper interval is the exact sufficient bound (5.9m), and its lower interval is (5.9p) for the specified integer \(s=3d+1\). The two absent cone degrees establish (5.9q). The reproducible drawing source supplies the original CC0 diagram.

#### Closed embeddings, bounded complexes and the generator hypothesis

For a closed holomorphic embedding \(i:X\hookrightarrow Z\), the right forward transfer has its global order-zero section \(1\) and locally is the normal-derivative polynomial basis over the source. The canonical map
\[
i_*N\longrightarrow i_+N,
\qquad m\longmapsto m\otimes1
\tag{5.9r}
\]
is holomorphically linear and injective: in adapted coordinates it is the constant normal-derivative coefficient, and multiplication by a target function on it is multiplication by its source restriction. Normal derivative PBW proves that \(i_+\) is exact here. If \(L\subset N\) is a coherent global holomorphic generator, its image \(i_*L\) in (5.9r) is coherent and generates \(i_+N\) under the target operators. Indeed the other normal monomials are obtained by target right derivatives, and a source tangential operator acting on \(m\) is, in the tensor balance relation, a corresponding local target operator acting on \(m\otimes1\). These local generation assertions determine the global one. The finite source presentation with the normal-coordinate relations proves target operator coherence, explicitly: lift a finite source operator presentation to tangential ambient operators and add the normal-coordinate annihilation relations for its finite generators. Right normal ordering identifies the resulting finite quotient with the normal-derivative expansion of the transfer module. Analytic operator coherence gives target coherence. In right-module form the order-zero submodule has no extra normal determinant; the left-module conversion produces exactly that section's transfer line \(\det N_{X/Z}\).

Therefore factoring a locally projective holomorphic map of complex manifolds through a closed embedding and projection proves:

> If a coherent analytic operator module has a coherent global holomorphic generator on a neighbourhood of the chosen compact fibre, its actual derived locally projective direct image has bounded coherent operator cohomology near the chosen target point.

The property of having that generator is local near the fibre in this assertion; it is not assumed on all of the manifold. A bounded complex whose finitely many cohomology modules have such generators is covered after a common finite base shrinking, using its finite Postnikov tower and the coherent-category closure just proved. The general transfer/topological bound gives the sufficient range \([a-d_X,b+2d_X]\) for input cohomology in \([a,b]\).

This proves coherence for singular-support source modules once their generators exist. It does not prove the filtered characteristic transport inequality or holonomicity, nor the radical regular-filtration condition, canonical global lattice for arbitrary regular holonomic modules, convergent infinite-order extension, its faithful flatness, regularization, regular-pair Hom comparison, or the unit-compatible infinite-order projective comparison. The arbitrary proper holomorphic and support-proper theorem, including nonprojective maps, remains the active full obligation.


### A coherent cotangent image and its analytic transported support

Let \(p:Z=\mathbf P^n\times D\to D\), \(m=\dim D\), \(d=n+m\), and set
\[
\begin{aligned}
W&=Z\times_D T^*D,\\
j&:W\hookrightarrow T^*Z,\\
j(z,\eta)&=(z,(dp_z)^t\eta),\\
q&:W\to T^*D,\\
q(z,\eta)&=\eta.
\end{aligned}
\tag{5.10a}
\]
The image of \(j\) consists exactly of the covectors whose vertical components vanish. The projection \(q\) is proper, and in base coordinates is the projective product projection \(\mathbf P^n\times T^*D\to T^*D\).

**Theorem 5.10 (the cotangent image).** For every coherent analytic sheaf \(G\) on \(T^*Z\), the actual object
\[
B=Rq_*Lj^*G
\tag{5.10b}
\]
has bounded coherent holomorphic cohomology on \(T^*D\), with sufficient range \([-n,n]\). Moreover
\[
\operatorname{Supp}\mathcal H^a(B)
\subset q\bigl(j^{-1}\operatorname{Supp}G\bigr)
\qquad\text{for every }a,
\tag{5.10c}
\]
and the set on the right is a closed analytic subset.

**Proof.** Locally choose fibre coordinates \(x_1,\ldots,x_n\) and base coordinates \(y_1,\ldots,y_m\). The embedding \(j\) is defined by the vertical cotangent coordinates \(\xi_{x_1},\ldots,\xi_{x_n}\). These form a regular sequence in the local convergent function ring: adjoining each coordinate as an independent analytic variable makes multiplication by it injective, and successive quotients remove it. Their Koszul complex is a length-\(n\) locally free resolution of \(j_*\mathcal O_W\). Exactness follows by the one-variable two-term injection and induction on the number of independent variables, or equivalently by the successive polynomial-coordinate contraction. Tensoring this finite resolution with \(G\) computes \(Lj^*G\). Oka's coherent-category theorem makes each kernel and cohomology coherent. The Koszul homotopies show that the vertical coordinates kill its cohomology, which is therefore coherent over \(\mathcal O_W\), in degrees \(-n,\ldots,0\).

Lemma 5.9.1 proves the locally projective coherent holomorphic image theorem. Apply it to \(q\), on a coordinate polydisc about any point of \(T^*D\). It applies to each coherent cohomology sheaf of \(Lj^*G\) without flatness or smooth-support restrictions, and gives higher-image range \([0,n]\). Its finite cohomology spectral sequence proves that (5.10b) is coherent and lies in \([-n,n]\).

Every derived-restriction cohomology sheaf is supported on \(S=j^{-1}\operatorname{Supp}G\): away from that set the input \(G\) vanishes on a neighbourhood and so does its tensor Koszul complex. Properness makes \(q(S)\) closed. The image complex restricted to its open complement is zero, since every source cohomology sheaf there vanishes. Open restriction of the same derived image gives (5.10c).

To prove analyticity without importing a general proper analytic image theorem, use the possibly nonreduced coherent ideal on \(W\) obtained by restricting and generating from the annihilator of \(G\). The annihilator is coherent: locally it is the kernel of the action map from \(\mathcal O\) to the coherent internal Hom sheaf \(\mathcal Hom(G,G)\), whose coherence follows from finite presentations and Oka. Its zero set is \(\operatorname{Supp}G\) by the finite-module annihilator criterion; restricting its finite generators gives the zero set \(S\) on \(W\). The quotient \(\mathcal O_S\), viewed as a coherent sheaf on \(W\), has coherent \(q_*\mathcal O_S\) by the just-proved projective image theorem. Its support is exactly \(q(S)\). Outside that image it vanishes. At a point of the image, its globally defined unit has a nonzero stalk in \(q_*\mathcal O_S\): if that unit became zero on the inverse image of some target neighbourhood, its source germ at a chosen point of the nonempty fibre would be zero, contradicting the unit in the nonzero analytic local quotient. Thus the coherent support of \(q_*\mathcal O_S\) is the stated set, making it analytic by a finite local presentation. \(\square\)

For a coherent operator module \(M\) with a global good filtration, apply the lemma to its analytified coherent symbol sheaf
\[
G=\mathcal O_{T^*Z}\otimes_{\operatorname{gr}\mathcal D_Z}\operatorname{gr}M.
\tag{5.10d}
\]
This symbol sheaf is coherent from finite presentations: a locally finite presentation of \(\operatorname{gr}M\) over \(\operatorname{Sym}T_Z\) gives a finite matrix with fibre-polynomial holomorphic entries; viewing those entries on \(T^*Z\) presents \(G\) as the cokernel of a finite holomorphic free matrix. Oka proves coherence. Then \(S\) in (5.10c) is the intersection of its actual characteristic support with zero vertical covectors. The maps (5.10a) are the cotangent correspondence for this projection. The lemma proves that the natural derived **candidate** symbol image is coherent on the target cotangent space and has the expected support containment, including at singular-support points.

The characteristic bound for the **actual** operator image still requires a good output filtration and a filtered derived comparison with this symbol object, allowing the necessary spectral-sequence subquotients and filtration strictness. Holonomicity additionally requires the geometric dimension estimate for the transported characteristic set. Formula (5.10c) alone proves neither of those facts. This makes the surviving filtered/microlocal leaf smaller and explicit while preserving full arbitrary-proper regularity as the ultimate goal.

![The actual cotangent embedding and proper projection in the symbol-image lemma](assets/projective-symbol-image-leaf.png)

The diagram displays the defined objects (5.10a), the finite codimension-\(n\) Koszul restriction and the actual projective holomorphic image (5.10b). The support statement is the proved inclusion (5.10c); the bottom line lists the additional output-symbol and dimension steps beyond Theorem 5.10. Theorems 5.11–5.12 supply those proofs; the actual output symbol is a subquotient of the candidate. The reproducible drawing source is original CC0 material. Every input is proved above or in the linked earlier programme lessons.


### Singular pieces and isotropic projective cotangent images

**Theorem 5.11.** Let \(D\) be a complex coordinate polydisc of dimension \(m\), let \(Z=\mathbf P^n\times D\), and let \(\Lambda\subset T^*Z\) be a closed reduced conic analytic set, pure of dimension \(d=n+m\), whose smooth locus is Lagrangian. Set
\[
\begin{aligned}
W&=Z\times_DT^*D,\\
j(z,\eta)&=(z,(dp_z)^t\eta),\\
q(z,\eta)&=\eta,\\
B&=q(j^{-1}\Lambda).
\end{aligned}
\tag{5.11a}
\]
Then \(B\) is a closed conic analytic subset of \(T^*D\). Every local irreducible component of \(B\) is generically isotropic, and
\[
\dim B\le m.
\tag{5.11b}
\]
The proof includes components of \(j^{-1}\Lambda\) lying wholly inside \(\Lambda_{\mathrm{sing}}\). It does not assume Whitney stratification, an analytic resolution theorem, RH, or regularity preservation.

Theorem 5.12 below constructs the actual output filtration and characteristic inclusion. The present theorem proves the cotangent dimension bound needed for that conclusion.



#### Analytic foundations

The earlier programme proof of local parametrization provides the following precise data for an irreducible analytic germ \(A\) of dimension \(a\) in a coordinate ambient manifold. On a sufficiently small representative there is a proper finite projection
\[
\pi:A\longrightarrow V\subset\mathbf C^a
\tag{5.11c}
\]
and a nonzero holomorphic discriminant \(\delta\) on \(V\). Over \(V\setminus\{\delta=0\}\) it is a finite covering by regular points of \(A\), locally the graphs of holomorphic functions. This locus is dense. This is local parametrization, Theorem 4.1.

The same lesson proves finite local decomposition, Proposition 1.2, and dense regular points, dimension and strict dimension drop, Theorem 5.4 and Proposition 5.5. We use these statements locally, with countable coordinate covers.

The inverse, implicit and constant-rank theorems are the actual earlier coordinate proofs (HC9)–(HC12). The Riemann extension theorem, Theorem 4.2, and Corollary 4.3 prove bounded removal and connectedness of hypersurface complements. The coherent vanishing ideal is supplied by Cartan’s theorem, Theorem 1.1. These are exact earlier programme proofs.

Lemma 5.9.1 proves coherent projective holomorphic images. Theorem 5.10 gives the analytic-image unit argument, repeated below for the set in (5.11a).

#### The singular restriction lemma, proved by finite branches

**Lemma 5.11.1.** Let \(A\) be a reduced analytic subset of a complex manifold \(H\). Let \(\alpha\) be an ambient holomorphic differential form whose pullback to \(A_{\mathrm{reg}}\) is zero. For every locally closed analytic subset \(S\subset A\), its pullback to \(S_{\mathrm{reg}}\) is zero. This holds even when \(S\subset A_{\mathrm{sing}}\).

It is enough to prove the assertion locally for an irreducible component of \(S\) inside an irreducible component of \(A\). If \(A\) is locally reducible, its regular points away from the other components are dense in the regular locus of each component; holomorphic continuity therefore makes the hypothesis valid on each component's regular locus. Zero-dimensional \(S\) has no positive-degree forms and the degree-zero assertion is ordinary continuity. If \(\dim S=\dim A\), it is the same component locally and the assertion follows from the hypothesis. We now treat \(0<r=\dim S<a=\dim A\).

##### A finite projection restricts to an embedding generically on \(S\)

Take (5.11c) for an irreducible representative containing the chosen \(S\). The restriction to a regular source patch of \(S\) has maximum rank \(r\). Indeed, if its maximum rank were \(s<r\), a nonzero maximal minor gives a constant-rank neighbourhood. HC12 would supply local fibres of dimension \(r-s>0\), contradicting finiteness of \(\pi\). The rank-\(r\) patches are dense: the common vanishing of their minors is a proper analytic subset in each regular patch.

At such a point, HC12 identifies \(\pi|_S\), after shrinking, with a holomorphic embedding onto a smooth local submanifold \(S'\subset V\) of dimension \(r\). Flatten it in target coordinates \((s,v)\in\mathbf C^r\times\mathbf C^{a-r}\), so \(S'=(v=0)\). Here \(s\) is a parameter vector, not a source operator.

The inverse image \(\pi^{-1}S'\) is analytic. Every one of its irreducible components has dimension at most \(r\). To see this without a finite-image dimension theorem, take the regular locus of such a component of dimension \(k\); its map to the smooth \(r\)-manifold \(S'\) has maximum rank at most \(r\). Constant rank would give positive-dimensional fibres if \(k>r\), again contradicting finiteness. Thus \(k\le r\). Since our \(S\) has dimension \(r\), it is one irreducible component of this inverse image.

There are finitely many components near a given point. Remove their proper intersections with \(S\), and choose a point on the remaining open part of \(S\). In a neighbourhood of that point the underlying set of \(\pi^{-1}S'\) is exactly \(S\), and \(\pi|_S\) is an embedding. The other points of the finite fibre can be separated by disjoint ambient neighbourhoods. Properness shrinks the target so its entire inverse image lies in those neighbourhoods: otherwise a sequence outside them over base points tending to the chosen fibre would have a limit in that fibre. The distinguished neighbourhood consequently gives a proper finite restriction
\[
\pi:A_0\longrightarrow V_0,
\qquad
\pi^{-1}(s,0)=\{\sigma(s)\},
\tag{5.11d}
\]
set-theoretically, with \(\sigma:S'\to S\) the holomorphic inverse. Scheme multiplicities in this fibre are allowed. No normality, normalization or resolution is used.

The projection over the discriminant complement remains a finite covering. Every distinguished source part contains regular covering points by density. Since a hypersurface complement in a small connected polydisc is connected, that covering part has nonzero constant sheet number there.

##### A ramified transverse family has a bounded holomorphic lift

In the flattened coordinates expand the nonzero discriminant:
\[
\delta(s,v)=\sum_{\gamma\in\mathbf N^{a-r}}c_\gamma(s)v^\gamma.
\]
Let \(\ell\) be the least total degree with some coefficient not identically zero along a connected \(s\)-polydisc. Choose a constant vector \(b\in\mathbf C^{a-r}\) for which the polynomial
\[
c(s)=\sum_{|\gamma|=\ell}c_\gamma(s)b^\gamma
\]
is not identically zero. Such a vector exists because a nonzero finite polynomial cannot vanish for all constant vectors. At a generic \(s\)-point \(c(s)\ne0\). After a further shrink,
\[
\delta(s,tb)=t^\ell h(s,t),\qquad h(s,0)=c(s),\qquad h(s,t)\ne0.
\tag{5.11e}
\]
This is just normally convergent Taylor expansion and a finite leading coefficient; it assumes no Puiseux theorem with parameters. It also covers \(\ell=0\).

For \(t\ne0\), the map \((s,t)\mapsto(s,tb)\) avoids the discriminant. Pull back the finite covering in (5.11d) to the product of this \(s\)-polydisc and a punctured \(t\)-disc. Its monodromy is one finite permutation: the parameter polydisc contracts radially, and the punctured disc retracts to its circle. Path lifting through the local covering charts identifies sheets; continuation around one circle permutes the finite fibre. If the total covering degree is \(q_0\), the permutation has order dividing \(q_0!\).

Choose any positive integer \(e\) divisible by that order, for example \(e=q_0!\), and substitute \(t=u^e\). The resulting cover has trivial monodromy and a global holomorphic sheet
\[
F(s,u)\in A_0,\qquad
\pi F(s,u)=(s,b u^e),\qquad u\ne0.
\tag{5.11f}
\]
To spell out the elementary covering argument, continue one inverse chart along paths from a fixed base point. Two paths differ by loops homotopic to integral powers of the circle loop; the power substitution makes their sheet permutation the identity. Thus the continuation is independent of the path. On every covering chart it is a holomorphic inverse, so the resulting sheet is holomorphic. The radial contraction and angular circle homotopies also prove the required homotopy invariance by subdividing a homotopy square into finitely many covering charts.

Choose all parameter closures inside \(V_0\). Properness puts the sheet inside a compact subset of its ambient source chart, hence all its coordinates are uniformly bounded near \(u=0\). Bounded removal extends each coordinate holomorphically across \(u=0\). In elementary Laurent terms, each negative coefficient satisfies \(|c_{-k}(s)|\le M\rho^k\) on fixed smaller parameter compacts; radius independence and \(\rho\to0\) kill it. The coefficients are holomorphic in \(s\) by their circle integrals. This proves joint holomorphic extension, not merely pointwise limits.

The source equations remain zero by continuity, and properness keeps the limit in the distinguished neighbourhood. Equation (5.11d) forces
\[
F(s,0)=\sigma(s),\qquad
F(s,u)\in A_{\mathrm{reg}}\quad(u\ne0).
\tag{5.11g}
\]
This family is the required singular-stratum bridge.

##### Pull back the form and pass to \(u=0\)

The form \(F^*\alpha\) is holomorphic on the entire parameter product. It is zero for \(u\ne0\), because (5.11g) maps that open set into \(A_{\mathrm{reg}}\). Its holomorphic coefficients are therefore identically zero. Restricting to \(u=0\) gives \(\sigma^*\alpha=0\) on a nonempty open part of \(S_{\mathrm{reg}}\).

The construction can be restarted in every nonempty regular patch of \(S\). Each such patch therefore contains a nonempty open on which the restricted form vanishes. The zero set is dense in \(S_{\mathrm{reg}}\), and the restricted holomorphic coefficients are continuous, so they vanish everywhere. This proves Lemma 5.11.1. There is no assumption that tangent spaces at singular points are ordinary limits of tangent spaces; the actual holomorphic family supplies the needed restriction map. \(\square\)



#### The conic Lagrangian one-form vanishes on every singular piece

Fix the sign convention
\[
\theta_Z=\sum_{i=1}^n\xi_i\,dx_i+
         \sum_{a=1}^m\eta_a\,dy_a,
\qquad
\omega_Z=d\theta_Z,
\qquad
E=\sum_i\xi_i\partial_{\xi_i}+\sum_a\eta_a\partial_{\eta_a}.
\tag{5.11h}
\]
Thus \(\iota_E\omega_Z=\theta_Z\). Reversing the symplectic sign would leave the isotropy assertions unchanged, but (5.11h) states the sign actually used.

Conicity makes \(E\) tangent to \(\Lambda_{\mathrm{reg}}\): its fibre-scaling flow preserves \(\Lambda\) and the regular locus. Lagrangianity gives \(\omega_Z(E,v)=0\) for every tangent vector \(v\) there. Therefore
\[
\theta_Z|_{\Lambda_{\mathrm{reg}}}=0.
\tag{5.11i}
\]
Lemma 5.11.1 now makes the pullback of \(\theta_Z\) zero on the regular locus of **every** locally closed analytic subset of \(\Lambda\). In particular it is zero on every smooth local analytic piece of \(j^{-1}\Lambda\), after applying \(j\), including pieces entirely over \(\Lambda_{\mathrm{sing}}\).

In the correspondence (5.11a), \(j\) sets precisely the vertical covectors to zero. With
\(\theta_D=\sum_a\eta_a\,dy_a\), the actual coordinate identity is
\[
j^*\theta_Z=q^*\theta_D.
\tag{5.11j}
\]
It glues intrinsically because tautological cotangent one-forms are defined by evaluation on the derivative of the cotangent projection. Consequently \(q^*\theta_D\) vanishes on every smooth local analytic piece of \(j^{-1}\Lambda\).

#### The proper image is analytic, using the proved coherent theorem

Put \(A=j^{-1}\Lambda\) as a reduced analytic subset of \(W\). Its ideal is coherent by the exact Cartan coherence theorem. The proper projective map \(q:W\simeq\mathbf P^n\times T^*D\to T^*D\) has coherent \(q_*\mathcal O_A\) by the projective analytic coherent-image theorem already proved in Lemma 5.9.1.

Its support is exactly \(q(A)\). Outside that closed image it is zero. At \(b\in q(A)\), choose a point of the nonempty fibre in \(A\). The global unit of \(\mathcal O_A\) cannot have zero direct-image stalk at \(b\): that would mean zero on the inverse image of one neighbourhood, contradicting its nonzero source germ at the chosen point. Thus the coherent support is \(B=q(A)\), and support of a coherent analytic sheaf is analytic by a finite local presentation. This is the same unit argument independently checked in Theorem 5.10. Properness supplies closedness. Fibre scaling commutes with \(j\) and \(q\), so \(B\) is conic.

No general proper analytic-image theorem or analytic stratification theorem is used at this step.

#### A countable rank cover supplies image tangent vectors

We need a consequence of surjectivity onto \(B\), including when all sources over a component lie in singular loci. The following elementary argument avoids an assumed stratification.

##### Analytic sets admit the needed countable smooth covers

Every local irreducible analytic set of positive dimension \(k\) has, by (5.11c), a dense smooth open whose complement is contained in the proper analytic subset \(\{\delta\circ\pi=0\}\). The latter has dimension at most \(k-1\). Decompose that complement into its finitely many local irreducible components, apply the same construction to each, and continue. The dimension strictly drops, so after at most \(k\) levels the remaining pieces are zero-dimensional and smooth.

Using a countable ambient chart cover and countable relatively compact subcharts makes this a **countable cover** by smooth locally closed analytic pieces. Overlap is allowed. No Whitney condition, frontier condition or local finiteness of a stratification is claimed or needed. Every piece is locally an analytic subset of the original analytic set, so the tautological-form vanishing proved above applies to it.

For a holomorphic map on one such smooth piece, take its maximum differential rank on a coordinate patch. The locus of that rank is open and has constant rank locally. The remaining locus is cut out by the maximum-rank minors and is a proper analytic subset. Its local components have lower dimension. Apply the same smooth-cover and rank argument recursively. Zero-rank maps on connected patches are constant. This gives a countable cover by smooth pieces on which \(q\) has constant rank, locally in the normal form HC12. The recursion again terminates by strict dimension drop.

##### Surjectivity forces full-rank image pieces somewhere in every target ball

Choose a target open on which \(B\) itself is a smooth manifold of dimension \(r\), avoiding the other local components. Such opens are dense in each component by the proved local parametrization and component-dimension statements. Restrict the source to its inverse image. Its images cover this target open.

If all the constant-rank pieces above had rank smaller than \(r\), HC12 would place every local image inside a smooth target submanifold of smaller dimension. Exhaust each sufficiently small source rank chart by countably many compact subsets. Their images are compact, hence closed, and have empty interior in the \(r\)-dimensional target manifold because they lie in those smaller-dimensional submanifolds. They are consequently nowhere dense.

A small closed target coordinate ball cannot be covered by countably many such closed nowhere-dense sets. Here is the precise elementary Baire argument: inside the ball choose a smaller closed ball disjoint from the first set; inside its interior choose a closed ball disjoint from the second, of radius at most \(1/2\); continue with radii tending to zero. At each stage nowhere density allows the choice. Nested compact balls have a common point, obtained equivalently by the Cauchy limit of their centres. That point misses every listed set, a contradiction. Intersecting the compact image sets with the target ball keeps them closed and nowhere dense; its boundary has no relative interior and causes no exception.

Thus some piece has rank \(r\); HC12 makes its image contain a target open and its differential surjective there. Pullback of \(\theta_D\) is zero on that piece by (5.11j) and Lemma 5.11.1. Surjectivity of its differential makes \(\theta_D|_B=0\) on its image open.

In fact the form is zero throughout each smooth target component patch. If it were nonzero at one point, one coefficient in a local tangent frame would remain nonzero on a smaller open neighbourhood. On the inverse image of that neighbourhood no rank-\(r\) piece could exist, because such a piece would force that coefficient zero. The same Baire argument contradicts the covering. Therefore
\[
\theta_D|_{B_{\mathrm{smooth}}}=0
\tag{5.11k}
\]
on the dense smooth portions of every local component. A zero-dimensional component satisfies this automatically.

This is the explicit image-rank step; it does not silently discard source pieces in \(\Lambda_{\mathrm{sing}}\).

#### Isotropy and the dimension bound

Take the exterior derivative of (5.11k) on a smooth target component patch:
\[
\omega_D|_B=d(\theta_D|_B)=0,
\qquad \omega_D=d\theta_D.
\tag{5.11l}
\]
Its tangent space \(V\) is isotropic in the \(2m\)-dimensional symplectic tangent space of \(T^*D\). Nondegeneracy gives
\(\dim V+\dim V^\perp=2m\), while isotropy gives \(V\subset V^\perp\). Hence \(2\dim V\le2m\). Dimension of an analytic component is the dimension of its dense smooth locus, proving (5.11b). Empty images and dimension-zero cases are included.

The singular restriction lemma and the countable rank cover are the two necessary steps beyond the familiar smooth correspondence calculation. Both have been proved here from finite parametrization, bounded removal, constant rank and dimension drop. In particular no unresolved Whitney-stratification package is used as a hidden provider.

![The finite-branch singular bridge and exact cotangent one-form identity](assets/projective-isotropic-image-proof.png)

The upper panel is the local family of Lemma 5.11.1, with \(e\) a finite covering-monodromy exponent and \(b\) a chosen transverse parameter vector. It is a schematic, not an analytic resolution or a global parametrization. The middle panel is the actual correspondence (5.11a) with zero vertical covectors and the exact identity (5.11j). The bottom panel records (5.11k)–(5.11l) and the proved bound (5.11b), including source pieces wholly in the singular locus. The complete proof is above; the reproducible drawing source supplies the original CC0 diagram.


### Actual filtered images and analytic projective holonomicity

**Theorem 5.12.** Let \(p:Z=\mathbf P^n\times D\to D\), where \(D\) is a coordinate polydisc. Let \(M\) be a coherent analytic operator module with a global good order filtration near the compact fibre under consideration; an initial coherent global holomorphic generator suffices by Theorem 5.9. Every cohomology module of \(p_+M\) has an actual good output filtration, and
\[
\operatorname{Ch}\mathcal H^b(p_+M)\subset
q\bigl(j^{-1}\operatorname{Ch}M\bigr).
\]
If \(M\) is holonomic, every output cohomology module is holonomic. This conclusion includes every singular support and every derived degree. It extends to locally projective holomorphic maps of complex manifolds and bounded complexes whose cohomology modules have the stated generator or good filtration near the compact fibre. Right-module calculations transfer to the programme's left-module conventions by the density equivalence proved in Theorem 5.9.

The proof keeps the torsion in the derived Rees image until both specializations have been computed. Removing its locally bounded power torsion then constructs the good filtration of the actual output. Its symbol module is a quotient of a submodule of the coherent cotangent candidate of Theorem 5.10. Theorem 5.11 supplies the dimension bound, including singular intersections.

#### Objects, grading and the claimed characteristic containment

Work with right modules and let \(p:Z=\mathbf P^n\times D\to D\), \(d=\dim Z\), \(m=\dim D\). Let \(M\) be coherent over \(\mathcal D_Z\) with one globally defined good order filtration \(F\) near the tested compact fibre. The filtration is exhaustive, has coherent holomorphic pieces, is locally bounded below, and its graded module is locally finitely generated over \(\operatorname{Sym}T_Z\). A coherent global holomorphic generator supplies such a filtration by the construction in Theorem 5.9.

Use the central variable \(h\) of internal degree one and put
\[
\mathscr R_Z=\bigoplus_{a\ge0}\mathcal D_{Z,\le a}h^a,
\qquad\mathscr R_FM=\bigoplus_{a\in\mathbf Z}F_aM\,h^a.
\tag{5.12a}
\]
In coordinates write \(\vartheta_i=h\partial_i\). The relations are
\[
[\vartheta_i,a]=h\partial_i(a),\qquad
[\vartheta_i,\vartheta_j]=0,
\qquad h\text{ central},
\tag{5.12b}
\]
and both \(h,\vartheta_i\) have degree one. PBW gives the holomorphic coefficient basis \(h^k\vartheta^\alpha\), with \(k\ge0\). Consequently
\[
\mathscr R_Z/(h)=S_Z:=\operatorname{Sym}T_Z,
\qquad\mathscr R_Z/(h-1)=\mathcal D_Z.
\tag{5.12c}
\]
The Rees module embeds in \(M[h,h^{-1}]\). It is torsion-free, hence flat, over the coefficient PID \(\mathbf C[h]\). Here is the coefficient-flatness proof used throughout. Euclidean division by a nonzero ideal element of least degree proves that every ideal of \(\mathbf C[h]\) is principal. A finitely generated torsion-free module \(V\) over this PID injects into its finite-dimensional fraction-field span; choose a basis there and clear the finitely many generator denominators, embedding \(V\) into a finite free module. Every submodule \(N\subset A^r\) over a PID is free of rank at most \(r\): its first-coordinate image is a principal ideal \((a)\); if nonzero, choose a vector mapping to \(a\), subtract its multiples and split \(N\) as that free rank-one summand plus a submodule of \(A^{r-1}\). If the image is zero, it is already a submodule of the latter. Induction proves the assertion, including zero images. Thus the finitely generated \(V\) is free. An arbitrary torsion-free module is the filtered union of its finitely generated submodules, all free by this argument. Tensoring an exact sequence with each is exact, and a filtered union commutes with kernels and cokernels because every equality or preimage witness involves finitely many elements at one later index. Tensor with their union is therefore exact, proving flatness. The stalkwise argument proves sheaf coefficient-flatness. The Rees specializations are \(\operatorname{gr}_FM\) at zero and \(M\) at one, including negative filtration shifts.

Write
\[
\begin{aligned}
W&=Z\times_D T^*D,\\
j(z,\eta)&=(z,(dp_z)^t\eta),\\
q(z,\eta)&=\eta,\\
G&=\mathcal O_{T^*Z}\otimes_{S_Z}\operatorname{gr}_FM.
\end{aligned}
\tag{5.12d}
\]
The conclusion proved below is
\[
\operatorname{Ch}\mathcal H^b(p_+M)
\subset q\bigl(j^{-1}\operatorname{Supp}G\bigr)
\qquad\text{for every }b.
\tag{5.12e}
\]
These are analytic characteristic supports in the actual cotangent spaces. Theorem 5.10 proves that \(Rq_*Lj^*G\) is coherent and supported in the set on the right. The intervening Rees proof connects that object to the actual operator output by subquotients rather than an unsupported equality of symbols.

#### Graded analytic Rees coherence, with a finite termination argument

We need coherence for finite **graded** Rees presentations and degree-preserving maps. We do not assume a flat analytic infinite-order extension or import an unproved Rees coherence theorem.

At a holomorphic coefficient stalk \(H\), filter the Rees ring by the increasing vanishing order in the central variable \(h\), namely the decreasing filtration \(h^v\mathscr R\). Formula (5.12b) gives
\[
\operatorname{gr}_h\mathscr R=H[h,\xi_1,\ldots,\xi_d],
\tag{5.12f}
\]
a commutative Noetherian polynomial ring. Here this auxiliary filtration records the **least** power of \(h\), while the existing internal grading gives every \(h\) and \(\xi_i\) weight one. In a shifted finite free module, a homogeneous element of fixed internal degree has a finite upper bound on its possible \(h\)-order. Thus raising \(h\)-order at fixed internal degree terminates. This bound replaces any completeness or infinite-series assumption.

For a graded submodule \(I\) of a shifted finite free stalk module, its induced initial module is a bigraded submodule of (5.12f), hence has finitely many bigraded generators. Lift them to homogeneous \(g_i\in I\), retaining their initial \(h\)-orders. Match the first nonzero \(h\)-symbol of any homogeneous element by these lifts and subtract. The remainder has strictly higher \(h\)-order and the same internal degree. The preceding bound makes this process finite. It proves finite graded generation of \(I\) and strictness for this auxiliary filtration. It also proves the ascending-chain condition for graded submodules at every stalk.

The uniform neighbourhood statement follows by the same finite-identity argument as the earlier analytic operator Lemma 3.0c.4, with the auxiliary order now raised instead of lowered. Here are the details. For a degree-zero matrix \(\phi:A\to B\) between internally shifted finite free Rees modules with their usual coefficient \(h\)-order, choose at the point the lifts \(g_i\) of initial generators of its image. The replacement free module \(E\) must have **two explicit shifts**: its basis \(e_i\) has internal degree \(a_i=\deg(g_i)\) and auxiliary order \(v_i=\operatorname{ord}_h(g_i)\). Define
\[
F_h^vE=\bigoplus_i h^{\max(0,v-v_i)}\mathscr R\,e_i.
\]
Then \(\psi(e_i)=g_i\) is filtered for the auxiliary order and strict onto its induced image by the finite initial reduction. At fixed internal degree \(a\), every possible auxiliary order is at most \(\max_i(v_i+a-a_i)\); zero components with \(a<a_i\) are omitted. This gives the precise finite termination bound on the replacement module too. Its initial matrix is bigraded, with these two basis shifts. Internal shifts alone would be insufficient: multiplication by \(h\) would otherwise have a zero initial map. The finite reduction expresses the original columns through the \(g_i\), and every \(g_i\) already comes from a column combination. Therefore there are finite internally homogeneous matrices \(u:A\to E\), \(v:E\to A\) with
\[
\phi v=\psi,\qquad\psi u=\phi.
\tag{5.12g}
\]
Represent these finitely many identities on one neighbourhood. Choose finite bigraded generators \(\lambda_a\) of \(\ker\operatorname{gr}_h\psi\) at the point. Lift each to a vector of its internal degree and initial \(h\)-order. Its image under \(\psi\) has higher \(h\)-order. Auxiliary strictness expresses that image by a vector of the higher order; subtracting it gives an exact homogeneous relation \(r_a\) whose initial symbol is \(\lambda_a\).

The uniform homogeneous-polynomial-kernel argument, Lemma 3.0c.3, applies to the polynomial ring with \(d+1\) variables in (5.12f). Its proof does not depend on that number: extend the finite homogeneous matrix to the analytic variable space near the zero section; Oka makes its analytic kernel and the quotient by the chosen finite generators coherent. Relative analytic flatness identifies the kernel at the chosen stalk. A coherent quotient with zero stalk vanishes on a product neighbourhood. Faithful-flat contraction at each zero-section point and bounded-below graded zero-section detection then kill the polynomial quotient in every degree on one common neighbourhood. For clarity, the flatness input is the already proved Noetherian completion argument. Put \(A=H[u]_{(\mathfrak m,u)}\), \(B=\mathbf C\{y,u\}\). Their completed local rings agree over \(A\) as \(\widehat A=\widehat B=\mathbf C[[y,u]]\). Completion is faithfully flat on both rings by the exact earlier programme proof. Hence \(\widehat B\) is flat over \(A\), being its faithfully flat completion \(\widehat A\). If tensoring an injection with \(B\) had a nonzero kernel, faithful-flat tensoring with \(\widehat B\) would retain it, contradicting flatness over \(A\). Thus \(B\) is flat over \(A\); if a module tensoring with \(B\) is zero, tensoring with \(\widehat B=\widehat A\) is zero, and faithfulness of \(\widehat A\) makes that module zero. This proves faithful flatness directly. Graded detection follows from the finite graded module modulo \((u)\), Nakayama over \(H\), and induction on degree. This proves the needed uniform assertion for these \(d+1\) variables as well.

Represent the finitely many exact relations \(\psi(r_a)=0\) and their orders nearby. The same finite raising-order reduction now proves that the \(r_a\) generate \(\ker\psi\) on that neighbourhood and that \(\psi\) is auxiliary-strict onto its image. Its initial kernel and image are finite. Finally for \(z\in\ker\phi\),
\[
z=v\,u(z)+(1-vu)z;
\tag{5.12h}
\]
the first part is generated by the \(v(r_a)\), and the second by the finitely many columns of \(1-vu\). Both lie in \(\ker\phi\). This proves uniform graded Rees kernel and image coherence.

Finite presentation lifts and block matrices therefore prove that finite graded Rees modules form an abelian coherent category, closed under finite extensions. Every fixed internal-degree piece is holomorphically coherent: a graded finite presentation has finite holomorphic free pieces in that degree, and Oka applies. This is the exact graded coherence needed below. The proof uses finite polynomials throughout.

#### Strict induced twist segments from finitely many coherent order pieces

After shrinking the base, the global filtration is bounded below uniformly on the tested inverse image. Indeed a finite cover of the compact fibre supplies a common lower degree, and proper closedness shrinks the base into that cover. Likewise local good generation and a finite cover supply finitely many degrees \(a\) such that \(\operatorname{gr}_FM\) is generated by its pieces in those degrees on a smaller entire inverse image.

For each such degree apply GL13 [Theorem 5.8](#relative-analytic-generation-in-every-projective-dimension) to the **coherent holomorphic sheaf** \(F_aM\). Only finitely many degrees and shrinkings are used. It supplies a finite twist bundle \(E_a\) and a map \(E_a\twoheadrightarrow F_aM\). Compose with the quotient to \(\operatorname{gr}_aM\), and induce the original map to right operators, assigning \(E_a\) filtration degree \(a\). The resulting map
\[
A_0=\bigoplus_a E_a\otimes\mathcal D_Z\twoheadrightarrow M
\tag{5.12i}
\]
is a **strict** filtered epimorphism. Its symbol map is onto; lift a symbol, subtract its image, and lower filtration degree. The uniform lower bound makes reduction finite in every germ. Generating the actual \(F_aM\) avoids any unproved global lifting of a chosen section of its quotient.

The kernel has the induced global good filtration by the exact earlier uniform filtered-submodule proof. Repeat the same finitely-many-degree construction for that kernel. For any fixed integer \(s\) we obtain, after only finitely many base shrinkings, a strict filtered exact segment
\[
0\to K\to A_s\to\cdots\to A_0\to M\to0,
\quad A_i=\bigoplus_a E_{i,a}\otimes\mathcal D_Z
\text{ with the stated degree shifts}.
\tag{5.12j}
\]
Each \(E_{i,a}\) is a finite sum of projective twists. Rees formation makes this an actual exact graded sequence; specialization at zero makes it an actual exact symbol sequence. Thus no filtration-strictness or \(h\)-torsion error has been hidden at the source augmentation.

#### Actual Rees transfer and its independent bounds

The forward Rees transfer is
\[
\mathscr T=
\mathcal O_Z[h]\otimes_{p^{-1}\mathcal O_D[h]}p^{-1}\mathscr R_D.
\tag{5.12k}
\]
Locally its source action is \(\vartheta_x=h\partial_x\) on coefficients in a vertical direction and \(\vartheta_y=h\partial_y+\vartheta_{D,y}\) in a horizontal direction. These formulas respect (5.12b), glue for the product projection and commute with the right target Rees action. It is free over \(\mathcal O_Z[h]\) in the target \(\vartheta_D\)-monomials. Define the actual graded derived image
\[
\mathcal C=Rp_*\bigl(\mathscr R_FM\otimes^L_{\mathscr R_Z}\mathscr T\bigr).
\tag{5.12l}
\]

The absolute Rees Spencer model has length \(d\). Its terms are
\(\mathscr R_Z\otimes_{\mathcal O_Z[h]}\bigl((\bigwedge^rT_Z)[h]\bigr)(-r)\otimes_{\mathcal O_Z[h]}\mathscr T\). Here \((\bigwedge^rT_Z)[h]=\mathcal O_Z[h]\otimes_{\mathcal O_Z}\bigwedge^rT_Z\) is explicitly extended over the polynomial coefficient ring, and the internal shift \((-r)\) puts its wedge generators in degree \(r\), with \(V(-r)_a=V_{a-r}\). Its cochain degree is \(-r\). The differential is the ordinary action-difference differential with \(\partial\) replaced by \(\vartheta\), and bracket terms multiplied by \(h\). Explicitly its action terms omit \(\xi_i\) with sign \((-1)^{i-1}\) and use \(P\vartheta_{\xi_i}\otimes u-P\otimes\vartheta_{\xi_i}u\); its bracket term for \(i<j\) is \((-1)^{i+j}hP\otimes[\xi_i,\xi_j]\wedge\xi_1\wedge\cdots\widehat\xi_i\cdots\widehat\xi_j\cdots\wedge\xi_r\otimes u\). The relations \([\vartheta_\xi,a]=h\xi(a)\) and \([\vartheta_\xi,\vartheta_\zeta]=h\vartheta_{[\xi,\zeta]}\) make it well defined over the holomorphic-polynomial coefficient ring; paired action-commutator terms and the Jacobi identity give \(d^2=0\). Every term is left Rees-flat, since the coefficient transfer is holomorphically-polynomial flat. Filter the first Rees factor by its \(\vartheta\)-order, shifting by wedge degree. The leading differential is Koszul on the independent first-factor symbol variables with a flat coefficient. Successive polynomial variables act injectively even with arbitrary coefficient modules; the two-term cone induction proves Koszul exactness. Lowering the finite first-factor order proves exactness of the augmented Spencer complex. Its differentials preserve the internal grading and right target action. Consequently Rees transfer has amplitude \([-d,0]\) on arbitrary input modules, independently of output coherence.

The all-sheaf real-manifold bound, (D8)–(D10) gives proper sheaf-image amplitude \([0,2d]\). This also applies in the graded Rees category. Indeed evaluation at any internal degree has the exact left adjoint given by the free graded Rees module generated in that degree, tensored over \(\mathbf C\). It therefore takes graded injectives to injective complex sheaves. A bounded-below graded injective resolution computes image degreewise. Its underlying sum has c-soft terms, since each component is injective and coproducts of c-soft sheaves are c-soft; proper image commutes with these sums by the compact-fibre coproduct proof, (C1)–(C4) and (F4). Thus its underlying complex computes the ordinary sheaf-derived image, retaining every homogeneous target action. This also justifies passage to the ungraded category when later specializing at \(h=1\).

For any Rees input module \(U\),
\[
Rp_*(U\otimes^L_{\mathscr R_Z}\mathscr T)
\in D^{[-d,2d]}.
\tag{5.12m}
\]
For an induced shifted twist term of (5.12j), transfer has no higher Tor because its Rees module is locally free over \(\mathscr R_Z\). Proper coefficient projection and the holomorphic PBW basis \(h^k\vartheta_D^\alpha\) give its actual image
\[
Rp_*E_{i,a}\otimes_{\mathcal O_D}\mathscr R_D
\quad\text{with its internal degree shift }a.
\tag{5.12n}
\]
Each cohomology is finite locally free graded Rees, by the proved parameter twist calculation. The comparison respects right derivatives and functions: the product rule is
\[
\vartheta^\alpha b=
\sum_{\beta\le\alpha}\binom\alpha\beta
h^{|\beta|}\partial^\beta(b)\vartheta^{\alpha-\beta}.
\tag{5.12o}
\]
Its coefficients pull back on the source and act by the same target scalars after image. Thus (5.12n) is Rees-operator linear, not merely an underlying sheaf identity.

Take (5.12j) with \(s=3d+2\), and let \(A=[A_s\to\cdots\to A_0]\). The strict Rees augmentation has cone \(\mathscr R K[s+1]\). By (5.12m) its pushed cone lies in
\[
[-d-s-1,2d-s-1]=[-4d-3,-d-3].
\tag{5.12p}
\]
Thus the finite induced image computes every possible degree of \(\mathcal C\); graded coherence and finitely many cones prove that all \(H^b\mathcal C\) are finite graded Rees modules. This does not use an infinite resolution or assume that the image is strict.

#### Both central specializations are actual derived comparisons

The two-term free coefficient resolutions of \(\mathbf C[h]/(h-c)\), for \(c=0,1\), express derived specialization as the cone of multiplication by \(h-c\). Derived image commutes with that finite cone. On the source use the bounded absolute Spencer model from the absolute Rees-Spencer argument above, tensoring it with \(\mathscr R_FM\). Every resulting term is a sum of copies of that coefficient-flat input, with finite tangent-bundle factors and target \(\vartheta_D\)-monomials, so it is flat over \(\mathbf C[h]\). Ordinary specialization of this bounded flat complex computes derived specialization. At one its differential and terms are the ordinary absolute Spencer transfer tensor with \(M\). At zero they are the symbol Koszul differential on the vertical variables and the differences between source horizontal and target symbol variables. The independent polynomial-variable induction proves that this is a flat symbol-transfer resolution; equivalently its relative vertical Koszul resolution has length \(n\). This supplies the natural identities
\[
\mathcal C\otimes^L_{\mathbf C[h]}\mathbf C_1\simeq p_+M,
\tag{5.12q}
\]
and
\[
\mathcal C_0:=\mathcal C\otimes^L_{\mathbf C[h]}\mathbf C_0
\simeq Rp_*\left(
\operatorname{gr}_FM\otimes^L_{S_Z}
(\mathcal O_Z\otimes_{p^{-1}\mathcal O_D}p^{-1}S_D)
\right).
\tag{5.12r}
\]
At zero the transfer symbol is the quotient by the vertical covector coordinates; its finite relative Koszul model has length \(n\). Both displayed identities preserve the full ordinary or symbol target action. At \(h=1\) internal grading is forgotten; at zero it is retained.

Since (5.12m) bounds \(\mathcal C\), its zero specialization has sufficient range \([-d-1,2d]\). Specializing the pushed cone (5.12p) at zero changes its lower bound by at most one and keeps its upper bound \(-d-3\). It follows that the finite induced segment computes every possible zero-specialization degree \(b\ge-d-1\) as well. This is the reason for using \(s=3d+2\) here. The earlier finite unfiltered proof's \(3d+1\) remains sufficient for its own unchanged assertion.

#### Analytic cotangent comparison, justified through that finite segment

Let \(\mathfrak a_D\) denote analytic extension from polynomial symbol modules to \(T^*D\): pull back along the cotangent projection and tensor over \(S_D\) with \(\mathcal O_{T^*D}\). It is exact. At every cotangent point, translate the polynomial variables by that point's constant coordinates; the local polynomial and analytic rings have the same completion, so the same exact-completion/flatness proof used in the graded Rees-coherence argument above applies. The analogous extension on \(Z\) identifies the symbol-transfer restriction with \(Lj^*G\).

The required natural comparison is
\[
\mathfrak a_D(\mathcal C_0)\xrightarrow{\sim}Rq_*Lj^*G.
\tag{5.12s}
\]
It must not be inferred from a general analytic coherent-image comparison. Its natural map is obtained by pulling an image section to the cotangent product and multiplying its polynomial coefficient by the new analytic coefficient; resolutions and the flat coefficient extension give its derived version. Equivalently the adjunction unit to the pulled-back image, followed by this coefficient multiplication and image adjunction, defines that map. These operations are functorial in the source complex. Its being an isomorphism follows here from the finite strict segment. For each induced symbol term \(\operatorname{gr}A_i=\bigoplus_a E_{i,a}\otimes S_Z\) with its shifts, derived vertical restriction is ordinary restriction, giving \(\bigoplus_a E_{i,a}\otimes p^{-1}S_D\). Its analytic extension is \(\bigoplus_a E_{i,a}\otimes\mathcal O_W\). The parameter twist computation, first on \(D\) and then on coordinate polydiscs in \(T^*D\), gives the specific coefficient comparison
\[
\mathfrak a_D(Rp_*E_{i,a}\otimes S_D)
\simeq Rq_*(E_{i,a}\otimes\mathcal O_W).
\tag{5.12t}
\]
The parameter Laurent proof, Lemma 6.5, identifies this particular natural base-coefficient map, and its estimates commute with parameter restrictions and target coefficient multiplication. Functoriality of the constructed map, rather than a claim that a chosen contraction commutes with every projective-coordinate operation, makes it commute with the polynomial-symbol differentials between induced terms. Finite totalization proves the comparison for the entire finite segment.

Its zero-specialized Rees cone lies below \(-d-2\), as just proved. On the analytic side its strict symbol cone is \(\operatorname{gr}K[s+1]\); derived vertical restriction has amplitude \([-n,0]\), and the already proved projective coherent analytic image theorem has amplitude \([0,n]\). Hence its analytic pushed cone lies in \([-n-s-1,n-s-1]\), whose upper endpoint is at most \(-d-3\), since \(s=3d+2\) and \(n\le d\). The actual analytic candidate lies in \([-n,n]\) by Theorem 5.10. The natural finite comparison therefore gives an isomorphism in every potentially nonzero degree on both sides. This proves (5.12s), with no unbounded limit, general Grauert theorem or arbitrary analytical base-change assumption.

#### Image torsion is retained, then removed to produce a good output filtration

Put \(H_b=H^b\mathcal C\), a finite graded Rees module. Multiplication by \(h-1\) is injective on any graded module viewed as an ordinary module: in a finite sum of homogeneous components, its lowest nonzero degree has coefficient minus that component and cannot cancel. Formula (5.12q) and its two-term specialization sequence therefore give
\[
\mathcal H^b(p_+M)=H_b/(h-1)H_b.
\tag{5.12u}
\]

Let \(T_b\) be the \(h\)-power torsion in \(H_b\). It is locally killed by one uniform power and is graded coherent. Here are the neighbourhood details. At a stalk the graded ascending-chain condition proved in the graded Rees-coherence argument above makes this union of kernels finite and killed by some \(h^N\). The sheaf kernel \(T_N=\ker(h^N:H_b\to H_b(N))\) is graded coherent. At that stalk the quotient by \(T_N\) has injective multiplication by \(h\). Its multiplication kernel is coherent and has zero stalk, so vanishes on a smaller neighbourhood. On that neighbourhood the quotient is \(h\)-torsion-free; hence every power-torsion element belongs already to \(T_N\), proving \(T_b=T_N\) there. No intersection of infinitely many opens occurs.

Set \(\overline H_b=H_b/T_b\). Torsion vanishes under specialization at one, so (5.12u) is also its specialization. Its homogeneous degree pieces embed into that ordinary module. Indeed if \((h-1)u=v_a\ne0\) is homogeneous, the lowest degree of the finite sum \(u\) must be \(a\), since its negative contributes that same lowest degree. Its highest degree is therefore at least \(a\); multiplying that component by \(h\) produces a nonzero component of degree greater than \(a\), contradicting the displayed equality. Here injectivity of \(h\) in the quotient is essential. Define \(F'_a\mathcal H^b(p_+M)\) as the image of \((\overline H_b)_a\). These pieces are increasing since multiplication by \(h\) identifies their classes at one. They exhaust the module because any finite sum of homogeneous representatives can be raised to a common degree. They are locally bounded below and coherent holomorphic, by finite graded presentation and finite-order PBW. Finite homogeneous Rees generation proves goodness. Moreover
\[
\mathscr R_{F'}\mathcal H^b(p_+M)=\overline H_b,
\qquad
\operatorname{gr}_{F'}\mathcal H^b(p_+M)=\overline H_b/h\overline H_b.
\tag{5.12v}
\]
Thus an actual good output filtration has been constructed; proper-image strictness was not assumed.

The two-term zero-specialization sequence gives an injection
\[
H_b/hH_b\hookrightarrow H^b\mathcal C_0;
\tag{5.12w}
\]
its cokernel is the \(h\)-kernel in \(H_{b+1}\), with the corresponding internal shift. The natural map from \(H_b/hH_b\) onto \(\overline H_b/h\overline H_b\) is surjective. Exact analytic extension, (5.12s), (5.12v) and (5.12w) therefore make the actual analytic output symbol sheaf a **quotient of a submodule** of \(H^b(Rq_*Lj^*G)\). Its support is contained in that candidate's support. Theorem 5.10 now gives (5.12e).

This proves the characteristic-transport inequality for the actual projective operator image, including every derived degree, singular source support, torsion in filtered image and the genuine target action. It does not declare the image Rees module torsion-free or identify the output symbols with the whole candidate.

#### Locally projective maps and holonomicity

The closed-embedding transfer of Theorem 5.9 preserves coherent generators. In adapted normal coordinates \(z_1,\ldots,z_c\), its actual normal-derivative expansion is \(\bigoplus_{\alpha\in\mathbf N^c}i_*M\,\partial_z^\alpha\). The filtration is
\[
F_a(i_+M)=\bigoplus_{\alpha\in\mathbf N^c}i_*F_{a-|\alpha|}M\,\partial_z^\alpha.
\]
Only finitely many terms occur in each degree because the source filtration is bounded below. Those pieces are coherent holomorphic by the closed-embedding argument in Theorem 5.9. Normal multiplication lowers derivative degree, while normal differentiation raises it. PBW therefore identifies the symbol module with the source symbol module tensored with the polynomial normal covectors, at zero normal position. This proves goodness and the full normal characteristic extension. Density conversion changes the presentation by an invertible line and preserves the reduced support. Factoring a locally projective map of complex manifolds through such an embedding and projection extends the argument with the supplied generator or good filtration. Finite Postnikov towers extend the module conclusions to bounded complexes with that cohomology hypothesis: kernels, quotients and extensions have characteristic supports contained in the relevant unions by the exact-sequence proof in the good-filtrations lesson.

The needed analytic Lagrangian input is obtained as follows, rather than importing an algebraic-variety-only dimension statement. The full Noetherian Gabber theorem, Theorem 7.0, applies to every filtered Noetherian ring with commutative Noetherian rational associated graded, explicitly without finite type over a field. The analytic operator stalk \(\mathcal D_{Z,z}\), with coefficient ring \(H=\mathcal O_{Z,z}\), satisfies these hypotheses by its analytic PBW/Noetherian proof; \(S=H[\xi]\), and a good filtration gives a finite symbol module. For right modules use the opposite filtered ring; its commutator Poisson bracket changes sign, which leaves ideal bracket closure unchanged. Thus \(J=\sqrt{\operatorname{ann}_S\operatorname{gr}M}\) is homogeneous and involutive. This is an application of the full Noetherian theorem, not its algebraic geometric corollary.

The passage to the actual analytic cotangent ideal uses Theorem 3.0c and Lemmas 3.0c.1–3.0c.2, with the following argument repeated for this arbitrary module. At a zero covector, faithful-flat relative analytification and homogeneous reduction make \(J\mathcal O_{T^*Z}\) radical. Concretely its cotangent-variable completion is the product of the graded pieces of the reduced homogeneous quotient \(S/J\); the first nonzero homogeneous term of a hypothetical nilpotent has a nonzero power in that reduced quotient. The completion is therefore reduced, and faithful-flat Noetherian completion contracts reducedness. Flatness and finite generators identify the analytified symbol support with this ideal's zero set. Finite homogeneous generators of \(J\), their bracket relations and the analytic Leibniz rule give \(\{J\mathcal O,J\mathcal O\}\subset J\mathcal O\). The analytic Nullstellensatz and Cartan coherence extend these finite identities to a neighbourhood of the zero covector. Conic scaling transports the assertion to every covector, preserving the symplectic form up to a nonzero scalar.

At a smooth point of any analytic irreducible characteristic component, avoid the other components and use its holomorphic graph coordinates. Its ideal differentials span the conormal space; bracket closure says this conormal is isotropic for the inverse symplectic form. Thus the tangent space is coisotropic and the component has dimension at least \(d\), by the elementary inequality \(\dim V+\dim V^\perp=2d\). This proves the analytic componentwise Bernstein bound for every nonzero coherent input without an algebraic-only comparison. For holonomic \(M\), the defining upper bound \(\dim\operatorname{Ch}M\le d\) makes every component dimension exactly \(d\); equality in coisotropy makes its smooth tangent spaces Lagrangian. The reduced analytic characteristic support is therefore pure conic Lagrangian, including arbitrary singular support. The dense analytic smooth-locus and dimension statements here are the exact earlier local-parametrization and dimension proofs linked in Theorem 5.11.

For this holonomic source, Theorem 5.11 proves
\[
\dim q(j^{-1}\operatorname{Ch}M)\le m,
\tag{5.12x}
\]
Together with (5.12e), this proves every actual cohomology image holonomic. The same analytic componentwise bound just proved applies to a nonzero coherent target module, so the upper bound gives the required dimension; the zero module is included.

Theorem 5.11 proves (5.12x), including components of the restricted characteristic set contained wholly in singular characteristic strata. Its finite-branch restriction lemma and countable constant-rank proof provide the required dimension bound.

Section5.90, GG.A–GG.G proves the coherent ordinary generator near every compact set and over proper-support target neighbourhoods. Regularity of the arbitrary analytic proper image requires its further argument.

![The two actual central specializations and the subquotient producing the output characteristic support](assets/projective-filtered-characteristic-proof.png)

The upper comparisons are (5.12q) and (5.12s). The lower injection is (5.12w), and its quotient is the actual good output symbol module constructed in (5.12v), after removing locally uniform power torsion. The displayed analytic extension \(\mathfrak a_D\) is exact. This records a subquotient, rather than equality of the candidate and actual output symbols. The new strict segment uses \(s=3d+2\) and the cone bound (5.12p); the earlier unfiltered \(s=3d+1\) provider remains unchanged. The reproducible drawing source supplies the original CC0 diagram.


### Algebraic SNC boundary factors and every curve pullback

**Theorem 5.13.** Let \(P\) be smooth proper, let \(D_P\) be a reduced SNC divisor with smooth components, and let \(E\) be any finite-rank curve-regular connection on \(P\setminus D_P\). Its normalized algebraic logarithmic extension \(\bar E\) satisfies: \(\bar E(*D_P)\) is algebraic regular holonomic, every boundary simple factor is regular, and the same holds on every algebraic open \(Y\subset P\). For every algebraic map \(k:C\to Y\) from a smooth algebraic curve, the derived inverse image of this meromorphic module is regular holonomic. Arbitrary rank, Jordan blocks, tangency multiplicities, constant curves and curves lying in a boundary intersection are included.

We prove regularity in the algebraic composition-factor sense, including infinity. The proof extracts a coherent logarithmic lattice on each possible support with its intrinsic conormal determinant, then uses the already proved connection divisorial criterion. The inverse image calculation is explicit; it does not assume the general singular curve criterion.

All varieties below are smooth separated finite-type complex algebraic varieties. We use left differential-operator modules, the order filtration, and the programme convention
\[
f^!K=Lf_{\mathcal O}^*K[\dim X-\dim Y]
\quad(f:X\longrightarrow Y).
\tag{5.13a}
\]
Regularity of a connection means regularity on every smooth algebraic curve, at every point of the curve's smooth complete model, including infinity. Regular holonomicity means that every simple factor is an affine minimal extension of an irreducible curve-regular connection. These are the definitions fixed at the beginning of this section.

The exact earlier proofs are Theorem 5.1, Proposition 5.2.2, Lemma 5.2.3, Theorem 5.2, Lemma 1.0 and Theorem 1.1 of this lesson; PBW, Theorems 3.2 and 4.1; characteristic subquotients, Theorem 4.1, and the zero-section criterion, Proposition 6.1; finite connections, Theorem 3.1; Kashiwara’s equivalence and determinant inverse, Theorem 3.1; generic connections and finite length, Theorems 2.1–2.2; and localization, Proposition 1.1, minimal-extension uniqueness, Theorem 4.1, and simple-factor classification, Theorem 5.1. No general four-map or singular curve-test statement is used.

Free human source: Bernstein, [*Algebraic theory of D-modules*](https://www.math.columbia.edu/~khovanov/resources/Bernstein-dmod.pdf), printed pp. 31–33, especially the first step of the proof of Main Theorem B. The complete algebraic boundary-factor argument is supplied below. No Riemann–Hilbert statement or regularity-preservation theorem is an input.

#### Three elementary permanence facts

**Lemma 5.13.1.**

1. Regular holonomic modules form a Serre subcategory, and bounded complexes with regular holonomic cohomology form a triangulated subcategory.
2. Restriction to an arbitrary algebraic open preserves regular holonomicity.
3. If \(i:Z\hookrightarrow X\) is a smooth closed embedding, Kashiwara's exact equivalence restricts to an equivalence between regular holonomic modules on \(Z\) and regular holonomic modules on \(X\) supported on \(Z\). The same assertion holds for bounded complexes with supported cohomology.

**Proof.** For (1), intersect a composition series with a submodule. Each resulting successive quotient is zero or the corresponding simple factor. Quotient filtrations give the same assertion for quotients. Concatenating a composition series of a submodule with the inverse images of one in the quotient proves the extension assertion. A long cohomology sequence is assembled from kernels, images, cokernels and extensions, so proves the assertion about triangles.

For (2), consider a simple regular module \(S=j_{!*}E\), with \(j:U\hookrightarrow X\) affine and \(E\) irreducible and curve regular. On an open \(V\subset X\), its restriction is zero or the minimal extension from \(U\cap V\): restrict the canonical map defining the minimal extension, or apply its no-boundary uniqueness theorem. Restriction is exact and commutes with the transfers and the boundary map, so it commutes with its degree-zero image. The inclusion \(U\cap V\hookrightarrow V\) is the base change of \(j\), hence affine. If it is nonempty, the connection \(E|_{U\cap V}\) remains irreducible: a proper connection subobject there extends by the graph-closure Lemma 5.2.3, or its minimal extension gives a proper submodule of \(S\). It remains curve regular, because a curve mapping to this open also maps to \(U\). Thus the restriction is simple regular. Apply restriction to a composition series.

For (3), the exact closed direct image takes simple objects to simple objects. If \(S=j_{!*}E\) on \(Z\), its image is \((ij)_{!*}E\). Indeed, both restriction to \(U\) and boundary-supported subobjects and quotients are identified by the supported equivalence; minimal-extension uniqueness gives the asserted identification. The composite of two affine maps is affine. Conversely, if a simple \(i_*S\) has a regular presentation \(j_{!*}E\) on \(X\), the closure of the presenting stratum is its support, which lies in \(Z\). The stratum and its reduced locally closed immersion therefore factor through \(Z\). Its immersion into \(Z\) is the base change of the immersion into \(X\), hence affine. Kashiwara's inverse and the same uniqueness argument give a regular presentation of \(S\). Exactness transports composition series in both directions. On supported bounded complexes, the earlier Kashiwara derived equivalence is exact on each cohomology module; applying the module assertion in each degree proves (3). \(\square\)

The inverse assertion in (3) applies only to a complex already supported on \(Z\). It makes no assertion here about \(i^!K\) for a general regular complex on \(X\).

#### Algebraic finite logarithmic hulls

Let \(P\) be smooth proper, let \(D=\bigcup_{a=1}^sD_a\) be a reduced SNC divisor with globally smooth components, and put \(U=P\setminus D\). Let \(E\) be a curve-regular connection on \(U\), of any finite rank. Theorem 5.1 gives its normalized algebraic logarithmic extension \(\bar E\) on \(P\). Put
\[
M=\bar E(*D)=\bigcup_{m\geq0}\bar E(mD).
\tag{5.13b}
\]
Write \(\mathscr L_P\) for the algebra generated by \(\mathcal O_P\) and the vector fields preserving the ideals of all the components \(D_a\). In an adapted étale coordinate chart it is generated by
\[
z_1\partial_{z_1},\ldots,z_k\partial_{z_k},
\partial_{w_1},\ldots,\partial_{w_{n-k}}.
\tag{5.13c}
\]
Every \(\bar E(mD)\) is finite over \(\mathcal O_P\) and stable under \(\mathscr L_P\). This follows directly from the logarithmic Leibniz rule: acting on \((z_1\cdots z_k)^{-m}\) adds the integral residue shift \(-m\) in each normal direction and does not increase its pole order.

**Lemma 5.13.2 (subquotient hulls).** Every differential-operator subquotient \(F=A/B\) of \(M\) is a union of \(\mathcal O_P\)-coherent \(\mathscr L_P\)-stable subsheaves.

**Proof.** Define
\[
L_m=\operatorname{im}\bigl(A\cap\bar E(mD)\longrightarrow A/B\bigr).
\tag{5.13d}
\]
The intersection is coherent over \(\mathcal O_P\), since on an affine chart it is a submodule of a finite module over a Noetherian ring. Intersections and images commute with localization, so these modules define the stated coherent subsheaves. Both \(A\) and \(\bar E(mD)\) are stable under \(\mathscr L_P\); hence so are their intersection and its image. Every local section of \(A\) has a locally finite pole order in (5.13b), so belongs locally to one of the intersections. Thus \(\bigcup_m L_m=F\). \(\square\)

#### The possible supports of boundary factors

**Lemma 5.13.3.** The characteristic support of \(M\), and of each of its differential-operator subquotients, lies in the union of the conormals to the SNC intersections. Every nonzero simple factor of \(M\) has support equal to one connected component \(Z\) of an intersection \(D_I=\bigcap_{i\in I}D_i\), with \(I=\varnothing\) allowed. On
\[
S_I=Z\setminus\bigcup_{j\notin I}(D_j\cap Z),
\tag{5.13e}
\]
its Kashiwara inverse is a nonsingular finite-rank connection.

**Proof.** We give an algebraic order-filtration argument; an analytic constant-residue gauge is unnecessary. The finite lattice \(F_0M=\bar E(D)\) generates \(M\) as a differential-operator module. To see this, choose a logarithmic frame as a row \(e\) on an adapted étale chart, and write its normal logarithmic matrices as \(A_i\), so that \(z_i\nabla_{\partial_{z_i}}e=eA_i\). For a constant coefficient column \(v\) and \(a_i\geq1\),
\[
\partial_{z_i}(z^{-a}ev)
=z^{-(a+\mathbf e_i)}e(A_i-a_iI)v.
\tag{5.13f}
\]
At \(z_i=0\) the matrix on the right is invertible: all residue eigenvalues have real parts in \([0,1)\), whereas \(a_i\) is a positive integer. Its inverse exists in the algebraic local ring. Away from \(z_i=0\), multiplication by \(z_i^{-1}\) itself is allowed. Starting with \(a=(1,\ldots,1)\), these local statements inductively generate each lattice with any prescribed finite pole orders, and hence (5.13b). This is a sheaf-local argument for each finite multi-index; it does not assert that one algebraic affine neighborhood inverts infinitely many determinants simultaneously.

Set \(F_pM=\mathcal D_{P,\leq p}F_0M\) for \(p\geq0\), with zero negative terms. Each term is coherent over \(\mathcal O_P\); locally it is the image of a finite operator module tensored with a finite lattice, inside the finite lattice \(\bar E((p+1)D)\). PBW and (5.13f) show exhaustivity and that this is a good generated filtration.

Each operator in (5.13c) preserves \(F_0M\). Commuting it past a normally ordered derivative monomial of order \(p\) leaves an operator of order at most \(p\): specifically,
\[
[z_i\partial_{z_i},\partial_z^a\partial_w^b]
=-a_i\partial_z^a\partial_w^b,
\qquad
[\partial_{w_j},\partial_z^a\partial_w^b]=0,
\tag{5.13g}
\]
and differentiation of a coefficient does not increase its order. These operators therefore preserve every \(F_pM\). Their first-order symbols consequently annihilate \(\operatorname{gr}_F M\), because their actions into \(F_{p+1}/F_p\) are zero. In cotangent coordinates this gives
\[
\operatorname{Ch}(M)\subset
V(z_1\xi_1,\ldots,z_k\xi_k,\eta_1,\ldots,\eta_{n-k})
=\bigcup_{J\subset\{1,\ldots,k\}}T^*_{D_J}P.
\tag{5.13h}
\]
The last equality is set-theoretic: at each point, choose the indices for which the normal coordinate is zero; for every remaining index its covector coefficient must vanish. Each displayed component has dimension \(n\). Localization holonomicity and the standard strict induced good filtrations give the same characteristic-support inclusion for every subquotient. In detail, the induced graded submodule and quotient are finite over the Noetherian graded operator ring, so are good after finitely many degree shifts; the exact graded sequence proves the inclusion.

Let \(F\) be a nonzero simple factor, with irreducible support \(W\). At a general point of \(W\), exactly one SNC stratum \(S_I\) contains that point. The earlier simple-factor classification and Kashiwara calculation give the conormal of the smooth general part of \(W\) as a characteristic component of dimension \(n\). Formula (5.13h) restricts its normal fiber to the span of the \(|I|\) normals of the stratum. Hence
\[
n\leq\dim W+|I|\leq\dim Z+|I|=n.
\tag{5.13i}
\]
Thus \(\dim W=\dim Z\), and the closed irreducible support is the entire component \(Z\) of \(D_I\). This dimension argument is applied at a general point of \(W\); it does not infer a support from a nongeneric characteristic fiber.

On \(S_I\), the only cotangent directions allowed by (5.13h) are the normals to \(Z\). Kashiwara's inverse therefore has characteristic variety contained in its zero section. Such a coherent differential-operator module is \(\mathcal O\)-coherent: in a finite good graded presentation, the ideal generated by the positive-degree vector symbols has a power annihilating the graded module, so only finitely many symbol monomials are needed as structure-sheaf generators. Lifting those finitely many graded generators gives structure-sheaf generators of the original module by induction down its filtration. The earlier coherent-connection theorem makes it a finite-rank nonsingular connection. \(\square\)

In (5.13e), each \(Z\) is smooth and proper, and its remaining boundary is SNC. Empty intersections contribute nothing. The empty subset contributes the open stratum \(U\).

#### Extracting a logarithmic lattice with the correct determinant

**Lemma 5.13.4 (normal extraction).** Let \(F\) be a simple factor of \(M\), supported on the component \(Z\) from Lemma 5.13.3, and let \(Q\) be its Kashiwara inverse on \(Z\). Then the connection \(H=Q|_{S_I}\) is curve regular, including infinity.

**Proof.** Let \(\mathcal I\) be the ideal of \(Z\). The intrinsic Kashiwara inverse has underlying structure-sheaf module
\[
Q=\det(\mathcal I/\mathcal I^2)\otimes
\operatorname{ann}_{\mathcal I}F.
\tag{5.13j}
\]
The determinant factor cannot be dropped. Apply Lemma 5.13.2 and put
\[
\begin{aligned}
A_m&=\operatorname{ann}_{\mathcal I}L_m,\\
Q_m&=\det(\mathcal I/\mathcal I^2)\otimes A_m.
\end{aligned}
\tag{5.13k}
\]
The annihilator of the finitely generated ideal in a coherent module is coherent, so each \(Q_m\) is finite over \(\mathcal O_Z\); and \(\bigcup_mQ_m=Q\).

We check its logarithmic stability, including the twist. Work in the adapted coordinates \(Z=(z_1=\cdots=z_k=0)\), and write
\(\nu=[z_1]\wedge\cdots\wedge[z_k]\) for a frame of the determinant in (5.13j). A logarithmic vector field \(\xi\) on \(Z\), along its remaining SNC boundary, has a local lift \(\widetilde\xi\) to a logarithmic vector field on \(P\) preserving each selected normal divisor. Write \(\widetilde\xi z_i=a_i z_i\). The induced action is
\[
\begin{aligned}
\nabla^Q_\xi(\nu\otimes v)
&=\nu\otimes\left(\widetilde\xi v+\sum_{i=1}^k a_i v\right),\\
&\qquad v\in\operatorname{ann}_{\mathcal I}F.
\end{aligned}
\tag{5.13l}
\]
Indeed the lift acts on the conormal determinant by \(\sum_i a_i\). It preserves \(\operatorname{ann}_{\mathcal I}F\), since
\(z_i\widetilde\xi v=\widetilde\xi(z_iv)-a_iz_iv=0\).

The formula is independent of the lift. In a difference of two lifts, the terms with nonzero restriction in normal derivative directions are \(\sum_i z_i b_i\partial_{z_i}\). On an annihilated vector,
\[
z_i\partial_{z_i}v=-v,
\tag{5.13m}
\]
by \([\partial_{z_i},z_i]=1\); this contributes \(-\sum_i b_i v\), canceled by the determinant contribution \(+\sum_i b_iv\). Terms with coefficients in \(\mathcal I\) in the tangential directions act by zero on an annihilated vector. Thus (5.13l) is precisely the earlier intrinsic supported inverse. A lift preserving all divisors exists locally by lifting its coefficient functions in the adapted étale chart; these are exactly the logarithmic generators (5.13c). Coordinate changes give the conormal determinant transition and the Leibniz transformation, so the calculation glues.

Since \(L_m\) is stable under every such lift, (5.13l) preserves \(Q_m\). Put
\[
B_Z=\bigcup_{j\notin I}(D_j\cap Z).
\]
The complement immersion \(h:S_I\hookrightarrow Z\) is affine: locally its boundary is a single nonzero divisor equation, so its complement is a principal affine open; affineness of a morphism is local on its target. Underlying localization gives
\[
Q(*B_Z)=h_*H.
\tag{5.13n}
\]
This is torsion free as an \(\mathcal O_Z\)-module, since a section of a vector bundle on a dense open of the integral smooth \(Z\) cannot be killed by a nonzero regular function. The images
\[
\widetilde Q_m=
\operatorname{im}(Q_m\longrightarrow Q(*B_Z))
\tag{5.13o}
\]
are coherent torsion-free logarithmically stable modules. Their union has generic fiber \(H_{\mathbf C(Z)}\). Choose finitely many vectors spanning that fiber and then one index \(m\) containing all of them; the hull \(\widetilde Q_m\) has full generic rank.

At the generic point of any component of \(B_Z\), its stalk is a finite torsion-free module of full rank over the transverse DVR. It is therefore a full lattice, by Lemma 1.0. The generator \(t\partial_t\) of the logarithmic normal field preserves it, by (5.13l). It supplies a regular singular lattice for the transverse differential module. This proves regularity at every boundary divisor of the smooth proper SNC compactification \(Z\) of \(S_I\).

Apply the already proved connection divisorial criterion, Theorem 5.1, to the nonsingular algebraic connection \(H\) on \(S_I\). It makes \(H\) regular on every algebraic curve and at every completion point. In particular it tests curves lying in deeper boundary intersections after their own compactification; these are included by that earlier theorem, rather than omitted in the transverse DVR argument. \(\square\)

For \(I=\varnothing\), (5.13j) has no determinant twist, and the proof is the same. For a zero-dimensional \(Z\), there are no boundary divisors and the connection is a finite-dimensional vector space; the zero-rank case uses the zero lattice. No diagonalization, splitting of residue eigenspaces, or semisimplicity assumption is used in (5.13j)–(5.13o).

#### The full algebraic SNC extension theorem

**Conclusion of Theorem 5.13.** Let \((P,D_P)\) be any smooth proper SNC pair, let \(E\) be any finite-rank curve-regular connection on \(U=P\setminus D_P\), and let \(\bar E\) be its normalized algebraic logarithmic extension. Then \(\bar E(*D_P)\) is algebraic regular holonomic. The same conclusion holds on every algebraic open \(Y\subset P\) for the affine open direct image from \(V=Y\cap U\), with \(D=D_P\cap Y\):
\[
j_*E=E(*D),\qquad j:V\hookrightarrow Y,
\tag{5.13p}
\]
is an algebraic regular holonomic module. All its boundary simple factors, including factors on arbitrary nonempty SNC intersections, are regular. Every connection factor and every point factor is retained.

**Proof for the proper pair.** A simple factor \(F\) of \(M\) has the support and nonsingular connection \(H\) of Lemma 5.13.3. The connection is regular by Lemma 5.13.4. Its restriction to \(S_I\) is irreducible: the earlier simple-factor classification makes the restriction to some dense open irreducible; a connection subobject on \(S_I\) would restrict to one there, and a nonzero connection or quotient has positive generic rank. Thus it is either zero or the entire connection. Simplicity of \(F\) gives no nonzero submodule or quotient supported on the omitted boundary. Minimal-extension uniqueness gives
\[
F\simeq (S_I\hookrightarrow P)_{!*}H.
\tag{5.13q}
\]
The presenting immersion is affine: it is the composite of the affine complement immersion \(S_I\hookrightarrow Z\) and the closed immersion \(Z\hookrightarrow P\). Thus (5.13q) is exactly a regular presentation under the course definition. Every simple factor has this property, so \(M\) is regular holonomic. This proof establishes algebraic regularity, including infinity, rather than merely the analytic filtration condition of Theorem 5.3.

**Restriction to an arbitrary open of this pair.** The restriction of \(\bar E(*D_P)\) to \(Y\) is exactly \(j_*(E|_V)\): its local boundary equation is the restriction of that of \(D_P\), and localization sheafifies the given connection on its complement. Lemma 5.13.1(2) proves that the restriction is regular holonomic. The localization is concentrated in degree zero because \(j\) is affine and its transfer is localization. \(\square\)

The starting data are a proper SNC pair and its open restrictions. Theorem 5.1 supplies such proper SNC compactifications for regular connections. No stronger assertion about preserving a pre-existing SNC divisor during compactification is used.

#### Every algebraic curve map tests an SNC meromorphic extension

**Corollary 5.13.5.** For \(M=j_*(E|_V)\) on \(Y\) as in Theorem 5.13 and every algebraic map \(k:C\to Y\) from a smooth algebraic curve, the bounded holonomic complex \(k^!M\) is regular. This includes \(Y=P\). The map need not be an immersion, noncharacteristic, dominant onto its image, or transverse to \(D\).

**Proof.** Work on a connected component of \(C\), which is integral, and on a connected component of \(Y\). Locally \(M\) is a finite locally free logarithmic lattice tensored with \(\mathcal O_Y[1/q]\), where \(q\) is a local equation for \(D\); hence it is flat over \(\mathcal O_Y\). This flatness is intrinsic, since localizations are flat and the lattice is locally free. Consequently the transfer formula (5.13a) has no higher structure-sheaf Tor terms:
\[
k^!M\simeq
k_{\mathcal O}^*M[1-\dim Y].
\tag{5.13r}
\]
If \(k(C)\subset D\), the inverse image is zero. At each point, some boundary equation pulled back to the integral curve is identically zero; the localization tensor inverts that zero and is the zero module. This covers constant maps into the boundary and maps lying in any deeper SNC intersection.

Otherwise \(T=k^{-1}D\) is finite, and \(W=C\setminus T\) is dense. On \(W\), (5.13r) is the usual flat-bundle pullback \((k|_W)^*E\). It is curve regular, including infinity, because every curve mapping to \(W\) composes with \(k|_W\) to a curve mapping to \(V\). In particular its own identity map supplies the logarithmic lattice at every point of the complete model of \(W\). Across \(T\), (5.13r) is its meromorphic localization; its operator action is the chain-rule pullback connection. This follows in a local logarithmic frame from
\[
k^*(A_i\,d\log z_i)=
k^*A_i\left(m_i\,\frac{dt}{t}+\frac{du_i}{u_i}\right)
\quad\text{if }k^*z_i=t^{m_i}u_i,
\tag{5.13s}
\]
where \(m_i\geq0\) and \(u_i\) is a unit. Thus all tangency multiplicities, residues and nilpotent parts are retained.

For clarity, a holonomic module on a smooth curve with a regular generic connection is regular holonomic under the composition-factor definition. Its point-supported simple factors are the point delta modules, regular presentations from points. Every full-support simple factor restricts to a subquotient of the generic connection on a common dense open; this is a regular connection by Proposition 5.2.2. Its presenting open is affine (the complement is a finite divisor), and minimal-extension uniqueness gives its regular presentation. There are no other supports on a curve. This proves regularity of the meromorphic localization in (5.13r). The fixed dimension shift preserves regularity, and the zero case was included. \(\square\)

The logarithmic frame used for the flatness calculation exists on \(Y\) by the proper-pair construction of Theorem 5.13; equivalently it is the restriction of the normalized lattice on \(P\). At infinity of \(C\), regularity is supplied by the intrinsic regularity of \((k|_W)^*E\), even when \(k\) does not extend as a map to \(Y\). Equation (5.13s) is a local check at points of \(C\), and is not substituted for that infinity assertion.

![Normal extraction and the transverse lattice](assets/snc-boundary-extraction.png)

*Figure. The upper route is the actual subquotient-hull construction (5.13d). The middle row shows the sign cancellation (5.13l)–(5.13m) that makes the intrinsic supported inverse independent of a normal lift. The lower row is the full-rank DVR lattice (5.13o), followed by the previously proved connection divisorial criterion. The figure records a proof mechanism, not a splitting of the subquotient or a decomposition of the logarithmic connection. The reproducible drawing source supplies the original CC0 figure.*


### Arbitrary inverse transfer and singular supported graph base change

Dimension shifts are taken componentwise. A smooth finite-type variety has finitely many disjoint open-and-closed smooth integral components, each of pure dimension. A connected source maps into one target component. Finite component sums extend all statements below to varieties whose component dimensions differ, without assigning a single incorrect global dimension.

Theorem 5.13 is used with its proved scope: every proper smooth SNC pair, every boundary simple factor of its meromorphic extension and every open restriction of that pair.

The exact earlier proofs are inverse transfer and composition, Proposition 1.1 and Theorem 2.1; supported localization, product base change and absolute Spencer transfer, Lemmas 1.1 and 3.3 and Theorem 2.1; and the PBW, Kashiwara and holonomic proofs linked in Theorem 5.13. The finite Čech argument below extends direct composition to every smooth separated finite-type variety. Theorem 5.6 supplies full generic Gauss–Manin regularity.

#### The inverse transfer for every map

Let \((P,D)\) be any smooth proper SNC pair over \(\mathbb C\), let \(E\) be any finite-rank curve-regular connection on \(U=P\setminus D\), and let \(\bar E\) be its normalized logarithmic extension. Put \(M=\bar E(*D)\). Let \(f:Y\to P\) be **any** map from a smooth separated finite-type variety. Work componentwise, so \(P\) and \(Y\) have pure dimensions \(n\) and \(d\).

**Theorem 5.14.1.** On each connected component of \(Y\) whose image lies in \(D\), \(f^!M=0\). On every other component, \(B=f^*D\) is an effective Cartier divisor, allowed to have arbitrary multiplicities and singular reduced support, and
\[
\begin{gathered}
f^!M\simeq N_f[d-n],\\
N_f=(f^*\bar E)(*B).
\end{gathered}
\tag{5.14a}
\]
The only possible nonzero cohomology is \(H^{n-d}(f^!M)=N_f\). The connection on the localization is the actual inverse-transfer connection. If locally
\[
\nabla^{\bar E}=d+\sum_i A_i\,\frac{dz_i}{z_i}
                  +\sum_j C_j\,dw_j,
\]
then, on the localization where the expressions are defined,
\[
\begin{gathered}
\nabla^{N_f}=d+\sum_i f^*A_i\,\frac{dh_i}{h_i}\\
+\sum_j f^*C_j\,d(f^*w_j),\\
h_i=f^*z_i.
\end{gathered}
\tag{5.14b}
\]
The unshifted module \(N_f\) is \(\mathcal O_Y\)-flat and holonomic. For every map \(k:C\to Y\) from a smooth algebraic curve, \(k^!N_f\) is a regular holonomic bounded complex, including every point at infinity. This is a proved **curve-testing assertion**; regularity of \(N_f\) under the composition-factor definition is not inferred from it.

**Proof.** On a logarithmic frame chart, \(M\) is \(\bar E\otimes\mathcal O_P[1/q]\), where \(q\) is a local equation for the Cartier divisor \(D\). It is flat over \(\mathcal O_P\): the lattice is locally free and localization is flat. The actual transfer/structure-sheaf comparison in the inverse-image provider therefore gives
\[
\begin{gathered}
\mathcal D_{Y\to P}\otimes^L_{f^{-1}\mathcal D_P}f^{-1}M\\
\simeq\mathcal O_Y\otimes_{f^{-1}\mathcal O_P}f^{-1}M.
\end{gathered}
\tag{5.14c}
\]
This conclusion does not require \(f\) to be flat. Tensor associativity and the universal property of localization identify the right side with
\(f^*\bar E\otimes\mathcal O_Y[1/f^*q]\).

If the image of an integral component of \(Y\) is contained in the divisor, \(f^*q=0\) on that component, and this tensor is zero. Otherwise a nonzero \(f^*q\) is a nonzerodivisor in each smooth local domain. Its local equations differ by units and define the effective Cartier divisor \(B\). Localization is independent of its multiplicities, but the differential in (5.14b) retains them. The pulled-back lattice is locally free and its localization is \(\mathcal O_Y\)-flat. Adding the programme dimension shift proves (5.14a), including the degree assertion.

The source's chain-rule formula identifies the action on a pulled-back coefficient with the ordinary differential of the pulled-back functions; applied to the logarithmic frame, it is exactly (5.14b). The coefficient-derivative term is included, so this is a balanced connection formula and not tensor pullback of a non-\(\mathcal O\)-linear connection map. Its flatness is the already proved Lie-bracket calculation for the transfer. The old arbitrary holonomic inverse-image theorem applies to \(M\), which is holonomic by Theorem 5.13, and therefore proves holonomicity of \(N_f\). No regularity-preservation theorem is used.

Now let \(k:C\to Y\), with \(C\) connected. If \(f k(C)\subset D\), its pullback is zero by the same localization calculation. Otherwise the properness of \(P\) extends \(f k\) uniquely to
\(\bar h:\bar C\to P\), where \(\bar C\) is the smooth complete model of \(C\). The extension uses the earlier valuative criterion at each curve DVR, followed by uniqueness from separatedness. Near each point \(p\in\bar C\), choose the local boundary equations through \(\bar h(p)\). Since the curve is not contained in the boundary,
\[
\begin{gathered}
\bar h^*z_i=t^{m_i}u_i,\\
m_i\geq0,\quad u_i\in\mathcal O_{\bar C,p}^{\times},\\
\frac{d(\bar h^*z_i)}{\bar h^*z_i}\\
=m_i\frac{dt}{t}+\frac{du_i}{u_i}.
\end{gathered}
\tag{5.14d}
\]
Thus the full lattice \(\bar h^*\bar E\) is logarithmic at \(p\); its residue is \(\sum_i m_i\bar h^*A_i(p)\). There is no assertion that this sum remains normalized. All multiplicities and nilpotent parts are present. This proves regularity of the generic pullback connection at every point, including infinity, by the one-variable lattice criterion. Its meromorphic localization on \(C\) is regular holonomic by the curve argument in 5.13.5: full-support factors inherit a regular connection and point factors are delta modules. Since \(N_f\) is flat, its own extraordinary pullback has the single shift \([1-d]\). This completes the curve-testing assertion. \(\square\)

**Functoriality with all degrees.** For \(g:Z\to Y\), the flatness of \(N_f\) gives
\[
\begin{gathered}
N_{f,g}=(g^*f^*\bar E)(*g^*B),\\
g^!N_f=N_{f,g}[\dim Z-d].
\end{gathered}
\tag{5.14e}
\]
with zero again on boundary-contained components. Applied to \(N_f[d-n]\), the total shift is
\((\dim Z-d)+(d-n)=\dim Z-n\), the shift of \((fg)^!M\). The chain-rule transfer composition identifies these localization maps and their operator actions. This includes nonflat \(g\), singular \(g^{-1}B\), and every pure-dimensional component separately.

The calculation here proves curve testing of the singular meromorphic extension. Corollary 5.17.1 below supplies its regular simple-factor reconstruction after the full singular curve criterion has been proved.

![The singular pulled-back divisor, two exact curve tests, and supported graph base change](assets/arbitrary-inverse-transfer.png)

*Figure. The upper plot is the actual real slice of the divisor pulled back by \(f(x,y)=(x^2-y^3,x^2+y^3)\) in the affine chart of \(P=(\mathbb P^1)^2\), whose SNC boundary contains zero and infinity in both factors. For the commuting local residues \(A_1=\frac15I+J\), \(A_2=\frac3{10}I+J\), with \(J=\left(\begin{smallmatrix}0&1\\0&0\end{smallmatrix}\right)\), the boundary-contained curve has zero inverse image, and the diagonal curve has residue \(I+4J\) at zero. This residue is logarithmic but is not normalized; the finite multiplicities and nilpotent parts are exactly (5.14d). The lower panel is the arbitrary-map formula (5.14j), proved below, and retains the singular fiber product only as a support inside a smooth product. The figure displays the actual transfer and curve tests. The later Corollary 5.17.1 supplies simple-factor reconstruction. The reproducible drawing source supplies the original CC0 figure.*

#### Direct composition without quasi-projectivity

**Lemma 5.14.2.** Differential-operator direct image and its composition isomorphism exist on bounded quasi-coherent differential-operator complexes for every map of smooth separated finite-type varieties. For \(X\xrightarrow aY\xrightarrow bZ\),
\[
(ba)_*K\simeq b_*a_*K,
\tag{5.14f}
\]
naturally and associatively. This lemma concerns transfer, boundedness and composition; it asserts no regularity preservation.

**Proof.** Work on right modules for the tensor notation. Put \(T_a=\mathcal D_{X\to Y}\) and \(T_b=\mathcal D_{Y\to Z}\). Lemma 3.3 of the adjunctions lesson gives the augmented absolute Spencer resolution of \(T_a\), with terms
\[
\begin{gathered}
\mathcal D_X\otimes_{\mathcal O_X}\bigwedge^rT_X\\
\otimes_{\mathcal O_X}T_a,\\
\text{in degree }-r,\\
0\leq r\leq\dim X.
\end{gathered}
\tag{5.14g}
\]
Its proof uses the bracket/Leibniz differential, PBW, the polynomial Koszul sequence, and descent in the finite order of the first operator factor. Its terms are flat as left \(\mathcal D_X\)-modules, because \(T_a\) is \(\mathcal O_X\)-flat. Hence it computes transfer tensor with an arbitrary right quasi-coherent module; the resulting terms are quasi-coherent over \(\mathcal O_X\), and the amplitude is bounded by \([-\dim X,0]\). A bounded input complex has the corresponding finite band.

On an affine open \(V\subset Y\), \(a^{-1}V\) is quasi-compact and separated, and therefore has a finite affine cover with affine intersections. The usual finite affine Čech complex of the transferred quasi-coherent terms computes their sheaf direct image and supplies a finite additional degree bound. This requires separated finite type, not quasi-projectivity or a global free resolution.

We check the interchange needed for composition locally on \(Y\). Shrink to a coordinate affine chart whose image under \(b\) lies in an operator-coordinate chart on \(Z\). PBW on that target chart makes \(T_b\) a locally free \(\mathcal O_Y\)-module with possibly infinitely many basis elements. The same absolute Spencer resolution (5.14g), now for \(T_b\) on \(Y\), is bounded; on this chart its terms are direct sums of free left \(\mathcal D_Y\)-modules. This uses the trivialized finite exterior bundles of \(T_Y\) and the PBW basis of \(T_b\), so avoids any global-dimension assertion for arbitrary infinitely generated modules.

For a free term \(\mathcal D_Y^{(I)}\), tensoring on either side of
\[
\begin{gathered}
C_a=A\otimes^L_{a^{-1}\mathcal D_Y}a^{-1}T_b,\\
Ra_{\mathrm{sh},*}A\otimes^L_{\mathcal D_Y}T_b\\
\longrightarrow Ra_{\mathrm{sh},*}C_a,\\
A=K\otimes^L_{\mathcal D_X}T_a.
\end{gathered}
\tag{5.14h}
\]
gives the same direct sum of the finite Čech model for \(Ra_{\mathrm{sh},*}A\). Sections of quasi-coherent sheaves on each affine intersection commute with these direct sums, and the cover is finite. Thus (5.14h) is an isomorphism for each free term and for the bounded Spencer total. It is a natural sheaf comparison and hence glues from these local coordinate charts. No claim about arbitrary sheaf pushforward commuting with infinite sums is needed.

Inverse transfer composition from the inverse-images lesson gives the derived identity
\[
T_a\otimes^L_{a^{-1}\mathcal D_Y}a^{-1}T_b
\simeq T_{ba}.
\tag{5.14i}
\]
Its higher Tor vanishes because the underlying \(\mathcal O_Y\)-module of \(T_b\) is flat; its underived action is the chain rule. Substitute (5.14h) and (5.14i) in the two direct-image definitions and use composition of derived sheaf pushforward to obtain (5.14f). The bounded Spencer and finite Čech bands justify the totalizations. Tensor, chain-rule and pushforward composition are their actual associative maps, so the result is compatible with three maps. Side changing at \(X,Y,Z\) cancels at the middle variety and gives the left-module formula. \(\square\)

#### Base change through a possibly singular fiber product

Let \(f:X\to Y\) and \(g:Y'\to Y\) be arbitrary maps of smooth separated finite-type varieties. The scheme \(X\times_Y Y'\) may be singular or nonreduced. Use the smooth ambient product
\[
\begin{gathered}
T=Y'\times X,\\
t:T\to X,\qquad s:T\to Y',\\
F=\mathrm{id}_{Y'}\times f:T\to A,\\
A=Y'\times Y.
\end{gathered}
\]
Let \(i:Y'\hookrightarrow A\) be the closed graph of \(g\), and let \(W=F^{-1}(i(Y'))\subset T\) be the closed fiber product locus. Local cohomology depends only on its closed support, so allows its given nonreduced structure.

**Theorem 5.14 (supported graph formula).** For every bounded holonomic \(K\) on \(X\),
\[
g^!f_*K\simeq s_*R\Gamma_W(t^!K).
\tag{5.14j}
\]
Here the supported complex is on the **smooth** variety \(T\). No smooth transfer formula is assigned to \(W\). All complexes in (5.14j) are bounded holonomic, and the comparison is natural.

**Proof.** The closed-support local cohomology on \(T\) is the finite augmented Čech localization complex for local generators of the ideal of \(W\). Each term is a finite localization of a bounded holonomic complex, so is bounded holonomic by the earlier localization and holonomicity results. This construction glues and gives the triangle
\[
\begin{gathered}
Q=R\Gamma_W(t^!K)\\
\longrightarrow t^!K\\
\longrightarrow j'_*j'^!(t^!K)\longrightarrow.
\end{gathered}
\tag{5.14k}
\]
where \(j':T\setminus W\hookrightarrow T\). A product inverse image preserves holonomicity, so all assertions here precede regularity.

Let \(r:A\to Y\) and \(\pi:A\to Y'\) be the projections. The product case of the already proved smooth-ambient base-change theorem is
\(r^!f_*K\simeq F_*t^!K\). Since \(g=ri\), inverse composition gives
\[
g^!f_*K\simeq i^!F_*t^!K.
\tag{5.14l}
\]
Let \(j:A\setminus i(Y')\hookrightarrow A\). The restriction of \(F\) to \(T\setminus W\) maps into this complement, so composition identifies the last term of \(F_*\)(5.14k) with \(j_*H\) for a bounded holonomic complex \(H\) on that smooth open. The supported smooth graph calculation gives \(i^!j_*H=0\). Thus the map
\(i^!F_*Q\to i^!F_*t^!K\) is an isomorphism.

The output \(F_*Q\) has cohomology supported on \(i(Y')\): its restriction to the complement is the direct image of the zero restriction of \(Q\). This open-restriction calculation follows either from transfer tensor and finite Čech sheaf pushforward, or from the open case of composition. Kashiwara's smooth supported equivalence therefore gives
\[
F_*Q\simeq i_*i^!F_*Q.
\tag{5.14m}
\]
Apply \(\pi_*\). Since \(\pi i=\mathrm{id}_{Y'}\), 5.14.2 identifies
\(\pi_*F_*Q\simeq s_*Q\) and
\(\pi_*i_*i^!F_*Q\simeq i^!F_*Q\). Combining this with (5.14l) and (5.14k) proves (5.14j). The maps are localization augmentations, the product comparison and the intrinsic supported counit. They are natural, not chosen identifications. Holonomicity and boundedness of the outputs use only the earlier arbitrary holonomic direct/inverse preservation and 5.14.2. \(\square\)

If \(W\) happens to be smooth, the earlier supported formula identifies \(R\Gamma_W t^!K\) with its closed direct image of the appropriate intrinsic inverse image, recovering the usual smooth-square formula with its shifts. For singular \(W\), (5.14j) is the usable replacement, with the operator directions retained.

#### Curve testing commutes with closed support

**Lemma 5.14.3.** For any algebraic map \(a:V\to T\) of smooth varieties and closed subset \(W\subset T\),
\[
a^!R\Gamma_W K\simeq R\Gamma_{a^{-1}W}(a^!K).
\tag{5.14n}
\]
Consequently, if every smooth-curve inverse image of a bounded holonomic \(K\) is regular, then the same holds for \(R\Gamma_W K\).

**Proof.** On an affine chart choose finitely many equations \(u_1,\ldots,u_l\) for \(W\). Derived pullback commutes with each localization:
\[
\begin{gathered}
La_{\mathcal O}^*(K[1/u])\\
\simeq(La_{\mathcal O}^*K)[1/a^*u].
\end{gathered}
\tag{5.14o}
\]
Take a flat resolution of \(K\); termwise this is the universal tensor/localization identity, and flatness of localization justifies the derived comparison. The chain-rule action preserves the localization identity. The finite augmented Čech complex and the fixed dimension shift then prove (5.14n); refinements give the same augmentations, so it glues.

For a connected smooth curve \(C\), its inverse image of \(W\) is empty, the whole curve, or a finite closed subset. In the first case local cohomology is zero; in the second it is the original inverse complex. In the finite case every cohomology module is holonomic and supported on finitely many points, hence regular by Kashiwara and the point presentation. This proves the final assertion for every curve map, without asserting Serre closure of the curve-testing property on the original ambient variety. \(\square\)

#### A converse criterion for nonsingular cohomology on a smooth support

**Lemma 5.14.4.** Suppose \(i:Z\hookrightarrow X\) is an arbitrary smooth closed embedding, and \(L\in D_h^b(\mathcal D_Z)\) has finite-rank nonsingular connections as all its cohomology modules. Then \(i_*L\) is regular holonomic if and only if every smooth locally closed curve in \(X\) has regular extraordinary restriction of \(i_*L\). Equivalently, one may test every map from a smooth algebraic curve to \(X\).

**Proof of the connection test.** For a nonsingular connection \(A\) on a smooth variety, testing its pullback to smooth locally closed embedded curves suffices for the all-map definition. Given a nonconstant curve map \(k:C\to Z\), choose a dense smooth locally closed open \(B\) of its irreducible curve image. Its embedding in \(Z\) is one of the tests. On the dense open \(k^{-1}B\), the pullback is the pullback of the tested regular connection on \(B\). The map between function fields extends to a finite nonconstant map of smooth complete curve models. Pulling a stable logarithmic lattice through a ramification \(t=u^e v\), \(v\) a unit, gives a stable logarithmic lattice: its coefficient is multiplied by \(e\) and has the additional holomorphic term \(dv/v\), exactly as in (5.14d). This supplies regularity at every completion point. At a point of \(C\) omitted from \(k^{-1}B\), the full pullback connection on \(C\) is nonsingular and supplies its own lattice. Constant maps give a trivial finite-rank connection. This proves the all-map connection test, including infinity and arbitrary ramification.

On a bounded complex with flat cohomology, the finite hyper-Tor spectral sequence for structure-sheaf inverse image has only its zero Tor row. Thus, for \(k:C\to Z\),
\[
\begin{gathered}
H^q(k^!L)\\
=k^*H^{q+1-\dim Z}(L).
\end{gathered}
\tag{5.14p}
\]
The same formula follows from finitely many standard truncation triangles and flatness of their cohomology pieces; no abstract derived-category-of-the-heart identification is needed. Regularity of the curve complex therefore tests exactly those connection modules. The connection/regular-holonomic identification of Theorem 5.2 and the preceding embedded-curve argument prove the criterion for \(L\).

If \(i_*L\) passes the embedded-curve tests on \(X\), every smooth locally closed curve \(C\hookrightarrow Z\) also embeds locally closed in \(X\). Inverse composition and \(i^!i_*L=L\) identify its test with that of \(L\), so the proved connection criterion makes \(L\) regular. 5.13.1's exact supported equivalence then makes \(i_*L\) regular.

Conversely let \(L\) be regular and let \(h:C\to X\) be arbitrary. If \(h(C)\subset Z\), reducedness of the smooth curve makes the map factor through the reduced smooth closed subvariety \(Z\). Composition and supported Kashiwara identify \(h^!i_*L\) with the intrinsic curve inverse image of \(L\), regular by (5.14p). Otherwise \(h^{-1}Z\) is finite, and the earlier arbitrary holonomic inverse theorem makes \(h^!i_*L\) a bounded holonomic complex supported on those finitely many points. It is regular. This includes constant maps and nontransverse intersections. \(\square\)

This is a genuine converse on every smooth closed support with nonsingular cohomology; it does not assert a converse for an arbitrary singular extension or a complex with singular cohomology on that support.

#### The support-dimension reduction and low images

**Lemma 5.14.5.** For an SNC meromorphic module, image dimension at most one gives regular direct image. For higher image dimension, lower-support curve and direct-image theorems give its curve tests by the supported graph construction.

Let \(P\) be integral of dimension \(n\), let \(M=\bar E(*D)\) as in 5.14.1, and let \(f:P\to Y\) be arbitrary. For a nonconstant \(g:C\to Y\), 5.14 realizes the curve inverse image of \(f_*M\) as the direct image of
\[
\begin{gathered}
Q=R\Gamma_W(t^!M),\\
W\subset C\times P.
\end{gathered}
\tag{5.14q}
\]
The product complex \(t^!M\) is curve-tested by 5.14.1 and inverse composition. It is also regular: it is the open restriction of the SNC meromorphic bundle \(\mathcal O_{\bar C}\boxtimes\bar E\) on the proper pair \((\bar C\times P,(\bar C\setminus C)\times P+\bar C\times D)\), to which 5.13 applies. 5.14.3 makes \(Q\) curve-tested independently of its simple-factor regularity.

If \(\dim f(P)\geq2\), the locus \(f^{-1}(\overline{g(C)})\) is a proper closed subset of \(P\), of dimension at most \(n-1\). The projection \(W\to P\) is quasi-finite, since every fiber of a nonconstant algebraic curve map is finite. Hence
\[
\begin{gathered}
\dim\operatorname{Supp}Q\leq\dim W\\
\leq n-1.
\end{gathered}
\tag{5.14r}
\]
This is a dimension calculation on the actual possibly singular fiber product, not an assertion that it is a smooth variety. If the full converse curve criterion and regular direct-image preservation have already been proved for source supports of dimension less than \(n\), they apply in that order to \(Q\), and (5.14j) proves that \(g^!f_*M\) is regular. For a constant curve map, factor it through a point: holonomic inverse image at the point is a bounded finite-dimensional complex, and its pullback to the curve is a complex of trivial connections with the correct shift. Thus constant maps cause no induction increase.

If \(\dim f(P)\leq1\), \(f_*M\) is already regular without that conditional induction. Its support has dimension at most one. In dimension zero it has only point factors. In dimension one, choose a dense smooth curve open in its image. Write \(f_*M=(f|_U)_*E\) by 5.14.2 and affine localization. After shrinking the image curve, generic smoothness in characteristic zero makes \(f|_U\) smooth there; the elementary derivation/Jacobian proof used in Theorem 5.6 gives this shrink by removing the finitely many nondominating critical images. The full generic Gauss–Manin Theorem 5.6 then makes every generic cohomology connection regular, with its actual transfer action and shift. Every curve-support simple factor inherits a subquotient of one of these regular generic connections; every remaining simple factor has point support. Choose a further affine dense curve open for its presenting immersion and apply minimal-extension uniqueness. All these simple factors are regular under the definition, so every cohomology module is regular.

For image dimension at least two, this is the lower-support step used in the ordered induction below. Its same-dimensional converse is proved by Theorem 5.16.1 after the forward assertion at that dimension; curve testing is not assumed closed under arbitrary heart subquotients.

### Dual SNC Spencer and every curve test

Let \((P,D)\) be any smooth proper SNC pair over \(\mathbb C\), with finitely many globally smooth components \(D_i\). Fix their order for exterior signs. Let \(a:U=P\setminus D\hookrightarrow P\), let \(E\) be any curve-regular finite-rank connection, and let \(\bar E\) be its normalized algebraic logarithmic extension, with residue real parts in \([0,1)\). Put \(M=a_*E=\bar E(*D)\). The dual connection has local normal residues \(-A_i^t\).

The exact earlier proofs are PBW and holonomic duality; the normalized logarithmic and constant-residue calculations in Theorems 5.0 and 5.3; the faithful-flat analytic local comparison linked there; Theorem 5.13; and the inverse-transfer convention of Proposition 1.1 in the inverse-images lesson. Analytic normalization verifies an algebraic presentation by faithful flatness. The proper-curve lattice argument below proves regularity at infinity.

#### Logarithmic induction and its dual

Write \(T_P(-\log D)\) for the locally free logarithmic tangent bundle and \(\mathscr L_P\) for its operator algebra. If \(L\) is a locally free logarithmic integrable connection, its induced module has the augmented logarithmic Spencer complex
\[
\begin{gathered}
\operatorname{Sp}_{\log}^{-r}(L)\\
=\mathcal D_P\otimes_{\mathcal O_P}
\bigwedge^rT_P(-\log D)\\
\otimes_{\mathcal O_P}L,\qquad0\leq r\leq n,\\
n=\dim P.
\end{gathered}
\tag{5.15a}
\]
with multiplication-minus-connection and bracket differential. Its augmentation is to
\(\mathcal D_P\otimes_{\mathscr L_P}L\).

**Lemma 5.15.1.** The complex (5.15a) resolves its augmented induced module in degree zero. In particular
\[
M\simeq\mathcal D_P\otimes_{\mathscr L_P}\bar E(D).
\tag{5.15b}
\]
Its dual is
\[
\mathbb D_PM\simeq
\mathcal D_P\otimes_{\mathscr L_P}\bar E^{\vee}.
\tag{5.15c}
\]
Every statement retains the full connection matrices and their nilpotent parts.

**Proof of exactness.** On adapted étale coordinates \(z_1,\ldots,z_k,w_1,\ldots,w_{n-k}\), the logarithmic generators are \(\theta_i=z_i\partial_{z_i}\) and \(\partial_{w_j}\). Give the term in degree \(-r\) the order filtration shifted by \(r\). Its leading differential is the Koszul differential on
\[
\begin{gathered}
z_1\xi_1,\ldots,z_k\xi_k,\\
\eta_1,\ldots,\eta_{n-k}
\quad\text{in }\mathcal O_P[\xi,\eta].
\end{gathered}
\tag{5.15d}
\]
This is a regular sequence. Over the polynomial coordinate ring, after quotienting the preceding products, multiplication by the next independent coordinate \(z_i\), the new variable \(\xi_i\), or a new \(\eta_j\) is injective: the quotient has a monomial decomposition involving only the preceding coordinate pairs. Thus each new product is a nonzerodivisor. Tensoring its polynomial Koszul resolution through the flat étale coordinate algebra preserves exactness. Tensoring with the free coefficient bundle \(L\) does so as well. Subtract a lift of a leading graded boundary from a cycle and descend its finite order; the lower bound terminates the process. This proves exactness in negative degrees. The degree-zero relations are exactly the logarithmic tensor relations, because functions and the displayed logarithmic vector fields generate \(\mathscr L_P\). Hence the augmented module is as asserted. The intrinsic bracket differential makes the resolutions glue.

For \(L=\bar E(D)\), its local normal matrices are \(A_i-I\), and the generation proof 5.13.3 gives the natural surjection from this induced module to \(M\). It is injective as well. After analytic stalk extension, the proved normalized logarithmic local model changes the lattice by a holomorphic invertible matrix to the commuting constant-residue model. Its exact presentation is Theorem 5.3, equation (5.3j), whose relations are precisely \(\theta_i-(A_i-I)\) and the tangential connection relations. Thus the analytic extension of the surjection is an isomorphism. The analytic stalk ring is faithfully flat over the algebraic local ring; the underlying operator extension is its tensor extension by PBW. Faithfulness therefore kills the algebraic kernel. No analytic regularity theorem is being used to infer algebraic regularity.

Dualize (5.15a). Its terms are finite locally free over \(\mathcal D_P\), so it computes operator duality. Wedge pairing identifies the reversed exterior complex with the logarithmic Spencer complex for
\[
\begin{gathered}
L^\vee\otimes\bigl(\bigwedge^n\Omega_P^1(\log D)\bigr)\\
\otimes\omega_P^{-1}\\
=L^\vee\otimes\mathcal O_P(D).
\end{gathered}
\tag{5.15e}
\]
The degree reversal is canceled by the duality shift \([n]\). The density action in (5.15e) has normal logarithmic coefficient \(-1\) on the local frame \((z_1\cdots z_k)^{-1}\). Explicitly \(\theta_i^t=-\theta_i-1\), whereas \(\partial_{w_j}^t=-\partial_{w_j}\) in the coordinate volume. Thus a normal matrix \(R_i\) becomes \(-R_i^t-I\), and a tangential matrix becomes its negative transpose. These are exactly the connection and bracket terms in the paired Spencer differential. This coordinate verification on the generating vector fields, with Leibniz, verifies the full differential; the wedge and density pairing is intrinsic and hence glues.

In particular, the local source presentation has generators related by \(\theta_i-(A_i-I)\) and \(\partial_{w_j}-C_j\); its density-transposed presentation has the relations \(\theta_i+A_i^t\) and \(\partial_{w_j}+C_j^t\). The reversed exterior bases have the usual Koszul complement signs; wedge pairing fixes them, and multiplying a whole generator relation by \(-1\) does not change its quotient. Thus the sign calculation includes the scalar divergence term as well as the matrix transpose. The variable coefficient derivatives in higher wedges obey the same Leibniz rule; the curvature commutators vanish by integrability.

For \(L=\bar E(D)\), (5.15e) is \(\bar E^{\vee}\), and its normal matrices are \(-A_i^t\), since \(- (A_i-I)^t-I=-A_i^t\). Exactness of the new logarithmic Spencer complex proves (5.15c), in degree zero. For the curve argument below it is (5.15c) that is used; relabeling \(E^{\vee}\) gives all !-extensions. \(\square\)

A logarithmic lattice with residues in \([0,1)\) is generally not the lattice dual to the normalized lattice of \(E^{\vee}\). For every \(E\), write \(\bar F\) for the normalized lattice of \(F=E^{\vee}\), and use the exact formula
\[
\begin{gathered}
a_!E=\mathbb D_P(a_*F)\\
\simeq\mathcal D_P\otimes_{\mathscr L_P}\bar F^\vee.
\end{gathered}
\tag{5.15f}
\]
This keeps all exponents and avoids an invalid normalization claim. The proof and theorems below use (5.15c)/(5.15f) only.

#### Normal inverse image and the finite residue complex

Let \(H=\mathbb D_PM\), presented by (5.15c). Its lattice \(L=\bar E^{\vee}\) has normal matrices \(B_i=-A_i^t\), with eigenvalue real parts in \((-1,0]\). For an intersection component \(Z\) of \(D_I\), remove the other boundary components and call the resulting smooth stratum \(S\). Write \(c=|I|\), and write \(i:S\hookrightarrow P\setminus\bigcup_{j\notin I}D_j\) for its closed immersion.

**Lemma 5.15.2.** On \(S\), the cohomology connections of \(i^!H\) are the cohomology of the normal chain Koszul complex
\[
\begin{gathered}
K_I^{-r}=L|_S\otimes\bigwedge^r\mathbb C^I,\\
d=\operatorname{Koszul}((-B_i)_{i\in I}),\\
-B_i=A_i^t,\qquad0\leq r\leq c.
\end{gathered}
\tag{5.15g}
\]
shifted by \([-c]\). Thus the degrees are \(0,\ldots,c\). The full commuting matrices \(A_i^t\), including zero eigenvalues and Jordan blocks, are retained. On the proper \(Z\), the same residue complex on \(L|_Z\) has coherent cohomology with a logarithmic connection along the remaining boundary, and restricts to these intrinsic cohomology connections on \(S\).

**Proof of the normal calculation.** Resolve \(H\) by (5.15a) for \(L\), and tensor its terms with \(\mathcal O_S\). They are \(\mathcal O_P\)-flat because they are finite free operator terms with locally free coefficients. This computes \(Li_{\mathcal O}^*H\). Use PBW to put every normal derivative to the left of the tangential derivatives, and filter this tensor complex by its total normal derivative degree.

The differential preserves that filtration. Moving a coefficient matrix through a normal derivative only introduces terms of smaller normal degree. On the graded piece labeled by a normal multi-index \(b\in\mathbb N^I\), right multiplication by \(\theta_i\) gives
\[
\partial_z^b\theta_i
=\theta_i\partial_z^b+b_i\partial_z^b.
\tag{5.15h}
\]
The first term vanishes after structure-sheaf tensor with \(\mathcal O_S\), since its left coefficient is \(z_i\). Consequently its normal differential is the Koszul operator
\[
b_iI-B_i|_S=b_iI+A_i^t|_S.
\tag{5.15i}
\]
The tangential differential is the tangential Spencer differential for \(L|_S\). Integrability makes the restricted residues commute and horizontal for that tangential connection, so these differentials commute with the required total-complex signs.

If \(b\ne0\), choose \(i\) with \(b_i\geq1\). The matrix in (5.15i) is invertible at every stalk, because every eigenvalue of \(A_i\) has real part in \([0,1)\). Its inverse is horizontal as well. Exterior multiplication by the corresponding basis vector, composed with this inverse and its Koszul sign, contracts that normal Koszul factor and commutes with the tangential differential. Hence the entire positive-normal-degree graded piece is acyclic. Every section has finite normal degree. Descending its leading degree proves that the quotient by the degree-zero subcomplex is acyclic.

That degree-zero subcomplex has exactly the normal Koszul differential \(-B_i\) and the tangential Spencer resolution. Resolving its tangential terms leaves (5.15g). Finally the programme closed inverse image appends \([-c]\), placing its cohomology in degrees \(0,\ldots,c\). This is an algebraic filtration calculation; no constant-residue gauge is used for this calculation.

**The connection and its boundary lattice.** The normal residue endomorphisms are canonical endomorphisms of \(L|_Z\), and commute by integrability, so (5.15g) defines a finite complex of algebraic vector bundles on all of \(Z\). A logarithmic tangent field on \(Z\) lifts locally to a logarithmic field on \(P\) preserving the components \(D_i\), \(i\in I\). Acting on \(L|_Z\) with this lift commutes with the residue differential. Two lifts differ, after restriction, by \(\sum_i u_i\theta_i\); their coefficient actions differ by \(\sum_i u_iB_i\). Multiplication by \(B_i\) is null-homotopic on the residue Koszul complex, using the corresponding exterior multiplication or contraction with its sign. Thus the induced action on its cohomology is independent of the lift. The Leibniz rule is inherited; the bracket/curvature error is likewise a normal residue action and hence null-homotopic. Its cohomology consequently has an intrinsic integrable logarithmic connection.

On \(S\) this is an ordinary coherent connection, hence a finite-rank bundle by the earlier coherent-connection theorem. The normal-degree-zero comparison above intertwines its tangential action with the actual inverse-transfer action: tangential left operators pass through the normal PBW terms, and the tangential Spencer augmentation leaves precisely the coefficient connection. The lift ambiguity is the just-computed residue homotopy. This identifies the connections, not only their vector spaces.

At the remaining boundary on \(Z\), the cohomology of the finite residue-bundle complex remains coherent and logarithmically stable. Torsion may occur there; no assertion of local freeness at that boundary is needed. These coherent cohomology modules are the boundary lattices used in 5.15. \(\square\)

#### Every curve inverse image of the dual SNC module is regular

**Theorem 5.15.** For every algebraic map \(k:C\to P\) from a smooth algebraic curve, \(k^!\mathbb D_PM\) is a regular holonomic bounded complex. The same holds on every open restriction of the proper pair and for every SNC !-extension (5.15f). The curve can lie entirely in any intersection component, be constant, or have arbitrary tangencies.

**Proof.** Work on a connected curve. Let \(I\) consist of those global boundary components containing its entire image. If \(I\) is empty, on a dense open of \(C\) the inverse image is the pullback of the regular connection \(E^{\vee}\), with the fixed dimension shift; its generic cohomology is regular at every compactification point by the all-curve definition. The remaining finite-point factors are regular, so the entire bounded curve complex is regular. This argument also covers curves meeting the boundary only at finitely many points, without asserting that their full inverse image is a flat localization.

If \(I\ne\varnothing\), properness of \(P\) extends the map to \(\bar k:\bar C\to P\). Its image still lies in one connected component \(Z\) of \(D_I\), since the selected boundary equations vanish on the dense original curve. It is not contained in any remaining boundary component. Away from the finitely many inverse images of the remaining boundary, the map factors through the smooth stratum \(S\) in 5.15.2. Composition and its normal calculation give its generic inverse cohomology by pullback of the connections (5.15g)[\(-c\)], with the additional shift \([1-\dim S]\). These are the actual inverse-image degrees and connections.

Pull the finite vector-bundle residue complex (5.15g) on \(Z\) to all of \(\bar C\). Its cohomology agrees on that dense open with pullback of the cohomology on \(S\): its terms and cohomology are flat there, so the bounded flat-complex base-change argument proved in Gauss–Manin Theorem 5.6 applies to any such curve map. At a completion point choose a parameter \(t\). Each remaining boundary equation has the form \(t^m u\) with \(u\) a unit. The logarithmic tangent vector \(t\partial_t\) therefore maps to an \(\mathcal O_{\bar C}\)-linear combination of the logarithmic tangent generators on \(Z\). The same local lift construction in 5.15.2 defines a logarithmic connection on the pulled residue complex and on its coherent cohomology. Normal lift ambiguities remain null-homotopic residue actions.

For each such coherent cohomology module, divide out its torsion. The torsion is logarithmically stable: if \(t^r v=0\), the Leibniz rule gives \(t^r(\Theta v+r v)=0\), so \(\Theta v\) is torsion too. The resulting finite torsion-free module at a curve DVR is a free lattice. It spans precisely the generic inverse cohomology connection, and \(t\nabla_{\partial_t}\) preserves it. The one-variable criterion proves regularity at every point of \(\bar C\), including infinity and points where additional boundary components meet the curve. Thus every generic connection of the actual inverse complex is regular. All its other cohomological simple factors on \(C\) have point support and are regular. Holonomic inverse preservation supplies boundedness and holonomicity, so the full complex is regular.

Constant maps are included: the residue complexes give finite-dimensional connections pulled back from a point; alternatively factor through that point and use holonomic inverse image there, followed by the trivial curve pullback. Open restrictions are tested by composing the curve map into \(P\), so the same proof applies. Finally apply the result to the normalized lattice of \(E^{\vee}\) in (5.15f), obtaining the theorem for \(a_!E\). \(\square\)

The theorem establishes the forward curve test of !-extensions, independently of the desired singular criterion. The finite residue complex rather than a diagonalized residue eigenspace is essential: eigenvalue zero and all nilpotent Jordan maps determine its kernels and cokernels.


### Lowest-cohomology detection and bounded socle peeling

Call a complex **curve-tested** if every extraordinary inverse image by a map from a smooth algebraic curve is regular holonomic. Call it **embedded-curve-tested** if the same holds for every smooth locally closed curve immersion. Either class is triangulated, because inverse image is triangulated and the proved composition-factor regular category is a Serre category. Neither class is assumed to be closed under heart subquotients or standard truncations.

The exact earlier proofs are closed-embedding Koszul inverse image and Kashiwara, holonomic inverse preservation, generic connection structure and finite length, Theorem 5.2 and Proposition 5.2.2. Lemma 5.14.4 below the supported graph formula proves the nonsingular embedded-curve criterion. No singular curve criterion or regularity stability is an input.

#### Detecting every simple subobject in the lowest cohomology

**Theorem 5.16.** Let \(K\) be a bounded holonomic complex, with \(H^qK=0\) for \(q<a\). If \(K\) is embedded-curve-tested, every simple subobject of \(H^aK\) is regular holonomic. This also applies when \(H^aK=0\), and requires no smoothness of a simple subobject's closed support.

**Proof.** Let \(S\subset H^aK\) be a nonzero simple subobject, with irreducible reduced support \(W\). Choose an ambient open \(V\) where a dense open \(W_0\) of \(W\) is smooth and closed in \(V\), and where the supported inverse of \(S|_V\) is a nonsingular irreducible connection \(E\) on \(W_0\). These choices are the earlier simple-factor classification and generic-connection theorem; no regularity is asserted by them. Write \(i:W_0\hookrightarrow V\).

For a module in degree zero, the normal Koszul formula for a closed embedding of codimension \(c\) places \(i^!\) in degrees \(0,\ldots,c\). Indeed \(Li_{\mathcal O}^*\) lies in \([-c,0]\), and the shift \([-c]\) moves those terms to \([0,c]\). Consequently \(i^!\) preserves the lower standard cohomological bound of a bounded complex, and \(H^0i^!\) is left exact on modules. Apply it to the lowest truncation triangle to obtain
\[
H^a(i^!K|_V)\simeq H^0i^!(H^aK|_V).
\tag{5.16a}
\]
The subobject \(S\) therefore gives an injection
\[
\begin{gathered}
E=H^0i^!(S|_V)\\
\hookrightarrow H^a(i^!K|_V).
\end{gathered}
\tag{5.16b}
\]

Shrink \(W_0\) further so that all the finitely many cohomology modules of \(i^!K|_V\) are nonsingular finite-rank connections. Holonomic inverse preservation makes these modules holonomic, so the generic-connection theorem permits this common shrink. It remains a smooth locally closed subset of \(X\), and its immersion into a suitable smaller ambient open is closed.

For every smooth locally closed curve \(C\hookrightarrow W_0\), composition identifies its extraordinary restriction of \(i^!K|_V\) with the given restriction of \(K\) to that same locally closed curve in \(X\). It is regular by hypothesis. All cohomology pieces on \(W_0\) are flat, so the bounded hyper-Tor calculation in 5.14.4 gives, with \(w=\dim W_0\),
\[
\begin{gathered}
H^r(k^!i^!K|_V)\\
\simeq k^*H^{r+1-w}(i^!K|_V).
\end{gathered}
\tag{5.16c}
\]
Thus the embedded-curve test proves that each nonsingular cohomology connection on \(W_0\), and in particular that on the right of (5.16b), is curve regular. The exact connection test in 5.14.4 extends this statement from embedded curves to every curve map and every compactification point. The connection Serre property now makes its connection subobject \(E\) curve regular.

Choose a nonempty affine open \(U\subset W_0\). Its immersion into \(X\) is affine: for an affine open \(A\subset X\), its inverse image is the graph closed in the affine product \(U\times A\), because \(X\) is separated. The simple-factor classification and no-boundary uniqueness recover
\[
S\simeq (U\hookrightarrow X)_{!*}(E|_U).
\tag{5.16d}
\]
Its irreducible connection is curve regular, and its presenting immersion is affine. This is precisely a regular presentation under the composition-factor definition. Singular points of \(W\) were removed only to obtain the presenting connection; no regularity criterion on the singular closure was assumed. \(\square\)

#### Peeling replaces the same-dimensional converse

Fix a nonnegative integer \(n\). Let \(R_n\) be the assertion that every regular simple module with support dimension at most \(n\), on every smooth ambient variety, is curve-tested for **all** curve maps. This is a forward assertion, and is not assumed true without proof in the later induction.

**Theorem 5.16.1.** If \(R_n\) has been proved, a bounded holonomic complex whose cohomological supports all have dimension at most \(n\) is regular if and only if it is embedded-curve-tested; equivalently if and only if it is curve-tested for all curve maps. In particular the bounded-complex converse needs no further same-dimensional reconstruction theorem once \(R_n\) is available.

**Proof of the forward implication under the stated input.** Every regular module has a finite composition series of regular simples. Its successive short exact sequences are distinguished triangles. \(R_n\) and triangulated closure of the curve-tested class therefore make the whole module curve-tested. The finitely many standard truncation triangles of a bounded regular complex give the same conclusion for that complex. This proves testing under every curve map and hence under embedded curves.

**Proof of the converse.** For a nonzero bounded embedded-curve-tested \(K\), choose its lowest nonzero degree \(a\), and a simple subobject \(S\subset H^aK\). Finite length supplies this subobject, and Theorem 5.16 proves that it is regular. Its support dimension is at most \(n\); \(R_n\) therefore makes \(S[-a]\) curve-tested, and hence embedded-curve-tested. Compose its inclusion with the canonical lowest-cohomology map
\[
S[-a]\longrightarrow H^aK[-a]\longrightarrow K,
\]
and let \(K'\) be its cone. The cohomology sequence gives
\[
\begin{gathered}
H^aK'=H^aK/S,\\
H^qK'=H^qK\quad(q\ne a).
\end{gathered}
\tag{5.16e}
\]
with all lower cohomology zero. The cone is embedded-curve-tested by triangulated closure. The integer
\[
\ell(K)=\sum_q\operatorname{length}H^qK
\tag{5.16f}
\]
is finite and has decreased by one. Induction on \(\ell(K)\), with the zero complex as base, proves that \(K'\) is regular. Then (5.16e), the regular simple \(S\), and the proved Serre property make \(K\) regular. Every step retains the original support-dimension bound. The proof assumes neither standard-truncation closure of the testing class nor subquotient closure of that class. \(\square\)

This theorem isolates the remaining forward input exactly. It applies to all singular closed supports, all bounded holonomic complexes, all cohomological degrees and every smooth ambient dimension. The later induction must prove \(R_n\); it cannot silently replace that input by the full regularity-stability theorem.


### Four-map regularity on every singular support

The original definitions are retained. A regular module is one whose simple factors have affine minimal-extension presentations by irreducible all-curve regular connections; a regular bounded complex has such modules as every cohomology module. The Serre and triangulated properties follow from that definition, as proved in Lemma 5.13.1. “Curve-tested” and “embedded-curve-tested” are temporary testing properties and acquire no assumed Serre or truncation closure.

#### The three induction assertions

For a nonnegative integer \(n\), use the following assertions on **all** smooth ambient varieties, with no restriction on their dimension:

- \(R_n\): every regular simple module with support dimension at most \(n\) is curve-tested for every smooth algebraic curve map.
- \(C_n\): every bounded holonomic complex with cohomological supports of dimension at most \(n\) is regular if and only if it is embedded-curve-tested, equivalently curve-tested for all smooth curve maps.
- \(P_n\): every algebraic direct image preserves regular bounded complexes whose source cohomological supports have dimension at most \(n\).

The order at dimension \(n\) is
\[
\begin{gathered}
(R_{n-1},C_{n-1},P_{n-1})\\
\Longrightarrow R_n\\
\Longrightarrow C_n\\
\Longrightarrow P_n.
\end{gathered}
\tag{5.17a}
\]
Thus a same-dimensional converse is never used to prove its own forward assertion. All lower-support complexes occurring below may live on smooth ambient varieties of larger dimension.

#### A curve test for proper images of the two SNC standard modules

Assume \(C_{n-1}\) and \(P_{n-1}\), with \(n\geq2\). Let \(P\) be a smooth proper integral variety of dimension \(n\), let \(D\) be SNC, let \(Q^+=\bar E(*D)\), and let \(Q^-=a_!E\) be the corresponding !-extension. Theorem 5.13 and regular duality prove that both are regular modules. Proposition 5.14.1 and Theorem 5.15 prove, independently, that both are curve-tested on \(P\) and on every open restriction.

Let \(F:P\to\bar Y\) be proper, where \(\bar Y\) is smooth, and let \(Y\subset\bar Y\) be open. Write \(P_Y=F^{-1}Y\) and \(F_Y:P_Y\to Y\). Suppose \(\dim F(P)\geq2\). We claim
\[
\begin{gathered}
F_{Y,*}(Q^\pm|_{P_Y})\\
\text{is curve-tested.}
\end{gathered}
\tag{5.17b}
\]

For a nonconstant curve map \(g:C\to Y\), the supported graph formula 5.14 is
\[
\begin{gathered}
g^!F_{Y,*}(Q^\pm|_{P_Y})\\
\simeq s_*R\Gamma_Wt^!(Q^\pm|_{P_Y}),\\
W=P_Y\times_YC\subset C\times P_Y.
\end{gathered}
\tag{5.17c}
\]
Inverse composition makes \(t^!Q^\pm\) curve-tested, and 5.14.3 makes its supported local cohomology curve-tested. The projection \(W\to P_Y\) is quasi-finite: its fibers are fibers of the nonconstant curve map, hence finite. Its image lies in
\(F_Y^{-1}(\overline{g(C)})\), a proper closed subset of the integral \(n\)-dimensional \(P_Y\). Therefore
\[
\dim W\leq n-1.
\tag{5.17d}
\]
The scheme \(W\) need not be smooth or reduced; it is used only as support inside \(C\times P_Y\). \(C_{n-1}\) makes the supported complex in (5.17c) regular, and \(P_{n-1}\) makes its direct image under \(s\) regular. This proves the test. If \(P_Y\) is empty there is no output; otherwise it is a dense integral open, so the dimension assertion applies. A constant curve map factors through a point. Arbitrary holonomic inverse image at that point is a bounded finite-dimensional complex, and its inverse image on the curve is a shifted complex of trivial connections. It is regular. Thus (5.17b) includes every curve map.

For the standard immersions used in the forward simple-module argument, \(\dim F(P)=n\), so (5.17b) applies directly. For a general map whose image has dimension zero or one, only \(Q^+\) is needed in the direct-image induction. 5.14.5 proves its output regular by full generic Gauss–Manin and the curve-support simple-factor argument, independently of the same-dimensional criterion.

#### Base dimensions zero and one

Every simple point-supported module is a delta module for a one-dimensional vector space. A curve inverse image has finite support unless the curve maps constantly to the point; in that case factor through the point and obtain a shifted trivial connection. Thus \(R_0\) holds. 5.16.1 gives \(C_0\). Every point-supported holonomic complex is recovered by exact supported Kashiwara on a finite disjoint union of points. Under any map its image is again a bounded complex of point modules, by direct composition, so \(P_0\) holds.

For \(R_1\), let \(S\) be regular simple with curve support \(Z\), possibly singular. Its regular presentation is a connection \(E\) on a dense smooth curve open \(U\subset Z\). A constant curve map is handled as above. A nonconstant curve map either has image outside \(Z\) generically, in which case its inverse image is supported on finitely many points; or has image in \(Z\), in which case on a dense source open it factors through \(U\). Inverse composition and \(j^!S=E\) identify its generic inverse connection with the pullback of \(E\). That connection is regular at every completion point by the all-curve property of \(E\). Every other simple factor of the inverse complex has point support. Hence the whole inverse complex is regular. This includes curve maps meeting the singular points of \(Z\) nontransversely. Thus \(R_1\) holds, and 5.16.1 gives \(C_1\).

The proof of \(P_n\) in the direct-image induction, used at \(n=1\), invokes \(P_0,C_0,C_1\) and the low-image Gauss–Manin clause of 5.14.5; there is no image of dimension at least two. It therefore establishes \(P_1\) without a missing induction input. These are the full bases of (5.17a).

#### Forward testing of every regular simple at dimension n

Let \(n\geq2\), and assume the three assertions at dimensions less than \(n\). Take a regular simple \(S\) of support dimension \(n\), on any smooth separated \(X\). Choose its defining affine immersion
\[
j:U\hookrightarrow X,\qquad S=j_{!*}E,
\tag{5.17e}
\]
where \(E\) is irreducible and curve regular on the smooth integral \(n\)-dimensional \(U\). The two standard extensions are modules in degree zero. Any immersion factors as closed followed by open, which gives \(j_*E\) degrees \(\geq0\). Because \(j\) is affine, on every affine target open its source is affine; the bounded flat transfer resolution has nonpositive degrees and quasi-coherent terms, whose affine sheaf pushforward is acyclic. Thus \(j_*E\) also has degrees \(\leq0\), and is a module. This is the earlier affine-direct-image bound, with 5.14.2's transfer/Čech construction. It does not require the chosen open factor itself to be affine. The !-extension is the holonomic dual of the corresponding *-extension of \(E^{\vee}\), and ambient holonomic duality is exact on modules.

We prove that both \(j_*E\) and \(j_!E\) are curve-tested without claiming their simple factors regular first. Take a smooth proper compactification \(\bar X\) of \(X\). Independently compactify \(U\), close the graph of its map into \(\bar X\), resolve that graph while preserving the smooth open \(U\), and principalize its boundary outside \(U\). The exact compactification and resolution proofs linked in Theorem 5.1 give a smooth proper SNC pair \((P,D)\), with \(P\setminus D=U\), and a proper map
\[
F:P\to\bar X
\tag{5.17f}
\]
extending \(j\). This preserves the smooth connection open only; it makes no claim of identity on a pre-existing SNC crossing. Its image is the closure of \(U\), of dimension \(n\).

Over \(X\), put \(P_X=F^{-1}X\), and \(a:U\hookrightarrow P_X\). By 5.14.2,
\[
j_*E=F_{X,*}a_*E.
\tag{5.17g}
\]
The source is the restriction of the proper SNC module \(Q^+\). For the !-extension, the exact proper duality proper operator-duality, Theorem 4.7 and its definition give
\[
\begin{gathered}
j_!E=\mathbb D_Xj_*E^\vee\\
\simeq F_{X,*}\mathbb D_{P_X}(a_*E^\vee)\\
=F_{X,*}a_!E.
\end{gathered}
\tag{5.17h}
\]
The source is a restriction of the dual SNC module (5.15f). Equation (5.17b), using only \(C_{n-1},P_{n-1}\), proves that both standard modules in (5.17g)–(5.17h) are curve-tested.

Let \(\phi:j_!E\to j_*E\) be the actual canonical boundary map. Its image is \(S\). Both modules restrict to \(E\), and \(\phi\) restricts to its identity; hence its kernel and cokernel are supported on \(\overline U\setminus U\), of dimension at most \(n-1\). Its cone is curve-tested by the triangulated property, and its only cohomology is
\[
\begin{gathered}
H^{-1}\operatorname{Cone}\phi=\ker\phi,\\
H^0\operatorname{Cone}\phi=\operatorname{coker}\phi.
\end{gathered}
\tag{5.17i}
\]
Apply \(C_{n-1}\) to this cone. Its two cohomology modules are regular. The forward part of the lower criterion makes them curve-tested. In the short exact sequence
\[
0\to\ker\phi\to j_!E\to S\to0,
\tag{5.17j}
\]
the first two terms are curve-tested, so triangulated closure makes \(S\) curve-tested. This proves \(R_n\). This argument never assumes subquotient closure of the testing class: it uses the cone with strictly lower support first, and only then the short exact sequence.

Theorem 5.16.1 now proves \(C_n\) for all bounded complexes and every singular support at this dimension. The socle peeling there supplies the same-dimensional converse; it is not imported from a stability statement.

#### Direct images of every regular singular object

Assume \(R_n,C_n\) have just been proved and \(P_{n-1}\) is available. Let \(f:X\to Y\) be arbitrary and let \(S\) be regular simple of support dimension \(n\), with presentation (5.17e). Set \(B=\overline U\setminus U\), a closed subset of \(X\), of dimension at most \(n-1\). The localization triangle on \(X\) is
\[
R\Gamma_BS\longrightarrow S\longrightarrow j_*E\longrightarrow.
\tag{5.17k}
\]
To check this when \(U\) is merely locally closed in \(X\), use the open \(X\setminus B\). There \(U\) is closed in the support, and supported Kashiwara identifies restriction of \(S\) with its closed direct image of \(E\); composing its open direct image gives \(j_*E\). Thus (5.17k) is the ordinary closed-support localization triangle, not an assumed localization triangle for a singular ambient support.

By \(R_n\), \(S\) is curve-tested. 5.14.3 makes \(R\Gamma_BS\) curve-tested. Its support dimension is at most \(n-1\), so \(C_{n-1}\) makes it regular, and \(P_{n-1}\) makes
\(f_*R\Gamma_BS\) regular. For \(n=0\), the boundary is empty and this term is zero.

We prove regularity of
\[
f_*j_*E=(fj)_*E.
\tag{5.17l}
\]
Compactify the graph of the map \(fj:U\to Y\) in a proper SNC model of \(U\) and a smooth proper compactification \(\bar Y\) of \(Y\), resolve while preserving \(U\), and principalize its complement. This yields \((P,D)\) and \(F:P\to\bar Y\) as in the proper-SNC image argument, and composition identifies (5.17l) with
\(F_{Y,*}(Q^+|_{P_Y})\).

If the image dimension is zero or one, 5.14.5 proves this output regular by the full generic Gauss–Manin theorem with arbitrary coefficients, followed by the curve-support simple-factor argument. If its image dimension is at least two, the proper-SNC image argument proves it curve-tested using only \(C_{n-1},P_{n-1}\). Its cohomological supports have dimension at most \(n\): the proper map \(F\) confines them to the closed image of the \(n\)-dimensional proper model, and restriction confines them to its intersection with \(Y\). Therefore \(C_n\) makes this output regular.

Apply \(f_*\) to (5.17k). Its first and third terms have just been proved regular. The regular category is triangulated, so \(f_*S\) is regular. Finite composition series extend this assertion from regular simples to every regular module with the stated support bound. Standard truncation triangles extend it to every bounded regular complex. This proves \(P_n\), completing (5.17a). The argument does not assume the ambient varieties affine, proper or quasi-projective, and never assigns a smooth transfer to a singular fiber product.

#### The full theorem and singular-divisor reconstruction

**Theorem 5.17.** For every map \(f:X\to Y\) of smooth separated finite-type complex varieties, the four algebraic functors
\[
f_*,\quad f_!,\quad f^!,\quad f^*
\tag{5.17m}
\]
preserve bounded regular holonomic complexes. A bounded holonomic complex \(K\) on \(X\) is regular if and only if \(k^!K\) is regular for every map \(k:C\to X\) from a smooth algebraic curve. It is enough to test smooth locally closed curve immersions. Every simple factor and every singular support is included.

**Proof.** Every bounded holonomic complex has a finite maximum support dimension. The complete induction proves \(C_n,P_n,R_n\) at that bound. Thus \(f_*\) preserves regularity and both versions of the curve criterion hold. For inverse image, let \(L\) be regular on \(Y\). For every curve map \(k:C\to X\), inverse composition gives
\[
k^!f^!L=(fk)^!L,
\tag{5.17n}
\]
regular by the forward all-map curve criterion on \(Y\). The already proved arbitrary holonomic inverse theorem makes \(f^!L\) bounded holonomic. The converse criterion on \(X\) now makes it regular. This invokes the criterion only after its support-dimension induction is complete. The independent full regular duality Theorem 5.7, and the definitions
\(f_!=\mathbb D_Y f_*\mathbb D_X\),
\(f^*=\mathbb D_X f^!\mathbb D_Y\), prove the two remaining operations. \(\square\)

**Corollary 5.17.1 (singular boundary reconstruction).** Let \(B\) be any effective Cartier divisor on smooth separated \(Y\), with arbitrary singular reduced support and multiplicities, let \(j:Y\setminus B\hookrightarrow Y\), and let \(E\) be any curve-regular finite-rank connection on its complement. Then \(j_*E\) is regular holonomic and every boundary simple factor is regular. Every module \(N_f\) in 5.14.1 is therefore regular holonomic, with precisely its recorded flat transfer action and degrees.

**Proof.** The connection/regular-holonomic identification makes \(E\) regular. Apply the just-proved direct-image theorem to \(j\). Its affine localization is the degree-zero meromorphic extension, and the Serre definition proves the assertion about every factor. Alternatively 5.14.1 makes \(N_f\) curve-tested and 5.17's now-proved converse applies. This is a consequence of the completed induction, not an input to its SNC dual calculation. \(\square\)

All dimension formulas act on the finitely many open-and-closed pure-dimensional smooth components; connected maps land in one component. Finite component sums cover different component dimensions. The proof preserves every cohomological shift through the transfer formulas, the graph local cohomology formula, (5.15g)'s \([-c]\), and (5.16c)'s \([1-\dim W_0]\). Shifts do not alter regularity and have not been silently changed.

![The residue calculation, lower-support standard cone, socle peeling and ordered induction](assets/full-regularity-induction.png)

*Figure. The top left panel is the exact dual density (5.15e), positive-normal-degree contraction (5.15i), and uncontracted residue complex (5.15g). The top right and central panels show why the standard-extension cone has strictly smaller support before its kernel is removed, (5.17c)–(5.17j). The lower left is the actual integer-valued socle-peeling argument (5.16a)–(5.16f); the lower right is the localization triangle (5.17k). The bottom arrow is the order of proof (5.17a), not an assumption of all three assertions at the current dimension. Source reading: [Bernstein, Algebraic theory of D-modules](https://www.math.columbia.edu/~khovanov/resources/Bernstein-dmod.pdf), Main Theorem B. Every mechanism is proved above. The reproducible drawing source supplies the original CC0 figure.*



Write $D^b_{\rm rh}(\mathcal D_X)$ for bounded complexes with regular holonomic cohomology. This is a triangulated subcategory: a composition series for a submodule and quotient concatenates to one for the middle module. Thus regular modules form a Serre subcategory, and the long cohomology sequence of a triangle preserves it.

**Stability and testing (proved).** For a morphism $f:X\to Y$ of smooth separated complex algebraic varieties, the four functors
\[
f_*,\ f_!,\ f^!,\ f^*
\]
of Adjunctions, base change and the projection formula, and holonomic duality $\mathbb D$, preserve the bounded regular holonomic categories. No properness hypothesis is imposed in this algebraic statement. A bounded holonomic complex $M$ on $X$ is regular if and only if $k^!M$ is regular for every map $k:C\to X$ from a smooth algebraic curve. It is enough to test smooth locally closed curves. Theorem 5.17 proves the four-map and both curve-test assertions, in the notation $f^!$ and $f^*$ used here; Theorem 5.7 proves duality preservation. The statements apply to every singular support and bounded complex. The analytic proper direct-image assertion likewise remains a proof obligation; Kashiwara's freely accessible §8 is reading material. Neither citation establishes an input used below.

### External product and both tensor operations

**Theorem 5.18 (external product and the two tensor operations).** External products, $\otimes^!$, and the dual tensor/Hom operation formed using $\mathbb D$ preserve bounded regular holonomic complexes. **Proof.** We use the four-map and curve-test assertions of Theorem 5.17, together with duality preservation in Theorem 5.7. External products of holonomic modules are holonomic: a product good filtration has associated graded the external product of the two associated graded modules, so its characteristic support is the product of the two Lagrangian supports. The dimension is the sum of the dimensions. Derived external products and tensor products are consequently holonomic by bounded dévissage and the already proved inverse-image theorem; the internal tensor is the diagonal inverse image with its dimension shift.

On a curve, two regular holonomic complexes restrict outside finitely many points to complexes of regular connections. A tensor of two regular connections is regular, since tensor products of logarithmic lattices are logarithmic lattices, including at the compactification boundary. Every simple factor supported at an omitted point is a delta module and is regular. The simple-factor description therefore shows that the tensor complex is regular on the whole curve. Restrictions to a dense open detect the positive-dimensional simple factors, and any such factor with nonzero restriction inherits a regular generic connection by the stable sublattice and image argument following (1.5).

For an arbitrary curve map $k:C\to X\times Y$, the transfer-module tensor identity gives
\[
k^!(M\boxtimes N)\simeq
(\operatorname{pr}_X k)^!M\otimes^!
(\operatorname{pr}_Y k)^!N.                                   \tag{5.2}
\]
By Theorem 5.17, both factors are regular, and their tensor is regular by the curve argument. The curve criterion of Theorem 5.17 makes $M\boxtimes N$ regular. Diagonal inverse image then gives regularity of $\otimes^!$ on $X$. Theorem 5.7 then gives the dual tensor/Hom operation formed using $\mathbb D$. This proves the theorem. Recall that for left modules $\otimes^!$ is the derived $\mathcal O_X$ tensor shifted by $[-\dim X]$; shifts do not affect regularity.

For instance, $\mathcal O_{\mathbb A^1}$ and $\mathcal O.e^x$ both have zero-section characteristic variety. The first is regular algebraically and the second is not. Characteristic variety records the cotangent directions of propagation, not the exponential rate at infinity.

### Actual analytic descent from a supplied Euler-stable lattice

The next eight sections prove an actual convergence and descent result on the dummy cotangent chart. Let \(M\) be a coherent analytic left \(\mathcal D_X\)-module and \(j_0(x)=(0,x;1,0)\). The conclusion applies to a **supplied** coherent order-zero lattice \(L\subset\Phi(M)\) which generates \(\Phi(M)\) under finite-order operators and satisfies \(tL\subset hL\), where \(h=D_t^{-1}\). It produces one ordinary neighbourhood on which every actual intersection \(j_0^{-1}h^{-k}L\cap M\) is coherent and the resulting filtration is locally good. No canonical lattice is being assumed to have been constructed for every regular holonomic module.

The argument proceeds through actual common-domain factorial symbols, Banach division, finite-stalk exact completion, fixed neighbourhood kernel sections, bounded actual ambient modules, and formal Euler projection followed by faithful recovery of actual sections. Completion is used at the dummy zero-section stalks \(j_0(y)\); at other points of the chart the ideal \((h,t,z)\) can be the whole ring. Coherence of the operator sheaf, proved in §5.24, holds throughout the fixed chart.

The coefficient inputs are the complete earlier Cauchy and Taylor proofs, Theorems 1.2, 2.1 and 2.3, bounded Weierstrass preparation/division and analytic Noetherianity, Theorems 1, 2 and 6, Oka coherence and finite coherent kernels, Theorems 2.1 and 3.1, and commutative completion, Theorems 3.1–3.3. Ordinary polynomial Noetherianity and relative coefficient flatness are proved in GL06, the analytic local-algebra calculation and Lemmas 3.0c.1–3.0c.2; Lemmas 3.0c.3–3.0c.4 and their finite-presentation proof give the uniform polynomial kernels and ordinary local good filtrations. The graded Rees argument, with both generator shifts retained, is also proved in [§5.12, graded analytic Rees coherence](#graded-analytic-rees-coherence-with-a-finite-termination-argument). These are earlier full proofs with the displayed hypotheses. The operator-ring division, nearby kernels and completion below are proved separately.

### 5.19. Actual factorial-growth symbols and ordered jets

#### 5.19.1. The chart and the convergence class

Use base coordinates \(u=(t,x_1,\ldots,x_d)\), cotangent coordinates \((\tau,\xi_1,\ldots,\xi_d)\), and the chart \(\tau\ne0\). Put \(z_i=\xi_i/\tau\), \(h=\tau^{-1}\), and work near \((t,x,z)=(0,x_0,0)\). The coefficient symbols of order zero are sequences
\[
 P=\sum_{n\geq0}\tau^{-n}p_n(u,z).
 \tag{5.19a}
\]
All \(p_n\) are holomorphic on one common neighbourhood in \((u,z)\). For every compact subset \(K\) of that neighbourhood there exist \(A,C>0\) such that
\[
\begin{gathered}
\sup_K|p_n|\leq AC^n n!,\\ n\geq0.
\end{gathered}
\tag{5.19b}
\]
The series is a symbol expansion; it is not a convergent power series at a numerical nonzero value of \(h\). A germ means an equivalence class of these sequences under restriction to a common neighbourhood. In particular it cannot be defined by unrelated holomorphic germs with shrinking coefficient domains.

This is the order-zero factorial-growth class in the free primary source Schapira, [*Une introduction à l'étude des systèmes d'équations microdifférentielles*](https://www.numdam.org/item/AST_1981__89-90__45_0/), §2.1, printed pp. 67–69. It supplies human context for the class. Its multiplication, division and Noetherian assertions are proved here by the displayed estimates and later sections.

Denote the sheaf defined by (5.19a)–(5.19b) in this fixed chart by \(\mathcal A_{\rm gv}\). Restriction is coefficientwise. The sheaf gluing assertion is elementary: coefficient functions glue holomorphically; a compact set is covered by finitely many neighbourhoods on which bounds (5.19b) hold, and taking the maximum of their finitely many \(A,C\) gives a bound on the compact set. This proves the actual sheaf condition for this convergence class.

#### 5.19.2. Multiplication, with all cotangent derivatives retained

For \(\alpha=(a,\beta)\in\mathbf N\times\mathbf N^d\), write \(|\alpha|=a+|\beta|\), \(\alpha!=a!\beta!\), and \(\partial_\rho^\alpha=\partial_\tau^a\partial_\xi^\beta\), \(\partial_u^\alpha=\partial_t^a\partial_x^\beta\). The product coefficients are defined below.

For the next formulas write \(F_n(u,\rho)=\tau^{-n}p_n(u,\xi/\tau)\) and \(\operatorname{ev}_0f=f(u,1,z)\). The evaluation takes place after the indicated original cotangent derivatives. The expressions \(F\) in (5.20m) and \(F_{n,b,\gamma}\) in (5.21f) below are the displayed homogeneous symbols, with their cotangent dependence retained.

\[
\begin{aligned}
c_N(u,z)&=\sum_{\substack{n,m\geq0,\ \alpha\geq0\\n+m+|\alpha|=N}}\\ &\quad\frac{\operatorname{ev}_0(\partial_\rho^\alpha F_n)}{\alpha!}\\ &\quad\cdot\partial_u^\alpha q_m(u,z).
\end{aligned}
\tag{5.19c}
\]
Every sum for a fixed \(N\) is finite. Cotangent homogeneity gives a term of degree \(-n-|\alpha|\) in the first derivative, so the product is \(\sum_{N\geq0}\tau^{-N}c_N\). The ordinary microdifferential Leibniz product is therefore fully specified in these coordinates.

For clarity, no \(t\)-dependent term has been removed. If \(E_z=\sum_i z_i\partial_{z_i}\), the first derivative in (5.19c) is
\[
\begin{gathered}
\left[\prod_{\ell=0}^{a-1}(-n-|\beta|-\ell-E_z)\right]\\ \cdot\partial_z^\beta p_n(u,z).
\end{gathered}
\tag{5.19d}
\]
Indeed \(\partial_\xi^\beta\) first gives
\(\tau^{-n-|\beta|}\partial_z^\beta p_n\); each subsequent \(\tau\)-derivative decreases homogeneity by one and applies the indicated Euler factor. This proves (5.19d), including its signs and all integer factors.

#### 5.19.3. A bounded product estimate on explicit nested domains

Translate \(x_0\) to zero. Let \(0<r<R\), and suppose all coefficient functions extend holomorphically beyond the closed polydisc
\[
 K_R=\{|u_i|\leq R,\ |z_i|\leq R\},
\]
with bounds \(A_P C_P^n n!\), \(A_Q C_Q^m m!\) there. Put
\[
\begin{gathered}
\varepsilon=\frac{R-r}{2},\\ \delta=\min\left(\frac14,\frac{R-r}{4(1+R)}\right),\\ D=d+1,\\ K_*=\frac{C_P}{1-\delta}+C_Q+\frac D{\delta\varepsilon}.
\end{gathered}
\tag{5.19e}
\]
All constants are strictly positive and finite.

**Theorem 5.19.1.** The product (5.19c) is defined on \(K_r\) and satisfies
\[
 \sup_{K_r}|c_N|\leq A_PA_Q\,K_*^N N!.
 \tag{5.19f}
\]

**Proof.** Fix \((u,z)\in K_r\). On the cotangent polydisc
\(|\tau-1|\leq\delta,\ |\xi_i-z_i|\leq\delta\), one has
\[
\begin{gathered}
|\tau|\geq1-\delta,\\ \left|\frac{\xi_i}{\tau}\right|\leq\frac{r+\delta}{1-\delta}\\ =r+\frac{(1+r)\delta}{1-\delta}\\ \leq r+\frac{R-r}{3}<R.
\end{gathered}
\tag{5.19g}
\]
Thus the coefficient function in the first derivative of (5.19c) is holomorphic beyond this closed cotangent polydisc and is bounded there by
\(A_P(C_P/(1-\delta))^n n!\). Iterated Cauchy estimates give
\[
\begin{aligned}
|\operatorname{ev}_0(\partial_\rho^\alpha F_n)|\\ {}\leq\alpha!\delta^{-|\alpha|}A_P\\ {}\cdot(C_P/(1-\delta))^n n!.
\end{aligned}
\tag{5.19h}
\]
The base polydisc with coordinate radius \(\varepsilon\) around \(u\) lies strictly inside the radius-\(R\) polydisc. The same Cauchy estimate gives
\[
\begin{aligned}
|\partial_u^\alpha q_m(u,z)|\\ {}\leq\alpha!\varepsilon^{-|\alpha|}A_QC_Q^m m!.
\end{aligned}
\tag{5.19i}
\]
These are the earlier proved several-variable Cauchy estimates, obtained by differentiating the iterated Cauchy integral. No estimate for an operator of infinite order is used.

Set \(s=|\alpha|\) and \(N=n+m+s\). Dividing the product of (5.19h)–(5.19i) by \(\alpha!\) leaves \(\alpha!n!m!\), which is at most \(s!n!m!\leq N!\). The number of multiindices \(\alpha\) of length \(D\) and sum \(s\) is at most \(D^s\): each such multiindex is the count vector of at least one word of length \(s\) in \(D\) letters, and there are \(D^s\) words. Consequently
\[
\begin{aligned}
|c_N|&\leq A_PA_QN!\sum_{n+m+s=N}\\ &\quad\left(\frac{C_P}{1-\delta}\right)^nC_Q^m\\ &\quad\cdot\left(\frac D{\delta\varepsilon}\right)^s.
\end{aligned}
\]
The sum is bounded by \(K_*^N\). Expand the \(N\)-fold product defining that power; it contains the displayed monomials with positive integer multinomial coefficients, all at least one. This proves (5.19f). \(\square\)

For any compact set strictly inside the common coefficient domain, use finitely many such nested polydiscs and take maxima of the constants. Hence (5.19c) preserves the actual class (5.19b).

#### 5.19.4. Associativity and the dummy Ore relations

**Proposition 5.19.2.** The product (5.19c) is associative, has unit one, and makes \(\mathcal A_{\rm gv}\) a sheaf of complex algebras. In this algebra the elements \(h=\tau^{-1}\), \(\theta_i=z_i\), \(t\), and holomorphic functions of \(x\) satisfy
\[
\begin{gathered}
{}[\theta_i,b(x)]=h\partial_i b,\\ {}[\theta_i,\theta_j]=0,\\ {}[h,\theta_i]=[h,b]=0,\\ {}[t,h]=h^2,\\ {}[t,\theta_i]=h\theta_i,\quad[t,b]=0.
\end{gathered}
\tag{5.19j}
\]

**Proof.** On three factors in original \((u,\rho)\) variables put
\(A_{ij}=\sum_{\ell=0}^d\partial_{\rho_\ell}^{(i)}
                                  \partial_{u_\ell}^{(j)}\).
These ordinary partial-derivative operators commute. The Leibniz rule identifies both parenthesizations of a triple product with
\[
\begin{gathered}
\mu_3\exp(A_{12}+A_{13}+A_{23})\\ \cdot(P\otimes Q\otimes S).
\end{gathered}
\tag{5.19k}
\]
This is a formal homogeneous-order identity: for any fixed output order, there are only finitely many coefficient indices and derivative orders. Thus the exponential notation expresses finite coefficient identities, and does not require convergence of an infinite-order operator. For the first parenthesization the inner derivatives are \(A_{12}\), and differentiation of its two factors in the outer product gives \(A_{13}+A_{23}\). The other parenthesization gives \(A_{23}\) and \(A_{12}+A_{13}\). Their expansions agree since the operators commute. Theorem 5.19.1 puts both products in the convergence class. The constant one has no nonzero derivatives, so it is a unit.

To verify (5.19j), apply (5.19c) to the displayed coordinate symbols. For example \(h\circ t=ht-h^2\), whereas \(t\circ h=th\). Likewise \(\theta_i\circ t=z_it-hz_i\), \(t\circ\theta_i=tz_i\), and \(\theta_i\circ b=z_ib+h\partial_i b\). All other displayed commutators have no correction terms. This proves the signs, including the genuine \(t\)-action. \(\square\)

The polynomial subring generated by these elements is exactly the graded Ore ring \(A=R[t;h\delta]\) of formal Lemma 5.23.4. To see the injectivity as well as the relations, polynomial normal symbols have finite total degree and ordered polynomial normal form; the reordering from \(t\)-left to \(t\)-right is triangular in its \(t\)-degree with unchanged leading coefficient. Thus no nonzero ordered polynomial is the zero symbol. The two presentations are inverse using their normal forms.

#### 5.19.5. The positive-degree formal map

Let \(H=\mathcal O_{X,x_0}\), and let \(I_{\rm alg}=(h,t,\theta_1,\ldots,\theta_d)\) in the polynomial Ore ring \(A\). The degree product
\(\widehat A=\prod_{k\geq0}A_k\) is its positive-degree completion, by the already proved degree-one generation of \(A\).

Expand each \(p_n(x,t,z)\) at \(t=z=0\). A term
\[
\begin{gathered}
f_{n,\beta,b}(x)h^nz^\beta t^b\\ \text{has total degree}\\ n+|\beta|+b.
\end{gathered}
\tag{5.19l}
\]
At a fixed total degree there are finitely many such indices. Their coefficient germs belong to \(H\). Reorder each finite homogeneous polynomial using (5.19j) into \(t\)-right Ore normal form. This defines
\[
 T:\mathcal A_{{\rm gv},\,j_0(x_0)}\longrightarrow\widehat A.
 \tag{5.19m}
\]

**Lemma 5.19.3.** The map \(T\) is an injective ring map, and multiplication preserves the lowest total degree.

**Proof.** On monomials of (5.19l), (5.19c) preserves the sum of their total degrees. A \(\xi\)-derivative lowers \(z\)-degree by one and raises negative \(\tau\)-order by one. A \(\tau\)-derivative raises that order by one while preserving \(z\)-degree; its paired \(t\)-derivative lowers the \(t\)-degree of the other factor by one. Derivatives in \(x\) change only the degree-zero holomorphic coefficient. Thus every nonzero contribution has precisely the sum of the input total degrees. In particular it cannot decrease their sum.

Every fixed total-degree coefficient of the product therefore depends on finite homogeneous Taylor polynomials of the two input symbols. On these finite polynomials (5.19j) is exactly the Ore product, by Proposition 5.19.2. This proves multiplicativity of (5.19m).

If \(T(P)=0\), the triangular finite reordering in every total degree gives zero for every original coefficient in (5.19l). For each \(n\), the Taylor series in \((t,z)\) of \(p_n\) is zero, with holomorphic coefficient germs in \(x\). On a smaller common product neighbourhood the function \(p_n\) is therefore zero by its Taylor expansion. This conclusion concerns the germ of each coefficient; to obtain zero as a common-domain symbol, use the identity theorem on the connected coefficient domain: a function with zero germ is identically zero there. The original common connected neighbourhood then works for every \(n\). Hence \(P\) is the zero symbol germ. \(\square\)

For \(r\geq1\), let \(J_r=\{P:T(P)\text{ has no degree below }r\}\). The lemma gives \(J_rJ_s\subset J_{r+s}\); in particular \(J_1\) is a two-sided ideal.

#### 5.19.6. Actual ordered-jet division, without a formal convergence substitution

**Theorem 5.19.4.** In the actual convergent stalk, let \(I=(h,t,\theta_1,\ldots,\theta_d)\) be the two-sided ideal generated by these elements. Then
\[
\begin{gathered}
I^r=J_r,\quad r\geq1,\\ \mathcal A_{{\rm gv},j_0(x_0)}/I^r\\ {}\simeq A/I_{\rm alg}^r.
\end{gathered}
\tag{5.19n}
\]
Consequently the completion of this actual convergent ring at \(I\) is canonically \(\widehat A\).

**Proof.** The generators have degree one, and Lemma 5.19.3 gives \(I^r\subset J_r\). For the converse let \(P\in J_r\). The inverse triangular reordering in each degree shows that all Taylor terms of \(p_n\), \(n<r\), of degree less than \(r-n\) in \((t,z)\) are zero. There are only finitely many such coefficients; their germ vanishings hold on one smaller common neighbourhood.

For these finitely many \(n\), write the actual holomorphic Taylor factorization
\[
\begin{gathered}
p_n(x,t,z)=\sum_{b+|\beta|=r-n}\\ t^bz^\beta a_{n,b,\beta}(x,t,z).
\end{gathered}
\tag{5.19o}
\]
All \(a_{n,b,\beta}\) are holomorphic on one smaller common product neighbourhood. Here is an explicit construction, rather than an ideal-membership assertion. Order the finitely many monomials of total degree \(r-n\). Assign each monomial of the Taylor series of degree at least \(r-n\) to the first such monomial dividing it, and remove that factor. Each resulting coefficient series converges normally on any smaller polydisc: Cauchy bounds for the original coefficients, multiplied by a fixed inverse radius power for the removed degree, bound the assigned subseries. There are finitely many groups and finitely many \(n<r\), so one common smaller domain suffices.

Each term of \(\tau^{-n}p_n\) in (5.19o) equals the actual symbol product
\[
 t^b\circ a_{n,b,\beta}\circ(\theta^\beta h^n).
 \tag{5.19p}
\]
The leftmost factor has no cotangent derivatives; the rightmost has no base derivatives. Thus (5.19c) gives precisely their ordinary symbol product, with no omitted correction. It belongs to \(I^{b+|\beta|+n}=I^r\), since \(I\) is two-sided and the middle coefficient is an actual order-zero symbol.

The remaining tail is also an actual element of \(I^r\). Put
\[
 Q=\sum_{k\geq0}\tau^{-k}p_{k+r}(u,z).
 \tag{5.19q}
\]
Its coefficients have factorial growth on every compact set. In fact
\[
 (k+r)!\leq 2^{k+r}k!r!
\]
because \(\binom{k+r}{r}\leq2^{k+r}\), so (5.19b) for \(P\) gives
\[
\begin{aligned}
\sup_K|p_{k+r}|\\ {}\leq(AC^r2^rr!)(2C)^kk!.
\end{aligned}
\tag{5.19r}
\]
Hence \(Q\in\mathcal A_{\rm gv}\) on the original coefficient domain. The right factor \(h^r\) has no base derivatives, so \(Q\circ h^r\) is exactly the tail \(\sum_{n\geq r}\tau^{-n}p_n\). This puts the tail in \(I^r\), proving \(J_r\subset I^r\). The case \(r=1\) also verifies directly that the candidate ideal is the full kernel of the degree-zero map.

The map \(T\) followed by truncation is surjective onto \(A/I_{\rm alg}^r\): every finite ordered polynomial with coefficient germs in \(H\) is represented by an actual finite symbol after shrinking for its finite list of coefficients. Its kernel is \(J_r=I^r\). This proves (5.19n). The quotient maps are compatible as \(r\) varies, so their inverse limits give exactly \(\widehat A\). This is completion of actual stalk modules at an actual two-sided ideal; no sheaf-stalk/inverse-limit interchange is used. \(\square\)

### 5.20. One-norm multiplication, units and dummy division

In this section \(\circ\) denotes the actual Leibniz product. Juxtaposition of homogeneous normal symbols, Taylor coefficients or the weight variables in a norm calculation denotes ordinary coefficient multiplication. These operations agree only when the derivative corrections shown below vanish.

#### 5.20.1. The actual homogeneous-symbol norm

Use the actual convergence class in §5.19. Put \(D=d+1\), \(u=(t,x)\), \(\rho=(\tau,\xi)\), \(z=\xi/\tau\), and
\[
\begin{gathered}
F_n(u,\rho)=\tau^{-n}p_n(u,\xi/\tau),\\ P=\sum_{n\geq0}F_n.
\end{gathered}
\tag{5.20a}
\]
Every \(F_n\) is homogeneous of cotangent degree \(-n\). Work on one common open subset of \(\tau\ne0\); a compact \(K\) below is contained in that subset. For \(\mu,\nu\in\mathbf N^D\), use total lengths \(|\mu|,|\nu|\), but retain the multiindices in the derivatives. Let \(\lambda=2D\). Define the nonnegative majorant
\[
\begin{gathered}
\mathsf N(P,K,\epsilon)=2\sum_{n,\mu,\nu}\\ \frac{\lambda^{-n}n!}{(n+|\mu|)!(n+|\nu|)!}\\ \cdot\sup_K|\partial_u^\mu\partial_\rho^\nu F_n|\\ \cdot\epsilon^{2n+|\mu|+|\nu|}.
\end{gathered}
\tag{5.20b}
\]
The numerical value may initially be infinite. Formal comparison of these majorants always means coefficientwise comparison of their nonnegative coefficients. For a fixed exponent there are finitely many indices. In particular \(\mathsf N(1,K,\epsilon)=2\).

**Lemma 5.20.1 (norm detects exactly the factorial class).** For an actual symbol, \(\mathsf N(P,K,\epsilon)\) is finite for some \(\epsilon>0\), for every compact \(K\) strictly inside its common coefficient domain. Conversely, finiteness gives on \(K\)
\[
\begin{aligned}
\sup_K|F_n|\\ {}\leq\tfrac12\mathsf N(P,K,\epsilon)\\ {}\cdot(\lambda/\epsilon^2)^n n!.
\end{aligned}
\tag{5.20c}
\]

**Proof.** For the converse retain the term \(\mu=\nu=0\) of (5.20b). For the forward implication, thicken \(K\) by a sufficiently small common polydisc radius \(\eta>0\). This thickening stays in the cotangent chart and the common open domain. The normalized factorial bounds on its compact image under \((u,\rho)\mapsto(u,\xi/\tau)\), and the positive lower bound for \(|\tau|\), give constants \(A,C>0\) such that \(\sup|F_n|\le AC^n n!\) on the thickening. Iterated Cauchy integrals give
\[
 \sup_K|\partial_u^\mu\partial_\rho^\nu F_n|
 \le AC^n n!\,\mu!\nu!\,\eta^{-|\mu|-|\nu|}.
\]
Put \(p=|\mu|,q=|\nu|\). Since \(\mu!\le p!\), \(\nu!\le q!\), and \((n!)^2p!q!\le(n+p)!(n+q)!\), each weighted term is at most
\(2A(C\epsilon^2/\lambda)^n(\epsilon/\eta)^{p+q}\).
There are at most \(D^p\) multiindices of total length \(p\): assign to each multiindex a word having those coordinate counts. The same holds for \(q\). Thus, if both ratios in the following expression are less than one,
\[
\begin{gathered}
\mathsf N(P,K,\epsilon)\\ {}\leq\frac{2A}{1-C\epsilon^2/\lambda}\\ {}\cdot\frac1{(1-D\epsilon/\eta)^2}.
\end{gathered}
\tag{5.20d}
\]
Choosing a sufficiently small \(\epsilon\) proves finiteness. Restriction to the normalized slice \(\tau=1\) in (5.20c) gives the original factorial bound for \(p_n\). \(\square\)

#### 5.20.2. A combinatorial estimate with the actual cotangent derivatives

The actual symbol multiplication is
\[
\begin{aligned}
(P\circ Q)_k&=\sum_{n+m+|\alpha|=k}\\ &\quad\frac{\partial_\rho^\alpha F_n\,\partial_u^\alpha G_m}{\alpha!}.
\end{aligned}
\tag{5.20e}
\]
Each output is homogeneous of degree \(-k\). The preceding actual symbol proof establishes this product and associativity; (5.20e) retains the \(\tau\)-derivatives paired with \(t\)-derivatives.

**Theorem 5.20.2 (one-norm product inequality).** Coefficientwise as majorants, and numerically whenever the right side is finite,
\[
\begin{gathered}
\mathsf N(P\circ Q,K,\epsilon)\\ {}\preccurlyeq\mathsf N(P,K,\epsilon)\mathsf N(Q,K,\epsilon).
\end{gathered}
\tag{5.20f}
\]

**Proof.** Differentiate one summand of (5.20e). Allocate its external \(u\)-derivatives and \(\rho\)-derivatives between the two factors. Fix the resulting derivative indices of \(F_n\) to be \((\mu,\nu)\), and those of \(G_m\) to be \((\mu',\nu')\). An internal derivative index \(\alpha\) satisfies \(\alpha\le\nu,\mu'\). The external indices are
\[
\begin{gathered}
a=\mu+\mu'-\alpha,\\ b=\nu+\nu'-\alpha,\\ k=n+m+|\alpha|.
\end{gathered}
\]
The Leibniz coefficient is exactly
\(\binom a\mu\binom b{\nu'}/\alpha!\).
Put \(p=|\mu|,q=|\nu|,r=|\mu'|,v=|\nu'|,s=|\alpha|\), and \(A=p+r-s,B=q+v-s\). Compare the weight of this term, including its Leibniz coefficient, to the product of the two weights of the fixed derivative pair in (5.20b), with the initial factors 2 removed. Multiindex binomial coefficients are at most their total-length binomial coefficients: choosing fixed coordinate counts is a subset of choosing the specified number among all entries. The ratio is therefore at most
\[
 \lambda^{-s}\frac{s!}{\alpha!}\,
 \frac{\dfrac{k!}{n!m!s!}\binom A p\binom B v}
 {\binom{k+A}{n+p}\binom{k+B}{m+v}}.
 \tag{5.20g}
\]
This follows by cancellation of the four input factorials and the two output factorials; \(k+A=n+m+p+r\), \(k+B=n+m+q+v\).

The fraction on the second line is at most one. Indeed, choosing \(p\) entries from a block of size \(A\) and \(n\) from a block of size \(k\) is one contribution to choosing \(n+p\) from their union. Thus
\[
\begin{gathered}
\binom{k+A}{n+p}\geq\binom Ap\binom kn,\\ \binom{k+B}{m+v}\geq\binom Bv\binom km.
\end{gathered}
\]
Finally
\(\binom k n\binom k m\ge
\binom k n\binom{k-n}m=k!/(n!m!s!)\), since \(k=n+m+s\). This proves the asserted bound in (5.20g).

For the fixed derivative pair, sum (5.20g) over all permitted \(\alpha\). Enlarging to all multiindices and expanding \((1+\cdots+1)^s\) gives
\[
\begin{gathered}
\sum_\alpha\lambda^{-|\alpha|}\frac{|\alpha|!}{\alpha!}\\ {}\leq\sum_{s\geq0}(D/\lambda)^s=2.
\end{gathered}
\tag{5.20h}
\]
Also the power of \(\epsilon\) is exactly preserved:
\[
\begin{gathered}
2k+|a|+|b|\\ {}=2n+|\mu|+|\nu|\\ {}+2m+|\mu'|+|\nu'|.
\end{gathered}
\]
Hence the unscaled sums of weights for the product are at most twice the product of the unscaled sums for the inputs. Multiplying by the initial factor 2 in (5.20b) gives exactly the two initial factors 2 on the right of (5.20f). All summands are nonnegative, so these reindexings are valid first for finite truncations and then by increasing limits. They prove both the coefficientwise and the numerical statement. \(\square\)

The use of \(\lambda=2D\) precisely controls the possible internal derivative allocations. No repeated loss of the coefficient domain or repeated enlargement of a factorial-growth constant is used in this inequality.

#### 5.20.3. Actual elliptic inverses and the stalk radical

**Theorem 5.20.3 (elliptic inversion in the actual class).** If \(p_0\) has no zero on a common chart neighbourhood, \(P\) has an actual two-sided inverse there. In particular a germ is invertible if and only if its leading coefficient is nonzero at the point.

**Proof.** Let \(B\) be the actual symbol whose only homogeneous coefficient is \(p_0^{-1}\). Put \(R=1-P\circ B\). Its degree-zero homogeneous coefficient is zero. Its higher coefficients are holomorphic on the common neighbourhood by the actual product theorem. Fix any compact \(K\) therein and choose \(\epsilon_0>0\) for which its norm is finite, by Lemma 5.20.1. Since every term of \(\mathsf N(R,K,\epsilon)\) has exponent at least two,
\[
\begin{gathered}
\mathsf N(R,K,\epsilon)\\ {}\leq(\epsilon/\epsilon_0)^2\mathsf N(R,K,\epsilon_0),\\ 0<\epsilon\leq\epsilon_0.
\end{gathered}
\tag{5.20i}
\]
Choose \(\epsilon\) so the right side is \(q<1\). The formal Neumann symbol \(S=\sum_{j\ge0}R^{\circ j}\) has well-defined holomorphic coefficients on the original common domain: the coefficient of homogeneous order \(-n\) receives terms only from \(j\le n\). Theorem 5.20.2 and triangle inequalities give
\[
 \mathsf N(S,K,\epsilon)
 \le 2+\sum_{j\ge1}q^j=2+q/(1-q).
\]
This inequality is justified first on finite sums, then coefficientwise by increasing nonnegative bounds. Lemma 5.20.1 proves the actual factorial growth of \(S\) on every compact of the common domain. Thus \(Q=B\circ S\) is actual and \(P\circ Q=1\) by finite homogeneous coefficient identities. Multiplication by \(P\) is injective on formal symbols: the first nonzero homogeneous coefficient of \(W\) would give the nonzero product \(p_0w_n\). Consequently \(P\circ(Q\circ P-1)=0\) implies \(Q\circ P=1\). No numerical convergence at \(h\ne0\) was asserted. For a germ with nonzero leading value, shrink so that \(p_0\) has no zero. Conversely the degree-zero product of a germ and its inverse has value one, so its leading value cannot be zero. \(\square\)

At the actual stalk \(\mathcal A=\mathcal A_{{\rm gv},j_0(x_0)}\), evaluation of the leading coefficient is a surjective algebra map to \(\mathbf C\). Its kernel \(\mathfrak m\) consists exactly of the nonunits, by Theorem 5.20.3. Every proper left or right ideal is contained in \(\mathfrak m\), because an element outside it is a unit. Since the quotient by \(\mathfrak m\) is a field, \(\mathfrak m\) is the unique maximal left and right ideal, and the Jacobson radical. In particular the actual two-sided ideal
\[
\begin{gathered}
I=(h,t,\theta_1,\ldots,\theta_d),\\ I\subset\mathfrak m.
\end{gathered}
\tag{5.20j}
\]

**Corollary 5.20.3.1 (finite-module zero detection).** For a finite left \(\mathcal A\)-module \(V\),
\[
\begin{gathered}
\varprojlim_r V/I^rV=0\\ {}\Longrightarrow V=0.
\end{gathered}
\tag{5.20k}
\]
This conclusion needs neither Noetherianity nor exactness of completion.

**Proof.** The transition maps are surjective. Starting with any element of \(V/IV\), successively choose lifts along them; this gives a compatible family. The projection of the inverse limit onto \(V/IV\) is therefore surjective. A zero inverse limit implies \(V=IV\). Take a finite generating list \(v_1,\ldots,v_r\). If \(r>0\), write \(v_r=\sum_i a_iv_i\) with \(a_i\in I\): use \(V=IV\) and the fact that \(I\) is two-sided. Since \(1-a_r\) is a unit, this expresses \(v_r\) in the preceding generators. Repeating removes every generator and proves \(V=0\). This is the noncommutative Nakayama argument in full. \(\square\)

The corollary detects zero objects. It does not assert that completion is exact or injective on arbitrary finite-module morphisms. For the two coherent quotients in Proposition 5.26.1, it supplies the zero-detection step once their vanishing completions have been genuinely proved by exactness and a valid coefficient model.

#### 5.20.4. Bounded actual division by \(t-ah\)

Let \(a\in\mathbf Z\) be a fixed shift. The proof below also works for any fixed complex \(a\), using a branch of \(\tau^{-a}\) on a small disc about 1; only integer shifts are used for the ordinary lattice applications. Define on symbols independent of \(t\)
\[
 \mathscr L_a=-\partial_\tau+a\tau^{-1}.
 \tag{5.20l}
\]
It raises negative homogeneous order by one. It is the differential operator \(\tau^a(-\partial_\tau)\tau^{-a}\), as the product rule directly verifies. In the normalized chart,
\[
\begin{gathered}
F=\tau^{-n}f(x,\xi/\tau),\\ \operatorname{ev}_0(\mathscr L_a^bF)\\ {}=\prod_{j=0}^{b-1}(n+a+j+E_z)\\ {}\cdot f(x,z),\\ E_z=\sum_i z_i\partial_{z_i}.
\end{gathered}
\tag{5.20m}
\]
For \(b=0\) the product is the identity. One \(-\partial_\tau\) gives \(\tau^{-n-1}(n+E_z)f\); the extra \(a\tau^{-1}\) gives the stated first factor, and induction proves every factor and sign.

Assume the coefficients \(p_n(t,x,z)\) are holomorphic beyond a closed product domain with \(|t|\le T\), and satisfy \(|p_n|\le AC^n n!\) there. Fix \(0<s<T\), smaller compact \(x,z\) ranges, and \(0<\delta<1\) such that the cotangent disc \(|\tau-1|\le\delta\), at fixed \(\xi=z\) in those smaller ranges, remains in the common normalized coefficient domain. This is always possible; the explicit choice and ratio estimate of 5.19e–5.19g may be used. Put
\[
\begin{gathered}
H_a=\max_{|\tau-1|\leq\delta}|\tau^{-a}|,\\ C_*=C/(1-\delta),\\ K_*=C_*+(\delta T)^{-1}.
\end{gathered}
\tag{5.20n}
\]
The branch has value one at \(\tau=1\), and \(H_a\ge1\).

**Theorem 5.20.4 (actual bounded dummy division).** There are unique actual symbols \(Q\) and \(R\), with \(R\) independent of \(t\), such that
\[
 P=Q\circ(t-ah)+R.
 \tag{5.20o}
\]
On the specified smaller compact product with \(|t|\le s\), their normalized coefficients satisfy
\[
\begin{gathered}
|q_N|\leq\frac{AH_a}{T-s}K_*^NN!,\\ |r_N|\leq AH_aK_*^NN!.
\end{gathered}
\tag{5.20p}
\]
In particular one common smaller holomorphic neighbourhood serves for all coefficients.

**Proof.** Taylor-expand in the actual base variable:
\[
\begin{gathered}
p_n(t,x,z)=\sum_{b\geq0}t^bp_{n,b}(x,z),\\ |p_{n,b}|\leq AC^nn!T^{-b}.
\end{gathered}
\tag{5.20q}
\]
This is an ordinary convergent Taylor expansion on the common domain. Write \(F_{n,b}(x,\rho)=\tau^{-n}p_{n,b}(x,\xi/\tau)\). For a fixed pair \((n,b)\), the normal-symbol identity
\[
\begin{gathered}
t^bF_{n,b}=\\ \left(\sum_{j=0}^{b-1}t^{b-j-1}\mathscr L_a^jF_{n,b}\right)\\ \circ(t-ah)\\ {}+\mathscr L_a^bF_{n,b}.
\end{gathered}
\tag{5.20r}
\]
holds, with the sum empty for \(b=0\). Indeed, for every symbol \(W\),
\[
\begin{gathered}
W\circ(t-ah)\\ {}=tW+\partial_\tau W-ahW\\ {}=tW-\mathscr L_aW.
\end{gathered}
\]
The first equality is the actual Leibniz product: \(t\) has only one nonzero base derivative, and the right factor \(h\) has no base derivatives. The two finite sums telescope in (5.20r); \(\mathscr L_a\) does not differentiate \(t\).

Define the homogeneous coefficients of the remainder by the finite sums
\[
\begin{aligned}
R_N(x,\rho)&=\sum_{n+b=N}\\ &\quad\mathscr L_a^bF_{n,b}(x,\rho),
\end{aligned}
\tag{5.20s}
\]
and those of the quotient by
\[
\begin{gathered}
Q_N(t,x,\rho)=\sum_{n+j=N}\sum_{b\geq j+1}\\ t^{b-j-1}\mathscr L_a^jF_{n,b}(x,\rho).
\end{gathered}
\tag{5.20t}
\]
Every inner sum here is an actual holomorphic series in \(t\). To verify its convergence and the factorial bounds, apply the Cauchy derivative estimate in \(\tau\) to \(\tau^{-a}F_{n,b}\). Equations (5.20l) and (5.20q) give, on the normalized slice,
\[
\begin{gathered}
|\mathscr L_a^jF_{n,b}|\\ {}\leq AH_aC_*^nn!T^{-b}j!\delta^{-j}.
\end{gathered}
\tag{5.20u}
\]
The factor \(\tau^a\) in (5.20l) equals one at that slice. The \(\tau\)-disc remains in the normalized coefficient domain by the explicit hypothesis preceding (5.20n). Thus the bound includes the actual Euler/cotangent derivative factors, rather than replacing them with formal Ore coefficients.

For \(|t|\le s\),
\[
 \sum_{b\ge j+1}T^{-b}s^{b-j-1}
 =T^{-j}/(T-s).
\]
The inner series of (5.20t) therefore converges normally on the smaller product and is bounded by
\(AH_a C_*^n n!j!(\delta T)^{-j}/(T-s)\).
Sum over \(n+j=N\), use \(n!j!\le N!\), and bound the sum of monomials by \((C_*+(\delta T)^{-1})^N\). This proves the quotient bound in (5.20p). The same argument on the finite sum \(n+b=N\) gives the remainder bound. All coefficients are holomorphic on the same smaller open product; the estimates on its compact subsets prove that both are actual symbols.

Identity (5.20r), summed by homogeneous degree and ordinary \(t\)-Taylor degree, proves (5.20o). At a fixed total degree there are only finitely many input Taylor terms; alternatively normal convergence in (5.20t) allows coefficientwise holomorphic differentiation on smaller compact products, and the sums telescope. Neither interpretation interchanges unrelated germs with infinite sums.

For uniqueness, suppose \(Q\circ(t-ah)+R=0\), with \(R\) independent of \(t\). At homogeneous degree zero, the equation is \(tq_0+r_0=0\). Evaluate at \(t=0\) to obtain \(r_0=0\), hence \(q_0=0\). If all previous coefficients are zero, the degree-\(-N\) equation is again \(tq_N+r_N=0\), since \(\mathscr L_a\) raises negative homogeneous order by one. The same argument proves their vanishing. Induction gives uniqueness. \(\square\)

The theorem proves division by these actual degree-one dummy operators. It does not prove division by a general operator regular in an arbitrary cotangent or base direction.

#### 5.20.5. The convergent free dummy module and its completion

Let \(\mathcal R_{\rm gv}\subset\mathcal A\) be the actual symbols independent of \(t\). It is a subring, because every term with a \(t\)-derivative of the right factor vanishes in (5.20e). Its variable \(h\) is central, and \([\theta_i,b(x)]=h\partial_i b\). The bounded division proves an actual left-module identification
\[
\begin{gathered}
\mathcal A/\mathcal A(t-ah)\simeq\mathcal R_{\rm gv}u_a,\\ t(Su_a)=h(a+\delta)Su_a,\\ \delta\left(\sum_n h^ns_n(x,z)\right)\\ {}=\sum_n h^n(n+E_z)s_n(x,z).
\end{gathered}
\tag{5.20v}
\]
The right side means a free rank-one left \(\mathcal R_{\rm gv}\)-module endowed with the displayed actual \(t\)-action. It is not a statement that \(\mathcal A\) is commutative or that this quotient is a ring.

**Proof of (5.20v).** Theorem 5.20.4 gives one and only one \(t\)-independent representative of every coset. Left multiplication by a \(t\)-independent symbol preserves \(t\)-independence, and preserves the left ideal generated by \(t-ah\). Thus the representative map is \(\mathcal R_{\rm gv}\)-linear. For a representative \(S\), the relation \([t,S]=-\partial_\tau S\), proved directly from the product, and \(tu_a=ah u_a\) give
\(tS u_a=(ahS-\partial_\tau S)u_a=h(a+\delta)S u_a\).
These are actual symbols: the derivatives preserve factorial growth on smaller compact products by Cauchy estimates, and multiplying the \(n\)-th coefficient by \(n\) preserves such growth after increasing its exponential constant. The same conclusion follows from the actual remainder of \(tS\) in Theorem 5.20.4. \(\square\)

Put \(J=(h,\theta_1,\ldots,\theta_d)\) in \(\mathcal R_{\rm gv}\). The positive-degree Taylor map of Lemma 5.19.3 restricts injectively to
\[
 \mathcal R_{{\rm gv},x_0}
      \longrightarrow\widehat R,
 \qquad R=\bigoplus_{k\ge0}\mathcal D_{\le k,x_0}h^k.
\]
Its degree-\(k\) coefficient is the finite Taylor polynomial with \(n+|\beta|=k\). The common-domain identity argument of Lemma 5.19.3 applies unchanged. The actual ideal identity
\[
\begin{gathered}
J^r=\\ \left\{S:\begin{gathered}\text{all Taylor terms}\\ \text{of total degree}\\ \text{below }r\text{ vanish}\end{gathered}\right\}.
\end{gathered}
\tag{5.20w}
\]
has the following full local proof. The forward inclusion follows from degree additivity of the actual product. For the reverse inclusion, for the finitely many coefficients \(n<r\), factor their convergent \(z\)-Taylor series as finite sums of monomials \(z^\beta\) of degree \(r-n\) times holomorphic functions on a common smaller polydisc. This is obtained by assigning each Taylor monomial to the first monomial of degree \(r-n\) dividing it; each assigned subseries converges normally on smaller polydiscs by Cauchy bounds. The resulting symbols are products \(b(x,z)\circ z^\beta h^n\), with no corrections because the right factor has no base derivatives. They lie in \(J^r\). The tail factors on the right by \(h^r\); its shifted coefficients are actual because
\((k+r)!\le2^{k+r}k!r!\), by the binomial theorem. This proves (5.20w).

Consequently finite truncation gives
\[
\begin{gathered}
\mathcal R_{{\rm gv},x_0}/J^r\simeq R/R_{\geq r},\\ \varprojlim_r\mathcal R_{{\rm gv},x_0}/J^r\simeq\widehat R.
\end{gathered}
\tag{5.20x}
\]
Surjectivity of each finite truncation follows by representing its finitely many holomorphic coefficient germs by one finite actual symbol on their common neighbourhood. The kernel is exactly (5.20w). This is an inverse limit of actual stalk quotients, with no stalk/sheaf-limit interchange.

The \(I\)-adic filtration of the cyclic \(\mathcal A\)-module in (5.20v) equals this \(J\)-adic filtration of its representative module. For the forward containment, lift an element of \(I^r\) to its actual positive-degree Taylor model, supplied by Theorem 5.19.4. Each generator of \(I\) raises total degree by one in the module: \(h\) and \(\theta_i\) do so directly, and \(t=h(a+\delta)\) does so by (5.20v). The action of every actual symbol has nonnegative degree, and the leading-degree preservation of Lemma 5.19.3 extends to the module relation, which is homogeneous. Thus the representative has no terms below \(r\); by (5.20w) it belongs to \(J^r\). For the reverse containment, the finite factorizations in the proof of (5.20w), and the tail factor by \(h^r\), express it as the action on \(u_a\) of actual elements of the two-sided ideal \(I^r\). Hence
\[
\begin{gathered}
\widehat{\mathcal A/\mathcal A(t-ah)}^{\,I}\\ {}\simeq\widehat R\,u_a,\\ tu_a=ah u_a,\\ t(Su_a)=h(a+\delta)Su_a.
\end{gathered}
\tag{5.20y}
\]
One may also see the forward containment explicitly from (5.20s)–(5.20t): Taylor terms have degrees \(n+b+|\beta|\); the remainder replaces each \(t^b\) by \(b\) applications of \(\mathscr L_a\), preserving that total degree.

For integral \(a\), identify \(u_a\) with \(h^a u_0\) inside the finite-order localization of the free dummy transfer. The actual Weyl relation gives \(t h^a u_0=ah^{a+1}u_0\), also for negative \(a\). Thus (5.20y) is exactly the shifted formal free ordinary Rees model \(h^a\widehat R\) in Lemma 5.23.4. The construction respects any finite direct sum of such shifts: finite truncations and the inverse limit act coordinatewise. This proves the actual free coefficient model and its bounded presentation, the free case of the general actual ambient presentation proved in §5.25.

### 5.21. Actual simultaneous division and finite stalk relations

#### 5.21.1. An absolute Taylor version of the product norm

Translate the ordinary base point to zero. Put \(w=(u,z)=(t,x,z)\); there are \(q=2d+1\) variables. Choose positive polyradii \(R=(R_1,\ldots,R_q)\). For a holomorphic Taylor series \(f=\sum_\gamma c_\gamma w^\gamma\), define
\[
 |f|_R=\sum_\gamma |c_\gamma|R^\gamma.
 \tag{5.21a}
\]
This norm is finite on a smaller closed polydisc than any polydisc of holomorphy. Indeed Cauchy bounds on a larger polyradius \(S\) give \(|c_\gamma|\le A S^{-\gamma}\), and the geometric product \(A\prod_j(1-R_j/S_j)^{-1}\) bounds (5.21a). It is submultiplicative by coefficient convolution and the triangle inequality.

For the actual homogeneous coefficients \(F_n=\tau^{-n}p_n(u,\xi/\tau)\), replace the compact supremum in (5.20b) by the norm (5.21a) after evaluation at \(\tau=1\), \(\xi=z\):
\[
\begin{gathered}
\|P\|_{R,\epsilon}=2\sum_{n,\mu,\nu}\\ \frac{\lambda^{-n}n!\epsilon^{2n+|\mu|+|\nu|}}{(n+|\mu|)!(n+|\nu|)!}\\ \cdot|\operatorname{ev}_0(\partial_u^\mu\partial_\rho^\nu F_n)|_R,\\ \lambda=2(d+1).
\end{gathered}
\tag{5.21b}
\]
Every actual germ has a representative for which this norm is finite, with sufficiently small \(R,\epsilon\). To prove this, use Cauchy estimates for the original homogeneous functions on a fixed thickening of the normalized slice at polyradius \(S>R\), and then the geometric bound preceding (5.21b). The same calculation as (5.20d) bounds the derivative sum, with a further constant \(\prod_j(1-R_j/S_j)^{-1}\). Conversely, the zero-derivative terms imply
\[
 |p_n|_R\le\tfrac12\|P\|_{R,\epsilon}
                     (\lambda/\epsilon^2)^n n!.
 \tag{5.21c}
\]
Thus a finite norm supplies a common polydisc of holomorphic coefficients and actual factorial growth on it.

The proof of Theorem 5.20.2 used only a submultiplicative coefficient norm, the Leibniz rule and the exact allocation inequality. Consequently it applies without change to (5.21b):
\[
 \|P\circ Q\|_{R,\epsilon}
       \le\|P\|_{R,\epsilon}\|Q\|_{R,\epsilon}.
 \tag{5.21d}
\]
For ordinary multiplication of homogeneous normal symbols, retain only the allocation \(\alpha=0\) in that proof. There is no internal-allocation factor 2. Hence
\[
 \|PQ\|_{R,\epsilon}
       \le\tfrac12\|P\|_{R,\epsilon}\|Q\|_{R,\epsilon}.
 \tag{5.21e}
\]
These assertions include the actual \(\tau\)-derivatives and the mixed base/cotangent derivatives.

For clarity about completeness and support projections, write a coefficient monomial as \(c\,h^n u^b z^\gamma\). If \(\nu=(a,\beta)\), its normalized cotangent derivative is
\[
\begin{gathered}
F_{n,b,\gamma}=\tau^{-n}u^b(\xi/\tau)^\gamma,\\ \operatorname{ev}_0(\partial_\tau^a\partial_\xi^\beta F_{n,b,\gamma})\\ {}=(-1)^a(n+|\gamma|)^{\overline a}\\ {}\cdot\gamma^{\underline\beta}u^bz^{\gamma-\beta}.
\end{gathered}
\tag{5.21f}
\]
The expression is zero if \(\beta\nleq\gamma\); the rising factorial with initial value zero is zero for \(a>0\). Equation (5.21f) follows directly from 5.19d, since the Euler degree after the \(\xi\)-derivatives is \(|\gamma|-|\beta|\). Base derivatives add the falling factorial \(b^{\underline\mu}\).

For fixed derivative indices, distinct original monomials remain distinct after the derivatives. Therefore (5.21b) is exactly a weighted \(\ell^1\)-norm on the original coefficients, with positive weights. Each monomial has finite weight for sufficiently small \(\epsilon<1\): the base and \(\xi\) derivative sums are finite on a monomial, and the remaining \(\tau\) derivative sum is a geometric series times a fixed polynomial in its derivative order, by the factorials in (5.21b). The resulting coefficient space is Banach. A coefficientwise support restriction is a closed subspace and has norm at most one.

Ordinary division of monomials by a fixed \(w^\eta\), restricted to terms divisible by it, has norm at most \(R^{-\eta}\). In fact it decreases each base falling factorial in (5.21f), each cotangent falling factorial and the nonnegative initial value \(n+|\gamma|\) of each rising factorial. Its radius factor changes by exactly \(R^{-\eta}\). This proves the bound term by term in the weighted \(\ell^1\)-norm, including both \(u\)- and \(z\)-variables. No formal division convergence is presumed.

Finally, let \(\partial_{\rho_j}\) act on the homogeneous series, moving its coefficient of index \(n\) to index \(n+1\). In (5.21b), compare its term to the input term with one extra \(\rho_j\)-derivative. The ratio of weights is
\[
 \frac\epsilon\lambda\frac{n+1}{n+|\mu|+1}
 \le\frac\epsilon\lambda.
\]
Consequently
\[
 \|\partial_\rho^\alpha P\|_{R,\epsilon}
          \le (\epsilon/\lambda)^{|\alpha|}
                      \|P\|_{R,\epsilon}.
 \tag{5.21g}
\]

#### 5.21.2. Initial exponents and the finite diagram

Let \(\mathcal A\) be the actual stalk and \(L\subset\mathcal A^s\) a left submodule, with \(s\) finite. A nonzero vector has a first nonzero homogeneous coefficient \(h^n\). Among that coefficient's nonzero \(w\)-Taylor monomials, choose the least total degree, then the least lexicographic exponent, and finally the least component index. Denote the resulting leading exponent by \((n,\gamma,j)\). This order is well ordered and compatible with addition of exponents; component comparison is used only at an identical exponent.

The set \(\Delta(L)_j\subset\mathbf N^{q+1}\) of leading exponents in component \(j\) is upward closed. Indeed multiplication on the left by \(h^m w^\beta\) adds \((m,\beta)\) to the leading exponent. Every nonzero internal derivative correction in the actual product has strictly larger homogeneous index \(n\), and the degree-zero ordinary Taylor product has the usual multiplicative leading monomial. Thus the leading exponent is exactly translated as stated.

Each upward closed subset of \(\mathbf N^k\) has finitely many minimal elements. Here is the combinatorial proof. Every infinite sequence in \(\mathbf N^k\) has a pair of terms in their sequence order with the first componentwise at most the second. Induct on \(k\). For the first coordinate, either some value occurs infinitely often, giving a constant subsequence, or its values are unbounded, giving a strictly increasing subsequence. Apply induction to the remaining coordinates of that subsequence. An infinite collection of incomparable minimal elements is therefore impossible. Every exponent in an upward closed set dominates a minimal one, by choosing a minimal element in the finite coordinate box below it. This proves finite generation of the upward closed set.

Choose vectors \(P_1,\ldots,P_\ell\in L\) representing all the finitely many minimal exponents \((n_i,\gamma_i,j_i)\). Scale each by its nonzero complex leading coefficient so that the leading coefficient is one. Their exponent orthants cover \(\Delta(L)\). The argument below proves they generate the actual submodule.

Remove the finite initial homogeneous shift on the left:
\[
 B_i=h^{-n_i}\circ P_i.
 \tag{5.21h}
\]
This is an actual order-zero vector even though the intervening symbol \(h^{-n_i}=\tau^{n_i}\) has positive order. Its product has only finitely many internal derivatives, since \(\tau^{n_i}\) is a polynomial. Explicitly, if \(r=n_i\) and \(p_n=0\) for \(n<r\), its normalized coefficient of index \(N\) is
\[
\begin{aligned}
(B_i)_N&=\sum_{a=0}^{\min(r,N)}\\ &\quad\binom ra\partial_t^a p_{N+r-a}.
\end{aligned}
\tag{5.21i}
\]
There are finitely many derivative orders \(a\le r\). Cauchy estimates on one smaller common domain and
\((N+r)!\le2^{N+r}N!r!\) prove the required factorial bounds for every coefficient. Its principal vector is exactly \(p_r\), so its principal leading Taylor monomial remains \(w^{\gamma_i}e_{j_i}\).

#### 5.21.3. One domain and radii for all the divisors

Choose positive rational weights \(v_1,\ldots,v_q\), as close to one as necessary, so that \(\gamma_i\) is the unique minimal weighted exponent among all the different Taylor monomials in the principal vector of every \(B_i\). Terms at that identical exponent may remain only in later components.

To justify this simultaneous choice, let \(K\) be the largest of the finitely many degrees \(|\gamma_i|\). Among exponents of degree at most \(K\), use a sufficiently small hierarchical perturbation \(v_j=1+\delta^j\) to make lexicographic comparison agree with weighted comparison. The finitely many relevant differences are preserved by sufficiently small positive rational \(\delta\). The least-degree/lexicographic definition of the heads then gives the desired strict inequalities at degree at most \(K\). Take the perturbation still smaller so that \((K+1)\min v_j>K\max v_j\); every larger-degree monomial has larger weight than every head. This also controls all infinitely many higher Taylor terms.

Give component \(j\) the positive norm factor \(c_j=\kappa^j\). Choose \(\kappa>0\) sufficiently small that, for every \(i\), the finitely many coefficients at the identical head exponent in later components have total weighted norm less than \(c_{j_i}/64\). There are no earlier-component terms at that exponent. Fix these factors. Set \(R_j=r^{v_j}\) for sufficiently small \(r>0\), inside the common coefficient domains. The other monomials in the principal vectors then have total weighted Taylor norm divided by \(c_{j_i}R^{\gamma_i}\) tending to zero. For completeness, this follows by domination by their convergent absolute Taylor series at one fixed larger polyradius: every exponent has a strictly positive weight gap, and removing the head radius power leaves a summable majorant for small \(r\). Thus one \(r\) works for the finite list.

Define the vector norm \(\|F\|_c=\sum_j c_j\|F_j\|_{R,\epsilon}\). Put
\[
 A_i=B_i-w^{\gamma_i}e_{j_i}.
 \tag{5.21j}
\]
The remaining positive-homogeneous-index coefficients of \(A_i\) have norms tending to zero as \(\epsilon\to0\); this follows by domination by their finite norm at an earlier \(\epsilon_0\), since their powers of \(\epsilon\) are at least two. The derivative terms of the principal coefficient similarly tend to zero, and the zero-derivative principal term has norm exactly twice its absolute Taylor norm. It follows that, after the above choices of \(c,R\), one can choose a sufficiently small common \(\epsilon\) making
\[
\begin{gathered}
\frac{\|A_i\|_c}{c_{j_i}R^{\gamma_i}}\\ \text{arbitrarily small up to}\\ \text{the already fixed}\\ \text{head-component tolerance.}
\end{gathered}
\tag{5.21k}
\]
In particular the ratio can be made less than \(1/8\) for every \(i\). All these choices concern a finite list of actual coefficient domains and divisors.

#### 5.21.4. Bounded simultaneous division

Order the finite orthants of the heads. Partition their union by assigning each exponent \((n,\beta,j)\) to the first head with \(j=j_i\), \(n\ge n_i\), \(\beta\ge\gamma_i\). Denote its assigned set by \(\Delta_i\), and their complement by \(\Delta_0\). For any coefficient vector \(F\), let \(D_iF\) be the terms in \(\Delta_i\), taken from component \(j_i\) and divided ordinarily by \(w^{\gamma_i}\). No \(h\)-shift is removed; all its homogeneous indices are at least \(n_i\). Let \(V_i\) be the closed coefficient-support subspace occupied by these divided monomials. Give \(V=\bigoplus_i V_i\) the norm
\[
 \|Q'\|_V=\sum_i c_{j_i}R^{\gamma_i}
                              \|Q'_i\|_{R,\epsilon}.
 \tag{5.21l}
\]
§5.21.1 proves that these are Banach spaces. Support restrictions and monomial division give
\[
\begin{gathered}
\|DF\|_V\leq\|F\|_c,\\ D=(D_1,\ldots,D_\ell).
\end{gathered}
\tag{5.21m}
\]
The estimate is for the sum, because the assigned supports are disjoint before division. Let
\(T_0Q'=\sum_i Q'_iw^{\gamma_i}e_{j_i}\), using ordinary multiplication. By the support assignment, \(DT_0\) is the identity of \(V\), and the kernel of \(D\) is exactly the coefficient vectors supported in \(\Delta_0\).

Actual multiplication differs from \(T_0\) by a small bounded operator. Define
\[
\begin{gathered}
TQ'=\sum_iQ'_i\circ B_i,\\ E=T-T_0.
\end{gathered}
\tag{5.21n}
\]
For the term \(Q'_i\circ A_i\), (5.21d) gives norm at most \(\|Q'_i\|\|A_i\|_c\). For the difference between actual and ordinary multiplication by the monomial \(w^{\gamma_i}=u^b z^\eta\), only finitely many internal derivatives of the right factor can occur, namely \(0<\alpha\le b\). Equations (5.21e) and (5.21g) give the bound
\[
\begin{gathered}
\|Q'_i\circ w^{\gamma_i}-Q'_iw^{\gamma_i}\|\\ {}\leq C_i(\epsilon,R)\|Q'_i\|,\\ C_i=\tfrac12\sum_{0<\alpha\leq b}\binom b\alpha\\ {}\cdot(\epsilon/\lambda)^{|\alpha|}\\ {}\cdot\|u^{b-\alpha}z^\eta\|_{R,\epsilon}.
\end{gathered}
\tag{5.21o}
\]
For fixed \(R\), this constant tends to zero as \(\epsilon\to0\); each monomial norm has a finite limit. Its component norm is multiplied by \(c_{j_i}\). Decrease the common \(\epsilon\) once more so that
\[
\begin{gathered}
\vartheta=\max_i\\ \frac{\|A_i\|_c+c_{j_i}C_i(\epsilon,R)}{c_{j_i}R^{\gamma_i}},\\ \vartheta<\tfrac12.
\end{gathered}
\tag{5.21p}
\]
This is possible by the explicit choices in §5.21.3. Then (5.21m)–(5.21p) show \(\|DE\|\le\vartheta\) on \(V\).

For any \(F\) with finite actual norm, solve
\[
\begin{gathered}
Q'=(1+DE)^{-1}DF\\ {}=\sum_{k\geq0}(-DE)^kDF.
\end{gathered}
\tag{5.21q}
\]
The sum converges in the specified Banach space, with norm at most \((1-\vartheta)^{-1}\|F\|_c\). Its coefficients are actual symbols on one common smaller polydisc by (5.21c) and completeness. Define
\(S=F-TQ'\). Equations (5.21n) and (5.21q) give \(DS=0\), so every coefficient of \(S\) is supported in \(\Delta_0\). This gives an actual finite division expression
\[
 F=\sum_i Q'_i\circ B_i+S.
 \tag{5.21r}
\]
No closure of the original submodule was required to construct the coefficients.

An arbitrary actual germ \(F\) need not be defined on the initially selected divisor domain. Repeat the finite domain/radius choices on the intersection with its own common coefficient domain. The same fixed heads and principal Taylor coefficients are used; the radii and norm parameter may be smaller. This gives actual division for every such germ, without requiring unrelated coefficient domains or a uniform domain for every germ of the stalk.

#### 5.21.5. Noetherianity of the actual stalk

**Theorem 5.21.1.** Every left submodule of \(\mathcal A^s\), for finite \(s\), is generated by finitely many actual germs. In particular the actual order-zero factorial-growth dummy stalk is left Noetherian.

**Proof.** Use the finite head vectors \(P_i\in L\) and the actual division (5.21r) for \(F\in L\). Every \(Q'_i\) has homogeneous indices at least \(n_i\). Put
\(Q_i=Q'_i\circ h^{-n_i}\); the right factor has no base derivatives, so this simply shifts its coefficient sequence down by \(n_i\). The resulting symbol is actual: the fixed shift preserves factorial growth by \((k+n_i)!\le2^{k+n_i}k!n_i!\), on the same common coefficient domain. Associativity of the finite homogeneous Leibniz identities gives
\[
 Q_i\circ P_i=Q'_i\circ(h^{-n_i}\circ P_i)
              =Q'_i\circ B_i.
\]
The temporarily positive-order polynomial \(h^{-n_i}\) has finite derivative order; all displayed endpoints are order-zero actual symbols. Thus (5.21r) becomes \(F=\sum_iQ_i\circ P_i+S\). The finite sum belongs to \(L\), so \(S\in L\). If \(S\ne0\), its leading exponent must belong to \(\Delta(L)\), by the definition of that set. But every coefficient of \(S\) lies outside its covering orthants, by \(DS=0\). This contradiction gives \(S=0\), and the \(P_i\) generate \(L\). Taking \(s=1\) proves left Noetherianity. \(\square\)

For every finite left stalk module, its submodules are finite: pull back a submodule along a finite free surjection and apply Theorem 5.21.1, then take its finite image. Every finite module also has a finite presentation because the kernel of its finite free surjection is finite. Thus the finite stalk-module category has finite kernels, images and cokernels. These are actual convergent algebra statements.

### 5.22. Exact faithful positive-degree completion at the dummy zero section

#### 5.22.1. The actual normal element

Write \(\mathcal A\) for the actual order-zero stalk, \(h=\tau^{-1}\), and \(I=(h,t,z)\). Let \(T=\mathcal O_{x,t,z,0}\), \(H=\mathcal O_{x,0}\). Right multiplication by \(h^r\) is ordinary normal-symbol multiplication, so \(\mathcal Ah^r\) consists exactly of the symbols whose first \(r\) homogeneous coefficients vanish. The reverse inclusion uses the fixed coefficient shift and \((n+r)!\le2^{n+r}n!r!\), preserving the actual common domain and factorial class.

The element \(h\) is normal. Direct actual multiplication gives
\[
\begin{gathered}
h^{-1}\circ P\circ h=P+h\partial_tP,\\ h\circ P\circ h^{-1}\\ {}=\sum_{j\geq0}(-1)^jh^j\partial_t^jP.
\end{gathered}
\tag{5.22a}
\]
The first product has only the zero and first derivatives of \(h^{-1}=\tau\). The second identity has finite sums at each homogeneous index. It is actual: on any smaller compact coefficient domain, Cauchy estimates bound its index-\(N\) coefficient by
\(A\sum_{n+j=N}C^n n!\,j!\eta^{-j}\le A(C+\eta^{-1})^NN!\).
The two operators in (5.22a) are inverse, by cancellation of their coefficientwise geometric series. Associativity of the homogeneous product makes them inverse algebra automorphisms. Denote \(P\mapsto h\circ P\circ h^{-1}\) by \(\sigma\). Thus
\[
\begin{gathered}
hP=\sigma(P)h,\\ (h)=\mathcal Ah=h\mathcal A,\\ \sigma(P)\equiv P\pmod h.
\end{gathered}
\tag{5.22b}
\]
Consequently the \(h\)-adic graded ring is the genuinely commutative ring \(T[h]\). Finite truncation of actual coefficients also gives
\[
 B:=\varprojlim_r\mathcal A/(h^r)
       =\prod_{n\ge0}Th^n
 \tag{5.22c}
\]
as coefficient sequences, with the actual homogeneous Leibniz product. This is completion of actual stalk quotients, not an interchange of a sheaf stalk and an inverse limit. The same automorphism, defined coefficientwise, makes \(h\) normal in \(B\).

#### 5.22.2. Normal-element Artin–Rees and exact completion

**Lemma 5.22.1.** Let \(R\) be left Noetherian and let \(h\) be a normal element with \(ha=\sigma(a)h\) for an algebra automorphism \(\sigma\). If \(N\subset M\) are finite left modules, there is \(c\) such that
\[
\begin{gathered}
N\cap h^rM\\ {}=h^{r-c}(N\cap h^cM),\\ r\geq c.
\end{gathered}
\tag{5.22d}
\]

**Proof.** Put \(N_r=\{m\in M:h^rm\in N\}\). These are left submodules because \(h^r(am)=\sigma^r(a)h^rm\), and they form an ascending sequence. The finite module \(M\) is Noetherian, so they stabilize at \(c\). Now \(N\cap h^rM=h^rN_r\), directly by writing an element in the image of multiplication by \(h^r\). For \(r\ge c\), this is \(h^rN_c=h^{r-c}(N\cap h^cM)\). No assumption about \(h\)-torsion of \(M\) is used. \(\square\)

**Lemma 5.22.2.** In the setting of Lemma 5.22.1, \(h\)-adic completion is exact on finite left modules, and is naturally \(\widehat R^{\,h}\otimes_R M\).

**Proof.** In a short exact sequence \(0\to N\to M\to P\to0\), use the exact finite quotients
\(0\to N/(N\cap h^rM)\to M/h^rM\to P/h^rP\to0\).
The transitions of the left terms are surjective. Their inverse limit is exact: injectivity and the middle kernel are coefficientwise; to lift a compatible element of the right limit, choose a lift at the first stage, then at each successive stage choose any lift and correct its discrepancy by a lift from the preceding left kernel. Surjectivity of the left transitions supplies that correction. The induced filtration on \(N\) is cofinal with its intrinsic \(h\)-filtration by (5.22d), so the left limit is \(\widehat N^{\,h}\). This proves exactness. A finite presentation of \(M\) exists by left Noetherianity. Its completed cokernel and its tensor cokernel are the same map of finite free \(\widehat R^{\,h}\)-modules, proving the natural tensor identity. \(\square\)

Apply these lemmas first to the actual \(\mathcal A\), using actual Noetherianity and (5.22b). Thus \(B\otimes_\mathcal A-\) is exact on finite left \(\mathcal A\)-modules. It detects zero objects, since \((h)\) lies in the actual Jacobson radical by Theorem 5.20.3, and the successive-lift/Nakayama proof of Corollary 5.20.3.1 applies unchanged.

The formal coefficient ring \(B\) is left Noetherian, and its finite modules are \(h\)-adically complete. Here is a proof without importing a completion theorem for operators. Its graded ring \(T[h]\) is Noetherian. For a submodule of a finite free \(B\)-module, choose finitely many lifts of homogeneous generators of its graded initial module. Successively cancel its first \(h\)-coefficient by those lifts. The coefficient series converge in the complete product (5.22c), and a finite sum of the resulting products equals the given element. The same iteration for an element of the closure proves the submodule closed. This is exactly the full coefficientwise argument of formal Lemma 5.23.1, now at the \(h\)-filtration. It proves left Noetherianity. Lemma 5.22.2 applied to \(B\) then says that every finite \(B\)-module is its own \(h\)-completion, since \(\widehat B^{\,h}=B\).

#### 5.22.3. The two formal rings and the coefficient comparison

Let
\[
 S=H[[t,z]],\qquad
 C=\widehat{\mathcal A}^{\ I}.
 \tag{5.22e}
\]
The actual ring-jet theorem 5.19.4 identifies \(C\) with the formal positive-degree Ore ring, and regrouping its degree coefficients gives
\[
 C=\prod_{n\ge0}S h^n
 \tag{5.22f}
\]
with the same homogeneous Leibniz product. In these coefficient-sequence identities, a written factor \(h^j\partial_t^jP\) means ordinary normal-symbol multiplication by \(h^j\); it does not mean the Leibniz product \(h^j\circ\partial_t^jP\). The maps in (5.22a) are defined by the displayed \(\circ\)-products, then expanded coefficientwise. Each formal coefficient in \(S\) has actual holomorphic \(x\)-germs, as required by the earlier degree completion; there is no additional completion in \(x\). Finite Taylor quotients show that \(S\) is exactly the \((t,z)\)-adic completion of \(T\). Thus commutative completion, Theorem 3.2, gives the faithfully flat coefficient map \(T\to S\).

The ring \(S\) is Noetherian by commutative completion, Theorem 3.3. The element \(h\) is normal in \(C\), by the same coefficient identities (5.22a). The ring is \(h\)-adically complete, with commutative graded ring \(S[h]\). The first-coefficient reduction and closed-submodule argument of §5.22.2 proves that \(C\) is left Noetherian. Lemma 5.22.2 therefore proves that every finite \(C\)-module is \(h\)-adically complete and separated. The coefficient map defines a compatible ring map \(B\to C\).

#### 5.22.4. Flatness at every finite normal-element quotient

**Lemma 5.22.3 (finite filtered flatness).** Let \(R\to D\) be a filtered ring map with exhaustive, separated, bounded decreasing multiplicative filtrations, and suppose \(\operatorname{gr}R\) is left Noetherian and \(\operatorname{gr}D\) is flat as a right \(\operatorname{gr}R\)-module. Then tensoring with \(D\) is exact on finite left \(R\)-modules.

**Proof.** Give a finite module its quotient filtration from a finite free module. Lift a finite homogeneous generating list of its graded module to a shifted filtered finite free module. Successive leading-filtration cancellation proves the resulting surjection is strict: the filtration has finitely many levels, so each reduction ends after finitely many steps. Its kernel has the induced bounded filtration, and its graded module is finite over the Noetherian graded ring. Repeat to construct a filtered free resolution with finite shifted free terms and strict differentials onto successive kernels. Its graded complex is a free resolution of the graded module.

Tensor this resolution with \(D\), giving each tensor free term the induced shifted filtration. Its graded term is precisely the tensor with \(\operatorname{gr}D\) of the corresponding graded free term; this is immediate on each free generator and its shift. Its graded complex is exact in positive degrees by graded flatness. The tensor complex itself is exact in positive degrees: for a cycle, solve its leading graded equation, lift that boundary and subtract, then solve the higher leading degree. At any fixed homological degree, the finite free terms have bounded filtrations, so this reduction terminates. In particular its first homology is zero. The degree-zero tensor cokernel is the ordinary tensor product by the tensor universal property. This proves vanishing of the first derived tensor obstruction for every finite module. Equivalently, apply the resolution argument to a finite module quotient in a short exact sequence to obtain the required tensor injection; right exactness always holds. Thus tensoring is exact on finite modules. \(\square\)

For each \(r\ge1\), apply the lemma to
\(R=B/(h^r)\), \(D=C/(h^r)\), filtered by their nilpotent powers of \(h\). Their graded rings are
\[
 T[h]/h^r,
 \qquad S[h]/h^r.
\]
The second is a flat right module over the first: its tensor functor is the coefficient-base extension \(S\otimes_T-\), with the same truncated \(h\)-action, and this is exact by coefficient flatness. Both filtrations have exactly \(r\) nonzero levels. Therefore \(C/(h^r)\otimes_{B/(h^r)}-\) is exact on finite modules. This conclusion is proved for the actual noncommutative quotients, not just their commutative graded rings.

#### 5.22.5. Lifting exactness from the quotients

**Proposition 5.22.4.** The functor \(C\otimes_B-\) is exact on finite left \(B\)-modules and detects zero objects.

**Proof.** Let \(N\subset M\) be finite \(B\)-modules, and set \(N_C=C\otimes_BN\), \(M_C=C\otimes_BM\). They are finite \(C\)-modules and hence \(h\)-adically complete and separated. Choose the constant \(c\) in (5.22d) for \(N\subset M\). For \(r\ge c\), the kernel of
\(N/h^rN\to M/h^rM\)
is \((N\cap h^rM)/h^rN\), and (5.22d) puts it inside \(h^{r-c}N/h^rN\).

If \(v\in N_C\) maps to zero in \(M_C\), its reduction modulo \(h^r\) belongs, by finite-quotient exactness of §5.22.4, to the tensor image of that kernel. This image is contained in \(h^{r-c}N_C/h^rN_C\). To verify the last containment with the normal element, rewrite \(d\otimes h^{r-c}n\) as \(dh^{r-c}\otimes n=h^{r-c}\sigma^{-(r-c)}(d)\otimes n\), using the compatible automorphism on \(C\). Thus \(v\in h^{r-c}N_C\) for every \(r\ge c\), and separatedness gives \(v=0\). Right exactness of tensoring now proves exactness on finite modules.

If \(C\otimes_BM=0\), reduction modulo \(h\) gives \(S\otimes_T(M/hM)=0\). Coefficient faithful flatness implies \(M/hM=0\). The ideal \((h)\) is in the Jacobson radical of the \(h\)-complete ring \(B\): for every \(a\), the geometric series for \(1-ah\) exists in its \(h\)-adic topology. The finite-generator elimination argument of Corollary 5.20.3.1 gives \(M=0\). This proves zero-object detection. \(\square\)

Combining §5.22.2 and Proposition 5.22.4 shows that \(C\otimes_\mathcal A-\) is exact and detects zero objects on finite actual stalk modules. This proves the noncommutative lifting; ordinary coefficient completion was only its base layer.

#### 5.22.6. Identification with actual positive-degree module completion

The degree formal ring \(C\) has \(C/I^rC\simeq\mathcal A/I^r\), by Theorem 5.19.4. Its degree-at-least-\(r\) part is exactly \(I^rC\): in each ordered monomial, put the \(x\)-coefficient on the left and split off an ordered right factor of degree \(r\). There are finitely many degree-\(r\) monomials in \(h,t,z\), and collecting the remaining coefficients gives formal elements of \(C\). The reverse inclusion follows from degree additivity. Thus this is the actual ideal-power quotient, not merely a notation for a formal filtration.

Every finite \(C\)-module is \(I\)-adically complete. For a finite free presentation \(C^s\to V\), formal Lemma 5.23.1, applied to the graded Ore ring of Lemma 5.23.4, proves that its kernel is closed in the degree topology. The quotient is separated. It is complete: choose successive lifts of a compatible sequence of quotient classes to the finite free module, correcting discrepancies by the surjective transitions of the kernel's induced finite quotients, as in Lemma 5.22.2. The free module is complete, so the resulting compatible lifts have a limit. The quotient filtration is precisely \(I^rV\). This proves the assertion without an assumed \(I\)-Artin–Rees theorem.

For finite actual \(M\), put \(M_C=C\otimes_\mathcal A M\). It is finite over \(C\), and
\[
 M_C/I^rM_C
 \simeq(C/I^rC)\otimes_\mathcal A M
 \simeq M/I^rM.
\]
Taking the compatible inverse limits and using its just-proved completeness yields
\[
 \widehat M^{\ I}\simeq C\otimes_\mathcal A M.
 \tag{5.22g}
\]
Consequently actual positive-degree completion is exact and detects zero objects on finite left actual stalk modules. An exact additive functor with this zero-detection property is faithful: if its image of a morphism is zero, its exactness identifies the image of the morphism with a zero completed object, so that image was zero. This supplies exact and faithful completion on finite actual stalk modules at the dummy zero section.

### 5.23. Formal Euler projection and homogeneous ordinary generators

#### 5.23.1. The positively graded ordinary operator ring

Let \(H=\mathcal O_{X,x}\), let \(D=\mathcal D_{X,x}\), and work with left modules. Right modules are obtained by the already proved density equivalence; no change of signs is needed inside the following left-module calculation. Write
\[
\begin{gathered}
R=\bigoplus_{a\geq0}D_{\leq a}h^a,\\ R_0=H,\quad\theta_i=h\partial_{x_i}.
\end{gathered}
\tag{5.23a}
\]
Here \(h\) is central, \([\theta_i,b]=h\partial_i b\), and \(h,\theta_i\) have internal degree one. Thus \(R_a\) is finite free over \(H\), with basis \(h^{a-|\alpha|}\theta^\alpha\), \(|\alpha|\leq a\). The graded ring is generated in degrees zero and one.

The ring \(R\) is left Noetherian. Here is the ungraded stalk argument, so that graded coherence is not incorrectly substituted for this assertion. Filter by the number of factors \(\theta_i\), giving \(h\) auxiliary order zero. PBW and the displayed commutator give the commutative auxiliary symbol ring \(H[h,\xi_1,\ldots,\xi_d]\). It is Noetherian by the earlier GL06 proof of polynomial Noetherianity, the complete leading-coefficient argument in GL06, over the Noetherian analytic local ring \(H\). For a left submodule of a finite free \(R\)-module, take its auxiliary leading-symbol module and choose finitely many homogeneous generators. Lift them to actual elements of the submodule. Match the leading symbol of any element by these lifts and subtract; the auxiliary order strictly decreases, so the process terminates after finitely many steps. The lifted elements generate the original submodule. Applying this to left ideals proves the asserted Noetherianity, and applying it to finite free modules proves the version used below. This argument uses polynomials in \(h\), not a microlocal completion.

The uniform graded analytic Rees coherence used here is proved in [§5.12](#graded-analytic-rees-coherence-with-a-finite-termination-argument). Its auxiliary \(h\)-adic symbol argument terminates at each fixed internal degree and retains both shifts \(a_i,v_i\) on every lifted free generator. The larger actual ring requires the independent arguments in §§5.19–5.22 and 5.24.

For a finitely generated graded \(R\)-module
\[
\begin{gathered}
Q=\bigoplus_{a\geq0}Q_a,\\ \widehat Q=\prod_{a\geq0}Q_a,\\ \widehat R=\prod_{a\geq0}R_a.
\end{gathered}
\tag{5.23b}
\]
products mean formal products. Each product coefficient is a finite sum, since all degrees are nonnegative. Put \(V^r\widehat Q=\prod_{a\geq r}Q_a\). An infinite sum is used only when its summands have degrees tending to infinity. It then converges in this formal topology. In particular, nothing in (5.23b) asserts analytic or microdifferential convergence.

The Euler operators are
\[
\begin{gathered}
\delta\left(\sum_a r_a\right)=\sum_a ar_a,\\ \delta_Q\left(\sum_a q_a\right)=\sum_a aq_a.
\end{gathered}
\tag{5.23c}
\]
The first is a continuous derivation and the second satisfies
\(\delta_Q(rq)=\delta(r)q+r\delta_Q(q)\). These identities follow coefficient by coefficient from addition of the internal degrees.

#### 5.23.2. Finite generation and closedness in the formal product

**Lemma 5.23.1.** Every \(\widehat R\)-submodule \(L\subset\widehat Q\) is generated by finitely many elements and is closed for the topology \(V^\bullet\widehat Q\).

**Proof.** For nonzero \(q\in\widehat Q\), let \(v(q)\) be its smallest nonzero degree. Define
\[
\begin{gathered}
\operatorname{in}(L)_a\\ {}=\{q_a:q\in L\cap V^a\widehat Q\}\\ {}\subset Q_a.
\end{gathered}
\tag{5.23d}
\]
The direct sum of these submodules is a graded \(R\)-submodule of \(Q\). To verify stability, multiply a representative of degree at least \(a\) by a homogeneous element of \(R_b\); its degree \(a+b\) coefficient is exactly the product of the two indicated coefficients. Since \(R\) is Noetherian and \(Q\) is finite, choose finitely many homogeneous generators of \(\operatorname{in}(L)\), say \(s_i\in Q_{a_i}\). Each is, by definition, the first coefficient of an element \(\ell_i\in L\cap V^{a_i}\widehat Q\).

Given \(q\in L\) with first degree \(a\), express its first coefficient as
\(q_a=\sum_i r_{i,a-a_i}s_i\), where \(r_{i,a-a_i}\in R_{a-a_i}\), and negative-degree terms are zero. Subtract \(\sum_i r_{i,a-a_i}\ell_i\). The result is zero or has strictly larger first degree and remains in \(L\). Repeat. At every fixed degree there are only finitely many corrections. Consequently the sums \(r_i=\sum_{b\geq0}r_{i,b}\) belong to \(\widehat R\), and completeness of the ambient product gives
\[
 q=\sum_i r_i\ell_i.
 \tag{5.23e}
\]
This is a finite sum of products; it belongs to \(L\) by the submodule property. It proves finite generation without requiring \(L\) to have been known closed.

If instead \(q\) lies in the closure of \(L\), any nonzero first coefficient of \(q\) belongs to \(\operatorname{in}(L)\): choose an element of \(L\) with the same finite truncation through that degree. After a correction by the chosen \(\ell_i\), the remainder still lies in the closure. The identical iteration proves (5.23e) for \(q\), so \(q\in L\). This proves closedness. The zero case is immediate. \(\square\)

This proof also proves that \(\widehat R\) is left Noetherian, by taking \(Q=R\). It does not invoke an Artin–Rees theorem for a noncommutative completion. Notice that no norm estimate or bound on the coefficients \(r_{i,b}\) was used; that distinction is resolved by the actual comparison and recovery in §§5.25–5.26.

#### 5.23.3. Euler stability gives actual homogeneous generators formally

**Lemma 5.23.2.** Suppose \(L\subset\widehat Q\) is a \(\widehat R\)-submodule and \(\delta_Q(L)\subset L\). Then
\[
\begin{gathered}
L=\prod_{a\geq0}L_a,\\ L_a=L\cap Q_a.
\end{gathered}
\tag{5.23f}
\]
and \(L\) is generated over \(\widehat R\) by finitely many homogeneous elements of the direct sum \(\bigoplus_aL_a\). Moreover that direct sum is a finitely generated graded \(R\)-submodule of \(Q\).

**Proof.** Fix \(q=\sum_aq_a\in L\), a degree \(j\geq0\), and \(r\geq j\). The scalar polynomial
\[
 P_{r,j}(z)=\prod_{\substack{0\leq k\leq r\\k\ne j}}
                      \frac{z-k}{j-k}
 \tag{5.23g}
\]
has value one at \(j\) and zero at every other integer from zero to \(r\). Its denominators are nonzero integers. Stability under \(\delta_Q\) implies \(P_{r,j}(\delta_Q)q\in L\). Its coefficient through degree \(r\) is exactly \(q_j\). Hence these elements converge formally to \(q_j\), and Lemma 5.23.1 gives \(q_j\in L\). Conversely, if each \(q_a\in L_a\), the finite partial sums belong to \(L\) and converge to \(\sum_aq_a\); closedness gives (5.23f).

Now \(\bigoplus_aL_a\subset Q\) is a graded \(R\)-submodule and is finitely generated because \(R\) is Noetherian. Choose homogeneous generators \(g_i\in L_{a_i}\). For every \(a\), generation in the graded module gives
\[
 L_a=\sum_i R_{a-a_i}g_i.
 \tag{5.23h}
\]
Applying these finite expressions separately in each degree, and collecting their coefficients into formal series, proves that the same \(g_i\) generate the product in (5.23f) over \(\widehat R\). Conversely every such product has its coefficients in the right side of (5.23h). \(\square\)

Thus the formal homogeneous projection is not asserted to be an element of the ring: it is obtained as a limit of ordinary scalar polynomials in the Euler action, inside a separately proved closed submodule. This removes a possible circular use of a convergent exponential \(e^{z\delta_Q}\) or an unproved circle average.

#### 5.23.4. The precise ordinary filtration obtained at a stalk

Choose an actual local good filtration \(F_aM\) of a coherent analytic \(D\)-module \(M\), bounded below, and shift it so that \(F_aM=0\) for \(a<0\). Its existence and coherent order pieces are the earlier ordinary analytic operator-presentation theorem; this is a local filtration and is not the global generator sought. Let
\[
 Q=\bigoplus_{a\geq0}h^aF_aM.
 \tag{5.23i}
\]
The formal product embeds coefficientwise into \(M[[h]]\); this notation uses only the underlying complex vector space of \(M\). For a bounded shift of \(\widehat Q\), an element has finitely many possible negative degrees. Extend the Euler action by \(\delta_Q(h^a m)=a h^a m\).

**Theorem 5.23.3 (formal ordinary descent).** Let \(L\subset h^{-B}\widehat Q\) be a \(\widehat R\)-submodule, where \(B\geq0\), stable under \(\delta_Q\), and assume
\[
\begin{gathered}
L[h^{-1}]=\widehat Q[h^{-1}]\\ \text{inside }M((h)).
\end{gathered}
\tag{5.23j}
\]
Here \(M((h))\) permits finitely many negative powers and arbitrary nonnegative formal coefficients; the displayed equality concerns these two particular submodules. Put
\[
 G_a=\{m\in M:h^a m\in L\}.
 \tag{5.23k}
\]
Then \((G_a)\) is a bounded-below exhaustive good \(D\)-filtration at the stalk, and \(L=\prod_a h^aG_a\). In particular every \(G_a\) is a finite \(H\)-module. There exist finitely many \(m_i\in G_{a_i}\) such that
\[
\begin{gathered}
G_a=\sum_iD_{\leq a-a_i}m_i,\\ D_{\leq b}=0\quad(b<0).
\end{gathered}
\tag{5.23l}
\]

**Proof.** Multiply by \(h^B\). This identifies the ambient shifted module with \(\widehat Q\), and changes the Euler action by the constant \(B\). Stability is therefore preserved. Lemma 5.23.2 applies, and translating its conclusion back gives finitely many homogeneous generators \(h^{a_i}m_i\) of \(L\). Each homogeneous coefficient lies in \(F_{a+B}M\), so \(G_a\subset F_{a+B}M\) and is zero for \(a<-B\). It is an \(H\)-submodule of a finite \(H\)-module and is finite, since the analytic local ring \(H\) is Noetherian. The homogeneous action formula in (5.23a) says precisely that (5.23h) is (5.23l): powers of \(h\) fill the difference between the differential order of an operator and its internal degree. This also proves \(G_a\subset G_{a+1}\) and \(D_{\leq b}G_a\subset G_{a+b}\).

For \(m\in M\), choose \(a\geq0\) with \(m\in F_aM\). Then \(h^am\in\widehat Q\). Equality (5.23j), in the localization by the central non-zero-divisor \(h\), gives an integer \(b\geq0\) with \(h^{a+b}m\in L\). Thus \(m\in G_{a+b}\), proving exhaustion. Finally (5.23l) is a finite graded generating presentation of the ordinary Rees module. It proves goodness, and the product assertion is (5.23f) with the same shift. \(\square\)

The statement is deliberately a stalk statement. Finite generation of every stalk of an arbitrary subsheaf is insufficient to deduce analytic sheaf coherence. In particular, Theorem 5.23.3 by itself does not prove that the proposed global \(F_kM=j_0^{-1}N_{<0}(k)\cap M\) is coherent on any neighbourhood. It first constructs finite ordinary generators of the completed lattice at each point.

#### 5.23.5. Dummy-variable signs and the algebraic half of descent

On the actual dummy transfer \(\mathcal B=\mathcal E_{\mathbf C_t\times X}/\mathcal E_{\mathbf C_t\times X}t\), write \(u_0\) for the image of one and \(h=D_t^{-1}\). The following calculations require only \([D_t,t]=1\) and the existence of \(D_t^{-1}\) on \(\tau\ne0\). For every integer \(a\),
\[
\begin{gathered}
th^au_0=ah^{a+1}u_0,\\ (tD_t+1)h^au_0=ah^au_0.
\end{gathered}
\tag{5.23m}
\]
For \(a\geq0\), the first follows by induction from \([t,D_t^{-1}]=D_t^{-2}\); for negative \(a\), use \([t,D_t^b]=-bD_t^{b-1}\). The second follows by applying the first with \(a-1\). Coefficients and ordinary \(D_X\)-operators commute with \(t,D_t\). Consequently the action of \(tD_t+1\) on the formal dummy expansion is exactly the Euler action (5.23c), including on negative degrees.

For any supplied lattice \(L\) with \(tL\subset hL\), the operator \(h^{-1}t=D_tt=tD_t+1\) preserves \(L\), since \(h^{-1}(hL)=L\). This obtains the Euler stability required by Theorem 5.23.3. The order-shift convention is \(L(-1)=hL\). This calculation supplies stability of an existing lattice and does not construct a canonical cutoff.

Theorem 5.23.3 supplies finite ordinary homogeneous generators of the completed lattice once the actual bounded ambient comparison and localized equality in §5.25 are used. Recovering equality of the actual convergent sheaves is the additional exact-faithful two-quotient argument in §5.26.

Here is also the formal elimination of the \(t\)-dependent coefficients; it is not delegated to an unproved normal form.

**Lemma 5.23.4 (formal dummy Ore presentation).** Let \(\Delta=h\delta\) on \(R\), and define the graded ring
\[
\begin{gathered}
A=R[t;\Delta],\\ tr=rt+\Delta(r),\quad\deg t=1.
\end{gathered}
\tag{5.23n}
\]
Its underlying left \(R\)-module is \(\bigoplus_{j\geq0}Rt^j\). It is Noetherian on the left. The module \(Q\) in (5.23i), with action \(tq=h\delta_Q(q)\), has a finite graded \(A\)-presentation. Its degree completion is exactly \(\widehat Q\), with the stated action of \(t\). If \(q_i=h^{a_i}m_i\) are finite homogeneous \(R\)-generators, its \(A\)-presentation is given by their finite \(R\)-relations and
\[
\begin{gathered}
(t-a_ih)q_i=0,\\ t^jq_i=\\ a_i(a_i+1)\cdots(a_i+j-1)\\ \cdot h^jq_i.
\end{gathered}
\tag{5.23o}
\]
The empty product for \(j=0\) is one.

**Proof.** The derivation identity for \(\delta\), centrality of \(h\) in \(R\), and (5.23c) show that \(\Delta\) is a derivation of \(R\), raising degree by one. The identity
\[
 t^jr=\sum_{k=0}^j\binom jk\Delta^k(r)t^{j-k}
 \tag{5.23p}
\]
defines multiplication in the displayed free \(R\)-module. Iterated Leibniz and the binomial coefficient identity verify associativity: in a triple product, apply
\(\Delta^n(rs)=\sum_{k=0}^n\binom nk\Delta^k(r)\Delta^{n-k}(s)\) and collect each power of \(t\); the coefficients agree. The case \(j=1\) gives the required relation.

Filter \(A\) by the number of factors \(t\). Its symbol ring is the central polynomial ring \(R[T]\). This is left Noetherian: the leading coefficients of a left ideal form a left ideal of \(R\); choose finite generators represented by polynomials, cancel all sufficiently high leading degrees, and use finite generation of the remaining bounded-degree coefficient submodule of a finite free left \(R\)-module. This is exactly the finite leading-coefficient proof, which needs only left Noetherianity, not commutativity of \(R\). Leading-order reduction now proves that \(A\) is left Noetherian, just as in §5.23.1.

The proposed \(t\)-action satisfies the Ore relation since
\[
 h\delta_Q(rq)=\Delta(r)q+r\,h\delta_Q(q).
\]
Choose finite homogeneous generators of \(Q\) over \(R\). Their \(R\)-relations are generated by finitely many homogeneous relations by Noetherianity. Add the first relation in (5.23o). It reduces every \(A\)-linear expression in the generators to an \(R\)-linear expression: induction using \([t,h]=h^2\) proves the second formula in (5.23o). The quotient by these relations therefore maps onto \(Q\); the ordinary \(R\)-presentation maps back to it. The maps are inverse on generators, and the same formula verifies the \(t\)-action. This proves the finite \(A\)-presentation.

Complete by total degree. All reductions in (5.23o) preserve that degree, and only finitely many monomials \(h,\theta,t\) contribute to any one coefficient. Their infinite sums are formal degree sums. Coefficientwise reduction gives exactly \(\widehat Q\). Equivalently, the kernels in this finite graded presentation are finite graded modules; collecting their homogeneous expressions shows that their formal products are the actual images of the completed finite presentation. Thus no completion flatness for a larger analytic ring is assumed. \(\square\)

The factorial coefficient in (5.23o) explains why this formal calculation cannot assert convergent coefficient estimates without proof. The actual ambient comparison in §5.25 identifies the completion of a bounded dummy lattice with this formal model. Its image is automatically a \(\widehat R\)-submodule. Lemma 5.23.1 supplies its finiteness and closedness; an independent assertion of finiteness over the convergent \(t\)-independent subalgebra is unnecessary.

### 5.24. Fixed kernel sections on one actual neighbourhood

The full coefficient-kernel input is GL06, Lemma 3.0c.3. Its proof represents finitely many homogeneous relations, uses the relative commutative coefficient-flatness calculation at the analytic zero section, kills their coherent analytic cokernel on one product neighbourhood by Oka coherence, and then descends its vanishing at every nearby zero-section stalk by faithful flatness and graded zero detection. In the present case there is one polynomial variable \(h\) and the holomorphic base variables \(w=(t,x,z)\); shifted free bases are retained. No operator completion is substituted for that coefficient calculation.

#### 5.24.1. The filtration used here

Write \(\mathscr A\) for the actual factorial-growth order-zero sheaf on the chart \(\tau\ne0\), \(h=\tau^{-1}\), and \(w=(t,x,z)\). Its decreasing filtration for this argument is the homogeneous-index filtration
\[
\begin{gathered}
H^r\mathscr A=\mathscr Ah^r\quad(r\geq0),\\ \operatorname{gr}_H\mathscr A=\mathcal O_w[h].
\end{gathered}
\tag{5.24a}
\]
Right multiplication by \(h^r\) just shifts coefficients. The reverse description in (5.24a) follows from the fixed-shift factorial estimate in Theorem 5.19.4. Every nonzero Leibniz correction raises homogeneous index, so the associated graded multiplication is ordinary multiplication of holomorphic coefficients. This filtration is not the positive-degree ideal filtration in \(h,t,z\).

The bounded division at a stalk works at every point of this chart. To reduce a point with normalized covector \(z_0\ne0\) to the coordinates used in its proof, make the linear base change
\[
 t'=t+\sum_i(z_0)_i x_i,\qquad x'_i=x_i.
\]
The cotangent coordinates are then \(\tau'=\tau\), \(\xi'_i=\xi_i-(z_0)_i\tau\), so \(z'=z-z_0\) and \(h'=h\). Translate the base coordinates to their values at the point. The chain rule preserves the bilinear contraction \(\sum\partial_{\rho_i}\otimes\partial_{u_i}\), hence each finite homogeneous coefficient of the Leibniz product. The coefficient functions retain a common neighbourhood and the factorial bound after this fixed linear change. Thus 5.21a–5.21r apply locally in these coordinates, with the same homogeneous-index filtration. This is a local coordinate change inside the chart, not an assertion about the global dummy hypersurface.

#### 5.24.2. An additional consequence of the actual division

**Lemma 5.24.1 (strict image generators at one stalk).** For a left submodule \(J\subset\mathscr A_p^b\), the standard generators \(P_i\) selected in Theorem 5.21.1 can be chosen with homogeneous orders \(n_i\ge0\) so that every \(F\in J\cap H^r\mathscr A_p^b\) has an actual expression
\[
\begin{gathered}
F=\sum_iQ_i\circ P_i,\\ Q_i\in H^{\max(0,r-n_i)}\mathscr A_p.
\end{gathered}
\tag{5.24b}
\]
Consequently their principal vectors generate \(\operatorname{gr}_H J\) over \(\mathcal O_{w,p}[h]\), retaining the shifts \(n_i\).

**Proof.** Use precisely the division operator of 5.21l–5.21r. Its support projections and ordinary monomial divisions do not change homogeneous index. Its operator \(DE\) does not lower that index: every normalized divisor \(B_i=h^{-n_i}\circ P_i\) is order zero, ordinary multiplication preserves index, and actual derivative corrections raise it. Thus the closed subspace of quotient vectors whose coefficients below index \(r\) vanish is invariant under \(DE\). The Neumann series in 5.21q stays in it whenever its input does. Its assigned supports also require the \(i\)-th quotient to have no index below \(n_i\). Therefore \(Q'_i\) there has no index below \(\max(r,n_i)\). Its right shift \(Q_i=Q'_i\circ h^{-n_i}\), exactly as in the proof of Theorem 5.21.1, is an actual symbol of order at least \(\max(0,r-n_i)\). The remainder is zero by that proof. This proves (5.24b). Taking its first homogeneous coefficient proves the assertion about the graded image. Conversely the principal vectors of the \(P_i\) belong to that graded image by definition. \(\square\)

All convergence in this lemma is the actual Banach convergence from 5.21q, on one common domain for each division problem. An arbitrary infinite homogeneous cancellation was not used.

#### 5.24.3. Lift finitely many principal relations

Let \(\phi:\mathscr A^a\to\mathscr A^b\) be any finite matrix of actual sections near a point \(p\). A left-module matrix acts as \(\phi(q)=\sum_i q_i\circ C_i\), where \(C_i\) are its columns. At \(p\), apply Lemma 5.24.1 to its image and obtain \(P_1,\ldots,P_s\), with orders \(n_i\).

Each \(P_i\) is a finite actual combination of the original columns, and each original column is a finite combination of the \(P_i\). Extend this finite list and its finite identities to one neighbourhood. If \(\psi:\mathscr A^s\to\mathscr A^b\) has columns \(P_i\), there are matrices \(u,v\) there with
\[
 \psi u=\phi,
 \qquad \phi v=\psi.
 \tag{5.24c}
\]
They show that the represented columns generate the image at every nearby stalk.

Give the source of \(\psi\) the filtration shift \(n_i\) on its \(i\)-th generator. Its principal matrix is a graded map
\[
\begin{gathered}
g:\bigoplus_i\mathcal O_w[h]e_i\longrightarrow\mathcal O_w[h]^b,\\ \deg e_i=n_i,\\ g(e_i)=h^{n_i}p_i(w).
\end{gathered}
\tag{5.24d}
\]
where \(p_i\) is the principal holomorphic vector of \(P_i\). The coefficient polynomial-kernel theorem gives finitely many homogeneous generators \(\nu_1,\ldots,\nu_m\) of \(\ker g_p\). They can be represented so that the same sections generate \(\ker g_q\) for every point \(q\) of one smaller neighbourhood.

Each \(\nu_l\), of degree \(r_l\), has an actual kernel lift \(K_l\) of source order \(r_l\). Indeed replace its coefficient of \(e_i\) by the actual finite symbol \(a_{li}(w)h^{r_l-n_i}\), omitting a component when \(r_l<n_i\). Its image under \(\psi_p\) has homogeneous order at least \(r_l+1\). Lemma 5.24.1 gives an actual expression for that image with coefficient orders at least \(\max(0,r_l+1-n_i)\). Subtract that coefficient vector from the proposed lift. The difference is an actual relation and has exactly principal vector \(\nu_l\). These finitely many exact equations extend to one smaller neighbourhood.

This uses strictness only at the original stalk. The following argument proves both relation generation and strictness nearby; it does not presume either one.

#### 5.24.4. Why the lifted relations generate at all nearby points

The shifts can be handled inside an ordinary finite free module, without applying the division norm to negative coefficient indices. Define
\[
\begin{gathered}
\mathscr J=\bigoplus_i\mathscr Ah^{n_i}f_i\subset\mathscr A^s,\\ \iota(q)_i=q_i\circ h^{n_i}.
\end{gathered}
\tag{5.24e}
\]
The map \(\iota\) is a left-module isomorphism onto \(\mathscr J\), since it multiplies on the right. Its inverse on this image is the actual finite coefficient shift; no negative indices remain. The source's shifted order becomes ordinary homogeneous order in \(\mathscr J\).

Put \(B_i=h^{-n_i}\circ P_i\). These are actual order-zero vectors by 5.21h–5.21i. The matrix
\[
\begin{gathered}
\bar\psi:\mathscr A^s\to\mathscr A^b,\\ \bar\psi(F)=\sum_iF_i\circ B_i.
\end{gathered}
\tag{5.24f}
\]
satisfies \(\bar\psi\iota=\psi\). Its principal matrix on \(\operatorname{gr}_H\mathscr J\) is precisely (5.24d), under the embedding \(e_i\mapsto h^{n_i}f_i\). The represented \(\iota(K_l)\) are actual relations in \(\mathscr J\); their principal vectors generate that principal kernel at every nearby point.

Fix such a point \(q\). Let \(\mu_q\) denote this graded principal kernel inside \(\bigoplus_i h^{n_i}\mathcal O_{w,q}[h]f_i\). Its homogeneous vectors have leading exponents obtained by first comparing the power of \(h\), then the total degree and lexicographic exponent of the holomorphic Taylor monomial, then the component. Their diagram is upward closed and has finitely many minimal exponents, by the Dickson argument proved in the division leaf. Choose homogeneous vectors of \(\mu_q\) realizing these minimal exponents. Since the represented principal relations generate \(\mu_q\), each chosen vector is a finite homogeneous polynomial combination of them. Lift those combinations to actual finite-symbol combinations of \(\iota(K_l)\), obtaining \(T_1,\ldots,T_k\in\mathscr J_q\cap\ker\bar\psi_q\). Their leading exponents are the chosen ones; their higher homogeneous coefficients may be arbitrary actual coefficients.

Apply the bounded actual division 5.21l–5.21r to these finite divisors in the unshifted module \(\mathscr A_q^s\). For \(F\in\mathscr J_q\cap\ker\bar\psi_q\), it gives
\[
 F=\sum_j A_j\circ T_j+S,
 \tag{5.24g}
\]
where the actual remainder's coefficient support avoids their leading orthants. Since \(\mathscr J_q\) is a left submodule and all \(T_j\) are actual relations, \(S\) still belongs to \(\mathscr J_q\cap\ker\bar\psi_q\). If it were nonzero, its first homogeneous coefficient would belong to \(\mu_q\). Its leading exponent would consequently lie in one of the chosen orthants, contradicting the remainder condition. Hence \(S=0\).

Each \(T_j\) was a finite combination of the represented \(\iota(K_l)\). Applying \(\iota^{-1}\) proves that the same finitely many \(K_l\) generate \(\ker\psi_q\) for every nearby \(q\). The division domain and norms may depend on the input germ at \(q\); the generating sections and the neighbourhood do not. This distinction is what the stalk-only proof had not supplied.

The same argument also proves nearby strictness. For any \(F\in\mathscr J_q\) whose image has homogeneous order at least \(r\), perform (5.24g). If the nonzero remainder had first homogeneous degree less than \(r\), its principal vector would lie in \(\mu_q\), again contradicting its support. Thus the remainder lies in \(H^r\mathscr J_q\), while having the same image as \(F\). Its inverse under \(\iota\) has source orders at least \(\max(0,r-n_i)\). This proves strictness onto the image nearby as well.

#### 5.24.5. Coherence of the actual sheaf

**Theorem 5.24.2 (actual neighbourhood coherence).** By the actual bounded division proved in §5.21, the sheaf \(\mathscr A\) is left coherent. Kernels of finite actual matrices are generated by finitely many actual sections on one neighbourhood. Finite presented left modules form an abelian coherent category; finite-type submodules of coherent modules are coherent.

**Proof.** §5.24.4 gives a finite neighbourhood kernel list for \(\psi\). From (5.24c), for every \(z\in\ker\phi\),
\[
 z=v\,u(z)+(1-vu)z.
\]
The first term is generated by the finite sections \(v(K_l)\), because \(u(z)\in\ker\psi\). The columns of \(1-vu\) lie in \(\ker\phi\), since \(\phi(1-vu)=\phi-\psi u=0\); they generate the second term. Thus the finite list \(v(K_l)\) and these columns generates \(\ker\phi\) on the common neighbourhood. This proves ring-sheaf coherence.

For completeness, the extension to presented modules uses finite block matrices. The kernel of a map between two finite presentations is the quotient of the finite-type kernel of the matrix imposing its target relation, with the source relation added. Coherence of the finite-free matrix kernels makes that quotient finite presented. Cokernels are directly given by adjoining the two finite relation lists; images are the quotient of the source by its coherent kernel. A finite-type submodule is the image of a finite free module into a coherent module and is therefore coherent. These constructions give the asserted abelian category on the same local neighbourhoods. They use no commutativity of \(\mathscr A\). \(\square\)

No stalk/inverse-limit interchange, formal convergence substitution, ordinary operator-completion theorem, generic smoothness of a characteristic set, or infinite sequence of neighbourhood shrinkings enters this argument. Singular principal-symbol loci are included: the only coefficient kernel theorem is Oka's theorem for an arbitrary finite holomorphic matrix.

### 5.25. General bounded Rees ambient modules and actual lattice bounds

#### 5.25.1. The finite actual presentation

Let \(M\) be a coherent analytic left \(\mathcal D_X\)-module. Choose a local ordinary good filtration, shift it to have \(F_aM=0\) for \(a<0\), and write at the ordinary point \(x\)
\[
\begin{gathered}
R=\bigoplus_{a\geq0}h^aD_{\leq a},\\ Q=\bigoplus_{a\geq0}h^aF_aM,\\ A=R[t;h\delta].
\end{gathered}
\tag{5.25a}
\]
Use the same objects as sheaves over one ordinary coordinate neighbourhood. Uniform graded Rees coherence gives a finite homogeneous \(R\)-presentation of \(Q\) there. Choose its generators \(q_i=h^{a_i}m_i\), with \(a_i\ge0\). Formal Lemma 5.23.4 gives the finite homogeneous \(A\)-presentation consisting of those ordinary Rees relations and
\[
 (t-a_i h)q_i=0.
 \tag{5.25b}
\]

Map \(A\) into the actual order-zero chart ring \(\mathscr A\) by the actual symbols \(h,\theta_i=z_i,t\) and coefficient functions. Their Ore relations and injective finite normal form are Proposition 5.19.2. Define the actual module by that finite presentation, equivalently
\[
 V=\mathscr A\otimes_A Q.
 \tag{5.25c}
\]
Here the algebraic presentation coefficients are represented by finite actual symbols on one common neighbourhood. Theorem 5.24.2 makes \(V\) a coherent actual module there. No operator flatness of \(A\to\mathscr A\) is being presumed in this definition.

**Lemma 5.25.1.** At \(j_0(x)=(0,x;1,0)\), its actual positive-degree completion has the natural coefficientwise model
\[
\begin{gathered}
\widehat V^{\,I}\simeq\prod_{a\geq0}h^aF_aM=\widehat Q,\\ I=(h,t,\theta_1,\ldots,\theta_d).
\end{gathered}
\tag{5.25d}
\]
The distinguished generator \(q_i\) maps to \(h^{a_i}m_i\), and the completed action is the Ore action of Lemma 5.23.4.

**Proof.** Write \(C=\widehat{\mathscr A_{j_0(x)}}^{\ I}=\widehat A\), as proved by the actual ring-jet theorem 5.19.4. The finite-module completion theorem gives
\[
 \widehat V^{\ I}=C\otimes_{\mathscr A}V
                  =C\otimes_A Q.
\]
This is the cokernel of the completed finite homogeneous presentation (5.25b). The coefficientwise finite-degree reduction in Lemma 5.23.4 identifies that cokernel exactly with the product \(\widehat Q\). Every finite-degree coefficient is an actual holomorphic germ; the product has no additional completion in \(x\). Its map into the vector-space coefficient product \(M[[h]]\) is injective since each \(F_aM\) is an actual submodule of \(M\). Thus (5.25d) is an actual completed-presentation calculation, including arbitrary ordinary relations, rather than only the free-module model. \(\square\)

#### 5.25.2. Unit maps and the actual finite-order localization

**Lemma 5.25.2.** If an extension functor \(C\otimes_S-\) is exact and detects zero objects on finite left modules over a left Noetherian ring \(S\), the unit map \(T\to C\otimes_ST\) is injective for every finite \(T\).

**Proof.** Its kernel \(K\) is finite. Exactness makes \(C\otimes_SK\to C\otimes_ST\) injective. Its image is zero: it is generated as a left \(C\)-module by \(1\otimes k\) for \(k\in K\), and each such tensor is zero by the definition of \(K\). Zero-object detection gives \(K=0\). \(\square\)

Apply this to (5.25d). Multiplication by \(h\) is injective in the coefficient product \(\widehat Q\), so it is injective on \(V_{j_0(x)}\). The actual element \(h\) is normal, with the actual coefficient automorphism \(\sigma(P)=h\circ P\circ h^{-1}\), established in 5.22a–5.22b of the proved completion proof. Hence the kernel of multiplication by \(h\) is an \(\mathscr A\)-submodule. Regard this semilinear multiplication as a linear map into the module with action twisted by \(\sigma\); twisting by a sheaf algebra automorphism preserves a finite presentation. Neighbourhood coherence therefore makes its kernel coherent. Its zero stalk at \(j_0(x)\) makes it zero on one neighbourhood: finitely many local generators have zero germs and all vanish on their common smaller neighbourhood. Thus \(V\) is actually \(h\)-torsion-free there.

Let
\[
 \mathscr E=\mathscr A[h^{-1}].
 \tag{5.25e}
\]
This is the actual finite-order factorial-growth ring on the fixed chart. Indeed an actual finite-order symbol has finitely many possible positive orders; right multiplication by a sufficiently large \(h^B\) shifts all of them to order zero and preserves the common coefficient domain and factorial bounds. Conversely finite multiplication by \(h^{-B}=\tau^B\) preserves finite order and the actual factorial class by its finite derivative formula and the fixed-shift estimate. The normal-element relation moves all denominators to either side, so (5.25e) describes the actual ring, including its products.

Its localization of a module can be constructed by equivalence classes of fractions with finite denominators. A fraction from \(V\) is zero exactly when a power of \(h\) kills its numerator; this follows directly by moving denominators with \(\sigma\) and taking a common finite power. Therefore \(h\)-torsion-freeness gives an actual injection \(V\hookrightarrow\mathscr E\otimes_{\mathscr A}V\). The same fraction construction proves exactness of this localization: lift a numerator for surjectivity; if an image fraction is zero, clear one further denominator and its numerator belongs to the preceding kernel. No analytic infinite-order extension is involved.

#### 5.25.3. Identification with the actual dummy module

Put
\[
\begin{gathered}
\Phi(M)=(\mathscr E/\mathscr Et)\otimes_D M,\\ u_0=1\bmod\mathscr Et.
\end{gathered}
\tag{5.25f}
\]
The right \(D\)-action exists since \(t\) commutes with ordinary \(D_X\). In (5.25c), send \(q_i\) to \(h^{a_i}u_0\otimes m_i\). The ordinary relations hold, and the actual Weyl identity
\[
 t h^a u_0=a h^{a+1}u_0
 \tag{5.25g}
\]
makes (5.25b) hold, including for every integral \(a\).

**Lemma 5.25.3.** This map gives a natural isomorphism
\[
 V[h^{-1}]\simeq\Phi(M).
 \tag{5.25h}
\]

**Proof.** Algebraically \(R[h^{-1}]=D[h,h^{-1}]\), by \(\partial_i=h^{-1}\theta_i\), and \(Q[h^{-1}]=M[h,h^{-1}]\): exhaustion puts every \(m\) in one \(F_aM\); injectivity follows from the coefficient embedding of the Rees module into \(M[h,h^{-1}]\). These are algebraic Laurent modules, with finite sums.

Write \(A_*=A[h^{-1}]\). The actual finite Ore normal form gives an algebraic bimodule identification
\[
 Q[h^{-1}]
   \simeq (A_*/A_*t)\otimes_D M.
 \tag{5.25i}
\]
To verify it, reduce all powers of \(t\) on the right of the cyclic generator. The remaining vector space is \(D[h,h^{-1}]\otimes_D M=M[h,h^{-1}]\). Its \(t\)-action is \(h\delta\), because \(t h^a=h^at+a h^{a+1}\) and \(t\) commutes with \(D\); this is the action on the left. The two maps are inverse on every finite monomial. This proves (5.25i) without a flatness assumption.

Tensor (5.25i) with \(\mathscr E\) over \(A_*\). Extension of a presentation is right exact by the tensor universal property, so \(\mathscr E\otimes_{A_*}(A_*/A_*t)=\mathscr E/\mathscr E t\). Tensor associativity now gives exactly (5.25h), with the displayed generator map. \(\square\)

Combining Lemma 5.25.1, Lemma 5.25.2 and Lemma 5.25.3 gives an actual coherent ambient lattice
\[
\begin{gathered}
V\subset\Phi(M),\\ \mathscr EV=\Phi(M),\\ \widehat V=\widehat Q\subset M[[h]].
\end{gathered}
\tag{5.25j}
\]
on one micro-neighbourhood of \(j_0(x)\). The ordinary embedding is also genuine. If \(u_0\otimes m=0\), choose \(a\) with \(m\in F_aM\). Then the image of \(h^am\in V\) is zero in its localization. Its injection and (5.25d) give \(h^am=0\) in \(M[[h]]\), hence \(m=0\). Apply the same presentation at each ordinary point to obtain the sheaf embedding \(M\hookrightarrow j_0^{-1}\Phi(M)\).

#### 5.25.4. Shifted ambient modules and common finite bounds

For any integer \(b\), the actual submodule \(h^bV\subset\Phi(M)\) is coherent. Multiplication by \(h^b\) is a semilinear isomorphism onto its image; the normal-element automorphism makes it a finite-presentation-preserving twist. Its finite presentation is the shifted version of (5.25b): its generators are \(h^{a_i+b}u_0\otimes m_i\), its ordinary homogeneous relations are shifted by \(b\), and its dummy relation is
\[
 (t-(a_i+b)h)h^bq_i=0.
 \tag{5.25k}
\]
The same finite-degree presentation proof, with lower degree \(b\) allowed, yields
\[
 \widehat{h^bV}^{\ I}
       \simeq h^b\widehat Q\subset M((h)).
 \tag{5.25l}
\]
The ideal-power topology is cofinal with degree truncation: every degree-one generator raises degree, and a module with finitely many homogeneous generators has a finite largest generating degree. Factoring sufficiently long monomials gives the opposite cofinal bound. Thus finitely many negative degrees cause no change to the completion calculation. These identifications commute with the actual inclusions of shifted lattices, by checking the finite distinguished generators and then their finite-degree coefficients.

**Lemma 5.25.4 (two actual bounds).** Let \(L\subset\Phi(M)\) be a coherent \(\mathscr A\)-submodule that generates \(\Phi(M)\) over \(\mathscr E\). Then, after one local shrinking, there are \(B,C\ge0\) with
\[
 h^C V\subset L\subset h^{-B}V.
 \tag{5.25m}
\]
Every specified finite collection of shifted dummy ordinary sections can be put in the same coherent ambient \(h^{-B}V\), increasing \(B\) if necessary.

**Proof.** Choose finite local generators of \(L\). By (5.25h), each is a fraction from \(V\) with a finite denominator. Move denominators to the left using normality and take their maximum exponent. These finitely many stalk identities extend on one common neighbourhood and give the upper bound.

Conversely each of the finite distinguished generators of \(V\) is a finite \(\mathscr E\)-linear combination of lattice generators, since \(\mathscr E L=\Phi(M)\). Choose \(C\) so that every coefficient in those finitely many expressions becomes order zero after multiplication on the left by \(h^C\). Then \(h^Cq_i\in L\). For any coefficient \(a\in\mathscr A\), normality gives \(h^C(aq_i)=\sigma^C(a)h^Cq_i\in L\), proving the lower bound. All equations used are finite actual equations on a common neighbourhood. A finite list of other sections is bounded by precisely the same fraction argument. \(\square\)

#### 5.25.5. The formal localized equality is a consequence of those actual bounds

**Theorem 5.25.5 (actual ambient model and bounds).** Every coherent lattice \(L\) as in Lemma 5.25.4 has a finite completed image \(\Lambda\) inside \(h^{-B}\widehat Q\), and
\[
\begin{gathered}
h^C\widehat Q\subset\Lambda\subset h^{-B}\widehat Q,\\ \Lambda[h^{-1}]=\widehat Q[h^{-1}]\\ \text{inside }M((h)).
\end{gathered}
\tag{5.25n}
\]
The common actual coherent ambient contains \(L\) and any fixed finite list of shifted ordinary dummy sections. Completion is exact and faithful on its coherent submodules and quotients. Its model carries the ordinary sections and the completed operator action to the coefficientwise Ore model.

**Proof.** Use the coherent ambient \(h^{-B}V\) and complete the two inclusions (5.25m) at its actual stalk. The exact completion of §5.22 identifies the completed inclusions with actual injections. The compatible shifted-presentation identifications (5.25l) identify their two extremes exactly with the two extremes in (5.25n). This proves the bounds. Localizing these two bounds by the central \(h\) of the formal ordinary Rees ring makes both extremes equal to \(\widehat Q[h^{-1}]\); the middle must then equal it. This is an equality of these submodules of the coefficient space \(M((h))\), and follows from finite uniform actual denominator bounds. It is not a purported interchange of analytic localization and an arbitrary inverse limit.

Coherence of the ambient and Theorem 5.24.2 give coherence of its finite submodules, sums and quotients. The proved completion theorem applies to all their finite actual stalk modules. Lemma 5.25.1 and formula (5.25l) give the stated coefficient identification and generator actions. \(\square\)

If additionally \(tL\subset hL\), the actual operator \(h^{-1}t=D_tt=tD_t+1\) preserves \(L\). Its commutator with an actual symbol is the actual coefficient derivation
\[
\begin{gathered}
\sum_{n\geq0}h^np_n(t,x,z)\\ {}\longmapsto\\ \sum_{n\geq0}h^n(t\partial_t+E_z+n)p_n.
\end{gathered}
\tag{5.25o}
\]
This follows directly from the two first-derivative corrections for the symbol \(t\tau\). Derivatives of order at least two separately in the base variables, and separately in the cotangent variables, vanish; the mixed derivative \(\partial_t\partial_\tau(t\tau)=1\) does not enter either of these two-factor Leibniz products. Cauchy estimates on a smaller common domain and \(n\le2^n\) keep (5.25o) in the actual factorial class. Each Taylor monomial in \(h,t,z\) is multiplied by its total degree, so this derivation preserves every positive-degree ideal power. Consequently the preserving operator on \(L\) is continuous for that filtration and extends to its completion. By (5.25g) it acts in the coefficient model as \(h^am\mapsto a h^am\), exactly the Euler action required by Theorem 5.23.3. There is no analytic diagonalization assertion.

### 5.26. Actual coherent ordinary intersections for a supplied lattice

**Proposition 5.26.1.** Let \(M\) be coherent over \(\mathcal D_X\), and let \(L\subset\Phi(M)\) be a supplied coherent \(\mathcal A_{\rm gv}\)-lattice which generates \(\Phi(M)\) over the finite-order localization and satisfies \(tL\subset hL\). Then the actual ordinary intersections
\[
 B_kM=j_0^{-1}L(k)\cap M
 \tag{5.26a}
\]
are holomorphically coherent on a neighbourhood of each ordinary point. They form an exhaustive locally good filtration. These statements concern the actual intersections, not merely modules isomorphic to their completions.

**Proof.** Fix \(x\). The relation \(tD_t=D_tt-1\) shows that \(tD_t+1\) preserves \(L\); its completed action is the Euler operator. Theorem 5.23.3 in §5.23 therefore supplies finitely many homogeneous formal generators \(h^{a_i}m_i\) of the completed lattice, where \(m_i\) are actual germs of \(M\). Extend their finite list to sections on a common ordinary neighbourhood, and write
\[
\begin{gathered}
\ell_i=h^{a_i}(u_0\otimes m_i),\\ L'=\sum_i\mathcal A\ell_i.
\end{gathered}
\tag{5.26b}
\]
§5.25 supplies a coherent ambient \(V\) containing both \(L\) and these finitely many sections. The submodule \(L'\) is a finite type submodule of the coherent module \(V\), hence coherent by §5.24. The sum \(L+L'\) and both quotients by \(L,L'\) are coherent.

The completions of \(L\) and \(L'\) have equal images in the completed \(V_x\). Indeed Theorem 5.23.3 says that the homogeneous \(\ell_i\) generate the completed \(L\) over \(\widehat R\). The completed \(\mathcal A\)-action on them adds nothing: Lemma 5.23.4 gives
\[
\begin{gathered}
t^j\ell_i=\\ a_i(a_i+1)\cdots(a_i+j-1)\\ \cdot h^j\ell_i,
\end{gathered}
\tag{5.26c}
\]
and its coefficientwise reduction proves that the \(\widehat A\)-span is the \(\widehat R\)-span, including infinite formal degree sums. Exactness in §5.22 identifies the completion of the sum with this common image, and the completions of both coherent quotients are therefore zero. Faithfulness gives
\[
\begin{gathered}
((L+L')/L)_{j_0(x)}=0,\\ ((L+L')/L')_{j_0(x)}=0.
\end{gathered}
\tag{5.26d}
\]
A coherent sheaf with zero stalk is zero on a neighbourhood: use finitely many local generators, whose zero germs vanish on the intersection of finitely many neighbourhoods. Consequently \(L=L'\) as actual convergent coherent sheaves near \(j_0(x)\). Shrink the ordinary neighbourhood so its image under \(j_0\) lies in this common micro-neighbourhood. This is the step that cannot be replaced by formal equality alone.

For every \(y\) in the smaller ordinary neighbourhood define the actual finite-order submodule
\[
\begin{gathered}
G_k=\sum_i\mathcal D_{\leq k-a_i}m_i,\\ \mathcal D_{\leq b}=0\quad(b<0).
\end{gathered}
\tag{5.26e}
\]
It is holomorphically coherent. To check this, put the finitely many \(m_i\) in finitely many pieces of the chosen local good filtration of \(M\). For fixed \(k\), all images in (5.26e) lie in one coherent local order piece. The source is a finite sum of coherent finite-order operator sheaves. Oka's coherent kernel/image theorem therefore makes (5.26e) coherent, on the same ordinary neighbourhood. This argument applies to every \(k\); it does not require one uniform bound on \(k\).

We prove \(B_kM=G_k\) as subsheaves there. If \(m\in(B_kM)_y\), then \(h^k(u_0\otimes m)\in L_{j_0(y)}\). In the injective coefficientwise model of §5.25 its image is the homogeneous element \(h^km\) in the completion of \(L'\). The degree-\(k\) part of the latter is the finite sum
\(\sum_i R_{k-a_i}h^{a_i}m_i\). Formula (5.23a) identifies this sum exactly with \(h^k(G_k)_y\). Thus \(m\in(G_k)_y\).

Conversely, for \(m=\sum_iP_im_i\) with \(\operatorname{ord}P_i\leq k-a_i\), the section
\[
 u_0\otimes m=\sum_i(P_ih^{-a_i})\ell_i
 \tag{5.26f}
\]
belongs to \(L(k)\): \(P_i\) commutes with \(h\), and \(P_ih^{-a_i}\) has microlocal order at most \((k-a_i)+a_i=k\). Hence \(m\in(B_kM)_y\). Equality at every stalk proves equality of the actual subsheaves. In particular all intersections in (5.26a) are coherent on this common neighbourhood.

The finite shifted generating expression (5.26e) proves local goodness and the ordinary filtration compatibility. Exhaustion follows either from Theorem 5.23.3 with faithfulness, or directly from the actual lattice property: a germ \(u_0\otimes m\) is a finite sum of finite-order microlocal operators applied to lattice germs, so lies in \(L(k)\) for one common finite bound \(k\). This is exactly membership in (5.26a). Boundedness below and finite graded generation are also immediate from (5.26e). \(\square\)

The proof above gives the actual identity \(L=\sum_i\mathcal A_{\rm gv}h^{a_i}(u_0\otimes m_i)\) on one neighbourhood and every actual ordinary piece \(B_kM=\sum_i\mathcal D_{\le k-a_i}m_i\) there. This proves actual descent for a supplied coherent generating Euler-stable lattice. The intrinsic strict real-order cutoff through every singular characteristic stratum, often denoted \(N_{<0}\), is constructed in Section5.89, CK.A–CK.K. Section5.90, GG.A–GG.G proves the initial coherent ordinary generator near every compact set. The full arbitrary analytic proper/support-proper regularity theorem requires its further argument. The algebraic four-map theorem above retains its full stated scope.

![Actual bounded division, exact completion and faithful recovery of a supplied Euler-stable lattice](assets/actual-convergent-lattice-descent.png)

*Figure. The actual norm and finite-head division in §§5.20–5.21 give convergent coefficients and finite actual relations. §§5.22 and 5.24 supply exact faithful stalk completion and fixed neighbourhood kernels. The two actual bounds in §5.25 identify the formal localized lattice. The finite-node Euler projections in §5.23 are applied after the common grading shift makes the degrees nonnegative; they give homogeneous generators. The two vanishing coherent quotients in §5.26 recover actual sheaf equality and every ordinary order piece. The canonical cutoff and its everywhere characteristic stability are proved in Section5.89, CK.A–CK.K. All arrows are proved in the displayed sections. Human context: Schapira, the freely accessible article cited in §5.19, and Kashiwara–Kawai, [Holonomic systems III](https://www.kurims.kyoto-u.ac.jp/~kenkyubu/kashiwara/HolIII.pdf), Appendix A.8, printed pp. 966–968; no external descent assertion is used as a proof. The reproducible original drawing source supplies the CC0 illustration.*

### 5.27. An actual coherent strict coefficient lattice on a singular divisor

#### 5.27.1. Hypotheses, conventions and earlier analytic proofs

Let $X$ be a complex manifold, $Y\subset X$ a reduced analytic hypersurface, and $\mathscr E$ a coherent $\mathcal O_X(*Y)$-module with an integrable connection. Here coherence over $\mathcal O_X(*Y)$ means locally a finite presentation by actual meromorphic matrices with a finite pole order on one neighbourhood. All sections, matrices, derivatives and coordinate changes in this leaf are convergent analytic or meromorphic. No completion of a sheaf stalk is used.

Assume that, on a nonempty relatively open part of the smooth locus of **each local irreducible component** of $Y$, the connection is regular on a nonempty open family of transverse analytic discs. The assumption is local at every $p\in X$, with the local components through $p$. It is an analytic divisorial hypothesis, not a test on algebraic curves. The zero-rank module is allowed.

Put $\Sigma=\{\lambda\in\mathbf C:0\leq\operatorname{Re}\lambda<1\}$. Our residue convention is $\nabla=d+B\,dq/q+\cdots$; horizontal monodromy is $\exp(-2\pi iB)$. For a horizontal *dual* section paired with a coefficient section, the scalar powers have exponent $+\lambda$. This explains the sign in the strict Nilsson statement below.

The full local proofs of [Fuchs's criterion, Theorem 3.3](#3-1-fuchs-s-criterion-in-every-order), and [the normalized local analytic theorem, Theorem 5.0](#normalization-and-the-local-theorem), give the following inputs. The parameter finite-pole proof is [Lemmas 5.0.1–5.0.3](#finite-poles-from-growth-and-transverse-families); the [meromorphic uniqueness proof](#meromorphic-maps-and-normalized-uniqueness) and [analytic disc criterion](#the-local-analytic-disc-criterion) include all nilpotent blocks. We use precisely these proved local analytic portions:

1. For a convergent scalar cyclic equation of rank $r$, regularity is equivalent to $q^j a_j$ holomorphic, $1\leq j\leq r$. Its proof supplies the monic falling-factorial relation, with coefficients on the left.
2. On a smooth divisor chart, a nonempty open family of regular transverse analytic discs supplies an **actual** free meromorphic frame and a free normalized logarithmic lattice. The finite-pole step is the proved parameter Laurent/Baire argument, not a formal-to-sheaf assertion.
3. A normalized logarithmic bundle is uniquely identified with the constant residue model. All nilpotent blocks are retained. Uniqueness follows from the invertibility of the leading-pole Sylvester operator, because differences of real parts in $\Sigma$ belong to $(-1,1)$.

The analytic local-ring proof, Theorem 3.1 and Lemmas 2.3, 2.4, 3.3, proves that $H=\mathcal O_{X,p}$ is a factorial Noetherian domain. Its convergent preparation and division inputs are actual analytic proofs. Theorem 3.4 in the same local-ring lesson proves persistence of coprimality on an actual neighbourhood. The full Oka proof, Lemma 2.2 and Theorem 2.1, gives actual neighbourhood finite generation of the kernel of a finite row of holomorphic coefficients. The finite-module consequences can be obtained without importing the general proposition cited there: induct on the number of rows of a finite matrix. A finite generating family for the kernel of the first rows turns the last row into a finite row on those generators, whose kernel is finite; its image generates the full matrix kernel. For a map between two finitely presented modules, lift its finite generator matrix to the two free presentations. The kernel of the finite block matrix consisting of that lift and the negative target relation matrix is finite by the preceding result; its projection to the source module generates the desired kernel. Finite cokernels and images then have finite presentations. This proves precisely the neighbourhood kernels, images and duals used below. A finite presentation also suffices for the localization and dual calculations in §5.27.8.

For completeness, the polynomial-factorial step used in the earlier convergent factoriality proof has the following elementary argument. Let $A$ be a factorial domain and $K_A$ its fraction field. The content of a nonzero polynomial is the product of the minimum prime valuations of its coefficients. Contents multiply: after dividing out contents, reduction modulo any prime of $A$ gives two nonzero polynomials over the domain $A/(\pi)$, whose product is nonzero. Thus the product of primitive polynomials is primitive. The ring $K_A[T]$ has division with remainder; division decreases degree, so every ideal is principal and Bézout proves that irreducibles are prime. Factoring there and clearing the denominators of each factor gives primitive polynomials in $A[T]$. If two primitive polynomials differ by a scalar in $K_A$, the minimum valuation of their coefficients is zero at every prime, so that scalar has valuation zero at every prime and is a unit of $A$. It follows that a primitive polynomial in $A[T]$ factors into primitive lifts of the irreducible factors over $K_A$, up to a unit. Each such lift is prime in $A[T]$: divisibility of a product over $K_A$ selects a factor, and the primitive denominator-clearing argument in the earlier Gauss Lemma 3.3 puts its quotient back in $A[T]$. A prime of $A$ remains prime in $A[T]$ since the quotient is $(A/(\pi))[T]$, a domain. Factoring the content and primitive part therefore proves that $A[T]$ is factorial. Starting with $A=\mathbf C$ and using the earlier convergent Weierstrass Lemmas 2.3–2.4 proves the factoriality step of its Theorem 3.1 without using the external polynomial-ring citation as a proof.

We also record the height-one and DVR facts used below, without invoking a principal ideal theorem. In a factorial domain $H$, an irreducible $q$ generates a prime ideal. If a nonzero prime $\mathfrak r$ is contained in $(q)$, factor a nonzero element of $\mathfrak r$; primality puts one irreducible factor $a$ in $\mathfrak r$. Since $a\in(q)$, irreducibility makes $a$ associate to $q$, whence $\mathfrak r=(q)$. There is therefore no prime strictly between $0$ and $(q)$, so $(q)$ has height one. Conversely a height-one prime $\mathfrak p$ contains an irreducible factor $q$ of any of its nonzero elements. The inclusions $0\subsetneq(q)\subset\mathfrak p$ force $\mathfrak p=(q)$. In $H_{(q)}$ every nonzero fraction is $q^m$ times a unit, by unique factorization; the minimum valuation of a nonzero ideal generates that ideal. Thus it is a DVR. Finally a reduced fraction with a nonunit denominator fails membership at a denominator-prime localization, whereas a fraction in $H$ belongs to all of them. This proves $H=\bigcap_{\operatorname{ht}\mathfrak p=1}H_\mathfrak p$ inside its fraction field, including the precise prime indexing.

**Elementary polynomial, field and radical details.** The following arguments make explicit the algebra used in the divisorial identity argument and in the earlier analytic Nullstellensatz.

Let \(K\) be a field of characteristic zero. A nonconstant irreducible polynomial \(P\in K[T]\) is separable: its derivative is nonzero and has smaller degree, so a common divisor of \(P,P'\) would have to be \(P\), an impossibility. Euclidean division therefore gives \(\gcd(P,P')=1\). In an algebraic closure a repeated root is precisely a common root with the derivative, so all roots are distinct. For a finite extension obtained by adjoining finitely many algebraic generators, every embedding of the preceding field extends in as many ways as the degree of the next minimal polynomial, by sending that generator to each of its distinct roots. Induction and the tower-degree formula give exactly \([L:K]\) embeddings of a finite characteristic-zero extension \(L/K\).

For the primitive-element step, take generators \(b_1,\ldots,b_s\) of \(L/K\). The finitely many embeddings differ on at least one generator. A complex linear combination \(u=\sum c_i b_i\), when \(K\) contains \(\mathbf C\), has pairwise distinct conjugates after excluding finitely many proper complex linear subspaces of the \(c_i\). Its minimal polynomial then has at least \([L:K]\) roots and degree at most \([L:K]\), so \(K(u)=L\). Equivalently the generators can be recovered by Euclidean gcds. For two generators \(a,b\), choose a nonzero scalar \(c\) so that \(a+cb\) differs from every other sum \(a_i+cb_j\) of roots of their minimal polynomials. Only finitely many scalars are excluded. In \(K(a+cb)[T]\), the minimal polynomial of \(b\) and the polynomial obtained from that of \(a\) by substituting \(a+cb-cT\) have exactly the one common root \(b\). They are separable, so their monic gcd is \(T-b\). Its coefficients put \(b\), and then \(a\), in \(K(a+cb)\); induction handles any finite list.

For nonzero polynomials \(P,G\in K[T]\) of degrees \(m,n\), respectively, the Sylvester resultant is the determinant, up to a fixed sign, of the linear map
\[
\begin{gathered}
\{A:\deg A<n\}\oplus\{B:\deg B<m\}\\ \longrightarrow\{C:\deg C<m+n\},\\ (A,B)\longmapsto AP+BG .
\end{gathered}
\]
The spaces have the same finite dimension. A common factor \(D\) of positive degree gives the nonzero kernel pair \((G/D,-P/D)\). Conversely, if \(AP+BG=0\) with the displayed degree bounds and \(\gcd(P,G)=1\), Bézout's identity makes \(P\mid B\), whence \(B=0\), and then \(A=0\). Thus the determinant vanishes exactly when the polynomials have a common factor. A nonzero constant \(G\) gives the same nonvanishing conclusion directly; the zero polynomial is treated separately. The matrix entries are polynomial expressions in the coefficients, so for holomorphic coefficient families the resultant is holomorphic. Apply this with \(G=P'\) to obtain the nonzero discriminant in the separability case. Apply it with a Weierstrass remainder to justify the divisorial identity argument below, including the case of a constant remainder.

A factorial domain is integrally closed by the same prime valuations: if a reduced fraction \(a/b\) satisfies a monic equation of degree \(m\), multiplication by \(b^m\) shows that any prime dividing \(b\) divides \(a^m\). That contradicts coprimality. Hence \(b\) is a unit. This supplies the normality used for the coefficient algebra in the earlier finite parametrization proof.

Finally, for any commutative ring \(A\) and ideal \(J\),
\[
\sqrt J=\bigcap_{\mathfrak p\supset J,\ \mathfrak p\ {\rm prime}}\mathfrak p.
\]
Every prime containing \(J\) contains its radical. For the converse take \(a\notin\sqrt J\); the powers \(1,a,a^2,\ldots\) are disjoint from \(J\). The union of a chain of ideals containing \(J\) and disjoint from these powers still has that property. The maximality principle therefore gives a maximal such ideal \(\mathfrak p\). If \(u,v\notin\mathfrak p\) but \(uv\in\mathfrak p\), maximality gives \(a^r=p_1+bu\) and \(a^s=p_2+cv\), with \(p_1,p_2\in\mathfrak p\). Multiplying puts \(a^{r+s}\) in \(\mathfrak p\), a contradiction. Thus \(\mathfrak p\) is prime and excludes \(a\). This proves the formula, including the unit-ideal case with an empty intersection. With the earlier full prime-ideal vanishing proof, it gives the analytic Nullstellensatz used below.

Finite linear avoidance is also elementary here. If an ideal is not contained in any of finitely many divisor primes, choose one element outside each prime and take their finite complex linear span. Each prime cuts out a proper linear subspace of that span. Finitely many proper linear subspaces cannot cover a finite-dimensional complex vector space: choose a nonzero linear functional vanishing on each, and their product is a nonzero polynomial, which cannot vanish at every point over an infinite field, by induction on the number of variables. Hence one linear combination avoids every divisor prime.

One analytic consequence needed repeatedly can also be proved directly from those inputs. If $q\in H$ is irreducible and $g\in H$ vanishes on a nonempty smooth open part of the local hypersurface $q=0$, then $q\mid g$. After a linear coordinate change, replace $q$ by its irreducible Weierstrass polynomial $P(z',z_n)$, and divide $g$ by $P$ to obtain a polynomial remainder $G$ of degree less than $\deg P$. The discriminant of $P$ is nonzero: over $\operatorname{Frac}\mathbf C\{z'\}$, an irreducible characteristic-zero polynomial is separable. Choose an unramified point in the given open part. On its projected open set, one root of $P$ is also a root of $G$, so the resultant $\operatorname{Res}(P,G)$ is zero there. The holomorphic identity theorem makes the resultant identically zero. Gauss's lemma makes $P$ irreducible over the fraction field, hence $P\mid G$, which forces $G=0$. This proves the assertion. In particular a finite list of nonzero meromorphic functions has its claimed $q$-valuations on a nonempty smooth open part of $q=0$, avoiding the zeros of the finitely many residual numerators and denominators.

This coefficient construction uses actual analytic finite presentations and sheaf duals. The bounded ambient comparison in §5.25 does not identify an arbitrary microlocal module with the meromorphic connection here; that realization remains a separate obligation in §5.27.12.

#### 5.27.2. Statement

**Theorem 5.27.** Under all the hypotheses of §5.27.1, there is a unique coherent reflexive $\mathcal O_X$-submodule
\[
\mathscr L_\Sigma\subset\mathscr E,\qquad
\mathscr L_\Sigma(*Y)=\mathscr E,                              \tag{5.27a}
\]
whose restriction on every smooth divisor chart is the normalized logarithmic lattice. This restriction exists at **every** smooth point of $Y$, including points outside the regular open sets in the hypothesis. At a singular point, $\mathscr L_\Sigma$ is allowed to be reflexive rather than locally free.

For a local meromorphic section $s\in\mathscr E$, membership in $\mathscr L_\Sigma$ is detected by strict Nilsson membership on the smooth divisor locus. Namely, pair with a basis of horizontal dual sections and use the local constant-residue model; the resulting multivalued scalar expressions are finite sums
\[
\sum_{\lambda\in\Sigma}\sum_{j=0}^{r-1}
q^\lambda(\log q)^j a_{\lambda,j}(q,w),                      \tag{5.27b}
\]
with holomorphic coefficient functions. The intrinsic condition is membership in the normalized lattice; (5.27b) is its expression in that model. No choice of a single-valued branch on all of a punctured neighbourhood is asserted.

The following statements hold at every point of $Y$, including all singular strata:

* **Saturation.** If $g$ is holomorphic and nonzero on every local irreducible component of $Y$, $s\in\mathscr E$, and $g^m s\in\mathscr L_\Sigma$, then $s\in\mathscr L_\Sigma$.
* **Logarithmic stability.** If $\eta$ is a holomorphic vector field satisfying $\eta(I_Y)\subset I_Y$, then $\nabla_\eta\mathscr L_\Sigma\subset\mathscr L_\Sigma$. The algebra generated by holomorphic functions and these logarithmic covariant derivatives also preserves it.

This is a coefficient theorem on a singular hypersurface, with no projectivity hypothesis on $X$ or $Y$.

#### 5.27.3. A finite actual initial lattice and projectivity off the divisor

Fix $p\in X$, $H=\mathcal O_{X,p}$, a reduced equation $f$, $R=H[f^{-1}]$, and $K=\operatorname{Frac}H$. A finite $R$-presentation of $E=\mathscr E_p$, after clearing finitely many denominators, is the localization of a finite $H$-presentation $C$. Thus $C[f^{-1}]=E$. The kernel of $C\to E$ is precisely the union of the kernels of multiplication by $f^k$. That ascending chain stabilizes, since $C$ is Noetherian. Take $N$ for which it stabilizes and set $F=C/\ker(f^N:C\to C)$. Then $F\subset E$ and $F[f^{-1}]=E$.

This is a neighbourhood construction. Represent the finite presentation and $f^N$ near $p$, take its coherent kernel, and form the coherent quotient. Multiplication by $f$ on the quotient has coherent kernel and zero stalk at $p$. A coherent sheaf with zero stalk has zero on a neighbourhood: write finitely many local generators, each zero on some neighbourhood, then intersect those finitely many neighbourhoods. Shrink once. The quotient is now $f$-torsion-free throughout that neighbourhood and embeds into its localization $\mathscr E$. Thus we have an actual coherent initial lattice $\mathscr F\subset\mathscr E$, not merely finite stalk generators.

We shall need $E$ to be projective over $R$. Here is a proof. The ring $R$ is Noetherian and its only ideals stable under every coordinate derivation $\partial_i$ are $0,R$. For a nonzero such ideal, multiply a nonzero meromorphic member by a power of $f$, obtaining $0\ne u\in H$ in the ideal. A derivative corresponding to a nonzero Taylor coefficient of least total degree has nonzero value at $p$, so it is a unit of $H$. That derivative is in the ideal by stability.

Take a finite presentation $R^a\xrightarrow{\phi}R^b\to E\to0$. For a coordinate derivative $\partial$, lift the connection on the $b$ chosen generators to a matrix $B\in M_b(R)$, so $\partial+B$ on $R^b$ induces the connection on $E$. It preserves the relation module. For the $a$ chosen relation generators, choose preimages of their derivatives in $R^a$; their columns form $A\in M_a(R)$, and
\[
\partial\phi+B\phi=\phi A.                                  \tag{5.27c}
\]
By the determinant multilinearity formula, the derivative of each minor of $\phi$ is a sum of minors of the same size multiplied by entries of $A,B$. Consequently all Fitting ideals of $E$ are differential ideals. Put $r=\dim_K(E\otimes_RK)$. At the generic point, $\operatorname{Fitt}_{r-1}E=0$ and $\operatorname{Fitt}_r E=K$. The first ideal is zero already in the domain $R$; the second is nonzero and hence $R$.

For completeness these conditions imply projectivity of rank $r$. If $r=0$, $\operatorname{Fitt}_0E=R$ makes the presentation surjective at every maximal ideal and hence $E=0$; every subsequent lattice is zero and all claims follow. Otherwise, at a maximal ideal of $R$, one $(b-r)$-minor is a unit. Elementary invertible row and column operations split that block off as an identity and reduce the presentation to $\psi:R^{a-b+r}\to R^r$. All $(b-r+1)$-minors vanish by $\operatorname{Fitt}_{r-1}=0$, so all entries of $\psi$ vanish. The cokernel is free of rank $r$ there. The unit minor works on an open neighbourhood in $\operatorname{Spec}R$, and these neighbourhoods cover; a finite locally free module is projective. This last assertion can be seen by taking a finite cover by principal opens and using a partition of $1$ by sufficiently high powers of their defining functions to patch local lifts of a map from the module through a surjection. Thus $E$ is projective and embeds in $E_K=E\otimes_RK$. In particular the initial $F$ is torsion-free over $H$.

#### 5.27.4. Actual stable lattices at the finitely many divisor primes

Factor $f=u\prod_{i=1}^s q_i$ into distinct irreducibles. Put $H_i=H_{(q_i)}$. Factoriality makes $H_i$ a discrete valuation ring, with uniformizer $q_i$, fraction field $K$, and residue field $\kappa_i=\operatorname{Frac}(H/(q_i))$. These are localizations of actual analytic germs, not formal coefficient rings.

Choose a coordinate derivative $\partial_j$ for which $d_i=\partial_jq_i\notin(q_i)$, and put $v_i=d_i^{-1}\partial_j$. Such a derivative exists since a characteristic-zero reduced Weierstrass polynomial is generically smooth. Then $v_i(q_i)=1$, and $v_i$ preserves $H_i$: both $d_i$ and any denominator outside $(q_i)$ are units there. Write $D=\nabla_{v_i}$, $q=q_i$, and $\Theta=qD$.

We first prove cyclicity without importing a formal cyclic-vector theorem. For a $K$-basis $e_0,\ldots,e_{r-1}$ of $E_K$, set
\[
\begin{gathered}
b_i=\sum_{k=0}^i(-1)^k\binom ik D^ke_{i-k},\\ c(a)=\sum_{i=0}^{r-1}\frac{(q-a)^i}{i!}b_i,
\end{gathered}
\tag{5.27d}
\]
where $a$ is a constant indeterminate under $D$. Apply $D^j$, then substitute $a=q$. The result is
\[
\begin{gathered}
\sum_{i=0}^j\binom jiD^{j-i}b_i=e_j\\ (0\leq j<r).
\end{gathered}
\tag{5.27e}
\]
Indeed the coefficient of $D^{j-\ell}e_\ell$ is $\binom j\ell\sum_{k=0}^{j-\ell}(-1)^k\binom{j-\ell}k$, zero unless $\ell=j$. Hence the determinant of $c(a),Dc(a),\ldots,D^{r-1}c(a)$, a polynomial in $a$ over $K$, is not zero. It has only finitely many roots; choose $a\in\mathbf C$ avoiding them. This gives an actual cyclic vector $c\in E_K$ and a unique scalar relation
\[
\begin{gathered}
D^rc+\sum_{j=1}^r a_jD^{r-j}c=0,\\ a_j\in K.
\end{gathered}
\tag{5.27f}
\]

On a nonempty smooth open part of $q=0$, avoid the finitely many residual numerator/denominator zeros of the basis, the cyclic determinant and the coefficients. This avoidance is justified by the divisorial identity proof in §5.27.1. There $q$ is an actual transverse coordinate, $v_i=\partial_q$ in coordinates $(q,x_1,\ldots,\widehat{x_j},\ldots,x_n)$, and (5.27f) is a convergent meromorphic cyclic equation on each transverse disc. Pairing $c$ with horizontal dual solutions gives its full scalar solution space; multiplication by actual meromorphic coefficients preserves regular growth. The assumed analytic regularity and the earlier Fuchs criterion therefore imply that every $q^ja_j$ has no pole on this open part. Detection of the $q$-valuation there gives
\[
\begin{gathered}
\nu_{q_i}(a_j)\geq-j,\\ q^ja_j\in H_i.
\end{gathered}
\tag{5.27g}
\]
This argument uses actual discs to prove finite valuation bounds on actual germs. It does not replace the analytic sheaf by its formal completion.

Let $F_m(T)=T(T-1)\cdots(T-m+1)$. The identity $q^mD^m=F_m(\Theta)$, proved by the Leibniz rule, changes (5.27f) to the monic relation
\[
\left(\begin{gathered}F_r(\Theta)+\\\sum_{j=1}^r q^ja_jF_{r-j}(\Theta)\end{gathered}\right)c=0.
\tag{5.27h}
\]
All coefficients are on the left. The $H_i$-span of $c,\Theta c,\ldots,\Theta^{r-1}c$ is finite free and stable under $\Theta$, by (5.27g)–(5.27h) and $qv_i(H_i)\subset H_i$. Call it $M_i$.

#### 5.27.5. Constant residue spectra, normalization, and nilpotent blocks

Since $qv_i(H_i)\subset qH_i$, the residue $B_0$ of $\Theta$ on $M_i/qM_i$ is $\kappa_i$-linear. We show that its characteristic polynomial has constant complex roots; this fact must not be assumed for a matrix over $\kappa_i$.

Work on one of the smooth regular open charts chosen in §5.27.4. The earlier local analytic proof of Theorem 5.0 gives a normalized logarithmic lattice $L^{\rm can}$ and a constant-residue model $A\in M_r(\mathbf C)$. The meromorphic basis of $M_i$ gives an actual stable lattice on a smaller such chart. It is bounded above and below by two powers of $qL^{\rm can}$, uniformly on that chart, after avoiding its determinant zeros. Filter $M_i/qM_i$ by
\[
\begin{gathered}
F^k={}\\ (M_i\cap q^kL^{\rm can}+qM_i)/qM_i.
\end{gathered}
\tag{5.27i}
\]
There are finitely many nonzero associated quotients. Each is a subquotient of $q^kL^{\rm can}/q^{k+1}L^{\rm can}$, on which the residue is $A+kI$. To check the subquotient claim, first quotient $M_i\cap q^kL^{\rm can}$ by $M_i\cap q^{k+1}L^{\rm can}$, which embeds in the latter canonical quotient, and then quotient by the image of $qM_i\cap q^kL^{\rm can}$. All intersections are stable under $\Theta$.

Thus all residue eigenvalues belong to a **finite** set of constants $\lambda+k$, $\lambda\in\operatorname{Spec}A\subset\Sigma$, with integers $k$ bounded by the two lattice bounds. The characteristic-polynomial coefficients are holomorphic on the connected chart, and take values in the finite set of elementary symmetric functions of those finite possible multisets. Continuity makes them constant there. The divisorial identity proof in §5.27.1 then makes those coefficients constant in $\kappa_i$. In particular $B_0$ splits into generalized eigenspaces over $\kappa_i$, with complex eigenvalues.

Normalize $M_i$ by finite elementary modifications. The generalized-eigenspace projectors are polynomials in $B_0$ with constant coefficients, using the Euclidean algorithm on its distinct factors $(T-\lambda)^{m_\lambda}$. Lift a basis of each projector image to an $H_i$-basis of $M_i$; the lifted determinant is a unit. The residue matrix is block diagonal and all off-block entries of the full matrix are divisible by $q$.

If a block has eigenvalue with real part below $0$, replace its basis vectors by $q$ times them. Its residue eigenvalue changes by $+1$. If its real part is at least $1$, replace those vectors by $q^{-1}$ times them; its eigenvalue changes by $-1$. Stability persists: conjugating an off-block entry divides or multiplies it by $q$, and the entry that is divided was already divisible by $q$. The new residue matrix is block triangular, its diagonal blocks are the designated shift and the unshifted blocks, and its eigenvalue multiset is exactly the described one. Recompute generalized blocks if shifted eigenvalues meet other eigenvalues. All vectors with any given eigenvalue have the same membership in the strip, so this recomputation does not force a normalized eigenvalue to be shifted as an outlier.

The sum, counted with multiplicity, of the nonnegative integer distances required to move the real parts into $[0,1)$ decreases at every modification. After finitely many steps we obtain a finite free $H_i$-lattice $L_i\subset E_K$, stable under $\Theta$, whose residue has spectrum in $\Sigma$. Generalized blocks and their nilpotents are never discarded. This is a finite construction over actual analytic localizations.

On a nonempty regular smooth chart, the resulting meromorphic frame is normalized and agrees with $L^{\rm can}$. One may apply the earlier normalized uniqueness to the identity map, or repeat its leading-pole proof over the DVR: if the identity from one normalized stable lattice to another had a pole of order $m>0$, its leading matrix would be killed by $-mI+B_0(\cdot)-(\cdot)A_0$, whose eigenvalues have nonzero real part. On generalized Hom blocks the nilpotent remainder is nilpotent, so a finite geometric series supplies the inverse. Repeat for the reverse identity. This proves equality and also independence of all choices at the divisor prime.

#### 5.27.6. Tangential stability at a divisor prime

Write $q=q_i$, $v=v_i$ and use the coordinate vector fields
\[
w_k=\partial_{x_k}-(\partial_{x_k}q)v\quad(k\ne j).
\]
They satisfy $w_k(q)=0$ and $[v,w_k]=0$: they are the coordinate partial derivatives in the actual coordinates $(q,x_1,\ldots,\widehat{x_j},\ldots,x_n)$ on the smooth chart, and equality of the corresponding rational germs follows from equality there. They preserve $H_i$.

In a basis of the normalized $L_i$, write
$\nabla_{qv}=qv+B$ and $\nabla_w=w+C$, where $B\in M_r(H_i)$, $C\in M_r(K)$. Flatness and $[qv,w]=0$ give
\[
qv(C)-w(B)+[B,C]=0.                                         \tag{5.27j}
\]
If $C$ had a pole of order $m>0$, write its nonzero leading coefficient as $C_{-m}=q^mC\bmod q\in M_r(\kappa_i)$. Since $v(H_i)\subset H_i$, reducing $q^m$ times (5.27j) modulo $q$ gives
\[
(-mI+\operatorname{ad}B_0)C_{-m}=0.                         \tag{5.27k}
\]
The eigenvalues of this operator are $-m+\beta-\alpha$, with $\alpha,\beta\in\Sigma$, and none is zero. On a generalized Hom block the remaining operator is a sum of commuting nilpotents and is nilpotent; its finite geometric inverse proves invertibility, including all Jordan blocks. This contradicts $C_{-m}\ne0$. Thus $C\in M_r(H_i)$ and $\nabla_wL_i\subset L_i$.

Every holomorphic vector field $\eta$ logarithmic for $Y$ satisfies $\eta(q_i)\in q_iH_i$, because all other factors of $f$ are units in $H_i$. In the above basis of vector fields it is an $H_i$-linear combination of $q_iv_i$ and the $w_k$. Therefore $L_i$ is stable under every such $\nabla_\eta$.

#### 5.27.7. One finite coherent lattice with all the divisorial germs

We now build an actual finite $H$-module $F_0\subset E$ such that
\[
\begin{gathered}
(F_0)_{(q_i)}=L_i\\ (1\leq i\leq s),\\ F_0[f^{-1}]=E.
\end{gathered}
\tag{5.27l}
\]
For each of the finitely many basis vectors of $L_i\subset E_K$, choose a nonzero denominator outside $(q_i)$ that makes that vector lie in $E$. This is possible by writing it in the original finite $R$-generators and separating all denominator factors into powers of $q_i$ and factors not divisible by $q_i$; powers of $q_i$ are already invertible in $R$. Multiplication by that denominator is a unit of $H_i$, so the resulting vectors still generate $L_i$ there. Multiply each such vector by sufficiently high nonnegative powers of every $q_j$, $j\ne i$, to make it belong to every $L_j$. These multipliers are units of $H_i$ and hence do not destroy generation at $i$. All chosen vectors now lie in $E$ and in every $L_j$.

The finitely generated initial $F$ of §5.27.3 has finitely many generators. Choose $B\geq0$ for which $f^BF\subset L_i$ for every $i$, using the finite DVR valuations of those finitely many generators in the finitely many bases. Let $F_0$ be generated over $H$ by the chosen divisor vectors and $f^BF$. It is a finite torsion-free $H$-module contained in $E$. Each of its divisor localizations contains a generating basis of $L_i$, and all its generators belong to $L_i$, proving the first assertion of (5.27l). The second follows from $f^BF[f^{-1}]=E$.

Choose representatives of the finitely many meromorphic vectors and finite relations on one neighbourhood. The earlier $\mathcal O$-coherence theorem makes their image a coherent subsheaf $\mathscr F_0\subset\mathscr E$. Its localization is $\mathscr E$, after shrinking once. All comparisons at the divisor primes involve finitely many fractions with denominators coprime to the respective $q_i$; coprimality persists by the earlier analytic Theorem 3.4. Thus the same comparisons hold on dense smooth open subsets of every nearby branch of the divisor representatives. We shall obtain the statement at the other points from the coherent sheaf double dual, not by treating the intersection of infinite families of stalk modules as a sheaf.

#### 5.27.8. The reflexive intersection formula, proved inside the actual fraction module

For a finite torsion-free $H$-module $G\subset E_K$ spanning $E_K$, define $G^*=\operatorname{Hom}_H(G,H)$. A finite presentation proves that $G^*$ is finite and that duals commute with localization: dualize the presentation to a kernel between finite free modules, and localize that finite kernel, which is exact. Its $K$-span is $E_K^*$. The double dual $G^{**}$ is thus naturally a submodule of $E_K$, with
\[
G^{**}=\{s\in E_K:\ell(s)\in H\text{ for every }\ell\in G^*\}.
\]
Factoriality gives $H=\bigcap_{\operatorname{ht}\mathfrak p=1}H_\mathfrak p$ inside $K$: in a reduced fraction, a denominator prime of positive multiplicity is detected in precisely its height-one localization. The height-one classification and DVR assertion were proved directly from factoriality in §5.27.1. At such a prime $G_\mathfrak p$ is finite torsion-free over a DVR and is free. An elementary proof of this last fact chooses a nonzero element of least valuation in one coordinate, clears that coordinate using its unit coefficient, and inducts on the rank. Hence $G_\mathfrak p=(G_\mathfrak p)^{**}$. Applying the displayed dual description and the intersection formula for $H$ proves
\[
\begin{gathered}
G^{**}=\bigcap_{\operatorname{ht}\mathfrak p=1}G_\mathfrak p\\ \text{inside }E_K.
\end{gathered}
\tag{5.27m}
\]
This proves reflexivity of $G^{**}$ as well: its height-one localizations agree with those of $G$, and applying (5.27m) again gives the same module.

Set $L=F_0^{**}$. It is finite. At primes dividing $f$, $L_\mathfrak p=L_i$. At primes not dividing $f$, $(F_0)_\mathfrak p=E_\mathfrak p$, by (5.27l). Since $E$ is projective over $R$, it is reflexive over $R$. Applying the same proof of (5.27m) to the factorial localization $R$ shows that the intersection of the latter prime-localizations is $E$. Consequently
\[
\begin{gathered}
F_0\subset L\subset E,\\ L[f^{-1}]=E.
\end{gathered}
\tag{5.27n}
\]

On the chosen neighbourhood define
$\mathscr L=\mathscr F_0^{**}$, using **sheaf** duals over $\mathcal O_X$. They are coherent finite kernels. At each point sheaf duals have the ordinary finite-module stalk duals, by a finite presentation. Thus (5.27m) holds independently at every nearby actual stalk. The embedding $\mathscr L\to\mathscr E$ is obtained by localizing the finite dual construction and using (5.27n); its finite kernel is zero at every stalk. This is the actual neighbourhood lattice required here.

#### 5.27.9. Stability everywhere, and regularity at all smooth divisor points

Let $\eta$ be a local holomorphic logarithmic vector field and $s\in\mathscr L$. Then $\nabla_\eta s\in\mathscr E$. At every height-one prime not in $Y$, the lattice equals $E$ and the connection preserves $E$. At a divisor prime the lattice is $L_i$, and §5.27.6 proves preservation. Formula (5.27m) therefore gives $\nabla_\eta s\in L$ at $p$.

This also holds at every nearby point. Here is a finite-stalk comparison which makes that assertion precise. At a nearby point $p'$, carry out §5.27.3–§5.27.8 using the local components through $p'$ and obtain $L'$. Such a construction is available by the stated local hypothesis at every point. The finite comparisons and persistence of coprimality in §5.27.7 make the original sheaf $\mathscr F_0$ agree, at each divisor prime of $\mathcal O_{X,p'}$, with the normalized prime lattice of the original finite meromorphic frames. Those frames are normalized on a dense smooth open of that local branch: their finitely many denominators are coprime to it, and the residue characteristic-polynomial identity in §5.27.5 extends there. DVR normalized uniqueness in §5.27.5 identifies that prime lattice with the one used for $L'$. At every other height-one prime both sheaves equal $E$. Sheaf duals commute with stalks by their finite presentations, so both $\mathscr L_{p'}$ and $L'$ have the intersection formula (5.27m), with exactly the same height-one modules. They are equal. The first paragraph of §5.27.9, applied to the construction at $p'$, proves logarithmic preservation there. This uses finite presentation comparisons followed by the proved reflexive formula, without a uniform infinite cancellation argument.

We next prove local freeness at a smooth point of $Y$, including an initially exceptional point. Use coordinates $(q,w)$ with $Y=(q=0)$. The finite module $T=L/qL$ has a flat connection in all the $w$-directions: the logarithmic vector fields $\partial_w$ preserve $L$, commute and preserve $qL$. Their induced connection has holomorphic coefficients in finite presentations. The Fitting-ideal proof of §5.27.3, now over $H/(q)=\mathbf C\{w\}$, shows that $T$ is free. Its rank is $r$, since at the divisor generic point it is $L_i/qL_i$ of dimension $r$.

Lift a free basis of $T$ to $L$. Nakayama gives a surjection $H^r\to L$, with finite kernel $K_0$. Multiplication by $q$ is injective on $L\subset E_K$, so the tensor exact sequence modulo $q$ has zero $\operatorname{Tor}_1^H(H/(q),L)$, as is seen directly from the two-term resolution $H\xrightarrow qH\to H/(q)$. The lifted map is an isomorphism modulo $q$, hence $K_0/qK_0=0$. Nakayama makes $K_0=0$. Thus $L$ is free. Nakayama here follows from the elementary determinant trick: if a finite module $M=qM$, a generator matrix $I-qA$ annihilates its generator column and has unit determinant.

Its logarithmic normal residue has spectrum in $\Sigma$. The characteristic polynomial is the constant normalized polynomial on a dense smooth open set; its holomorphic coefficients are therefore those constants at the initially exceptional point too, by §5.27.1. We have obtained a normalized free logarithmic lattice at every smooth point. The earlier normalized uniqueness now identifies it with the local constant-residue model there. In particular regularity has extended to all smooth points; it was not added to the hypothesis.

The same argument applied to a second construction shows uniqueness. They agree at every height-one prime dividing $f$, by normalized DVR uniqueness in §5.27.5, and at every other height-one prime, because both localize to $E$. Their reflexive intersection formula makes them equal. On overlapping neighbourhoods the same argument applies to every point. Hence the local coherent lattices glue uniquely to $\mathscr L_\Sigma$ in (5.27a), and the construction is independent of the initial presentation, cyclic vectors, coordinates and divisorial bases.

#### 5.27.10. Strict Nilsson membership, including singular points

In a smooth normalized constant-residue chart, write $A=\lambda I+N$ on each generalized eigenspace, $N$ nilpotent. A horizontal dual frame contributes
\[
q^A=q^\lambda\sum_{j=0}^{r-1}\frac{N^j(\log q)^j}{j!}.       \tag{5.27o}
\]
Pairing with a coefficient section whose vector is holomorphic gives (5.27b). Conversely this generalized-power criterion implies that the meromorphic coefficient vector in the normalized frame is holomorphic. Here is the needed independence argument. Continuation $T$ around $q=0$ fixes every single-valued meromorphic coefficient, multiplies $q^\lambda$ by $e^{2\pi i\lambda}$, and replaces $\log q$ by $\log q+2\pi i$. Distinct classes in $\Sigma$ have distinct multipliers. Polynomial projectors in $T$, using the relatively prime powers of $T-e^{2\pi i\lambda}$, isolate each exponent class. On one class repeated finite differences lower the log degree and extract the nonzero constant multiple of its top coefficient. Descending induction proves independence of all log powers over single-valued meromorphic coefficients. In (5.27o), the log-degree-zero coefficient of each generalized block is the original meromorphic coefficient vector. If the same scalar pairing has an expansion with holomorphic coefficients, independence makes this vector holomorphic. This proves the converse. Changes of horizontal dual basis are constant invertible matrices and preserve the criterion. The finite nilpotent logarithm matrices retain every Jordan block. This proves the smooth strict Nilsson coefficient criterion with all nilpotent blocks. It asserts no microlocal order formula.

**Strict growth equivalence on the actual meromorphic object.** In this smooth chart the preceding coefficient criterion is also equivalent to a bound by a power of \(1+\log(1/|q|)\), for all scalar pairings with a horizontal dual basis on bounded-angle sectors and compact parameter sets. If the coefficient vector is holomorphic, it is bounded on a smaller closed product domain. For \(0\leq\operatorname{Re}\lambda<1\), the modulus of \(q^\lambda\) is bounded on a sector of bounded argument as \(q\to0\). The nilpotent exponential is a finite logarithmic polynomial. Thus every pairing has the stated bound, uniformly on smaller compact parameter sets.

Conversely suppose the meromorphic coefficient vector has a pole. Project to a generalized block with a pole of order \(k\geq1\), using the constant residue projectors, and fix a parameter value where its leading vector \(v\) is nonzero. In that block the leading term after multiplication by \(q^A\) is
\[
q^{\lambda-k}
\sum_{j=0}^{r-1}\frac{N^jv}{j!}(\log q)^j .
\]
This vector polynomial is nonzero, since its constant coefficient is \(v\). A constant dual row has a nonzero leading logarithmic coefficient. On a fixed radial direction, that row grows as a nonzero constant times
\[
|q|^{\operatorname{Re}\lambda-k}
(\log(1/|q|))^j
\]
for some \(j\geq0\). Higher Taylor terms are smaller by a positive power of \(|q|\), times a bounded-degree logarithmic polynomial, and cannot cancel this leading term. Since \(\operatorname{Re}\lambda-k<0\), the leading expression exceeds every fixed logarithmic power. This contradicts the assumed bound. Constant changes of the horizontal dual basis preserve a common bound and allow the block and row to be isolated. A nonzero leading coefficient germ is nonzero on an open parameter set, so this argument excludes every pole germ. It proves the strict logarithmic-growth meaning of the coefficient criterion and gives no microlocal order or half-order formula.

Let $s\in\mathscr E_p$ be meromorphic on an actual neighbourhood and satisfy this strict condition on its smooth divisor locus. Then $s\in L_i$ at each divisor prime: on a nonempty smooth open part the condition says precisely membership in the normalized lattice, and a negative $q_i$-valuation would be detected on such a part. At other height-one primes $s\in E=(F_0)_\mathfrak p$. Formula (5.27m) gives $s\in L$. Conversely a section of $L$ belongs to the normalized logarithmic lattice at every smooth point by §5.27.9, and therefore satisfies (5.27b). This proves strict membership at every singular point by actual finite coefficient dual inequalities, without assuming a coherent $j_*$ extension.

In particular the intrinsic sheaf
\[
\{s\in\mathscr E:\ \begin{gathered}s\text{ is strict on the}\\\text{smooth divisor locus}\end{gathered}\}
\]
is the coherent sheaf $\mathscr L_\Sigma$. The left hand side is not defined as arbitrary functions on the complement: its sections are already actual finite-pole meromorphic sections of $\mathscr E$. This distinction prevents an unwarranted meromorphic-extension assertion.

#### 5.27.11. Local saturation and singular-stratum stability

Take $g\in H$ nonzero on every local component of $Y$, $s\in E$, and $g^ms\in L$. At each divisor prime $q_i$, $g$ is a unit, by §5.27.1 and $q_i\nmid g$. Hence $s\in L_i$. At every other height-one prime, $s\in E=(F_0)_\mathfrak p$ already. Formula (5.27m) gives $s\in L$. This proves the saturation assertion for all actual meromorphic sections of $E$, a stronger coefficient statement than an adjacent-lattice version restricted to $h^{-1}L/L$.

Let $Q\subset E/L$ be a coherent $\mathcal O$-**submodule** whose support lies in a closed analytic subset $W\subset Y$ containing no local component of $Y$. Choose $g\in I_W$ avoiding the finitely many divisor primes by finite linear avoidance. For a local class $u\in Q$, its cyclic submodule $\mathcal O u$ is coherent. Its support is the zero set of its coherent annihilator ideal, by the finite-presentation support proof, Theorem 3.1. Since that support is contained in $W$, the analytic Nullstellensatz, Theorem 5.1, whose finite parametrization and prime-vanishing proofs are in §§3–4, makes some power $g^m$ belong to its annihilator. Represent $u$ by an actual section $s\in E$. Then $g^ms\in L$, and the proved saturation makes $s\in L$. Thus $u=0$. Applying this to the finite generators gives $Q=0$.

The same conclusion holds for an actual class $s\bmod L$ supported in $W$. Its cyclic $\mathcal O$-module is coherent, as can be checked without assuming $E/L$ coherent. Projectivity of $E$ over $R$ in §5.27.3 gives a split embedding into a finite free $R$-module; its finitely many matrices and identities extend on a neighbourhood. Represent the finitely many generators of $L$ and the single $s$ in that embedding and clear one common finite pole denominator. The resulting holomorphic matrix into a finite free $\mathcal O$-module has coherent kernel by the earlier actual Oka proof. Hence $L+\mathcal O s$ is coherent, and $(L+\mathcal O s)/L$ is the coherent cyclic module of the class. The preceding annihilator argument applies.

This assertion is restricted to submodules and actual supported classes. Arbitrary coherent **subquotients** of $E/L$ can have support in $W$. For example, with $H=\mathbf C\{x,y\}$, $Y=(y)$, $E=H[y^{-1}]$ and $L=H$, the coherent first-pole submodule $y^{-1}H/H\simeq H/(y)$ has the nonzero quotient $H/(x,y)$ supported at $W=\{(0,0)\}$. Saturation of $L$ under $x$ does not make this quotient vanish. The corrected submodule/class assertion is a coefficient support-vanishing consequence of the constructed lattice, rather than a hypothesis used to construct it.

§5.27.9 has already proved stability at **each singular point**, with no smoothness or generic-stratum qualification. Thus every finite composition of holomorphic multiplication and logarithmic covariant derivatives preserves $\mathscr L_\Sigma$. This assertion concerns the actual logarithmic differential-operator algebra for the divisor. Identifying it with the relevant generators of a microlocal $\mathscr E_\Lambda$-algebra is an additional step, not a change of notation.

#### 5.27.12. Characteristic cutoffs and proper images

Theorem 5.27 constructs the coefficient lattice under the analytic divisorial hypotheses of §5.27.1, at every point of a reduced analytic hypersurface. Its initial finite coefficient lattice in §5.27.3 is an actual local $\mathcal O_X$-lattice in a supplied finite meromorphic connection. Section5.90, GG.A–GG.G constructs the initial ordinary generator on a neighbourhood of every compact set.

The characteristic cutoff across singular strata uses the following mathematical comparisons, proved in Sections5.28,5.87 and5.89:

1. The actual generic-position contact map of Section5.28 and the faithful finite D-type realization of Section5.87 retain the entire module on a common neighbourhood.
2. Section5.87 supplies the finite meromorphic target and generic divisorial regularity needed by §5.27.1.
3. Section5.89, CK.A–CK.K proves the exact source/function cutoff correspondence, including every resonant logarithm and the half-order sign.
4. CK.J transfers the coefficient saturation to the adjacent characteristic quotient $h^{-1}L_0/L_0$ and CK.K proves preservation by every generator of the characteristic operator algebra.

Section5.89, CK.A–CK.K constructs coherent characteristic cutoffs at every real bound and the everywhere stable generating lattice. Section5.90, GG.A–GG.G supplies the initial analytic generators near compact sets. The full arbitrary analytic proper/support-proper direct-image theorem requires its further argument.

The figure shows the exact real slice $Y:\ y(y^2-x^3)=0$ in $\mathbf C^2$. Its residue shift, finite double dual, height-one membership, saturation and logarithmic stability have their full proofs in §§5.27.4–5.27.11. The curve picture illustrates singularities and component intersections; the theorem applies in arbitrary complex dimension. The final panel identifies the realization and order comparisons in Sections5.28,5.87 and5.89.

![Finite normalized divisorial lattices and their coherent strict coefficient lattice](assets/singular-divisor-strict-coefficient-lattice.png)

The reproducible drawing source retains the exact coordinates, shift signs and hypotheses. Human context is Kashiwara–Kawai, *Holonomic Systems III* (1981), §§1–2 and Theorem 5.1.6, [freely available](https://www.kurims.kyoto-u.ac.jp/~kenkyubu/kashiwara/HolIII.pdf). The complete coefficient proof above and the earlier linked analytic proofs establish the statements here; the reference accompanies the complete programme proofs; the arbitrary analytic proper-image theorem requires its further argument.


### 5.28. Convergent contact quantization and whole-germ generic position

This leaf proves an actual finite-order, factorial-growth symbol quantization of a particular homogeneous contact transformation. It also proves the geometric generic-position reduction, under an explicit tangent-cone condition which is verified below for a reduced analytic Legendrian germ. The proof includes the whole germ and all its branches. It does not establish the faithful finite D-type embedding of an arbitrary regular holonomic microlocal module. Section Q.8 records the exact analytic obligations still preventing that conclusion.

The actual symbol, product and finite-order shift proofs are [§5.19](#5-19-actual-factorial-growth-symbols-and-ordered-jets) and [§5.22](#5-22-exact-faithful-positive-degree-completion-at-the-dummy-zero-section). The elementary separability, primitive-element, Gauss, normality and radical arguments needed by the analytic parametrization below are proved in [§5.27.1](#5-27-1-hypotheses-conventions-and-earlier-analytic-proofs). The remaining local analytic inputs are finite irreducible decomposition, Proposition1.2, adapted projection and cone bound, Corollary2.3, holomorphic minimal-polynomial coefficients, Lemma3.2, finite parametrization, Theorem4.1, and Nullstellensatz/dimension, Theorems5.1/5.4 and Proposition5.5. Their full proofs, rather than external contact or curve-selection theorems, are used here.

#### Q.1. Coordinates and the contact transformation

Let the base dimension be $d+1$, with coordinates $u=(t,x_1,\ldots,x_d)$, covectors $\rho=(\tau,\xi_1,\ldots,\xi_d)$, and $\tau\ne0$. Set $z=\xi/\tau$ and $h=\tau^{-1}$. The distinguished point is $p_0=(u,z,\tau)=(0,0,1)$. A linear cotangent coordinate change and a translation put any nonzero covector in this form. Their symbol maps and factorial bounds follow from the actual linear coordinate proof in [§5.55, DP.5](#dp-5-actual-finite-order-straightening-by-a-holomorphic-time-shift).

For a symmetric complex $d\times d$ matrix $C$, put

\[
\begin{gathered}
a_C(\rho)=\frac{\xi^{\mathsf T}C\xi}{2\tau},\\ b_C(\rho)=\partial_\rho a_C(\rho)\\ =\left(-\frac{\xi^{\mathsf T}C\xi}{2\tau^2},\frac{C\xi}{\tau}\right).
\end{gathered}
\tag{Q.1}
\]

Define $\kappa_C(u,\rho)=(u+b_C(\rho),\rho)$. Thus

\[
\begin{gathered}
t'=t-\tfrac12 z^{\mathsf T}Cz,\\ x'=x+Cz,\\ z'=z,\qquad\tau'=\tau .
\end{gathered}
\tag{Q.2}
\]

This is a holomorphic germ with inverse $\kappa_{-C}$, fixes $p_0$, and commutes with covector dilation. If $\lambda=\tau\,dt+\xi^{\mathsf T}dx$, direct differentiation gives

\[
\begin{gathered}
dt'=dt-z^{\mathsf T}C\,dz,\\ dx'=dx+C\,dz,\\ \kappa_C^*\lambda=\lambda.
\end{gathered}
\tag{Q.3}
\]

Consequently it is a homogeneous canonical transformation, preserving the contact form $\alpha=dt+z^{\mathsf T}dx$ on the section $\tau=1$. There is no sign or scale factor in (Q.3).

#### Q.2. A linear incidence lemma

Let $K\subset\mathbf C_x^d\oplus\mathbf C_z^d$ be a closed complex cone whose projectivization is a closed complex analytic set of dimension at most $d-1$. Then one can choose $C=C^{\mathsf T}$, as small as desired, with

\[
\begin{gathered}
K\cap\{(x,z):x+Cz=0\}\\ =\{0\}.
\end{gathered}
\tag{Q.4}
\]

Here no smoothness or isotropy of $K$ is required. Let $s=d(d+1)/2$. Consider the pairs $([x:z],C)$ with $[x:z]\in\mathbf P(K)$ and $x+Cz=0$. If $z=0$, there are no such pairs. If $z\ne0$, the linear map on symmetric matrices $C\mapsto Cz$ is surjective. Indeed choose $j$ with $z_j\ne0$. For a desired vector $v$, set $C_{ij}=C_{ji}=v_i/z_j$ for $i\ne j$, set the entries outside row and column $j$ to zero, and set

\[
 C_{jj}=\frac{v_j-\sum_{i\ne j}(v_i/z_j)z_i}{z_j}.
 \tag{Q.5}
\]

The bad matrices have real $2s$-dimensional measure zero. The following proves this without a projected-incidence algebraicity or elimination assertion.

Cover $\mathbf P(K)$ by countably many smooth complex manifold pieces of dimension at most $d-1$. Here is the actual construction using the earlier analytic-germs proof. In a sufficiently small coordinate neighbourhood, decompose its reduced germ into finitely many irreducible components, by Proposition 1.2. For each component the complete finite-parametrization Theorem 4.1 supplies a nonzero discriminant whose complement is a smooth manifold of the component's dimension. The leftover is a proper analytic subset, and Proposition 5.5 proves its strictly smaller dimension. Apply the same construction to each of its finitely many components. Dimension drops strictly, so at most $d$ levels suffice. Use a countable coordinate cover at each level, since projective space and all its subsets are second countable. This produces countably many smooth pieces covering the set; disjointness is unnecessary. These are the actual discriminant and proper-subset dimension proofs at the stated earlier locators, rather than an imported stratification theorem.

On a piece of complex dimension $k\le d-1$, restrict to the finitely many charts $z_j\ne0$. Formula (Q.5) solves the $d$ equations $Cz=-x$; the remaining $s-d$ entries are free. The incidence is locally the product of that piece with $\mathbf C^{s-d}$. Its real coordinate dimension is $r=2k+2(s-d)\le2s-2$. The matrix projection is smooth and thus Lipschitz on each compact coordinate cube. Countably many such cubes cover this incidence: exhaust each coordinate domain by closed cubes contained in it, and exhaust the free affine coordinates by bounded cubes.

A Lipschitz image of an $r$-dimensional real cube in $\mathbf R^{2s}$, for $r<2s$, has $2s$-dimensional outer measure zero. Subdivide a cube of fixed side into at most $A\varepsilon^{-r}$ cubes of side at most $\varepsilon$. For Lipschitz constant $L_0$, each image lies in a $2s$-cube of side at most $2L_0\sqrt r\,\varepsilon$. The sum of covering volumes is at most $A(2L_0\sqrt r)^{2s}\varepsilon^{2s-r}$, which tends to zero. A constant image when $r=0$ has measure zero directly. Countable subadditivity here follows directly: for the $n$th null set choose a cover of total volume less than $\eta/2^n$, and combine the covers to get total volume less than $\eta$. Thus the entire bad matrix set has measure zero. Every nonempty matrix ball has positive volume, and hence contains a good $C$. This proves the arbitrarily small choice. If $d=0$, the projective direction set is empty and $K=\{0\}$, so no choice is required.

#### Q.3. The entire analytic germ, including singular points

Let $A\subset(T^*X\setminus0,p_0)$ be a reduced homogeneous analytic Lagrangian germ. On $\tau=1$, let $L\subset(\mathbf C^{2d+1}_{t,x,z},0)$ be its reduced Legendrian germ. Its components have dimension $d$, and $\alpha|_{L_{\rm reg}}=0$.

Here is the slicing calculation. The covector Euler field $\mathcal E=\tau\partial_\tau+\sum_i\xi_i\partial_{\xi_i}$ is tangent to the regular locus of the conic germ, by differentiating dilation. Direct contraction gives $\iota_{\mathcal E}(d\lambda)=\lambda$. Lagrangian isotropy says that $d\lambda$ vanishes on the tangent space; since $\mathcal E$ is in that space, $\lambda$ vanishes there too. On $\tau\ne0$, the coordinates $(u,z,\tau)$ and conicity identify $A$ with the product of its $\tau=1$ slice and a nonzero $\tau$ disc. Thus its pure dimension $d+1$ gives pure dimension $d$ for $L$, and the product identifies regular loci. The pullback of $\lambda/\tau$ to the slice is $\alpha=dt+z^{\mathsf T}dx$, so $\alpha$ vanishes on $L_{\rm reg}$. Its projective tangent directions form a closed analytic set of dimension at most $d-1$, and their cone lies in $\{t=0\}$. We prove exactly these analytic facts below; tangent-cone algebraicity is unnecessary.

For an analytic germ $Z\subset(\mathbf C^N,0)$ of dimension at most $d$, construct its projective graph closure by finite equations. On the chart with projective coordinate $j$ equal to one, put $q_j=\ell$, $q_i=\ell v_i$. Substitute into finite equations of $Z$, radicalize their ideal, and saturate that radical ideal $J$ by $\ell$. The exact analytic Nullstellensatz, Theorem 5.1, and finite irreducible decomposition, Proposition 1.2, identify $J$ with the finite intersection of the component prime ideals. Saturation commutes with this finite intersection. For a component prime $\mathfrak p$, its saturation is $\mathfrak p$ if $\ell\notin\mathfrak p$, and the whole ring if $\ell\in\mathfrak p$. Thus its saturated zero set is exactly the union of the components not contained in $\ell=0$. No primary-decomposition theorem is used. The ascending saturation chain stabilizes by the earlier holomorphic Noetherian proof, so these are finite analytic equations near each specified chart point.

The nonzero-$\ell$ part is the graph of $q\mapsto[q]$. It is open and dense in each retained component: its complement is a proper analytic subset, which has smaller dimension and empty interior by the complete earlier Proposition 5.5. That open part is isomorphic to an open part of $Z$, and hence the retained component has dimension at most $d$, by Theorem 5.4, which identifies local finite-projection dimension with the dimension of each regular piece. Proposition 5.5 again bounds its intersection with $\ell=0$ by $d-1$. These constructions agree on chart overlaps because each describes the closure of the same graph, so they define a closed analytic set in the projective fibre. Every rescaled limit direction is in that set: the finite saturated equations vanish on the nonzero graph and persist under limits. The converse follows from the curve construction below. Compactness of projective space then gives a finite coordinate cover around the closed direction set.

Fix an exceptional point $(0,v_0)$ on a retained reduced irreducible component $B$. Apply the earlier finite-parametrization Theorem 4.1 to this component at that exact point, translated to zero. Its adapted projection $\pi_B$ to a polydisc has central fibre equal to the chosen point: the cone bound in Corollary 2.3 says that all complementary coordinates are bounded by a constant times the base coordinates. The projection is proper, and off its nonzero discriminant it is a finite holomorphic covering.

The nonzero function $\ell$ in the component domain has a monic minimal polynomial over the fraction field of the projection base, with holomorphic coefficients by Lemma 3.2. Its constant coefficient is nonzero, since a nonzero element of a field cannot have an irreducible minimal polynomial divisible by $T$. Denote that coefficient by $g_\ell$. At every point with $\ell=0$, $g_\ell$ vanishes on the base. We can exclude any additional proper analytic subset in the same way: choose a holomorphic function nonzero on $B$ that vanishes on that subset, and use the nonzero constant coefficient of its minimal polynomial. For the regular-locus condition, the nonzero graph in $B$ is an open part of one irreducible component of $Z$. Pull back its finite-parametrization discriminant, and also exclude its intersections with the finitely many other components. Each intersection is proper, since the distinct component prime ideals are incomparable. For each intersection choose an equation of the other component which is nonzero on the chosen component, and pull it back to $B$. The resulting finite product is nonzero on the graph and hence on $B$. Outside its zero set the original point belongs to exactly one component and is regular there, so it belongs to $Z_{\rm reg}$.

Choose a complex line through the projection base origin which avoids the initial homogeneous zero set of the product of these finitely many nonzero holomorphic functions and the projection discriminant. Such a direction exists since a nonzero polynomial has a nonempty complement. On a sufficiently small punctured disc of this line, all those functions are nonzero. The finite covering over that punctured disc has a finite sheet permutation as its monodromy. Replace the disc parameter by $s^m$, with $m$ a multiple of the permutation orders; each sheet is then a single-valued holomorphic map.

The covering argument is elementary. Continue a local inverse along a path by a finite subdivision into evenly covered discs; uniqueness on overlaps gives a unique lift. Subdividing a homotopy into small rectangles shows that its endpoint depends only on the path class. A punctured disc retracts radially to a circle. Local continuous arguments lift a circle path to a real angle; its endpoint change is $2\pi$ times an integer, and linear interpolation of angles shows that this integer determines the homotopy class. Monodromy is therefore generated by one finite permutation. The substitution $r=s^m$ changes this generator to its $m$th power, which is the identity for the chosen $m$. Continuation of each sheet is single valued, and local holomorphic inverses make the sheet holomorphic. Properness and the cone bound keep its complementary coordinates bounded and force their limit to the chosen central point. The bounded one-variable removable-singularity argument from the Cauchy integral extends it across zero.

For completeness, if $F$ is a bounded holomorphic coordinate on a punctured disc, put $H(s)=s^2F(s)$ there and $H(0)=0$. Then $H$ is continuous and complex differentiable at zero, with derivative zero, since $H(s)/s=sF(s)\to0$. It is holomorphic on the disc. Its Taylor expansion has zero constant and linear coefficients, so $H=s^2G$ for a holomorphic $G$, which equals $F$ off zero. This proves the required extension for every coordinate. The analytic equations persist under the extension by the holomorphic identity theorem. Thus it gives a holomorphic curve through the specified $(0,v_0)$, with $\ell(s)\ne0$ for $s\ne0$, and with original image in $Z_{\rm reg}$ off zero. There is no assumption that a curve chosen elsewhere passes through the assigned exceptional point.

Apply this to $Z=L$. Since $\ell(s)$ is nonzero off zero, it has a finite positive vanishing order $k$, and its image curve in $L$ has the expansion

\[
\begin{gathered}
(t(s),x(s),z(s))\\ =s^k(t_k,x_k,z_k)+O(s^{k+1}),\\ k\ge1.
\end{gathered}
\tag{Q.6}
\]

It lies in the regular Legendrian locus off zero. Hence $t'(s)+z(s)^{\mathsf T}x'(s)=0$. The second term has order at least $2k-1$, whereas a nonzero $t_k$ gives order $k-1$ in the first term. Thus $t_k=0$. Every exceptional direction has just been realized by such a curve, proving that the whole tangent-direction cone lies in $\{t=0\}$. Conversely that curve, divided by its leading scalar, realizes its exceptional direction as a rescaled limit, completing the identification with the tangent directions.

Identify these directions with a projective analytic set in $\mathbf P(\mathbf C_x^d\oplus\mathbf C_z^d)$ and apply Q.2 to its cone $K$. Then $\kappa_C(L)\cap\{t'=0,x'=0\}$ has the origin as an isolated point. Otherwise a sequence of nonzero points in this intersection tends to zero. Normalize their preimages by Euclidean length and take a convergent subsequence on the unit sphere. Its nonzero limit $v$ is a tangent direction. Differentiating (Q.2) at zero gives $t(v)=0$ and $x(v)+Cz(v)=0$, contradicting (Q.4). Shrink one neighbourhood to remove all other fibre points. In the conic cotangent space this is

\[
\begin{gathered}
\kappa_C(A)\cap\pi^{-1}(0)=\mathbf C^*p_0\\ \text{near }p_0 .
\end{gathered}
\tag{Q.7}
\]

The whole characteristic germ is in the generic position of HolIII Definition 1.6.3. Every component and branch is included. This is not a restriction to a smooth generic stratum. A finite union of additional smooth Legendrian germs can be included in the direction set, and the same choice works for all of them.

#### Q.4. The resummed symbol formula

Let $\mathscr E(0)$ be the actual ring of symbols

\[
\begin{gathered}
P(u,\rho)=\sum_{n\ge0}\tau^{-n}p_n(u,z),\\ \sup_K|p_n|\le B_P C_P^n n!,
\end{gathered}
\tag{Q.8}
\]

with all coefficients holomorphic on one common neighbourhood and with the bounds on every compact set, as in [§5.19.1–§5.19.3](#5-19-actual-factorial-growth-symbols-and-ordered-jets). Denote its proved Leibniz product by $\circ$. Finite-order symbols are $\mathscr E=\bigcup_m\tau^m\mathscr E(0)$ as normal symbols; this is the localization by the invertible normal element $h$, whose normality and actual inverse-conjugation estimates are proved in C.1–C.2 of §5.22. One can instead check each finite shift directly by $(n+m)!\le2^{n+m}n!m!$. We use no infinite-order symbol ring here.

Write $a=a_C$, $b=b_C$, $D=d+1$, and set

\[
\begin{gathered}
a_\gamma(z)=\left.\frac{\partial_\rho^\gamma a(\rho)}{\gamma!}\right|_{\tau=1,\xi=z}\\ (|\gamma|\ge2),\\ R_a=\sum_{|\gamma|\ge2}h^{|\gamma|-1}a_\gamma(z)\partial_u^\gamma.
\end{gathered}
\tag{Q.9}
\]

Define $T_C P$ by

\[
\begin{gathered}
T_C P=\sum_{n\ge0}h^n\\ \exp(R_a)\,p_n(u+b(1,z),z).
\end{gathered}
\tag{Q.10}
\]

The exponential in (Q.10) is a notation for finite sums at each homogeneous index. Each occurrence of a multiindex $\gamma$ raises that index by $|\gamma|-1\ge1$. The index-$N$ coefficient therefore has $n\le N$, at most $N-n$ factors, and total base differentiation at most $2(N-n)$. All coefficient functions are defined on a common smaller neighbourhood on which $u+b(1,z)$ stays inside the original domain. Since $b(1,0)=0$, such a neighbourhood exists for every fixed $C$.

#### Q.5. An explicit factorial estimate

Choose compact nested coefficient polydiscs so that, at every shifted point $u+b(1,z)$ in the smaller one, the base-coordinate polydisc of radius $\epsilon>0$ remains in the coefficient domain. Choose $\delta>0$ so that the cotangent polydisc of radius $\delta$ around $(1,z)$ avoids $\tau=0$ and remains in one bounded set for all $z$ in the smaller compact. Let $A_a\ge0$ bound $|a|$ there. Iterated Cauchy estimates give

\[
\begin{gathered}
|a_\gamma(z)|\le A_a\delta^{-|\gamma|},\\ |\partial_u^\beta p_n(u+b,z)|\\ \le B_P C_P^n n!\,\beta!\epsilon^{-|\beta|}.
\end{gathered}
\tag{Q.11}
\]

Put

\[
\begin{gathered}
L=32D^2\max(1,A_a)\\ \max\bigl(1,(\delta\epsilon)^{-1}\bigr)^2.
\end{gathered}
\tag{Q.12}
\]

**Theorem Q.1.** The index-$N$ coefficient $q_N$ of $T_C P$ satisfies

\[
 \sup|q_N|\le B_P(C_P+L)^N N! .
 \tag{Q.13}
\]

**Proof.** In a term with $l\ge1$ ordered factors, write $k_i=|\gamma_i|\ge2$, $k=\sum k_i$, and $r=\sum(k_i-1)=k-l\ge1$. It contributes to $N=n+r$. The product of coefficient bounds and the derivative bound in (Q.11), including $1/l!$ from the exponential, is at most

\[
 B_P C_P^n n!\,
 \frac{k!}{l!}\,A_a^l(\delta\epsilon)^{-k}.
 \tag{Q.14}
\]

Indeed $\beta!\le k!$ for $\beta=\sum\gamma_i$. Since $l\le r$ and $k=r+l\le2r$,

\[
 \frac{k!}{l!}=\binom{k}{l}r!\le 2^k r!\le4^r r! .
 \tag{Q.15}
\]

For fixed $r,l$, the number of ordered length lists $k_i\ge2$ with $\sum(k_i-1)=r$ is $\binom{r-1}{l-1}\le2^r$. For each such list, the number of multiindex lists is at most $D^k\le D^{2r}$, by the word-count argument in S.1. Summing over $1\le l\le r$ gives at most $r2^rD^{2r}\le(4D^2)^r$ lists. This slightly sharper count would give $16D^2$ in (Q.12); the stated $32D^2$ is a valid uniform upper bound. The remaining factors in (Q.14) are at most $\max(1,A_a)^r\max(1,(\delta\epsilon)^{-1})^{2r}$. Thus all terms of total loss $r$ are bounded by $B_PC_P^n n!L^r r!$. For $r=0$, the sole term is $p_N(u+b,z)$ and has bound $B_PC_P^N N!$. Sum over $n+r=N$, use $n!r!\le N!$, and use the nonnegative expansion of $(C_P+L)^N$, whose monomial coefficients are at least one. This proves (Q.13). The finitely shifted case $\tau^mP$ has the same base-derivative estimate with its fixed normal prefactor and is treated identically. $\square$

Hence (Q.10) maps the actual symbol class to itself, with a common domain and a verified factorial estimate. It is not an argument that a formal series converges for a numerical value of $h$, and it does not assert that $\exp(a_C(\partial))$ belongs to $\mathscr E$.

#### Q.6. Multiplication and the inverse

**Theorem Q.2.** $T_C$ is a sheaf algebra isomorphism on the corresponding germs of the finite-order symbol ring. It preserves order, has inverse $T_{-C}$, fixes $h$ and $\tau$, and its principal-symbol map is pullback by $\kappa_C$.

**Proof.** We first verify multiplication using the analytic generating functions $e^{v\cdot u}f(\rho)$, where $v$ is an auxiliary parameter and $f$ is a holomorphic homogeneous symbol independent of $u$. For these, the Leibniz formula [5.19c](#5-19-2-multiplication-with-all-cotangent-derivatives-retained) and (Q.9)–(Q.10) give the coefficientwise identities

\[
\begin{gathered}
(e^{v\cdot u}f)\circ(e^{w\cdot u}g)\\ =e^{(v+w)\cdot u}f(\rho+w)g(\rho),\\ T_C(e^{v\cdot u}f)\\ =e^{v\cdot u}e^{a(\rho+v)-a(\rho)}f(\rho).
\end{gathered}
\tag{Q.16}
\]

Translations in the covector are expanded in homogeneous order. The second exponential has leading term $e^{b(\rho)\cdot v}$; its remaining exponent has strictly negative homogeneous order, so each negative-order coefficient is finite. Its formula is exactly (Q.10), not a product of unbounded positive-order operator series. Using (Q.16) twice, the scalar factors in the transformed product telescope:

\[
\begin{gathered}
e^{a(\rho+w+v)-a(\rho+w)}\\ e^{a(\rho+w)-a(\rho)}\\ =e^{a(\rho+v+w)-a(\rho)}.
\end{gathered}
\tag{Q.17}
\]

Thus $T_C(P\circ Q)=T_C(P)\circ T_C(Q)$ for these generating functions, coefficient by coefficient. Differentiating in $v,w$ at zero gives the identity for polynomial coefficient functions in $u$, with arbitrary holomorphic covector coefficients and arbitrary finite order shifts. For general holomorphic coefficients, take their Taylor polynomials on a larger base polydisc. They and each fixed finite number of their derivatives converge uniformly on a smaller compact polydisc by the iterated Cauchy formula. At any fixed homogeneous index the product and transformation formulas use finitely many coefficient indices and finitely many derivatives, evaluated at points in a fixed shifted compact polydisc. The polynomial identity therefore passes to the uniform limit. This proves multiplication for every actual symbol. The estimates S.1 and Q.1 put both sides in the actual ring, so equality of their coefficients is equality of germs.

For the inverse, $b$ and every coefficient in $R_a$ are independent of $u$. The base translation and all base-derivative operators in (Q.9) commute with one another. Replacing $C$ by $-C$ changes $b,R_a$ to $-b,-R_a$. Thus $T_{-C}T_C=T_CT_{-C}=1$, by cancellation of their finite homogeneous coefficients and the opposite base translations. Each composition uses a common smaller domain and the estimate Q.1, so this is an actual germ identity. The leading coefficient of (Q.10) is $p_0(u+b,z)$, proving the asserted principal-symbol pullback. Symbols independent of $u$, including $h$ and $\tau$, are fixed. Order preservation follows from the leading coefficient and the actual inverse. Restriction commutes with every construction, giving the sheaf germ isomorphism. $\square$

Its values on coordinate operators are

\[
\begin{gathered}
T_C(x_i)=x_i+\\ \sum_j C_{ij}\partial_{x_j}\partial_t^{-1},\\ T_C(t)=t-\tfrac12\sum_{i,j}C_{ij}\\ \partial_{x_i}\partial_{x_j}\partial_t^{-2},\\ T_C(\partial_{x_i})=\partial_{x_i},\\ T_C(\partial_t)=\partial_t .
\end{gathered}
\tag{Q.18}
\]

These values also follow from the exact commutators with the constant-coefficient order-one expression $a_C(\partial)$; the higher commutators vanish on the coordinate generators. That finite check by itself would not have established the convergent sheaf isomorphism. Q.1 and Q.2 supply the missing analytic argument.

#### Q.7. Transport of actual finite microlocal modules

Use the convention that $T_C$ pulls symbols back by $\kappa_C$. To transport a left $\mathscr E$-module $M$ to the image germ $\kappa_C(A)$, give its underlying germ the action

\[
 P\cdot_{\rm new}m=T_C(P)\cdot_{\rm old}m.
 \tag{Q.19}
\]

A finite presentation is transported by applying $T_{-C}$ to each old relation; its inverse is (Q.19) with $-C$. Indeed the new annihilator is $T_C^{-1}(\operatorname{Ann}_{\rm old}M)$, whose symbols vanish on $\kappa_C(A)$. Kernels and exact sequences are preserved since this is restriction along a ring isomorphism. A finite order-zero lattice and its principal relation ideal are transported in the same way; their symbol support becomes $\kappa_C(A)$. No flatness theorem is needed for this step.

Definition 1.1.11(i) of HolIII uses, on an involutive characteristic germ $V$, the subalgebra generated by order-one operators whose first symbol vanishes on $V$, and a finite order-zero lattice stable under that subalgebra. Since $T_{-C}$ preserves the order filtration and sends the vanishing ideal of $V$ to that of $\kappa_C(V)$, it transports precisely these operators, their subalgebra, and each finite invariant lattice. Thus regularity *along $V$* is preserved wherever its actual defining lattice is given. Definition 1.1.16 of a regular holonomic module only requires that this property hold outside a nowhere dense set in its characteristic Lagrangian. The biholomorphism $\kappa_C$ preserves that condition as well. We do not strengthen that definition to an invariant lattice at every singular point: HolIII reserves that strengthening for V.1.7. This is a transport statement; it neither constructs the canonical cutoff nor supplies a previously absent invariant lattice on a singular stratum.

The geometric reduction Q.3 together with Q.2–Q.7 therefore removes the need to import an unspecified quantized contact transformation for this generic-position reduction. It does not require a global contact transform, and it makes no assertion about infinite-order rings, infinite-order flatness, or a finite-pole embedding after the reduction.

#### Q.8. Exact remaining realization obligations

HolIII Theorem 4.1.1, printed p. 906 (PDF page 94), starts with a holonomic finite-order $\mathscr E$-module in generic position and constructs a $\mathscr D^\infty$-linear map

\[
\begin{gathered}
\phi:M_{p_0}^{\infty}\longrightarrow N_{q_0}^{\infty},\\ N=L/\mathcal O_X^r,\\ M_{p_0}^{\infty}\longrightarrow\\ \mathscr E_{p_0}^{\infty}\otimes_{\mathscr D_{q_0}^{\infty}}N_{q_0}^{\infty}\\ \simeq\mathscr E_{p_0}^{\infty}\otimes_{\mathscr D_{q_0}}N_{q_0},\\ s\longmapsto1\otimes\phi(s),
\end{gathered}
\tag{Q.20}
\]

with the latter map injective and $\mathscr E^\infty$-linear; $L$ is a finite D-type system. This statement alone is neither a map from the actual finite module into a finite-pole coefficient space nor a proof of its faithfulness. The further regularity argument in V.1.1–V.1.4 is essential. The finite and infinite operator rings in this target map are distinct.

The realization and regularity inputs are as follows. Each result is used only with its complete programme proof.

1. **Finite generic-position ordinary realization.** Under the supplied actual finite order-zero lattice and finite principal fibre of G.1, [§5.29, G.1–G.5](#5-29-actual-finite-generation-over-the-coordinate-commuting-ring) proves finite generation over the ring commuting with the $x_i$; [§5.30, F.1–F.7](#5-30-actual-polynomial-flatness-and-reconstruction-after-finite-realization) proves actual polynomial flatness, whole-stalk reconstruction and polynomial-submodule stability; and [§5.31, L.1–L.6](#5-31-a-bounded-actual-generic-position-complex-with-polynomial-operator-matrices) proves a bounded free resolution over the actual finite-order microlocal rings with polynomial matrices on one base neighbourhood. The actual bounded complex is not claimed exact over its polynomial coefficient ring. The equality of a characteristic support defined through an infinite-order extension with the actual principal support remains a separate prerequisite if that convention is used. Section5.86, EV.A--EV.F uses the actual special-ring reconstruction of ISR.3 for the whole infinite extension. The commuting ring is distinct from the full order-zero ring bearing the same letter in earlier dummy leaves. No ordinary $h=1$ specialization supplies those remaining inputs. For every coherent actual holonomic germ, §5.32, J.1–J.3 constructs that actual lattice and proves the finite principal fibre after the contact reduction; the actual finite-order support convention is explicit. Section 5.33, K.1–K.3 supplies a convergent ordinary negative-order Volterra kernel complex, with the endpoint relation stated in K.2.

2. **Sectorial complex-analytic operator action and relative vanishing.** HolIII IV.4.4.1 imports Kashiwara–Schapira, *Micro-hyperbolic systems*, Theorem 4.5.1. That theorem is freely available, but its proof reduces to its own Theorem 4.1.1 and Corollary 3.2.5, which require the convergent complex-domain kernel action and division estimates. HolIII weakens its condition (c3) to exactness on $\Omega_1\setminus\Omega_0$. The exact weakened statement, the cone conditions, and every cohomological shift still require a complete proof. A smooth real-symbol propagation or Egorov argument cannot replace the complex factorial-growth action.

3. **Continuation on the entire hypersurface complement.** Section5.82, CF.A--CF.H proves multivalued holomorphic continuation on one common neighbourhood for every prescribed finite list of actual holonomic function solutions, including every characteristic branch. Section5.83, FD.A--FD.F proves finite dimensionality of the entire directional solution space and vanishing of positive Ext. Section5.85, MC.A--MC.G proves actual finite E-linear solution monodromy and a common finite-determination ideal for the chosen generator functions and all their branches. Section5.86, EV.A--EV.F proves continuation of every infinite-extension section, a stable finite function algebra, and a faithful full infinite-order tensor-unit realization along the original hypersurface.

4. **Separation of sections at singular support.** IV.4.6.1 and IV.4.6.2 assert finite dimensionality and injectivity of evaluation against sectorial solutions on $M^\infty$. Section5.84, SE.A--SE.F proves separation for every section of the whole infinite extension, including singular germs and all coefficient tails. The proof uses actual normal-form identity, finite-dimensional cap Hom, injective evaluation-target restriction and purity on a fixed analytic exceptional set. Section5.85, MC.A--MC.G proves actual solution monodromy and common finite function determination. Section5.86, EV.A--EV.F proves continuation of every infinite-extension section, a stable finite function algebra, and a faithful full infinite-order tensor-unit realization along the original hypersurface.

5. **Infinite-order linearity and faithful tensor maps.** IV.4.7.1 explicitly treats $\mathscr E^\infty$-linearity as a substantive step. It uses III.4.9 to enlarge a finite-codimensional monodromy ideal under the full $\mathscr A^\infty$-action, and III.5.5 for the $\mathscr E^\infty\otimes_{\mathscr A^\infty}$ comparison. The finite-order isomorphism in Q.6–Q.7 and the actual positive-degree completion prove neither this kernel action nor infinite-order flatness. Section5.86, EV.A--EV.F supplies the full infinite-order tensor-unit injection using an actual normal-parameter module and finite primitive witnesses, without a mixed-ring flatness premise.

6. **Passage to faithful finite poles for a regular module.** HolIII V.1.1 uses the embedding above and the regular-singular-to-Nilsson implication at generic divisors, then extension to the whole finite D-type system. V.1.2–V.1.4 recover a finite $\mathscr D$-module and an actual finite-pole injection. Section5.87, RF.A--RF.I proves the generic regularity test, finite-pole quotient injection, entire-stalk ordinary finite generation, actual finite and ordinary-infinite reconstruction, and extension on one common neighbourhood with the whole-fibre support bound. The coefficient lattice of §5.27 starts after this passage; using it to assume the passage would be circular.

Section5.89, CK.A–CK.K proves the source/function order criterion, coherent singular cutoffs and everywhere characteristic stability. The arbitrary analytic proper/support-proper theorem requires its further argument. Section5.86, EV.A--EV.F proves Q.20 and the finite D-type tensor realization along the original projected hypersurface; Section5.87, RF.A--RF.I proves finite-pole recovery and both actual reconstructions for regular modules.

#### Q.9. Illustration and reproducibility

The figure below shows a stated real coordinate slice of the exact singular Legendrian germ

\[
\begin{gathered}
x=s^3,\qquad z=s^2,\\ t=-\tfrac35s^5,\qquad C=1,\\ x'=s^3+s^2,\\ t'=-\tfrac35s^5-\tfrac12s^4.
\end{gathered}
\tag{Q.21}
\]

The contact equation is $dt+z\,dx=0$. Its tangent-cone line is the $z$-axis; $x+z=0$ misses that line except at zero. The plotted interval is explicitly $|s|\le0.4$; the other root of the projected equation $x'=0$ at $s=-1$ lies outside the chosen local interval, so the plot asserts only the local germ result. The figure separately displays the symbol-class bound (Q.13), exact inverse, and the unproved finite D-type arrow. It is an illustration of the original contact/quantization argument, not evidence of full realization. The displayed projection and interval illustrate the local mechanism; the proof includes every complex branch.

Free human source context: M. Kashiwara and T. Kawai, *On the holonomic systems of microdifferential equations. III. Systems with regular singularities*, [author-hosted full PDF](https://www.kurims.kyoto-u.ac.jp/~kenkyubu/kashiwara/HolIII.pdf), I.6 and IV.1–IV.7, V.1.1–V.1.4; M. Kashiwara and P. Schapira, *Micro-hyperbolic systems*, [author-hosted full PDF](https://perso.imj-prg.fr/pierre-schapira/wp-content/uploads/schapira-pub/Microhyp.pdf), III–IV. The exact remaining targets are in Q.8. Their citations do not replace any required proof.

![Whole-germ generic position and actual factorial-growth contact quantization](assets/quadratic-contact-shear.png)

*The plotted real slice is the projection of (Q.21), for $|s|\le0.4$ and $C=1$. The entire complex-germ reduction is proved in Q.2–Q.3, and the actual convergent symbol isomorphism in Q.4–Q.7. Section5.86, EV.A--EV.F proves the tensor-unit realization denoted by the dashed finite D-type arrow; Section5.87, RF.A--RF.I proves the regular-module finite-pole passage in Q.8. Free human source: [Kashiwara–Kawai, author-hosted HolIII](https://www.kurims.kyoto-u.ac.jp/~kenkyubu/kashiwara/HolIII.pdf), I.6 and IV.1–IV.7.*


### 5.29. Actual finite generation over the coordinate-commuting ring

This leaf proves a convergent generic-position finite-generation statement. Its precise premise is an actual finite order-zero lattice with finite principal fibre, as in (G.1). Its conclusion includes the entire actual stalk and its singular-support sections. It does not prove the infinite-order extension or faithful finite D-type embedding.

The exact earlier proofs are [§5.19, S.1–S.4](#5-19-actual-factorial-growth-symbols-and-ordered-jets) for the actual symbol class and product; [§5.20, B.1–B.2](#5-20-one-norm-multiplication-units-and-dummy-division) for the complete derivative norm and product estimate; [§5.21, N.1–N.18](#5-21-actual-simultaneous-division-and-finite-stalk-relations) for absolute Taylor weights, completeness, support projection, Dickson heads, monomial division and convergent simultaneous division; and [§5.22, C.1–C.2](#5-22-exact-faithful-positive-degree-completion-at-the-dummy-zero-section) for actual normality and finite order shifts. The analytic Nullstellensatz used in G.1 is Theorem5.1, with its elementary algebra proved in [§5.27.1](#5-27-1-hypotheses-conventions-and-earlier-analytic-proofs). The relative remainder supports and contraction needed here are proved below.

#### G.1. The exact premise and smaller ring

At $p_0=(u,z,\tau)=(0,0,1)$, put $u=(t,x_1,\ldots,x_d)$, $z=\xi/\tau$, $h=\tau^{-1}$. Let $E_0$ be the actual order-zero factorial-growth stalk and $E$ its finite-order localization. Let $M$ be a left $E$-module and let $M_0\subset M$ be generated by finitely many $m_1,\ldots,m_b$ over $E_0$, with $EM_0=M$. Suppose

\[
\begin{gathered}
V=(M_0/hM_0)\otimes_{\mathbf C\{u,z\}}\mathbf C\{z\}\\ \text{is finite dimensional over }\mathbf C,\\ u\longmapsto0 .
\end{gathered}
\tag{G.1}
\]

The image $hM_0$ is a left submodule because $h$ is normal. Its quotient is finite over the commutative symbol ring $E_0/(h)=\mathbf C\{u,z\}$. If the actual principal-symbol support of this lattice meets the base fibre only at $z=0$, (G.1) follows: its fibre is finite over $\mathbf C\{z\}$, the analytic Nullstellensatz puts a power of $(z)$ in its annihilator, and the finite generating list is then spanned by finitely many $z$-monomials over $\mathbf C$. Nilpotents and nonreduced principal modules are included.

Define $A_0\subset E_0$ by requiring every homogeneous coefficient to be independent of $z$. All $\xi$-derivatives in the Leibniz product vanish, so this is a subring; its only internal contractions use $\tau,t$. Put $A=A_0[h^{-1}]$. It is exactly the subring of $E$ commuting with all $x_i$, because

\[
 [P,x_i]=\partial_{\xi_i}P.
 \tag{G.2}
\]

Indeed the right factor $x_i$ has only its zero and first base derivative, while the left factor $x_i$ has no cotangent derivative. Equation (G.2) says that each coefficient has zero $z_i$-derivative. The normal-element formulas of §5.22, C.1–C.2, remain actual in this subring. Here $A$ denotes precisely the coordinate-commuting ring; the full actual order-zero ring is $E_0$.

#### G.2. Finite pure-covector heads

Let $I\subset E_0^b$ be the kernel of $e_j\mapsto m_j$. Let $\bar I\subset\mathbf C\{u,z\}^b$ be its principal image modulo $h$, and let $J\subset\mathbf C\{z\}^b$ be the image of $\bar I$ under $u=0$. Then

\[
 \mathbf C\{z\}^b/J\simeq V.
 \tag{G.3}
\]

The principal-kernel lifting in this assertion is important. If a vector has image $hm$ in $M_0$, lift $m$ to $w\in E_0^b$ and subtract $hw$. The difference lies in $I$ and has the same principal vector. Thus the principal kernel is exactly the principal image of $I$. Tensoring the resulting cokernel presentation with $\mathbf C\{z\}$ is right exact and proves (G.3).

Use the least total $z$-degree, then the least lexicographic exponent, then the least component as the head of a nonzero vector in $J$. Its exponent diagram is upward closed under multiplication by $z$-monomials. The complete Dickson argument of §5.21, N.2 gives finitely many minimal heads $(\gamma_i,j_i)$. Select $v_i\in J$ having those heads, normalized to head coefficient one. Since (G.3) has finite complex dimension $m$, every $z_i$ acts nilpotently: $z_i-\lambda$ is an invertible analytic germ for every nonzero scalar $\lambda$, so its matrix $Z$ has characteristic polynomial $T^m$. The cofactor identity $(TI-Z)\operatorname{adj}(TI-Z)=\det(TI-Z)I$ is a polynomial identity; the adjugate coefficients commute with $Z$, so evaluation at $T=Z$ gives $Z^m=0$. Thus $z_i^mV=0$. The commuting actions then make $(z)^{dm+1}V=0$. This is also immediate when $V=0$. Thus $(z)^k$ annihilates it for some $k$, and every exponent of total degree at least $k$ lies in that head diagram. Consequently

\[
\begin{gathered}
B=\{(\beta,j):\beta\notin\\ \bigcup_{i:j_i=j}(\gamma_i+\mathbf N^d)\}\\ \text{is finite.}
\end{gathered}
\tag{G.4}
\]

Each $v_i$ is the specialization of the principal vector of an actual relation $P_i\in I$: express it as a finite $\mathbf C\{z\}$-combination of the specialized principal images that define $J$, lift those scalar coefficients as symbols depending only on $z$, and take the finite actual left combination. Therefore

\[
 P_i=z^{\gamma_i}e_{j_i}+A_i,
 \tag{G.5}
\]

where the principal specialization of $A_i$ at $u=0$ has only terms after the selected head. We do not assume that this pure-$z$ head is first under the original total-degree ordering in all $(u,z)$ variables.

#### G.3. One actual norm makes the errors small

Use the full absolute Taylor Banach norm N.1–N.7 in variables $(u,z)$, including all $\tau$-derivatives. For a vector use $\|F\|_c=\sum_jc_j\|F_j\|$, with positive component factors. There are fixed positive $u,z$ polyradii, fixed $c_j$, and one derivative parameter $\epsilon>0$ such that

\[
 \|A_i\|_c<\tfrac18c_{j_i}R_z^{\gamma_i}
 \quad\text{for every }i .
 \tag{G.6}
\]

The order of choices proves this precisely. First restrict the principal vectors to $u=0$. The finite weighted-head construction in N.3, now only in $z$, chooses hierarchical rational weights preserving their least-degree/lexicographic heads and making each other exponent have strictly larger weight. Decreasing positive component factors make terms at the same exponent in later components small. With $R_{z,j}=r^{w_j}$, their remaining Taylor-tail norms divided by head radii tend to zero as $r\to0$, by domination by one fixed convergent Taylor series. Fix a sufficiently small $r$.

Next shrink the $u$-radii while holding these $z$-radii fixed. Every additional principal term has positive $u$-degree, since it vanishes at $u=0$. Its absolute Taylor norm tends to zero on shrinking $u$-polydiscs by domination by its common convergent series. The head radii and component factors are now fixed positive numbers. Thus all these ratios become as small as required. This also controls terms linear in $u$ having smaller ordinary total degree than the pure-$z$ head.

Finally shrink $\epsilon$. Positive homogeneous indices in N.2 have positive powers $\epsilon^{2n}$, and each nonzero derivative of a principal coefficient has a positive derivative power. Their total norms tend to zero by domination by the previously finite norm at an earlier $\epsilon_0$. The zero-derivative principal term is twice its absolute Taylor norm, already made sufficiently small. One common $\epsilon$ works for the finite list. This proves (G.6).

The exact norm estimates used in the next step are

\[
\begin{gathered}
\|Q\circ A_i\|_c\le\|Q\|\|A_i\|_c,\\ \|\text{division by }z^{\gamma_i}\|\\ \le R_z^{-\gamma_i}.
\end{gathered}
\tag{G.7}
\]

They are N.4 and the termwise falling/rising factorial estimate N.6. Coefficient support restrictions have norm at most one. No $t$-term or cotangent derivative is removed.

#### G.4. Actual relative division

Partition the union of the head orthants in (G.4) into disjoint assigned supports. Require $Q_i z^{\gamma_i}e_{j_i}$ to occupy its assigned support, and a remainder $S$ to occupy its complement $B$ in the $z$-exponent and component only. All $u$-Taylor exponents and homogeneous indices remain allowed. The ordinary head map is

\[
\begin{gathered}
L_0(Q,S)=\sum_iQ_i z^{\gamma_i}e_{j_i}+S,\\ D=L_0^{-1}.
\end{gathered}
\tag{G.8}
\]

Use the quotient/remainder norm

\[
\begin{gathered}
\|(Q,S)\|_*=\\ \sum_i c_{j_i}R_z^{\gamma_i}\|Q_i\|+\|S\|_c.
\end{gathered}
\tag{G.9}
\]

These are Banach spaces: their support subspaces are closed in the weighted coefficient $\ell^1$ spaces proved complete in N.1. The assigned supports are disjoint. Before division their absolute weighted coefficient contributions add. Dividing by a pure-$z$ monomial only decreases the base falling factors, cotangent falling factors, and nonnegative initial values of the cotangent rising factors in N.6. Therefore

\[
 \|DF\|_*\le\|F\|_c .
 \tag{G.10}
\]

No factor equal to the number of heads occurs.

Set $L(Q,S)=\sum_iQ_i\circ P_i+S$. Since each right head is independent of every base variable, $Q_i\circ z^{\gamma_i}=Q_i z^{\gamma_i}$ exactly. Hence $L=L_0+E$, with $E(Q,S)=\sum_iQ_i\circ A_i$, and

\[
\begin{gathered}
\|E(Q,S)\|_c\le\tfrac18\|(Q,S)\|_*,\\ \|DE\|\le\tfrac18 .
\end{gathered}
\tag{G.11}
\]

Consequently

\[
\begin{gathered}
(Q,S)=(1+DE)^{-1}DF\\ =\sum_{\ell\ge0}(-DE)^\ell DF
\end{gathered}
\tag{G.12}
\]

converges in the specified Banach space and solves $F=\sum_iQ_i\circ P_i+S$. The zero-derivative estimate N.3 gives a common coefficient domain and actual factorial bounds for the resulting symbols. This is convergent division, not formal cancellation.

For an arbitrary input germ, intersect its representative domain with the finite divisor domains and repeat the finite radius choices of G.3 there. Thus every germ has an actual division. The finite divisors, complement $B$, and eventual generators do not depend on the input. Its particular norm and division radii may depend on it, as a stalk argument permits.

#### G.5. Finite generation of the entire actual stalk

Every remainder has the finite form

\[
\begin{gathered}
S=\sum_{(\beta,j)\in B}r_{\beta,j}(u,h)z^\beta e_j,\\ r_{\beta,j}\in A_0.
\end{gathered}
\tag{G.13}
\]

Indeed extracting each of the finitely many $z$-Taylor coefficients on a fixed smaller polydisc is bounded by a fixed Cauchy factor $R_z^{-\beta}$. Its common $u$-domain and homogeneous factorial bound remain. Also $r_{\beta,j}\circ z^\beta=r_{\beta,j}z^\beta$, since the right factor has no base derivative.

The $P_i$ are actual relations in $I$. Dividing a representative of an arbitrary $m\in M_0$ therefore proves

\[
 M_0=\sum_{(\beta,j)\in B}A_0\,z^\beta m_j .
 \tag{G.14}
\]

Finally every finite-order symbol is a left shift $h^{-k}$ of an order-zero symbol, for a finite $k$, by the actual normal-element shift formulas. Since $EM_0=M$, every element of $M$ is a finite sum of $h^{-k}m$ with $m\in M_0$. These shifts belong to $A$, yielding

\[
 M=\sum_{(\beta,j)\in B}A\,z^\beta m_j .
 \tag{G.15}
\]

**Theorem G.1.** Under the actual-lattice premise and finite principal fibre (G.1), $M_0$ is finite over $A_0$, and the entire actual stalk $M$ is finite over $A$, with the explicit finite lists (G.14)–(G.15). Forgetting to the smaller ring keeps the underlying module and every section, so this stalk realization is faithful. Singular-support sections are included.

No regularity assumption was used. This theorem cannot convert arbitrary infinite-growth solutions into finite-pole solutions.

#### G.6. Exact interface and unproved boundaries

For every coherent actual holonomic microlocal germ, [§5.32, J.1–J.3](#5-32-coherent-actual-lattices-and-principal-support-for-the-whole-germ) constructs a finite order-zero lattice, proves equality of its reduced principal support with the actual module support, and proves the finite principal fibre after generic-position reduction. The generic-position geometry and actual convergent lattice transport are [§5.28, Q.3 and Q.6–Q.7](#5-28-convergent-contact-quantization-and-whole-germ-generic-position). Thus (G.1) holds for the entire coherent actual holonomic germ, and Theorem G.1 supplies its finite coordinate-commuting-ring realization. If the starting convention defines characteristic support instead by an infinite-order or microfunction extension, equality with the support of $M_0/hM_0$ remains the separate extension-faithfulness prerequisite stated in J.2. It is not inferred from infinite-order flatness.

G.1 alone does not give a bounded finite free resolution over $A$ or its polynomial differential extension $R$. The finite-order comparison is proved in [§5.30, F.1–F.7](#5-30-actual-polynomial-flatness-and-reconstruction-after-finite-realization), and [§5.31, L.1–L.6](#5-31-a-bounded-actual-generic-position-complex-with-polynomial-operator-matrices) constructs a bounded free resolution over the actual microlocal rings, with polynomial matrices and one base neighbourhood, under the same supplied-lattice and finite-principal-fibre premise. Infinite-order base change, a convergent sectorial operator action, relative solution vanishing, continuation and separation at singular support, faithful finite D-type injection, ordinary finite-pole recovery and microlocal order comparison remain the obligations in Q.8. Earlier positive-degree completion is a different functor.

The characteristic-stable lattice is proved in Section5.89, CK.A–CK.K. Arbitrary analytic proper/support-proper regularity requires its further argument. Free human context: Kashiwara–Kawai, [HolIII](https://www.kurims.kyoto-u.ac.jp/~kenkyubu/kashiwara/HolIII.pdf), III.5.5. Its statement is not used as the proof of G.1. The full argument above proves the stated finite-stalk theorem.

#### G.7. Exact illustration and a noncommutative division check

The figure below uses the two selected heads $z_1^2,z_2^3$ in one component. The first orthant is assigned to the first head; the portion of the second orthant outside the first is assigned to the second. The complement is the six-element list $1,z_1,z_2,z_1z_2,z_2^2,z_1z_2^2$. They are candidate module generators, not a claim of linear independence or free rank. The figure displays the actual norm bound $1/8$, the Banach inverse (G.12), the entire-stalk conclusion, and the still unproved finite D-type arrow. Its axes show the two pure-covector exponents; every base exponent and homogeneous index remains allowed.

Here is an exact one-$z$ example retaining the $\tau,t$ contractions. For the divisor $P=z^2+ht$ and input $F=z^4$, put

\[
\begin{gathered}
Q=z^2-ht+2h^2,\\ S=h^2t^2-3h^3t+4h^4.\\ \text{Then }Q\circ P+S=F .
\end{gathered}
\tag{G.16}
\]

Here the only nonzero internal contraction in $Q\circ(ht)$ is the first $\tau,t$ contraction, because $ht$ has base degree one. Specifically $\partial_\tau z^2=-2hz^2$, $\partial_\tau(-ht)=h^2t$, and $\partial_\tau(2h^2)=-4h^3$, and multiplication by $\partial_t(ht)=h$ gives $-2h^2z^2+h^3t-4h^4$. Adding the ordinary product leaves $z^4-h^2t^2+3h^3t-4h^4$, which (G.16) corrects by $S$. This is an exact identity with no truncation tail. This check illustrates the retained contractions and signs, while the general proof remains G.1–G.5.

![Finite covector remainder supports and actual whole-stalk generation](assets/finite-commuting-realization.png)

*The ordered heads $z_1^2,z_2^3$ have the exact six-monomial complement (G.4). The convergent division is G.3–G.4 and the entire-stalk generation theorem is G.5. The input is the actual finite-lattice/finite-principal-fibre premise (G.1), including nonreduced principal modules. The infinite-order and finite D-type arrows remain open as stated in G.6. Free human source: [Kashiwara–Kawai, author-hosted HolIII](https://www.kurims.kyoto-u.ac.jp/~kenkyubu/kashiwara/HolIII.pdf), III.5.5.*


### 5.30. Actual polynomial flatness and reconstruction after finite realization

This argument proves the finite-order polynomial-extension comparison needed after Theorem G.1. It retains the entire actual stalk. It proves neither an infinite-order extension nor a bounded terminal free resolution. All rings and tensor products in F.1–F.7 below are finite-order or expressly $h$-adic formal rings.

The actual symbol and product are [§5.19, S.1–S.4](#5-19-actual-factorial-growth-symbols-and-ordered-jets); the complete derivative norm and actual units are [§5.20, B.1–B.3.1](#5-20-one-norm-multiplication-units-and-dummy-division); the convergent simultaneous-division and finite-head proofs are [§5.21, N.1–N.18](#5-21-actual-simultaneous-division-and-finite-stalk-relations). We use the full [§5.22.1–§5.22.5, Lemmas 5.22.1–5.22.3 and Proposition 5.22.4](#5-22-2-normal-element-artin-rees-and-exact-completion), including exact normal-element Artin–Rees, finite completion, bounded filtered flatness and the separated kernel lift. The commutative input is Completion, §§1–3 and Theorems 3.1–3.3, with the Rees Artin–Rees and Krull-intersection proofs, §§5–6. The coefficient comparison is proved in full in F.3. The actual finite realization used below is [§5.29, Theorem G.1 and G.1–G.5](#5-29-actual-finite-generation-over-the-coordinate-commuting-ring), with its supplied actual lattice and finite principal fibre explicit. The actual analytic quotient in F.5 uses bounded Weierstrass division, preparation and uniqueness.

#### F.1. The polynomial rings and all commutation formulas

Retain $u=(t,x)$, $z=\xi/\tau$, $h=\tau^{-1}$, $E_0,E,A_0,A$ from G.1. Let

\[
\begin{gathered}
R_0=\bigoplus_{\beta\in\mathbf N^d}A_0z^\beta\ \subset E_0,\\ R=R_0[h^{-1}]=\bigoplus_\beta A\,\partial_x^\beta .
\end{gathered}
\tag{F.1}
\]

The direct sums mean finite polynomial degree, uniformly across the coefficient sequence; the actual coefficients in each $A_0$ still have a common holomorphic domain and factorial growth. Right multiplication by $z^\beta$ is ordinary normal multiplication. Consequently the displayed bases are independent, and every finite polynomial represents an actual symbol. The second basis follows from $\partial_{x_i}=z_i h^{-1}$ and the fact that the cotangent symbols $z_i,h$ commute.

Put $\sigma(a)=h\circ a\circ h^{-1}$ and $\delta_i(a)=h\circ\partial_{x_i}a$. Direct multiplication, or $\xi_i\circ b=b\circ\xi_i+\partial_{x_i}b$, gives

\[
\begin{gathered}
z_i\circ a=\sigma(a)\circ z_i+\delta_i(a),\\ z_i\circ z_j=z_j\circ z_i,\\ \sigma(z_j)=z_j,\quad\delta_i(z_j)=0 .
\end{gathered}
\tag{F.2}
\]

In fact $z_i=\xi_i\circ h$, so


\[
\begin{gathered}
z_i\circ a=\xi_i\circ\sigma(a)\circ h\\ =\sigma(a)\circ z_i\\ {}+(\partial_{x_i}\sigma(a))\circ h
\end{gathered}
\]

The last term equals $h\circ\partial_{x_i}a$. The normal conjugation formulas C.1 show that $\sigma$ and $\sigma^{-1}$ preserve $A_0$. Base differentiation preserves the actual class on a smaller common domain. Thus $\delta_i$ preserves $A_0$. Associativity gives its $\sigma$-derivation rule
$\delta_i(ab)=\sigma(a)\delta_i(b)+\delta_i(a)b$.
The same formulas show $\delta_i\sigma=\sigma\delta_i$ and $\delta_i\delta_j=\delta_j\delta_i$. Alternatively these identities follow by expanding the commuting $z_i,z_j$ in the independent polynomial basis.

Adjoining the $z_i$ successively is therefore a skew polynomial extension with an invertible coefficient automorphism. At every stage $\sigma$ fixes the previously adjoined $z_j$, and $\delta_i$ annihilates them. No completion or unproved PBW assertion is used: the actual symbol embedding proves the independence and consistency of every finite normal polynomial.

The normal element $h$ stays normal in $R_0$. Finite homogeneous truncation gives

\[
\begin{gathered}
R_0/(h)=H[z],\quad E_0/(h)=T,\\ H=\mathbf C\{u\},\quad T=\mathbf C\{u,z\}.
\end{gathered}
\tag{F.3}
\]

The same truncation at $h^r$ is exact: coefficients of $h$-index below $r$ can be chosen arbitrarily, with the finitely many polynomial and holomorphic domains intersected. The kernel is $h^rR_0$ by the actual fixed-shift formulas. There is no assertion that $h$ belongs to the Jacobson radical of $R_0$.

#### F.2. Noetherianity of the smaller rings

The complete actual division argument N.1–N.18 also proves that $A_0$ is left Noetherian. Here are the adjustments that make that invocation exact. For a submodule $I\subset A_0^b$, select heads in $(h,u)$ alone. Multiplication by $h^nu^\alpha$ raises its head by that exponent; every internal contraction has higher $h$-index. Dickson's finite-head argument therefore applies. Use the same absolute Taylor norm N.2 inside $E_0$, with support restricted to zero $z$-degree. This is a closed coefficient subspace. Division by a head in $u$ and every product of elements independent of $z$ preserve that subspace. The norm, support projection, monomial-division and Neumann estimates are exactly N.1–N.18, including all $\tau$-derivatives. The quotients consequently belong to $A_0$. For an input in $I$, its remainder belongs to $I$; a nonzero remainder would have a head in its diagram outside all selected head orthants, a contradiction. Thus the finitely selected elements generate $I$ over $A_0$. This proves finite generation of every submodule of $A_0^b$. Elliptic inversion B.3 also preserves this closed subring; in particular $(h)$ is in its Jacobson radical.

For completeness, the skew polynomial Noetherian argument can be proved directly. Let $B$ be left Noetherian, let $\sigma$ be an automorphism, and let $B[y;\sigma,\delta]$ have the independent left coefficient basis $y^n$. For a left ideal $I$, define

\[
\begin{gathered}
J_n=\{\sigma^{-n}(\operatorname{coeff}_{y^n}P):\\ P\in I,\ \deg P\le n\}\subset B .
\end{gathered}
\tag{F.4}
\]

These are left ideals: multiplying by $a\in B$ changes the normalized coefficient by $\sigma^{-n}(a)$. Moreover $J_n\subset J_{n+1}$, by multiplying $P$ on the left by $y$. They stabilize at some $N$. Select finitely many coefficient generators and polynomial lifts for every $J_n$, $0\le n\le N$. For $P$ of degree $m>N$, express its normalized leading coefficient in $J_N$. If a selected lift has degree $N$, its left multiple $y^{m-N}$ has the same normalized leading coefficient. To multiply that normalized coefficient by $a$, multiply the polynomial on the left by $\sigma^m(a)$. Subtract those finite multiples; the degree decreases. At a degree at most $N$, use the corresponding selected list and again decrease degree. The process terminates. These finitely many lifts generate $I$. Finite-free submodules are finite by induction on the number of components: project to the first component, lift a finite generating list of its ideal image, and add generators for the kernel in the remaining components. Thus every successive skew polynomial extension is left Noetherian.

Applying this proof to (F.2) proves that $R_0$ is left Noetherian. Normal localization proves the same for $A,R$: for an ideal in a localization, contract it to the original ring. Every localized element becomes an element of this contraction after multiplying on the left by a large enough power of $h$. A finite generating list of the contraction therefore generates the localized ideal.

#### F.3. The exact commutative coefficient comparison

The map $H[z]\to T$ is flat. Put
$B=H[z]_{(\mathfrak m_H,z)}$.
The maximal-ideal completions of $B$ and $T$ are the same ring $\mathbf C[[u,z]]$. This equality is verified at every finite order: Taylor polynomials supply the same quotient, and any denominator outside the maximal ideal has a finite geometric inverse there. The earlier commutative Artin–Rees proof makes the induced intersection filtrations cofinal. Surjective quotient transitions then give exact completion on finite modules by successive compatible lifting, and a finite presentation identifies it with tensoring by the completed ring. Here is the flatness and faithfulness consequence explicitly. Every injection of finite modules remains injective after that tensor functor. For an injection $U\hookrightarrow V$ of arbitrary modules over a Noetherian ring, write $V$ as the filtered union of its finite submodules $V_j$. Each intersection $U\cap V_j$ is finite; its injection into $V_j$ remains injective after tensoring. Tensor commutes with this filtered union: every tensor and every witness to equality uses finitely many elements and therefore occurs at a finite stage. The resulting injection remains injective, proving flatness on arbitrary modules. For a Noetherian local ring, the completion has the same nonzero residue field. If $M\ne0$, a nonzero element generates a cyclic injection $D/J\hookrightarrow M$, where $D$ denotes that local ring and $J$ is a proper ideal. Flatness preserves this injection, and the tensor of $D/J$ is nonzero because its quotient modulo the maximal ideal is the residue field: $J$ lies in that ideal. Completion is therefore faithful. Apply these arguments to the genuinely commutative local rings $B$ and $T$.

To prove flatness of $T$ over $B$, tensor an injection of $B$-modules with $T$. Any resulting kernel survives tensoring with the faithfully flat completion of $T$ if it was nonzero; but the completed tensor is tensoring with the flat completion of $B$, and has zero kernel. This contradiction proves flatness. Localization makes $B$ flat over $H[z]$, so $T$ is flat over $H[z]$. The local map is faithful as well: a nonzero $B$-module remains nonzero after its faithfully flat completion, which is the completion of $T$; it therefore could not have become zero after tensoring with $T$. Faithfulness is used only after localization. The unlocalized polynomial map is asserted flat.

#### F.4. Normal formal comparison and actual finite flatness

Let

\[
\begin{gathered}
\widehat R_0^{\,h}=\prod_{n\ge0}H[z]h^n,\\ \widehat E_0^{\,h}=\prod_{n\ge0}Th^n .
\end{gathered}
\tag{F.5}
\]

These are $h$-adic completions of the actual rings, by F.1 and the earlier exact coefficient-quotient calculation C.3. Individual formal coefficients are holomorphic germs; an infinite formal sequence is not claimed to have a common coefficient domain or factorial bounds. The formal multiplication has finite sums at each index and is the same normalized Leibniz product.

Both formal rings are left Noetherian. Their $h$-graded rings are $H[z,h]$ and $T[h]$, which are Noetherian by the polynomial proof in F.2 with $\sigma=1,\delta=0$. For a submodule of a finite free formal module, choose finitely many lifts of its graded initial generators. Successive leading-$h$ cancellation expresses each element by those lifts: each coefficient multiplier is a convergent series in the complete $h$ topology, and equality follows from separatedness. This is the complete coefficientwise proof in Section 2 of the earlier normal-completion leaf, applied to these exact graded rings. It proves Noetherianity without assuming a submodule closed in advance.

The normal-element Artin–Rees proof C.1 and exact completion proof C.2 apply to $R_0$ and to both formal rings. They imply exact completion and its tensor identity on finite modules. Every finite module over either formal ring is complete and separated, since that ring equals its $h$-completion. At each truncated level, (F.3) and F.3 above give

\[
\begin{gathered}
\operatorname{gr}_h(\widehat E_0^{\,h}/h^r)=T[h]/h^r\\ \text{flat over}\\ \operatorname{gr}_h(\widehat R_0^{\,h}/h^r)=H[z,h]/h^r .
\end{gathered}
\tag{F.6}
\]

The coefficient functor is $T\otimes_{H[z]}-$, so graded flatness here follows directly from F.3. The bounded-filtration lifting Lemma C.3 is within its exact hypotheses: there are only $r$ nonzero levels, all terms have finite shifted free resolutions, and solving a leading equation raises order and terminates. It proves flatness on finite modules at each truncated level.

The full lift is exactly Proposition C.4's kernel argument. For $N\subset M$ finite over $\widehat R_0^{\,h}$, its normal Artin–Rees constant $c$ puts the kernel of $N/h^rN\to M/h^rM$ in $h^{r-c}N/h^rN$. By (F.6), any vector in the kernel of the completed tensor map belongs to $h^{r-c}(\widehat E_0^{\,h}\otimes N)$ for every $r$. That finite module is separated, so the vector is zero. Moving $h^{r-c}$ across a tensor factor uses the compatible normal automorphism $\sigma$, exactly as in C.4. Thus $\widehat E_0^{\,h}\otimes_{\widehat R_0^{\,h}}-$ is exact on finite modules. Composing with exact normal completion over $R_0$ proves exactness of $\widehat E_0^{\,h}\otimes_{R_0}-$ on finite $R_0$-modules.

Finally, for a finite $R_0$-module $V$, construct a free resolution with finite free terms, using F.2 at every kernel. Tensor it with $E_0$. Its homology modules are finite over the left Noetherian $E_0$. Exact $h$-completion on finite $E_0$-modules commutes with the cycles, boundaries and these homology modules. The completed complex is $\widehat E_0^{\,h}\otimes_{R_0}$ of the original resolution, hence is exact in positive degrees by the preceding paragraph. Exact $h$-completion over $E_0$ detects zero: a zero completed finite module has zero quotient modulo $h$, and finite-generator Nakayama applies since $(h)\subset\operatorname{Jac}(E_0)$ by B.3. Consequently the actual tensor complex has zero positive homology. In particular

\[
\begin{gathered}
E_0\otimes_{R_0}-\\ \text{is exact on finite left }R_0\\ \text{-modules}.
\end{gathered}
\tag{F.7}
\]

This proves actual flatness in the finite scope required here. It does not identify an infinite-order extension with either formal ring in (F.5). Normal localization of the target is exact: it is a filtered union of right copies of $E_0$, with the normal automorphism used for the coefficient actions. Tensoring with the union preserves exactness because every relation and preimage uses finitely many elements. Thus

\[
\begin{gathered}
E\otimes_R-\\ \text{is exact on finite left }R\\ \text{-modules}.
\end{gathered}
\tag{F.8}
\]

Equivalently apply (F.7) to the underlying localized modules, by filtered unions of their finite $R_0$-submodules. The tensor identity $E\otimes_{R_0}V=E\otimes_RV$ for an $R$-module follows from its universal balanced-map property: $h$ is already invertible on both tensor factors.

#### F.5. The principal finite-module identity

Let $W$ be a finite $H[z]$-module, finite over $H$, such that every $z_i$ acts nilpotently on $W/\mathfrak m_HW$. Then

\[
 T\otimes_{H[z]}W\simeq W .
 \tag{F.9}
\]

Here the right side has an actual analytic $T$-action, which we now construct, rather than assume. Take minimal $H$-generators $w_1,\ldots,w_b$; their residues form a basis modulo $\mathfrak m_H$. Write $z_iw=A_i(u)w$ with holomorphic matrices $A_i$. The matrices $A_i(0)$ are the given nilpotent actions. Cayley–Hamilton therefore gives distinguished monic polynomials

\[
\begin{gathered}
p_i(u,Z)\\ =\det(Z-A_i(u)),\\ p_i(0,Z)=Z^b,\\ p_i(u,z_i)W=0 .
\end{gathered}
\tag{F.10}
\]

The last equality follows by applying each matrix power to the vector of generators, so it does not require the matrices to commute before passing to $W$. Each $p_i$ involves only $u,z_i$. Successive proved Weierstrass divisions in the individual $z_i$ show that $T/(p_1,\ldots,p_d)$ is free over $H$ in the monomials with $0\le\beta_i<b$: each division has a holomorphic remainder and commutes with the other variables, and uniqueness gives uniqueness of the final remainder. Ordinary monic polynomial division gives the identical coefficient basis and relations for $H[z]/(p_1,\ldots,p_d)$. Thus the natural map between these two quotient rings is an isomorphism. Tensoring with $W$, which is annihilated by those polynomials, proves (F.9) and defines its analytic action. The case $W=0$ is immediate. Every division has an actual common neighbourhood because this is a finite list of convergent Weierstrass polynomials.

#### F.6. Reconstruction of the actual generic-position module

Use the exact premise of G.1 and its conclusion that $M_0$ is finite over $A_0$. The finite module $\bar M_0=M_0/hM_0$ is therefore finite over $H=A_0/(h)$, and is also a finite $H[z]$-module. Its fibre at $u=0$ is the finite local $\mathbf C\{z\}$-module (G.1). Every $z_i$ has only the eigenvalue zero there, by G.2. Hence F.5 applies. The canonical multiplication map

\[
\begin{gathered}
\mu_0:E_0\otimes_{R_0}M_0\longrightarrow M_0,\\ P\otimes m\longmapsto Pm
\end{gathered}
\tag{F.11}
\]

is surjective and reduces modulo $h$ to the isomorphism (F.9). Its source is finite over $E_0$, since $M_0$ is finite over $R_0$, so its kernel $K$ is finite. Moreover $h$ acts injectively on $M_0$, since $M_0$ is a submodule of the $E$-module $M$, on which $h$ is invertible.

If $k\in K\cap h(E_0\otimes_{R_0}M_0)$, write $k=hp$. Then $0=\mu_0(k)=h\mu_0(p)$ forces $\mu_0(p)=0$, so $p\in K$. Thus this intersection is exactly $hK$. The principal isomorphism in (F.11) now says $K/hK=0$, and actual finite Nakayama gives $K=0$. Localizing by $h$ yields

\[
\begin{gathered}
E_0\otimes_{R_0}M_0\simeq M_0,\\ E\otimes_RM\simeq M .
\end{gathered}
\tag{F.12}
\]

Every map is the stated multiplication map and includes the whole actual stalk. The localization of $M_0$ is $M$: finite-order left shifts give $M=\bigcup_kh^{-k}M_0$, as in G.5. No equality on generic-stratum cohomology modules is substituted for these module maps.

#### F.7. Every polynomial submodule is an actual microlocal submodule

Let $N\subset M$ be an $R$-submodule. It is finite over $A$, since $M$ is finite over the left Noetherian $A$. Put $N_0=N\cap M_0$. It is finite over $A_0$, by F.2 and the finite $A_0$-module $M_0$, and is stable under $R_0$. It is $h$-saturated in $M_0$: if $hm\in N$ then $m=h^{-1}(hm)\in N$. Hence $N_0\cap hM_0=hN_0$, and $\bar N_0$ injects into $\bar M_0$.

Set $P_N=E_0\otimes_{R_0}N_0$. Actual flatness (F.7) embeds it into $E_0\otimes_{R_0}M_0=M_0$. Tensoring the $h$-injective quotient $Q_0=M_0/N_0$ and its normal-action twist also shows that this image $P_N$ is $h$-saturated in $M_0$. More precisely, multiplication by $h$ is an injective $R_0$-linear map $\sigma^{-1}Q_0\to Q_0$, where the source action is $a\cdot q=\sigma^{-1}(a)q$. After tensoring, $E_0\otimes_{R_0}\sigma^{-1}Q_0$ identifies with $\sigma^{-1}(E_0\otimes_{R_0}Q_0)$ by $e\otimes q\mapsto\sigma^{-1}(e)\otimes q$. The tensor map is then multiplication by $h$, because $h\sigma^{-1}(e)=eh$. Flatness makes this map injective. Hence $M_0/P_N$ has no $h$-torsion, which is exactly the claimed saturation. Thus
$P_N/hP_N=T\otimes_{H[z]}\bar N_0$ injects into $\bar M_0$.
The annihilating distinguished polynomials (F.10) for $\bar M_0$ also annihilate its submodule $\bar N_0$, which is finite over $H$. The proof of (F.9), using those same quotient rings, gives $P_N/hP_N\simeq\bar N_0$.

The actual finite $E_0$-module $P_N\subset M_0$ has finite principal fibre, by this equality; after localizing by $h$ it supplies an actual lattice in an $E$-module. Theorem G.1 therefore makes $P_N$ finite over $A_0$. The natural image of $N_0$ in $P_N$ has the same principal quotient, so
$P_N=N_0+hP_N$.
The quotient $P_N/N_0$ is a finite $A_0$-module equal to its $h$-multiple. Since $(h)\subset\operatorname{Jac}(A_0)$, finite Nakayama gives $P_N=N_0$. In particular $N_0$ is stable under all of $E_0$, without having presumed this during the tensor construction. Localizing gives

\[
 E\otimes_RN\simeq N,\qquad EN=N .
 \tag{F.13}
\]

Here $N=\bigcup_kh^{-k}N_0$, because every element of $N\subset M$ can be shifted into $M_0$ and the shifts belong to $R$. This proves the actual finite-order polynomial-submodule assertion on the entire stalk.

#### F.8. The exact remaining scope

F.2 supplies finite free resolution segments of any requested finite length over $R_0,R$; F.7–F.8 make their extension to $E_0,E$ exact, and F.12 identifies the augmentation for the modules in G.1. The last kernel remains in such a segment. [§5.31, L.1–L.6](#5-31-a-bounded-actual-generic-position-complex-with-polynomial-operator-matrices) proves a bounded free resolution over the actual finite-order microlocal rings with polynomial matrices and one base neighbourhood. Its explicit syzygy stabilization concerns the principal polynomial complex; it does not claim that the actual bounded complex is a free resolution over the polynomial ring itself.

No ring in this argument is $\mathscr E^\infty$ or $\mathscr A^\infty$. The identities (F.12)–(F.13) therefore do not prove HolIII (3.5.8), the sectorial kernel action, continuation, singular-support separation, faithful finite D-type embedding, regular finite-pole recovery, intrinsic microlocal order correspondence, C.1, or arbitrary proper analytic regularity. They supply actual finite-order prerequisites to those still binding obligations. The supplied actual lattice and finite-principal-fibre premise in G.1 remains explicit throughout.

The figure below displays the exact rings in (F.3) and (F.5), the finite-module flatness lift, the principal distinguished-polynomial quotient, the saturated kernel $K/hK=0$, and the open infinite-order arrow. Free human reading: Kashiwara–Kawai, [author-hosted HolIII](https://www.kurims.kyoto-u.ac.jp/~kenkyubu/kashiwara/HolIII.pdf), III.5.1–III.5.8.

![Actual polynomial flatness and whole-stalk reconstruction](assets/actual-polynomial-reconstruction.png)

*The polynomial and actual coefficient rings are (F.3); the explicitly formal rings and finite exact-completion comparison are (F.5)–(F.8). The finite distinguished quotient retains nilpotent multiplicities in (F.9)–(F.10). The saturated-kernel argument proves the whole-stalk multiplication identities (F.11)–(F.13). G.1 supplies the actual lattice and finite principal fibre; the infinite-order arrow remains open as stated in F.8. Free human source: [Kashiwara–Kawai, HolIII](https://www.kurims.kyoto-u.ac.jp/~kenkyubu/kashiwara/HolIII.pdf), III.5.1–III.5.8.*

For every coherent actual holonomic germ whose holonomicity uses its reduced actual finite-order support, [§5.32, J.1–J.3](#5-32-coherent-actual-lattices-and-principal-support-for-the-whole-germ) constructs the actual lattice and proves the finite principal fibre after the contact reduction. The separate infinite-order support convention retains the interface stated there.

### 5.31. A bounded actual generic-position complex with polynomial operator matrices

This argument continues the exact finite-lattice and finite-principal-fibre premise of G.1. It proves a bounded free resolution over the actual finite-order microlocal ring whose matrices lie in the actual polynomial operator ring $R_0$. It also gives one base neighbourhood for that complex and its entire finite-support realization. It does not claim a bounded free resolution over $R_0$ itself, or an infinite-order comparison.

The actual finite realization is [§5.29, Theorem G.1 and G.1–G.5](#5-29-actual-finite-generation-over-the-coordinate-commuting-ring). Actual polynomial Noetherianity, coefficient flatness, finite-order reconstruction and exact normal-completion detection are [§5.30, F.1–F.7](#5-30-actual-polynomial-flatness-and-reconstruction-after-finite-realization). All coefficient identities below concern actual convergent representatives or expressly designated positive h-adic formal modules.

The full uniform homogeneous kernel proof is GL06, Lemma 3.0c.3, with the coefficient comparison proved locally in F.3. Its scalar analytic relation input is Oka Theorem 2.1 and Lemma 2.2, using the local monic division and nearby Weierstrass splitting, Lemmas 2.3 and 3.2. L.5 supplies the scalar-row and finite block-matrix consequences explicitly. Actual neighbourhood coherence and translated-chart coordinates are [§5.24, Lemma 5.24.1 and Theorem 5.24.2](#5-24-fixed-kernel-sections-on-one-actual-neighbourhood), proved by the bounded division of §5.21.

#### L.1. A finite free resolution over the holomorphic coefficient ring

Put $n=d+1$, $q=n+d=2d+1$, $H=\mathbf C\{u_1,\ldots,u_n\}$ and $S=H[z_1,\ldots,z_d]$. By G.1, $\bar M_0=M_0/hM_0$ is finite over $H$. We first prove that every finite $H$-module has a finite free resolution of length at most $n$.

The coordinate sequence $u_1,\ldots,u_n$ is regular: each successive quotient is the convergent power-series ring in the remaining coordinates, and multiplication by the next coordinate is injective there by its Taylor coefficients. The two-term-complex cone induction therefore proves that its Koszul complex is a free resolution of the residue field $\mathbf C$, of length $n$. Its differential on an ordered wedge is

\[
\begin{gathered}
d(e_{i_1}\wedge\cdots\wedge e_{i_k})\\ =\sum_{a=1}^k(-1)^{a-1}u_{i_a}\\ e_{i_1}\wedge\cdots\widehat e_{i_a}\cdots\wedge e_{i_k}.
\end{gathered}
\tag{L.1}
\]

Indeed for one coordinate the complex is injective multiplication followed by the quotient. Adjoining the next coordinate tensors with that two-term complex; on the previously resolved quotient its kernel is zero and its cokernel is the next quotient. This proves the induction, including the displayed signs.

Construct a minimal free resolution of a finite $H$-module $W$ by selecting at each stage generators whose residues are a basis of the finite kernel modulo the maximal ideal. Kernels are finite since $H$ is Noetherian. The kernel of each such minimal cover lies inside the maximal ideal times its free source. Hence every differential in this resolution has all entries in the maximal ideal, and tensoring it with $\mathbf C$ makes all differentials zero.

The homology of this tensor complex is also computed by the Koszul resolution (L.1) tensored with $W$. Here is the required balanced-resolution argument. Tensor the two free resolutions to form a first-quadrant double complex, with the Koszul term in one degree and the minimal-resolution term in the other, and use total differential $d_K\otimes1+(-1)^a1\otimes d_F$ on Koszul degree $a$. Resolving first in either direction gives respectively those two tensor complexes, because a free tensor factor preserves each augmentation's exactness. At any fixed total degree there are finitely many terms. Successive row or column reduction of a cycle and of a boundary consequently proves that both augmentations induce the same total homology; there is no infinite totalization or limit exchange. The Koszul side has no terms above degree $n$. Thus the minimal resolution has zero free term modulo the maximal ideal above degree $n$, and its ranks there are zero. Its finite next kernel is zero by finite Nakayama. The minimal free resolution terminates at degree $n$.

#### L.2. Adjoining each commuting covector variable gives a finite polynomial resolution

Suppose $B$ is a commutative ring, $V$ is a $B[y]$-module, and $V$ has a finite free $B$-resolution of length $a$. Write $Y$ for the $B$-linear action of $y$ on $V$. There is an exact sequence

\[
\begin{gathered}
0\longrightarrow B[y]\otimes_B V\\ \xrightarrow{D}B[y]\otimes_BV\\ \xrightarrow{\mu}V\longrightarrow0,\\ D(y^j\otimes v)=y^{j+1}\otimes v\\ {}-y^j\otimes Yv,\\ \mu(y^j\otimes v)=Y^jv.
\end{gathered}
\tag{L.2}
\]

The tensor module is the direct sum of copies of $V$ indexed by powers of $y$. A highest nonzero coefficient proves injectivity of $D$. For $f=\sum_{j=0}^b y^j\otimes v_j$, define
$g=\sum_{j=1}^b\sum_{k=0}^{j-1}y^{j-1-k}\otimes Y^kv_j$.
The terms telescope to $Dg=f-1\otimes\mu(f)$. This proves exactness at the middle, and $\mu$ is surjective.

Lift $Y$ to a chain endomorphism $\widetilde Y$ of the finite free $B$-resolution. At degree zero lift the image of each free basis element through its augmentation. At degree $j$, the already lifted image of its boundary is a cycle, hence has a preimage in degree $j$; choose that preimage for each of its finitely many basis elements. These finite choices give a chain map. Induction reaches the last degree because exactness there includes the required cycle image. Every coefficient is in $B$.

Tensor the resolution with the free coefficient ring $B[y]$, and take the cone of the chain map $y-\widetilde Y$. In homological degree $j$ its term is $L'_j\oplus L'_{j-1}$ and its differential is

\[
\begin{gathered}
d(a,b)=(d_{L'}a+(y-\widetilde Y)b,\\ {}-d_{L'}b).
\end{gathered}
\tag{L.3}
\]

The chain-map identity gives $d^2=0$ with these exact signs. The induced map on degree-zero homology is $D$ in (L.2); both induced resolutions have zero higher homology. The cone's homology is therefore $V$ in degree zero and zero elsewhere. One can verify this directly by solving a cone cycle first in the source resolution and then the target resolution, or by the short exact cone sequence. It is a finite free $B[y]$-resolution of length at most $a+1$.

Start with the finite free $H$-resolution of $\bar M_0$ in L.1. Apply this construction successively to $z_1,\ldots,z_d$. At each stage the next variable acts linearly over the preceding polynomial ring, because their actions commute on $\bar M_0$. Chain lifts need not commute before passing to homology; only the single next chain map is used at that stage. We obtain a finite free $S$-resolution

\[
\begin{gathered}
0\longrightarrow L_q\longrightarrow L_{q-1}\\ \longrightarrow\cdots\longrightarrow L_0\\ \longrightarrow\bar M_0\longrightarrow0,\\ q=2d+1.
\end{gathered}
\tag{L.4}
\]

If it has shorter length, use zero free terms to fill the declared bound. All matrices have finite polynomial degree and holomorphic coefficient germs. No general projective-module or stable-free theorem has been imported.

#### L.3. Comparing the actual polynomial syzygy with that finite resolution

By F.2, $R_0$ is left Noetherian, and $M_0$ is finite over $R_0$. Choose an actual finite free $R_0$-resolution segment with successive surjections onto its kernels through degree $q-1$:

\[
\begin{gathered}
0\longrightarrow K_q\longrightarrow F_{q-1}\\ \longrightarrow\cdots\longrightarrow F_0\\ \longrightarrow M_0\longrightarrow0.
\end{gathered}
\tag{L.5}
\]

Its matrices are in $R_0$; its kernels are finite. Every module in the segment has injective multiplication by $h$: this holds for $M_0\subset M$ and for each kernel inside a finite free module. In an exact pair $0\to K\to F\to V\to0$ with $h$ injective on $V$, one has $K\cap hF=hK$, by writing $k=hf$ and cancelling $h$ in its image. Reduction modulo $h$ is therefore exact on that pair. Repeating gives a free $S$-resolution segment of $\bar M_0$ with last kernel $\bar K_q=K_q/hK_q$.

Its last kernel is stably free, and an elementary comparison proves this exact conclusion. If $0\to A\to F\to W\to0$ and $0\to B\to G\to W\to0$ have finite free middle terms, their fibre product over $W$ has split sequences with kernels $A,B$ and quotients $G,F$. Thus $A\oplus G\simeq B\oplus F$. Apply this successively to the reduced segment (L.5) and the finite free resolution (L.4). More explicitly, if their $j$-th kernels satisfy $A_j\oplus U_j\simeq B_j\oplus V_j$ with $U_j,V_j$ finite free, compare the free covers $F_j\oplus U_j$ and $L_j\oplus V_j$ of that same module. The fibre-product argument gives
$A_{j+1}\oplus(L_j\oplus V_j)\simeq B_{j+1}\oplus(F_j\oplus U_j)$.
Begin at $A_0=B_0=\bar M_0$, and iterate $q$ times. Since $B_q\simeq L_q$ is free, this constructs finite integers $s,r$ and an actual $S$-linear isomorphism

\[
 \bar K_q\oplus S^s\simeq S^r.
 \tag{L.6}
\]

Add $R_0^s$ to $F_{q-1}$ with zero outgoing differential, so its kernel becomes $K_q\oplus R_0^s$. Lift each basis vector of the free principal kernel (L.6) to this actual kernel. The resulting map
$P_q:R_0^r\to F_{q-1}\oplus R_0^s$
is an actual relation map with principal image equal to the entire principal kernel and with injective principal map. All preceding differentials remain unchanged apart from the declared zero columns. We have constructed a bounded actual polynomial-matrix complex whose principal complex is the exact resolution (L.4)'s syzygy comparison.

This does not assert that $P_q$ is an isomorphism onto the last kernel over $R_0$. In particular $(h)$ was not declared a Jacobson ideal of $R_0$, and no such Nakayama argument is valid there.

#### L.4. Exactness over the actual microlocal ring

Tensor the bounded free complex with $E_0$ and use its multiplication augmentation to $M_0$. Its principal complex is the tensor of the exact $S$-complex with $T=\mathbf C\{u,z\}$, which is exact by F.3. Its $h$-adic formal complex is exact: for a cycle solve its first nonzero homogeneous equation in that principal exact complex, lift a boundary, subtract, and repeat at each higher $h$-order. In a finite free formal term, the sum of these lifts exists and equality follows from separatedness. The augmentation module is finite and $h$-complete after completion by the exact normal-completion argument, so the same successive lifting applies there. At the last term, an element in the kernel has a first nonzero principal coefficient, which would contradict principal injectivity; its kernel is zero.

All homology modules of the actual $E_0$ complex are finite, because $E_0$ is left Noetherian. Its exact finite $h$-completion commutes with homology, by the normal-element Artin–Rees proof in F.4. Their completed modules are zero by the previous paragraph. The exact completion detects zero on finite $E_0$-modules, using $(h)\subset\operatorname{Jac}(E_0)$. Hence the actual complex is exact, including its last term. Localizing by $h$ preserves exactness and identifies its target with $M$. Consequently

\[
\begin{gathered}
0\longrightarrow E^{r_q}\xrightarrow{P_q}E^{r_{q-1}}\\ \longrightarrow\cdots\xrightarrow{P_1}E^{r_0}\\ \longrightarrow M\longrightarrow0,\\ P_j\text{ has entries in }R_0,\\ q=2d+1 .
\end{gathered}
\tag{L.7}
\]

This is the finite-order conclusion. Its proof made no assertion that an arbitrary formal lift has factorial growth: every actual matrix came from finite actual $R_0$-relations; formal completion was used only as an exact detecting functor on the finite actual homology modules.

#### L.5. A common base neighbourhood for the polynomial principal complex

The earlier uniform homogeneous kernel proof yields the following nonhomogeneous polynomial assertion: for a finite polynomial matrix over $H[z]$, its kernel is generated by finitely many polynomial sections on one base neighbourhood. To prove the reduction, choose source shifts $a_j$ at least as large as every entry degree in its $j$-th column, and give the target shift zero. Introduce a variable $v$ and homogenize each entry to degree $a_j$. If a polynomial vector $c_j(z)$ is a relation, choose $b\ge\max_j(a_j+\deg c_j)$ and homogenize it as $v^{b-a_j}c_j(z/v)$. The homogeneous matrix times this vector is $v^b$ times the original zero relation. Conversely evaluation at $v=1$ sends any homogeneous relation to a relation. The finitely many uniform homogeneous generators therefore dehomogenize to generators of every nearby polynomial kernel. Their coefficients remain actual holomorphic functions on the same base neighbourhood.

Here the full homogeneous proof uses the coherent analytic kernel on the zero-section product, the faithfully flat polynomial-to-analytic coefficient map of F.3, and finite graded zero-section detection. The matrix-kernel consequence of the scalar Oka relation theorem follows by induction on the number of target rows. For one row it is exactly that theorem. For $r>1$ rows, let the finite columns of a matrix $G$ generate the first-row kernel on one neighbourhood. Apply the induction hypothesis to the remaining $r-1$ rows composed with $G$, obtaining finitely many kernel columns $c_j$. Their images $Gc_j$ generate the full original kernel: every original relation is $Gc$ by its first row, and the remaining rows say precisely that $c$ is in the composed kernel. The relation sheaf of the resulting finite generator matrix is again a finite free matrix kernel, so this kernel has a finite presentation.

Here are the required presented-module consequences. Write two finite presentations with relation matrices $A$ and $B$, and represent a morphism by a finite free matrix $D$. The finite block matrix $(D,-B)$ imposes the equation $Dx=By$ on pairs $(x,y)$. Its kernel is generated by the preceding finite-row argument. The first coordinates of those generators, together with the source relation columns of $A$, describe the kernel of the presented morphism. Its relation matrix is again a finite free matrix kernel, so the presented kernel is coherent. The cokernel is presented by adjoining the finite columns of $D$ to $B$. The image is the source presentation modulo that coherent kernel. In particular the quotient of a coherent kernel by a finite generator list is coherent, using those same finite block presentations. If a finite-type quotient has zero stalk at a point, choose its finite generators on a neighbourhood. Each generator has zero germ, hence vanishes on some smaller neighbourhood; intersect this finite list of neighbourhoods. The quotient vanishes there. These finite scalar-row and block-matrix constructions give every analytic kernel, quotient and zero-stalk consequence used in this paragraph.

The graded zero-section detection is also elementary: a finite graded module bounded below whose localization at the zero-section maximal ideal is zero has zero quotient modulo the polynomial variables, since that quotient is finite over the local coefficient ring. Equality with its polynomial-variable multiple then kills its first possible degree and, inductively, every degree. Thus the earlier proof's zero-section descent has exactly the needed hypotheses and includes arbitrary shifted free matrices.

Apply this uniform polynomial kernel result to the finitely many principal maps in L.3. At the base origin they form an exact complex. Each represented kernel generator there is a finite polynomial combination of columns of the preceding map. Represent those finitely many identities on one smaller base neighbourhood. The same sections generate every nearby kernel, and hence prove exactness there in every polynomial degree. At the injective last map the initial kernel list is empty, so its nearby kernel is zero. Thus one base neighbourhood $U$ makes the entire principal polynomial complex exact over $\mathcal O_{U,u}[z]$ for every $u\in U$. There is no sequence of shrinking neighbourhoods indexed by a relation degree.

#### L.6. The actual complex on the whole conic chart over that base neighbourhood

The finitely many actual $R_0$ matrix entries have coefficients in $A_0$, all represented on one smaller $U$ with common coefficient domains and factorial bounds. Their finite polynomial $z$-dependence defines them on all of

\[
\begin{gathered}
\mathcal E_U=\{(u;\tau,\xi):u\in U,\ \tau\ne0\},\\ z=\xi/\tau\in\mathbf C^d.
\end{gathered}
\tag{L.8}
\]

On a compact $z$-set a fixed polynomial degree contributes a fixed factor to each homogeneous coefficient; finite shifts and derivatives have the actual factorial estimates already proved. Thus these are actual sheaf sections everywhere on (L.8). Their finitely many differential-product identities also hold there on one common smaller base neighbourhood: equality of actual germs is simultaneous equality of their coefficient sequences on a common domain, as in S.1. Each coefficient identity is polynomial in the covector variables, so equality on a covector neighbourhood implies equality for every covector value. The actual product preserves finite polynomial degree for these finitely many expressions. The represented matrices therefore form one complex on the entire chart.

Fix any $(u_*,z_*)$. The coefficient map
$\mathcal O_{U,u_*}[z]\to\mathcal O_{(u_*,z_*),U\times\mathbf C^d}$
is flat by precisely F.3's local completion proof, now localizing the polynomial ring at $(\mathfrak m_{u_*},z-z_*)$. The principal complex is therefore exact at this point. Actual stalk Noetherianity, elliptic inversion, and exact finite $h$-completion hold there: the earlier U.1 coordinate change translates a normalized covector to zero while preserving $h$ and the actual product, and the normal-element completion proof applies to its holomorphic coefficient ring. Repeating L.4 proves exactness at every such point in positive degrees, including terminal injectivity. No uniform norm over all unbounded $z$ is required: the finite matrices and $U$ are fixed; their compact coefficient bounds may depend on the compact set.

Let $M'_0$ be the cokernel of this actual order-zero complex and $M'=M'_0[h^{-1}]$. It is a coherent actual module on (L.8) by the earlier U.2 matrix-kernel proof. Localization preserves that coherent category: the finitely many entries of any finite-order matrix can be shifted into order zero using finite powers of $h$ on its free bases; its order-zero kernel is generated on one neighbourhood by U.2, and every finite-order relation can itself be shifted into that kernel. Localizing those fixed generators therefore generates every finite-order kernel on the same neighbourhood. The finite block-matrix proof of U.2 then gives coherence of the localized cokernels as well. At the distinguished stalk the augmentation gives $M'_0\simeq M_0$ and $M'\simeq M$. Moreover $h$ acts injectively on $M'_0$ at every point. Indeed multiplication by $h$ is the normal-action twisted injection of the free complex into itself; its quotient is the principal complex. The long exact homology sequence and the vanishing of its degree-one principal homology make multiplication by $h$ injective on the degree-zero cokernel. Its principal module is exactly the polynomial principal cokernel analytically extended, not an assumed equality with an infinite-order support.

For completeness this realization has bounded $z$-support on one smaller $U$. The distinguished polynomials $p_i(u,z_i)$ constructed in F.10 annihilate the principal module at the base germ. That finite annihilation is witnessed by finitely many polynomial column identities for the principal presentation; extend them to one smaller $U$. If
$p_i(u,Z)=Z^b+\sum_{k<b}a_{ik}(u)Z^k$,
choose $\delta>0$ and shrink $U$ so that

\[
\begin{gathered}
\sum_{k<b}|a_{ik}(u)|\,\delta^{k-b}<\tfrac12\\ \text{for all }u\in U\text{ and every }i.
\end{gathered}
\tag{L.9}
\]

This is possible since $a_{ik}(0)=0$. For $|Z|\ge\delta$, the left sum with $|Z|$ in place of $\delta$ is no larger, so $|p_i(u,Z)|>|Z|^b/2$. Thus every root has $|Z|<\delta$. At any point outside that $z$-polydisc, at least one annihilating principal polynomial is a unit and the principal module is zero. Finite actual Nakayama then makes $M'_0$ and $M'$ zero there. When $d=0$ the support statement has no covector coordinate to bound.

If the starting object is an actual coherent sheaf germ, the distinguished-stalk isomorphism to $M'$ extends on a product neighbourhood of that point. This uses only finite presentations: represent its finitely many generator images, relations, inverse generator images and inverse identities on one smaller neighbourhood; the two maps then are inverses there. Choose $\delta$ in (L.9) inside that product's covector radius and shrink the base accordingly. The support bound then places the whole realized support inside the comparison product. Hence $M'$ is a representative of the whole original germ on the conic chart over one base neighbourhood, with no discarded branch or supported section. The bounded resolution (L.7), with its fixed polynomial matrices, is exact throughout that chart.

#### L.7. What this closes and what remains

Under G.1's actual premise, L.1–L.6 prove the actual finite-order bounded generic-position realization and polynomial-matrix resolution, with the exact bound $q=2d+1$ and one base neighbourhood. The finite-ring submodule comparison is F.13. These supply the finite-order clauses corresponding to HolIII III.5's ordinary realization. Equality of a characteristic support defined through an infinite-order extension with the given principal support remains the explicit G.6 interface, if that convention is used.

The proof contains no $\mathscr E^\infty$ or $\mathscr A^\infty$ tensor identity. Canonical kernel action is proved in Section5.43. The all-degree complex relative vanishing for this bounded actual complex, with acyclicity required only over the closed changing carrier, is proved in Section5.80, CP.F--CP.G. Continued solutions, singular-support separation, infinite-order linearity, faithful D-type embedding, regular finite-pole recovery, intrinsic order correspondence, C.1 and arbitrary proper analytic regularity require further arguments. A finite free actual resolution is a prerequisite for those arguments, not their proof.

The figure below records the base bound, the polynomial bound, the cone signs (L.3), the explicit syzygy stabilization (L.6), actual exact-completion detection, and the root bound (L.9) on the entire conic chart. Free human reading: Kashiwara–Kawai, [author-hosted HolIII](https://www.kurims.kyoto-u.ac.jp/~kenkyubu/kashiwara/HolIII.pdf), III.5.3–III.5.8.

![Bounded polynomial-matrix resolution over the actual microlocal ring](assets/bounded-polynomial-complex.png)

*L.1–L.2 prove the lengths and signed cones. L.3 constructs the stable principal syzygy and actual relation lifts; L.4 detects exactness over the actual finite-order ring. L.5–L.6 give one base neighbourhood and the full-chart root bound, including every supported section. The nonreduced example and its actual correction term are (L.10)–(L.11). G.1 supplies the actual lattice and finite fibre. The diagram makes no claim of a free resolution over the actual polynomial ring. Free human source: [Kashiwara–Kawai, HolIII](https://www.kurims.kyoto-u.ac.jp/~kenkyubu/kashiwara/HolIII.pdf), III.5.3–III.5.8.*

#### L.8. An exact nonreduced principal example with the actual correction term

For $d=1$, let $H=\mathbf C\{t,x\}$, and set

\[
\begin{gathered}
J=\begin{pmatrix}0&1\\0&0\end{pmatrix},\\ A=\begin{pmatrix}0&0\\0&1\end{pmatrix},\\ T_*=tI-hA,\quad Z_*=zI-J .
\end{gathered}
\tag{L.10}
\]

Their actual products include the $\tau,t$ contraction $[z,t]=-hz$, and $[J,A]=J$. Therefore

\[
\begin{gathered}
Z_*\circ T_*-T_*\circ Z_*=-hZ_*,\\ Z_*\circ T_*-(T_*-hI)\circ Z_*=0 .
\end{gathered}
\tag{L.11}
\]

Use row vectors for free left modules, and let a row of operator coefficients act on the right by these matrices. The maps
$d_1(v,w)=v\circ T_*+w\circ Z_*$
and $d_2(u)=(u\circ Z_*,-u\circ(T_*-hI))$
satisfy $d_1d_2=0$ by (L.11). Their principal complex is the cone resolution in L.2 for the $H$-module $(H/(t))^2$ with nilpotent $z$-action $J$: its first differential is $(v,w)\mapsto vt+ w(zI-J)$ and its second is $u\mapsto(u(zI-J),-ut)$. This is exact in positive degrees, by the explicit cone proof; its cokernel has both principal equations $t=0$ and $z^2=0$ and retains its nonreduced nilpotent part. L.4–L.6 consequently prove that the displayed actual length-two free complex resolves its actual cokernel and that this cokernel has no $h$-torsion. The generic bound for this dimension is $q=3$, so length two fits it.

The term $T_*-hI$ in $d_2$ is essential: replacing it by $T_*$ gives $d_1d_2=-hZ_*$, not zero. The right factor has base degree at most one, so every higher Leibniz contraction vanishes identically. Thus (L.11) is an exact actual identity. The example retains the principal nilpotent support and the actual differential correction separately.

[§5.32, J.1–J.3](#5-32-coherent-actual-lattices-and-principal-support-for-the-whole-germ) supplies G.1 for every coherent actual holonomic germ, and [§5.33, K.1–K.3](#5-33-a-convergent-negative-order-polynomial-kernel-complex) constructs the ordinary negative-order Volterra kernel complex. Its ordinary analytic action has the explicit endpoint relation stated there and is not a full localized-ring action.

### 5.32. Coherent actual lattices and principal support for the whole germ

Work with the actual finite-order factorial symbol sheaf $\mathscr E$ of §5.28 and its order-zero sheaf $\mathscr E(0)$; write them as $E$ and $E_0$. The actual Noetherianity, matrix-kernel coherence, normal localization and finite Nakayama arguments are proved in §§5.20–5.24 and §5.31, L.6. The following construction supplies the finite-lattice and finite-principal-fibre hypotheses of §5.29 for every coherent actual holonomic germ, including all singular branches and nonreduced fibres.

#### J.0. The coefficient normality used by the analytic Nullstellensatz

The finite-fibre argument below uses analytic parametrization and the Nullstellensatz, Theorem 5.1. Its prime-case proof uses normality of the coefficient ring of holomorphic germs. Here is a direct analytic proof of that step. Suppose a fraction $g=a/b$ of holomorphic germs, with $b\ne0$, satisfies

\[
\begin{gathered}
g^r+c_1g^{r-1}+\cdots+c_r=0,\\ c_j\in\mathbf C\{u\}.
\end{gathered}
\tag{J.0}
\]

Choose one small polydisc on which all coefficients are bounded, and put $R=\max(1,2\max_j\sup|c_j|^{1/j})$. At a point where $b\ne0$, the inequality $|g|>R$ would give $1\le\sum_j|c_jg^{-j}|<\sum_j2^{-j}<1$. Thus $g$ is bounded by $R$ off the denominator's zero set. The Riemann extension theorem, Theorem 4.2 extends it holomorphically. The identity $bg=a$ then holds everywhere by density, so the fraction belongs to the holomorphic germ ring. This proves its normality, including the zero-dimensional coefficient case. In the cited prime-case argument, coefficients of the minimal polynomial are integral symmetric functions of the conjugates and lie in the coefficient fraction field; this proof puts them in the holomorphic ring. Finite integrality, closure under sums and products, and the determinant argument are proved in integral extensions, Section 1; separability and primitive elements are proved in §5.27.1 above.

For clarity, the radical step needs no decomposition theorem. If $f$ has no power in an ideal $I$, an ideal maximal among those containing $I$ and disjoint from $\{1,f,f^2,\ldots\}$ exists by unions of chains. It is prime: if $a,b$ are outside it but $ab$ is inside, each enlarged ideal contains a power of $f$, and multiplying those two expressions puts a power of $f$ in the original ideal, a contradiction. Therefore the radical is the intersection of the primes above $I$. The cited generic-sheet parametrization proves the prime vanishing-ideal equality, and these two arguments prove the exact Nullstellensatz used below.

#### J.1. A finite actual order-zero lattice exists on a neighbourhood

Let $M$ be an actual coherent left $E$-module germ at $p_0$, with $h=\tau^{-1}$. Choose finitely many actual sections $m_1,\ldots,m_b$ whose germs generate $M_{p_0}$ over $E_{p_0}$, and define the stalk submodule

\[
 M_{0,p_0}=\sum_{j=1}^bE_{0,p_0}m_j\subset M_{p_0}.
 \tag{J.1}
\]

It spans $M_{p_0}$ over $E_{p_0}$ and has injective multiplication by $h$, since $h$ is invertible on $M_{p_0}$. Its presentation kernel in $E_{0,p_0}^b$ is finite by the actual left Noetherianity proved in Theorem 5.21.1. Choose a finite generating list of that kernel and represent its entries and its identities on one neighbourhood of $p_0$. Define the coherent $E_0$-module $P_0$ by that finite presentation there. The actual matrix-kernel arguments in Theorem 5.24.2 and §5.31, L.5 prove its coherence. The chosen sections give a map $P_0\to M$, and at $p_0$ identify $P_{0,p_0}$ with (J.1).

Localize that map:
$E\otimes_{E_0}P_0\to M$.
At $p_0$ it is an isomorphism. Surjectivity is the chosen $E$-generation, and injectivity follows from the injective $h$ action on (J.1): the normal localization is the union of finite left shifts $h^{-k}M_{0,p_0}$, and an element maps to zero precisely when a finite shift of its numerator is zero. A numerator in (J.1) cannot acquire such $h$-torsion.

Both modules in this localized map are coherent over $E$. The localized coherence statement is proved by finite matrix shifts and the kernel list of Theorem 5.24.2 and §5.31, L.5, as detailed in §5.31, L.6. Its inverse at the distinguished stalk extends to one smaller neighbourhood: lift its finitely many free presentation generators, extend their finitely many relation identities, and extend the finitely many identities saying that its two compositions are the identity. Thus the localized map is an isomorphism on that neighbourhood.

There is also one neighbourhood on which $h$ is injective on $P_0$. Regard it as the $E_0$-linear map $\sigma^{-1}P_0\to P_0$, where $\sigma(a)=hah^{-1}$ and the source action is $a\cdot s=\sigma^{-1}(a)s$. Its kernel is coherent by Theorem 5.24.2 and the finite block-matrix argument in §5.31, L.5. That kernel has zero distinguished stalk, by (J.1). A finite-type sheaf with zero stalk vanishes near that point: each of its finitely many local generators vanishes on some neighbourhood, and their finite intersection works. Shrink accordingly.

Now $P_0$ embeds in its normal localization at every nearby stalk, because no finite power of $h$ kills a nonzero element. Via the just-constructed localized isomorphism, $P_0$ is an actual coherent $E_0$-submodule $M_0\subset M$ on one neighbourhood, with

\[
\begin{gathered}
EM_0=M,\quad h:M_0\longrightarrow M_0\\ \text{injective as a }\\ \text{normal-action twisted map}.
\end{gathered}
\tag{J.2}
\]

Every construction used finitely many generators, relations, inverse identities and neighbourhood shrinkings. No arbitrary stalk submodule was assumed to extend coherently without this presentation and torsion check.

#### J.2. The principal support is exactly the actual module support

At every point of this neighbourhood, $\bar M_0=M_0/hM_0$ is finite over the commutative coefficient ring $T=E_0/(h)$. Its actual presentation modulo $h$ makes it a coherent analytic module. The underlying reduced support is analytic: a finite local presentation $T^a\to T^b\to\bar M_0$ has zero cokernel at a point exactly when its constant coefficient matrix is surjective; finite commutative Nakayama proves that equivalence, and the $b\times b$ minors give its analytic complement.

At a stalk,

\[
\begin{gathered}
\bar M_{0,p}=0\\ \Longleftrightarrow M_{0,p}=0\\ \Longleftrightarrow M_p=0 .
\end{gathered}
\tag{J.3}
\]

The first forward implication is the actual finite noncommutative Nakayama argument in Corollary 5.20.3.1, since $M_{0,p}$ is finite and $(h)\subset\operatorname{Jac}(E_{0,p})$ by actual elliptic inversion. Its reverse is immediate. For the second equivalence, $M_0$ spans $M$, and the injective localization map in J.1 prevents a nonzero $M_{0,p}$ from becoming zero. Therefore

\[
 \operatorname{Supp}M
       =\operatorname{Supp}(M_0/hM_0).
 \tag{J.4}
\]

It is equality of reduced sets, not a claim that the two modules or their nonreduced structures coincide. In particular all nilpotent principal relations and all actual supported sections remain in the module; none was discarded to prove this equality. A different finite actual order-zero lattice gives the same reduced support by the same proof.

This is the support of an actual finite-order $E$-module. For an ordinary $\mathscr D$-module, the characteristic support defined as the support of its finite-order $E\otimes_{\mathscr D}-$ realization falls under the same statement after that realization is supplied. If some other convention first forms an infinite-order or microfunction extension, equality with (J.4) is a separate extension-faithfulness assertion; this argument does not prove it.

#### J.3. An arbitrary actual holonomic germ meets the finite principal-fibre premise

Suppose now that the given coherent $E$-module is holonomic, meaning that its reduced actual support $A$ is a homogeneous analytic Lagrangian germ. Regular holonomicity is allowed, but no regularity is needed for this paragraph. Apply Q.1–Q.7 to this whole reduced germ and to its actual module: the explicitly proved contact shear and actual symbol isomorphism put $A$ in generic position, and transport the actual lattice (J.2). By (J.4), its principal support is the transformed $A$.

On $\tau=1$, the fibre $\{u=0\}$ meets that principal support only at $z=0$ in a sufficiently small neighbourhood. The finite analytic module

\[
\begin{gathered}
V=(M_0/hM_0)\\ {}\otimes_{\mathbf C\{u,z\}}\mathbf C\{z\}
\end{gathered}
\tag{J.5}
\]

therefore has support contained in that one point. This follows directly by tensoring the finite presentation: at any other fibre point the original module is zero, so its tensor fibre is zero. If $V=0$ the conclusion is immediate. Otherwise take a finite presentation with $b$ target generators. Its maximal-minor ideal has zero set equal to that support by the rank/Nakayama argument in J.2. Each such minor annihilates $V$: the adjugate of its selected square relation matrix puts its determinant times every target basis vector in the relation image. The analytic Nullstellensatz in J.0 consequently puts a power of the maximal ideal $(z)$ in this minor ideal and hence in $\operatorname{Ann}V$. To justify the power step explicitly, each $z_i$ has some power in that ideal; a monomial of degree greater than the sum of those exponents minus $d$ must contain one of them. Every convergent series of that degree or higher is a finite sum of the degree-bound monomials times convergent series: group each Taylor monomial by a chosen degree-bound divisor; absolute convergence is retained by taking a subseries and a fixed monomial shift. Hence the stated power of $(z)$ annihilates $V$. Its finite generating list multiplied by the monomials below that degree spans it over $\mathbf C$, by finite Taylor remainder. If $d=0$, the coefficient ring is $\mathbf C$ already and finite generation gives the conclusion directly. Thus $V$ is finite dimensional. Nonreduced fibres are included.

The precise actual premise of §5.29, G.1 has now been established for this arbitrary coherent holonomic germ, after the proved contact reduction. Consequently §5.29, G.1 gives its faithful finite coordinate-commuting-ring realization, §5.30, F.1–F.7 give the actual finite-order tensor reconstruction and polynomial-submodule stability, and §5.31, L.1–L.6 give its bounded polynomial-matrix resolution of length $2d+1$ on one whole conic chart over a common base neighbourhood. These are actual finite-order conclusions for the entire assigned germ, rather than a supplied-lattice special case.

#### J.4. Scope retained

J.1 constructs a finite order-zero lattice. It does not make that lattice stable under the characteristic-vanishing order-one subalgebra, and it makes no assertion about the canonical order cutoff. Q.7 transports a regular invariant lattice wherever the actual definition supplies one; this is separate from the general lattice existence above.

The full actual infinite comparison, continuation and separation are proved in Sections5.43–5.86. Section5.87 proves the faithful regular finite-pole realization, and Section5.89, CK.A–CK.K proves coherent source cutoffs and characteristic stability. These results supply the finite target to which the coefficient theorem §5.27 applies.

![The finite lattice construction and exact reduced-support comparison](assets/actual-lattice-support-interface.png)

*The two finite neighbourhood checks construct an actual coherent lattice inside the module. The support equality is an equality of reduced sets; the entire module and its nilpotent principal structure remain. Proof: §5.32, J.1–J.3 and equations (J.2)–(J.5). Free reading: Kashiwara–Kawai, [HolIII](https://www.kurims.kyoto-u.ac.jp/~kenkyubu/kashiwara/HolIII.pdf), I.1.13 and III.5.5.*


### 5.33. A convergent negative-order polynomial kernel complex

The bounded actual polynomial-matrix resolution of §5.31 gives a common-domain ordinary analytic Volterra kernel complex. We prove its signs, radii and multiplicativity explicitly. [§§5.34–5.37](#5-34-full-current-resolutions-and-controlled-proper-image-comparison) supply the canonical finite polynomial line-cone ring and its action on the unrestricted degree-one directional quotient, with complete current, cup/excision and collar proofs.

#### K.1. One exact rescaling makes every differential negative

Use homological degree $j$, with a differential $P_j$ from the $j$-th free term to the $(j-1)$-st. All entries of $P_j$ lie in $R_0$ and have total symbol order at most zero. Replace every free basis in degree $j$ by $h^j$ times that basis; over $E$ these are invertible basis changes. The new differential is

\[
\begin{gathered}
Q_j=h^j\circ P_j\circ h^{-(j-1)}\\ =\sigma^j(P_j)\circ h,\\ \operatorname{ord}Q_j\le-1 .
\end{gathered}
\tag{K.1}
\]

The normal automorphism $\sigma$ is the actual one in §5.22.1 and §5.30, F.1 and preserves $R_0$. Every entry can consequently be written as a finite polynomial

\[
\begin{gathered}
Q=\sum_{\beta\in B}a_\beta(u,h)z^\beta h\\ =\sum_{\beta\in B}\sum_{n\ge0}\\ a_{\beta,n}(t,x)h^{n+|\beta|+1}\partial_x^\beta,\\ |a_{\beta,n}|\le B_\beta C_\beta^n n! .
\end{gathered}
\tag{K.2}
\]

The first sum has finite $B$, the common coefficient domain is actual, and right multiplication by the cotangent polynomial is ordinary normal multiplication. The inverse basis changes show exactness of the rescaled free $E$-complex. Moreover
$Q_j\circ Q_{j-1}=h^j\circ(P_j\circ P_{j-1})\circ h^{-(j-2)}=0$,
with row-vector map composition understood as in §5.31, L.8. If $Q_j$ is an $r_j\times r_{j-1}$ matrix, its action on column function vectors runs from length $r_{j-1}$ to length $r_j$. Thus the function complex runs in the opposite direction to the free resolution, and $\mathcal V_{Q_j}\mathcal V_{Q_{j-1}}$ is well typed. No differential sign has been changed.

#### K.2. A convergent analytic kernel with an explicit radius

For the coefficient of $\partial_x^\beta$, define

\[
\begin{gathered}
K_\beta(t,s,x)=\\ \sum_{n\ge0}a_{\beta,n}(t,x)\\ \frac{(t-s)^{n+|\beta|}}{(n+|\beta|)!}.
\end{gathered}
\tag{K.3}
\]

On a fixed compact coefficient domain and for $C_\beta|t-s|<1$, this converges absolutely and uniformly on smaller compacts, with

\[
 |K_\beta(t,s,x)|
 \le \frac{B_\beta|t-s|^{|\beta|}}
                       {1-C_\beta|t-s|}.
 \tag{K.4}
\]

Indeed $n!/(n+|\beta|)!\le1$ and the geometric sum proves the bound. On a compact with $C_\beta|t-s|\le r<1$, termwise holomorphy and uniform convergence make the sum holomorphic in $(t,s,x)$. Fixed derivatives have bounds on a smaller common product by the Cauchy formula. The finite number of coefficients in all $Q_j$ permits one coefficient-domain intersection and one radius $\eta>0$ with $C_\beta\eta<1/2$ for every one of them.

Choose a convex $t$-disc of diameter less than $\eta$ and a fixed anchor $a$ in it, independent of $x$. For a holomorphic function $f$ on the product with a smaller $x$-polydisc, define

\[
\begin{gathered}
\mathcal V_Qf(t,x)\\ =\sum_{\beta\in B}\int_a^t\\ K_\beta(t,s,x)\,\partial_x^\beta f(s,x)\,ds .
\end{gathered}
\tag{K.5}
\]

The path is the straight segment in that convex disc. Bounds (K.4), finite $B$ and fixed Cauchy derivative bounds for $f$ give absolute uniform convergence on smaller compacts. The integral is holomorphic, either by its segment parametrization and differentiation on compacts or the primitive theorem for a holomorphic integrand in $s$ with holomorphic parameters. The anchor contributes no $x$-derivative terms.

This normalization agrees with
$h^m f=\int_a^t(t-s)^{m-1}f(s)/(m-1)!\,ds$
for a pure negative integer power, with positive orientation from $a$ to $t$. In particular $\partial_t h f=f$. There is no residue normalization or $(2\pi i)$ factor in this ordinary Volterra action. The relation $h\partial_t f=f-f(a,x)$ explains why this action is asserted for the negative-power algebra, not as an action of the full localized ring containing $\partial_t^{-1}$ as a two-sided inverse.

#### K.3. The analytic kernel product is exactly the actual symbol product

For two actual symbols with finite polynomial $\partial_x$ dependence and only strictly negative $\partial_t$ powers, the action (K.5) preserves their product on a sufficiently small common convex product. We prove this, rather than regard it as a formal operator manipulation.

First consider individual terms
$P=p(t,x)h^m\partial_x^\beta$ and
$Q=q(t,x)h^l\partial_x^\gamma$,
where $m,l\ge1$. Distribute the finite $\partial_x^\beta$ derivative in the iterated integral, giving terms indexed by $\nu\le\beta$ with coefficient $\binom{\beta}{\nu}\partial_x^\nu q(s,x)$ and derivative $\partial_x^{\beta-\nu+\gamma}f$. On a smaller coefficient domain expand that coefficient at $t$:

\[
\begin{gathered}
\partial_x^\nu q(s,x)\\ =\sum_{k\ge0}\frac{(-1)^k}{k!}\\ \partial_t^k\partial_x^\nu q(t,x)(t-s)^k .
\end{gathered}
\tag{K.6}
\]

Choose the convex $t$-disc still smaller than a fixed Taylor radius of the larger coefficient domain. This is uniformly convergent for all segments at issue. Absolute convergence justifies interchanging the integrals, putting their ordered triangle in the real parameters $0\le r\le q\le1$, with $v=a+r(t-a)$ and $s=a+q(t-a)$. This specifies an oriented complex segment without ordering complex numbers. The exact integer beta integral is

\[
\begin{gathered}
\int_v^t\frac{(t-s)^{m+k-1}(s-v)^{l-1}}{(m-1)!(l-1)!k!}\,ds\\ =\binom{m+k-1}{k}\\ \frac{(t-v)^{m+l+k-1}}{(m+l+k-1)!}.
\end{gathered}
\tag{K.7}
\]

For justification, substitute $s=v+\theta(t-v)$; the scalar real integral
$\int_0^1(1-\theta)^{m+k-1}\theta^{l-1}d\theta$
equals $(m+k-1)!(l-1)!/(m+l+k-1)!$ by integration by parts and induction on the two positive integers. Thus (K.7) holds for every complex segment with the declared orientation.

Multiplying by the sign in (K.6), its coefficient is
$(-1)^k\binom{m+k-1}{k}=\binom{-m}{k}$.
Together with $\binom{\beta}{\nu}$, these are precisely
$\partial_\tau^k(\tau^{-m})/k!$ and the finite $\xi$-derivative coefficients of $\xi^\beta$ in the actual normalized Leibniz product. Consequently the convolution kernel is the kernel of $P\circ Q$, with the correct derivative $\partial_x^{\beta-\nu+\gamma}$ and negative $\partial_t$ exponent $m+l+k$. Every contraction, including all $\tau,t$ contractions, is retained.

For arbitrary actual coefficients in (K.2), the kernels converge uniformly on a common smaller product by K.2, and their finite base derivative orders have the same property by Cauchy estimates. Their iterated integrals are uniformly absolutely convergent, so convolution passes to the infinite negative-power sums. Alternatively compare their Taylor coefficients in $t-v$: at a fixed power only finitely many homogeneous indices and $k$ occur, so the calculation above proves each coefficient identity. The complete actual product theorem §5.19 puts the symbol product in the factorial class, whose kernel is analytic on a smaller product by K.2. Two analytic kernels with those identical Taylor coefficients coincide there. It follows that

\[
 \mathcal V_P\mathcal V_Q=\mathcal V_{P\circ Q}
 \tag{K.8}
\]

on the stated holomorphic functions, with their finite $\partial_x$ derivatives, on a common smaller convex product. The same proof applies to matrices, with coefficient order retained.

In particular the finitely many rescaled matrices (K.1) give an actual holomorphic kernel complex:
$\mathcal V_{Q_j}\mathcal V_{Q_{j-1}}=0$.
This is a convergent matrix identity, not a claim that its holomorphic solution evaluation is faithful on $M$.

#### K.4. What remains at the next analytic interface

The common-domain convergent kernels above supply the negative-order integral matrices of the finite polynomial construction. [§§5.34–5.37](#5-34-full-current-resolutions-and-controlled-proper-image-comparison) prove their canonical classes in $\mathcal E(G;D)$ for the actual line cone, ring composition and action on every unrestricted-growth degree-one directional germ class, with the exact residue/trace signs and all sheaf support maps. Identification with the full kernel ring $E_R$, the full convergent-$z$ $E$ ring and arbitrary derived relative action remain required. Sectorial propagation, continuation and separation at singular support remain separate obligations.

One finite algebraic observation may reduce that later proof's flatness requirements. Off the actual support, the bounded free $E$-complex in §5.31 is acyclic. A bounded acyclic complex of projective modules is split contractible: beginning at its lowest nonzero degree, split the surjection onto that projective term; its kernel is a direct summand of the next projective term, hence projective, and repeat finitely. The chosen splittings give a contracting homotopy. Any ring extension preserves that homotopy after tensoring. Thus, once the actual ring-to-kernel-stalk map is proved, that particular bounded complex remains acyclic off support after extension without requiring a general flatness theorem. This observation neither proves the ring map nor proves extension-faithfulness or the infinite-order comparison on $M$ itself.

Sections5.86–5.87 prove the faithful D-type and regular finite-pole realizations; Section5.89, CK.A–CK.K proves the exact cutoff and everywhere characteristic stability. Arbitrary analytic proper regularity requires its further argument.

![The oriented parameter triangle and exact negative-order kernels](assets/negative-polynomial-kernels.png)

*The triangle uses real parameters along the oriented complex segment from $a$ to $t$. Every mixed derivative contraction and factorial denominator is retained. Proof: §5.33, K.1–K.3 and equations (K.1)–(K.8). The ordinary action has no assigned residue or trace normalization. Free reading: Kashiwara–Kawai, [HolIII](https://www.kurims.kyoto-u.ac.jp/~kenkyubu/kashiwara/HolIII.pdf), III.1–III.3 and IV.3–IV.4.*



### 5.34. Full current resolutions and controlled proper-image comparison

#### R.1. Complexes, supports and the coefficient convention

Let M be a Hausdorff second-countable complex manifold of complex dimension m, and let L be a holomorphic vector bundle of finite rank. In the applications, M is an open product in two or three sets of N complex coordinates and L is the holomorphic line of input top forms. Dolbeault degree refers to **all complex coordinates of M**, including output coordinates. A complex constrained to remain holomorphic in output variables is not the complex considered here.

Write A_L^q for smooth (0,q)-forms with coefficients in L and D_L^q for distributional (0,q)-forms with coefficients in L, q=0,…,m. Distributional means that each coefficient is a continuous linear functional on compactly supported smooth density test sections; in a coordinate box and a holomorphic frame it is the ordinary scalar distribution on that box. All sheaves are sheaves of complex vector spaces. Put

    δ(∑_I u_I dbar z_I ⊗η)=∑_{j,I}(∂_{bar z_j}u_I)dbar z_j∧dbar z_I⊗η,
    ∂_{bar z_j}=(∂_{a_j}+i∂_{b_j})/2,  z_j=a_j+ib_j.

The ordered multiindices I are increasing. A holomorphic change of frame commutes with δ, so this defines a global differential and δ²=0. There are augmented complexes

    0→O(L)→A_L^0→…→A_L^m→0,
    0→O(L)→D_L^0→…→D_L^m→0,                         (R.1)

and their inclusion j:A_L^*→D_L^*. The map O(L)→D_L^0 sends a holomorphic section to its locally integrable coefficient distribution. R.2–R.4 prove exactness and a local homotopy for j.

The coefficient frame η is ungraded. If it is a holomorphic differential form, its full exterior realization is

    J_q(α⊗η)=α∧η.                                   (R.2)

This is the antiholomorphic-first convention and satisfies Jδ=∂bar J. This section does not put η in a shifted degree and does not define fibre integration. R.8 records the precise conversion to full exterior products.

For a continuous map p:M→B between the manifolds occurring here, define p_!F as the sheaf whose sections over U⊂B are sections of F on p^{-1}U with support closed in p^{-1}U and proper over U. Proper means that the inverse image in that support of every compact subset of U is compact. Restriction to a **base open subset** preserves this property. Restriction to an arbitrary source open subset need not preserve it; R.7 instead uses controlled extension by zero. The functor p_! is left exact: kernels and finite limits retain their closed support inside a proper support.

#### R.2. The one-coordinate fundamental solution and its parameter version

On C, set E(z)=1/(πz), interpreted as a locally integrable distribution. It is locally integrable because the integral of |z|^{-1} on a radius-r disc is 2πr. For a compactly supported smooth test function φ,

    ⟨∂bar E,φ⟩=−(1/π) lim_{ε→0}∫_{|z|>ε}(1/z)∂bar φ da db=φ(0).   (R.3)

Indeed, Green's formula ∫∂bar f da db=(1/(2i))∮f dz follows by expanding ∂bar=(∂a+i∂b)/2 and applying the one-real-variable fundamental theorem to rectangles, then to a domain by subdivision. Apply it to φ/z off the deleted disc. The outer boundary is zero. The inner boundary is clockwise and tends to −2πiφ(0), whereas its difference from −2πiφ(0) is O(ε) by the mean-value bound for φ. Thus the integral in (R.3) tends to −πφ(0). This proves ∂bar E=δ_C with the indicated π and sign.

Choose discs P_j⊂C and smaller concentric discs Q_j with closure contained in P_j. Pick χ_j∈C_c^∞(P_j), equal to 1 on an open neighbourhood of closure Q_j. Let P=∏P_j and Q=∏Q_j. For a distribution u on P, define partial convolution

    T_j u=E *_j(χ_j u),     P_j u=E *_j((∂bar_jχ_j)u).             (R.4)

The notation P_j for an operator is distinguished by its argument from the coordinate disc P_j. Multiplication by χ_j cuts off only the j-th coordinate; the other coordinates remain distributional parameters. Extend by zero only in that coordinate before convolution. The result is defined on C×∏_{k≠j}P_k and can be restricted to Q_j×∏_{k≠j}P_k.

To make (R.4) precise, transpose the partial integral against a compact test function φ(z_j,z_hat). For example, T_j pairs u with

    χ_j(ζ_j)∫_C E(z_j−ζ_j)φ(z_j,z_hat) da_j db_j.

This is smooth in ζ_j and z_hat, has compact support in ζ_j from χ_j, and has compact support in z_hat from φ. Smoothness follows by moving every derivative to φ in convolution with the locally integrable E. Thus it is a valid distributional test function. The same construction with ∂bar_jχ_j defines P_j. This also proves continuity on bounded test-function sets and defines the operators for arbitrary distributional parameters. Finite iterated convolutions in different coordinates are legitimate: their transposed tests are compact and smooth in the integrated coordinates, and repeated integration gives the same product kernel. Fubini here concerns smooth compact tests paired with locally integrable kernels, with absolute integrability near their independent singularities.

On Q_j×∏_{k≠j}P_k, the following identities hold distributionally:

    ∂bar_j T_j u=u,       T_j∂bar_j u=u−P_j u,       ∂bar_jP_j u=0.       (R.5)

The first follows from (R.3) and χ_j=1. For the second, use

    χ_j∂bar_j u=∂bar_j(χ_ju)−(∂bar_jχ_j)u

and (R.3) under convolution. For the third, use (R.3) again: its derivative equals (∂bar_jχ_j)u, which vanishes on Q_j. These calculations follow on tests by integration by parts, so they do not presume that u is smooth. For k≠j, T_j and P_j commute with ∂bar_k. The operators in distinct coordinates commute, with restrictions to the corresponding smaller boxes understood.

Both T_j and P_j preserve smoothness. For T_j, every derivative can be transferred to the compactly supported smooth coefficient χ_ju in the j-th variable; derivatives in the other variables are ordinary derivatives of that coefficient. For P_j, when the output coordinate is in Q_j, the support of ∂barχ_j is a positive distance from it. The kernel E(z_j−ζ_j) is then smooth and holomorphic in z_j. A distribution in the remaining coordinates may still remain, which is exactly what the parameter construction allows.

#### R.3. The explicit multivariable contraction

Let ε_j mean left exterior multiplication by dbar z_j and let ι_j be its standard contraction:

    ι_j(dbar z_{i_1}∧…∧dbar z_{i_q})
      =∑_{r=1}^q(−1)^{r−1}δ_{j,i_r}
         dbar z_{i_1}∧…∧omit(dbar z_{i_r})∧…∧dbar z_{i_q}.

Then ε_jι_j+ι_jε_j=1 and, for j≠k, ε_kι_j+ι_jε_k=0. Put

    h_j=T_jι_j,       π_j=(1−ε_jι_j)P_j.                       (R.6)

Here P_j acts coefficientwise and 1−ε_jι_j deletes every component containing dbar z_j. On the appropriate partially shrunken box,

    δh_j+h_jδ=1−π_j,       δπ_j=π_jδ.                         (R.7)

For verification, write a form as a+ε_jb with neither a nor b containing dbar z_j. The j-th contribution to δh_j+h_jδ is

    T_j∂bar_j a+ε_j∂bar_jT_jb=a−P_ja+ε_jb,

by (R.5). Every k≠j contribution cancels by ε_kι_j+ι_jε_k=0 and commutation of ∂bar_k with T_j. This proves the first formula. The second follows because π_j annihilates ε_j-components, its surviving coefficients are ∂bar_j-closed by (R.5), and it commutes with every other coordinate differential. The maps π_i and π_j commute, including their exterior deletion operators.

Define

    Π=π_1…π_m,
    H=h_1+π_1h_2+π_1π_2h_3+…+π_1…π_{m−1}h_m.               (R.8)

Every term is applied from P to a partially smaller product and finally restricted to Q. The cutoffs depend on separate coordinates, so no unannounced common-domain choice is involved. Telescoping (R.7) gives the actual chain identity

    δH+Hδ=restriction_{P→Q}−Π.                              (R.9)

The image of Π is zero in every positive form degree: every nonempty multiindex contains a coordinate deleted by one π_j. In degree zero its coefficients are given by the full product kernel

    Πu(z)=⟨u(ζ), ∏_{j=1}^m E(z_j−ζ_j)∂barχ_j(ζ_j)⟩.        (R.10)

All ζ-coordinates have compact support in their respective cutoff annuli; all output coordinates are separated from those annuli. The paired test is therefore smooth, compactly supported in P, and holomorphic in every z_j∈Q_j. Pairing a distribution with this parameterized test is smooth, with derivatives obtained by differentiating the test. It is holomorphic because all its ∂bar_j derivatives vanish. Consequently Πu is an ordinary smooth holomorphic function on Q, without any growth hypothesis on u. The same statements apply componentwise in a holomorphic bundle frame.

For a positive-degree closed current ω, (R.9) gives ω|Q=δHω. For a degree-zero current with δω=0, Hω=0 and (R.9) gives ω|Q=Πω, which is smooth and holomorphic by (R.10). Conversely every holomorphic coefficient is δ-closed. This proves exactness of both complexes (R.1), including identification of degree-zero distributional solutions with O(L). The proof is local on arbitrary coordinate products and hence proves sheaf exactness. It never assumes a distributional version of the assertion being proved.

#### R.4. Local comparison of smooth forms and currents

The inclusion j commutes with δ and with restriction. The operator Π takes currents to smooth forms (only degree zero survives), is a chain map, and is the identity on holomorphic closed sections after restriction. Equation (R.9), applied to currents and separately to smooth forms, states

    jΠ−restriction=−(δH+Hδ),
    Πj−restriction=−(δH+Hδ).                               (R.11)

Thus j has the explicit local homotopy inverse Π. These are local maps from an outer product to a smaller product, rather than global support-preserving maps. They suffice for exactness and the sheaf quasi-isomorphism. Their spreading of supports is harmless for that claim. Proper-image comparisons below use a different support argument and never silently apply H to a fixed supported class.

#### R.5. A self-contained acyclicity proof for C∞-module sheaves

Let A=C_M^∞. Each A_L^q and D_L^q is an A-module by smooth multiplication. Multiplication need not commute with δ; acyclicity is an assertion about the individual sheaves, not A-linearity of the Dolbeault differential.

We first give enough injectives in the category of sheaves of complex vector spaces. For any such sheaf F, put

    G(F)=∏_{x∈M} i_{x*}F_x,
    G(F)(U)=∏_{x∈U}F_x.                                   (R.12)

The natural map a_F:F→G(F) sends a section to its germs and is injective. Each i_{x*}V is injective: a sheaf map F→i_{x*}V is precisely a linear map F_x→V; stalks preserve exact sequences, and a linear map from a subspace extends to the whole vector space by extending a basis. Products of injectives are injective because maps into the product are the product of the individual maps, and products of surjective linear maps are surjective. Thus G(F) is injective.

The functor G is exact: for every U its sections are a product of the exact stalk sequences. Define Q(F)=G(F)/a_F(F), Q^0(F)=F, and

    I^r(F)=G(Q^r(F)),
    d_I:I^r(F)→Q^{r+1}(F)→G(Q^{r+1}(F)).                  (R.13)

The first arrow is the quotient and the second is its germ embedding. The augmented complex 0→F→I^0(F)→I^1(F)→… is exact by this construction and has injective terms. Q is exact: in a short exact sequence, apply the exact G to obtain a diagram with injective vertical germ maps; taking cokernels is exact by a direct diagram chase. In detail, a cokernel class mapping to zero lifts to the middle G-term; its image is an original section in the right term; lift that original section, subtract its germ vector, and the resulting class comes from the left term. Surjectivity follows from surjectivity in the two right columns. Hence every functor I^r=GQ^r is exact.

When F is an A-module, the stalk products in (R.12) are naturally A-modules, by the germ of a smooth function acting at each point. The augmentation, quotient maps and the differentials in (R.13) are A-linear. The underlying injective assertion remains the assertion in the category of complex-vector sheaves. This furnishes an actual injective resolution whose terms and column differentials carry the needed A-module structure.

Here is the support-preserving lifting argument. In an exact complex of A-modules with A-linear differentials, let c be a cocycle section with compact support K. At each x∈K, stalk exactness gives a primitive v_x on a neighbourhood U_x. For any prescribed open O containing K, shrink U_x to relatively compact coordinate neighbourhoods in O and choose a finite subcover of K. Choose smooth bumps ψ_i supported compactly in these U_i and positive on a covering of K. Put ψ=∑ψ_i, and choose a smooth compactly supported η in {ψ>0} equal to 1 near K. Such bumps are obtained in a real coordinate ball from exp(−1/(1−|r|²)) inside the ball and zero outside, followed by rescaling and finite sums. Then

    ρ_i=ηψ_i/ψ,       v=∑_i extension_by_zero(ρ_i v_i)        (R.14)

are well defined, smooth-multiplication expressions. Every ρ_i has compact support inside U_i. The differential is A-linear, so dv=(∑ρ_i)c=ηc=c. The primitive has compact support contained in O. This is the exact support assertion: the primitive can enlarge K inside a chosen compact neighbourhood, but it need not stay inside K or a fixed singular support.

For an unrestricted section c, use a locally finite version of the same construction. To obtain it, choose a countable precompact coordinate cover and a compact exhaustion K_n⊂int K_{n+1}. Cover each compact shell K_n\int K_{n−1} by finitely many coordinate neighbourhoods with closures in int K_{n+1}\K_{n−2}. The resulting collection is locally finite. Choose positive bumps on a covering refinement; their locally finite sum is positive everywhere. Division by that sum gives a smooth partition of unity. The locally finite sum in (R.14) then gives a global primitive. This proves Γ-acyclicity of A-module sheaves using the actual injective resolution (R.13), rather than an unproved fine-sheaf assertion.

For p_! acyclicity assume p is smooth, as all projections in the application are, and take a stalk b∈B and a positive-degree cocycle c represented in p_!I^r(F) over a base neighbourhood U. Its support S is closed and proper over U. Choose V containing b with compact closure in U and choose θ∈C_c^∞(U), θ=1 near closure V. The section θ∘p times c has compact support, because it is supported in S∩p^{-1}(supp θ), which is compact by properness. Apply (R.14) on p^{-1}U to obtain a compactly supported primitive v. Restrict it to p^{-1}V. Its support is closed there and proper over V: over a compact C⊂V it is the closed subset supp(v)∩p^{-1}C of a compact support in p^{-1}U. The differential of this restricted primitive is c on p^{-1}V. Thus every positive-degree stalk cohomology class of p_!I^*(F) is zero; degree zero is p_!F by left exactness. Therefore

    R^r p_!F=0 for r>0, for every C∞_M-module F.           (R.15)

Only a base shrink is used, which is permitted in a sheaf assertion. A support remains proper; no growth, fibre-radius estimate or globally uniform support bound is presumed. The same proof applies to any smooth p between the local manifolds here, not only projections, because θ∘p is smooth. Ordinary global Γ corresponds to the unrestricted partition argument; compact Γ_c corresponds to (R.14).

There is also an actual Čech homotopy preserving a **fixed common** closed support S. Let (U_i) be a locally finite cover with subordinate smooth partition (ρ_i). In alternating Čech degree r use cochains whose every component on U_{i_0…i_r} is supported in S∩U_{i_0…i_r}, with augmented degree -1 equal to Γ_S(M,F). Define

    (h_Cech c)_{i_0…i_{r−1}}=∑_j extension_by_zero(ρ_j c_{j i_0…i_{r−1}}).

The extension is inside U_{i_0…i_{r−1}}; its support is closed inside the j-th open set because supp ρ_j⊂U_j. Local finiteness makes the sum a section, and each summand vanishes outside S. The alternating differential Δ gives Δh_Cech+h_CechΔ=1: expand

    (Δc)_{j i_0…i_r}=c_{i_0…i_r}+∑_{k=0}^r(-1)^{k+1}c_{j i_0…omit(i_k)…i_r};

after multiplication by ρ_j and summation the first term is c, and the remaining terms cancel Δh_Cech c. In augmented degree -1, h_CechΔ=1. This works for every C∞ module, including currents. For the family of proper supports, take the filtered union over common proper S; finite unions are proper and the homotopy preserves each S. An unrestricted product of components with merely individually proper supports is different and is not used: their supports can escape to infinity even for B a point. This supplementary homotopy is not needed to identify Čech cohomology with derived cohomology; the injective proof above establishes acyclicity independently.

#### R.6. The acyclic-resolution comparison, with chain maps

Let C^q be either complex A_L^q or D_L^q, 0≤q≤m. Apply the exact functors I^r of R.5 to its terms and differential, forming

    B^{q,r}=I^r(C^q),
    d_T|B^{q,r}=I^r(δ)+(-1)^q d_I,
    T^k=⊕_{q+r=k}B^{q,r}.                                 (R.16)

The sum is finite for each k. The two differentials commute before insertion of (-1)^q, hence d_T²=0. Every T^k is injective. Define the chain augmentation

    a_C:C^q→T^q,   a_C(c)=a_{C^q}(c) in B^{q,0}.            (R.17)

It commutes with δ by naturality and its vertical derivative is zero. There is also a chain map a_L:I^*(O(L))→T^* induced by O(L)→C^0, in the column q=0.

Before applying p_!, vertical exactness shows that a_C is a quasi-isomorphism. Horizontal exactness after applying each exact I^r to (R.1) shows that a_L is a quasi-isomorphism. Both assertions can be checked by the following finite elimination, without invoking a spectral-sequence theorem. In a cocycle of total degree k, start with the smallest q that occurs. Its component of vertical degree r=k−q is vertically closed, since the preceding q-component is zero. If r>0, solve its vertical equation, subtract the total boundary of that solution with the factor (-1)^q, and thereby remove that component. Only the next q-component can be introduced. Repeat; there are at most m+1 q-components. The survivors lie in vertical degree zero and in the augmented source. For the horizontal comparison, run the same procedure with r first; the relevant diagonal has only finitely many components. Injectivity is proved by applying the same removal to a purported boundary primitive. Thus T is an injective complex representing the same holomorphic sheaf as either resolution.

After p_!, each vertical column p_!I^*(C^q) has cohomology p_!C^q in degree zero and zero above by (R.15). Consequently the actual chain map

    p_!a_C:p_!C^*→p_!T^*                                (R.18)

is a quasi-isomorphism. To verify this explicitly, apply the finite elimination just described to a total cocycle over a base neighbourhood. When a vertical component of positive degree is closed, R.5 provides a primitive after shrinking the base around the stalk under consideration, with proper support. Only finitely many components are removed; intersect the finitely many smaller base neighbourhoods. Multiplication by the column sign (-1)^q yields the required boundary correction. If k>m all components disappear. If k≤m, the last possible component has r=0; vertical closure puts it in the image of p_!C^k by left exactness. The total cocycle condition says its C-differential is zero. For injectivity, if a_C(z)=d_Tt, remove the least-q positive-r components of t by the same procedure; the right side has only q=k,r=0, so those least-q components are vertically closed. The remaining t lies at q=k−1,r=0 and vertical closure puts it in a_C(p_!C^{k−1}); hence z is a C-boundary. This proves (R.18) on every stalk, including all support conditions.

The injective augmentation I^*(O(L))→T provides the usual derived comparison, so (R.18) identifies p_!A_L^* and p_!D_L^* with Rp_!O(L). Here Rp_! is defined using injective complexes; the explicit injective model (R.13) makes that definition concrete. For completeness, a bounded-below exact complex K of injectives has the following actual contraction. Write Z^n=ker d^n. Starting in the bottom degree with Z^n=0, injectivity of Z^n gives a retraction K^n→Z^n. Its complementary summand is isomorphic under d^n to Z^{n+1}, so Z^{n+1} is injective too. Inductively choose sections s^n:Z^{n+1}→K^n and write K^n=Z^n⊕s^nZ^{n+1}. Then

    h^n(z+s^nw)=s^{n−1}z,       dh+hd=1.                  (R.18a)

Apply this to the exact cone of a quasi-isomorphism f:X→Y between bounded-below injective complexes. In the convention Cone(f)^n=Y^n⊕X^{n+1}, d(y,x)=(d_Yy+fx,−d_Xx), the block g:Y^n→X^n of its contraction is a chain map. The equations dh+hd=1 say fg is homotopic to 1 via the Y-to-Y block, and gf is homotopic to 1 via the X-to-X block. Thus f is a chain homotopy equivalence, and applying p_! preserves those homotopies. This gives the claimed inverse for I^*(O(L))→T. No canonical splitting is asserted or needed. It also verifies independence of the injective model without importing a separate acyclic-resolution theorem.

All constructions G,Q,I,a and the induced total maps are natural in C-linear sheaf maps. In particular j:A_L^*→D_L^* gives the commutative square with the two maps (R.18); it induces the identity on O(L) and hence on Rp_!O(L). The cone of j is a bounded exact sheaf complex whose terms are A-modules. Applying the same double-complex elimination to that cone shows that p_!j is a quasi-isomorphism. The support choices in elimination affect a primitive, not the chain maps or the induced derived identification.

The same statement holds for Γ and Γ_c. For Γ, use the unrestricted locally finite partition proof in R.5. For Γ_c, use its compact version. In particular the complexes of global smooth forms and currents compute the same derived ordinary sections, and their inclusion gives the comparison. No fixed-domain higher-cohomology vanishing is asserted for O(L).

#### R.7. Relative supports and controlled open excision

Fix a closed subset S⊂M and let k:M∖S→M be its open inclusion. Define the sheaf complex

    C_S^q(D)=D_L^q ⊕ k_*D_{L|M∖S}^{q−1},
    d(a,b)=(δa, a|M∖S−δb),                               (R.19)

with negative-degree terms zero; use the same definition with A in place of D. This is the explicit shifted-cone convention used throughout this section. This section only proves the resolution/excision comparison for that convention, rather than its connecting-map or cup normalization.

Here is the missing elementary identification with the right-derived support functor, so no cone definition substitutes for a comparison. An injective complex-vector sheaf I is flabby. Indeed, for opens V⊂U, let C_V^! and C_U^! be their constant complex sheaves extended by zero to M. The stalkwise injection C_V^!→C_U^! induces, by injectivity,

    Hom(C_U^!,I)→Hom(C_V^!,I) surjective.

These two spaces are Γ(U,I) and Γ(V,I): a map from the constant sheaf is determined by the image of 1, and the extension-by-zero adjunction preserves this description. Thus restriction is surjective. The open restriction k^{-1} also sends injectives to injectives, since Hom(F,k^{-1}I)=Hom(k_!F,I) and k_! is exact on stalks. For an injective resolution I^* of O(L), there is consequently a degreewise exact sequence

    0→Γ_S I^*→I^*→k_*k^{-1}I^*→0,

where Γ_S is the subsheaf of sections vanishing off S. The first complex is the definition of RΓ_S O(L), and the third computes Rk_*O(L|M∖S). The map from Γ_S I^* to the shifted cone of the last restriction sends T to (T,0). Its quotient is the cone of the identity on k_*k^{-1}I^*, with contraction (j,b)↦(b,0); hence the map is a quasi-isomorphism. The same degreewise surjective restriction on global sections identifies the right-derived global supported-section functor with the global relative cone. This proves the local and global support comparisons from the injective definition.

Every sheaf k_*D^{q−1} is a C∞_M-module: multiply by the restriction of a smooth function on M. For an open U⊂M, Γ(U,k_*D)=Γ(U∖S,D). The functor k_* sends injectives to injectives: Hom(F,k_*I)=Hom(k^{-1}F,I), and k^{-1} is exact, as verified on stalks. To check k_*-acyclicity of an individual term, use its injective resolution on M∖S. The stalk at x of its k_*-image complex is the filtered colimit over U containing x of the section complexes on U∖S. Filtered colimits of vector spaces preserve kernels and images: each finite equation or witness to an image already appears at one common later index. Their higher cohomology therefore vanishes by the Γ-acyclicity proved in R.5 on each open U∖S. This proves the individual terms are k_*-acyclic. The same finite double-complex elimination as in R.6 now proves that k_*D^* computes Rk_*O(L|M∖S); it need not be an exact resolution of a sheaf concentrated in degree zero. Thus (R.19) represents the canonical local-support object

    RΓ_S O(L)=Cone(O(L)→Rk_*O(L|M∖S))[-1].               (R.20)

The injective comparison just proved shows this is the right-derived support object, with its support long exact sequence furnished by the short exact sequence of cone complexes. Every term of (R.19) is a C∞ module and is p_!-acyclic by R.5. Applying R.6's bounded-complex comparison therefore gives

    p_!C_S^*(D) represents Rp_!RΓ_S O(L),                  (R.21)

and the smooth/current inclusion is a quasi-isomorphism also in (R.21). For ordinary sections the global cone is exactly

    Γ(M,D^q) ⊕ Γ(M∖S,D^{q−1}),

with the differential displayed in (R.19); it computes supported cohomology via (R.20). A closed current T supported in S maps to (T,0). Neither this map nor (R.21) asserts that every supported-cohomology class admits a current supported in S.

Away from S, (R.19) is the cone of the identity on D^*. It has the explicit contracting homotopy

    h(a,b)=(b,0),   dh+hd=(a,b).                           (R.22)

The same formula works for A. This proves directly that its cohomology sheaves are supported in S.

Now let i:W→M be an open inclusion with S⊂W. Restriction of (R.19) as a sheaf complex is C_S(D)|W=C_S(D|W). Let i_! be extension by zero. On sections, i_!F consists of sections on W whose support is closed in the ambient open set and contained in W; such sections vanish near its boundary and can be extended by zero. It is an exact sheaf functor, as can be checked on stalks: inside W it is the identity and outside W its stalk is zero. Define the literal chain map

    e_W:i_!(C_S(D)|W)→C_S(D)                              (R.23)

by extension by zero. On stalks in W it is the identity. Outside W, S is absent and the target is contractible by (R.22), while the source is zero. Thus e_W is a quasi-isomorphism. Both source and target terms are C∞_M-modules, and its cone is bounded and exact. R.6's elimination, applied to that cone, proves that

    p_!e_W:p_!i_!(C_S(D)|W)→p_!C_S(D)                    (R.24)

is a quasi-isomorphism. Moreover p_!i_!= (p|W)_! on sections. In one direction, extension by zero retains the same closed support and properness. Conversely, a section with proper support T in W vanishes near every boundary point z∈M∖W over the current base open U. To see this, choose a compact base neighbourhood C⊂U of p(z). The set T∩p^{-1}C is compact in W and hence a compact closed subset of M disjoint from z. A neighbourhood of z avoids that set; after also restricting to p^{-1}(int C), it avoids T. Thus zero extension is defined, its support is closed in p^{-1}U, and it remains proper. The identity does not claim that an arbitrary properly supported source section can be restricted to W. It identifies the properly supported extension-by-zero domain.

Equations (R.23)–(R.24) are the controlled excision roof. They apply whenever the cohomological support is a closed S contained in W; S itself need not be globally compact, and the p_! representatives retain their own proper support. If S is already proper over B, it is the support condition used by a support-to-proper-image morphism. For two neighbourhoods W_1,W_2 of S, put W_12=W_1∩W_2. The extension-by-zero maps from W_12 into W_i and then into M compose literally to e_{W_12}. Each map is a quasi-isomorphism by the same stalk argument. Consequently the two excision identifications agree as morphisms of the injective/derived models from R.6. This supplies the actual comparison maps, rather than presuming that ordinary restriction of proper supports is allowed.

The projection (a,b)↦a is a chain map C_S(D)→D. It defines the support-forgetting morphism and, after p_!, the supported-to-proper-image morphism when the selected representatives have proper support. It commutes literally with (R.23). A closed supported current is sent to the same current. Maps obtained on a smaller base neighbourhood agree on a common base restriction because all sheaf maps here commute with restriction. This is the precise interface needed when the line-cone intersection is proper only after the output has been shrunk.

There is no extension assumption at a singular boundary. If f is holomorphic only on M∖S, even with unrestricted growth toward S, it is a smooth coefficient and a distribution on **that open set**. It is a valid second component of (R.19). This does not extend it as a distribution on M. Any collar used subsequently must stay inside its holomorphic domain; changes of collar and their cup/trace comparisons are the explicit chain statements of [§5.37, W.1–W.7](#5-37-interior-collars-and-the-canonical-directional-action).

#### R.8. Coefficient products and scope

For the [§5.36, U.1–U.5 construction](#5-36-canonical-finite-polynomial-line-cone-kernels), take m=2N on the kernel product or m=3N on the triple product, and take L to be the appropriate holomorphic input top-form line. The local full-normal currents use all output antiholomorphic components, exactly as required by D_L^*. There is no restriction to antiholomorphic fibre variables. The compact segment geometry of U.1 or U.5 is to be checked for the selected closed intersection S and output neighbourhood. Once checked, (R.18), (R.21) and (R.24) are available for the full smooth/current resolution and that controlled support.

Our ungraded coefficient product and full exterior product differ by a declared factor. If η_A has holomorphic exterior degree a and β has antiholomorphic degree q, then

    J((α∧β)⊗(η_A∧η_B))
       =(-1)^{aq}J(α⊗η_A)∧J(β⊗η_B).                    (R.25)

This follows by moving η_A across β. For two N-dimensional normal-block kernel factors at degree N, the factor is (-1)^{N²}. A raw coefficient integration defined by conjugating actual real fibre integration with J acquires this factor when applied to the ordinary coefficient product rather than the full-block wedge. The factor is explicit. [§5.35, V.3–V.6](#5-35-relative-cones-unshifted-cup-and-the-normalized-trace) proves the coefficient-line identification, fibre orientation and degree-dependent trace map. In the holomorphic-first realization η∧α, the full-form differential is (-1)^aδ, and the tensor permutation gives the usual shifted tensor sign. That is an equivalent convention when those signs are retained throughout.

This section establishes exact full Dolbeault and current resolutions; explicit positive-degree homotopies and degree-zero distributional holomorphicity; their smooth/current comparison; ordinary, compact and properly supported acyclicity; the injective-double-complex comparison; and relative support and controlled excision by extension by zero. The normalized cup and trace are proved in [§5.35, V.1–V.6](#5-35-relative-cones-unshifted-cup-and-the-normalized-trace), the actual finite polynomial kernels in [§5.36, U.1–U.5](#5-36-canonical-finite-polynomial-line-cone-kernels), and the unrestricted-input collar comparison in [§5.37, W.1–W.8](#5-37-interior-collars-and-the-canonical-directional-action). The full convergent-\(z\) \(E\) map, infinite-order and arbitrary derived \(E\) actions, propagation, module separation, finite D-type, finite poles, intrinsic order, full C1 and arbitrary analytic proper regularity remain required.

![Proper neighbourhoods and controlled excision](assets/proper-support-control.png)

*The exact real model has \(p(x,y)=x\), \(S=\{y=x\}\), and closed neighbourhood \(F_r=\{|y|\le\sqrt{1+x^2}+1\}\). Its intersection over every compact base set is compact. Restricting \(S\) to \(W=\{y>0\}\) gives over \([-1,1]\) the noncompact set \(\{(x,x):0<x\le1\}\). Thus arbitrary source restriction need not preserve properness; the extension-by-zero maps (R.23)–(R.24) supply the controlled excision used here. This is an exact model of the support mechanism. Free human comparison: [Kashiwara–Schapira, Micro-hyperbolic systems](https://perso.imj-prg.fr/pierre-schapira/wp-content/uploads/schapira-pub/Microhyp.pdf), §3.1; the complete support proof is R.5–R.7 above.*

### 5.35. Relative cones, unshifted cup and the normalized trace

The full smooth/current and proper-support comparisons are [§5.34, R.1–R.7](#5-34-full-current-resolutions-and-controlled-proper-image-comparison). This section fixes the relative differential, connecting signs, ordered normal calibration and all-degree trace. The input coefficient lines are ungraded before the trace shift. Free human comparison: [Kashiwara–Schapira, Micro-hyperbolic systems](https://perso.imj-prg.fr/pierre-schapira/wp-content/uploads/schapira-pub/Microhyp.pdf), §3.1. Every chain identity used here is proved below.

#### V.1 The actual relative differential and connecting homotopy

Let I be a cochain complex on a manifold M, with differential δ, δ²=0,
and let r:I(M)→I(M∖Z) be restriction. Define

\[
\begin{gathered}
C_Z^q(I)=I^q(M)\\
\oplus I^{q-1}(M\setminus Z),\qquad\\
d(a,b)=(\delta a,ra-\delta b).
\end{gathered}
\tag{V.1}
\]

Both components of d² are zero: the second is
rδa−δra+δ²b=0. This is the fixed model of Cone(r)[−1]; writing merely
“shifted cone” would not specify its connecting sign. The inclusion of
supported cochains is T↦(T,0). This is a chain map because rT=0.

For a degree-zero δ-closed section f on the complement, its **positive**
connecting representative in this convention is c_f=(0,−f), of degree1.
If a distribution F on M restricts to f, then

\[
 c_f+d(F,0)=(\delta F,0).
 \tag{V.2}
\]

This gives the actual chain homotopy identifying the connecting representative
with the supported current δF. If F is replaced by F+G, with rG=0, the current
changes by δG and the relative representative changes by d(G,0). Thus the
comparison is independent of the extension, whenever such an extension
exists. We do not assert that an arbitrary holomorphic function of unrestricted
boundary growth has a distributional extension.

The relative cone's ordinary projection to I(M∖Z)[−1] is (a,b)↦b. Its positive
connecting convention above corresponds to taking the negative of that
projection when identifying a complement function with its connecting class.
That sign is fixed by (V.2), rather than inferred from an exterior permutation.

#### V.2 Tensoring the normal connecting classes

For ordered independent normal coordinates w₁,…,w_N, let C_i be the relative
complex for the corresponding one-coordinate closed support. Tensor the
underlying complexes over C, not over the holomorphic coefficient ring. Use the ordinary
cochain tensor differential

\[
\begin{gathered}
d(a\otimes b)=da\otimes b\\
+(-1)^{|a|}a\otimes db.
\end{gathered}
\tag{V.3}
\]

Suppose the one-coordinate functions f_i have extensions F_i, and put
c_i=(0,−f_i), T_i=(δF_i,0), e_i=(F_i,0). Then T_i−c_i=de_i, and c_i,T_i
are closed of degree1. The finite telescoping identity is

\[
\begin{gathered}
T_1\otimes\cdots\otimes T_N\\
-c_1\otimes\cdots\otimes c_N=dH,
\end{gathered}
\tag{V.4}
\]

where

\[
\begin{gathered}
H=\sum_{i=1}^N(-1)^{i-1}\\
T_1\otimes\cdots\otimes T_{i-1}\otimes e_i\\
\otimes c_{i+1}\otimes\cdots\otimes c_N.
\end{gathered}
\tag{V.5}
\]

Indeed, every differential on a c or T factor is zero. The differential on
e_i has the preceding tensor sign(−1)^{i−1}, which cancels the displayed
coefficient. The resulting ith term replaces e_i by T_i−c_i. Adjacent
telescoping terms cancel, leaving exactly(V.4). This proves the product
comparison for every N, including changes of an individual extension.
Multiplying by a holomorphic coefficient preserves it: δg=0, so the same
H multiplied by g has differential g dH.

The tensor-relative top-complement entry of c₁⊗⋯⊗c_N is
(−1)^N f₁⋯f_N. Thus a statement using the unsigned ordered holomorphic
representative f₁⋯f_N must declare its identification with this tensor entry;
its sign is(−1)^N. It may not silently use the ordinary projection sign of
the cone in(V.1).

Products of the particular currents in U exist because the singular normal
variables of the two factors are independent. Here is the distributional
construction. For a tensor of distributions S(a),T(b), choose cutoffs equal
to1 near the projections of the compact support of a test function φ(a,b).
Define(S⊠T)(φ)=S_a(T_b(φ)). The inner evaluation is a smooth compactly
supported function of a. To check this, derivatives in a may be moved inside
T by continuity in the test-function topology: difference quotients converge
there with every derivative on one compact set. A continuity bound for T
and then one for S bounds the value by finitely many derivatives of φ on
that compact product, so this is a distribution. It is independent of the
cutoffs. On a third independent variable the two evaluations agree by the
same construction; testing finite sums of separated test functions and
approximating a compactly supported smooth function with its rectangular
Fourier partial sums gives equality on general tests. The Fourier coefficients
decay faster than every polynomial by repeated integration by parts, so the
distributional seminorm bounds justify the limit and both evaluation orders.
The rectangular Fourier expansion is applied inside a box after extending
the compactly supported test by zero to a larger periodic box. This also
proves the two-factor order interchange. Leibniz differentiation follows
by testing and integration by parts. Consequently the exterior tensor map
commutes with(V.3). Holomorphic parameter coefficients and the finite spatial
derivatives of U are legitimate under the same seminorm argument.

This proves equality in the specified tensor-relative model. Identifying
that model with the derived relative support cup still requires the analytic
resolution/comparison interface; it is not supplied by the existence of the
distribution tensor product alone.

#### V.3 The ungraded coefficient line and its full-form realization

Let L have holomorphic frame η of holomorphic degree l. Its coefficient
Dolbeault complex is I_L^q={α⊗η: α has antiholomorphic degree q}, with
δ(α⊗η)=(∂bar α)⊗η. L is an ordinary degree-zero coefficient sheaf.
The anti-first realization is

\[
 J_q(\alpha\otimes\eta)=\alpha\wedge\eta.
 \tag{V.6}
\]

It commutes with ∂bar because η is holomorphic and occurs after α. If a has
antiholomorphic degree p and holomorphic coefficient degree l, and b has
antiholomorphic degree q, define the unshifted coefficient cup by

\[
 c(a,b)=(\alpha\wedge\beta)\otimes(\eta_A\wedge\eta_B).
 \tag{V.7}
\]

Moving η_A past β gives the exact comparison

\[
 J(c(a,b))=(-1)^{lq}J(a)\wedge J(b).
 \tag{V.8}
\]

The ungraded-line differential satisfies
δc(a,b)=c(δa,b)+(−1)^p c(a,δb). This is verified by applying the ordinary
exterior Leibniz rule to α∧β. Formula(V.8) records the extra full-form sign;
it must not be suppressed by calling a complete block “even”.

Within a kernel, grouping the N normal one-coordinate blocks
∏_i(∂bar F_i∧dv_i) puts all antiholomorphic factors first with sign

\[
 (-1)^{N(N-1)/2}.
 \tag{V.9}
\]

There are precisely(N−1)+(N−2)+⋯+1 crossings. Together with the ordered
connecting map(V.2), this specifies the coefficient current of the ordered
holomorphic input form dv₁∧⋯∧dv_N without a hidden connecting or regrouping
choice. For a pair of kernels the two(V.9) factors multiply to1, while the
between-kernel sign(V.8) is(−1)^{N²}.

#### V.4 Proper fibre integration and the all-degree shifted trace

Let p:B×F→B, dim_C F=N, with the complex orientation on the real2N fibre.
For a smooth full form whose support is proper over B, real fibre integration
is defined by first inserting all ordered real fibre tangent vectors and
integrating the coefficient. On a compact subset K of B, properness puts
the relevant support in a compact subset of K×F; finitely many fibre charts
and a smooth partition of unity suffice. Derivatives in the base commute
with integration by differentiation under a compactly supported integral.
Fibre derivatives integrate to zero by ordinary one-coordinate integration
by parts. These statements prove, in the vertical-first real orientation,

\[
 I\,\bar\partial=\bar\partial\,I.
 \tag{V.10}
\]

For currents, define the same integral by duality against a compactly
supported test on B, multiplying its pullback by a cutoff equal to1 near
the properly supported current over that test's compact support. Such a
cutoff exists inside finitely many coordinate charts. Changing it does not
change the value, since the difference vanishes near the current support.
The distributional differentiation identity follows by this definition and
integration by parts; it is(V.10). There is no assertion that restriction
of an arbitrary p!-supported section to an arbitrary smaller open set
remains p!-supported.

In coefficient notation set I=J_target^{-1} p_* J_source, where the source
coefficient form orders the intermediate holomorphic block before the
remaining holomorphic block. The differential on I_source[N] is(−1)^Nδ.
The trace in degree q of the original complex is

\[
\begin{gathered}
\operatorname{Tr}_q=(-1)^{N(q-N)}I_q,\\
\qquad I_q=0\quad(q<N).
\end{gathered}
\tag{V.11}
\]

It is a chain map I_source[N]→I_target. For q≥N, the left-hand coefficient
in Tr_{q+1}(−1)^Nδ is
(−1)^{N(q+1−N)+N}=(−1)^{N(q−N)}, the right-hand coefficient in δTr_q.
The q<N cases are zero. At the boundary q=N−1, I_N δ is also zero:
the degree-N vertical contribution is a compactly supported fibre
antiholomorphic derivative, whose integral vanishes by the just proved
integration-by-parts identity. At q=N the sign is1, fixing the positive
one-fibre Cauchy residue normalization.

The definition is invariant under a holomorphic change of intermediate
coordinates. The holomorphic coefficient block acquires detJ, its
antiholomorphic block acquires the conjugate determinant, and the real
complex-oriented integration acquires exactly|detJ|². This follows by
writing the real linear matrix of a complex matrix; its determinant is
detJ·conj(detJ), first for triangular matrices and then for an invertible
matrix by Gaussian elimination. The ordinary real change-of-variables
formula therefore identifies the two integrals, including the sign.
Duality extends this equality to the currents just defined. No new residue
scalar is introduced by a coordinate change.

If the fibre-coordinate change depends holomorphically on the base, a
full-form lift of its relative holomorphic top block also acquires terms
containing base holomorphic differentials. Every such term has fibre
holomorphic degree strictly less than N. The fibre antiholomorphic degree
is at most N, so its total real fibre degree is strictly less than 2N;
its fibre integral is zero. This also proves independence of the chosen
full-form lift of a relative top block. The surviving fibre-top term is
exactly the determinant term above. The same conclusion holds for
currents by transposition on compactly supported tests.

For consecutive fibres of dimensions N and M, raw integrations agree with
integration over the ordered product fibre by repeated integration of the
compactly supported coefficient. The scalar trace signs satisfy

\[
\begin{gathered}
(-1)^{N(q-N)}(-1)^{M(q-N-M)}\\
=(-1)^{NM}\\
(-1)^{(N+M)(q-N-M)}.
\end{gathered}
\tag{V.12}
\]

This is a direct expansion of the exponents modulo2. The extra(−1)^{NM}
is the tensor/shift identification of the two ordered holomorphic fibre
coefficient blocks; it is not a second residue normalization. The full
shift calculation is recorded next, so(V.12) does not silently identify
two different tensor presentations.

#### V.5 Cup first, then shift the intermediate trace coefficient

The operation in U.3 is an **unshifted** cohomological cup of two degree-N
classes, followed by the trace which lowers degree by N. Thus the cup
current has q=2N and intermediate holomorphic coefficient degree N. By
(V.8) and(V.11),

\[
\begin{gathered}
\operatorname{Tr}_{2N}c(a,b)\\
=(-1)^{N^2} I(c(a,b))\\
=J_{\rm target}^{-1}\\
p_*\bigl(J(a)\wedge J(b)\bigr).
\end{gathered}
\tag{V.13}
\]

The two parities cancel. This is the positive full-normal-block convolution,
with precisely the normalization of U.5–U.8. The ordinary coefficient cup
and the full-form cup are different before applying that trace.

To make the shifts checkable, for complexes A,B and cochain degrees p,q,
the standard tensor suspension identification is

\[
\begin{gathered}
A[N]\otimes B[M]\\
\longrightarrow(A\otimes B)[N+M],\qquad\\
a\otimes b\longmapsto(-1)^{Mp}a\otimes b.
\end{gathered}
\tag{V.14}
\]

On a differential in A, p increases by1, so the displayed sign increases
by(−1)^M. This converts the domain sign(−1)^N to the target sign
(−1)^{N+M}. On a differential in B, the domain sign is
(−1)^{p−N+M}, equal to the target sign(−1)^{N+M+p}. This proves(V.14)
in every degree.

Only A, the intermediate trace coefficient, is shifted at the trace step:
A[N]⊗B→(A⊗B)[N] has M=0 and therefore has no tensor sign. The input
classes at this stage have degrees0 and N. Applying(V.11) then gives(V.13).
If one instead first writes **both** kernel classes as degree-zero classes
in A[N] and B[N], the coordinate identification(V.14) contributes
(−1)^{Np}; to express the same unshifted cup one must invert that
identification before its trace. At p=N that inverse has sign(−1)^{N²}.
Adding(V.14) as an extra product sign to a definition already using the
unshifted cup changes the operation in odd N and is incorrect.

The equivalent trace-first regrouping
A[N]⊗B[M]→(A[N]⊗B)[M] has sign(−1)^{M(p−N)}. Its differential check
is the same two computations just given, with the first factor's degree
p−N. Composing it with the first-factor-only shift identifies the total
shift with(V.14) multiplied by(−1)^{NM}. This is precisely the regrouping
factor in(V.12). Both presentations of consecutive ordered traces thus
have the same sign after their declared tensor identifications.

For degree-N kernel classes, equality(V.13) reduces either association of
three transverse factors to the ordered full-form wedge and the same two
real fibre integrations. Both have compact support over compact endpoints;
the distributional tensor construction in V.2 and duality in V.4 give the
same repeated evaluation. Consequently the two associations agree in this
current model, with positive normalized diagonal unit. For the unit, the
normal Cauchy current has support v=u and integrates a holomorphic test
coefficient to that coefficient: in one coordinate U.6 proves its scalar1;
iterating in the fixed coordinate order gives scalar1 in every dimension.
Convolution on either side therefore restricts to the diagonal with scalar1.

These computations establish the signs of the canonical relative-support operation: [§5.34, R.1–R.7](#5-34-full-current-resolutions-and-controlled-proper-image-comparison) supplies its full resolution/support comparison and controlled excision. For unrestricted directional input, [§5.37, W.1–W.8](#5-37-interior-collars-and-the-canonical-directional-action) supplies the actual collar-to-relative-cone homotopy with the complement term retained. The keyhole evaluation is consistent with that complete chain comparison.

#### V.6 The complement cube and the normal-class calibration

For N normal supports Z_i in a common manifold M put U_i=M∖Z_i,
Z=∩_i Z_i, U_A=∩_{i∈A}U_i and U_empty=M. The actual finite cube is

\[
\begin{gathered}
K^k=\bigoplus_{A\subset\{1,\ldots,N\}}\\
I^{k-|A|}(U_A),\qquad\\
d_K=d_C+(-1)^{|A|}\delta.
\end{gathered}
\tag{V.15}
\]

At an ordered nonempty set A={i₀<⋯<i_r}, the d_C component is
∑_{j=0}^r(−1)^j c_{A\setminus{i_j}} restricted to U_A. In particular
the M component restricts to each singleton with sign+. Every pair of
deleted indices occurs twice in d_C² with opposite signs. Since δ commutes
with restriction, the cross terms of d_K² have coefficients(−1)^{|A|}
and(−1)^{|A|+1}; hence d_K²=0.

The complement U=M∖Z is covered by the U_i. Give its Čech–Dolbeault
complex differential d_C+(−1)^rδ in Čech degree r. The identification
of Cone(I(M)→Tot Čech(I,U))[−1] with(V.15) takes a complement component
of Čech degree r to the cube component of cardinality r+1 with factor
(−1)^r, leaving the M component unchanged. This is a chain isomorphism:
the cone negates the Čech differential while the target factor changes by−1
on increasing r; these signs cancel. Both Dolbeault signs are(−1)^{r+1}.
Restriction from M has the r=0 factor1.

The finite Čech comparison with I(U) needs no Leray-cover assumption.
Choose a smooth partitionρ_i on U subordinate to U_i. For each smooth
or current module I^q set
(hc)_{i₀…i_{r−1}}=∑_jρ_j c_{j i₀…i_{r−1}}, extending terms by zero
whereρ_j vanishes. Expansion gives d_C h+h d_C=1: the terms omitting j
sum to∑_jρ_j c=c, and all remaining terms occur with opposite signs.
Thus the horizontal rows of the augmented comparison cone are exact.
The total cone is acyclic as well. Filter by Dolbeault degree; its finite
associated horizontal complexes have zero cohomology. Equivalently, kill
a total closed element's lowest nonzero Dolbeault row with h and subtract
its total boundary. The remaining error lies in higher Dolbeault degree;
iteration terminates because that complex is bounded. The same procedure
applies to boundaries. The ordinary acyclic-resolution augmentation of I is proved in [§5.34, R.5–R.7](#5-34-full-current-resolutions-and-controlled-proper-image-comparison); thus this cube computes the relative derived support group.

The tensor-to-cube sign is explicit. A normal factor has complement degree
r_i∈{0,1}, antiholomorphic degree q_i and total degree p_i=q_i+r_i.
Its image is the coefficient exterior product on U_A, A={i:r_i=1}, times

\[
 \epsilon=(-1)^{\sum_{i<j}q_i r_j}.
 \tag{V.16}
\]

This moves each complement-degree symbol before preceding antiholomorphic
factors. A Dolbeault differential on factor i supplies the tensor exponent
∑_{l<i}(q_l+r_l)+r_i, while increasing q_i changesε by∑_{j>i}r_j.
Their sum is∑_l r_l+∑_{l<i}q_l, the cube's overall Dolbeault sign plus
its exterior-product sign. A complement restriction on factor i increases
r_i from0 to1, changingε by∑_{l<i}q_l. Added to the preceding tensor
degrees, this leaves∑_{l<i}r_l, its position in the ordered cube subset.
This proves the chain-map identity in every degree.

For smooth normal cochains the products are ordinary coefficient products.
For the separated current cochains of V.2 use its distributional tensor
construction. Holomorphic joint coefficients can be multiplied afterwards;
their differential is zero. No product of arbitrary distributions with a
shared singular parameter is asserted. Applying(V.16) to(V.5) gives the
actual cube homotopy. Its current endpoint is∂bar F₁∧⋯∧∂bar F_N and
its top-complement endpoint is(−1)^N f₁⋯f_N. By the cone-to-cube factor
(−1)^{N−1}, the latter is−f₁⋯f_N in the full cone's ordinary Čech
coordinate. This specifies the connecting augmentation.

A separate holomorphic-line calibration remains. Treat the ordered
one-coordinate classes with coefficients dv_i first inΩ¹[1]. Repeated
standard tensor suspension(V.14) gives

\[
 r_N=(-1)^{N(N-1)/2}.
 \tag{V.17}
\]

At each step the preceding unsuspended degree is the number of preceding
degree-one classes, so the exponent is1+⋯+(N−1). The resulting anti-first
coefficient current is r_N∂bar F₁∧⋯∧∂bar F_N with coefficient
dv₁∧⋯∧dv_N. By(V.9) its full form is exactly the positive ordered
complete-normal-block current. Its top cube entry is
r_N(−1)^N f₁⋯f_N. Thus an unsigned ordered holomorphic expression
f₁⋯f_N denotes this normalized class only with that declared identification;
it is not literally its unsigned ordinary cube entry. N=2 already detects
the distinction.

This calibration of a single N-normal class precedes the between-kernel
unshifted cup(V.13). Adding double-suspension there would count a different
operation. The one-coordinate positive logarithmic/Cauchy distributions
U.5–U.7 fix the scalar, and(V.17) extends that ordered input-frame
normalization to N coordinates. [§5.37, W.1–W.8](#5-37-interior-collars-and-the-canonical-directional-action) proves the collar trace for unrestricted directional input with this calibration.

![Relative connecting and trace signs](assets/cone-trace-signs.png)

*The actual relative cone is (V.1)–(V.2). The unshifted coefficient cup has the regrouping sign (V.8); the all-degree trace (V.11) cancels its between-kernel parity at degree \(2N\), as proved in (V.13). The single-normal-class calibration (V.17) gives ordinary top cube entry \(r_N(-1)^N\varphi\). Proof: V.1–V.6, with the full resolution/excision proof in §5.34 and collar proof in §5.37. Free human comparison: [Kashiwara–Schapira, Micro-hyperbolic systems](https://perso.imj-prg.fr/pierre-schapira/wp-content/uploads/schapira-pub/Microhyp.pdf), §3.1.*

### 5.36. Canonical finite polynomial line-cone kernels

This section realizes the actual finite polynomial ring \(R\) of [§5.30, F.1–F.7](#5-30-actual-polynomial-flatness-and-reconstruction-after-finite-realization) and the negative polynomial matrices of [§5.33, K.1–K.3](#5-33-a-convergent-negative-order-polynomial-kernel-complex) by canonical supported-cohomology classes. The full resolutions and support comparison are §5.34; the unshifted cup and normalized trace are §5.35; §5.37 proves the action on arbitrary degree-one directional input with its residual complement term retained. The actual operator product and normal localization are [§5.19, S.1–S.4](#5-19-actual-factorial-growth-symbols-and-ordered-jets) and [§5.22.1](#5-22-1-the-actual-normal-element), repeated in F.1. The bounded actual resolution, coherent lattice/support comparison and negative kernels are §§5.31–5.33. Finite spatial degree is explicit; the full convergent-\(z\) \(E\) ring and the infinite-order, propagation, separation, D-type and analytic proper-regularity obligations remain distinct.

#### U.1. Coordinates, properness and what the canonical trace means

Put \(N=d+1\), write output \(u=(t,x)\), input \(v=(s,y)\), and keep \(h=\tau^{-1}\), \(z=\xi/\tau\). Use the closed proper convex line cone
\[
\begin{gathered}
G_0=\{(r,0)\in\mathbf C\times\mathbf C^d:\\
r\in\mathbf R_{\le0}\},\qquad\\
Z_0=\{(u,v):v-u\in G_0\}.
\end{gathered}
\tag{U.1}
\]
Thus the kernel variable has its cut on \(s-t\le0\), whereas the directional quotient at \(p=(0;dt)\) has forbidden limiting directions \(\operatorname{Re}t\ge0\). The two sets live in different spaces. The normal coordinate of the diagonal used for the covector is output minus input; its pairing with \(dt\) is \(t-s\ge0\) on (U.1). No substitution \(\tau\mapsto-\tau\) is made. HolIII IV.2, printed p.908, explicitly defines \(G_0\) with \(\operatorname{Re}t\le0\); IV.5, printed p.917, explicitly uses \(\operatorname{Re}t\ge0\) for the directional support. 

Let \(D\) be a sufficiently small convex product of discs in \((t,x)\). It is \(G_0\)-round: if \(u,v\in D\) and \((u,w),(w,v)\in Z_0\), then the intermediate point \(w\) lies on the real segment from \(v\) to \(u\), with unchanged spatial coordinates, and hence lies in \(D\). The projection
\[
\begin{gathered}
p_{13}:p_{12}^{-1}Z_0\cap p_{23}^{-1}Z_0\\
\cap D^3\longrightarrow Z_0\cap D^2
\end{gathered}
\tag{U.2}
\]
is proper. Over a compact set of endpoints its intermediate points lie in the union of their compact segments. That union is compact inside \(D\): it is the continuous image of the compact endpoint set times \([0,1]\), and convexity puts it inside \(D\). This verifies properness itself, rather than only boundedness in an open disc. The same argument applies to any finite string of intermediate points.

Write \(\Omega_2^N\) for the holomorphic top forms in the input coordinates, ordered as \(ds\wedge dy_1\wedge\cdots\wedge dy_d\). The canonical multiplication under discussion is
\[
\begin{gathered}
\mathcal E(G_0;D)\\
=H^N_{Z_0}(D^2;\Omega_2^N),\qquad\\
a*b=\operatorname{Tr}_{p_{13}}\\
(p_{12}^*a\smile p_{23}^*b).
\end{gathered}
\tag{U.3}
\]
The support morphism in (U.3) is defined because of (U.2). The trace integrates the intermediate holomorphic top form and lowers cohomological degree by \(N\). It has no extra scalar factor; the factors \((2\pi i)^{-1}\) belong to the kernel representatives below.

The full current and smooth resolutions, their actual injective comparisons, and the ordinary/proper image and controlled extension-by-zero maps are proved in [§5.34, R.1–R.7](#5-34-full-current-resolutions-and-controlled-proper-image-comparison). We use its anti-first realization \(J(\alpha\otimes\eta)=\alpha\wedge\eta\), with the input top-form line ungraded before trace. [§5.35, V.1–V.6](#5-35-relative-cones-unshifted-cup-and-the-normalized-trace) supplies the explicit relative cone \(d(a,b)=(\bar\partial a,ra-\bar\partial b)\), the positive connecting class \((0,-f)\), the separated-current tensor construction, the finite complement cube augmentation, and the canonical unshifted cup. The all-degree input trace is \(\operatorname{Tr}_q=(-1)^{N(q-N)}I_q\), with \(I=J_{\rm out}^{-1}p_*J_{\rm source}\), and is zero below \(q=N\). It is a chain map on the source shifted by \([N]\), with positive one-coordinate residue.

For one ordered \(N\)-normal class the calibration is \(r_N=(-1)^{N(N-1)/2}\), proved in (V.17). Complete coordinate blocks below already incorporate it. Between two kernel classes of antiholomorphic degree \(N\), the unshifted coefficient cup contributes \((-1)^{N^2}\) relative to their full-block wedge, while the trace at degree \(2N\) contributes the same factor. Their product is \(1\), as proved in V.5. In triple coordinates \((u,a=v-u,b=w-v)\), the two singular normal variable lists are independent; V.2 and V.6 therefore identify their distributional tensor with the actual supported cup. A holomorphic joint coefficient can multiply that tensor. This does not define a product of arbitrary currents in shared singular variables.

The compact-segment geometry (U.2), R.7's controlled excision and V.4's proper fibre integration now make (U.3) an actual canonical operation. V.4 proves compactly supported Stokes, complete output-variable differentiation and base-coordinate invariance; V.5 proves the successive-trace signs. The normalized trace has no extra residue scalar.

This description also proves associativity wherever the support maps are proper: on four factors both associations use the same cup-product, the same proper support and the same double trace; Fubini identifies the traces. The normalized diagonal Cauchy class below is the unit. These are cohomology-class assertions; equality of contour actions is not used to infer equality of kernel classes.

#### U.2. The one-variable connecting classes and exact normalization

For a complex variable \(w=a+ib\), let \(\log w\) be the principal logarithm, with cut \(\mathbf R_{\le0}\). It is locally integrable, and its upper-minus-lower jump on \(a<0\) is \(2\pi i\). Its distributional antiholomorphic derivative is
\[
 \bar\partial\log w
     =-\pi\,\mathbf1_{a<0}\delta(b)\,d\bar w.
 \tag{U.4}
\]
For a direct proof, integrate against a compactly supported smooth test form, split the two half-planes along the negative ray, and integrate by parts. The two boundary values contribute \((i/2)(2\pi i)\delta(b)=-\pi\delta(b)\). The boundary of a deleted disc of radius \(\epsilon\) contributes at most a constant times \(\epsilon(1+|\log\epsilon|)\), which tends to zero. Off the cut the antiholomorphic derivative is zero. There is no additional point mass at the endpoint.

Since \(d\bar w\wedge dw=2i\,da\wedge db\), (U.4) gives
\[
\begin{gathered}
-\frac1{2\pi i}\bar\partial\log w\wedge dw\\
=\mathbf1_{a<0}\delta(b)\,da\wedge db.
\end{gathered}
\tag{U.5}
\]
The right side integrates a holomorphic coefficient along the ray with increasing real coordinate. In particular its truncation between \(s=a_0\) and \(s=t\), where \(a_0<t\) on a real line, is \(\int_{a_0}^{t}\), with positive orientation. The value assigned to the step function at its endpoint is irrelevant to this current.

The Cauchy–Pompeiu identity gives
\[
\begin{gathered}
\frac1{2\pi i}\bar\partial\bigl(\frac1w\bigr)\wedge dw\\
=\delta(a)\delta(b)\,da\wedge db.
\end{gathered}
\tag{U.6}
\]
One may prove its scalar coefficient by Stokes on a deleted disc: the positive boundary integral of \(dw/(2\pi i w)\) equals one. For \(k\ge0\), define the extension of \(k!/w^{k+1}\) by \((-1)^k\partial_w^k(1/w)\). Differentiating (U.6) then gives
\[
\begin{gathered}
\frac1{2\pi i}\bar\partial\bigl(\frac{k!}{w^{k+1}}\bigr)\wedge dw\\
=(-1)^k\partial_w^k\delta_{\mathbf C}(w).
\end{gathered}
\tag{U.7}
\]
On a holomorphic test function \(f\), the last distribution has value \(\partial_w^k f(0)\). Thus each spatial coordinate contributes precisely its ordinary derivative, with no surviving sign and with precisely one \((2\pi i)^{-1}\). The \(d\)-fold product gives the spatial diagonal derivative
\(\Delta_\beta(x,y)=\prod_j(-1)^{\beta_j}\partial_{y_j}^{\beta_j}\delta_{\mathbf C}(y_j-x_j)\).
It satisfies
\[
 \int_y\Delta_\beta(x,y)f(y)=\partial_x^\beta f(x).
 \tag{U.8}
\]

The classes in (U.5)–(U.7) are the connecting classes of their displayed holomorphic representatives. For one coordinate this follows from the relative-cone differential of the chosen distributional extension. For several coordinates take the ordered external product of these connecting maps. Before the holomorphic-line calibration, its top cube entry is \((-1)^N\) times the ordered product of the holomorphic representatives. Applying (V.17) gives \(r_N(-1)^N\) times that product in the ordinary cube, or \(-r_N\) times it in the full cone’s ordinary complement Čech coordinate. This argument supplies the class without assuming that a particular complement covering is Leray.

There is an equivalent contour verification useful for unrestricted holomorphic input functions. For a keyhole starting below the negative cut at a negative anchor \(a_0\), going positively around \(s=t\), and ending above that cut at the same anchor, set \(F(s)=\int_t^s A(t,r)f(r)\,dr\). Integration by parts on the two sides of the cut gives
\[
\begin{gathered}
-\frac1{2\pi i}\int_\Gamma\\
\log(s-t)A(t,s)f(s)\,ds\\
=-F(a_0)+\frac1{2\pi i}\\
\oint\frac{F(s)}{s-t}\,ds\\
=\int_{a_0}^{t}A(t,s)f(s)\,ds.
\end{gathered}
\tag{U.9}
\]
The last closed-contour integral is zero because \(F(t)=0\). Positive spatial circles evaluate (U.8) by the ordinary iterated Cauchy formula. The ordinary contour identity (U.9) is consistent with the positive residue normalization. Its equality with the canonical cup/excision/trace action on unrestricted-growth input is proved by the actual relative-cone and full-neighbourhood coboundaries in [§5.37, (W.24)–(W.27) and (W.32)–(W.33)](#5-37-interior-collars-and-the-canonical-directional-action). That proof retains the complement component and requires no distributional extension of the input across its boundary.

#### U.3. Actual analytic polynomial kernels define canonical classes

An element of \(R\) has a finite \(\partial_x\)-degree and finite positive \(\partial_t\)-order. Remove that finite positive time order using a left power of \(h\). Thus it suffices first to treat
\[
\begin{gathered}
P=\sum_{\beta\in B}\sum_{m\ge0}\\
b_{m,\beta}(t,x)h^m\partial_x^\beta,\quad\\
B\text{ finite},\quad\\
|b_{m,\beta}|\le B_\beta C_\beta^m m!.
\end{gathered}
\tag{U.10}
\]
All coefficients have one common holomorphic base domain. Changing from F.1's \(h^n z^\beta\) to (U.10) only makes a fixed finite shift of \(m\) for each \(\beta\). The factorial estimate is preserved by \((m+k)!\le2^{m+k}m!k!\), with enlarged constants. Terms with a negative \(m\) are part of the removed finite differential operator.

Separate the \(m=0\) terms and set
\[
\begin{gathered}
A_\beta(t,s,x)=\sum_{m\ge1}\\
b_{m,\beta}(t,x)\frac{(t-s)^{m-1}}{(m-1)!}.
\end{gathered}
\tag{U.11}
\]
For \(C_\beta|t-s|<1\), normal convergence follows from
\[
\begin{gathered}
|A_\beta(t,s,x)|\\
\le \frac{B_\beta C_\beta}{(1-C_\beta|t-s|)^2}.
\end{gathered}
\tag{U.12}
\]
The finite list \(B\) permits one actual product \(D^2\) with all \(C_\beta\operatorname{diam}D_t<1/2\). Fixed derivatives converge on one smaller common product by the Cauchy formula. This proves convergence at the time diagonal as well; the class is not defined by a merely formal Borel series.

The canonical kernel is represented on the ordered complement intersection by
\[
\begin{gathered}
K_P(u,v)=\frac1{(2\pi i)^{d+1}}\\
\sum_{\beta\in B}\\
\frac{\beta!}{\prod_j(y_j-x_j)^{\beta_j+1}}\\
\bigl\{\frac{b_{0,\beta}(t,x)}{s-t}\\
-\log(s-t)A_\beta(t,s,x)\bigr\}\\
ds\wedge dy_1\wedge\cdots\wedge dy_d.
\end{gathered}
\tag{U.13}
\]
The complement covering first uses \(s-t\notin\mathbf R_{\le0}\), then \(y_j\ne x_j\), in that order. Its union is \(D^2\setminus Z_0\). The unsigned expression (U.13) denotes the normalized ordered class: its literal top entry in the ordinary complement cube is \(r_N(-1)^N K_P\), and its literal top full-cone Čech entry is \(-r_N K_P\), by (V.15)–(V.17). It is not an unsigned ordinary cube entry. With this declared calibration, equations (U.5)–(U.8) show that (U.13) defines a class \(\kappa(P)\in\mathcal E(G_0;D)\). In complete coordinate blocks its supported-current representative is
\[
\begin{gathered}
T_P(u,v)=\sum_{\beta\in B}\\
\bigl\{b_{0,\beta}(t,x)\delta_{\mathbf C}(s-t)\\
+A_\beta(t,s,x)\mathbf1_{\operatorname{Re}(s-t)<0}\\
\delta(\operatorname{Im}(s-t))\bigr\}\\
\Delta_\beta(x,y).
\end{gathered}
\tag{U.14}
\]
Equation (U.14) is a density notation for the complete normal coordinate blocks from (U.5)–(U.7). In particular the time block contains \(d\overline{(s-t)}\), and the spatial blocks contain \(d\overline{(y_j-x_j)}\). Their output antiholomorphic components are retained. The holomorphic factors are the relative input forms; in coordinates \((u,v-u)\) they are the corresponding normal holomorphic forms modulo output holomorphic forms. Thus (U.14) denotes the full closed Dolbeault current, not merely its top fibre component. Every distribution in it has finite local order. This is where finite \(B\) is essential. Normal convergence in (U.11) gives a holomorphic smooth coefficient for the ray current, and its multiplication by the finite diagonal distributions is defined.

For K's rescaled matrices specifically, \(b_{m,\beta}=0\) until \(m=n+|\beta|+1\), and (U.11) is exactly K.3. Its sharper common-domain bound K.4 therefore remains unchanged. There is no loss of the \(|\beta|\) factorial and no change in the rescaling \(Q_j=h^jP_jh^{-(j-1)}\).

#### U.4. Composition is equality of canonical classes

First take strictly negative monomials
\(P=p(t,x)h^m\partial_x^\beta\), \(Q=q(t,x)h^l\partial_x^\gamma\), with \(m,l\ge1\). Their current product is a product in independent difference coordinates as explained in U.1. Under the proper intermediate trace, the two ray currents impose
\(\operatorname{Im}t=\operatorname{Im}s=\operatorname{Im}v\) and
\(\operatorname{Re}v\le\operatorname{Re}s\le\operatorname{Re}t\).
The spatial delta derivatives give the finite ordinary Leibniz sum. The resulting current has ray coefficient
\[
\begin{gathered}
p(t,x)\sum_{\nu\le\beta}\binom\beta\nu\\
\int_v^t\frac{(t-s)^{m-1}(s-v)^{l-1}}{(m-1)!(l-1)!}\\
\partial_x^\nu q(s,x)\,ds\\
\ \Delta_{\beta-\nu+\gamma}(x,w).
\end{gathered}
\tag{U.15}
\]
The integration has the positive orientation \(v\to t\) by (U.5), and lies on the actual compact segment required by (U.2). It is already the canonical cup/trace current, not an action evaluated on test solutions.

Expand \(\partial_x^\nu q(s,x)\) at \(t\) on a common smaller domain. Its coefficient of \((t-s)^k\) is \((-1)^k\partial_t^k\partial_x^\nu q(t,x)/k!\). The integer beta integral is exactly
\[
\begin{gathered}
\int_v^t\frac{(t-s)^{m+k-1}(s-v)^{l-1}}{(m-1)!(l-1)!k!}\,ds\\
=\binom{m+k-1}{k}\\
\frac{(t-v)^{m+l+k-1}}{(m+l+k-1)!}.
\end{gathered}
\tag{U.16}
\]
It follows by parametrizing the complex segment by a real variable and integrating by parts; it holds including the declared orientation. Thus the coefficient of the canonical convolution is
\(\binom\beta\nu\binom{-m}{k}\), since
\((-1)^k\binom{m+k-1}{k}=\binom{-m}{k}\).
These are precisely the actual \(\xi,x\) and \(\tau,t\) Leibniz coefficients of \(P\circ Q\), with exponent \(m+l+k\). All time contractions are retained. By (U.11) the current in (U.15) is therefore \(T_{P\circ Q}\).

For infinite negative time tails but finite spatial degree, their holomorphic coefficients and finitely many derivatives converge normally on one smaller product by (U.12). The trace integrates a compact segment, so the convolution is normally convergent there. K.3's Taylor-coefficient proof, or the same bound and (U.16), gives equality with the analytic coefficient of the actual symbol product. At a fixed time-difference power only finitely many indices \(m,l,k\) contribute. The actual product theorem puts the product in the factorial class, so its convergent Borel coefficient is uniquely determined by these Taylor coefficients. This proves equality of the supported currents, hence equality of their relative-cone classes. A function-space faithfulness argument is unnecessary for this step.

Terms with \(m=0\) are supported on the time diagonal. Their trace acts by the finite indicated derivative. The distributional Leibniz rule in (U.8) gives exactly their finite spatial contractions; multiplying by a holomorphic coefficient on the input differentiates that coefficient when the derivative is on its left. A negative time term composed on the right with a multiplying coefficient \(b(v,x)\) has coefficient \((t-v)^{m-1}b(v,x)/(m-1)!\). Taylor expansion of \(b(v,x)\) at \(t\) gives \((-1)^k/(k!(m-1)!)\), equal to \(\binom{-m}{k}/(m+k-1)!\), after retaining the factor \((t-v)^{m+k-1}\). Thus it gives every time contraction also in this diagonal case; the finite spatial Leibniz sum is taken at the same time. Uniform Taylor convergence and (U.12) justify the actual negative tails. For the time derivative, differentiation of the holomorphic representative gives
\[
\begin{gathered}
\partial_t\bigl(-\frac{\log(s-t)}{2\pi i}\bigr)\\
=\frac1{2\pi i(s-t)},\qquad\\
-\partial_s\bigl(-\frac{\log(s-t)}{2\pi i}\bigr)\\
=\frac1{2\pi i(s-t)}.
\end{gathered}
\tag{U.17}
\]
The first is composition with \(\partial_t\) on the output; the second is composition with its input diagonal derivative, by integration by parts. These are identities of holomorphic representatives on the complement and hence of their connecting classes, rather than literal derivatives of a chosen logarithmic current extension. To check the distinction, let \(G\) be the distributional logarithmic extension and \(F\) the normalized Cauchy extension. The displayed holomorphic derivative says that \(H=\partial_tG-F\) is supported on the cut. Explicitly, for \(w=s-t\), the jump calculation of (U.4) applied to the holomorphic derivative gives \(H=(2i)^{-1}\mathbf1_{\operatorname{Re}w<0}\delta(\operatorname{Im}w)\). The two supported-current representatives differ by \((\bar\partial H,0)=d(H,0)\), an actual relative-cone boundary; for the input derivative use \(H=-\partial_sG-F\), which has the same value. In \(N\) normals, let \(S_{\rm sp}\) be the ordered product of the normalized closed spatial Cauchy coefficient currents and put \(H_N=r_N H S_{\rm sp}\), with coefficient \(ds\wedge dy_1\wedge\cdots\wedge dy_d\). It has antiholomorphic degree \(N-1\), and the complete-current difference is \((\bar\partial H_N,0)=d(H_N,0)\), because \(S_{\rm sp}\) is closed. This boundary persists after the proper intersection cup and trace by the chain maps of R.7/V.4. Thus any extra supported term in the distributional derivative is retained as an exact current. It is not discarded as a pointwise identity. It follows as an equality of canonical classes that
\[
\begin{gathered}
\kappa(\partial_t)*\kappa(h)\\
=\kappa(h)*\kappa(\partial_t)=1.
\end{gathered}
\tag{U.18}
\]
The diagonal product of (U.6) over all coordinates is the unit: tracing against it restricts every coefficient to the diagonal, with positive sign. The ordinary relations of all finite differential operators follow by distributional differentiation of that same diagonal. Combining them with the preceding negative-tail calculation and the exact normal localization of F.1 yields a unital homomorphism
\[
\begin{gathered}
\kappa:R\longrightarrow\\
\underset{D\ni0}{\operatorname{colim}}\mathcal E(G_0;D),\qquad\\
\kappa(P\circ Q)=\kappa(P)*\kappa(Q).
\end{gathered}
\tag{U.19}
\]
If the two inputs need different coefficient domains, intersect those domains and shrink \(D\) once before the calculation. This is a germ statement. It is not a uniform domain assertion for all elements of \(R\) at once.

Consequently the rescaled K matrices form a canonical kernel complex:
\[
 \kappa(Q_j)*\kappa(Q_{j-1})=0.
 \tag{U.20}
\]
This includes the entire matrices and their actual analytic tails. It holds on a common \(D\) for the finite matrix list and its products, with row-vector composition in the order used in L.8. No reduced-fibre replacement or generic-support restriction occurs.

#### U.5. The canonical action on the directional degree-one quotient

Define \(C_p\) as the filtered colimit of
\(\mathcal O(V\setminus Z)/\mathcal O(V)\), where \(Z\) is a closed germ whose limiting directions satisfy \(\operatorname{Re}t\ge0\). It is the degree-one supported-cohomology colimit: the support exact sequence gives the quotient, and the colimit over neighbourhoods of the remaining \(H^1(V,\mathcal O)\) vanishes by exactness of stalks. This uses no degree-zero or higher-degree vanishing assertion for an arbitrary fixed \(V\).

Every such \(Z\) can be enlarged to one with convex complement
\[
\begin{gathered}
\Omega_g=\\
\{\operatorname{Re}t<-g(|(\operatorname{Im}t,x)|)\},\\
\qquad g\text{ convex, nondecreasing},\\
\quad g(r)=o(r).
\end{gathered}
\tag{U.21}
\]
Here is the construction. On a sufficiently small ball, define
\[
\begin{gathered}
e(r)=\sup\bigl(\{0\}\cup\\
\{-\operatorname{Re}t/|v|:\\
(t,x)\in Z,\operatorname{Re}t<0,\\
0<|v|\le r\}\bigr)
\end{gathered}
\tag{U.21a}
\], with \(v=(\operatorname{Im}t,x)\). The support direction condition first bounds this supremum and then gives \(e(r)\to0\). Replace a bound by one if necessary after shrinking. Put \(g(r)=\int_0^{2r}e(a)\,da\). Monotonicity gives convexity, \(g(r)\ge re(r)\) and \(g(r)\le2re(2r)=o(r)\). The radial composition is convex by the triangle inequality and monotonicity. Thus its strict hypograph in (U.21) is convex, its complementary support contains \(Z\), and that support has exactly the required limiting direction condition. Unions of two allowed support germs remain allowed, so this construction is cofinal and compares representatives.

The domain \(\Omega_g\) is \(G_0\)-open: decreasing the real time coordinate while fixing \(\operatorname{Im}t,x\) preserves the inequality. For a sufficiently small output neighbourhood inside a larger product \(D\), the kernel support intersected with the input support \(D\setminus\Omega_g\) is proper over the output. In fact its input real coordinate lies between \(-g(|v|)\) and the output real coordinate, with the other coordinates fixed. For output near zero this entire segment lies compactly inside the chosen larger \(D\), by continuity of \(g\) and \(g(0)=0\). This is the exact local support condition for the cup/trace action on the degree-one relative group. At output points of \(\Omega_g\), all lower real input points are in \(\Omega_g\); hence its support maps into the output support. Excision gives the action on the germ, independently of the larger product.

Take a sufficiently small negative real anchor \(a_0\). For output \((t,x)\) in a smaller convex part of \(\Omega_g\), the segment from \((a_0,x)\) to \((t,x)\) lies in the input domain. On a compact output subset that union of segments is compactly inside the input domain, and thus has a fixed spatial Cauchy radius. Only finitely many spatial derivatives occur. Equations (U.11)–(U.12) give the actual canonical action
\[
\begin{gathered}
\rho(P)[f]=\bigl[\\
\sum_{\beta\in B}\\
b_{0,\beta}(t,x)\partial_x^\beta f(t,x)\\
+\sum_{\beta\in B}\int_{a_0}^t\\
A_\beta(t,s,x)\partial_x^\beta f(s,x)\,ds\bigr].
\end{gathered}
\tag{U.22}
\]
Its equality with the canonical action is proved by [§5.37, W.1–W.8 and (W.24)–(W.33)](#5-37-interior-collars-and-the-canonical-directional-action), including the actual smooth/current augmentation, proper localization, shifted relative trace and full-output collar primitive. For the finite matrix list, select one anchor tube, one smaller product and one time radius before applying the formula. The integrals and their derivatives converge uniformly on every compact output subset; no derivative-radius choice depends on the tail index.

Changing the anchor changes (U.22) by integrals over a compact segment at negative real time. The input is holomorphic on a full tube around that segment. For \(t,x\) in a full sufficiently small neighbourhood of zero, (U.12) holds on that tube, so the difference extends holomorphically across zero. An extending input produces an extending output by the same bound. The cofinal common supports in (U.21) compare any two input representatives. Therefore the formula descends to \(C_p\).

The finite time derivative acts by its ordinary derivative. In particular
\(\partial_t\rho(h)f=f\) and
\(\rho(h)\partial_tf=f-f(a_0,x)\). The anchor term is holomorphic on a full neighbourhood of zero and is zero in \(C_p\). Thus (U.18) and (U.22) agree with the localized \(R\)-action, and \(\partial_t,h\) are two-sided inverses in this quotient. This does not assert a two-sided inverse on ordinary holomorphic functions before taking the quotient.

The action, and hence (U.19), is faithful on the ring \(R\). Here is a finite-polynomial proof. After a finite left shift by \(h\), take \(P\) in (U.10). For \(\lambda\notin\mathbf Z\) and \(v_0\in\mathbf C^d\), use
\(f=e^{v_0\cdot x}t^\lambda\), with its cut on the positive real time ray. Modulo extending functions, its \(m\)-th primitive is
\(e^{v_0\cdot x}t^{\lambda+m}/(\lambda+1)_m\).
The anchor corrections converge holomorphically near zero: for fixed \(\lambda\),
\(|(\lambda+1)_m|^{-1}\le A(m+1)^k/m!\), as follows by summing
\(\log|1+\lambda/j|=\operatorname{Re}\lambda/j+O(j^{-2})\).
The Taylor polynomial at the anchor of that primitive is bounded by a fixed polynomial in \(m\) times
\((|a_0|+|t-a_0|)^m/m!\). The factorial estimate in (U.10) and a sufficiently small anchor make its sum normally convergent. The set of spatial indices is finite.

If \(\rho(P)=0\), the nonextending part is
\[
\begin{gathered}
e^{v_0\cdot x}t^\lambda\\
\sum_{m\ge0,\beta\in B}\\
b_{m,\beta}(t,x)\frac{t^m v_0^\beta}{(\lambda+1)_m}=0.
\end{gathered}
\tag{U.23}
\]
The inner sum is holomorphic at zero. Continuation around a small circle multiplies \(t^\lambda\) by \(e^{2\pi i\lambda}\ne1\) and leaves a holomorphic extension fixed, so the inner sum must vanish. At the coefficient of \(t^q\) this is a finite rational identity
\(\sum_{m=0}^q R_{m,q}(v_0,x)/(\lambda+1)_m=0\).
The pole at \(\lambda=-q\) isolates \(R_{q,q}\); descending through the remaining poles and the constant term isolates every \(R_{m,q}\). The polynomial monomials \(v_0^\beta\) then isolate every Taylor coefficient of every \(b_{m,\beta}\). Hence \(P=0\). Invertibility of \(\rho(h)\) proves the same after localization. This is ring faithfulness; it is not a theorem that solution functionals separate every holonomic module.

#### U.6. Exact consequence and remaining interfaces

The actual finite-polynomial kernel complex of K has canonical line-cone classes, its positive residue normalization, its ring products and its degree-one directional action. Its matrices have one actual coefficient domain and the original factorial bounds. All singular-support sections of the original finite modules remain in the supplied algebraic complex; the construction has imposed no generic-support or reduced-fibre restriction. Off-support algebraic contractibility from K.4 is preserved after any already defined ring map, including the restricted map (U.19) wherever a contracting homotopy has entries in its domain. A homotopy with entries in arbitrary \(E\) cannot be transferred through (U.19) without the full \(E\) comparison.

The precise still-open steps are:

1. The analogous proper-cone cup/trace comparison for every convergent \(z=\xi/\tau\) coefficient of \(E\), beyond finite polynomial degree. An unbounded spatial derivative sum cannot be treated as the finite current (U.14); its proper-cone product and trace require a separate complete proof.
2. The ring action on the derived relative complexes needed by Microhyp 3.2.5 and 4.5.1, with its exact supports, cone hypotheses and shifts, and the weakened off-support exactness condition used in HolIII IV.4.4.1. A degree-one quotient action does not supply that derived assertion.
3. Sectorial propagation, continuation across the discriminant and separation for an arbitrary holonomic module, including singular support. Ring faithfulness of (U.19) proves none of these solution-space assertions.
4. Actual \(E^\infty\)-linearity, \(A^\infty\) monodromy-ideal stability, the \(E^\infty\otimes_{A^\infty}\) comparison and extension-faithfulness. Neither the finite-polynomial canonical action nor the actual positive-degree formal completion is such an extension.
5. The resulting finite D-type embedding, ordinary finite-pole recovery for regular holonomic systems, intrinsic microlocal order, full C1 and arbitrary proper/support-proper analytic regularity.

Free human source material, distinguished from this original proof: Kashiwara–Kawai, [HolIII](https://www.kurims.kyoto-u.ac.jp/~kenkyubu/kashiwara/HolIII.pdf), III.1 (printed pp.869–882), III.2–III.3 (pp.883–889), IV.2 (p.908) and IV.5 (p.917); Kashiwara–Schapira, [Micro-hyperbolic systems](https://perso.imj-prg.fr/pierre-schapira/wp-content/uploads/schapira-pub/Microhyp.pdf), §3.1, printed pp.11–13. Their statements are proof targets and normalization checks; no omitted SKK theorem, infinite-order flatness, finite D-type theorem or analytic proper theorem is imported.

The figure below illustrates precisely (U.1)–(U.2), (U.5)–(U.8), (U.15)–(U.18) and (U.20). It is a real slice of the stated complex supports and not a depiction of all higher-dimensional cotangent geometry.

![Canonical polynomial kernels, proper supports and positive residues](assets/canonical-line-cone-kernels.png)

*The exact real slice shows the compact intermediate segment of (U.2), the independent negative normal coordinates of (U.15), positive ray/Cauchy residues (U.5)–(U.8), and the convergent product (U.11)–(U.20). Derivative inverse identities in (U.17)–(U.18) hold in canonical cohomology with the exact supported correction stated there. The full complex and spatial constraints are retained in U.1–U.5. Free human comparisons are HolIII and Micro-hyperbolic systems at the locators above.*

### 5.37. Interior collars and the canonical directional action

The theorem proved here is the equality missing from U.9/U.22: the **unshifted** supported cup of an actual finite polynomial line-cone kernel with an arbitrary degree-one directional input, followed by the normalized shifted input trace, is represented by the anchored Volterra formula U.22. There is no boundary-growth assumption on the input. The complement component of the relative cone is retained throughout.

The actual coefficients and kernels are [§5.36, U.1–U.5](#5-36-canonical-finite-polynomial-line-cone-kernels); the full smooth/current resolutions, acyclic/injective and controlled-excision comparisons are [§5.34, R.1–R.7](#5-34-full-current-resolutions-and-controlled-proper-image-comparison); and the coefficient cup, finite complement cube, ordered normal calibration and all-degree trace are [§5.35, V.1–V.6](#5-35-relative-cones-unshifted-cup-and-the-normalized-trace). Free human comparisons are [Kashiwara–Schapira, Micro-hyperbolic systems](https://perso.imj-prg.fr/pierre-schapira/wp-content/uploads/schapira-pub/Microhyp.pdf), §3.1, and [Kashiwara–Kawai, HolIII](https://www.kurims.kyoto-u.ac.jp/~kenkyubu/kashiwara/HolIII.pdf), III.1, IV.2 and IV.5. These identify the target operation and kernel/input cones; every comparison used here is proved below.

Every degree-one **germ-colimit relative class** has a representative of the kind treated below. Indeed R.6–R.7 allow the smooth input relative cone. A closed degree-one cochain \((a,b)\) has \(\delta a=0\) on the input neighbourhood and \(ra=\delta b\) on its complement. The explicit R.3 contraction, preserving smoothness, supplies \(F\) on a smaller input box with \(a=\delta F\). Subtracting \(d(F,0)\) gives \((0,b-rF)\); its degree-zero complement coefficient is \(\delta\)-closed. The degree-zero identification in R.3 makes \(f=rF-b\) holomorphic on that entire open complement, so the representative is \((0,-f)\). This requires only a neighbourhood shrink in the germ colimit. It imports neither fixed-domain Stein vanishing nor an extension of \(f\) across its excluded boundary. Enlarging the support to the cofinal U.21 family then gives the convex complement used below.

#### W.1. Relative cones and an explicit proper-support roof

Write \(\delta=\bar\partial\) in the full coefficient Dolbeault complex \(I\), with all output and input antiholomorphic variables included. For a closed \(S\subset M\) put
\[
\begin{gathered}
C_S^q(I)=I^q(M)\\
\oplus I^{q-1}(M\setminus S),\qquad\\
d(a,b)=(\delta a,ra-\delta b).
\end{gathered}
\tag{W.1}
\]
Thus a holomorphic complement function has positive connecting representative \(c_f=(0,-f)\). An interior cutoff \(F=\chi f\), with \(\chi\) smooth and compactly supported in the holomorphic domain of \(f\), extends smoothly by zero and gives
\[
 c_f+d(F,0)=(\delta F,F-f).
 \tag{W.2}
\]
Its second component is generally nonzero. No extension of \(f\) across its excluded boundary occurs in (W.2).

Let \(p:M\to B\) be the projection forgetting input variables over the output base, and suppose \(S\) is proper over \(B\). On a sufficiently small base open choose a smooth \(\eta\), equal to one on an open neighbourhood of \(S\), whose closed support is proper over \(B\) and contained in an open \(W\subset M\). In our application \(W=B\times D'\) with \(\overline{D'}\subset D\), and an explicit \(\eta\) is constructed in W.4. Define
\[
 \begin{split}
 L_\eta(a,b)&=(\eta a+\delta\eta\wedge b,\eta b),\\
 E_\eta(a,b)&=((1-\eta)b,0).
 \end{split}
 \tag{W.3}
\]
The first occurrence of \(\delta\eta\wedge b\), and the first component of \(E_\eta\), extend by zero across \(S\). They are defined because \(\delta\eta\) and \(1-\eta\) vanish on a neighbourhood of \(S\). This is valid even when \(b\) has unbounded growth toward \(S\).

Direct expansion, using restriction on the complement, proves
\[
\begin{gathered}
dL_\eta=L_\eta d,\qquad\\
1-L_\eta=dE_\eta+E_\eta d.
\end{gathered}
\tag{W.4}
\]
For example the first component of \(dE_\eta+E_\eta d\) is
\((1-\eta)a-\delta\eta\wedge b\); the second is \((1-\eta)b\). In the first component of \(dL_\eta\), the cross term is
\(\delta\eta\wedge(ra-\delta b)\), exactly the first cross term of \(L_\eta d\).

All components of \(L_\eta c\) have closed support inside \(\operatorname{supp}\eta\), so they are proper. A complement component is a section of \(k_*I\); it need not extend as a current across \(S\). Its sheaf support is still contained in the same proper set. The homotopy \(E_\eta\) also preserves a pre-existing proper support. Consequently inclusion
\[
 p_!C_S(I)\longrightarrow p_*C_S(I)
 \tag{W.5}
\]
has the actual local chain homotopy inverse \(L_\eta\). Here \(p_*\) means the section complex on the inverse image of the base open; it is not an assertion that its individual terms are \(p_*\)-acyclic. Formula (W.4) proves exactly the needed comparison of these two particular complexes.

Moreover \(L_\eta\) lands in the extension-by-zero complex on \(W\). The roof
\[
\begin{gathered}
p_*C_S(I)\ \longleftarrow\\
(p|_W)_!C_S(I|_W)\\
\ \xrightarrow{\,e_W\,}\ p_!C_S(I)
\end{gathered}
\tag{W.6}
\]
uses inclusion/extension by zero, with inverse to its left arrow supplied by \(L_\eta\). The two homotopies are (W.4). The right arrow is the controlled excision quasi-isomorphism (R.24). Thus (W.6) is an explicit support-to-proper-image roof, and not restriction of a proper section to an arbitrary source open.

For two such cutoffs, all terms of
\[
\begin{gathered}
(L_{\eta'}-L_\eta)c\\
=d\bigl(( (\eta'-\eta)b,0)\bigr),\\
\qquad dc=0,
\end{gathered}
\tag{W.7}
\]
have proper support in the union of their two proper neighbourhoods. The primitive extends by zero across \(S\), since \(\eta'-\eta\) vanishes there. The same formula, including the \(E d\) term in (W.4), is the chain homotopy on arbitrary cochains. It establishes agreement of the two roofs, also after intersecting excision neighbourhoods. This supplies change-of-neighbourhood and change-of-collar compatibility at the chain level.

#### W.2. The cup and its augmentation with a supported kernel

Let \(A\) be the kernel support, \(B_{\rm in}\) the inverse image of the input support, and \(S=A\cap B_{\rm in}\). Let \(T\) be a closed supported kernel coefficient current of antiholomorphic degree \(N\). Its coefficient is the **ungraded** input top-form line; the input has the ungraded coefficient \(\mathcal O\).

For a smooth input relative cochain \((a,b)\), the cup with this kernel is
\[
\begin{gathered}
m_T(a,b)=\bigl(T\wedge a,\;\\
(-1)^N T\wedge b\bigr)\in C_S.
\end{gathered}
\tag{W.8}
\]
The complement component is first defined on \(M\setminus B_{\rm in}\). On \(M\setminus A\) it is zero. These definitions agree on the overlap because \(T\) vanishes there, and hence glue on \(M\setminus S\). Smooth multiplication of a current is well defined, including its finitely many distributional derivatives. The input resolution is the smooth one; no product of arbitrary distributions is used.

The general version with a supported current of degree \(p\) has the sign \((-1)^p\). Expanding \(\delta(Ta)\) and \(\delta(Tb)\) proves that it is a chain map from the ordinary tensor complex. With \(T\) closed, \(d\,m_T=(-1)^N m_Td\).

Here is the precise identification of (W.8) with the canonical supported cup. Use the two-support finite cube of V.6, whose components are forms on \(M\), \(M\setminus A\), \(M\setminus B_{\rm in}\), and their intersection. Map the tensor of the two relative complexes to this cube by V.16. A supported kernel has only its \(M\) component, of Dolbeault degree \(N\). Moving the input complement symbol past it contributes \((-1)^N\). Thus the only complement component is exactly that of (W.8); its restriction to \(M\setminus A\) is zero, and its higher intersection component is zero. The augmentation from the full relative cone to the complement Čech complex is ordinary restriction in singleton degree, with no further sign (V.15). Therefore (W.8) maps literally to this cup cube.

The cube augmentation is a quasi-isomorphism: V.6 gives the finite Čech partition homotopy and the terminating row elimination, without a Leray assumption. R.1–R.7 identify the smooth/current augmentations with the holomorphic and derived-support objects. Multiplication of a current by a smooth form commutes with these augmentations and with the holomorphic coefficient multiplication. Tensoring these augmentations over \(\mathbf C\) preserves quasi-isomorphisms: at each stalk an exact complex of vector spaces splits by choosing complements to kernels; tensoring its contraction with a bounded complex preserves exactness, and the finite degree filtration gives the assertion for the total tensor. Thus this commutative augmentation/cube roof realizes the canonical unshifted cup, rather than only a formal current product.

For the input \(c_f=(0,-f)\), (W.8) gives
\[
 c_0=(0,(-1)^{N+1}Tf).
 \tag{W.9}
\]
The current \(Tf\) is only a complement current on \(M\setminus S\), glued as just described. With an actual interior \(F=\chi f\), the representative becomes
\[
\begin{gathered}
c_F=\bigl(T\wedge\delta F,\;\\
(-1)^N T(F-f)\bigr),\qquad\\
c_F-c_0=(-1)^N d(TF,0).
\end{gathered}
\tag{W.10}
\]
This follows from \(\delta(TF)=(-1)^N T\wedge\delta F\). After localization it is the proper coboundary
\[
\begin{gathered}
L_\eta c_F-L_\eta c_0\\
=(-1)^N d(\eta TF,0).
\end{gathered}
\tag{W.11}
\]
These formulas keep the residual \(F-f=(\chi-1)f\) explicitly.

#### W.3. The relative shifted trace

Let \(Z\subset B\) be the output support and suppose \(p(S)\subset Z\). The open \(p^{-1}(B\setminus Z)\) lies in \(M\setminus S\). In the anti-first coefficient convention put
\[
\begin{gathered}
I_q=J_{\rm out}^{-1}p_*J_{\rm in},\qquad\\
\operatorname{Tr}_q=(-1)^{N(q-N)}I_q.
\end{gathered}
\tag{W.12}
\]
The input holomorphic coefficient block is ordered before any remaining block. The trace is zero below \(q=N\). V.4 proves by compactly supported Stokes/duality that \(I\delta=\delta I\), including the zero boundary case, and hence that \(\operatorname{Tr}:p_!I[N]\to I_{\rm out}\) is a chain map with positive residue scalar at \(q=N\).

The map on relative cones is **not** componentwise the same sign. It is
\[
\begin{gathered}
Q_N(a,b)=\bigl(\operatorname{Tr}_q a,\;\\
(-1)^N\operatorname{Tr}_{q-1}\\
(b|_{p^{-1}(B\setminus Z)})\bigr),\qquad\\
(a,b)\in p_!C_S^q.
\end{gathered}
\tag{W.13}
\]
It is a map \(p_!C_S[N]\to C_Z\). For its second component the chain-map identity is
\[
\begin{gathered}
(-1)^N\operatorname{Tr}_q\\
((-1)^N(ra-\delta b))\\
=r\operatorname{Tr}_qa\\
-\delta((-1)^N\operatorname{Tr}_{q-1}b).
\end{gathered}
\]
The first component is the identity already proved for (W.12). This verifies all degrees and the cone shift. In particular, for \(N=1,q=2\) the map is \((a,b)\mapsto(-I_2a,-I_1b)\). Omitting the second minus would change the connecting class.

Equations (W.6), (W.8) and (W.13) are the cup/augmentation/excision/trace roofs used here. Their maps commute with the holomorphic augmentations, with support forgetting and with restriction to the output complement. R.6–R.7 consequently identify their composite with the canonical support-to-proper-image morphism followed by the normalized holomorphic trace. This is exactly the operation specified by Microhyp Proposition 3.1.4 and Theorem 3.1.6. Its sign is fixed by the chain maps and the positive one-coordinate residue, not by the evaluation of a contour.

If proper source representatives change by \(de\) in the unshifted cone, their outputs change by
\[
 Q_N(de)=(-1)^N d\,Q_N(e).
 \tag{W.14}
\]
This shift sign will be used for collar changes.

#### W.4. Complete one-normal-coordinate geometry

First take \(N=1\), with output \(t\), input \(s\), kernel support
\(A=\{s-t\in\mathbf R_{\le0}\}\), and input holomorphic domain
\[
\begin{gathered}
\Omega=\{\,\operatorname{Re}s\\
<-g(|\operatorname{Im}s|)\,\},\qquad\\
g(0)=0,\quad\\
g\text{ convex and}\\
\text{ nondecreasing}.
\end{gathered}
\tag{W.15}
\]
We choose \(g\) from the convex, \(o(r)\) cofinal family of U.21; in particular it is continuous and \(\Omega\) is convex. Intersect with the actual coefficient disc \(D\). Let \(B\) be a much smaller disc about zero and set \(Z=B\setminus\Omega\), \(B_{\rm in}=p_s^{-1}(D\setminus\Omega)\). This convexity is retained when using the straight-segment formula (W.21).

On \(S=A\cap B_{\rm in}\), the input imaginary part equals \(\operatorname{Im}t\) and
\[
 -g(|\operatorname{Im}t|)\le\operatorname{Re}s\le\operatorname{Re}t.
 \tag{W.16}
\]
After shrinking \(B\), these intervals form a closed proper subset inside \(B\times D'\), \(\overline{D'}\subset D\). The inverse image of a compact output set is the closed set of these bounded intervals with matching imaginary part, contained in a compact subset of \(D'\). Also \(p(S)\subset Z\): if \(t\in\Omega\), all lower real input points at the same imaginary part are in \(\Omega\).

Choose real numbers \(L<a_0<0\), all inside \(D\), and a small \(\epsilon>0\), with the full closed tube
\[
\begin{gathered}
\{L-\epsilon\le\operatorname{Re}s\le a_0+\epsilon,\\
\ |\operatorname{Im}s|\le\epsilon\}\Subset D\cap\Omega.
\end{gathered}
\tag{W.17}
\]
This is possible because the real compact segment is strictly negative and \(g(0)=0\). Shrink \(B\) to have \(|t|<\epsilon'\ll\min(\epsilon,|a_0|)\). Take a smooth real function \(\theta\) which is zero for \(r\le L\), equals one for \(r\ge a_0\) in the relevant part of \(D\), and whose transition is contained in \((L,a_0)\). Multiply \(\theta(\operatorname{Re}s)\) by compact cutoffs in the input imaginary and far-right directions, equal to one on an open neighbourhood of the portion of \(A\) over \(B\) with \(\operatorname{Re}s\ge L\). A further left cutoff can be placed below \(L\), where \(\theta\) and its derivatives are zero. The resulting \(\eta(s)\) has compact support in \(D'\), equals one near \(S\), and, on \(A\) over \(B\), is exactly \(\theta(\operatorname{Re}s)\). Thus every occurrence of \(\delta\eta\) on \(A\) is in the strictly interior tube (W.17). No cutoff touches the singular input boundary.

Let \(A_0(t,s)\) be an actual holomorphic kernel coefficient on the common product, and let
\[
\begin{gathered}
T=A_0(t,s)\,\\
\bigl(-\frac1{2\pi i}\bar\partial\log(s-t)\bigr)\\
\otimes ds.
\end{gathered}
\tag{W.18}
\]
Its full realization is the positive ray density of U.5, with the complete antiholomorphic normal \(d\overline{(s-t)}\). It is a closed degree-one current supported on \(A\). The Cauchy kernel will be considered separately below.

#### W.5. One-coordinate trace and the actual coboundary

Let \(f\) be **any** holomorphic function on \(D\cap\Omega\). On the output complement define the smooth function
\[
\begin{gathered}
G(t)=I_1(\eta Tf)\\
=\int_{L+i\,\operatorname{Im}t}^{t}\\
\theta(\operatorname{Re}s)\,A_0(t,s)f(s)\,ds,
\end{gathered}
\tag{W.19}
\]
where the path is horizontal. The integral exists for each allowed output and is smooth there: every local compact set of outputs has its entire integration set compactly inside the input domain. This assertion needs no estimate as the output approaches its excluded boundary.

Localizing (W.9) gives \((\delta\eta\wedge Tf,\eta Tf)\). Formula (W.13) and compactly supported fibre differentiation give its output
\[
 Q_1L_\eta c_0=(-\delta G,-G).
 \tag{W.20}
\]
Here the first component is a current on **all** of \(B\). It only uses \(f\) on (W.17), since \(\delta\eta\) on the ray is supported there. The expression \(-\delta G\) initially denotes its restriction to \(B\cap\Omega\).

Let the anchored holomorphic Volterra function be
\[
 V(t)=\int_{a_0}^{t}A_0(t,s)f(s)\,ds.
 \tag{W.21}
\]
The straight segment is contained in the convex input domain. It can equivalently be replaced by \(a_0\to a_0+i\,\operatorname{Im}t\to t\); the small vertical segment is in (W.17), and a primitive on the convex domain proves equality of these paths.

The following function is defined smoothly on the **full** output neighbourhood:
\[
\begin{gathered}
E(t)=\\
\int_{a_0}^{a_0+i\,\operatorname{Im}t}\\
A_0(t,s)f(s)\,ds\\
-\int_{L+i\,\operatorname{Im}t}^{a_0+i\,\operatorname{Im}t}\\
\theta(\operatorname{Re}s)A_0(t,s)f(s)\,ds.
\end{gathered}
\tag{W.22}
\]
All its input points are in the fixed interior tube (W.17); \(A_0\) is holomorphic on the fixed product. Parametrizing the two compact segments gives smoothness with every derivative obtained on a compact subset of that tube. On the complement, \(V=G+E\), so \(\delta E=-\delta G\).

For completeness this equality of the **full** first component can be checked without density or continuation across the unknown boundary. Put \(t=r+iy\), \(H(t,s)=A_0(t,s)f(s)\). For the second integral of (W.22), differentiation along \(y\) and integration by parts gives
\[
\begin{gathered}
\bar\partial_t\int_L^{a_0}\\
\theta(u)H(t,u+iy)\,du\\
=-\tfrac12H(t,a_0+iy)\\
+\tfrac12\int_L^{a_0}\\
\theta'(u)H(t,u+iy)\,du .
\end{gathered}
\]
The first integral of (W.22) has derivative \(-\tfrac12H(t,a_0+iy)\). Their difference is therefore
\[
\begin{gathered}
\bar\partial_t E(t)\\
=-\tfrac12\int_L^{a_0}\\
\theta'(u)H(t,u+iy)\,du .
\end{gathered}
\tag{W.23}
\]
The upper endpoint of (W.19), with \(\theta=1\), contributes zero to \(\bar\partial_tG\), and the same integration by parts gives
\(\bar\partial_tG=\tfrac12\int_L^{a_0}\theta'H(t,u+iy)\,du\). Direct normal-block integration of \(-I_2(\delta\eta\wedge Tf)\) gives the negative of this expression: moving \(d\bar t\) before the fibre normal and using \(d\bar s\wedge ds=2i\,d\operatorname{Re}s\wedge d\operatorname{Im}s\) gives the same factor \(1/2\). Thus (W.23) identifies the first component on every output of \(B\), using only the compact tube.

Combining (W.20)–(W.23) yields the requested equality of relative classes with an actual primitive:
\[
\begin{gathered}
Q_1L_\eta c_0=(\delta E,E-V)\\
=(0,-V)+d(E,0).
\end{gathered}
\tag{W.24}
\]
This is a cone/cup/excision/trace equality. The ordinary keyhole evaluation is unnecessary for its proof.

Now use an arbitrary interior collar \(F=\chi f\) from (W.2). Define
\(J_\chi=I_1(\eta TF)\), a smooth function on all of \(B\). Its input support is compactly inside \(\Omega\); parametrizing the ray against the smooth compact \(F\) proves smoothness, including at its endpoint. Applying (W.11), (W.14) gives
\[
\begin{gathered}
Q_1L_\eta c_F\\
=(0,-V)+d(E+J_\chi,0).
\end{gathered}
\tag{W.25}
\]
In particular its complement component is \(-G+J_\chi\), the trace of \(T(f-F)\) with the relative minus sign. Dropping that term would not produce (W.25). The input need never extend as a distribution.

For the normalized Cauchy kernel \(T_{\rm C}\), the cutoff is one with all derivatives zero near \(s=t\), for every \(t\in B\). Therefore \(L_\eta(0,T_{\rm C}f)=(0,T_{\rm C}f)\), its first component is zero, and (W.13) gives \((0,-f)\). Finite derivatives of its diagonal current give \((0,-\partial_t^kf)\) by U.7. The same collar correction is (W.25) with the Volterra term replaced by \(f\) or its indicated derivative.

#### W.6. Changes of cutoffs, collars, anchors, and representatives

Equation (W.7), followed by the relative trace, gives for \(c_0\)
\[
\begin{gathered}
Q_N(L_{\eta'}-L_\eta)c_0\\
=-d\bigl(I_N((\eta'-\eta)Tf),0\bigr).
\end{gathered}
\tag{W.26}
\]
The function in this primitive is smooth on all of the base. Over a compact base set, the part of \(A\) where the difference is nonzero is compact and avoids \(S\); it therefore has a finite covering by interior input tubes on which \(f\) is smooth. A partition on those tubes makes its coefficient a finite sum of smooth compact coefficients near the normal-current support. Parametrizing the translated ray and evaluating the finite diagonal derivatives proves smoothness of each fibre integral; the result is independent of the coefficient extensions away from \(A\). This includes arbitrary changes of the compact excision collar, not only the special \(\theta\)'s of W.4.

For two interior \(F,F'\), their localized cups differ by
\((-1)^Nd(\eta T(F'-F),0)\); their outputs differ by
\[
 d\bigl(I_N(\eta T(F'-F)),0\bigr).
 \tag{W.27}
\]
This primitive is again smooth on the full base. Combined changes follow by adding these two proper primitives. There is no term left unaccounted for on the complement.

Changing \(a_0\) to another strictly negative anchor \(a_1\) changes \(V\) by
\(\int_{a_1}^{a_0}A_0(t,s)f(s)\,ds\), defined holomorphically on a full smaller output neighbourhood by the fixed negative-time tube. If this difference is \(H_0\), then \(c_{V+H_0}-c_V=d(-H_0,0)\). An extending input \(f_0\) has \(c_{f_0}=d(-f_0,0)\); (W.8), localization and (W.14) preserve this exactness. Its anchored output also extends holomorphically by the same fixed-product bound. These are the concrete quotient and germ compatibilities. Enlarging to a common cofinal U.21 support and restricting to common smaller coefficient products makes the same constructions compare two arbitrary input representatives.

#### W.7. Ordered multi-normal tensor and trace

Return to \(N=d+1\), with input \(v=(s,y)\), output \(u=(t,x)\), normal coordinates \(w_0=s-t\), \(w_j=y_j-x_j\), and kernel support \(A=Z_0\) of U.1. The input domain is the convex U.21 domain
\(\Omega_g=\{\operatorname{Re}s<-g(|(\operatorname{Im}s,y)|)\}\).

Take the ordered tensor of the one-normal negative-log/Cauchy classes and the finite spatial pole derivatives. Use the exact V.16 tensor-to-cube map and V.17 calibration
\[
 r_N=(-1)^{N(N-1)/2}.
 \tag{W.28}
\]
Consequently \(T_N\), in anti-first coefficient notation, includes \(r_N\); its full form is the positive ordered complete-normal-block current U.14. The top cube entry of an ordered unsigned holomorphic representative is \(r_N(-1)^N\) times that representative. No double suspension is added to the **unshifted** between-kernel or kernel/input cup.

All singular normal variables in this single kernel are independent. Their finite distributional derivatives and tensor product exist by V.2; holomorphic coefficients multiply them. The current contains every output antiholomorphic component \(d\overline{(v-u)}\) and is closed. In the action, its second factor is smooth. Thus this construction requires no product of distributions sharing a singular input boundary.

On the support intersection \(S\), \(y=x\), \(\operatorname{Im}s=\operatorname{Im}t\), and
\[
\begin{gathered}
-g(|(\operatorname{Im}t,x)|)\\
\le\operatorname{Re}s\le\operatorname{Re}t.
\end{gathered}
\tag{W.29}
\]
The proof of properness and \(p(S)\subset Z\) is the closed-interval argument of W.4, with the unchanged compact spatial coordinates. Choose the full tube (W.17) also with \(|x|<\epsilon\); continuity and \(g(0)=0\) put its closure inside \(\Omega_g\). Take
\(\eta(s,y)\) to be the time cutoff of W.4 times a compact spatial cutoff which is identically one on an open neighbourhood of every \(y=x\) over the smaller output product. All derivatives of this spatial cutoff vanish there. This \(\eta\) has support compactly inside the actual input product and equals one near \(S\).

Now (W.8)–(W.14) apply with kernel degree \(N\) and input degree one. In source degree \(q=N+1\), localization of (W.9) and the cone trace give, for every \(N\),
\[
\begin{gathered}
Q_NL_\eta c_0=(-\delta G,-G),\\
\qquad G=I_N(\eta T_Nf).
\end{gathered}
\tag{W.30}
\]
Indeed the source components are
\((-1)^{N+1}\delta\eta\wedge T_Nf\) and
\((-1)^{N+1}\eta T_Nf\).
The first trace sign is \((-1)^N\); the second is
\((-1)^N\operatorname{Tr}_N=(-1)^NI_N\). Both become a minus. Raw integration commutes with \(\delta\), so the first component is \(-\delta G\). These signs use \(r_N\) inside \(T_N\) once and the unshifted coefficient cup once.

Ordered spatial Cauchy integration is an equality of these currents, U.7–U.8, and evaluates the finite spatial derivatives. Since the spatial cutoff is flat on their support, it introduces no derivative terms. For a U.10 kernel its value on the output complement is
\[
\begin{gathered}
G(t,x)=\sum_{\beta\in B}\\
b_{0,\beta}(t,x)\partial_x^\beta f(t,x)\\
+\sum_{\beta\in B}\int_{L+i\,\operatorname{Im}t}^{t}\\
\theta(\operatorname{Re}s)A_\beta(t,s,x)\\
\partial_x^\beta f(s,x)\,ds.
\end{gathered}
\tag{W.31}
\]
Only the negative-time terms have cutoff tails. Define \(E(t,x)\) by the finite sum of (W.22), with
\(A_0(t,s)f(s)\) replaced by
\(A_\beta(t,s,x)\partial_x^\beta f(s,x)\).
The full negative-time tube is holomorphic also in \(x\), so this is smooth on the full output product and has no additional \(\bar\partial_x\) term. Formula (W.23) proves its full \(\bar\partial_t\) identity. Therefore (W.24) becomes
\[
\begin{gathered}
Q_NL_\eta c_0\\
=(0,-V_P)+d(E,0),\quad\\
V_P=\sum_{\beta}b_{0,\beta}\partial_x^\beta f\\
+\sum_{\beta}\int_{a_0}^{t}\\
A_\beta(t,s,x)\partial_x^\beta f(s,x)\,ds.
\end{gathered}
\tag{W.32}
\]
This is U.22 with its unchanged positive sign. For an arbitrary interior \(F=\chi f\), put \(J_\chi=I_N(\eta T_NF)\). It is smooth on the full output product: all inputs are in a compact interior set, spatial derivatives are finite, and the ray evaluation against a smooth compact coefficient can be differentiated by ordinary compact-segment calculus. Equations (W.11) and (W.14) again cancel the two factors \((-1)^N\), giving
\[
\begin{gathered}
Q_NL_\eta c_F\\
=(0,-V_P)+d(E+J_\chi,0).
\end{gathered}
\tag{W.33}
\]
This proves the full ordered tensor/trace extension and the actual collar coboundary with the conventions of §§5.34–5.35.

#### W.8. Actual analytic tails, finite positive orders, and scope

For every \(\beta\) the coefficient \(A_\beta\) is the actual normally convergent U.11 series on one common product, bounded by U.12. There are only finitely many \(\beta\). Choose \(B,D',L,a_0,\eta,\chi\) and the negative-time tube before evaluating that series. Its finitely many derivatives converge uniformly on every compact set used by the trace and the corrections. Multiplication by the fixed finite-order normal current is continuous on these bounded smooth-coefficient sets: by the definition of a finite-order distribution its value is bounded by finitely many coefficient derivatives on the compact support of each test. Consequently all the above identities hold for the actual negative tails and their trace, not merely termwise formal expansions. There is no boundary bound on \(f\); only compact interior bounds enter.

Finite positive time orders are normalized finite derivatives of the diagonal Cauchy current, just as U.7 treats the spatial derivatives. The cutoff is one with all derivatives zero near that diagonal, so their localization has no first component and their relative trace is the positive connecting class of the ordinary indicated derivative. Output holomorphic multipliers and finite output derivatives commute with the cup augmentations and trace: test-function transposition proves this for the currents, and proper support allows integration by parts without a boundary term. Derivatives commute literally with \(L_\eta\) for the chosen input-only cutoffs of W.4/W.7. For a general base-dependent cutoff and a first-order holomorphic coordinate derivative \(D\), the exact replacement is
\[
\begin{gathered}
{}[D,L_\eta]=dK_D+K_Dd,\qquad\\
K_D(a,b)=((D\eta)b,0).
\end{gathered}
\]
The coefficient \(D\eta\) vanishes near \(S\), so the primitive extends by zero and has proper support. Expansion gives first component \((D\eta)a+\delta(D\eta)\wedge b\), and second component \((D\eta)b\), which are precisely the commutator. Iterating this identity gives the corresponding homotopy for finite higher derivatives. Thus derivative compatibility with arbitrary controlled roofs holds on cohomology. This proves the same equality after the finite left time shift used to put an arbitrary element of the stated polynomial ring \(R\) into U.10. In particular the already identified ordinary \(h\)-primitive and \(\partial_t\) actions are inverse after quotienting the holomorphic anchor correction.

Thus every input in the stated unrestricted directional degree-one quotient, and every ordinary finite polynomial directional operator in the stated finite polynomial scope, has canonical action equal to U.22. Equations (W.24), (W.25), (W.32), (W.33), and the proper primitives (W.26)–(W.27) are the requested explicit comparison and independence data. The result uses the full-resolution and cup/trace proofs in §§5.34–5.35; it does not re-certify unrelated dependencies.

![Exact collar geometry and cone comparison](assets/collar-comparison.png)

**Figure W.1.** Exact illustrative slice \(g=0\), \(L=-1.8\), \(a_0=-0.8\), allowed output \(t=-0.20+0.10i\), and a second output \(t_+=0.20-0.08i\). The latter fibre has proper intersection \(S_{t_+}=[0,0.20]-0.08i\). The shaded finite transition window \([-1.6,-1.0]\times[-0.125,0.125]\) lies compactly inside the dashed negative-time tube; only that tube supplies the full-neighbourhood correction \(E\) in W.22. The right panel keeps \(F-f\) and records the cup, localization and shifted trace leading to W.33. The exact coordinates are retained in the reproducible figure source supplied with the lesson. It illustrates W.3, W.5 and W.7; it is not a numerical proof or a claim that the general \(g\) vanishes. Human-source context: Micro-hyperbolic systems §3.1 and HolIII III.1/IV.2/IV.5 above.

No full convergent-\(z\) \(E\) ring, infinite-order extension, arbitrary derived action, arbitrary \(E\)-valued off-support homotopy transfer, propagation, module separation, finite D-type theorem, pole recovery, intrinsic order, full C1, or analytic proper-regularity theorem follows here. Those interfaces remain outside this proof.


### 5.38. Convergent spatial symbols and supported thick-cone currents

This section constructs a normally convergent, canonically calibrated supported current for every actual order-zero symbol, with a normal distribution-order bound independent of the number of spatial Taylor terms. Its complete current construction is E.1–E.3. Normal-localization independence is proved in [§5.39,NL.1–NL.7](#5-39-canonical-normal-localization-of-convergent-spatial-symbols). The full canonical product and unrestricted-input action remain required in E.4; a limit of exact polynomial identities alone does not prove them.

The actual coefficient ring and factorial bounds are [§§5.19–5.22](#5-19-actual-factorial-growth-symbols-and-ordered-jets) and [§5.30, F.1](#5-30-actual-polynomial-flatness-and-reconstruction-after-finite-realization). The complete smooth/current resolutions and support comparisons are [§5.34, R.1–R.7](#5-34-full-current-resolutions-and-controlled-proper-image-comparison). The finite complement cube, connecting signs and ordered coefficient-line calibration are [§5.35, V.1–V.6](#5-35-relative-cones-unshifted-cup-and-the-normalized-trace). The finite-spatial polynomial kernel comparison is [§5.36, U.1–U.4](#5-36-canonical-finite-polynomial-line-cone-kernels); its unrestricted-growth input proof is [§5.37, W.1–W.8](#5-37-interior-collars-and-the-canonical-directional-action). Each is a complete earlier programme proof in its stated scope.

Free human comparison sources specifying the intended kernel/cohomology object and cone are Kashiwara–Schapira, [Micro-hyperbolic systems](https://perso.imj-prg.fr/pierre-schapira/wp-content/uploads/schapira-pub/Microhyp.pdf), §3.1, and Kashiwara–Kawai, [HolIII](https://www.kurims.kyoto-u.ac.jp/~kenkyubu/kashiwara/HolIII.pdf), III.1 and IV.2. Every comparison used in this section is proved below or at the exact earlier programme loci just listed.

#### E.1. The analytic coefficient and a genuinely thicker proper cone

Put \(N=d+1\), \(u=(t,x)\), \(v=(s,y)\), and use normal coordinates
\(w=s-t\), \(\zeta_j=y_j-x_j\). The input coefficient frame is
\(ds\wedge dy_1\wedge\cdots\wedge dy_d\), ungraded before the trace.
An actual order-zero symbol has, on one base neighbourhood and one cotangent polydisc,
\[
\begin{gathered}
P=\sum_{n\ge0,\ \beta\in\mathbf N^d}p_{n,\beta}(u)h^nz^\beta,\\
|p_{n,\beta}|\le B C^n n!r_z^{-|\beta|}.
\end{gathered}
\tag{E.1}
\]
These coefficient estimates hold on every smaller compact base polydisc, with fixed \(B,C,r_z>0\). Here is the Cauchy step explicitly. R.2 proves the Green formula and deleted-circle normalization. Apply that formula to a holomorphic function divided by its displacement from an interior point. The integral off that point is zero, while the inner positively oriented circle tends to \(2\pi i\) times its value there; this gives the Cauchy integral formula. Iterate it on a product of coordinate circles and expand each denominator geometrically on a smaller polydisc. The coefficient \(p_{n,\beta}\) is the integral of \(p_n\) times the ordered powers \(z_j^{-\beta_j-1}\), divided by \((2\pi i)^d\). The circle lengths cancel one inverse radius in each variable, giving the bound in (E.1) from the factorial estimate for \(p_n\). Differentiating the base Cauchy formula a fixed finite number of times on a strictly smaller product supplies the same estimate with a finite additional constant. All these choices use one common neighbourhood.

Define the unsigned ordered holomorphic coefficient
\[
\begin{gathered}
K_P=\frac1{(2\pi i)^N}\\
\left\{\begin{gathered}\frac{p_{0,0}(u)}{w\prod_j\zeta_j}\\-\log w\!\sum_{n+|\beta|\ge1}\\\frac{\begin{gathered}p_{n,\beta}(u)\beta!\\(-w)^{n+|\beta|-1}\end{gathered}}{\begin{gathered}(n+|\beta|-1)!\\\prod_j\zeta_j^{\beta_j+1}\end{gathered}}\end{gathered}\right\}.
\end{gathered}
\tag{E.2}
\]
The logarithm is principal, cut along the nonpositive real ray. Select
\(a_j>r_z^{-1}\). This coefficient is holomorphic on the intersection of
\[
\begin{gathered}
U_0=\{w\notin\mathbf R_{\le0}\},\\
U_j=\{|\zeta_j|>a_j|w|\}\\
(1\le j\le d).
\end{gathered}
\tag{E.3}
\]
Its complement support is the thin cone
\[
\begin{gathered}
G_a=\left\{\begin{gathered}w\in\mathbf R_{\le0},\\|\zeta_j|\le a_j(-\operatorname{Re}w)\end{gathered}\right\}.
\end{gathered}
\tag{E.4}
\]
In dimension \(d=0\), omit all spatial products and sets.

For clarity, normal convergence is proved here. If \(k=|\beta|\), then
\(\beta!\le k!\) and
\[
\begin{gathered}
\frac{n!\beta!}{(n+k-1)!}\le n+k\\
(n+k\ge1).
\end{gathered}
\tag{E.5}
\]
Indeed \((n+k)!/(n!k!)\) is a positive integer. After taking absolute values in (E.2), factor out
\(|w|^{-1}\prod_j|\zeta_j|^{-1}\). The remaining majorant is
\[
\begin{gathered}
B\sum_{n,\beta,\ n+|\beta|\ge1}\\
(n+|\beta|)\,q^n\prod_j\theta_j^{\beta_j},\\
q=C|w|,\\
\theta_j=\frac{|w|}{r_z|\zeta_j|}.
\end{gathered}
\tag{E.6}
\]
On a smaller time-difference disc, \(q<1/2\); on every compact subset of (E.3), each \(\theta_j<1\) uniformly. The sum is finite: multiply the geometric series and differentiate each one once. The same calculation with any fixed power of \(n+|\beta|\) proves locally uniform convergence of every fixed finite derivative. Thus (E.2) is an actual holomorphic coefficient.

Choose \(\epsilon>0\) and \(A_j>a_j\), and enlarge the support to
\[
\begin{gathered}
G=\left\{\begin{gathered}\operatorname{Re}w\le0,\\|\operatorname{Im}w|\le\epsilon(-\operatorname{Re}w),\\|\zeta_j|\le A_j(-\operatorname{Re}w)\end{gathered}\right\}.
\end{gathered}
\tag{E.7}
\]
This is a closed convex cone: it is the intersection of the halfspace and the epigraph inequalities of the convex absolute-value/norm functions. If a point and its negative belong to it, their real time is zero and every remaining coordinate is zero. It is therefore pointed. It satisfies
\[
\begin{gathered}
|(w,\zeta)|\le\left(1+\epsilon^2+\sum_jA_j^2\right)^{1/2}\\
(-\operatorname{Re}w)\quad\text{on }G.
\end{gathered}
\tag{E.8}
\]
In particular every nonzero cone point has strictly negative real time.
The angular set \(G_a\cap S^{2N-1}\) is a compact subset of the relative interior of \(G\cap S^{2N-1}\): on \(G_a\) the imaginary-time inequality is strict, all spatial inequalities have the strict margin \(A_j-a_j\), and the real-time coordinate cannot vanish on the sphere.

These inequalities also prove the required local properness for a two-kernel intersection. If \(a,b\in G\) and their sum ranges in a compact endpoint set, the two nonnegative numbers \(-\operatorname{Re}a\), \(-\operatorname{Re}b\) sum to the bounded number \(-\operatorname{Re}(a+b)\). Formula (E.8) bounds every intermediate coordinate. The intersection is closed in that bounded endpoint/fibre set, so it is compact in the full affine coordinate space. Choose the actual coefficient neighbourhood larger than this compact fibre family and shrink the endpoint neighbourhood once to place the whole family inside it. Compactness in that chosen source, rather than boundedness in an arbitrary open domain, proves properness. For a finite ordered list, the identical argument uses the sum of the nonnegative real-time lengths.

#### E.2. A homogeneous angular partition and the exact Čech sign

On the normal unit sphere, the open sets \(U_0,\ldots,U_d\) together with
\(U_G=\operatorname{int}G\) cover the sphere. Indeed a point outside all the \(U_i\) belongs to \(G_a\), and the final assertion of E.1 places it in \(U_G\).
Choose a smooth finite partition
\(\psi_0,\ldots,\psi_d,\psi_G\) subordinate to this cover, with each support compactly contained in its corresponding angular open set.

Here is the finite construction. At every sphere point select a sphere ball whose closed ball lies in one member of the cover and a smaller ball about the same point. Finitely many smaller balls cover the compact sphere. In a sphere coordinate chart use the bump
\(\exp(-1/(1-|q|^2))\) for \(|q|<1\) and zero otherwise, rescaled to the larger ball, and positive on the smaller ball. Sum the bumps assigned to each cover member and divide by the strictly positive sum of all the bumps. The smoothness of the bump at its boundary follows by differentiating: each derivative is a polynomial in inverse powers of \(1-|q|^2\) times the exponential, which tends to zero faster than those powers. Finite compactness bounds every fixed derivative. This constructs the asserted partition without a sheaf or acyclicity assumption.

Extend the partition by degree-zero homogeneity to the punctured normal space. If \(r=|(w,\zeta)|\), each normal derivative of order \(j\) is bounded by a constant times \(r^{-j}\). This follows by differentiating the radial projection to the sphere; its \(j\)-th derivatives have degree \(-j\) and bounded angular coefficients. Outside \(G\), \(\psi_G=0\), so \(\sum_{i=0}^d\psi_i=1\).

Set
\[
\begin{gathered}
\omega_\psi=\sum_{i=0}^{N-1}(-1)^i\psi_i\\
\bar\partial\psi_0\wedge\cdots\wedge\\
\widehat{\bar\partial\psi_i}\wedge\cdots\wedge\bar\partial\psi_{N-1},\\
F_P=r_N(N-1)!\,K_P\,\omega_\psi,\\
r_N=(-1)^{N(N-1)/2}.
\end{gathered}
\tag{E.9}
\]
For \(N=1\), the empty wedge is one, so \(F_P=\psi_0K_P\).
Attach the holomorphic input frame to \(F_P\) exactly once.

Every term in (E.9) is supported in a compact angular subset of the full intersection \(U_0\cap\cdots\cap U_d\): the undifferentiated factor has support in its own \(U_i\), and each differentiated factor has support in its own \(U_j\). Consequently the term, initially using (E.2) on that intersection, extends by zero smoothly across the other punctured normal points. This extension never evaluates a logarithm on its cut. Compact angular containment also supplies uniform strictly smaller bounds \(\theta_j<1\) in (E.6) on all these supports.

The factor in (E.9) is forced by the exact cone convention, rather than selected afterwards by its action. The unsigned \(K_P\) denotes ordinary top cube coefficient \(r_N(-1)^NK_P\) and ordinary full-cone complement Čech coefficient \(-r_NK_P\), by V.15–V.17. Write the usual alternating Čech contraction as
\((hc)_{i_0\ldots i_{r-1}}=\sum_j\psi_jc_{ji_0\ldots i_{r-1}}\), on the complement of \(G\).
With total differential \(d_C+(-1)^r\bar\partial\), subtracting the differential of \(hc\) sends a holomorphic Čech \(r\)-cocycle to
\((-1)^r\bar\partial hc\) in Čech degree \(r-1\).
Repeated descent therefore gives sign
\((-1)^{1+\cdots+(N-1)}=r_N\).
Applied to the complement coefficient \(-r_NK_P\), the resulting relative-cone complement coefficient is
\(-(\bar\partial h)^{N-1}K_P\).

Expanding the finite alternating sums gives
\[
\begin{gathered}
(\bar\partial h)^{N-1}K_P\\
=r_N(N-1)!\,K_P\omega_\psi.
\end{gathered}
\tag{E.10}
\]
The following local-entry calculation proves (E.10) in every normal dimension. Let the top alternating cochain be \(c_{0\ldots N-1}=K_P\) and put \(A^j=(\bar\partial h)^jc\). For an increasing index list \(I\) of length \(N-j\), let \(J\) be its increasing complement, and let \(\varepsilon(J,I)\) be the permutation sign of their concatenation. Then
\[
\begin{gathered}
A_I^j=(-1)^{j(j-1)/2}j!\,\\
\varepsilon(J,I)K_P\\
\bigwedge_{\ell\in J}\bar\partial\psi_\ell.
\end{gathered}
\tag{E.10a}
\]
For \(j=0\) this is the top cochain. To expand \(j\) consecutive contractions, choose the ordered list of the \(j\) deleted indices. Its coefficient has the alternating sign of that list followed by \(I\). Each new antiholomorphic differential acts on the left, so their wedge appears in the reverse deletion order. Reversing the list contributes \((-1)^{j(j-1)/2}\). Sorting the deleted indices contributes the same permutation sign to both coefficient and wedge, so those two signs cancel. Every one of the \(j!\) orderings therefore gives the same term displayed in (E.10a). This proves the formula by the full finite expansion, without extrapolating dimension checks.

At \(j=N-1\), \(I=(i)\) and \(\varepsilon(J,i)=(-1)^{N-1-i}\). Hence
\[
\begin{gathered}
A_i^{N-1}=r_N(-1)^i(N-1)!\\
K_P\bar\partial\psi_0\wedge\cdots\wedge\\
\widehat{\bar\partial\psi_i}\wedge\cdots\wedge\bar\partial\psi_{N-1}.
\end{gathered}
\tag{E.10b}
\]
On the complement of \(G\), the partition relations are \(\sum_i\psi_i=1\) and \(\sum_i\bar\partial\psi_i=0\). Substitute the second into \(\omega_\psi\); every summand becomes its coefficient \(\psi_j\) times the same signed wedge \((-1)^i\bigwedge_{j\ne i}\bar\partial\psi_j\). Their coefficients sum to one. Every local entry in (E.10b) is therefore \(r_N(N-1)!K_P\omega_\psi\), proving (E.10). For \(N=1\), \(\psi_0=1\) on the complement and the same identity holds immediately.

Thus the actual complement representative is \((0,-F_P)\). The positive connecting convention sends it to \((\bar\partial F_P,0)\) whenever \(F_P\) extends as a current to the normal origin. The following section supplies that extension with a quantitative bound.

#### E.3. Local integrability, supported current and actual tail convergence

On the compact angular supports in E.2, the geometric estimate (E.6) and the fixed angular separation from \(w=0\) and each \(\zeta_j=0\) imply, on every compact output base,
\[
\begin{gathered}
|K_P|\le C_0r^{-N}\\
(1+|\log r|),\\
|F_P|\le C_1r^{-(2N-1)}\\
(1+|\log r|).
\end{gathered}
\tag{E.11}
\]
The constants include the finite partition bounds and the actual coefficient bound. The multiplication term has the first bound without a logarithm. All terms with \(n\ge1\) are less singular; the \(n=0\), unbounded-\(\beta\) sum is controlled by the geometric factors in (E.6). There is no infinite derivative of a spatial delta distribution in this construction.

The normal space has real dimension \(2N\). Polar integration of the second bound is at most a constant times
\(\int_0^R(1+|\log r|)\,dr\), which is finite. Hence \(F_P\) is a locally integrable \((0,N-1)\)-coefficient form, and has its literal locally integrable extension to the origin. In particular
\[
 T_P=\bar\partial F_P
 \tag{E.12}
\]
is a distributional \((0,N)\)-coefficient current of order at most one in the normal variables, locally uniformly over each compact output base. For a compactly supported test form, its value is defined by transposing the differential onto that test form. The \(L^1\) bound on \(F_P\) bounds the value by a constant times its first derivatives. The derivative bound is independent of the number of spatial Taylor terms. The full antiholomorphic differential, including output variables, is used.

The current is closed because \(\bar\partial^2=0\) on test forms. It is supported in \(G\). Away from the origin all coefficients in E.9 are smooth, and direct differentiation gives
\(\bar\partial\omega_\psi=N\bar\partial\psi_0\wedge\cdots\wedge\bar\partial\psi_{N-1}\).
Outside \(G\), differentiating \(\sum_i\psi_i=1\) makes this wedge zero. The coefficient \(K_P\) is holomorphic wherever its multiplier is nonzero. Thus \(T_P\) vanishes outside \(G\), and the only new extension point, the origin, belongs to \(G\).

By E.2, the relative representative \((0,-F_P)\) on the complement has the calibrated full-cone Čech class \(-r_NK_P\). Adding the literal relative boundary \(d(F_P,0)\) gives \((T_P,0)\). The full resolution comparison R.1–R.7 consequently identifies (E.12) with the canonical supported class specified by (E.2), after enlargement from \(G_a\) to \(G\). This proves the class construction, rather than identifying it from a solution action.

For actual convergence, truncate (E.1) in \(n\) and \(|\beta|\), while retaining its holomorphic base coefficients. On the fixed angular supports, the geometric tail in (E.6) tends to zero uniformly after factoring the common radial bound in (E.11). Dominated polar integration therefore gives \(F_{P_m}\to F_P\) in \(L^1\) on each compact normal/output set. Transposing one differential gives \(T_{P_m}\to T_P\) as currents, with a common first-order test-function bound.

Output differentiation with the input held fixed includes normal derivatives. In the coordinates \((u,w,\zeta)\),
\[
\begin{gathered}
\left.\partial_t\right|_{s,y}=\left.\partial_t\right|_{w,\zeta}-\partial_w,\\
\left.\partial_{x_j}\right|_{s,y}=\left.\partial_{x_j}\right|_{w,\zeta}-\partial_{\zeta_j}.
\end{gathered}
\tag{E.13}
\]
The coordinate transformation is linear, so the constant normal differential forms and input holomorphic frame introduce no coefficient derivative. Expanding a fixed output derivative of order \(k\) gives finitely many mixed fixed-normal base and normal derivatives. For the base factors, a smaller common base and the coefficient Cauchy estimate retain the \(L^1\) convergence of the primitives. Transpose each normal factor onto the test form. This proves convergence of the actual output derivatives as currents, with a common normal distribution-order bound at most \(k+1\). It does not claim \(L^1\) convergence of their normal derivatives as functions.

This provides a uniform current representative for full convergent spatial order-zero symbols. Finite positive time orders can be treated by applying their finite output derivatives to these currents, which preserves closedness and support and gives a finite distribution order. The explicit supported normal-generator and presentation primitives in [§5.39,NL.2–NL.7](#5-39-canonical-normal-localization-of-convergent-spatial-symbols) prove independence of that presentation.

#### E.4. The remaining product and action step

The independent-normal distribution tensor of V.2 applies to the two full currents in (E.12), and E.1 supplies an actual locally proper thick-cone intersection. These facts define their canonical unshifted cup and normalized trace using R/V. They do not yet prove its equality with the symbol product.

The exact equality for every finite spatial polynomial follows from U.1–U.4. The convergent current construction in E.3 gives actual limits of those polynomial representatives. To deduce equality of canonical classes from that limit, one must provide a uniformly controlled proper primitive for their differences, or compute the full canonical finite Čech/trace roof on a fixed common angular cover and establish its normally convergent equality with (E.2) for \(P\circ Q\). Exactness of each approximating difference by itself is insufficient: no closed-range or continuity theorem for the relevant cohomology quotient has been proved. No such theorem is being silently used.

For unrestricted directional input, the chosen thick cone also changes the spatial input geometry. The finite-spatial-current argument of W.7 cannot be reused as an unbounded sum of spatial delta derivatives. A common-domain Cauchy contour on the thick cone, the explicit smooth-input relative augmentation and a full-output collar primitive remain necessary. The full-ring comparison must prove those estimates and the nested composition domains on a single neighbourhood before its action is accepted.

The full infinite comparison, propagation, separation and finite D-type results are proved in Sections5.43–5.87; Section5.89, CK.A–CK.K proves the intrinsic source cutoff. Arbitrary analytic proper regularity requires its further argument.

![Temporal and spatial sections of the precise thick cone](assets/thick-cone-sections.png)

*Exact sections of (E.7) with \(\epsilon=1/4\), \(a=2\), \(A=3\): the temporal section sets \(\zeta=0\); the spatial section sets \(\operatorname{Im}w=0\) and \(\operatorname{Im}\zeta=0\). The dashed thin support lies in the relative interior of the thick cone on every normal unit sphere. The depicted fibre interval uses real final normal length \(2\), so each intermediate nonnegative real-time length lies between \(0\) and \(2\). These are sections and a real slice, rather than the whole complex cone. Bounds and properness are proved in E.1; the angular-cutoff and distribution arguments are E.2–E.3. Reproducible source retains every coordinate.*


The exact figure has a full-resolution PNG, SVG, editable plotting source and coordinate data. Its mathematical proof is E.1–E.3; free human cone comparison is HolIII IV.2 and Micro-hyperbolic systems §3.1, linked above.


### 5.39. Canonical normal localization of convergent spatial symbols

This section proves both normal-generator products, the calibrated diagonal unit and two-sided inverse, and independence of every finite time-order presentation of an actual convergent-spatial symbol. All correction terms are literal supported relative-cone primitives. General symbol-pair multiplication and the canonical unrestricted-input action are separate requirements.

The actual coefficient algebra, common-domain growth and normal conjugation are proved in [§§5.19–5.22](#5-19-actual-factorial-growth-symbols-and-ordered-jets), with the precise finite-order coefficient class in [§5.28,Q.8](#5-28-convergent-contact-quantization-and-whole-germ-generic-position). The full current/smooth resolution and controlled proper support maps are [§5.34,R.1–R.7](#5-34-full-current-resolutions-and-controlled-proper-image-comparison); the calibrated complement cube, ungraded frame and all-degree cup/trace are [§5.35,V.1–V.6](#5-35-relative-cones-unshifted-cup-and-the-normalized-trace). The actual convergent-spatial current and complete derivative bounds are [§5.38,E.1–E.3](#5-38-convergent-spatial-symbols-and-supported-thick-cone-currents). Each cited programme locus contains its complete proof. The finite polynomial inverse correction is also explicit in [§5.36,U.17–U.18](#5-36-canonical-finite-polynomial-line-cone-kernels); the proof below supplies the full convergent-spatial and auxiliary-choice extension.

Free human comparison sources are Kashiwara–Schapira, [Micro-hyperbolic systems](https://perso.imj-prg.fr/pierre-schapira/wp-content/uploads/schapira-pub/Microhyp.pdf), §3.1, and Kashiwara–Kawai, [HolIII](https://www.kurims.kyoto-u.ac.jp/~kenkyubu/kashiwara/HolIII.pdf), III.1 and IV.2. Every additional argument used here is proved below.

#### NL.1 The actual shifts and their common domains

Keep \(N=d+1\), output \(u=(t,x)\), input \(v=(s,y)\), \(w=s-t\), \(\zeta =y-x\), \(h=\tau ^{-1}\), \(z=\xi /\tau\). The ungraded coefficient frame is \(\eta _{\mathrm{in}}=ds\wedge dy_1\wedge \cdots \wedge dy_d\). Write \(\delta\) for the full antiholomorphic differential, including output variables. The relative cone has

\[
 d(a,b)=(\delta a,ra-\delta b).
\tag{NL.1}
\]

Let \(P=\sum _{n,\beta }p_{n,\beta }(u)h^nz^\beta\) be actual order zero on one common coefficient domain, with \(|p_{n,\beta }|\le BC^n n!r_z^{-|\beta |}\) on a chosen compact base. Define

\[
\begin{gathered}
H_LP=h\circ P,\\
H_RP=P\circ h,\\
q_{0,\beta}^{L}=q_{0,\beta}^{R}=0.
\end{gathered}
\tag{NL.2}
\]

The exact §5.19 Leibniz rule gives, for \(r\ge 1\),

\[
\begin{gathered}
q^L_{r,\beta}=\sum_{n+a+1=r}\\
(-1)^a\partial_t^a p_{n,\beta},\\
q^R_{r,\beta}=p_{r-1,\beta}.
\end{gathered}
\tag{NL.3}
\]

Indeed \(\partial _\tau ^a h=(-1)^a a!h^{a+1}\); its paired base derivative is \(\partial _t^a\) and the \(a!\) cancels the Leibniz denominator. No \(\xi\) derivative of \(h\) occurs. Right multiplication by \(h\) has no base derivative correction. In particular

\[
 q^L_{r,\beta}+\partial_tq^L_{r-1,\beta}=p_{r-1,\beta}.
\tag{NL.4}
\]

Choose a base Cauchy margin \(\rho >0\). Then \(|\partial _t^a p_{n,\beta }|\le B C^n n!a!\rho ^{-a}r_z^{-|\beta |}\). Since \(n+a=r-1\) and \(n!a!\le (r-1)!\), the nonnegative binomial expansion bounds (NL.3) by

\[
\begin{gathered}
|q^L_{r,\beta}|\le B\\
(C+\rho^{-1})^{r-1}(r-1)!r_z^{-|\beta|}.
\end{gathered}
\tag{NL.5}
\]

Thus both shifts are actual order-zero symbols with one common spatial radius; both have zero index-zero coefficient. A fixed finite list of shifts and finite coefficient derivatives uses one smaller common base, one common spatial radius, and one larger time-growth constant. Shrink the time-difference disc once for that finite list. These choices precede all subsequent infinite sums.

In the actual finite-order ring, \(\tau =h^{-1}\). The exact rule gives

\[
\begin{gathered}
\tau\circ Q=\tau Q+\partial_tQ,\\
Q\circ\tau=\tau Q.
\end{gathered}
\tag{NL.6}
\]

Together with (NL.4) this proves \(\tau \circ H_LP=P\). The right identity \(H_RP\circ \tau =P\) follows immediately. The inverse and normality assertions in §5.22.1 hold in the actual common-domain factorial class, with the explicit Cauchy bounds stated there. We use that actual localization, not a formal completion.

#### NL.2 The supported principal-log derivative correction

In one normal variable \(w=a+ib\), let \(\log w\) be principal. Its upper-minus-lower jump at \(a<0\) is \(2\pi i\). Off the cut \(\partial _w \log w=1/w\) and \(\partial _{\bar w}\log w=0\). Integration by parts in the two half-planes gives the additional boundary terms

\[
\begin{gathered}
\partial_w\log w=\frac1w\\
+\pi\mathbf1_{a<0}\delta(b),\\
\partial_{\bar w}\log w\\
=-\pi\mathbf1_{a<0}\delta(b).
\end{gathered}
\tag{NL.7}
\]

The sign in the first formula is \((-i/2)(2\pi i)=+\pi\); the sign in the second is \((i/2)(2\pi i)=-\pi\). A deleted circle of radius \(\epsilon\) contributes \(O(\epsilon (1+|\log \epsilon |))\) and hence no origin point mass. The coefficient \(1/w\) is its literal locally integrable distribution. This proves both formulas by tests, including their endpoint behavior.

Put

\[
\begin{gathered}
L(w)=-\frac{\log w}{2\pi i},\\
C(w)=\frac1{2\pi i w},\\
B_{\log}(w)=\frac1{2i}\mathbf1_{a<0}\delta(b).
\end{gathered}
\tag{NL.8}
\]

Since \(D_{\mathrm{out}}=\partial _t|_s=-\partial _w\) and \(D_{\mathrm{in}}^{-}=-\partial _s=-\partial _w\) on this coefficient,

\[
\begin{gathered}
D_{\rm out}L=D_{\rm in}^{-}L=C+B_{\log},\\
D_{\rm out}(\delta L)-\delta C\\
=D_{\rm in}^{-}(\delta L)-\delta C\\
=\delta B_{\log}.
\end{gathered}
\tag{NL.9}
\]

\(B_{\log}\) is supported on the negative ray and is an order-zero measure coefficient. Therefore the difference in (NL.9) is the literal relative boundary \(d(B_{\log},0)\). Suppressing \(B_{\log}\) would be incorrect as an identity of currents.

For \(d\) spatial coordinates let \(C_j=(2\pi i\zeta _j)^{-1}\) and \(S_j=\delta C_j\), with their positive Cauchy normalization proved in R.2 and U.6. Let \(r_N=(-1)^{N(N-1)/2}\). The calibrated line-current representatives of \(h\) and \(1\) are

\[
\begin{gathered}
S_h=r_N\,\delta L\\
\wedge S_1\wedge\cdots\wedge S_d\otimes\eta_{\rm in},\\
I=r_N\,\delta C\\
\wedge S_1\wedge\cdots\wedge S_d\otimes\eta_{\rm in}.
\end{gathered}
\tag{NL.10}
\]

All independent distribution tensors exist by V.2. Their full differential forms contain \(d\bar w=d\bar s-d\bar t\) and \(d\bar \zeta _j=d\bar y_j-d\bar x_j\). Define the explicit degree-\((N-1)\) primitive

\[
\begin{gathered}
\mathcal B_{\log}=r_N B_{\log}\\
S_1\wedge\cdots\wedge S_d\otimes\eta_{\rm in}.
\end{gathered}
\tag{NL.11}
\]

It is supported in the line cone \(G_0\) and

\[
\begin{gathered}
D_{\rm out}S_h-I\\
=D_{\rm in}^{-}S_h-I=\delta\mathcal B_{\log},\\
(D_{\rm out}S_h-I,0)\\
=d(\mathcal B_{\log},0).
\end{gathered}
\tag{NL.12}
\]

The same equation holds after support enlargement to E.1's thick cone. The ordinary top complement coefficients of these line currents agree with the calibrated coefficients for \(h\) and \(1\) in E.2 by V.15–V.17. The relative resolution comparison thus identifies their classes with \(\kappa _0(h)\), \(\kappa _0(1)\). This comparison uses the exact cone/cube maps and the explicit V.4 telescoping homotopy, rather than a solution action.

#### NL.3 The spatial face left by a time derivative

Let \(c_N=(2\pi i)^{-N}\), \(S_\beta =\beta !/\prod _j\zeta _j^{\beta _j+1}\), and

\[
\begin{gathered}
M_P=\frac{p_{0,0}(u)}{\prod_j\zeta_j},\\
A_P=\sum_{n+|\beta|\ge1}p_{n,\beta}S_\beta\\
\frac{(-w)^{n+|\beta|-1}}{(n+|\beta|-1)!}.
\end{gathered}
\tag{NL.13}
\]

The accepted coefficient is \(K_P=c_N(M_P/w-\log w A_P)\). For \(Q=H_LP\) or \(H_RP\), \(q_{0,\beta }=0\), so \(K_Q=-c_N \log w A_Q\). Use respectively

\[
\begin{gathered}
D_L=\partial_t|_{s,y}\\
=\partial_t|_{w,\zeta}-\partial_w,\\
D_R=-\partial_s=-\partial_w.
\end{gathered}
\tag{NL.14}
\]

For \(D_L\), the coefficient of \((-w)^{n+|\beta |-1}/(n+|\beta |-1)!\) in \(D_LA_Q\) is \(q^L_{n+1,\beta }+\partial _tq^L_{n,\beta }=p_{n,\beta }\). For \(D_R\) it is \(q^R_{n+1,\beta }=p_{n,\beta }\). The excluded \(n=|\beta |=0\) term is constant before the normal derivative and contributes nothing. Normal convergence and the fixed derivative bounds in E.1 and (NL.5) justify this equality of actual holomorphic coefficients. Hence, for either ordered pair \((Q,D)\),

\[
\begin{gathered}
DA_Q=A_P,\\
DK_Q-K_P=R_{Q,P},\\
R_{Q,P}=c_N\frac{A_Q-M_P}{w}.
\end{gathered}
\tag{NL.15}
\]

\(R_{Q,P}\) is holomorphic on the spatial face \(V=U_1\cap \cdots \cap U_d\), including across the logarithm's time cut. For \(d=0\), \(V\) is the whole local normal space. To see the removable \(w\) factor termwise, \(q_{1,0}=p_{0,0}\); this is exactly \(A_Q\)'s single term with exponent zero. Every remaining term has \(r+|\beta |\ge 2\) and therefore contains a factor \(w\). After division the series is

\[
\begin{gathered}
R_{Q,P}=-c_N\\
\sum_{r+|\beta|\ge2}q_{r,\beta}S_\beta\\
\frac{(-w)^{r+|\beta|-2}}{(r+|\beta|-1)!}.
\end{gathered}
\tag{NL.16}
\]

The sum runs over \(r\ge 1\). In particular it has no logarithm and no negative power of \(w\).

Choose \(|\zeta _j|>a_j|w|\) with \(a_j>r_z^{-1}\), and \(C_Q|w|<1/2\). For \(r\ge 2\), (E.5) bounds the sum by a fixed multiple of

\[
\begin{gathered}
\prod_j|\zeta_j|^{-1} C_Q^2\\
\sum_{r'\ge0,\beta}(r'+2+|\beta|)\\
(C_Q|w|)^{r'}\prod_j\theta_j^{\beta_j},\\
\theta_j=\frac{|w|}{r_z|\zeta_j|}.
\end{gathered}
\]

For \(r=1\) and \(|\beta |\ge 1\), use \(\beta !/|\beta |!\le 1\). The sum is bounded by a constant times \(\prod |\zeta _j|^{-1}[(\prod (1-\theta _j)^{-1}-1)/|w|]\). Expanding the product one factor at a time bounds the bracket by a constant times \(\sum _j(r_z|\zeta _j|)^{-1}\) on compact angular subsets of \(V\). This expression remains bounded as \(w\to 0\) when the spatial denominators stay separated. These are geometric majorants for (NL.16), including all derivatives on smaller compacts, so the assertion of holomorphy on \(V\) is literal.

Write \(r=|(w,\zeta )|\). On any compact angular subset of \(V\) and a smaller output base,

\[
 |R_{Q,P}|\le C r^{-N}.
\tag{NL.17}
\]

On compact angular subsets of the full intersection, \(Q\)'s missing index-zero coefficient gives the stronger bound

\[
\begin{gathered}
|K_Q|\le C r^{1-N}\\
(1+|\log r|).
\end{gathered}
\tag{NL.18}
\]

The factor \(C_Q|w|\) in E.6's \(r\ge 1\) sum supplies the additional \(r\) in (NL.18). The bounds are uniform in actual spatial and time tails, on the chosen fixed domains.

#### NL.4 A literal supported primitive for both shifts

Use a single E.2 homogeneous partition \(\psi _0,\cdots ,\psi _d,\psi _G\) and the same thick cone \(G\) for the finite list under consideration. Write \(\omega =\omega _\psi\) and \(\lambda _N=r_N(N-1)!\). Then \(F_P=\lambda _NK_P\omega\) and \(T_P=\delta F_P\).

Assume first \(N\ge 2\). Put

\[
\begin{gathered}
\eta_\psi=\psi_1\bar\partial\psi_2\wedge\cdots\wedge\bar\partial\psi_d\\
(N=2:\ \eta_\psi=\psi_1),
\end{gathered}
\tag{NL.19}
\]

and define the degree-\((N-2)\) transgression

\[
\begin{gathered}
\Theta_D=\sum_{0\le i<j\le d}(-1)^{i+j+1}\\
(\psi_iD\psi_j-\psi_jD\psi_i)\\
\bar\partial\psi_0\wedge\cdots\wedge\\
\widehat{\bar\partial\psi_i}\wedge\cdots\wedge\\
\widehat{\bar\partial\psi_j}\wedge\cdots\wedge\bar\partial\psi_d.
\end{gathered}
\tag{NL.20}
\]

The wedge keeps its original increasing order. \(\eta _\psi\) is supported compactly in the spatial face \(V\) on the sphere. Every summand of \(\Theta _D\) is supported compactly in the full intersection: its two undifferentiated or \(D\)-differentiated indices and its \(N-2\) remaining differentiated indices cover all \(N\) indices. Therefore \(R\eta _\psi\) and \(K_Q\Theta _D\) extend by zero smoothly on the punctured normal space on their respective stated domains. The latter never evaluates \(\log w\) on its cut.

On the complement of \(G\), \(\sum \psi _i=1\), \(\sum D\psi _i=0\), \(\sum \delta \psi _i=0\). There \(\omega =\delta \psi _1\wedge \cdots \wedge \delta \psi _d\), whence \(\delta \eta _\psi =\omega\). Substituting \(\psi _0=1-\sum _{i\ge 1}\psi _i\), \(D\psi _0=-\sum _{i\ge 1}D\psi _i\), and \(\delta \psi _0=-\sum _{i\ge 1}\delta \psi _i\) into (NL.20) gives

\[
\begin{gathered}
\Theta_D=\sum_{j=1}^d(-1)^{j-1}D\psi_j\\
\bar\partial\psi_1\wedge\cdots\wedge\\
\widehat{\bar\partial\psi_j}\wedge\cdots\wedge\bar\partial\psi_d.
\end{gathered}
\tag{NL.21}
\]

To verify the substitution, fix the wedge omitting a given \(j\). The terms with either omitted index zero supply the coefficients involving \(\psi _0\) and \(\psi _j\); the remaining terms replace their \(\delta \psi _0\) by the unique nonrepeating \(\delta \psi\) factor. Their signed coefficients add to \((\sum \psi _i)D\psi _j-\psi _j\sum D\psi _i=D\psi _j\). Reordering to the original increasing wedge gives exactly \((-1)^{j-1}\). This checks every coefficient of (NL.21). Differentiating (NL.21) gives \(\delta \Theta _D=D\omega\), because \(D\) commutes with \(\delta\). Thus both forms \(\omega -\delta \eta _\psi\) and \(D\omega -\delta \Theta _D\) vanish on the complement of \(G\).

Define

\[
\begin{gathered}
\mathcal B_{Q,P,D}=\lambda_N\\
\{\begin{gathered}R_{Q,P}(\omega-\delta\eta_\psi)\\+K_Q(D\omega-\delta\Theta_D)\end{gathered}\}\\
\otimes\eta_{\rm in}.
\end{gathered}
\tag{NL.22}
\]

This is a literal supported degree-\((N-1)\) current, rather than an unnamed existence assertion. Its support is contained in \(G\). By homogeneity, \(|\eta _\psi |\le Cr^{-(N-2)}\), \(|\Theta _D|\le Cr^{-(N-1)}\), \(|\omega |\le Cr^{-(N-1)}\), and \(|D\omega |+|\delta \Theta _D|\le Cr^{-N}\). With (NL.17)–(NL.18), every coefficient of (NL.22) is bounded by \(Cr^{-(2N-1)}(1+|\log r|)\), hence is locally integrable exactly as in E.3. It has local order zero; its \(\delta\) derivative has local normal order at most one.

Also define

\[
\begin{gathered}
\mathcal H_{Q,P,D}=\lambda_N\\
(R_{Q,P}\eta_\psi+K_Q\Theta_D)\otimes\eta_{\rm in}.
\end{gathered}
\tag{NL.23}
\]

It has coefficient bound \(Cr^{-(2N-2)}(1+|\log r|)\). A deleted normal sphere has area proportional to \(r^{2N-1}\); its boundary term is consequently \(O(r(1+|\log r|))\to 0\). The classical derivative away from zero is locally integrable with bound \(Cr^{-(2N-1)}(1+|\log r|)\). Integration by parts therefore identifies its full distributional \(\delta\) with that classical derivative and introduces no origin correction. Since \(R\) and \(K_Q\) are holomorphic wherever their cutoffs occur,

\[
\begin{gathered}
DF_Q-F_P\\
=\mathcal B_{Q,P,D}+\delta\mathcal H_{Q,P,D}.
\end{gathered}
\tag{NL.24}
\]

Here \(DF_Q\) is also literal locally integrable: \(F_Q\) has the weaker singularity \(Cr^{-(2N-2)}(1+|\log r|)\), and the same deleted-sphere argument identifies its first derivative. The frame and normal differential basis are constant under the linear input/output coordinate transformation. All coefficients and all output antiholomorphic components are retained. Applying \(\delta\) to (NL.24) proves

\[
\begin{gathered}
DT_Q-T_P=\delta\mathcal B_{Q,P,D},\\
(DT_Q-T_P,0)\\
=d(\mathcal B_{Q,P,D},0).
\end{gathered}
\tag{NL.25}
\]

For \(N=1\), \(R\) is holomorphic on the full local product. Set

\[
\begin{gathered}
\mathcal B_{Q,P,D}=\\
\{\begin{gathered}R_{Q,P}(\psi_0-1)\\+K_QD\psi_0\end{gathered}\}\otimes ds.
\end{gathered}
\tag{NL.26}
\]

The first term is supported in \(G\) because \(\psi _0=1\) outside \(G\); the second is supported there because \(D\psi _0=0\) outside \(G\). Both are locally integrable. In this case \(DF_Q-F_P= B+R\otimes ds\), and \(\delta R=0\) on the full neighbourhood. Equation (NL.25) follows directly. This treats \(d=0\) without introducing a fictitious negative-degree form.

Taking classes in (NL.25) proves both actual canonical shift identities

\[
\begin{gathered}
D_{\rm out}\kappa_0(h\circ P)=\kappa_0(P),\\
-\partial_s\kappa_0(P\circ h)=\kappa_0(P).
\end{gathered}
\tag{NL.27}
\]

The spatial face \(R\) and the cutoff correction both remain explicitly in the supported primitive. No exactness argument is passed through a cohomology quotient limit.

#### NL.5 The canonical unit and products with the normal generator

The class of \(I\) in (NL.10) is the normalized diagonal unit. For the canonical cup followed by trace, V.13 converts the coefficient cup and shifted trace to the positive full-block distribution convolution. Tensor the independent normal variables \(a,b\) and apply \(I\) in the \(a\) variables. Its normalized delta evaluation sets the corresponding intermediate point to the output, and leaves \(T_P(u,b)\) with scalar one. On the other side it sets the intermediate point to the input and again leaves \(T_P\). This is an equality on arbitrary compact smooth tests, by the distribution tensor construction and the positive R.2 Cauchy normalization; it does not require holomorphic solution tests. The diagonal has a one-point proper fibre, so the controlled proper roof is available. Thus

\[
\begin{gathered}
{}[I]*\kappa_0(P)\\
=\kappa_0(P)*[I]=\kappa_0(P).
\end{gathered}
\tag{NL.28}
\]

Let \(J=D_{\mathrm{out}} I=-\partial _s I\), the normalized time derivative of the diagonal unit. Differentiating that same test evaluation gives

\[
\begin{gathered}
{}[J]*\kappa_0(P)=D_{\rm out}\kappa_0(P),\\
\kappa_0(P)*[J]=-\partial_s\kappa_0(P).
\end{gathered}
\tag{NL.29}
\]

For the right formula the delta derivative is transposed against the intermediate/input variable and gives the displayed minus sign. The input holomorphic frame is constant, and proper compact support permits the transposition without a boundary. For the left formula the surviving derivative is the derivative in the final output variable with input held fixed. The smooth parameter bounds of corrected E.13 justify every compact derivative of the full current families in these evaluations.

Equation (NL.12) now gives the two-sided inverse identities with their complete literal primitives:

\[
\begin{gathered}
{}[J]*\kappa_0(h)=[I]\\
=\kappa_0(h)*[J],\\
\kappa_0(1)=[I],\quad\kappa(\partial_t)=[J].
\end{gathered}
\tag{NL.30}
\]

Associativity here is the canonical current operation, not an asserted full-symbol product identity. For three factors use three independent normal variable sets, tensor their finite-order distributions with their smooth parameter coefficients, and the E.1 bound on all intermediate nonnegative time lengths. Over a compact endpoint set their support is compact inside the chosen outer source. V.2's repeated evaluation and V.4–V.5's ordered trace signs make the two repeated traces agree on tests. This proves the required associativity for the germ operation on these full current classes. No multiplication formula for general symbol pairs enters this argument.

Multiply the first identity in (NL.27) on the left by \(\kappa _0(h)\), using (NL.29)–(NL.30); multiply the second on the right. Associativity and the unit give

\[
\begin{gathered}
\kappa_0(h\circ P)\\
=\kappa_0(h)*\kappa_0(P),\\
\kappa_0(P\circ h)\\
=\kappa_0(P)*\kappa_0(h).
\end{gathered}
\tag{NL.31}
\]

These are special canonical products with the normal generator, proved for the full convergent spatial class. General \(P,Q\) products remain open.

For completeness, at fixed normal coordinates \((\partial _t|_{s,y}+\partial _s)=\partial _t|_{w,\zeta }\), so differentiation of \(F_P\) gives the exact current identity

\[
\begin{gathered}
D_{\rm out}T_P+\partial_sT_P\\
=T_{\partial_t^{\rm coeff}P}.
\end{gathered}
\tag{NL.32}
\]

The right member differentiates only the holomorphic base coefficients \(p_{n,\beta }\); the angular partition is fixed. This proves compatibility with (NL.6), including the genuine time contraction. It also supplies the right-normal identity when a right positive time factor is represented by a left presentation.

#### NL.6 Independence of finite time-order presentations

For an actual finite-order symbol \(P\) choose a finite left presentation

\[
\begin{gathered}
P=\tau^k\circ P_0,\\
k\ge0,\quad P_0\in E_0,\\
\kappa(P):=D_{\rm out}^{\,k}\kappa_0(P_0).
\end{gathered}
\tag{NL.33}
\]

The class is represented by the complete current \(D_{\mathrm{out}}^kT_{P_0}\), of common normal distribution order at most \(k+1\) on each chosen compact, by corrected E.13. Suppose also \(P=\tau ^l\circ Q_0\) and \(l\ge k\). Actual multiplication by \(h^l\) on the left and the actual inverse relation give \(Q_0=H_L^{l-k}P_0\). Put \(r=l-k\) and \(P_j=H_L^jP_0\). Choose one common domain, cone and angular partition for this finite list as in NL.1. Equation (NL.25) supplies \(B_j=B_{P_j,P_{j-1},D_{\mathrm{out}}}\) with

\[
 D_{\rm out}T_{P_j}-T_{P_{j-1}}=\delta B_j.
\]

The exact finite telescoping identity is

\[
\begin{gathered}
D_{\rm out}^{r}T_{P_r}-T_{P_0}\\
=\delta\left(\sum_{j=1}^{r}D_{\rm out}^{j-1}B_j\right).
\end{gathered}
\tag{NL.34}
\]

Expansion verifies it: consecutive terms \(D^{j}T_{P_j}\) and \(-D^{j-1}T_{P_{j-1}}\) cancel. After applying \(D_{\mathrm{out}}^k\), the difference between the two representatives in (NL.33) is therefore the literal relative boundary

\[
\begin{gathered}
d\left(\begin{gathered}D_{\rm out}^{k}\sum_{j=1}^{r}\\D_{\rm out}^{j-1}B_j,\ 0\end{gathered}\right).
\end{gathered}
\tag{NL.35}
\]

Every derivative preserves support in \(G\). The primitive is a finite-order current of normal order at most \(l-1\), since each \(B_j\) is locally integrable and \(j-1+k\le l-1\). A later two-kernel proper trace uses its intersection with the fixed endpoint support, whose properness is the E.1 bound. The case \(k\ge l\) is obtained by exchanging the presentations. This proves independence, including equality of actual full-current classes, without assuming a closed-range theorem or faithful solution action.

Linearity is immediate on a fixed common domain and partition, and different presentation orders may be raised to the same finite order by (NL.34). Thus (NL.33) is a well-defined complex-linear germ class map extending \(\kappa _0\). Equations (NL.29), (NL.31) and (NL.32) give its left and right normal-generator compatibility. Consequently the actual normal conjugation of §5.22.1 is carried to conjugation by the canonical two-sided inverse pair \([J],\kappa _0(h)\). This is the normal-localization interface needed to extend a future order-zero ring proof.

#### NL.7 Germ restriction and all auxiliary choices

For fixed coefficient representatives, cone, cover and partition, every displayed construction restricts literally on a smaller common base and normal neighbourhood. The Borel kernels converge normally and the primitives are actual locally integrable or finite-order currents. Distributional differentiation commutes with restriction. If two coefficient representatives define the same actual germ, their coefficient holomorphic functions agree on one smaller common connected coefficient domain for every \(n\); the identity theorem as used in §5.19 makes that one domain valid for the whole sequence. Their common Taylor coefficients and the actual normally convergent kernels then agree there.

Partition independence also has an explicit supported primitive. On one common cover and cone, let \(\psi (q)=(1-q)\psi +q\phi\), \(0\le q\le 1\), including the \(G\)-member. Use (NL.20) with \(D=\partial _q\) and call it \(\Theta _q\). On the complement, \(\sum \psi (q)=1\) and \(\sum \partial _q\psi (q)=0\), so the same direct calculation gives \(\partial _q\omega _q=\delta \Theta _q\). For \(N\ge 2\) put

\[
\begin{gathered}
B_{\psi,\phi}=\lambda_N\int_0^1\\
K_P(\partial_q\omega_q-\delta\Theta_q)\,dq\\
\otimes\eta_{\rm in},\\
H_{\psi,\phi}=\lambda_N\int_0^1K_P\Theta_q\,dq\\
\otimes\eta_{\rm in}.
\end{gathered}
\tag{NL.36}
\]

The first current is supported in \(G\). Here the \(q\)-variation has degree zero in normal coordinates, so \(\Theta _q\) has bound \(Cr^{-(N-2)}\). Thus \(H\) has coefficient bound \(Cr^{-(2N-2)}(1+|\log r|)\), while \(B\) has the integrable bound \(Cr^{-(2N-1)}(1+|\log r|)\). The deleted-sphere estimate used in NL.4 again removes any origin term, and \(F_\phi -F_\psi =B+\delta H\). Therefore

\[
\begin{gathered}
T_\phi-T_\psi=\delta B_{\psi,\phi},\\
(T_\phi-T_\psi,0)=d(B_{\psi,\phi},0).
\end{gathered}
\tag{NL.37}
\]

For \(N=1\) set \(B_{\psi ,\phi }=K_P(\phi _0-\psi _0)\otimes ds\); it is supported in \(G\) and gives the same literal boundary. This also proves partition independence for all finite output derivatives by transposition of the finite derivatives of (NL.37).

For two initially different cones choose \(\epsilon _*\) and \(A_{*,j}\) strictly larger than both original sets of constants. The resulting \(G_*\) is pointed and contains both angular closed cone sets in its interior. Both \(G\)-cutoffs are compactly supported in \(\operatorname{int} G_*\) on the sphere. Both families \(\psi _i\) are subordinate to the wider opens \(|\zeta _j|>\min(a_j,a'_j)|w|\) and the same principal slit. For two scalar spatial radii representing the same germ, the larger radius is available on a smaller common base from its original representative, and \(\min(a_j,a'_j)>1/\max(r_z,r'_z)\). Thus \(K_P\) converges on this wider common full intersection. Use the larger of the finitely many time growth constants and shrink the time disc once. The convex partition path above is now legitimate on the common wider cover and \(G_*\), and (NL.37) proves equality after support enlargement. Equivalently the calibrated finite complement cubes refine to the same top holomorphic coefficient; both descriptions use the exact relative maps of R/V.

This proves actual germ restriction and auxiliary-domain, cone and partition independence in the common supported kernel germ target. It is not an assertion that every symbol has one fixed domain or fixed cone at once. All finite localization presentations and their restriction maps agree after one common shrink and, when needed, one common support enlargement.

The complete normal-localization interface is proved. Once the full order-zero cup/product equality is proved, (NL.30)–(NL.35) provide the actual canonical extension to the full finite-order ring by its normal localization. Until then \(\kappa\) is a well-defined canonical linear class map with proved normal-generator compatibility; general multiplicativity is not asserted. An unrestricted-input action still requires the common-domain contour, smooth-input relative augmentation, proper collar and full-output primitive. Infinite-order, propagation, separation, finite \(D\)-type, finite-pole, intrinsic-order, C1 and arbitrary analytic proper-regularity obligations remain outside this proof.

![Principal-log ray correction and the exact canonical localization square](assets/localization-correction.png)

*Exact upper and lower principal-log limits at the normal point \(w=-1\) are \(+i\pi\) and \(-i\pi\). Their jump creates the retained ray coefficient \(B_{\log}=(2i)^{-1}1_{a<0}\delta (b)\); the open endpoint marks absence of an origin point mass, while its closed distributional support includes zero. The approach arrows are schematic. The right square is the actual identity \(\partial _t\circ (h\circ P)=P\) and its canonical class identity, proved through the supported relative primitive (NL.22)–(NL.27). The figure is an original exact-limit and class-map illustration; it supplies no numerical distribution proof. Reproducible source and coordinate data are retained. Proof locators and free human-source context appear in the figure.*


The figure has a full-resolution PNG, SVG, editable plotting source and coordinate data. Its exact proof locators are NL.2, (NL.7)–(NL.12), and NL.4, (NL.22)–(NL.27). The free human comparison sources are linked above.


### 5.40. Ordinary degree-one evaluation for actual convergent spatial symbols

Let \(p=(0;dt)\), with base coordinates \(u=(t,x)\in\mathbf C\times\mathbf C^d\), and put \(h=\tau^{-1}\), \(z=\xi/\tau\). We prove that every actual order-zero symbol defines a complex-linear endomorphism of the full directional degree-one holomorphic quotient, by an ordinary integral formula containing all spatial Taylor terms. The formula is independent of its anchor, domains and representatives. The ordinary integral operator for \(h\) has the ordinary time derivative as its two-sided inverse on that quotient.

The actual coefficient class is [§5.19.1, (5.19a)–(5.19b)](#5-19-1-the-chart-and-the-convergence-class); its absolute Taylor norm and coefficient estimate are [§5.21.1, (5.21b)–(5.21c)](#5-21-1-an-absolute-taylor-version-of-the-product-norm). The actual normal element is [§5.22.1, (5.22a)–(5.22b)](#5-22-1-the-actual-normal-element). For the degree-one relative convention, see [§5.34, R.1, R.3 and R.6–R.7](#5-34-full-current-resolutions-and-controlled-proper-image-comparison) and [§5.35, V.1](#5-35-relative-cones-unshifted-cup-and-the-normalized-trace). The positive one-variable orientation agrees with [§5.36, U.2, (U.5)](#5-36-canonical-finite-polynomial-line-cone-kernels). Comparison with the canonical cup and proper trace, and multiplication for general symbol pairs, require the further arguments given in the later sections.

#### PB.1. The full support quotient and cofinal convex complements

Write \(t=T+iT'\), \(v=(T',x)\in\mathbf R^{2d+1}\), using the Euclidean norm. A closed support germ \(Z\) at zero is admissible if every limiting direction of \(Z\) at zero satisfies \(T\ge0\). Define

\[
\begin{gathered}
C_p=\underset{Z,V}{\operatorname{colim}}\;\\
\mathcal O(V\setminus Z)/\mathcal O(V).
\end{gathered}
\tag{PB.1}
\]

The maps are restriction to smaller neighbourhoods and enlargement of the support. The union of two admissible closed germs is admissible: every sequence in their union has a subsequence in one member, so every limiting direction belongs to one of the two allowed sets. Intersecting the neighbourhoods and taking that union therefore makes the system filtered.

Every admissible support can be enlarged to one with a convex complement. First shrink a ball so that its points in \(Z\) with \(T<0\) satisfy \(-T\le |v|\). Otherwise a sequence tending to zero with \(-T>|v|\) has a normalized subsequential direction with strictly negative \(T\), contradicting admissibility. In particular there is no such point with \(v=0\). For sufficiently small \(r\), set

\[
\begin{gathered}
e(r)=\sup\\
\left(\begin{gathered}\{0\}\cup\\\left\{\begin{gathered}-T/|v|:(T,v)\in Z,\\T<0,\ 0<|v|\le r\end{gathered}\right\}\end{gathered}\right),\\
g(r)=\int_0^{2r}e(s)\,ds.
\end{gathered}
\tag{PB.2}
\]

The radius is small enough that \(e\) is defined up to \(2r\). It is nondecreasing and bounded by one. If it did not tend to zero, choose points with \(|v|\to0\) and \(-T/|v|\) bounded below by a fixed positive number. The preceding bound forces \(T\to0\), and a normalized subsequence again has strictly negative \(T\). Thus \(e(r)\to0\).

The integral makes \(g\) continuous, nondecreasing and convex, with \(g(0)=0\). Monotonicity gives

\[
\begin{gathered}
r e(r)\le g(r)\le 2r e(2r),\\
g(r)=o(r).
\end{gathered}
\tag{PB.3}
\]

Its one-sided slopes on a small interval are bounded by \(2e(2r)\) at that interval's outer radius. They can consequently be made arbitrarily small. The radial function \(v\mapsto g(|v|)\) is convex: use the triangle inequality, then monotonicity of \(g\), then its one-variable convexity. Hence

\[
\begin{gathered}
\Omega_g=\{T<-g(|v|)\},\\
Z_g=\{T\ge-g(|v|)\}
\end{gathered}
\tag{PB.4}
\]

are respectively convex open and closed. For a point of \(Z\) with \(T<0\), (PB.3) gives \(-T\le |v|e(|v|)\le g(|v|)\); points with \(T\ge0\) belong to \(Z_g\) too. Thus \(Z\subset Z_g\) as germs. Also \(Z_g\) is admissible: its negative \(T\)-coordinates satisfy \(-T\le g(|v|)=o(|v|)\), so every normalized limiting direction has \(T\ge0\). Intersecting \(\Omega_g\) with a smaller Euclidean ball preserves convexity. These complements form a cofinal family for every admissible support, rather than a single fixed wedge.

For any chosen small \(L>0\), restrict \(g\) to a radius at which its slopes are at most \(L\), and extend it beyond that radius by the linear ray of slope \(L\). The extension is convex, nondecreasing and globally \(L\)-Lipschitz; it has the same \(o(r)\) germ. It satisfies \(g(r)\le Lr\). This supplies a convenient representative whenever a global Lipschitz notation is used below; all evaluations of the original input function remain inside its given small ball.

The quotient (PB.1) is exactly the directional colimit of degree-one local support groups. Indeed the support long exact sequence contains

\[
\begin{gathered}
\mathcal O(V)\longrightarrow\mathcal O(V\setminus Z)\\
\longrightarrow H_Z^1(V,\mathcal O)\longrightarrow H^1(V,\mathcal O).
\end{gathered}
\]

The colimit of the last groups over neighbourhoods of zero is zero. For an injective resolution \(I^\bullet\) of \(\mathcal O\), the colimit of \(\Gamma(V,I^\bullet)\) is its stalk complex \(I^\bullet_0\); exactness of filtered colimits gives \(H^1(I^\bullet_0)=0\). Taking the colimit in the displayed exact sequence gives the asserted quotient, and then the filtered colimit over admissible supports gives (PB.1). A holomorphic section supported in an admissible \(Z\) is zero: it vanishes on a nonempty negative-direction open subset, and the identity theorem applies on a connected small ball. In the relative convention \(d(a,b)=(\bar\partial a,ra-\bar\partial b)\) of §5.35, V.1, a quotient representative \(f\) corresponds to \((0,-f)\). This degree-one identification uses no vanishing assertion for higher directional cohomology or for a fixed arbitrary neighbourhood.

#### PB.2. Actual coefficients and the order of choices

Write \(E_0\) for the actual order-zero class of §5.19. An element has coefficient functions on one common base and cotangent neighbourhood:

\[
\begin{gathered}
P=\sum_{n\ge0}h^n p_n(t,x,z),\\
p_n(t,x,z)=\sum_{\beta\in\mathbf N^d}p_{n,\beta}(t,x)z^\beta.
\end{gathered}
\]

Choose a closed base polydisc strictly inside that common neighbourhood and a cotangent polyradius \(r_z>0\) whose closed torus stays inside the coefficient domain. The factorial estimate of §5.19.1 and the Cauchy coefficient formula on that torus give

\[
\begin{gathered}
|p_{n,\beta}(t,x)|\le B C^n n! r_z^{-|\beta|}\\
(n\ge0,\ \beta\in\mathbf N^d)
\end{gathered}
\tag{PB.5}
\]

on the chosen base. Thus all functions \(p_{n,\beta}\) are defined on that same base neighbourhood. When \(d=0\), \(\beta=0\) is the only index, and \(r_z\) can be any positive constant for the following notation. This is a symbol expansion with factorial growth; it makes no assertion of numerical convergence of the series at a nonzero value of \(h\).

Let \(f\) be holomorphic on \(\Omega_g\cap B_R\), with this ball inside the coefficient base. Put

\[
\begin{gathered}
D=\max(1,d),\qquad\kappa=4/r_z,\\
b=R/(16\sqrt D).
\end{gathered}
\]

Choose \(a_0>0\), and then the small Lipschitz bound \(L\) as above, so that

\[
\begin{gathered}
\kappa(5a_0/4)<b/2,\\
C(5a_0/4)<1/2,\\
a_0<R/16,\quad L\le1/8,\\
L\sqrt D\,\kappa\le1/8,\quad LR<a_0/8.
\end{gathered}
\tag{PB.6}
\]

Here is a noncircular choice. Fix \(B,C,r_z,\kappa\) first. Choose a positive ratio \(\lambda=a_0/R\) with \(\lambda<1/16\) and \(5\kappa\lambda/4<1/(32\sqrt D)\). Require \(L<\min(1/8,1/(8\sqrt D\kappa),\lambda/8)\). Since the small-radius slopes of \(g\) tend to zero, shrink \(R\) until this requirement holds and extend \(g\) by slope \(L\) if needed. Shrink \(R\) further to arrange \(5C\lambda R/4<1/2\). Set \(a_0=\lambda R\). Every inequality in (PB.6) now holds. Fix the negative real anchor \(a=-a_0\).

#### PB.3. One domain for every spatial derivative

The target consists of points \((t,x)\in\Omega_g\) with

\[
 |t|<a_0/4,\qquad |x_j|<R/(8\sqrt D).
\]

At such a point define

\[
\begin{gathered}
\mu=-\operatorname{Re}t\\
-g(|(\operatorname{Im}t,x)|)>0,\\
\delta_t=\min\{b/2,\mu/(4L\sqrt D)\},\\
\delta_s=\delta_t+\kappa|t-s|,
\end{gathered}
\tag{PB.7}
\]

where \(s=a+\theta(t-a)\), \(0\le\theta\le1\). If \(L=0\), take \(\delta_t=b/2\). The time segment satisfies

\[
\begin{gathered}
-\operatorname{Re}s\ge-\operatorname{Re}t+\tfrac12|t-s|,\\
|\operatorname{Im}s-\operatorname{Im}t|\le |t-s|.
\end{gathered}
\]

For the first inequality, the ratio is \((a_0+\operatorname{Re}t)/|t-a|\ge(3a_0/4)/(5a_0/4)>1/2\). For every \(y\) in the closed complex polydisc of radius \(\delta_s\) about \(x\), the change in \(v\) has norm at most \(|t-s|+\sqrt d\,\delta_s\). Consequently

\[
\begin{gathered}
-\operatorname{Re}s-g(|(\operatorname{Im}s,y)|)\\
\ge\mu+\tfrac12|t-s|-L|t-s|\\
-L\sqrt d\,\delta_s\\
\ge\tfrac34\mu+\tfrac14|t-s|>0.
\end{gathered}
\tag{PB.8}
\]

We used \(L\sqrt d\,\delta_t\le\mu/4\), \(L\le1/8\), and \(L\sqrt d\,\kappa\le1/8\). The radii are at most \(b\), because \(|t-s|\le|t-a|<5a_0/4\). Moreover \(|s|\le a_0<R/16\), \(|x|\le R/8\), and \(|y-x|\le\sqrt d\,b\le R/16\); thus \(|y|\le3R/16\) and \((s,y)\) lies strictly inside \(B_R\). The Lipschitz estimate was therefore used on points belonging to the actual input domain.

For a compact target subset \(K\), \(\mu\) has a positive minimum and \(\delta_t\) has a positive lower bound. The union of the segments and their closed polydiscs is the continuous image of a compact set; (PB.8) and the preceding ball bounds place it compactly inside \(\Omega_g\cap B_R\). Let \(M_K\) bound \(f\) on that union. Cauchy's formula gives

\[
\begin{gathered}
|\partial_x^\beta f(s,x)|\le M_K\beta!\delta_s^{-|\beta|},\\
r_s=\frac{|t-s|}{r_z\delta_s}\le\frac14.
\end{gathered}
\tag{PB.9}
\]

These estimates use a single actual domain for all \(\beta\). No growth assumption is imposed near the input boundary. For \(d=0\) the spatial polydiscs and sums are absent; the same time-segment margin applies.

#### PB.4. The complete ordinary sum and normal convergence

Define, with the positive orientation from \(a\) to \(t\),

\[
\begin{gathered}
\rho_a(P)f(t,x)=\\
p_{0,0}(t,x)f(t,x)\\
+\sum_{\substack{n\ge0,\ \beta\in\mathbf N^d\\n+|\beta|\ge1}}p_{n,\beta}(t,x)\int_a^t\\
\frac{(t-s)^{n+|\beta|-1}}{(n+|\beta|-1)!}\,\\
\partial_x^\beta f(s,x)\,ds.
\end{gathered}
\tag{PB.10}
\]

The normal monomial \(h^n z^\beta\) here means \(n+|\beta|\) ordinary time primitives followed by the ordinary \(\beta\)-th spatial derivative, because \(z=\xi/\tau\). Repeated integration gives the displayed integral kernel: induction integrates \((t-s)^{m-1}/(m-1)!\) once more and gives \((t-s)^m/m!\). The only term with total index zero is multiplication by \(p_{0,0}\).

For total index \(m=n+|\beta|\ge1\),

\[
 \frac{n!\beta!}{(n+|\beta|-1)!}\le n+|\beta|.
 \tag{PB.11}
\]

Indeed \(\beta!\le|\beta|!\), and \(n!|\beta|!\le(n+|\beta|)!\). Put \(q=C|t-s|\). By (PB.5), (PB.9) and (PB.11), the sum of absolute integrands is bounded, for \(s\ne t\), by

\[
 \frac{BM_K}{|t-s|}\sum_{n+|\beta|\ge1}(n+|\beta|)q^n r_s^{|\beta|}.
\]

The identities \(\sum_\beta r^{|\beta|}=(1-r)^{-d}\) and \(\sum_\beta|\beta|r^{|\beta|}=dr(1-r)^{-d-1}\) follow by multiplying the \(d\) geometric series and differentiating. Separating the two parts of the last sum therefore gives

\[
\begin{gathered}
BM_K\\
\left\{\begin{gathered}\frac{C}{(1-q)^2(1-r_s)^d}\\+\frac{d}{\begin{gathered}(1-q)r_z\delta_s\\(1-r_s)^{d+1}\end{gathered}}\end{gathered}\right\}\\
\le BM_K\\
\left\{\begin{gathered}4C(4/3)^d\\+\frac{2d(4/3)^{d+1}}{r_z\delta_t}\end{gathered}\right\}.
\end{gathered}
\tag{PB.12}
\]

The apparent factor \(|t-s|^{-1}\) has cancelled against \(q\) or \(r_s\). The estimate extends to the endpoint \(s=t\); individual terms have nonnegative powers of \(t-s\). In particular the \(n=0,\ |\beta|\ge1\) terms have no endpoint divergence. We have \(q<1/2\), \(r_s\le1/4\) for every term, and the right side is uniform on \(K\).

For an explicit uniform tail majorant, put \(q_0=5Ca_0/4<1/2\) and \(\delta_*=\min_K\delta_t>0\). Terms with \(n\ge1\) are bounded by
\[
 BM_K(n+|\beta|)C q_0^{\,n-1}4^{-|\beta|};
\]
terms with \(n=0,\ |\beta|\ge1\) are bounded by
\[
 BM_K\,\frac{|\beta|}{r_z\delta_*}\,4^{-(|\beta|-1)}.
\]
Both bounds are summable over their indicated indices, independently of the point and segment parameter. Multiplication by the segment length, at most \(5a_0/4\), therefore supplies a summable uniform majorant for the integrated terms. Thus (PB.10) converges absolutely and uniformly on every compact target set, including convergence of its tails.

Each summand is holomorphic. To check this for its integral, parameterize \(s=a+\theta(t-a)\): on a compact target neighbourhood the segment and the required input polydiscs lie in a fixed interior compact, so the holomorphic integrand and its finite derivatives are bounded there uniformly in \(0\le\theta\le1\). Differentiation under the parameter integral is valid. Uniform convergence of holomorphic sums, and Cauchy's formula on a smaller target compact neighbourhood, give holomorphy and every finite termwise derivative of (PB.10). The spatial and time sums use the domains fixed in PB.2–PB.3 before either sum is taken.

There is also the norm continuity needed for subsequent common-domain arguments. Fix the absolute Taylor norm \(\|P\|_{R',\epsilon}\) of §5.21.1, (5.21b), with the base polyradii containing the selected base compact and cotangent polyradii at least \(r_z\). Equation (5.21c) and the absolute Taylor coefficient bound give (PB.5) with

\[
 B=\tfrac12\|P\|_{R',\epsilon},\qquad C=2(d+1)/\epsilon^2.
\]

Writing \(\delta_*=\min_K\delta_t>0\) and \(\ell=5a_0/4\), (PB.12) implies

\[
\begin{gathered}
\sup_K|\rho_a(P)f|\le\\
\tfrac12\|P\|_{R',\epsilon}M_K\\
\left[\begin{gathered}1+\ell\\\left\{\begin{gathered}4C(4/3)^d\\+\frac{2d(4/3)^{d+1}}{r_z\delta_*}\end{gathered}\right\}\end{gathered}\right].
\end{gathered}
\tag{PB.13}
\]

All constants and input compacts are fixed independently of the symbol's tails. The coefficient norm is the weighted absolute \(\ell^1\)-norm proved in §5.21.1 by (5.21f); finite coefficient-Taylor projections consequently converge in it. Equation (PB.13) gives continuity of their ordinary evaluations on these fixed compacts. It does not assert a composition-domain estimate for two operators; that estimate is proved separately in the later action section.

#### PB.5. Anchor, extending-input and representative independence

First consider two sufficiently small negative real anchors \(a,a'\) in one fixed input domain, with \(|a|,|a'|\le a_0\), and assume \(a<a'\). For target points on which both formulas are defined, convexity and path independence of holomorphic primitives give their difference by replacing \(\int_a^t-\int_{a'}^t\) with \(\int_a^{a'}\) in (PB.10). We prove that this entire infinite difference extends to a full neighbourhood of zero.

Put \(\alpha_*=\min(|a|,|a'|)>0\), and on the fixed negative segment use spatial radius \(\delta(s)=\kappa|s|=4|s|/r_z\). Choose the full target neighbourhood with \(|t|<\min(\alpha_*/2,a_0/4)\), \(|x|<\kappa\alpha_*\), and smaller if needed for the coefficient base. Every point on these spatial polydiscs satisfies

\[
\begin{gathered}
|y|\le\kappa(1+\sqrt d)|s|,\\
g(|y|)\le L\kappa(1+\sqrt d)|s|\le |s|/4
\end{gathered}
\]

when \(d\ge1\), by \(1+\sqrt d\le2\sqrt D\) and (PB.6). Their real input time is \(s<0\), so they lie strictly in \(\Omega_g\); their time and spatial bounds also put them compactly inside \(B_R\). For \(d=0\) the negative segment is already compactly inside the input domain. Furthermore

\[
\begin{gathered}
\frac{|t-s|}{r_z\delta(s)}<3/8,\\
C|t-s|<1/2,\qquad\delta(s)\ge\kappa\alpha_*>0.
\end{gathered}
\]

The second inequality follows from \(|t|\le a_0/4\), \(|s|\le a_0\), and (PB.6). The geometric calculation of (PB.12), with these fixed strict bounds, proves absolute uniform convergence of the difference on a full neighbourhood containing zero. Its coefficients are holomorphic there; its input values lie in the fixed interior negative-time tube. The segment integral and the sum are therefore holomorphic on that full neighbourhood. Reversing the anchors changes only the orientation.

An existing construction may consequently replace its anchor by any smaller negative anchor within its own input domain, changing the result only by an extending holomorphic function. For constructions initially using different small balls or cofinal majorants, first make this replacement in each original domain, then choose one common sufficiently small anchor for their comparison. This avoids requiring an old, more distant anchor to belong to a later restricted domain.

If \(f\) extends holomorphically to a full neighbourhood of zero, choose such a smaller anchor inside that neighbourhood. Select a fixed positive spatial Cauchy radius there, and a full target neighbourhood whose segments and spatial polydiscs stay compactly inside the extending domain. Make both \(C|t-s|\) and \(|t-s|/(r_z\delta)\) less than \(1/2\). The same geometric estimate now has the fixed radius \(\delta>0\); it proves absolute uniform convergence across zero. The multiplication term extends as well. Thus an extending input has an extending output. The anchor comparison just proved gives this assertion also for the original allowable anchor.

Finally, take two admissible support and input representatives of the same element of (PB.1). By the definition of the filtered quotient, after restriction to a common neighbourhood and enlargement to the union of the two supports, their difference is an extending holomorphic function. Enlarge that union to the cofinal convex support of PB.1. Replace their anchors by one common sufficiently small anchor as above and make one common choice of the coefficient base, cotangent radius and time constants. On this common complement (PB.10) is linear, so the extending-input result shows that the two outputs have the same quotient class. The Cauchy radii used in the proof do not occur in the formula; changing them or restricting the base gives the same formula on its common domain. Equivalent actual symbol representatives have all coefficient functions equal on one common neighbourhood by §5.19.1's definition, and their cotangent Taylor coefficients agree there by uniqueness of Taylor expansion. Their normally convergent evaluations therefore agree after the same common restriction.

It follows that

\[
\begin{gathered}
\rho(P)[f]=[\rho_a(P)f],\\
\rho:E_{0,p}\longrightarrow\operatorname{End}_{\mathbf C}(C_p)
\end{gathered}
\tag{PB.14}
\]

is a well-defined complex-linear map. It is linear in both \(P\) and \([f]\), since a finite list uses one common coefficient domain, cofinal complement and anchor. The identity symbol has only \(p_{0,0}=1\), so \(\rho(1)\) is the identity endomorphism.

#### PB.6. The elementary ordinary inverse

Ordinary differentiation \(\partial_t\) is well defined on \(C_p\): it preserves holomorphy on each input complement, commutes with restriction and takes extending holomorphic functions to extending functions. For the symbol \(h\), (PB.10) is exactly

\[
 \rho_a(h)f(t,x)=\int_a^t f(s,x)\,ds.
\]

The time slice of the convex input domain contains the anchor and target segment. The fundamental theorem for holomorphic primitives, with the stated positive orientation, gives

\[
\begin{gathered}
\partial_t\rho_a(h)f=f,\\
\rho_a(h)\partial_t f=f-f(a,x).
\end{gathered}
\tag{PB.15}
\]

The anchor belongs to a full spatial tube inside the input domain: \(g(0)=0\) and \(a<0\), so \(g(|x|)<-a/2\) for sufficiently small \(x\). Hence \(f(a,x)\) is holomorphic on a full spatial neighbourhood of zero and, being independent of \(t\), extends on a full base neighbourhood. It is zero in \(C_p\). Both equations in (PB.15) consequently descend to

\[
\begin{gathered}
\partial_t\,\rho(h)=1=\rho(h)\,\partial_t\\
\quad\text{on }C_p.
\end{gathered}
\tag{PB.16}
\]

This proves the two-sided inverse of the ordinary integral operator on the quotient. The general composition theorem, its same-anchor nested-domain estimate, finite-order symbol presentation independence, and the comparison with the canonical relative cup and proper trace are established in the subsequent sections. No faithfulness assertion is used in the present construction.

![Cofinal convex complement and the common Cauchy tube](assets/ordinary-action-prerequisites.png)

*Exact real sections of PB.1–PB.3, for \(d=1\), \(\operatorname{Im}s=\operatorname{Im}x=0\). The left panel uses the admissible sample support \(T\ge-\alpha|v|^2\), for which \(e(r)=\alpha r\) and the constructed majorant is \(g(r)=2\alpha r^2\), within the displayed small ball. Its convex complement lies below the lower parabola. The right panel uses \(R=1\), \(r_z=32\), \(\kappa=1/8\), \(C=2\), \(L=1/2048\), \(a_0=1/128\), \(a=-1/128\), \(t=-1/1024\), and \(x=1/20\); here \(\alpha=L/4\), \(b=1/16\), and \(\delta_t=1/32\). The shaded intervals are the real sections of the complex spatial polydiscs of radius \(\delta_s=\delta_t+\kappa|t-s|\) along the fixed segment. Every displayed constant satisfies (PB.6); (PB.8) proves the positive margin for the full complex polydiscs. The vertical axes explicitly rescale the real time coordinate for legibility and do not alter the Euclidean metric in the proof. This is a section, not the full complex support. The reproducible plotting source and exact rational parameter data accompany the figure. Human context: Kashiwara–Schapira, [Micro-hyperbolic systems](https://perso.imj-prg.fr/pierre-schapira/wp-content/uploads/schapira-pub/Microhyp.pdf), §3.1, and Kashiwara–Kawai, [HolIII](https://www.kurims.kyoto-u.ac.jp/~kenkyubu/kashiwara/HolIII.pdf), IV.5; the estimates and cofinal construction are proved above.*


The figure has a full-resolution PNG, SVG, editable plotting source and exact coordinate data.


### 5.41. Canonical products for full convergent spatial symbols

The actual canonical kernel of a convergent spatial symbol is compatible with the unshifted supported cup and normalized proper trace. Spatial Cauchy-shell representatives are constructed directly from the calibrated finite complement cube, and a locally integrable supported primitive controls their product difference. The proof retains every spatial Taylor coefficient and every mixed temporal and spatial contraction on one common neighbourhood. Explicit normal-localization and presentation primitives include finite positive time orders.

The actual product and common-domain estimates are §5.19, §5.20 and §5.21.1, (5.21b)–(5.21f); the normal automorphism is §5.22.1, (5.22a)–(5.22b). Full resolutions and controlled proper excision are §5.34, R.1–R.7; the finite cube, unshifted cup and normalized trace are §5.35, V.1–V.6; and the one-coordinate normalizations and beta integral are §5.36, U.2 and U.4. The angular current and its convergence estimates are §5.38, E.1–E.3. Both full spatial shifts and the presentation primitives are §5.39, NL.2–NL.7. The target operations are those of [Micro-hyperbolic systems, §3.1](https://perso.imj-prg.fr/pierre-schapira/wp-content/uploads/schapira-pub/Microhyp.pdf) and [HolIII, III.1 and IV.2](https://www.kurims.kyoto-u.ac.jp/~kenkyubu/kashiwara/HolIII.pdf); all arguments used here are proved in these proof loci.

#### CP.1. Fixed domains, cutoffs, and the spatial shells

Write \(N=d+1\), \(u=(t,x)\), \(v=(s,y)\), \(w=s-t\), \(\zeta=y-x\). The coefficient line is the **ungraded** ordered input frame
\(\eta=ds\wedge dy_1\wedge\cdots\wedge dy_d\). Put \(r_N=(-1)^{N(N-1)/2}\).

For the finite list \(P,Q,R=P\circ Q\), the actual symbol theorem supplies one coefficient product domain, after restriction, and constants
\[
\begin{gathered}
|p_{n,\beta}(u)|\le B C^n n! r^{-\lvert\beta\rvert},\\[2pt]
n\ge0.
\end{gathered}
\tag{CP.1}
\]
Use the largest necessary \(B,C\) and the smallest necessary \(r\) for this finite list. Every fixed finite base derivative has the same estimate with another constant on a smaller common base polydisc. Choose once
\[
 r^{-1}<a<b<c<B_*<A.
 \tag{CP.2}
\]
Choose the time-difference radius so that \(C|w|<1/2\). Choose a radial smooth function \(\chi\) on \(\mathbf C\), equal to one on \(|z|\le b\), zero on \(|z|\ge c\), and with \(0\le\chi\le1\). Such a function follows from the explicit smooth bump construction in §5.38, E.2. For \(w\ne0\), set
\[
\begin{gathered}
\chi_j(w,\zeta_j)=\chi(\zeta_j/|w|),\\[2pt]
f_{j,k}={1-\chi_j\over2\pi i}{k!\over\zeta_j^{k+1}}.
\end{gathered}
\tag{CP.3}
\]
The expression is zero near \(\zeta_j=0\), so it is smooth there. At \(w=0,\zeta_j\ne0\), its extension is the holomorphic Cauchy expression and is smooth. Only the joint origin in \((w,\zeta_j)\) needs separate interpretation. Throughout a full antiholomorphic differential is used, including the output variables.

For \(\sigma>0\), let \(s_{j,k}^{\sigma}\) denote the spatial coefficient of \(\bar\partial f_{j,k}\) at \(w=-\sigma\), before its input holomorphic differential. Thus it is
\[
\begin{gathered}
s_{j,k}^{\sigma}=-{1\over2\pi i}\\[2pt]
\bar\partial_{\zeta_j}\chi(\zeta_j/\sigma)\\[2pt]
{k!\over\zeta_j^{k+1}}.
\end{gathered}
\tag{CP.4}
\]
It is supported on \(b\sigma\le|\zeta_j|\le c\sigma\). The positive complete spatial block is \(s_{j,k}^{\sigma}\wedge d\zeta_j\). For a holomorphic \(g\) on a neighbourhood of \(|\zeta_j|\le c\sigma\), compactly supported Stokes on the annulus gives
\[
 \int s_{j,k}^{\sigma}\wedge d\zeta_j\,g(\zeta_j)=g^{(k)}(0).
 \tag{CP.5}
\]
Indeed the integral equals the positive outer Cauchy circle integral; its inner extension is zero. Taylor expansion of \(g\), uniformly on a slightly larger circle, selects coefficient \(k\) and gives the factor \(k!\). This proves the sign and scalar without identifying any class from a function-space action.

Define the spatial top coefficient form
\[
\begin{gathered}
A_P(u;\sigma,\zeta)=\\[2pt]
\sum_{n+|\beta|\ge1}p_{n,\beta}(u)\\[2pt]
{\sigma^{n+|\beta|-1}\over(n+|\beta|-1)!}\\[2pt]
s_{1,\beta_1}^{\sigma}\wedge\cdots\wedge s_{d,\beta_d}^{\sigma}.
\end{gathered}
\tag{CP.6}
\]
Spatial integration below means anti-first realization with the spatial holomorphic frame and its \(r_d\) calibration, equivalently the product of the positive blocks in (CP.5). With this convention the moment of (CP.6) is
\[
\begin{gathered}
\int A_P(u;\sigma,\zeta)\,g(\zeta)=\\[2pt]
\sum_{n+|\beta|\ge1}p_{n,\beta}(u)\\[2pt]
{\sigma^{n+|\beta|-1}\over(n+|\beta|-1)!}\partial_\zeta^\beta g(0).
\end{gathered}
\tag{CP.7}
\]
For a polynomial \(g\), only finitely many spatial indices survive. This will be used as a calculation of the Cauchy transform of compactly supported currents, not as faithfulness of an action on solutions.

If \(j\) is any fixed derivative order in \(\zeta\) and the base, differentiation of the fixed cutoffs and of their holomorphic coefficients gives
\[
\begin{gathered}
\|\partial^j A_P(u;\sigma,\cdot)\|_\infty\\[2pt]
\le K_j\sigma^{-2d-1-j},\\[2pt]
\int |A_P|\le K_0/\sigma,\\[2pt]
\left|\int A_P\right|\le K_0.
\end{gathered}
\tag{CP.8}
\]
Here and below coefficient norms in fixed coordinate frames suffice. To prove the first bound, differentiate a term of (CP.4): its coefficient is bounded by a fixed constant times
\(k!b^{-k}\sigma^{-k-2-j}\) times a fixed polynomial in \(k+1\). After inserting (CP.1), use
\[
 {n!\beta!\over(n+|\beta|-1)!}\le n+|\beta|
 \tag{CP.9}
\]
and sum the product of geometric series with ratios \(C\sigma<1/2\) and \((br)^{-1}<1\). Fixed powers of \(n+|\beta|\) are harmless: successive applications of \(q\partial_q\) to the geometric series give finite bounds on smaller ratios. The support volume is at most \((\pi c^2\sigma^2)^d\), proving the second bound. Formula (CP.5) with \(g=1\) annihilates every \(\beta\ne0\). The surviving terms have \(n\ge1\), and their geometric sum proves the last bound. All constants and radii are common to the finite list.

#### CP.2. The actual current and its finite-cube calibration

Let \(L(w)=-\log w/(2\pi i)\), with principal logarithm cut on the negative real ray. On the punctured normal space form
\[
\begin{gathered}
F_P^-=r_N\sum_{n+|\beta|\ge1}\\[2pt]
p_{n,\beta}(u){(-w)^{n+|\beta|-1}\over(n+|\beta|-1)!}\\[2pt]
L(w)\,\bar\partial f_{1,\beta_1}\wedge\cdots\wedge\\[2pt]
\bar\partial f_{d,\beta_d}\otimes\eta .
\end{gathered}
\tag{CP.10}
\]
Every term requires all spatial annuli, so \(|\zeta_j|\asymp|w|\) on its support. The same geometric bound proves
\[
\begin{gathered}
|F_P^-|\le K |(w,\zeta)|^{-(2N-1)}\\[2pt]
(1+|\log|(w,\zeta)||).
\end{gathered}
\tag{CP.11}
\]
The normal real dimension is \(2N\); polar integration leaves
\(\int_0^R(1+|\log r|)\,dr<\infty\). Give (CP.10) its literal locally integrable extension. Define
\[
 C_P=p_{0,0}(u)\Delta_N+\bar\partial F_P^- .
 \tag{CP.12}
\]
Here \(\Delta_N\) is the positive ordered normalized diagonal current of §5.36, U.6/§5.35, V.17. This is closed, of normal distribution order at most one. It is supported on
\[
\left\{\begin{gathered}
w\in\mathbf R_{\le0},\\[2pt]
|\zeta_j|\le c(-\operatorname{Re}w)
\end{gathered}\right\},
\tag{CP.13}
\]
and its negative-ray part is precisely the positive time-ray block of §5.36, U.5 times (CP.6).

The endpoint claim requires proof. Away from the origin, differentiating (CP.10) leaves only \(\bar\partial L\); the differential of each spatial factor is zero. Formula §5.36, U.4 makes it the negative-ray density. Delete \(|w|\le\epsilon\) and apply Stokes against a smooth compact test. On the cylindrical boundary, every term containing \(d\bar w\) vanishes on realization with \(dw\); the remaining term is the spatial-top part. Its spatial integral on a constant normal test is zero for \(\beta\ne0\). Replace the test by its difference from its value at the normal origin; on the spatial support this difference is at most \(K\epsilon\) times a first derivative seminorm. The bound (CP.8) then bounds the boundary integral by \(K\epsilon(1+|\log\epsilon|)\). For \(\beta=0\), necessarily \(n\ge1\), and the same bound follows from the extra positive time power. Both tend to zero, uniformly in the full series. Thus no additional endpoint current occurs. In particular the negative part has the direct, convergent test definition
\[
\begin{gathered}
\int_0^R\\[2pt]
\left\{\begin{gathered}
\int A_P(u;\sigma,\zeta)\\[2pt]
[\phi(-\sigma,\zeta)-\phi(0,0)]\\[2pt]
+\phi(0,0)\\[2pt]
\int A_P(u;\sigma,\zeta)
\end{gathered}\right\}\\[2pt]
\,d\sigma .
\end{gathered}
\tag{CP.14}
\]
Fixed compact cutoffs outside the normal origin are understood. Equations (CP.8) prove convergence and a first-order test bound. This is an actual distribution, rather than an infinite sum of spatial delta derivatives.

Here is the finite cube identifying (CP.12) canonically. Use the ordered complement opens
\(U_0=\{w\notin\mathbf R_{\le0}\}\),
\(U_j=\{|\zeta_j|>c|w|\}\). The unsigned holomorphic top expression is
\[
\begin{gathered}
K_P={1\over(2\pi i)^N}\\[2pt]
\left\{\begin{gathered}
{p_{0,0}\over w\prod_j\zeta_j}\\[2pt]
-\log w\sum_{n+|\beta|\ge1}\\[2pt]
{\begin{gathered}
p_{n,\beta}\beta!\\[2pt]
(-w)^{n+|\beta|-1}
\end{gathered}\over \begin{gathered}
(n+|\beta|-1)!\\[2pt]
\prod_j\zeta_j^{\beta_j+1}
\end{gathered}}
\end{gathered}\right\}.
\end{gathered}
\tag{CP.15}
\]
Its literal ordinary top cube coefficient is \(r_N(-1)^N K_P\), or \(-r_NK_P\) in the full-cone ordinary complement Čech coefficient, exactly §5.35, V.15–V.17. We prove this calibration for the shell current by a triangular complement homotopy; its intermediate entries live only on the complement, and require no distributional extension at a partial normal origin.

On the complement, \(F_P^-\) is smooth and closed. Its restriction to every spatial \(U_j\) is zero, since \(\bar\partial f_{j,\beta_j}=0\) there. On \(U_0\), \(L\) is holomorphic. Let the complement Čech–Dolbeault differential be \(D=d_C+(-1)^r\bar\partial\), with \(r\) the Čech degree, as in §5.35, V.6. For \(r=0,\ldots,d-1\), define a cochain of total degree \(d-1\), with its sole possibly nonzero ordered entry on \(U_0\cap\cdots\cap U_r\), by
\[
\begin{gathered}
b^r_{0\ldots r}=(-1)^r r_N\\[2pt]
\sum_{n+|\beta|\ge1}\\[2pt]
{p_{n,\beta}(-w)^{n+|\beta|-1}\over(n+|\beta|-1)!}\\[2pt]
L(w)\,f_{1,\beta_1}\cdots f_{r+1,\beta_{r+1}}\\[2pt]
\bar\partial f_{r+2,\beta_{r+2}}\wedge\cdots\wedge\\[2pt]
\bar\partial f_{d,\beta_d}.
\end{gathered}
\tag{CP.16}
\]
For \(r=0\) the domain is just \(U_0\). Give other ordered entries value zero, and antisymmetrize the Čech indices. Each of these domains contains \(U_0\); hence \(w=0\) is absent and every coefficient is smooth. Every spatial factor, differentiated or undifferentiated, requires \(|\zeta_j|\ge b|w|\), or is on the stronger complement domain. The strict geometric majorant in (CP.8) proves normal convergence with every fixed derivative on each compact subset of the specified domain. These are actual smooth complement cochains.

Start with \(c^0\), the Čech-degree-zero restriction of the global complement form \(F_P^-\); its only nonzero entry is at index \(0\). Subtract \(Db^0\). Its Dolbeault part cancels \(c^0\), and its remaining entry is positive on indices \(01\). Repeating, on indices \(0,\ldots,r\), the spatial factors \(f_1,\ldots,f_r\) are holomorphic; thus \(\bar\partial b^r\) differentiates only \(f_{r+1}\). The factor \((-1)^r\) in \(D\) cancels the factor in (CP.16). The only nonzero new Čech restriction is obtained by adding index \(r+1\): adding any larger index kills one of the still-differentiated spatial factors. Deleting the last new index contributes \((-1)^{r+1}\); the minus from subtracting \(Db^r\), and (CP.16)'s \((-1)^r\), make the next coefficient positive. Induction gives the exact total-complex identity
\[
 F_P^- -D\sum_{r=0}^{d-1}b^r=r_N K_P^-
 \tag{CP.16a}
\]
where the right side is the top Čech cochain, and \(K_P^-\) is the negative part of (CP.15). At that last intersection every \(f_j\) equals its holomorphic Cauchy function. For \(d=0\) the identity reads \(F_P^-=K_P^-\) on \(U_0\), with empty homotopy.

The relative current class has representative \((0,-F_P^-)\), because adding the literal boundary \(d(F_P^-,0)\) yields \((\bar\partial F_P^-,0)\). Equation (CP.16a) replaces its complement component by \(-r_NK_P^-\); the replacement boundary is exactly \(d(0,\sum b^r)=(0,-D\sum b^r)\) in the full cone over the Čech complex. Adding §5.36, U.6/§5.35, V.17's multiplication-class calibration gives (CP.15) in full. §5.35, V.6's finite Čech augmentation, with its explicitly proved row contraction, identifies this exact cone with the actual derived support cone. This proves the canonical class assertion.

Thus (CP.12) is the actual canonical class of (CP.15). It also supplies the explicit comparison to §5.38, E.2–E.3: both are compared with the same calibrated top cochain by their finite complement homotopies and their literal locally integrable global primitives. No polynomial-class limit is required. The finite complement homotopies (CP.16)–(CP.16a) and §5.38, (E.10), give their comparison on the same coefficient intersection.

#### CP.3. Smooth ray densities for the canonical convolution

The two kernel singularity lists are independent in \((u,a=v-u,b=z-v)\). Sections 5.34–5.35, §5.34, R.6–R.7 and §5.35, V.2, define their ordinary unshifted supported cup. The normalization is exact: its full-form realization has factor \((-1)^{N^2}\), and \(\operatorname{Tr}_{2N}\) has the same factor; they cancel by §5.35, V.13. The resulting complete-block convolution has positive orientation.

For two negative parts, at final real time length \(\tau>0\), the spatial density is
\[
\begin{gathered}
D_{P,Q}(u;\tau,\zeta)=\\[2pt]
\int_0^\tau\int_{\eta\in\mathbf C^d}\\[2pt]
A_P(u;\alpha,\eta)\,\\[2pt]
A_Q\left(\begin{gathered}
t-\alpha,x+\eta;\\[2pt]
\tau-\alpha,\zeta-\eta
\end{gathered}\right)\,d\eta\,d\alpha .
\end{gathered}
\tag{CP.17}
\]
The spatial notation denotes the positive ordered complete spatial block integral, including its usual real complex orientation. It has support \(|\zeta_j|\le c\tau\). Terms involving the diagonal are, respectively,
\(p_{0,0}(u)A_Q(u;\tau,\zeta)\) and
\(A_P(u;\tau,\zeta)q_{0,0}(t-\tau,x+\zeta)\). The diagonal×diagonal term is \(p_{0,0}q_{0,0}\Delta_N\).

We justify the endpoint integrals and their common bounds. If \(0<\alpha\le\tau/2\), the second density and all its fixed derivatives are smooth on scale \(\tau\). In its pairing with the first density subtract its value at \(\eta=0\). The mean-value bound gives a factor \(c\alpha\); (CP.8)'s \(K/\alpha\) mass bound cancels that factor. Its constant value is multiplied by the bounded zeroth moment of the first density. Thus for every fixed \(j\) the inner integral and its \(\zeta\)-derivatives are bounded by
\(K_j\tau^{-2d-2-j}\), uniformly as \(\alpha\downarrow0\). Base derivatives have the same estimate.

For \(0<\tau-\alpha\le\tau/2\), put \(\theta=\zeta-\eta\). Freeze every holomorphic coefficient of the second density at \(x+\zeta\). The difference of a coefficient from this frozen value is bounded by its common base Cauchy derivative estimate times \(|\theta|\le c(\tau-\alpha)\); summing retains exactly the geometric ratios used in (CP.8). This cancels the second density's mass factor \(1/(\tau-\alpha)\). Pair the frozen part with the first density's value at \(\theta=0\), and subtract that value in the remainder. The frozen zeroth moment is bounded by (CP.8), and the remainder gains the same \(|\theta|\) factor. Again the bound is \(K_j\tau^{-2d-2-j}\). Integrating over the two half intervals gives
\[
\begin{gathered}
\|\partial_\zeta^jD_{P,Q}\|_\infty\\[2pt]
\le K_j\tau^{-2d-1-j}.
\end{gathered}
\tag{CP.18}
\]
The same argument after fixed derivatives proves smoothness for \(\tau>0\) and holomorphic dependence on the base. It also applies to the diagonal-negative densities. The endpoint cancellation used in (CP.14), twice, identifies (CP.17) off the final normal origin with the tensor-current/proper-trace distribution: on tests the cutoff integrals converge, and the terms subtracted at the two endpoints are exactly the zeroth-moment terms retained above.

We check that this has no additional current at the final origin. Under the simultaneous dilation \((w,\zeta)\mapsto\lambda(w,\zeta)\), (CP.14) evaluated on a fixed rescaled test is uniformly bounded as \(\lambda\downarrow0\): its difference from the constant test gains \(\sigma/\lambda\), cancelling the mass factor \(1/\sigma\) on \(0<\sigma<O(\lambda)\). In the two-factor tensor use the two successive subtractions; their joint remainder gains \(\sigma_1\sigma_2/\lambda^2\), and their two masses are \(O(1/\sigma_1\sigma_2)\). The proper-cone geometry confines both lengths to \(O(\lambda)\). The remaining terms each include one or two bounded zeroth moments. Thus the convolution too is uniformly bounded on rescaled tests. Any distribution of finite order supported at a point is a finite linear combination of derivatives of its delta: subtract the finite Taylor polynomial of a test and use a cutoff shrinking to that point in its defining finite-order seminorm to prove that it depends only on that finite Taylor jet. Rescaled tests show that every nonzero derivative of delta is excluded by the uniform bound just proved. The only possible discrepancy is a scalar delta. First subtract the already identified diagonal×diagonal term \(p_{0,0}(u)q_{0,0}(u)\Delta _N\) from the canonical convolution. Compare this remaining distribution with the ray-density extension of (CP.17) and its two diagonal-negative terms. Test against a function constant in the spatial variables on the entire cone fibre and supported in time length at most \(O(\lambda )\). A diagonal-negative term has bounded zeroth moment: freeze its multiplying holomorphic coefficient and use (CP.8), with the remainder gaining one spatial length. For two negative terms the zeroth moment is bounded as \(\tau \downarrow 0\): in (CP.20) with \(g=1\), only \(\gamma =0\) and \(\nu =\beta\) survive, and the common Cauchy/geometric bounds give integrable nonnegative endpoint powers with both negative total indices positive. Equivalently, freeze the second holomorphic coefficient and use the two bounded zeroth moments and the spatial difference estimate. Both remaining distributions evaluate as \(O(\lambda )\). A scalar delta would give a constant nonzero value, so its coefficient is zero. Restoring the known diagonal×diagonal term now identifies the complete canonical convolution, including the final origin.


For the needed local properness, put all kernels and the forthcoming primitive inside the larger cone
\[
\begin{gathered}
G_A=\\[2pt]
\left\{\begin{gathered}
\operatorname{Re}w\le0,\\[2pt]
|\operatorname{Im}w|\le\epsilon(-\operatorname{Re}w),\\[2pt]
|\zeta_j|\le A(-\operatorname{Re}w)
\end{gathered}\right\}.
\end{gathered}
\tag{CP.19}
\]
For two differences in this cone, their nonnegative real time lengths add to the final length. Each intermediate normal length is bounded by a fixed multiple of that final length. Over a compact endpoint set the intermediate set is closed and bounded. Choose a coefficient source polydisc first, then a smaller endpoint polydisc so that this entire closed bounded fibre family has positive distance from the source boundary. It is then compact **inside** the selected source, giving the proper-support map required by §5.34, R.7/§5.35, V.4. No assertion that arbitrary polydiscs are cone-round is made. The same selection works for the finite list and its intermediate fibres.

#### CP.4. All holomorphic moments are the actual symbol product

Apply a polynomial test \(g(\zeta)\) to (CP.17). Formula (CP.7), the finite ordinary spatial Leibniz rule, and the stated orientation give for individual terms, writing \(m=n+|\beta|\), \(l=n'+|\gamma|\),
\[
\begin{gathered}
p(u)\sum_{\nu\le\beta}\binom\beta\nu\int_0^\tau\\[2pt]
{\alpha^{m-1}(\tau-\alpha)^{l-1}\over(m-1)!(l-1)!}\\[2pt]
\partial_x^\nu q(t-\alpha,x)\\[2pt]
\partial_x^{\beta-\nu+\gamma}g(0)\,d\alpha.
\end{gathered}
\tag{CP.20}
\]
Expand the holomorphic coefficient at \(t\): its \(k\)-th term is
\((-1)^k\alpha^k\partial_t^k\partial_x^\nu q(t,x)/k!\).
Repeated real integration by parts gives the integer beta integral
\[
\begin{gathered}
\int_0^\tau\\[2pt]
{\begin{gathered}
\alpha^{m+k-1}\\[2pt]
(\tau-\alpha)^{l-1}
\end{gathered}\over(m-1)!(l-1)!k!}\,d\alpha\\[2pt]
=\binom{m+k-1}{k}\\[2pt]
{\tau^{m+l+k-1}\over(m+l+k-1)!}.
\end{gathered}
\tag{CP.21}
\]
For completeness, start with
\(\int_0^1 a^{M-1}(1-a)^{L-1}da\). Integration by parts reduces \(L\) by one and increases \(M\) by one, multiplying by \((L-1)/M\); the last integral is \(1/(M+L-1)\). The product is \((M-1)!(L-1)!/(M+L-1)!\). Rescale by \(\tau\) and put \(M=m+k\). This proves (CP.21), including endpoints.

The coefficient in (CP.20) is consequently
\[
\begin{gathered}
\binom\beta\nu\binom{-m}{k}p\,\partial_t^k\partial_x^\nu q,\\[2pt]
h^{m+l+k}\partial_x^{\beta-\nu+\gamma}.
\end{gathered}
\tag{CP.22}
\]
These are precisely **all** mixed \((\tau,t)\) and \((\xi,x)\) contractions in the actual product of §5.19, with the complete estimates of §5.20–§5.21.1. Rewriting in \(h,z\) gives time index
\(n+n'+k+|\nu|\) and spatial exponent \(\beta-\nu+\gamma\). In particular none is a negative time index. A left multiplication coefficient gives its literal left product. A right multiplication coefficient is expanded at \((t,x)\) in \(( -\tau,\zeta)\), producing the same binomial coefficients with \(l=0,\gamma=0\); the equality
\((-1)^k/(k!(m-1)!)=\binom{-m}{k}/(m+k-1)!\)
is immediate from the factorial expression. The spatial finite Leibniz rule is unchanged. The only diagonal output is \(p_{0,0}q_{0,0}\).

We also need actual, rather than formal, equality. Fix a polynomial degree \(M\). In (CP.20), a nonzero derivative of \(g\) has \(\delta=\beta-\nu\ge0\), \(|\delta+\gamma|\le M\); thus \(\delta,\gamma\) run through finite sets, though \(\nu\) remains unbounded. At a fixed time power, (CP.22) has \(n+n'+k+|\nu|+|\delta+\gamma|\) fixed; hence all indices are finite. Take one larger base polydisc with spatial and temporal Cauchy margins \(R_x,R_t>0\). After the \(1/k!\) in the temporal Taylor expansion is included, the derivative of the second coefficient is bounded by
\(B C^{n'}n'!\nu! R_x^{-|\nu|}R_t^{-k}r^{-|\gamma|}\).
The spatial Leibniz factor satisfies the explicit inequality
\[
\begin{gathered}
{n!\over(n+|\beta|-1)!}\binom\beta\nu\nu!\\[2pt]
={n!\beta!\over\delta!(n+|\beta|-1)!}\\[2pt]
\le {n+|\beta|\over\delta!}.
\end{gathered}
\tag{CP.22a}
\]
The corresponding second factor satisfies
\(n'!/(n'+|\gamma|-1)!\le n'+|\gamma|\), since it is a negative term. Hence the absolute integrand is bounded, up to the finite \(g,\delta,\gamma\) constants, by a product of geometric series with fixed polynomial factors and ratios
\[
\begin{gathered}
C\alpha,\quad C(\tau-\alpha),\\[2pt]
{\alpha\over rR_x},\quad {\alpha\over R_t}.
\end{gathered}
\tag{CP.22b}
\]
Choose \(\tau\) once so that all these ratios are below \(1/2\), also retaining \((br)^{-1}<1\) for the shell series and putting the full shells in the base domain. The apparent factors \(\alpha^{-1}\) and \((\tau-\alpha)^{-1}\) in this majorant cancel: respectively \(n+|\nu|+|\delta|\ge1\) and \(n'+|\gamma|\ge1\). Differentiating geometric series bounds the fixed polynomial factors. This proves absolute uniform convergence and integrability of the moment sums at both endpoints, and of the temporal Taylor expansion, on one fixed smaller disc. Parametrize \(\alpha=\tau\lambda\); each term is an ordinary nonnegative integer power of \(\tau\), and the majorant proves that the moment is holomorphic at \(\tau=0\). Its Taylor coefficients are (CP.22). The factorial product theorem gives the same convergent moment series for \(R\). Uniqueness of ordinary holomorphic Taylor expansion gives equality on that disc.

This holds for every polynomial \(g\) on the **same** fixed domain; the domain was chosen from the coefficient constants and shell geometry, not from \(M\). Consequently, if \(S(u;\tau,\zeta)\) is the spatial density of the canonical convolution, including its two diagonal-negative terms, minus \(A_R\),
\[
\begin{gathered}
\int S(u;\tau,\zeta)\,\zeta^\alpha=0\\[2pt]
\text{for every }\alpha\in\mathbf N^d,\\[2pt]
\operatorname{supp}S\subset\{|\zeta_j|\le c\tau\},\\[2pt]
\|\partial^jS\|_\infty\le K_j\tau^{-2d-1-j}.
\end{gathered}
\tag{CP.23}
\]
This is a statement about the actual compactly supported current density and its Cauchy transform. The next section supplies the missing primitive explicitly.

#### CP.5. A finite Cauchy homotopy with support and norm control

Choose \(\chi_j^\tau\) equal to one on \(|\zeta_j|\le c\tau\), supported on \(|\zeta_j|<B_*\tau\), with a fixed positive scaled margin. Use \(E(z)=1/(\pi z)\), so \(\bar\partial E=\delta_{\mathbf C}\), proved in §5.34, R.2. Let \(\iota_j\) be the signed contraction with \(d\bar\zeta_j\), and let \(T_j=E*_j\) denote partial convolution in the \(j\)-th spatial variable. On a spatial top form \(U\), define
\[
\begin{gathered}
H_jU=\chi_j^\tau T_j\iota_jU,\\[2pt]
\Pi_jU=-d\bar\zeta_j\wedge\\[2pt]
(\partial_{\bar\zeta_j}\chi_j^\tau)T_j\iota_jU.
\end{gathered}
\tag{CP.24}
\]
Because a top form contains all the spatial antiholomorphic factors, every other spatial differential in \(\bar\partial H_jU\) repeats a factor and is zero. §5.34, R.2's fundamental-solution identity therefore gives
\[
 \bar\partial_{\rm sp}H_jU=U-\Pi_jU.
 \tag{CP.25}
\]
This identity holds whenever \(U\)'s \(j\)-th support is inside the region where \(\chi_j^\tau=1\). Previously applied \(\Pi_i\), \(i\ne j\), do not enlarge its \(j\)-th support. Define
\[
\begin{gathered}
H_\tau S=\sum_{j=1}^{d}\\[2pt]
H_j\Pi_{j-1}\cdots\Pi_1 S,\\[2pt]
\text{empty product }1.
\end{gathered}
\tag{CP.26}
\]
Telescoping (CP.25) gives
\(\bar\partial_{\rm sp}H_\tau S=S-\Pi_d\cdots\Pi_1 S\).
The last term is zero. In fact its coefficient, apart from its fixed exterior sign and the outer cutoff derivatives, is the simultaneous Cauchy transform
\[
 \int {S(\eta)\over\prod_j(\zeta_j-\eta_j)}\,d\eta
 \tag{CP.27}
\]
on the product of the outer annuli. Each \(|\zeta_j|\) is strictly larger than the original \(c\tau\)-support radius. Expand each denominator as its uniformly convergent geometric series in \(\eta_j/\zeta_j\). Every coefficient is a moment in (CP.23), and is zero. Fubini is justified by compact smooth coefficients and their bounded Cauchy denominators. Thus
\[
 \bar\partial_{\rm sp}H_\tau S=S.
 \tag{CP.28}
\]

Here are the quantitative bounds. On a disc of radius \(O(\tau)\), the integral of \(|E(z)|\) is \(O(\tau)\), by polar integration. Hence \(T_j\) multiplies a sup norm by at most \(K\tau\). Cutoff derivatives are \(O(\tau^{-1})\); therefore each \(\Pi_j\) preserves the estimate in (CP.23), and each \(H_j\) gains a factor \(\tau\). For derivatives transfer the convolution derivative to the smooth coefficient, or differentiate the fixed scaled cutoff; this gives
\[
\begin{gathered}
\operatorname{supp}H_\tau S\subset\{|\zeta_j|\le B_*\tau\},\\[2pt]
\|\partial^jH_\tau S\|_\infty\le K_j\tau^{-2d-j},\\[2pt]
\int |H_\tau S|\le K_0.
\end{gathered}
\tag{CP.29}
\]
All forms are smooth for \(\tau>0\) and holomorphic in the output base. No inverse with an unproved continuity property has been invoked.

Let \(R_t\) be the time coefficient current \(\bar\partial L(w)\), whose positive complete block is §5.36, U.5. In normal coordinates define the primitive, with the input frame attached once,
\[
\begin{gathered}
B_{P,Q}=-r_N R_t\wedge\\[2pt]
H_{-\operatorname{Re}w}S\otimes\eta.
\end{gathered}
\tag{CP.30}
\]
The ray coefficient of (CP.30) is locally integrable in \((\tau,\zeta)\) by (CP.29). It is an order-zero distribution on the ray, supported in \(G_A\) on the final kernel product. The proper map used in forming the convolution is \(\pi _{13}:S_{12}\cap S_{23}\to D_{\mathrm{end}}\), from the triple kernel support intersection to the selected final endpoint neighbourhood \((u,z)\). CP.3's outer-source/inner-endpoint choice makes its fibres over compact endpoint sets compact inside the intermediate source. The primitive \(B_{P,Q}\) itself lives on the final kernel product and supplies a supported relative-cone boundary there. For the degree-one action of §5.42 the proper support is obtained by intersecting with the input support. Its full differential is
\[
\begin{gathered}
\bar\partial B_{P,Q}\\[2pt]
=\operatorname{Tr}_{2N}(C_P\smile C_Q)-C_R.
\end{gathered}
\tag{CP.31}
\]
Indeed the time current is closed. Derivatives of the \(\tau\)-dependent cutoff or coefficients contribute \(d\bar w\), already in \(R_t\), so vanish. Output antiholomorphic derivatives vanish because the coefficients are holomorphic in the output base. The remaining spatial differential has the sign minus from passing the degree-one time factor; it cancels the minus in (CP.30), and (CP.28) gives (CP.31). This argument also holds on tests at \(\tau=0\): no time differential survives the repeated \(d\bar w\), and the spatial primitive has the literal integrable extension (CP.29). The diagonal terms have already cancelled, and (CP.23)'s zeroth moment eliminates an endpoint mass in the difference. Thus (CP.31) is a literal full-current equality, including the origin.

In the exact relative cone, (CP.31) says that the difference is \(d(B_{P,Q},0)\). It is a literal supported primitive in the final kernel target. The canonical cup and trace used the proper triple-to-endpoint map just specified; §5.34, R.7's controlled excision and §5.35, V.4's trace commute with the corresponding differentials. Equation (CP.31) gives the resulting boundary in the exact final relative cone and proves equality of the actual canonical classes on the selected common neighbourhood.
\[
\begin{gathered}
\kappa(P)*\kappa(Q)=\kappa(P\circ Q),\\[2pt]
P,Q\in E_0.
\end{gathered}
\tag{CP.32}
\]
For \(d=0\), omit the spatial shells and the homotopy: (CP.20)–(CP.22) are directly equality of the normally convergent time-ray densities, so (CP.31) holds with zero primitive.

#### CP.6. Finite positive time orders and preserved scope

The unit is the calibrated ordered diagonal class. The complete canonical normal-localization interface is §5.39, NL.2–NL.7. In particular, for the principal locally integrable logarithm, \(-\partial_wL=C+B_{\log}\), with \(B_{\log}=(2i)^{-1}\mathbf1_{\operatorname{Re}w<0}\delta(\operatorname{Im}w)\). Its derivative discrepancy is the literal supported boundary \(\bar\partial B_{\log}\); after spatial tensoring, §5.39, NL.11–NL.12 give the complete relative-cone primitive. The unit and derivative identities are established on arbitrary smooth current tests, and §5.39, NL.22–NL.27 give explicit supported primitives for both full convergent-spatial normal shifts. For \(P=\partial _t^k\circ P_0\), the complete representative is \(\partial _t^k T_{P_0}\); §5.39, NL.34–NL.35 give the literal finite supported telescoping primitive proving independence of every such presentation. §5.39, NL.36–NL.37 and §5.39, NL.7 prove actual germ restriction and partition/cone independence. Consequently the proved order-zero product identity (CP.32), together with the invertible normal-generator interface of §5.39, extends uniquely to the full finite-order ring \(E=E_0[h^{-1}]\) by the actual normal localization of §5.22.1, (5.22a)–(5.22b). The Ore localization property is applied after these current identities have been proved; no faithful solution-action inference or limit in a cohomology quotient is used.

The following extension argument retains all finite common-domain bounds and the supported presentation primitives of §5.39, (NL.22)–(NL.37).

##### CP.6.1 Actual algebra and common finite domains


Write \(E_0\) for the actual common-domain factorial order-zero class and \(h=\tau ^{-1}\). The actual §5.22.1 automorphism is

\[
\begin{gathered}
\sigma(P)=h\circ P\circ h^{-1},\\[2pt]
\sigma(P)=\sum_{a\ge0}\\[2pt]
(-1)^a h^a\partial_t^aP\\[2pt]
\text{as ordinary}\\[2pt]
\text{ normal-symbol}\\[2pt]
\text{ coefficients}.
\end{gathered}
\tag{CP.33}
\]

The second display is a coefficient expression, not a replacement of its ordinary \(h^a\) multiplication by another Leibniz product. At any fixed index it is a finite sum. A base Cauchy margin \(\rho\) gives the actual estimate

\[
\begin{gathered}
\sum_{n+a=m}BC^n n!a!\rho^{-a}\\[2pt]
\le B(C+\rho^{-1})^m m!.
\end{gathered}
\tag{CP.34}
\]

Its inverse is the actual coefficient expression \(P+h\partial _tP\). Cancellation at every coefficient gives inverse algebra automorphisms, using the already proved associativity of §5.19. The normal relation is

\[
 h\circ P=\sigma(P)\circ h.
 \tag{CP.35}
\]

The right \(h\) shift is ordinary coefficient shift and is injective; normality gives the corresponding left assertion. Finite shifts preserve the actual factorial class by \((n+k)!\le 2^{n+k}n!k!\). The finite-order class of §5.19 is the actual union of finite normal shifts; §5.21.2, (5.21h)–(5.21i), and §5.22.1 identify it with this normal localization. A finite-order symbol has a finite left product presentation

\[
\begin{gathered}
A=h^{-k}\circ P,\\[2pt]
k\ge0,\quad P\in E_0.
\end{gathered}
\tag{CP.36}
\]

For example choose \(k\) to remove its finite positive-order head and take \(P=h^k\circ A\). If \(A\) has ordinary normal coefficients with indices \(i\ge -m\) and \(k\ge m\), the normalized derivative of \(h^k\) has coefficient \((-1)^a\binom{k+a-1}{a}h^{k+a}\) for \(k\ge 1\). Hence its coefficient at index \(N\) is the finite sum over \(i+a+k=N\); no negative output index occurs. For \(i\ge 0\), the Cauchy bound contains \(i!a!\le (N-k)!\le N!\) and \(\binom{k+a-1}{a}\le2^{k+a-1}\). For the finitely many \(i<0\), one has \(a\le N\), and their Cauchy derivative \(a!\) is also bounded by \(N!\). A common enlarged exponential constant therefore gives actual factorial growth on a single smaller domain, including the spatial Cauchy radius. The \(k=0\) case is already order zero. This proves the stated finite left presentation directly and preserves the actual common domain; no arbitrary infinite formal coefficient lift occurs.

For the finite pair \(A=h^{-k}\circ P\), \(B=h^{-l}\circ Q\), (CP.35) gives

\[
\begin{gathered}
P\circ h^{-l}=h^{-l}\circ\sigma^l(P),\\[2pt]
A\circ B=h^{-(k+l)}\circ\\[2pt]
(\sigma^l(P)\circ Q).
\end{gathered}
\tag{CP.37}
\]

The power is \(\sigma ^l\), not \(\sigma ^{-l}\). Derive its first step by multiplying (CP.35) on the left and right by \(h^{-1}\), then iterate \(l\) times. At a fixed finite \(l\ge 1\), coefficientwise geometric multiplication gives \(\sigma^l(P)=\sum_a(-1)^a\binom{l+a-1}{a}h^a\partial_t^aP\) in ordinary normal-symbol coefficients. The bound \(\binom{l+a-1}{a}\le2^{l+a-1}\), with (CP.34), supplies another actual factorial estimate. Only a finite list of actual symbols and their finitely many \(\sigma\) iterates is used. Intersect their coefficient domains, take common factorial/Cauchy constants, and shrink once before applying the order-zero product proof.

##### CP.6.2 Actual canonical classes and presentation primitives

Let \(K\) denote the germ target of canonical supported kernel classes with its unshifted cup and normalized proper trace. For a finite list choose one common larger pointed thick cone and an outer source/inner endpoint neighbourhood. The sum of two such cone differences remains in the same cone, and their nonnegative real time lengths bound every intermediate point. §5.35, V.2/V.4/V.13, and the direct smooth-test proof in §5.39, NL.5, give an associative canonical operation with the positive diagonal unit on this germ target. This is a statement about actual finite-order distribution families and controlled proper traces, rather than a formal or solution-action algebra.

The product proof CP.1–CP.5 establishes the unital order-zero map

\[
\begin{gathered}
\phi:E_0\longrightarrow K,\\[2pt]
\phi(P\circ Q)=\phi(P)*\phi(Q),\\[2pt]
\phi(1)=I.
\end{gathered}
\tag{CP.38}
\]

The diagonal calibration in CP.2 gives \(\phi (1)=I\). Its extension does not assume a faithful map or a nonzero-kernel test.

Put \(U=\phi (h)\). Section 5.39, NL.4–NL.6, gives \(J\), the ordinary output derivative of the positive diagonal unit, with

\[
\begin{gathered}
J*U=I=U*J,\\[2pt]
J*\phi(P)=\partial_t^{\rm out}\phi(P).
\end{gathered}
\tag{CP.39}
\]

These inverse identities retain the principal-log ray term. For \(L=-\log (w)/(2\pi i)\) and \(C=(2\pi iw)^{-1}\),

\[
\begin{gathered}
-\partial_wL=C+B_{\log},\\[2pt]
B_{\log}=(2i)^{-1}\\[2pt]
\mathbf1_{\Re w<0}\delta(\Im w).
\end{gathered}
\tag{CP.40}
\]

§5.39, NL.7–NL.12 give the literal spatially tensored supported relative primitive for the discrepancy. §5.39, NL.22/NL.26 give the full convergent-spatial supported primitive for the normal shift identity; §5.39, NL.25 is an equality of full currents. These equations account for the spatial-face term, angular-cutoff term and full output components. Section 5.36, (U.17), retains the principal-log correction. The full convergent-spatial shift and presentation primitives are §5.39, (NL.22)–(NL.37).

Define the extension on the actual left presentation by

\[
\begin{gathered}
\Phi(A)=J^k*\phi(P),\\[2pt]
A=h^{-k}\circ P.
\end{gathered}
\tag{CP.41}
\]

It is represented by the complete current \((\partial_t^{\mathrm{out}})^kT_P\). Its common normal distribution order is at most \(k+1\) on each selected compact, by §5.38, (E.13) and derivative transposition. Suppose another presentation has order \(k+r\). Actual algebra gives \(Q=H_L^rP\), where \(H_LP=h\circ P\). For the finite list \(P_j=H_L^jP\), the shift primitive \(B_j\) of §5.39, NL.3, satisfies

\[
 \partial_t^{\rm out}T_{P_j}-T_{P_{j-1}}=\delta B_j.
\]

Hence the difference of the two complete representatives is the literal relative boundary

\[
d\left(\begin{gathered}
(\partial_t^{\rm out})^k\\[2pt]
\sum_{j=1}^{r}(\partial_t^{\rm out})^{j-1}B_j,0
\end{gathered}\right).
\tag{CP.42}
\]

Its primitive remains supported in the same chosen thick cone and has finite normal order at most \(k+r-1\). This proves presentation independence at chain level before any localization universal property is invoked. Finite addition can use a common raised order; restriction and auxiliary-choice compatibility are §5.39, NL.7's explicit partition homotopy and common-cone refinement. Thus \(\Phi\) is the actual canonical linear germ class map, not an abstract extension whose agreement with the original current map is unproved.

##### CP.6.3 Complete finite-order multiplication

Apply (CP.38) to both sides of the actual order-zero identity (CP.35):

\[
 U*\phi(P)=\phi(\sigma(P))*U.
\]

By (CP.39) this gives, for every finite \(l\),

\[
\begin{gathered}
\phi(\sigma^l(P))\\[2pt]
=U^l*\phi(P)*J^l,\\[2pt]
\phi(P)*J^l\\[2pt]
=J^l*\phi(\sigma^l(P)).
\end{gathered}
\tag{CP.43}
\]

For (CP.36)'s finite pair, associativity, (CP.43), the order-zero product theorem and (CP.37) now prove

\[
\begin{gathered}
\Phi(A)*\Phi(B)\\[2pt]
=J^k*\phi(P)*J^l*\phi(Q)\\[2pt]
=J^{k+l}*\phi(\sigma^l(P))*\phi(Q)\\[2pt]
=J^{k+l}*\phi(\sigma^l(P)\circ Q)\\[2pt]
=\Phi(A\circ B).
\end{gathered}
\tag{CP.44}
\]

This is the complete finite-order multiplication calculation. It uses actual finitely iterated \(\sigma\), common-domain order-zero multiplication, the separately proved literal inverse and presentation primitives, and actual canonical associativity. No unrestricted action, faithful representation, closed-range theorem, or limit of exact polynomial identities is needed. A homomorphism extending \(\phi\) and sending \(h^{-1}\) to \(J\) must obey (CP.41) on every actual finite left presentation, so it is unique. This proves the relevant normal Ore extension property directly; invoking its name introduces no missing theorem.

The exact literal primitives used to identify presentations are (CP.40)/(§5.39, NL.11), (§5.39, NL.22)/(§5.39, NL.26), and (CP.42)/(§5.39, NL.35). Order-zero multiplication has the separate literal primitive CP.30. The proper chain maps used by the canonical cup and trace preserve boundaries. Consequently the finite multiplication computation does not suppress a supported correction. The proper map is the triple support intersection over the final endpoint pair, as in CP.3 and CP.5.

This proves the complete canonical kernel ring-product interface for the actual finite-order convergent spatial symbol ring. Unrestricted-input action still requires its own actual relative roof, residual complement term and full-output primitive. Infinite-order linearity and extension-faithfulness, derived actions, propagation, separation of arbitrary modules, finite D-type, finite poles, intrinsic order, full C1, arbitrary analytic proper regularity, and the original BB/KL/GL scope remain substantive requirements. These further interfaces require their own proofs.

![Exact shell and primitive support geometry](assets/shell-homotopy-geometry.png)

*For one spatial complex coordinate these are exact sections at final time length \(\tau =1\). The orange ring is the centred first difference support \(b\alpha \le |\eta |\le c\alpha\), with \(\alpha =0.35\). Fixing the marked allowed point \(\eta =0.95\), the blue ring is the already translated second support \(b(\tau -\alpha )\le |\zeta -\eta |\le c(\tau -\alpha )\), with \(\tau -\alpha =0.65\). Its farthest possible radius from the origin is \(0.95+1.95=2.90<3=c\tau\). The union over all allowed first points lies in the \(c\tau\)-disc by \(|\zeta |\le |\eta |+|\zeta -\eta |\le c\alpha +c(\tau -\alpha )=c\tau\); equivalently, the Minkowski sum of the two centred difference supports lies in that disc. The green annulus is the outer cutoff transition used by the finite Cauchy homotopy; the full primitive has support in the \(B_*\tau\)-disc, with \(c<B_*<A\), inside the same larger cone. The time panel is the exact compact interval \(0\le \alpha \le \tau\). CP.5 gains one time length: density \(O(\tau ^{-2d-1})\) becomes primitive \(O(\tau ^{-2d})\), with bounded spatial \(L^1\) norm and integrability in \(d\tau\). These are sections and proved bounds, rather than a drawing of the full complex support. Proof locators are CP.17–CP.19 and CP.24–CP.31; reproducible plotting source and exact numeric parameters accompany the PNG/SVG.*


The figure has a full-resolution PNG, SVG, editable plotting source and exact coordinate data.


### 5.42. The canonical degree-one directional action

The canonical unshifted cup, controlled excision and normalized shifted proper trace act on arbitrary degree-one directional inputs by the complete ordinary convergent spatial evaluation. The comparison is an equality in the relative current complex with explicit supported and ambient primitives. Finite left time derivatives give the finite-order action.

The angular current is §5.38, E.1–E.3; write \(F^{\mathrm{root}}_P\) and \(T^{\mathrm{root}}_P=\bar\partial F^{\mathrm{root}}_P\) for its form and current. Its fixed-input output derivatives include the transposed normal derivatives of (E.13). Full smooth/current resolutions, finite Čech comparison and controlled excision are §5.34, R.1–R.7. The ungraded coefficient cup, ordered frame and all-degree trace are §5.35, V.1–V.6. Unrestricted-input relative roofs and the full-output collar calculation are §5.37, W.1–W.8. Cofinal input domains and complete ordinary evaluation are §5.40, PB.1–PB.5; the ordinary inverse is PB.6. CA.2 proves composition on one same-anchor nested domain. The separately proved canonical product and class-level localization are §5.41, CP.1–CP.6. The target operation agrees with [Micro-hyperbolic systems, §3.1](https://perso.imj-prg.fr/pierre-schapira/wp-content/uploads/schapira-pub/Microhyp.pdf) and [HolIII, III.1 and IV.2](https://www.kurims.kyoto-u.ac.jp/~kenkyubu/kashiwara/HolIII.pdf).

#### CA.1. The input, cone geometry and exact relative maps

Write \(N=d+1\), \(u=(t,x)\), \(v=(s,y)\), and normal coordinates \(w=s-t\), \(\zeta =y-x\). The input top-form line is ordered \(ds\wedge dy_1\wedge\cdots\wedge dy_d\), ungraded before tracing. Put
\[
\begin{gathered}
r_N=(-1)^{N(N-1)/2},\\[2pt]
G=\left\{\begin{gathered}
\operatorname{Re}w=-l\le0,\\[2pt]
|\operatorname{Im}w|\le\epsilon l,\\[2pt]
|\zeta_j|\le A_jl
\end{gathered}\right\},\\[2pt]
M=\sqrt{\epsilon^2+\sum_jA_j^2}.
\end{gathered}
\tag{CA.1}
\]
The symbol has actual coefficients on a common base and cotangent polydisc,
\[
\begin{gathered}
P=\sum_{n,\beta}p_{n,\beta}(u)h^nz^\beta,\\[2pt]
|p_{n,\beta}|\le B C^n n!r_z^{-|\beta|}.
\end{gathered}
\tag{CA.2}
\]
The cone constants satisfy \(A_j>a_j>1/r_z\). Shrinking to a common base gives the same bound for each fixed finite list of base derivatives, by the proved coefficient Cauchy formula.

Every degree-one germ-colimit input class has a representative \((0,-f)\), with \(f\) holomorphic on
\[
\begin{gathered}
\Omega_g\cap D=\\[2pt]
\left\{\begin{gathered}
\operatorname{Re}s<\\[2pt]
-g(|(\operatorname{Im}s,y)|)
\end{gathered}\right\}\cap D,\\[2pt]
g(0)=0,\\[2pt]
g\text{ convex, nondecreasing},\\[2pt]
g(r)=o(r).
\end{gathered}
\tag{CA.3}
\]
Indeed the smooth relative resolution sends a closed \((a,b)\) after shrinking to \(a=\bar\partial  F\) by §5.34, R.3. Subtracting \(d(F,0)\) leaves \((0,b-rF)\); degree-zero holomorphicity in §5.34, R.3 gives \(f=rF-b\). §5.40, PB.1’s cofinal convex majorants supply (CA.3). This uses no fixed-domain Stein vanishing and no extension of \(f\) as a distribution. Choose a sufficiently small radius so \(g\) has Lipschitz constant smaller than a chosen \(L\) with \(LM<1\) and the extra inequalities in CA.2. For a global notation in the following cone inclusion, extend this restricted \(g\) beyond that radius by the linear ray of slope \(L\). Its left slope is at most \(L\), so the extension is convex, nondecreasing and globally \(L\)-Lipschitz. It has exactly the same cofinal \(o(r)\) germ and input domain on the small ball used here.

For \(u \in  \Omega _g\) and \(a \in  G\), the input margin at \(u+a\) is at least the margin at \(u\) plus \((1-LM)l\). Thus \(\Omega _g+G \subset  \Omega _g\). If the input belongs to its excluded support instead, then
\[
\begin{gathered}
(1-LM)l\\[2pt]
\le\operatorname{Re}t+g(|(\operatorname{Im}t,x)|).
\end{gathered}
\tag{CA.4}
\]
Together with \(|(w,\zeta)|\le\sqrt{1+M^2}\,l\), this bounds the complete input fibre of \(S=G \cap  p_v^{-1}(D\setminus \Omega _g)\) over every compact small output set. It is closed, and an outer source product can be chosen before shrinking the output so that this entire bounded family lies compactly inside \(D\). This proves actual local properness and \(p_u(S) \subset  Z\), where \(Z=B\setminus \Omega _g\).

Choose negative real \(L_0<a<0\), inside the actual input and coefficient products. Take a smooth real \(\theta\) zero below \(L_0\), one above \(a\), with transition compactly contained in \((L_0,a)\). For sufficiently small full output \(B\), the part of \(G\) meeting that transition lies in one fixed compact negative-time tube contained in \(\Omega _g \cap  D\). This follows from (CA.1), the strictly positive time length on the transition, and \(g(r)=o(r)\) after shrinking \(L_0,a\) and \(B\). Make compact input cutoffs in the imaginary/spatial and far-right directions equal to one on an open neighbourhood of the clipped cone over \(B\); their derivatives vanish there. Their product with \(\theta(\operatorname{Re}s)\) gives an input-only cutoff \(\eta\), compactly supported in \(D\), equal to one near \(S\), whose derivative on \(G\) is just the negative-time transition derivative. All constructions below use this full tube, including outputs outside \(\Omega _g\).

For the full Dolbeault complex \(I\) put
\[
\begin{gathered}
C_S^q=I^q\oplus I^{q-1}(M\setminus S),\\[2pt]
d(a,b)=(\bar\partial a,ra-\bar\partial b),\\[2pt]
L_\eta(a,b)=\left(\begin{gathered}
\eta a+\bar\partial\eta\wedge b,\\[2pt]
\eta b
\end{gathered}\right).
\end{gathered}
\tag{CA.5}
\]
§5.37, W.3–W.7 prove the literal chain homotopy \(1-L_\eta =dE_\eta +E_\eta  d\), where \(E_\eta (a,b)=((1-\eta )b,0)\), and the controlled extension-by-zero excision roof. For a closed supported kernel current \(T\) of degree \(N\), the cup is
\[
m_T(a,b)=\left(\begin{gathered}
T\wedge a,\\[2pt]
(-1)^N T\wedge b
\end{gathered}\right).
\tag{CA.6}
\]
It is the canonical unshifted cup by the tensor-to-cube augmentation §5.35, V.15–V.17 and the smooth/current/injective comparisons §5.34, R.6–R.7. Only multiplication of a current by a smooth cochain is used.

Let \(I_q=J_{\mathrm{out}}^{-1}p_*J_{\mathrm{in}}\) denote raw coefficient integration. The exact shifted trace is
\[
\begin{gathered}
\operatorname{Tr}_q=(-1)^{N(q-N)}I_q,\\[2pt]
Q_N(a,b)=\\[2pt]
\left(\begin{gathered}
\operatorname{Tr}_qa,\\[2pt]
\begin{gathered}
(-1)^N\\[2pt]
\operatorname{Tr}_{q-1}b|_{p^{-1}(B\setminus Z)}
\end{gathered}
\end{gathered}\right).
\end{gathered}
\tag{CA.7}
\]
It is a chain map on \(C_S[N]\); in unshifted notation \(Q_N de=(-1)^N d Q_N e\). These are §5.37, W.12–W.14, with all-degree compact Stokes proved in §5.35, V.4. Consequently the actual output for \((0,-f)\) is
\[
\begin{gathered}
O_T(f)=Q_NL_\eta m_T(0,-f)\\[2pt]
=\left(\begin{gathered}
-I_{N+1}(\bar\partial\eta\wedge Tf),\\[2pt]
-I_N(\eta Tf)
\end{gathered}\right)\\[2pt]
=(A_T,-G_T).
\end{gathered}
\tag{CA.8}
\]
Here \(Tf\) is initially only a complement current. Its ambient first component uses \(f\) solely on the full negative-time transition tube, and is smooth on all of \(B\). To define it, multiply \(f\) by an interior bump equal to one near that compact tube and extend this smooth product by zero. Independence follows since \(\bar\partial  \eta  T\) vanishes outside the tube. On the output complement all clipped cone points lie in the holomorphic input domain, so \(G_T\) is defined without any boundary bound. Smoothness follows by compact current pairing in normal coordinates; §5.38, E.13 transposes each finite fixed-input output derivative onto the smooth coefficient. This also applies to the other finite-order kernel primitives constructed below.

#### CA.2. Composition on a common nested domain

Use the same anchor \(a=-a_0\) for both factors. Let \(D_0=\max (1,d)\), \(\kappa =4/r_z\), and reserve a spatial/base radius smaller than the base radius \(b\) of §5.40, PB.2 before choosing \(a_0/R\). Choose \(L\le 1/16\), \(L\sqrt{D_0}\,\kappa\le1/16\), \(C|t-a|<1/2\), and enough initial radius so that all spatial deviations below are less than that reserved radius. This is possible in the noncircular order of §5.40, PB.2. Fix a compact target set \(K \subset  \Omega _g\), with \(|t|<a_0/4\), and let \(\mu _*\) be its positive minimum input margin. Choose a fixed
\[
\begin{gathered}
0<\delta\le\\[2pt]
\min(b/8,\mu_*/(16L\sqrt{D_0})),
\end{gathered}
\tag{CA.9}
\]
omitting the second bound if \(L=0\). The outer and inner times are
\[
\begin{gathered}
s=a+\theta(t-a),\\[2pt]
r=a+\sigma(s-a),\\[2pt]
|t-s|+|s-r|=|t-r|\\[2pt]
(0\le\theta,\sigma\le1).
\end{gathered}
\tag{CA.10}
\]
Use outer spatial Cauchy radius \(\delta +\kappa |t-s|\) and inner radius \(\delta +\kappa |s-r|\). Every twice-used spatial input point differs from \(x\) by at most \(\sqrt d\,(2\delta+\kappa|t-r|)\). the ratio in §5.40, (PB.7), unchanged on this common segment, gives \(-\operatorname{Re}  r\ge -\operatorname{Re}  t+|t-r|/2\). The input margin therefore satisfies
\[
\begin{gathered}
-\operatorname{Re}r\\[2pt]
-g(|(\operatorname{Im}r,z)|)\\[2pt]
\ge\mu_*-2L\sqrt d\delta\\[2pt]
+(1/2-L-L\sqrt d\kappa)\\[2pt]
|t-r|\\[2pt]
\ge7\mu_*/8+3|t-r|/8>0.
\end{gathered}
\tag{CA.11}
\]
The spatial radii and time lengths are bounded before either symbol is approximated. All these nested sets are one compact subset of the original input domain. At \(s=a\) the inner integral has length zero, and its multiplication term is defined on the full fixed spatial anchor tube. Holomorphy near the anchor follows from that tube, so the outer Cauchy estimates include the anchor without a domain gap.

If \(M_f\) bounds \(f\) on the nested compact, §5.40, (PB.11)–(PB.13), bound the right action on every outer spatial Cauchy disc by a constant times \(\Vert Q\Vert  M_f\); the constant depends on \(\delta\) and the fixed geometric margins, not on finite Taylor projections. In that estimate the inner ratios are \(|s-r|/(r_z(\delta +\kappa |s-r|))\le 1/4\) and \(C|s-r|<1/2\). Applying the outer estimate gives a bound \(C_K \Vert P\Vert  \Vert Q\Vert  M_f\) for the composition. The actual absolute norm is the weighted \(\ell^1\)-norm of §5.21.1, (5.21b)–(5.21f); its product is submultiplicative by (5.21d), using §5.20’s complete allocation estimate. Thus finite monomial projections converge, their products converge, and these two actual operator estimates pass the compositions to the same-domain limit.

For precision the finite identity is also an actual identity. With \(m=n+|\beta |\ge 1\), differentiate the right term spatially by finite Leibniz, Taylor-expand its holomorphic coefficient at \(t\) on the common coefficient disc, and integrate on the nested triangle \(a\le r\le s\le t\). Uniform coefficient Taylor convergence and the preceding input bounds allow each exchange. The integer beta integral gives \(\binom{-m}{k}\) for the time contractions and \(\binom{\beta}{\nu}\) for the spatial contractions, exactly the normal product of §5.19, with the complete estimates of §5.20–§5.21.1; multiplication on the left is literal multiplication. No independent anchors are compared. Hence
\[
\begin{gathered}
\rho_a(P)\rho_a(Q)=\rho_a(P\circ Q),\\[2pt]
\rho_a(1)=1
\end{gathered}
\tag{CA.12}
\]
where the finite identities and limit are taken on smaller common target neighbourhoods. §5.40, PB.5’s normally convergent full-neighbourhood anchor corrections and extending-input argument descend this to the germ quotient. The intermediate points need not lie in the small target neighbourhood of §5.40, PB.3. The common nested estimates above prove the required input-domain bounds directly.

#### CA.3. An actual spatial-annulus representative

Choose \(0<\epsilon _1<\epsilon\) and numbers
\[
\begin{gathered}
a_j<c_j<b_j\\[2pt]
<A_j/\sqrt{1+\epsilon_1^2}.
\end{gathered}
\tag{CA.13}
\]
First make \(\epsilon _1\) small enough for these inequalities. Let \(\chi _0(w)\) be a smooth angular function zero on an angular neighbourhood of the negative ray and one outside a narrower temporal wedge contained strictly in \(|\operatorname{Im}  w|<\epsilon _1(-\operatorname{Re}  w)\). It extends smoothly over the negative cut as zero. Its intermediate values and derivatives have negative real time.

For a small positive \(e\), put
\[
\begin{gathered}
r=|(w,\zeta)|,\\[2pt]
R_e=\sqrt{|w|^2+e^2r^2},\\[2pt]
\chi_j=\chi_{c_j,b_j}(|\zeta_j|/R_e)\\[2pt]
(1\le j\le d).
\end{gathered}
\tag{CA.14}
\]
Here the flat step is zero below \(c\), one above \(b\), and in between is
\(q(v-c)/(q(v-c)+q(b-v))\), with \(q(h)=\exp (-1/h)\) for \(h>0\) and zero otherwise. It is smooth and flat at both ends, by direct differentiation of the exponential. Since \(R_e>0\) on the punctured full normal space and the step is zero near \(\zeta _j=0\), these spatial functions are smooth there. They have degree zero under real radial scaling.

Take \(e^2 \sum  b_j^2<1\). Wherever all spatial cutoffs are below one, or all their derivatives occur,
\[
\begin{gathered}
r^2\le |w|^2\\[2pt]
+(\sum b_j^2)(|w|^2+e^2r^2),\\[2pt]
r\le C_e|w|.
\end{gathered}
\tag{CA.15}
\]
Choose \(e\) still smaller so the upper bounds \(b_j\sqrt{1+e^2C_e^2}\sqrt{1+\epsilon_1^2}\) are strictly less than \(A_j\). Thus the intersection of the temporal and spatial transition regions lies compactly in the angular interior of \(G\). At \(w=0,r>0\), at least one spatial cutoff is identically one on a neighbourhood; this observation resolves the apparent singularity of the temporal angular function when multiplied by the spatial factors below.

Use the unsigned coefficient of §5.38, E.2, \(K_P\):
\[
\begin{gathered}
K_P={1\over(2\pi i)^N}\\[2pt]
\left[\begin{gathered}
{p_{0,0}(u)\over w\prod\zeta_j}\\[2pt]
-\log w\sum_{n+|\beta|\ge1}\\[2pt]
{\begin{gathered}
p_{n,\beta}(u)\beta!\\[2pt]
(-w)^{n+|\beta|-1}
\end{gathered}\over \begin{gathered}
(n+|\beta|-1)!\\[2pt]
\prod\zeta_j^{\beta_j+1}
\end{gathered}}
\end{gathered}\right].
\end{gathered}
\tag{CA.16}
\]
Define
\[
\begin{gathered}
F^{\rm ann}_P=r_N\chi_0K_P\,\\[2pt]
\bar\partial\chi_1\wedge\cdots\wedge\bar\partial\chi_d,\\[2pt]
T^{\rm ann}_P=\bar\partial F^{\rm ann}_P.
\end{gathered}
\tag{CA.17}
\]
For \(d=0\) the empty wedge is one. Every multiplier support in (CA.17) is compactly inside the complete coefficient intersection \(U_0\cap\cdots\cap U_d\) on the normal sphere: (CA.15) separates \(w\) from zero, \(\chi _0\) separates the logarithmic cut, and \(\bar\partial  \chi _j\) separates each spatial pole. Extension by zero is smooth on the punctured normal space. The §5.38, E.5–E.6 geometric sum gives
\[
\begin{gathered}
|F^{\rm ann}_P|\le C r^{-(2N-1)}\\[2pt]
(1+|\log r|).
\end{gathered}
\tag{CA.18}
\]
Polar integration proves its literal \(L^1\) extension to the origin. Its differential has order at most one. Away from the origin the differential is \(r_NK_P\,\bar\partial\chi_0\wedge\cdots\wedge\bar\partial\chi_d\), supported in \(G\); the origin belongs to \(G\). Thus this is a closed supported current with the same actual coefficient series and no infinite spatial delta derivatives.

#### CA.4. Explicit supported homotopy from the angular current

The following smooth partition is chosen to make the homotopy quantitative. Set
\[
\begin{gathered}
\vartheta_j=1-(1-\chi_j)^{1/j},\\[2pt]
\phi_j=\vartheta_j\prod_{k>j}(1-\vartheta_k)\\[2pt]
(1\le j\le d),\\[2pt]
Q=\prod_{k=1}^d(1-\vartheta_k),\\[2pt]
\phi_0=\chi_0Q,\\[2pt]
\phi_G=(1-\chi_0)Q.
\end{gathered}
\tag{CA.19}
\]
The displayed fractional powers are smooth: near each flat endpoint they are an exponential \(\exp (-1/(j h))\) times a smooth positive factor; away from that endpoint ordinary differentiation applies. All spatial factors are smooth on the punctured normal space. The factors involving \(\chi _0\) vanish on an open neighbourhood of every nonzero point with \(w=0\), by (CA.15). Hence all \(\phi\) are smooth homogeneous functions there. Their sum is one by telescoping. Each \(\phi _i\) is subordinate to \(U_i\), with compact angular margins, and \(\phi _G\) is compactly supported in the angular interior of \(G\).

For an angular partition \(q=(q_0,\cdots,q_d)\) write
\(\omega_q=\sum_i(-1)^iq_i\,\bar\partial q_0\wedge\cdots\wedge\widehat{\bar\partial q_i}\wedge\cdots\wedge\bar\partial q_d\).
On the complement of \(G\), either a spatial cutoff is locally one, making both \(\omega _\phi\) and \(\chi_0\bigwedge_j\bar\partial\chi_j\) zero, or \(\chi _0=1\) and \(\sum _i \phi _i=1\). In the latter region \(\omega _\phi =\bar\partial  \phi _1 \wedge  \cdots \wedge  \bar\partial  \phi _d\). The triangular Jacobian of \((\chi _1,\cdots,\chi _d)\to (\phi _1,\cdots,\phi _d)\) is
\[
\begin{gathered}
\prod_{j=1}^d\vartheta_j'\\[2pt]
\prod_{k=1}^d(1-\vartheta_k)^{k-1}\\[2pt]
=\prod_{j=1}^d{1\over j}={1\over d!}.
\end{gathered}
\tag{CA.20}
\]
Consequently
\[
\begin{gathered}
d!\,\omega_\phi=\chi_0\bar\partial\chi_1\wedge\cdots\\[2pt]
\wedge\bar\partial\chi_d\\[2pt]
\text{on }M\setminus G.
\end{gathered}
\tag{CA.21}
\]
Boundary points follow by smoothness and the flat plateau. This proves directly that the annular and partition representatives have the same complement normalization.

Let \(\psi\) be the angular partition of §5.38, E.2 and interpolate \(q(\lambda )=(1-\lambda )\psi +\lambda  \phi\), including its \(G\) component. In (CA.19) no angular partition is changed. Use the odd total differential \(D=\bar\partial+d\lambda\,\partial_\lambda\) and form \(\omega _q\) with \(Dq_i\). Put, for \(d\ge 1\),
\[
\begin{gathered}
b_P=r_Nd!K_P\int_0^1\\[2pt]
\iota_{\partial_\lambda}\omega_{q(\lambda)}\,d\lambda,\\[2pt]
H_P=F^{\rm root}_P-F^{\rm ann}_P\\[2pt]
+\bar\partial b_P.
\end{gathered}
\tag{CA.22}
\]
For \(d=0\) take \(b_P=0\) and the same formula for \(H_P\). On the complement \(\sum _i q_i=1\), hence \(\sum _i Dq_i=0\); expanding gives \(D\omega_q=N\bigwedge_iDq_i=0\). The coefficient of \(d \lambda\) says \(\partial_\lambda\omega_q=\bar\partial(\iota_{\partial_\lambda}\omega_q)\). Since \(K_P\) is holomorphic on all multiplier supports, integration proves \(\bar\partial b_P=F_\phi-F^{\mathrm{root}}_P\) there. Equations (CA.21)–(CA.22) therefore give \(H_P=0\) outside \(G\), while
\[
 \bar\partial H_P=T^{\rm root}_P-T^{\rm ann}_P.
 \tag{CA.23}
\]
For \(d=0\), both cutoffs are one outside \(G\), so the same support assertion is immediate.

This is a literal supported primitive for the full coefficient, not an assertion of cohomological continuity. Every term of the integrand has one partition factor or its \(\lambda\)-derivative for each normal open and \(N-2\) normal differential factors. Its angular support is compactly inside the full coefficient intersection. Thus \(|b_P|\le C r^{-(2N-2)}(1+|\log  r|)\). This is \(L^1\), as is each \(F\). Its distributional derivative exists by transposition. \(H_P\) has order at most one and is smooth away from the normal origin; its support assertion remains valid at the origin because that point belongs to \(G\). All fixed-normal base derivatives satisfy the same bounds. The weighted geometric tails in §5.38, E.3 bound \(F\) and \(b\) by one scalar tail tending to zero, and hence bound \(H\) with one common first-order test norm. This supplies controlled primitives for the entire convergent series.

The equality is also an actual canonical Čech roof. Section 5.38, E.10 identifies \(F^{\mathrm{root}}_P\) with the descent of ordinary full-cone complement coefficient \(-r_N K_P\); (CA.21) and (CA.22) give its explicit smooth homotopy to \(F^{\mathrm{ann}}_P\). Adding \(d(F^{\mathrm{ann}}_P,0)\) sends \((0,-F^{\mathrm{ann}}_P)\) to \((T^{\mathrm{ann}}_P,0)\), and (CA.23) is the supported ambient homotopy between the current endpoints. The augmentations of §§5.34–5.35, R.6–R.7 and §5.35, V.6, and §5.37’s cup/excision/trace maps therefore transport this same homotopy through the canonical operation. No Leray cover or polynomial-class limit is used.

For later use, a supported \(H\) of degree \(N-1\) with \(T'-T=\bar\partial  H\) changes (CA.8) by
\[
\begin{gathered}
O_{T'}(f)-O_T(f)\\[2pt]
=d(D_H,0),\\[2pt]
D_H=I_N(\bar\partial\eta\wedge Hf).
\end{gathered}
\tag{CA.24}
\]
Indeed on the output complement compact Stokes gives
\(I_N(\eta  \bar\partial  H f)=-I_N(\bar\partial  \eta  \wedge  Hf)\), since raw integration of a degree \(N-1\) compact coefficient is zero. On the full output neighbourhood,
\(-I_{N+1}(\bar\partial  \eta  \wedge  \bar\partial  H f)=\bar\partial  D_H\), using only the holomorphic negative-time tube. These two identities prove (CA.24) component by component with the shift signs of §5.35, V.4, and §5.37, (W.12)–(W.14). For (CA.22), \(D_H\) is smooth on all of \(B\) and all its actual tails converge with every fixed output derivative. The primitive has support proper before tracing. This gives the required controlled difference between the angular and annular actions.

#### CA.5. The spatial Cauchy trace is a full canonical trace

Fix \(w\ne 0\) in the temporal wedge, and a small allowed output. All spatial multiplier supports in (CA.17) lie in one polydisc contained in the holomorphic input domain, by (CA.13)–(CA.15) and the margin inequality in CA.1. The strict inequalities allow a slightly wider time wedge and slightly larger spatial polydiscs, still inside the same cone margins. Together with a small full normal neighbourhood of the diagonal for each allowed output, these give an actual open neighbourhood on which \(F^{\mathrm{ann}}_P f\) is defined and which contains the support needed in this calculation. Over compact subsets of the output complement these neighbourhoods can be chosen uniformly. The primitive and partial-current identities below are computed on this open neighbourhood, where \(f\) is holomorphic, and then transported by the controlled excision maps of §5.37, W.3–W.7, and §5.34, R.7. Outside the temporal support the traced supported current is zero; a globally defined \(F^{\mathrm{ann}}_P f\) at other positive-time input points is not used. For the ambient first component only the fixed full-output negative-time transition tube is used, as in CA.1 and CA.6.

Replace \(e\) in \(R_e\) continuously by \(v \in  [0,e]\). At fixed nonzero \(w\) the cutoffs are smooth, and their joint differential product has compact spatial support away from every coordinate pole. Its derivative is the spatial Dolbeault differential of
\[
\begin{gathered}
B_v=\sum_{j=1}^d(-1)^{j-1}\dot\chi_j^v\\[2pt]
\bar\partial_\zeta\chi_1^v\wedge\cdots\\[2pt]
\widehat{\bar\partial_\zeta\chi_j^v}\cdots\\[2pt]
\wedge\bar\partial_\zeta\chi_d^v.
\end{gathered}
\tag{CA.25}
\]
Every multiplier, including the dotted one, has annular support in its coordinate, so the holomorphic coefficient and \(f\) are defined on all these supports. Compact Stokes annihilates its spatial integral. Thus the trace is exactly its value at \(v=0\), where the cutoffs depend independently on \(|\zeta _j|/|w|\).

For one spatial coordinate, expanding the holomorphic Taylor series on a disc containing the annulus and integrating its angular terms gives
\[
\begin{gathered}
{1\over2\pi i}\int {\bar\partial\chi_j\over\zeta_j^{\beta_j+1}}\\[2pt]
f(s,x+\zeta)\wedge d\zeta_j\\[2pt]
={1\over\beta_j!}\partial_{x_j}^{\beta_j}f(s,x).
\end{gathered}
\tag{CA.26}
\]
The radial integral is \(\int\chi_j^{\prime}(r)\,dr=1\); the angular integral retains exactly exponent \(\beta _j\), with positive \(d\bar\zeta\wedge d\zeta=2i\,r\,dr\wedge d\theta\). Iteration in the ordered spatial coordinates gives the full Cauchy formula. Normal coefficient convergence is uniform on these annuli because every ratio \(|w|/(r_z|\zeta _j|)\) is bounded strictly below one. Hence the complete series, including \(n=0\) and unbounded \(\beta\), is integrated there as an actual normally convergent coefficient.

The holomorphic block and coefficient sign are explicit. Moving \(ds\) past the \(d\) spatial antiholomorphic factors, then grouping each spatial normal block, contributes \((-1)^d r_d\). Together with \(r_N\), this is one: \(r_N(-1)^d r_d=1\). The partial trace of \(F^{\mathrm{ann}}_P f\) is therefore the literal locally integrable one-time primitive
\[
\begin{gathered}
\chi_0(w)K_f(u,w),\\[2pt]
K_f={1\over2\pi i}\\[2pt]
\left[\begin{gathered}
{p_{0,0}(u)f(t+w,x)\over w}\\[2pt]
-\log w\,\Phi_P(u,t+w)
\end{gathered}\right],
\end{gathered}
\tag{CA.27}
\]
where
\[
\begin{gathered}
\Phi_P(u,s)=\sum_{n+|\beta|\ge1}\\[2pt]
p_{n,\beta}(u){(t-s)^{n+|\beta|-1}\over(n+|\beta|-1)!}\\[2pt]
\partial_x^\beta f(s,x).
\end{gathered}
\tag{CA.28}
\]
For outputs in a compact subset of \(\Omega _g\), \(f\) is holomorphic on a fixed neighbourhood of the diagonal, so (CA.28) is holomorphic and normally convergent near \(w=0\). Away from that diagonal the preceding annular margins apply on the specified open support neighbourhood. The normal \(L^1\) bound (CA.18), integrated spatially, is \(C|w|^{-1}(1+|\log|w||)\), locally integrable in the two real time dimensions. Thus the equality off \(w=0\) extends there as equality of \(L^1\) primitives; no point distribution can be added. A fixed spatial cutoff flat on their supports makes partial integration proper on a smaller product. It commutes with the complete differential by compact Stokes, so on that neighbourhood the partial trace of \(T^{\mathrm{ann}}_P f\) is exactly \(\bar\partial (\chi _0 K_f)\), including its diagonal and output antiholomorphic components. Controlled excision transports this identity of supported currents into (CA.8), keeping its complement component on the output complement and its ambient first component on the full transition tube. This proves a chain trace comparison, rather than identifying it from a contour action.

#### CA.6. One-time angular-to-ray comparison and the full ambient primitive

We use (CA.27) near \(w=0\) for outputs in \(\Omega _g\); its values on the negative-time transition tube are defined for every output in \(B\). The literal principal \(K_f\) is \(L^1\) in time on those domains. Its Cauchy differential gives \(p_{0,0}(u)f(u)\) on the diagonal, and its negative-log differential gives the positive ray density with holomorphic coefficient (CA.28), by the explicit one-coordinate §5.36, U.5–U.7 calculation. Define
\[
 H_f^{\rm time}=(\chi_0-1)K_f.
 \tag{CA.29}
\]
It is supported in the narrower negative-time wedge and is \(L^1\) there. The exact identity \(\bar\partial (\chi _0 K_f)-\bar\partial  K_f=\bar\partial  H_f^{\mathrm{time}}\) is valid as currents. Apply the one-time version of (CA.24): the two relative traced outputs differ by \(d(D_{\mathrm{time}},0)\), where
\[
D_{\rm time}=I_1\left(\begin{gathered}
\bar\partial\eta_{\rm time}\wedge\\[2pt]
H_f^{\rm time}
\end{gathered}\right).
\tag{CA.30}
\]
Only the fixed negative-time transition contributes. On that compact tube \(\Phi _P\) is an actual joint holomorphic series on a full smaller output neighbourhood: spatial Cauchy radii proportional to the strictly positive time length give all ratios below a fixed number less than one, and \(C|t-s|<1/2\). The logarithmic cut is a fixed \(L^1\) normal coefficient. Parametrizing in \(w\) and differentiating the smooth compact coefficient proves \(D_{\mathrm{time}}\) smooth on the full output neighbourhood. Every fixed derivative and every series tail has the same compact bounds. The imaginary input cutoffs are flat on this wedge, as arranged in CA.1. Consequently (CA.30) is a full ambient primitive, with no assertion about boundary growth.

The ray calculation of §5.37, W.19–W.24 now applies to the joint holomorphic function \(\Phi _P(u,s)\). Its anchored output is precisely
\[
\begin{gathered}
V_P(u)=p_{0,0}(u)f(u)\\[2pt]
+\int_a^t\Phi_P(u,s)\,ds.
\end{gathered}
\tag{CA.31}
\]
This is §5.40, (PB.10), with every spatial Taylor term. Its normal convergence and holomorphy on the common input/target domains are §5.40, PB.3–PB.4, (§5.40, PB.8)–(§5.40, PB.13). Points of the triangle between \(a\), \(a+i \operatorname{Im}  t\) and \(t\) have the form \(s=a+\theta (t-a)+i \nu (1-\theta )\operatorname{Im}  t\), with \(0\le \theta ,\nu \le 1\). They satisfy \(|t-s|\le (1-\theta )|t-a|\) and \(-\operatorname{Re}  s+\operatorname{Re}  t=(1-\theta )(\operatorname{Re}  t-a)\ge |t-s|/2\). Thus §5.40, (PB.8)’s complete spatial-disc margins hold on this triangle, including the vertical anchor edge. A holomorphic primitive changes its straight segment to the horizontal/vertical path. The full smooth correction is
\[
\begin{gathered}
E_{\rm ray}(u)=\\[2pt]
\int_a^{a+i\operatorname{Im}t}\Phi_P(u,s)\,ds\\[2pt]
-\int_{L_0+i\operatorname{Im}t}^{a+i\operatorname{Im}t}\\[2pt]
\theta(\operatorname{Re}s)\Phi_P(u,s)\,ds.
\end{gathered}
\tag{CA.32}
\]
All its input points belong to one fixed full negative-time tube. The geometric estimates for all \(\beta\) there converge uniformly with every fixed derivative. The §5.37, W.23 integration by parts gives its first component on every output, including the excluded output support:
\(\bar\partial_tE_{\mathrm{ray}}=-(1/2)\int\theta^{\prime}(r)\Phi_P(u,r+i\operatorname{Im}t)\,dr\), and there is no \(\bar\partial _x\) term. The ray output is exactly \((0,-V_P)+d(E_{\mathrm{ray}},0)\); its Cauchy multiplication part has zero first component and positive connecting representative \((0,-p_{0,0}f)\).

Combining (CA.24), (CA.27)–(CA.32), with \(H_P\) from (CA.22), proves the main literal equality
\[
\begin{gathered}
Q_NL_\eta m_{T^{\rm root}_P}(0,-f)\\[2pt]
=(0,-V_P)+d(E_P,0),\\[2pt]
E_P=E_{\rm ray}+D_{\rm time}\\[2pt]
+I_N(\bar\partial\eta\wedge H_Pf).
\end{gathered}
\tag{CA.33}
\]
Every term of \(E_P\) is smooth on a full output neighbourhood and is formed from the convergent coefficient and fixed interior input tubes. The canonical cube, augmentation, excision and shifted trace maps are (CA.5)–(CA.7); the supported homotopy is CA.22; the actual spatial trace homotopy is CA.25; and the temporal supported homotopy is CA.29. These explicit maps establish canonical equality, through the actual relative cup, excision and trace maps.

#### CA.7. Interior collars, changes, derivatives and remaining scope

For any smooth interior cutoff \(\chi\) compactly supported in the holomorphic input domain, \(F=\chi  f\) extends smoothly by zero, but its relative representative is
\[
\begin{gathered}
(\bar\partial F,F-f),\\[2pt]
F-f=(\chi-1)f.
\end{gathered}
\tag{CA.34}
\]
The residual term is retained. §5.37, W.10–W.14 are chain identities for any finite-order supported current, so they give
\[
\begin{gathered}
Q_NL_\eta\\[2pt]
m_{T_P}(\bar\partial F,F-f)\\[2pt]
=(0,-V_P)\\[2pt]
+d(E_P+J_\chi,0),\\[2pt]
J_\chi=I_N(\eta T_PF).
\end{gathered}
\tag{CA.35}
\]
\(J_\chi\) is smooth on the full output neighbourhood: in normal coordinates it pairs the smoothly parameterized order-one current with a smooth compact input coefficient. Each fixed-input output derivative is the finite combination of base derivatives and transposed normal derivatives in §5.38, E.13. Their bounded test norms prove the asserted smoothness and actual tail convergence. This argument uses one finite current order, independently of all spatial Taylor terms.

For a second collar \(F'\), the difference is \(d(I_N(\eta  T_P(F'-F)),0)\). For a second excision cutoff \(\eta '\), the output difference is \(-d(I_N((\eta '-\eta )T_Pf),0)\), exactly §5.37, W.26–W.27. The latter coefficient is defined on a compact subset of the input complement avoiding \(S\); an interior bump there gives its smooth zero extension, and compact finite-order current pairing gives a full smooth primitive. All proper supports and excision refinements are those in §5.37, W.4–W.7/§5.34, R.24. Changing angular partitions or annulus parameters gives the supported homotopies (CA.22)/(CA.25), so it changes the output by the displayed full-tube primitives. Changing negative anchors gives §5.40, PB.5’s normally convergent holomorphic full-neighbourhood correction. Extending inputs give extending outputs by that same estimate. These formulas prove the quotient, collar, domain, support and germ compatibilities for arbitrary original degree-one relative classes.

Finite output time derivatives commute with the canonical cup/trace by test-function transposition and proper Stokes, and with the selected input-only \(\eta\). For a general base-dependent cutoff, §5.37, W.8's exact commutator homotopy \([D,L_\eta ]=dK_D+K_Dd\), \(K_D(a,b)=((D \eta )b,0)\), supplies the replacement. Thus a finite left shift \(h^{-k} \circ  P\) acts canonically by \(\partial _t^k V_P\). CA.2 proves the order-zero action multiplicative; §5.40, PB.6, proves that the ordinary \(h\) primitive and \(\partial _t\) actions are inverse modulo the holomorphic anchor correction. The actual normal automorphism is fully proved in §5.22.1, equations (5.22a)–(5.22b): \(hP=\sigma (P)h\), with \(\sigma\) an actual algebra automorphism, and finite shifts preserve the common-domain factorial class. Each finite-order symbol is a finite left \(h^{-k}\) shift of an order-zero symbol by §5.19's order definition and the finite-shift calculation §5.21.2, (5.21h)–(5.21i). Equality of two such presentations is tested by multiplying by a common positive power of the regular normal \(h\); their actions then agree because \(\rho (h)\) is invertible. Multiplication of presentations uses \(hP=\sigma (P)h\) and its inverse, and the already proved action identity carries the same relation. This directly proves the presentation-independent full finite-order symbol action, without an extra localization theorem. It is action independence; it does not by itself identify different supported kernel representatives under all localized ring presentations. The canonical ring product and class-level localization are §5.41, CP.1–CP.6, using §5.39, NL.2–NL.7.

The closed result is the unrestricted-growth canonical degree-one action for the full convergent spatial symbol coefficients and their finite time shifts, with common domains, actual unbounded Taylor sums, residual complement, supported homotopies and full ambient primitives. It does not prove arbitrary derived-degree \(E\)-linearity, infinite-order extension/flatness, propagation, module separation, finite D-type embeddings, finite-pole recovery, intrinsic-order comparison, full C1, arbitrary analytic proper-image regularity, or any full GL/BB/KL/affine/critical/factorization completion. Those further interfaces require their own proofs.

![Proper thick-cone slice, nested Cauchy domains and the controlled primitive chain](assets/full-action-domain-and-primitives.png)

Figure 5.42.1 is an exact real slice of (CA.1)/(CA.4) with \(\epsilon =1/4\), \(A=3\), \(g(r)=r^2/20\), \(L=1/10\) on \(r\le 1\), and output \((\operatorname{Re}  t,x)=(1/10,1/20)\). It labels the complete local support-intersection bound. The middle panel has exact rational coordinates \(a=-3/20\), \(t=-3/200+i/50\), \(\theta =13/20\), \(\sigma =2/5\), \(\delta =1/250\), \(\kappa =2/5\); the two nested time lengths add exactly as in (CA.10). The right panel records the supported primitive (CA.22) and all three full-output corrections in (CA.33), retaining the collar residual (CA.34). This is a real section, a complex-time sample and a diagram of proved maps. Reproducible source and exact coordinate data are retained; the real section, time sample and primitive diagram use exactly these coordinates.


The figure has a full-resolution PNG, SVG, editable plotting source and exact coordinate data.


### 5.43. The canonical all-degree action and its derived module structure

The canonical kernel cup and normalized proper trace act in every relative degree. A proved auxiliary concentration theorem then constructs an actual module-derived supported object. Its underlying action is identified with the same canonical cup, controlled excision and shifted trace. Bounded finite-projective contractions transfer to this module model with explicit tensor and internal-Hom signs.

The actual factorial-growth symbol rings, finite common domains, normal automorphism and finite module relations are §§5.19–5.22. The complete smooth/current resolutions, injective comparisons and controlled proper excision are §5.34, R.1–R.7. The ungraded coefficient cup, ordered finite complement cube, tensor suspension and normalized all-degree trace are §5.35, V.1–V.6. The one-normal calibration is §5.36, U.1–U.4. Arbitrary-cochain collar roofs are §5.37, W.1–W.8. Actual full spatial currents and their fixed-normal base bounds and transposed normal derivatives are §5.38, E.1–E.3, especially (E.13). Finite time shifts and their supported primitives are §5.39, NL.1–NL.7. Actual directional input domains and ordinary evaluation are §5.40, PB.1–PB.6; the full canonical product is §5.41, CP.1–CP.6; and the calibrated degree-one action, full supported homotopies and common-domain geometry are §5.42, CA.1–CA.7.

Write \(R=E(G;D)\) for the canonical kernel cohomology ring of the full thick cone and round coefficient domain specified below. Sections 5.39 and 5.41 give an actual symbol-ring map to the filtered canonical kernel-germ ring, including finite positive time orders. They do not assert an isomorphism onto every canonical kernel class. Section 5.42 supplies actual bounds and calibration; its degree-one action is not used to infer the ring map or the derived module structure proved here.

The only global analytic input is proved completely in DLH.1–DLH.4 below, including the alternating coefficient expansion, weighted adjoint calculation, graph-domain approximation and unrestricted-input weight construction. The target local module category and geometric conditions are those of [Micro-hyperbolic systems, §§3.1–3.2.5](https://perso.imj-prg.fr/pierre-schapira/wp-content/uploads/schapira-pub/Microhyp.pdf) and [HolIII, IV.3–IV.4](https://www.kurims.kyoto-u.ac.jp/~kenkyubu/kashiwara/HolIII.pdf). Every argument used is supplied here or in the indicated earlier proof loci.

#### DL.1. The actual all-degree cup and trace

Put \(N=d+1\), \(u=(t,x)\), \(v=(s,y)\), \(w=s-t\), \(\zeta=y-x\). The coefficient frame \(ds\wedge dy_1\wedge\cdots\wedge dy_d\) is ungraded. With \(\delta=\bar\partial\), the fixed relative convention is
\[
\begin{gathered}
C_Z^q(I)=I^q\oplus k_*I^{q-1},\\
d(a,b)=(\delta a,ra-\delta b).
\end{gathered}
\tag{DL.1}
\]
R.3–R.7 prove that the smooth and current resolutions and this relative cone compute the actual holomorphic derived support objects, including proper supports. The input resolution is smooth. Complement cochains remain on their actual open complement.

For a supported current \(K\) of degree \(p\), with support \(A_K\), put \(S=A_K\cap B_{\rm in}\), where \(B_{\rm in}\) is the pulled-back input support. The cup is
\[
\begin{gathered}
m_K(a,b)=(Ka,(-1)^pKb),\\
d m_K=m_{\delta K}+(-1)^p m_Kd .
\end{gathered}
\tag{DL.2}
\]
Its complement entry is \(Kb\) on the input complement and zero on the kernel complement. They agree on their overlap and glue on the complement of \(S\). This multiplies a current only by a smooth cochain. The sign is V.16, which moves the input complement symbol past \(p\) antiholomorphic degrees. Direct Leibniz expansion proves the second identity: its second entry is
\((-1)^{p+1}\delta K\,b+K(ra-\delta b)\), exactly
\(r(Ka)-\delta((-1)^pKb)\).

A proper collar \(\eta=1\) near \(S\), supported in an actual excision neighbourhood, gives
\[
\begin{gathered}
L_\eta(a,b)=\\
(\eta a+\delta\eta\,b,\eta b),\\
E_\eta(a,b)=((1-\eta)b,0),\\
dL_\eta=L_\eta d,\\
1-L_\eta=dE_\eta+E_\eta d.
\end{gathered}
\tag{DL.3}
\]
The terms with \(1-\eta\) or \(\delta\eta\) vanish near \(S\) before extension by zero. Thus unrestricted growth of \(b\) at that boundary is allowed. R.23–R.24 and W.6 give the extension-by-zero excision roof; arbitrary restriction of proper supports is not used.

Let \(I_q\) be ordered raw fibre integration in the anti-first convention. V.10–V.11 prove compact Stokes and its zero boundary case. If \(p(S)\subset Z\), the relative trace is
\[
\begin{gathered}
\operatorname{Tr}_q=(-1)^{N(q-N)}I_q,\\
Q_N(a,b)=\\
\left(\begin{gathered}
\operatorname{Tr}_qa,\\
(-1)^N\\
\operatorname{Tr}_{q-1}b|_{p^{-1}(X\setminus Z)}
\end{gathered}\right),\\
Q_Nd=(-1)^NdQ_N .
\end{gathered}
\tag{DL.4}
\]
The trace is zero below degree \(N\). The first identity makes it a map out of the \(N\)-shifted complex; the additional second-component sign makes it a map on the shifted relative cone. Both identities follow by substituting (DL.1), including degree \(N-1\), where compact Stokes makes the vertical boundary integral zero.

For a closed degree-\(N\) kernel,
\[
\begin{gathered}
\mathcal A_T=Q_NL_\eta m_T,\\
d\mathcal A_T=\mathcal A_Td .
\end{gathered}
\tag{DL.5}
\]
The two degree-\(N\) signs cancel in every input degree. V.15–V.17 identify \(m_T\) with the canonical unshifted cup in the finite complement cube. Its Čech augmentation is a quasi-isomorphism by the actual partition contraction and terminating row elimination. R.6–R.7 identify its smooth/current and injective proper-support models. W.6 and (DL.4) then give the actual support-to-proper-image and shifted holomorphic trace roof. These maps supply the canonical derived morphism before any cohomological concentration is assumed.

#### DL.2. All-degree homotopies and germs

For a supported \(H\) of degree \(N-1\), \(\mathcal B_H=Q_NL_\eta m_H\) has degree \(-1\). Equations (DL.2)–(DL.4) give
\[
\begin{gathered}
\mathcal A_{T+\delta H}-\mathcal A_T\\
=d((-1)^N\mathcal B_H)\\
+((-1)^N\mathcal B_H)d .
\end{gathered}
\tag{DL.6}
\]
Indeed \(d\mathcal B_H+\mathcal B_Hd=(-1)^NQ_NL_\eta m_{\delta H}\). This is a homotopy on arbitrary cochains. For the actual full convergent root-to-annular change, \(H\) is the supported finite-order current CA.23 with its proved tails, not an unbounded derivative sum.

For two collars put \(K_{\eta',\eta}(a,b)=((\eta'-\eta)b,0)\). Then
\[
\begin{gathered}
L_{\eta'}-L_\eta\\
=dK_{\eta',\eta}+K_{\eta',\eta}d,\\
\mathcal A_T^{\eta'}-\mathcal A_T^\eta\\
=d\mathcal H+\mathcal Hd,\\
\mathcal H=(-1)^NQ_NK_{\eta',\eta}m_T .
\end{gathered}
\tag{DL.7}
\]
The primitive has proper support in the union of the two collar supports and vanishes near \(S\) before zero extension. An arbitrary input boundary \(de\) maps to \(d\mathcal A_T(e)\); this keeps its complement entry. For a base-dependent cutoff and holomorphic derivative \(D\),
\[
\begin{gathered}
{}[D,L_\eta]=dK_D+K_Dd,\\
K_D(a,b)=((D\eta)b,0).
\end{gathered}
\tag{DL.8}
\]
Finite output differentiation otherwise commutes by transposition and proper Stokes. §5.38, E.13 gives the actual fixed-input derivatives, including normal derivatives and their finite-order bounds.

With one collar, the maps commute literally with shrinking output opens on which their supports remain proper. Different collars are compared by (DL.7), and two excision neighbourhoods by R.23–R.24 on their intersection. No arbitrary restriction of a proper source support is asserted. A finite family of actual symbols and identities admits one common coefficient neighbourhood, one largest thick cone and one smaller output neighbourhood, by E.1–E.3 and CA.1–CA.2. Each filtered-germ equation is witnessed at one common later index because it involves finitely many representatives or witnesses. This proves the finite germ compatibility used below.

#### DL.3. Composition in every degree

Consecutive full kernel currents have a legitimate product. An order-zero current is one normal differential of an \(L^1\) primitive, with the same integrable bound for any finite list of fixed-normal base derivatives. Finite positive-order shifts add finitely many transposed normal derivatives. In consecutive normal variables \(a=v-u\), \(b=z-v\), the second primitive has base \(u+a\). Its base derivatives \(\partial_v^\alpha F_Q(v,b)\), evaluated at \(v=u+a\) with \(b\) held fixed, are \(L^1\) in \(b\), uniformly on compact \(u,a\) sets. The product of the two primitive coefficients is \(L^1\) in \((a,b)\), by Fubini and the separate uniform bounds. Transpose the finite normal derivatives; those in \(a\) falling on the second coefficient, with \(u,b\) held fixed, are exactly those base derivatives at fixed \(b\) and retain the bound. This defines the distribution product and its Leibniz rule. Equivalently, the second evaluation leaves a smooth test in the first normal variables, with finitely many controlled seminorms. The two descriptions agree by integration by parts. The same finite expansion of three integrable primitives is independent of association.

The support intersection is proper over compact endpoints: the nonnegative real-time lengths \(l_1,l_2\) add to the endpoint length, and \(|a|\le K_Gl_1\), \(|b|\le K_Gl_2\). This bounds the intermediate coordinates; the intersection is closed inside the chosen larger actual source domain. In the round diamond DL.13, the two strict cone margins at the endpoints persist at every intermediate point by the triangle inequality. Compact endpoint sets have positive uniform margins, so the compact fibre stays inside that source. The same proof works for a finite list.

For an input of degree \(q\), the coefficient cup has
\[
\begin{gathered}
J(c(T,c))=\\
(-1)^{Nq}J(T)\wedge J(c).
\end{gathered}
\tag{DL.9}
\]
The first trace sign in degree \(N+q\) is also \((-1)^{Nq}\), so it cancels. On a complement entry, the two additional signs from (DL.2) and (DL.4) are both \((-1)^N\), and cancel. Consecutive trace signs for fibre dimensions \(N,M\) differ from the product trace by \((-1)^{NM}\), exactly V.12. The ordered tensor suspension V.14 contributes that factor when the two shifted coefficient blocks are regrouped; its sign on ordinary degree \(p\) is \((-1)^{Mp}\), whose differential check is proved in V.5. Thus both associations of cup and trace agree in every degree. Raw proper integration is the repeated distributional Fubini calculation above. Common collars are obtained by (DL.7).

The diagonal unit has scalar one in every degree. Raw integration of its anti-first coefficient against a degree-\(q\) input gives \((-1)^{Nq}\) times that input, canceled by the trace sign; the complement signs cancel too. A collar equal to one on the whole small diagonal gives the identity literally; (DL.3) covers another choice.

For an arbitrary canonical kernel class, DLG.1–DLG.5 below constructs the full four-entry cup, its finite complement-cube inverse, the proper collar and all-degree normalized trace. Both kernel complement terms are retained. The explicit eight-entry triple-cube and projection-formula calculation proves multiplication and the diagonal unit. Thus the whole canonical ring \(R=E(G;D)\) acts in every relative degree and gives
\[
 R\longrightarrow
     \operatorname{End}_{D(\mathbf C)}(R\Gamma_Z\mathcal O).
 \tag{DL.10}
\]
This last statement is not yet a module-derived object. The auxiliary construction below supplies that stronger assertion.

#### DLG.1. Full kernel representatives and the intersection cube

The ring in DL.6 is the whole canonical ring \(R=E(G;D)\). A general element of this ring need not be the class of a current supported on the kernel support. The supported-current formulas (DL.2), (DL.5) and (DL.6) are special cases of the following construction. This construction supplies the general case used in (DL.10), DL.6 and DL.7. It leaves the analytic proof DLH.1–DLH.4 unchanged.

Use the full smooth Dolbeault resolution \(I=\mathcal A\), including the output antiholomorphic variables, with the ungraded coefficient lines of R.1. R.3–R.7, specifically (R.19)–(R.20) and their ordinary-section comparison, prove that an arbitrary kernel class of degree \(N\) has a **full relative representative**
\[
\begin{gathered}
k=(a,b)\in C_A^N(\mathcal A),\\
\delta a=0,\\
a|_{M\setminus A}=\delta b.
\end{gathered}
\tag{DLG.1}
\]
Here \(a\) is smooth on the whole kernel domain and \(b\) is smooth only on its actual open complement. This assertion is a consequence of the full-cone quasi-isomorphism, not of surjectivity of supported currents onto supported cohomology. No extension of \(b\) across \(A\) is used. Differences of representatives are boundaries in this same full cone. Pullback to a product by a projection preserves smoothness and commutes with \(\delta\); this follows in coordinates by differentiating the pulled-back coefficients.

On the common product let \(A\) be the pulled-back kernel support, let \(B\) be the pulled-back input support, and put
\(U_A=M\setminus A\), \(U_B=M\setminus B\),
\(U=U_A\cup U_B=M\setminus(A\cap B)\).
The two-support cube of (V.15) is
\[
\begin{gathered}
K^j=I^j(M)\\
\oplus I^{j-1}(U_A)\\
\oplus I^{j-1}(U_B)\\
\oplus I^{j-2}(U_A\cap U_B),\\
d_K(x,x_A,x_B,z)=\\
\left(\begin{gathered}
\delta x,\\
x|_{U_A}-\delta x_A,\\
x|_{U_B}-\delta x_B,\\
x_B-x_A+\delta z
\end{gathered}\right).
\end{gathered}
\tag{DLG.2}
\]
In the last entry the restrictions are understood. Let \(k=(a,b)\) have total degree \(p\) and let an arbitrary smooth input cochain \(e=(c,f)\) have total degree \(q\). Their unshifted coefficient cup in this cube is
\[
\begin{gathered}
F(k,e)=\\
\left(\begin{gathered}
a\wedge c,\\
b\wedge c,\\
(-1)^p a\wedge f,\\
(-1)^{p-1}b\wedge f
\end{gathered}\right).
\end{gathered}
\tag{DLG.3}
\]
The final entry is retained. The four signs follow from (V.16): the input complement symbol crosses respectively \(p\) or \(p-1\) kernel antiholomorphic degrees. Thus
\[
\begin{gathered}
d_KF(k,e)=\\
F(dk,e)\\
+(-1)^pF(k,de).
\end{gathered}
\tag{DLG.4}
\]
For completeness, the two singleton entries of the right side are
\(ac-\delta b\,c+(-1)^p b\,\delta c\) and
\(ac-(-1)^p\delta a\,f-a\,\delta f\).
The double entry is
\((-1)^p af-bc+(-1)^{p-1}\delta b\,f+b\,\delta f\),
which is exactly \(x_B-x_A+\delta z\) for (DLG.3). The whole-domain entry is ordinary Leibniz. This verifies (DLG.4) for nonclosed cochains too. All products are of smooth forms on the indicated open sets.

#### DLG.2. An explicit cube-to-full-cone inverse

Choose an ordinary smooth partition \(\rho_A+\rho_B=1\) on \(U\) subordinate to \(U_A,U_B\). R.5 constructs such partitions on arbitrary ordinary opens. Each partition support is closed inside its assigned open subset of \(U\), so a term multiplied by that partition, or its derivative, extends by zero in \(U\). This says nothing about extension across \(A\cap B\).

Define
\[
\begin{gathered}
\Phi_\rho(x,x_A,x_B,z)=\\
\left(\begin{gathered}
x,\\
\begin{gathered}
\rho_Ax_A+\rho_Bx_B\\
+\delta\rho_A\wedge z
\end{gathered}
\end{gathered}\right),\\
\iota(x,y)=\left(\begin{gathered}
x,y|_{U_A},\\
y|_{U_B},0
\end{gathered}\right),\\
H_\rho(x,x_A,x_B,z)=\\
(0,-\rho_Bz,\rho_Az,0).
\end{gathered}
\tag{DLG.5}
\]
The first two maps preserve degree; \(H_\rho\) has degree minus one. Their identities are
\[
\begin{gathered}
d\Phi_\rho=\Phi_\rho d_K,\\
\Phi_\rho\iota=1,\\
1-\iota\Phi_\rho=\\
d_KH_\rho+H_\rho d_K.
\end{gathered}
\tag{DLG.6}
\]
Indeed, both second components of the first identity equal
\[
\begin{gathered}
x|_U-\rho_A\delta x_A-\rho_B\delta x_B\\
+\delta\rho_A\wedge(x_B-x_A)\\
+\delta\rho_A\wedge\delta z.
\end{gathered}
\tag{DLG.7}
\]
For the last identity, the singleton components are
\(\rho_B(x_A-x_B)-\delta\rho_A\,z\) and
\(\rho_A(x_B-x_A)-\delta\rho_A\,z\);
the double component is \(z\), and the whole-domain component is zero.
These are precisely the components of \(1-\iota\Phi_\rho\).
Thus this is a deformation retraction, with no appeal to a Leray cover or to supported-current representability.

For another partition put \(\theta=\rho'_A-\rho_A\). Then
\[
\begin{gathered}
C_{\rho',\rho}(x,x_A,x_B,z)=\\
(0,-\theta z),\\
\Phi_{\rho'}-\Phi_\rho\\
=dC_{\rho',\rho}+C_{\rho',\rho}d_K.
\end{gathered}
\tag{DLG.8}
\]
The second component of this identity is
\(\theta(x_A-x_B)+\delta\theta\wedge z\), by direct expansion. This homotopy retains the double-complement entry. These maps also work for a current-valued cube, because they multiply currents only by smooth functions. The general cup itself will continue to use smooth representatives.

In particular the full intersection-cone cup with a degree-\(p\) kernel is
\[
\begin{gathered}
m_k(e)=(a\wedge c,\beta_k(e)),\\
\beta_k(e)=\rho_A b\wedge c\\
+(-1)^p\rho_Ba\wedge f\\
+(-1)^{p-1}\delta\rho_A\wedge b\wedge f.
\end{gathered}
\tag{DLG.9}
\]
It satisfies \(dm_k=m_{dk}+(-1)^p m_kd\). When the kernel is a supported current \((T,0)\), (DLG.3) is the image under \(\iota\) of (DL.2): its input-complement entry glues with zero on \(U_A\). Therefore \(\Phi_\rho\) returns exactly (DL.2), including its sign. Formula (DLG.9) adds the terms needed for all the other canonical kernel classes.

#### DLG.3. A finite-cover comparison with natural homotopies

Composition and iterated supports require more than two complements. Here is a uniform finite construction, including the maps that compare refinements. For a finite ordinary cover \(\mathcal U=(U_i)_{i=1}^n\) of an open \(U\), let \(T\) be its alternating Čech–Dolbeault total complex, with Čech degree \(r\ge0\), horizontal differential \(\Delta\), and vertical differential
\(v=(-1)^r\delta\). Write \(\iota_U:I(U)\to T\) for restriction to singleton entries. Let \(p\) be zero in positive Čech degree and equal to \(\sum_i\rho_i c_i\) in degree zero. Let
\[
\begin{gathered}
(hc)_{i_0\ldots i_{r-1}}\\
=\sum_j\rho_j c_{j i_0\ldots i_{r-1}}\\
\quad(r\ge1),\\
h=0\quad(r=0).
\end{gathered}
\tag{DLG.10}
\]
All extensions are within \(U\). Expanding the alternating sum proves
\(\Delta h+h\Delta=1-\iota_Up\), as in V.6. Antisymmetry gives \(h^2=0\) and \(ph=0\): in either double sum the coefficient \(\rho_i\rho_j\) is symmetric while the Čech entry is antisymmetric. Also \(h\iota_U=0\), \(p\iota_U=1\), \(p\Delta=0\), and
\(v\iota_U=\iota_U\delta\).

The following inverses are finite sums because \(vh\) lowers Čech degree:
\[
\begin{gathered}
A=(1+vh)^{-1}\\
=\sum_{j=0}^{n-1}(-vh)^j,\\
P=pA,\\
H=hA.
\end{gathered}
\tag{DLG.11}
\]
For a component in degree \(r\), terms with \(j>r\) are zero. Put \(D=\Delta+v\). Since \(\Delta v+v\Delta=0\) and \(v^2=0\),
\[
\begin{gathered}
D(1+vh)=\\
(1+vh)\Delta+\iota_U\delta p.
\end{gathered}
\tag{DLG.12}
\]
To verify this equality, expand its left side as
\(\Delta+v-v\Delta h\), and substitute
\(\Delta h=1-\iota_Up-h\Delta\).
Multiplying by \(A\) gives
\(AD=\Delta A+\iota_U\delta P\), since \(A\iota_U=\iota_U\).
Consequently
\[
\begin{gathered}
PD=\delta P,\\
P\iota_U=1,\\
DH+HD=1-\iota_UP.
\end{gathered}
\tag{DLG.13}
\]
For the last equality, use \(H=hA\) and the preceding formula to obtain
\((\Delta h+h\Delta+vh)A=(1+vh-\iota_Up)A\).
This proves every identity by finite algebra; no convergence or perturbation theorem is being cited.

Apply \(P\) to the complement coordinate of
\(\operatorname{Cone}(I(M)\to T)[-1]\), retaining the whole-domain coordinate. The map from this cone to the finite cube is the sign \((-1)^r\) on its Čech-degree-\(r\) complement entry, exactly (V.15). The resulting cube-to-full-cone map has inverse induced by \(\iota_U\), and its homotopy is
\((x,c)\mapsto(0,-Hc)\) in the cone coordinates. For two opens this is (DLG.5)–(DLG.6), including the derivative term.

More generally let \(R_T:T\to T'\) be a map of these total complexes, let \(f:I(U)\to I(U')\) be a map of the augmented complexes, and suppose
\(R_T\iota_U=\iota_{U'}f\). Then
\[
\begin{gathered}
P'R_T-fP\\
=\delta'(P'R_TH)\\
+(P'R_TH)D.
\end{gathered}
\tag{DLG.14}
\]
Indeed multiply \(1-\iota_UP=DH+HD\) by \(P'R_T\); its left side becomes the stated difference and use the chain identities for \(R_T,P'\). The corresponding full-cone homotopy again has complement coordinate minus \(P'R_THc\), and zero whole-domain coordinate.

For restriction, intersect each member of the cover with the smaller open and let \(R_T\) restrict every entry. For a cover refinement \((V_j)\), choose \(\alpha(j)\) with \(V_j\subset U_{\alpha(j)}\), and restrict the alternating component \(c_{\alpha(j_0)\ldots\alpha(j_r)}\) to \(V_{j_0\ldots j_r}\), with repeated-index entries zero. Alternation and the definition of \(\Delta\) show that this is a chain map and it commutes with the augmentation. Equation (DLG.14) applies. Two partitions, two refinement assignments, and two successive refinement paths consequently give the same map in the derived category: each comparison is homotopic to the same augmented map by (DLG.14). This assertion includes an explicit homotopy, not a choice of an unproved inverse.

For support-forgetting \(T_1\subset T_2\), the full-cone map is
\((a,b)\mapsto(a,b|_{M\setminus T_2})\). Restrict the old complement cover to \(M\setminus T_2\) and use a common refinement with the desired cover there. Formula (DLG.14) proves compatibility. The same argument applies to the cube for an intersection of supports: restrict all its entries and then refine its finite complement cover. Thus support inclusions and finite iterated support cubes are natural by specified chain homotopies. The maps preserve any fixed common closed support of their sheaf sections; multiplication and zero extension inside \(U\) cannot enlarge it. In particular they preserve a pre-existing proper support. A partition on the ordinary complement may have large derivatives near its excluded boundary; no bound there or extension across that boundary is required.

#### DLG.4. The proper collar and normalized trace for an arbitrary kernel

Suppose \(S=A\cap B\) is closed and proper over the output projection \(p:M\to X\), and \(p(S)\subset Z\) for the output support \(Z\). Choose a proper collar \(\eta=1\) near \(S\) supported in an actual excision neighbourhood \(W\). The collar exists locally without growth assumptions. Over a precompact output open \(V\Subset V'\), properness makes \(S\cap p^{-1}(\overline V)\) compact; finitely many coordinate bumps in \(W\) give a cutoff equal to one near that compact set. Its restriction over \(V\) has proper support. If a global proper collar is needed, take a locally finite subordinate output partition \(\lambda_i\), choose such a compact-source cutoff \(\eta_i\) near \(S\) over \(\operatorname{supp}\lambda_i\), and put \(\eta=\sum_i(p^*\lambda_i)\eta_i\). Every compact output set meets finitely many partition supports, so this collar is proper. Near a point of \(S\) all finitely many relevant \(\eta_i\) equal one; hence \(\eta=1\) there. Its support is contained in \(W\) and closed in the ambient inverse image. This is the ordinary locally finite bump construction of R.5.

Apply the full-cone collar **after** \(\Phi_\rho\):
\[
\begin{gathered}
L_\eta m_k(c,f)=\\
\left(\begin{gathered}
\begin{gathered}
\eta a\wedge c\\
+\delta\eta\wedge\beta_k(c,f)
\end{gathered},\\
\eta\beta_k(c,f)
\end{gathered}\right).
\end{gathered}
\tag{DLG.15}
\]
The term containing \(\delta\eta\) extends by zero across \(S\) because \(\delta\eta\) is zero on a neighbourhood of \(S\). The second component stays on the complement. No growth bound for either kernel or input complement is needed. Every component has proper sheaf support inside \(\operatorname{supp}\eta\). W.1, (W.3)–(W.6), prove
\(dL_\eta=L_\eta d\), \(1-L_\eta=dE_\eta+E_\eta d\), with
\(E_\eta(a,b)=((1-\eta)b,0)\).
This is the controlled zero-extension roof (R.23)–(R.24).

Let \(Q_N\) be the all-degree relative trace (DL.4), proved in V.4 and W.3. For the closed kernel (DLG.1) the actual operator is
\[
\begin{gathered}
\mathcal A_k=\\
Q_NL_\eta\Phi_\rho F(k,-),\\
d\mathcal A_k=\mathcal A_kd.
\end{gathered}
\tag{DLG.16}
\]
This follows from (DLG.4), (DLG.6) and
\(Q_Nd=(-1)^NdQ_N\). The two degree-\(N\) signs cancel. At an input of degree \(q\), the source-cone total degree is \(N+q\). Trace on its first component has sign \((-1)^{Nq}\), and on its second component the additional \((-1)^N\) of (DL.4) is present. These formulas also include the zero trace below degree \(N\) and compact-Stokes boundary case. Both kernel and input complement coordinates have entered (DLG.15) before tracing.

R.6–R.7 identify this roof with the derived holomorphic cup, support-to-proper-image map and normalized trace: (DLG.3) is the tensor-to-cube map (V.16), (DLG.6) is the inverse to its full-cone augmentation, and (DLG.15) is the actual controlled excision inverse. The holomorphic coefficient augmentation commutes with the cup. Tensoring its bounded comparison roofs over \(\mathbf C\) preserves quasi-isomorphisms: an exact stalk complex of vector spaces splits by choosing complements to its kernels, its contraction tensored with a bounded complex remains a contraction, and the finite total-degree filtration gives the assertion. This supplies the comparison with the holomorphic/injective models. It does not use any analytic concentration theorem.

If a degree-\(N-1\) full kernel cochain \(h=(a_h,b_h)\) changes the kernel by \(dh\), then
\[
\begin{gathered}
\mathcal A_{k+dh}-\mathcal A_k\\
=d\mathcal B+\mathcal Bd,\\
\mathcal B=\\
(-1)^N\\
Q_NL_\eta\Phi_\rho F(h,-).
\end{gathered}
\tag{DLG.17}
\]
Expanding (DLG.4) gives
\(d(Q_NL_\eta m_h)\) \(+(Q_NL_\eta m_h)d\) \(=(-1)^NQ_NL_\eta m_{dh}\), which proves the formula. It retains \(b_h\), and extends (DL.6).

Changing the partition gives the homotopy
\((-1)^NQ_NL_\eta C_{\rho',\rho}F(k,-)\) by (DLG.8). Changing the collar gives
\((-1)^NQ_NK_{\eta',\eta}\Phi_\rho F(k,-)\), where
\(K_{\eta',\eta}(x,y)=((\eta'-\eta)y,0)\), by (DL.7).
All their traced terms are proper, in the collar support or the union of the two collar supports. An input boundary is sent to the boundary of \(\mathcal A_k\) applied to its full boundary primitive. Restriction, support forgetting and finite complement refinement use (DLG.14), followed by one common collar and the same trace. R.23–R.24 compare two excision neighbourhoods on their intersection. This proves independence as a derived morphism, with proper homotopies and all complement entries included.

#### DLG.5. Multiplication and the unit of the whole canonical ring

The preceding construction also multiplies two arbitrary canonical kernel classes. On the triple product, pull their full smooth relative cochains back in the consecutive variables. Apply (DLG.3), then the full intersection-cone comparison and a proper collar, and finally trace the intermediate \(N\)-dimensional variable. This gives a full relative cochain of degree \(N\) with support in the endpoint kernel support. When both cochains are smooth, the two output components of this trace are smooth on their respective actual domains: over a compact base neighbourhood the proper support lies in finitely many compact fibre charts, so every finite base derivative passes under the integral. This is exactly the smooth calculation in V.4. Therefore this procedure can be iterated using smooth representatives. Arbitrary distributions with shared singular parameters are never multiplied.

Its properness is the actual round-domain properness, not a support-label assertion. Write consecutive differences as \(g_i\in G\) and their real-time lengths as \(l_i\ge0\). For fixed endpoints their sum is the endpoint difference, so
\(\sum_i l_i=L\), the endpoint length, and
\(|g_i|\le K_Gl_i\). All intermediate coordinates are bounded. If the endpoints lie in the round diamond (DL.13), each intermediate point lies in it: adding a member of \(G\) preserves the first open-cone inequality, and subtracting the remaining member of \(G\) preserves the second. More quantitatively, the strict endpoint margins
\(\epsilon l-|\operatorname{Im}t|\) and
\(A_jl-|x_j|\) remain at least their initial endpoint margins by the triangle inequality. Compact endpoint families have positive uniform margins. Thus the closed bounded fibre stays in a compact subset of the actual source domain. A closed subset of that compact set is compact (take any sequence, extract a convergent subsequence in the bounded Euclidean box, and use closedness); this proves properness. It also proves properness for a finite list of consecutive kernels.

Associativity of the coefficient cup at the finite cube level is literal. Expand three or more relative factors by (V.16). The resulting coefficient on the open indexed by complement symbols \((r_i)\) has sign
\((-1)^{\sum_{i<j}q_ir_j}\), independently of association: regrouping splits this sum into its within-block and between-block pairs. Wedge products of smooth forms and the ordered coefficient-line product are associative. Thus both associations give the same multi-support cube before the full-cone comparison.

Inserting a pairwise cube-to-cone inverse before the next cup changes this common cube only by the explicit homotopy (DLG.13): substitute
\(\iota P=1-DH-HD\) on that pair of factors and tensor \(H\) with the remaining factors, with the ordinary tensor differential signs. For a left factor of degree \(p\), a homotopy in a later factor is multiplied by \((-1)^p\); expansion of the two mixed differential terms gives cancellation. Multiplication by the subsequent cup map preserves that homotopy by (DLG.4). Different finite covers or refinement paths are compared by (DLG.14). Hence the full-cone cup is associative as the specified derived operation, including the higher complement entries.

Here is the projection formula used on the actual complement components. Let \(\alpha\) be a properly supported smooth coefficient of antiholomorphic degree \(h\), with the intermediate holomorphic top block of degree \(N\) ordered first, and let \(\beta\) have antiholomorphic degree \(j\) on the remaining variables. In local fibre coordinates only the component of \(\alpha\) containing all \(N\) fibre antiholomorphic differentials survives integration. Write that component with the fibre antiholomorphic block first, followed by its remaining block of degree \(h-N\). Moving the holomorphic fibre block past the remaining block and then past \(\beta\) gives
\[
\begin{gathered}
I_{h+j}(\alpha\wedge p^*\beta)\\
=(-1)^{Nj}I_h(\alpha)\wedge\beta,\\
\operatorname{Tr}_{h+j}(\alpha\wedge p^*\beta)\\
=\operatorname{Tr}_h(\alpha)\wedge\beta.
\end{gathered}
\]
The second equality follows because
\(N(h+j-N)+Nj\equiv N(h-N)\pmod2\).
Fibre components of smaller degree give zero on both sides. Each equality is the same compactly supported coefficient integral with \(\beta\) outside the integral; finite partitions of fibre charts give the equality globally. Derivatives in the remaining variables pass under that compact integral, as proved in V.4. For a factor on the other side, commute the antiholomorphic blocks first with their ordinary wedge sign; the coefficient blocks are then ordered as in (R.25). This gives the precise exterior sign in either order, without a separated-current premise.

The complement opens require an additional domain check. After the finite map \(P\), each term originating in an entry on \(U_{i_0}\cap\cdots\cap U_{i_r}\) has a factor \(\rho_{i_k}\) or a derivative of that factor for every one of its indices: each application of \(h\) supplies the removed index and the final \(p\) supplies the last index. Such a term extends by zero in \(U\), by the subordinate partition supports. It is not integrated as an unweighted cochain across an excluded face. Multiply by the common collar. On an output-complement open \(X\setminus Z\), the whole inverse image is in \(U\), since \(p(S)\subset Z\). Over a compact subset of that output open, the collar gives compact source support disjoint from \(S\). The partition factors therefore give compact support inside each term's actual complement domain. R.24 permits its zero extension there; its integral is the ordinary integral of this extended smooth coefficient. Thus the projection formula applies term by term to every finite complement entry, as well as to the whole-domain entry. The \(\delta\eta\) terms of the collar extend across \(S\) by their vanishing near \(S\) and obey the same calculation. In particular the second relative trace component is obtained by restricting this very calculation to the output complement, with the extra sign \((-1)^N\) of (DL.4).

For two kernels and an input, the common three-support cube has all eight entries. If their complement flags are \(r_1,r_2,r_3\in\{0,1\}\), and their corresponding antiholomorphic degrees are \(q_i=p_i-r_i\), its entry is the wedge of those three selected whole or complement forms, with sign \((-1)^{q_1r_2+q_1r_3+q_2r_3}\). In particular entries containing either kernel complement, the input complement, or any combination of them are present. Apply the finite map of DLG.3 and one final proper collar to this cube. Properness over the final base also gives properness over a base retaining an intermediate variable: a compact subset of the latter base projects to a compact subset of the final base, and its inverse image in the closed source support is a closed subset of the latter compact fibre set. Partial integration and then full integration are therefore defined. For every whole and partition-localized complement coefficient, the two integrations are the identical repeated integral over the same compact source support; Fubini is ordinary repeated integration of smooth compact coefficients. The projection formula just proved identifies the operation of tracing one pair and then cupping the remaining factor with that same three-factor integral. The complement restriction used by the first trace is legitimate because its projected support lies in its chosen intermediate support; its pulled-back output-complement open lies in the source complement. Restrict all cube entries to these actual opens and use (DLG.14) to compare their finite covers. This identifies both associativity roofs, including their intermediate complement entries.

All terms to be integrated can be given one final proper collar on the common multi-support intersection. An intermediate collar can be replaced by that common choice before the final integration: substitute
\(L_\eta=1-dE_\eta-E_\eta d\) inside the corresponding cup, using (DLG.4), and apply the final collar to the resulting homotopy. The final collar makes each homotopy term proper even when the original complement primitive is not proper. The equality is made on the whole intersection cone, so it also includes the derivative and complement parts of every \(L_\eta\). If an integration has already been made, represent this calculation on the product before that integration. Pulling back a base form, wedging and integrating a proper smooth coefficient commute by the coordinate definition of fibre integration; the signs are the exterior permutations in (R.25). Repeated integration agrees with integration over the ordered product fibre by the ordinary compactly supported coefficient calculation of V.4. This proves the same collar comparison for the sequential integration roof, rather than restricting a proper section to an arbitrary source open.

The precise normalization is unchanged. In input degree \(q\), moving the kernel's holomorphic coefficient block across input antiholomorphic degree contributes \((-1)^{Nq}\), which cancels the first trace sign. Repeated traces of fibre dimensions \(N,M\) have the discrepancy \((-1)^{NM}\) of (V.12); the trace-first versus total tensor suspension of (V.14) contributes exactly that factor. On complement entries the extra cone signs are those already checked in (DLG.4) and (DL.4). This proves associativity of multiplication and the module identity
\[
\begin{gathered}
{}[\mathcal A_k]\,[\mathcal A_l]=[\mathcal A_{k*l}]\\
\quad\hbox{in every relative}\\
\hbox{degree}.
\end{gathered}
\tag{DLG.18}
\]
Here \(k*l\) denotes the canonical unshifted kernel cup followed by its normalized intermediate trace, and the order agrees with consecutive input motion. This proof applies to the whole ring, without symbol-growth bounds or a supported-current representation of \(k,l\).

For the diagonal unit choose its supported current representative, whose positive ordered Cauchy scalar is proved in U.5–U.8 and V.5. This use of a supported representative is specific to the unit. A smooth full input cochain is a legitimate multiplier. The cube collapses to (DL.2), as proved after (DLG.9). Raw integration on a degree-\(q\) first component gives \((-1)^{Nq}\) times that component, canceled by \(\operatorname{Tr}_{N+q}\); the two complement signs cancel as well. A collar equal to one on the diagonal makes the identity literal. Thus the diagonal acts as the identity on every full relative cone. Applying this also to a kernel input gives both convolution units. Smooth/current comparison R.6–R.7 then identifies it with the same unit for arbitrary smooth full kernel representatives.

#### DLG.6. The strict auxiliary module and the all-degree comparison

Retain the exact geometry, exhaustion, concentration and injective module construction of DL.4–DL.6. For \(T=S\) or \(S_*\), put
\(A=A_G\), \(B=p_2^{-1}T\).
Over compact outputs in \(\Omega\), kernel-supported inputs lie in \(K+G\), have time at least \(-a\) for \(S_*\), and are bounded by the cone length estimate in DL.14. They stay in \(\Omega\) because \(\Omega+G\subset\Omega\), and in the coefficient domain by (DL.14). This gives the properness needed in DLG.4. The support-to-image condition is equally explicit. If an output is in \(\Omega_0\), every kernel-supported input lies in \(\Omega_0\), since \(\Omega_0\) is \(G\)-open. If the output is in \(\omega\), all such inputs are in \(\omega\). Hence
\[
\begin{gathered}
p_1(A_G\cap p_2^{-1}S)\subset S,\\
p_1(A_G\cap p_2^{-1}S_*)\subset S_*.
\end{gathered}
\tag{DLG.19}
\]
On an ordinary open \(V\cap D\) compute (DLG.16), then use the fixed output collar of DL.6, equal to one near \(\overline{S_*}\) and compactly supported in \(D\), to extend by zero to \(V\). The coefficient domain need not be \(G\)-open. This output excision is the full-cone operation of W.1 and R.23, so DLG.4 applies to a general \(b\) as well as to a supported current.

Here is the needed restriction comparison for arbitrary representatives. If \(V'\subset V\) is \(G\)-open, the total cohomological intersection over outputs in \(V'\) lies in the source open where the input is also in \(V'\): a kernel-supported input is in \(x+G\subset V'\). Choose the proper collar inside that source open by DLG.4. Restrict all four cube entries to it, apply the finite-cover comparison and this collar, and extend by zero. With a restricted common partition, the localized entries are literally the restrictions of (DLG.15), extended by zero; the source collar makes the extension legal. Equation (DLG.14) compares another partition, and DLG.4 compares another collar. Proper trace then commutes with output restriction. Thus the derived operation agrees with the operation using the restricted input. This proof does not pretend that a general kernel's whole-domain component \(a\) is supported on \(A_G\); its off-support contributions are removed by the full-cone excision homotopy, with \(b\) retained.

Consequently every \(r\in R\), represented by any (DLG.1), acts on
\(\mathcal F=H^1_{S_*}(q_{G*}\mathcal O_\Omega)\).
Boundary and choice independence are DLG.4; multiplication and the unit are DLG.5. They give strict identities of cohomology-sheaf maps. Additivity and scalar linearity follow by using a common partition and collar for finitely many representatives and adding their cochains in (DLG.3). Thus \(\mathcal F\) is a sheaf of left modules over the **whole** ring \(R\). The ring need not act strictly on these chosen full-cone representatives; the strict module action is on \(\mathcal F\). The module injective resolution in DL.6 and the tensor/Hom contractions of DL.8 are therefore valid with this whole ring.

There is also an action-compatible comparison in every degree for the smaller support \(S\). Use the same kernel representative, multi-support cube and collar on the intersections for \(S\subset S_*\). The support-forgetting maps and the restriction to \(\Omega_0\) are compatible by (DLG.14) and (DLG.19). The finite cube for the iterated supports is identified with its full intersection cone by (DLG.11)–(DLG.13); on injectives the corresponding support identity is literally
\(\Gamma_S\Gamma_{S_*}I=\Gamma_SI\), proved in DL.6. R.6–R.7's augmentation comparison identifies these same maps, since the cube maps commute with the holomorphic augmentations and the controlled extension maps. The resulting derived support operation is therefore the operation of \(r\) on
\(R\Gamma_S R\Gamma_{S_*}(q_{G*}\mathcal O_\Omega)\), not merely an unrelated action on its cohomology.

The concentration roof in DL.6 identifies
\(R\Gamma_{S_*}(q_{G*}\mathcal O_\Omega)\) with \(\mathcal F[-1]\).
After that identification a degree-zero morphism is uniquely the map on the single sheaf \(\mathcal F\): using a nonnegative injective resolution of the target, degree-zero cycles are exactly maps into its degree-zero kernel, and there is no degree-minus-one boundary. Hence this roof intertwines the arbitrary-kernel action with the strict map \(r:\mathcal F\to\mathcal F\). Apply the identical derived support functor \(R\Gamma_S\) to these morphisms and use the literal injective support identity. It follows that the underlying action on the actual module-derived object \(\mathcal Q_R\) of (DL.21) is precisely (DLG.16) on
\(R\Gamma_S(q_{G*}\mathcal O_\Omega)\), in every degree. This proves the general-kernel action comparison used in DL.6.

#### DLG.7. Finite common refinements and the retained scope

For halfspace changes, support inclusions and ordinary excision use DLG.3–DLG.4. The map \(\mathcal F_2\to\mathcal F_1\) of (DL.23) commutes with every \(r\in R\), since it is the map on \(H^1\) of the common natural full-cone roof. Applying \(R\Gamma_S\) identifies both sides with the same injective supported complex, exactly as in DL.6. This proves the required module quasi-isomorphism for arbitrary ring elements.

For a cone change \(G\subset G'\), the input and auxiliary complements must be \(G'\)-open, as in DL.7. Choose the common coefficient domain round for \(G'\), and choose the compact auxiliary-support geometry by (DL.14) using \(K_{G'}\). This domain is round for \(G\) as well. The shared auxiliary \(G'\)-open is also \(G\)-open; its compact truncated support lies in this domain. For the smaller cone the proof needs precisely those two facts: its cofinal basis translates inside the shared open have the concentration of DL.5, and its kernel intersection with the shared compact auxiliary support is proper by DLG.6. Thus the same auxiliary construction applies to both topologies. The envelope \(U+G'\) is used to prove the common compact containment; containment of that wider-cone-open inside a narrower infinite envelope is not presumed. On the ordinary injective model
\(q_*q_{G*}I=q_{G'*}I\) and support inclusions compose literally. On the smooth relative representatives the kernel-support enlargement restricts \(b\) to the smaller kernel complement. DLG.3 supplies its explicit comparison with any new complement cover; DLG.4 supplies a common proper collar; DLG.5 identifies the ring products and units. Thus the comparison on \(H^1\) is linear for the actual canonical ring homomorphism, and it commutes with the same all-degree operation. Exact additive vector-injective functors, as in the final paragraphs of DL.6, make these strict sheaf maps into module-derived maps. Two successive refinements give the same derived map because each agrees with the same ordinary injective map; (DLG.14) is the cochain comparison to that map.

Only finite diagrams are asserted. In a filtered system a representative, a boundary witness and a finite equation each occur at some index; finitely many such indices have a common later index. On it choose one admissible round domain, covers, collars and output neighbourhood for the finite diagram. Their ring and action identities follow from DLG.4–DLG.6. For symbol-germ entries the maps of NL.1–NL.7 and CP.1–CP.6 supply the actual representatives and common-domain symbol identities; these are maps into the canonical ring. They are not used to assert that every element of a fixed \(E(G;D)\) is a symbol class. The input opens must remain open for the common cone; existence for the directional family is the explicit CA.1 hypothesis. A new action on an inadmissible input open is not inferred.

The full-relative cup, finite-cube inverse, proper collar, normalized all-degree trace, whole-ring unit and multiplication, and finite naturality give the action required by DL.6–DL.7. DLH.1–DLH.4 proves analytic concentration on the specified domains. These local kernel and support statements are separate from infinite-order extension and flatness, extension-faithfulness, propagation, module separation, finite D-type realization, finite-pole recovery, intrinsic order, full C1, arbitrary analytic proper regularity, and the global GL, BB/KL, affine, critical and factorization assertions.

![Full relative-cube route for an arbitrary kernel](assets/whole-kernel-cube-route.png)

*Figure DLG.1. This is the exact two-support cochain diagram. Every displayed cube entry is (DLG.3); the map to the full intersection cone, including its derivative term, is (DLG.5). The proper collar is (DLG.15), and the final all-degree trace is (DLG.16). Both complement entries and the double-complement entry are retained. The unit/multiplication and finite-refinement mechanisms are DLG.3 and DLG.5–DLG.7. The complete earlier programme dependencies are §5.34, R.1–R.8, specifically (R.19)–(R.25); §5.35, V.4–V.6, specifically (V.10)–(V.16); §5.37, W.1–W.3, specifically (W.3)–(W.14); and §5.36, U.2, specifically (U.5)–(U.8) for the positive diagonal scalar. The reproducible source and exact formula data retain this diagram.*


The figure has a full-resolution PNG, SVG, editable plotting source and exact formula data.

#### DL.4. Cone topology and exact round domains

Fix
\[
\begin{gathered}
G=\left\{\begin{gathered}
\operatorname{Re}w=-l\le0,\\
|\operatorname{Im}w|\le\epsilon l,\\
|\zeta_j|\le A_jl
\end{gathered}\right\},\\
\epsilon,A_j>0,\\
K_G=\\
(1+\epsilon^2+\sum_jA_j^2)^{1/2}.
\end{gathered}
\tag{DL.11}
\]
The empty spatial list is allowed. This cone is closed, convex, pointed and has ordinary interior. All its nonzero points have strictly negative real time. Its invariant opens define the \(G\)-topology; \(q_G\) is the identity continuous map from the ordinary topology to it.

At \(x\) the opens
\[
\begin{gathered}
V_{x,a}=x+a e_t\\
+\operatorname{int}G,\qquad a>0,
\end{gathered}
\tag{DL.12}
\]
are a cofinal \(G\)-neighbourhood basis. They contain \(x\) and are \(G\)-open since \(\operatorname{int}G+G\subset\operatorname{int}G\). If \(B(x,r)\subset W\) and \(W+G\subset W\), then \(B(x,r)+G\subset W\); for \(a<r\), (DL.12) is contained in that set.

A bounded round coefficient domain inside any actual coefficient neighbourhood is
\[
\begin{gathered}
D_R=(R e_t+\operatorname{int}G)\\
\cap(-R e_t-\operatorname{int}G).
\end{gathered}
\tag{DL.13}
\]
It contains zero. If \(x,z\in D_R\) and \(v-x,z-v\in G\), the first condition puts \(v\) in the first open cone, and the second puts it in the second. Hence \(D_R\) is round. Its time lies between \(-R,R\) and its transverse norms are at most \(\epsilon R,A_jR\), so it lies in the ball of radius \(K_GR\). Ordinary Euclidean balls are not assumed round for this cone.

Choose a small \(U\subset B(0,r)\), and \(a>r\), such that
\[
\begin{gathered}
\overline{\begin{gathered}(U+G)\cap\\
\{\operatorname{Re}t\ge-a\}\end{gathered}}\\
\Subset D.
\end{gathered}
\tag{DL.14}
\]
To do so, write \(z=u+g\). Its cone length is at most \(a+r\); thus \(|z|\le r+K_G(a+r)\). Small \(r,a\) put this bound in a ball compactly inside \(D\).

#### DL.5. The required global analytic concentration input

Every domain (DL.12), and its intersection with a strict negative-time halfspace, has an explicit smooth strict plurisubharmonic exhaustion. Translate its vertex to zero, put \(l=-\operatorname{Re}t\), \(Y_0=\operatorname{Im}t\), \(Y_j=(\operatorname{Re}x_j,\operatorname{Im}x_j)\), \(a_0=\epsilon,a_j=A_j\), and set
\[
\begin{gathered}
B_j=a_j^2l^2-|Y_j|^2,\\
\psi(z)=|z|^2-\sum_{j=0}^{d}\log B_j .
\end{gathered}
\tag{DL.15}
\]
The function \(-\log(L^2-|Y|^2)\) is real convex on \(L>|Y|\). Rotate \(Y\) into its first axis. At a fixed point \((L_*,Y_*)\), the invertible real linear change
\[
\begin{gathered}
(L,Y_1)\mapsto\\
\left(\begin{gathered}
L_*L-|Y_*|Y_1,\\
-|Y_*|L+L_*Y_1
\end{gathered}\right)\\
/\sqrt{L_*^2-|Y_*|^2}
\end{gathered}
\tag{DL.16}
\]
preserves \(L^2-|Y|^2\) and sends that point to \((\sqrt{L_*^2-|Y_*|^2},0)\). Its Hessian there is \(2/(L_*^2-|Y_*|^2)\) times the identity, by two differentiations. Pullback preserves positive definiteness. Scaling \(L=a_jl\) proves that each logarithmic term in (DL.15) has nonnegative real Hessian in all its variables. The Levi form is one quarter the sum of the real Hessian in a direction and its \(i\)-multiple. The term \(|z|^2\) makes the total strictly plurisubharmonic.

This function is an exhaustion: the quadratic dominates logarithms at infinity since \(B_j\le C(1+|z|)^2\); on bounded sets a boundary face forces one \(B_j\) to zero and the others have bounded logarithms. Its sublevels are compact in the domain, and it is bounded below. Intersecting with \(\omega=\{\operatorname{Re}t<c\}\) adds \(-\log(c-\operatorname{Re}t)\), a real convex term, and gives the same properties.

The analytic appendix proves that a domain with such an exhaustion has a locally \(L^2\) primitive for every closed locally \(L^2\) form of positive degree, with no growth assumption. The locally \(L^2\) Dolbeault sheaves form an exact resolution: local existence follows by that proof on balls and degree zero by R.3 distributional holomorphicity. They are ordinary \(C^\infty\)-modules (both coefficients and differential remain locally \(L^2\) under smooth multiplication), and are ordinary-section acyclic by R.5. The R.6 double-complex comparison therefore gives
\[
\begin{gathered}
H^q(V_{x,a},\mathcal O)=\\
H^q(V_{x,a}\cap\omega,\mathcal O)=0\\
\quad(q>0).
\end{gathered}
\tag{DL.17}
\]
No global smoothness of the weak primitive is required.

Push an ordinary injective resolution of \(\mathcal O_\Omega\) under \(q_G\). It remains injective, since \(q_G^{-1}\) is exact and left adjoint to \(q_{G*}\). Its higher stalk cohomology is the filtered colimit of (DL.17) over the cofinal basis (DL.12). Hence
\[
\begin{gathered}
Rq_{G*}\mathcal O_\Omega=q_{G*}\mathcal O_\Omega\\
\quad\hbox{in degree zero}.
\end{gathered}
\tag{DL.18}
\]
This is the needed cone-topology assertion, not a consequence of ordinary local Dolbeault exactness alone. The actual smooth/current comparison also passes to this topology. Each ordinary smooth or current term is section-acyclic on every ordinary open by R.5; its positive \(q_G\)-direct images vanish on stalks, and its \(G\)-open section cohomology vanishes as well. For the latter assertion, push its ordinary injective resolution, then take sections on a \(G\)-open: those sections are literally the ordinary section complex on the same open, and the positive direct-image vanishings identify its augmentation. The finite-diagonal injective comparison is R.6. The same argument on the \(G\)-open complement gives the open-direct-image acyclicity of each term. Thus the \(q_G\)-images of the actual smooth/current relative cones represent the cone-topology derived supports, with the same R/V cube maps. This uses ordinary acyclicity, not a smooth partition subordinate to the cone topology.

#### DL.6. The genuine auxiliary module sheaf

Let \(\Omega_0\subset\Omega_1\) be \(G\)-open, with \(S=\Omega_1\setminus\Omega_0\) relatively compact in \(U\) from (DL.14). Intersect both opens with \(U+G\); this leaves \(S\) unchanged and the supported object unchanged by open excision. Put \(\Omega=\Omega_1\subset U+G\),
\(\omega=\{\operatorname{Re}t<-a\}\), and
\[
\begin{gathered}
S_*=\Omega\setminus\omega,\\
\mathcal F=H^1_{S_*}(q_{G*}\mathcal O_\Omega).
\end{gathered}
\tag{DL.19}
\]
Both supports are closed in \(\Omega_G\), \(S\subset S_*\), and (DL.14) puts the closure of \(S_*\) compactly inside the coefficient domain.

The full auxiliary object is concentrated in degree one:
\[
\begin{gathered}
R\Gamma_{S_*}(q_{G*}\mathcal O_\Omega)\\
\simeq\mathcal F[-1],\\
\mathcal F_x=\underset{b}{\operatorname{colim}}\,\\
\mathcal O(V_{x,b}\cap\omega)/\mathcal O(V_{x,b}).
\end{gathered}
\tag{DL.20}
\]
Indeed both opens in the relative long exact sequence have vanishing positive cohomology by (DL.17). Their holomorphic restriction is injective: \(V_{x,b}\) is connected and its intersection with \(\omega\) is nonempty by downward real-time translation. A holomorphic function zero on a nonempty open in a connected domain is zero by its convergent Taylor series and continuation along overlapping balls. Thus supported cohomology vanishes also in degree zero, and only degree one remains. Filtered colimits prove this on every \(G\)-stalk. This uses no fixed-domain vanishing on arbitrary \(\Omega_i\).

The whole-kernel operation DLG.1–DLG.6, extending the supported-current special case DL.1–DL.3, makes \(\mathcal F\) a genuine sheaf of left \(R=E(G;D)\)-modules. Over a basis domain the kernel intersection with the input support is proper: input points lie in \(x+G\), have real time at least \(-a\), and obey the cone length bound. For a compact output family \(K\subset\Omega\), the bounded relevant portion of \(K+G\) is closed and compact inside \(\Omega\), since \(\Omega+G\subset\Omega\). It is inside the coefficient domain by (DL.14). R/W proper excision therefore supplies the action. Ordinary excision through \(D\) is used here, since the auxiliary support lies inside \(D\); it is not claimed that \(D\) itself is \(G\)-open. On the actual \(q_G\)-relative models, compute the operation over \(V\cap D\), apply an output collar equal to one near \(\overline{S_*}\) and compactly supported in \(D\), and use controlled zero extension to obtain the roof over the full \(V\). That collar restricts to every \(V\), and DLG.4–DLG.6 proves its comparison with the identity, retaining the complete kernel complement. Restriction to a \(G\)-open smaller output is compatible because it is invariant under the input motion by \(G\); (DL.7) compares input collar choices. Multiplication and the unit are identities on the cohomology sheaf by DLG.5, and hence are strict identities of sheaf maps. Outside \(S_*\) the sheaf is zero. The action is consequently defined on all of \(\Omega_G\).

For completeness an actual injective module resolution can be constructed as follows. For a left \(R\)-module \(M\), \(J(M)=\operatorname{Hom}_{\mathbf C}(R,M)\), with \((r\phi)(s)=\phi(sr)\), is injective because
\(\operatorname{Hom}_R(A,J(M))=\operatorname{Hom}_{\mathbf C}(A,M)\) and vector-space exact sequences split. The natural injection is \(m\mapsto(s\mapsto sm)\). Apply \(J\) to each stalk and take the product of point pushforwards. The exact stalk functor has point pushforward as right adjoint, so those sheaves are injective; products are injective because their Hom is a product of exact vector sequences. The natural injection, iterated on the cokernel, gives a bounded-below resolution \(J_R^\bullet(\mathcal F)\). This works in the cone topology as well.

An injective \(R\)-module sheaf is injective after forgetting to vector sheaves: its left-adjoint induction \(R\otimes_{\mathbf C}-\) is exact on stalks. Thus
\[
\begin{gathered}
\mathcal Q_R=\Gamma_SJ_R^\bullet(\mathcal F)[-1]\\
\in D(R\text{-Mod}(\Omega_G))
\end{gathered}
\tag{DL.21}
\]
has underlying complex-derived object \(R\Gamma_S(q_{G*}\mathcal O_\Omega)\). To verify this comparison, a closed support functor \(\Gamma_T=i_*i^!\) preserves injectives: the respective left adjoints of \(i^!\) and \(i_*\) are \(i_*\) and \(i^{-1}\), both exact on stalks. The identity
\(\Gamma_S\Gamma_{S_*}I=\Gamma_SI\) on injectives is literal, since \(S\subset S_*\). Therefore
\[
\begin{gathered}
R\Gamma_S(\mathcal F)[-1]\\
\simeq R\Gamma_S R\Gamma_{S_*}\\
(q_{G*}\mathcal O_\Omega)\\
=R\Gamma_S(q_{G*}\mathcal O_\Omega).
\end{gathered}
\tag{DL.22}
\]

There is an actual chain roof for the concentration comparison. For
\(C^\bullet=\Gamma_{S_*}q_{G*}I^\bullet(\mathcal O_\Omega)\), the good truncation has \(C^0\), \(\ker(C^1\to C^2)\), and zero above. Its inclusion into \(C^\bullet\) and its degree-one projection to \(\mathcal F[-1]\) are quasi-isomorphisms by (DL.20). Apply the natural exact injective resolution functors R.5–R.6 to this roof and totalize. Their finite-diagonal elimination gives the injective comparison; R.18a makes a quasi-isomorphism between bounded-below injective complexes a homotopy equivalence. Applying \(\Gamma_S\) preserves those homotopies. This proves the actual roof underlying (DL.22), to which the smooth/current/cube/proper-trace model maps by R.6–R.7. The resolution functors use products of stalk pushforwards and are valid on any topological space; no smooth partition in the cone topology is asserted.

The comparison also identifies the **all-degree canonical operation**, not only the action on an auxiliary degree-one stalk. Fix \(r\in R\) and use its full relative representative and the common kernel cup, finite cube, excision collar and proper trace roof of DLG.1–DLG.6 for the two supports \(S\subset S_*\). The support-to-image condition for \(S\) is explicit: if an output \(x\) lies in the \(G\)-open \(\Omega_0\), every kernel-supported input \(y\in x+G\) lies in \(\Omega_0\); hence \(p_1(A_G\cap p_2^{-1}S)\subset S\). The analogous condition for \(S_*\) follows because \(\omega\) is \(G\)-open. The kernel operation on the auxiliary relative object therefore commutes with restriction to \(\Omega_0\) using that same roof. Its operation on the cone of that restriction is the full relative map DLG.3–DLG.4, with DL.4's trace sign; DL.2 is its supported-current special case. The finite cube with the kernel support, \(S_*\) and \(S\) realizes this iterated cone; V.16 and its augmentation give the same cup as the original support \(S\), since \(S\subset S_*\). On injectives this reduction is the literal identity \(\Gamma_S\Gamma_{S_*}I=\Gamma_SI\). The proper collar remains available after restricting the intersection: use controlled extension by zero on a common neighbourhood of \(A_G\cap p_2^{-1}S_*\), and DL.7 for a smaller chosen collar, rather than assuming arbitrary source restriction preserves properness. Consequently the underlying map of \(R\Gamma_S(r:\mathcal F\to\mathcal F)[-1]\) agrees with DLG.16's original full-relative canonical kernel action on \(R\Gamma_S(q_{G*}\mathcal O_\Omega)\) in every degree. The concentration roof commutes with the auxiliary operation: after shifting, its object is a single sheaf, and a degree-zero morphism between sheaves in the vector-derived category is an ordinary sheaf map. To see the last fact directly, resolve the target in nonnegative injective degrees; a degree-zero map from the source sheaf lands exactly in its degree-zero kernel, and no degree-minus-one homotopy can alter it. Hence the morphism is uniquely the map on \(H^1=\mathcal F\). This proves the action compatibility, rather than merely a vector-derived quasi-isomorphism.

There is a convenient common choice of the actual module complex. The exact vector-sheaf injective functors \(I_{\mathbf C}^r\) of R.5–R.6 are additive and natural in all vector-sheaf maps. A strict \(R\)-action on \(\mathcal F\) therefore gives a strict \(R\)-action on \(I_{\mathbf C}^\bullet(\mathcal F)\), with natural \(R\)-linear augmentation and differential. Each term is \(\Gamma_S\)-acyclic as an \(R\)-module sheaf: use an \(R\)-injective resolution and forget it to a vector-injective resolution; its positive support cohomology vanishes because that term is vector-injective. The bounded-below acyclic-resolution comparison, by finite-diagonal elimination, shows that
\(\Gamma_S I_{\mathbf C}^\bullet(\mathcal F)[-1]\) is another actual model for (DL.21). This common vector functor makes the strict maps compatible even under a ring homomorphism, since it acts on the underlying module maps without changing the functor when the ring changes.

Taking derived global sections of (DL.21) gives an object of \(D(R\text{-Mod})\). Its underlying global supported cohomology agrees with the ordinary one: \(q_{G*}I^\bullet\) is injective by (DL.18), and its global sections with support \(S=\Omega\setminus\Omega_0\) are literally the ordinary sections of \(I^\bullet\) vanishing on \(\Omega_0\). This proves the ordinary/cone global comparison directly.

#### DL.7. Choice independence and finite actual \(E\)-germs

For two auxiliary halfspaces with ordered levels, \(\omega_1\subset\omega_2\) gives \(S_{*,2}\subset S_{*,1}\) and the canonical support-forgetting map
\[
 \mathcal F_2\longrightarrow\mathcal F_1 .
 \tag{DL.23}
\]
It is \(R\)-linear by the naturality of cup, controlled excision and trace. After \(R\Gamma_S\), it is a quasi-isomorphism: both comparisons (DL.22) become the identical \(\Gamma_SI^\bullet\), and this map is the identity there. Module quasi-isomorphisms are detected on vector sheaves. Intersecting auxiliary halfspaces gives a common comparison for any two choices. Functorial module resolutions realize these maps on complexes.

Two neighbourhood choices are compared on their common \(G\)-open neighbourhood of \(S\), then by open excision. The auxiliary restriction and support maps are \(R\)-linear by the same canonical roof. Outside \(S\), the relative complex is a cone of the identity, contracted by \((a,b)\mapsto(b,0)\); extension by zero therefore gives the excision quasi-isomorphism on stalks, and injective comparison gives its derived map. This proves independence in the module-derived category with its specified underlying comparison.

For changing coefficient domains and cones, only actual finite common refinements are claimed. The input opens must remain open for the enlarged cone; an arbitrary \(G\)-open set is not assumed open for every larger cone. The directional \(\Omega_g\) family permits cofinal choices for each finite symbol family by the \(LM<1\) bound of §5.42, CA.1. A finite list of actual symbol entries, products, inverses and zero identities is represented on one actual domain and cone. By §5.41, CP.32/CP.38–CP.44 and NL.25/NL.35/NL.37, their canonical kernel identities hold at one common later index of the filtered kernel-germ colimit. Choosing a smaller round diamond there realizes the whole finite diagram in one constant ring \(R\).

For a common cone refinement \(G\subset G'\), suppose the input opens and halfspace are \(G'\)-open as required. The identity \(q:\Omega_G\to\Omega_{G'}\) is continuous. On the ordinary injective resolution the equality \(q_*q_{G*}I=q_{G'*}I\) is literal, and \(\Gamma_{S_*}\) commutes with that direct image because the complement \(\omega\) is \(G'\)-open. All these direct images preserve injectives. Equation DL.18 for both cones and the auxiliary concentration give
\(Rq_*\mathcal F_G=\mathcal F_{G'}\), in degree zero; the comparison on \(H^1\) is canonical. The cohomological cup/trace maps and support enlargement commute with it by the common finite cube and proper excision. Applying the common vector-injective functor above to the comparison roofs gives compatible module-derived maps. Successive refinements agree as derived maps because they come from the identical ordinary injective resolution and compose literally there. Halfspace changes are the strict maps DL.23. Together these prove the finite-diagram germ compatibility, rather than merely saying that choices are harmless.

This is the precise finite-diagram meaning of arbitrary full \(E\)-valued contraction transfer. It does not assert that every symbol germ has a representative on one preassigned domain, or an action on an input whose cone-open hypothesis fails. The canonical product and normal-localization proofs in §§5.39 and 5.41 supply the actual finite-order symbol map into the canonical kernel ring for every such finite diagram. DLG.1–DLG.7 proves the action of the whole canonical ring without assuming that this symbol map is surjective. The local derived objects are actual \(E(G;D)\)-module objects in the conventions of [Micro-hyperbolic systems, §3.2.5](https://perso.imj-prg.fr/pierre-schapira/wp-content/uploads/schapira-pub/Microhyp.pdf); a new literal action of every stalk germ on one fixed preassigned coefficient domain is not the statement.

#### DL.8. Actual off-support contractions and their total signs

A bounded exact cochain complex \(P^\bullet\) of finite projectives over a ring has an actual linear contraction. Start at its highest nonzero degree \(b\): exactness makes \(P^{b-1}\to P^b\) surjective, and projectivity splits it. Its kernel is a direct summand of \(P^{b-1}\), hence projective. Descend finitely. The decompositions give
\[
\begin{gathered}
P^i=Z^i\oplus s^iZ^{i+1},\\
h^i(z+s^iw)=s^{i-1}z,\\
d_Ph+hd_P=1 .
\end{gathered}
\tag{DL.24}
\]
For finite free terms these are finite matrices of actual ring germs. Their finitely many identities hold on a common neighbourhood and in one canonical ring \(R\) by the finite symbol-germ map of §5.41 and the whole-kernel refinement proof DLG.7. This transfers an arbitrary full \(E\)-valued contraction, including entries outside a polynomial subring.

Use the actual module complex \(\mathcal Q_R\) from (DL.21). For a right-module \(P\), its tensor total differential and contraction are
\[
\begin{gathered}
d(p\otimes c)=d_Pp\otimes c\\
+(-1)^i p\otimes d_Qc\\
\quad(p\in P^i),\\
H=h\otimes1,\\
dH+Hd=1 .
\end{gathered}
\tag{DL.25}
\]
The mixed signs are \((-1)^{i-1}+(-1)^i=0\). For a left-module \(P\), the internal Hom formula on degree \(k\) is
\[
\begin{gathered}
d\phi=d_Q\phi-(-1)^k\phi d_P,\\
H\phi=(-1)^k\phi h,\\
dH+Hd=1 .
\end{gathered}
\tag{DL.26}
\]
The terms \(d_Q\phi h\) cancel with signs \((-1)^k+(-1)^{k+1}\); the others give \(\phi(hd_P+d_Ph)=\phi\). All sums are finite in the \(P\) direction even when \(Q\) is unbounded above.

A bounded finite-projective complex computes derived Hom into \(Q\): each \(\operatorname{Hom}(P^i,-)\) preserves exactness, and a finite column elimination proves the total complex of an acyclic target is acyclic. Tensor has the analogous finite filtration proof. For projective summands, restrict these formulas using the corresponding split idempotents of finite free terms. Thus these are actual contractions of the derived models, without any general flatness or extension-faithfulness premise.

#### DL.9. The invalid shortcut and the scope retained

The need for the module lift has an exact elementary illustration. Let \(C^0=C^1=\mathbf C^2\), \(d_C=1\), and let two chain maps have both components
\[
\begin{gathered}
A=\begin{pmatrix}0&1\\0&0\end{pmatrix},\\
B=\begin{pmatrix}0&0\\1&0\end{pmatrix},\\
AB-BA=\operatorname{diag}(1,-1).
\end{gathered}
\tag{DL.27}
\]
The complex is contractible, so its endomorphism ring in \(D(\mathbf C)\) is zero and all ring relations hold there. But substituting these chosen lifts into the commutative Koszul differentials \((-y,x)^T\), \((x,y)\) gives the nonzero composition \(-AB+BA\). A naive total differential does not square to zero. This disproves that inference from derived endomorphisms to a strict total action; it does not claim that replacement is impossible. DL.19–DL.22 construct the needed replacement here.

The statement proved here is the genuine all-degree canonical \(R\)-module-derived supported object for the full thick cone and round/support geometry stated above, with its cup/cube/excision/trace comparison, all signs, analytic concentration and arbitrary finite-projective contraction transfer. §5.41, CP.1–CP.6 and §5.39, NL.1–NL.7 supply its full actual \(E\)-symbol-germ finite realizations.

Sections5.44–5.89 prove the full infinite extension and flatness, faithfulness, propagation, section separation, finite D-type realization, finite poles and coherent characteristic cutoffs. Arbitrary analytic proper-image regularity requires its further argument. The broader geometric representation-theoretic, affine, critical and factorization arguments are separate from this result.

![Exact module-lift and contraction mechanism](assets/derived-action-module-lift.png)

*Figure DL.1. The upper-left panel is the exact \(N=1\) complex-time section with \(\epsilon=1/4\), round diamond \(D_{1/2}\), \(\Omega=1/5+\operatorname{int}G\), \(\omega=\{\operatorname{Re}t<-1/10\}\), and \(S=\Omega\cap\{\operatorname{Re}t\ge-1/40\}\). Dashed diamond and green cone edges are excluded boundaries; the hollow green vertex is excluded. The shaded polygons display the closures of \(S_*\) and \(S\); their green edges remain excluded, while their vertical time boundaries are included inside \(\Omega\). These exact domains satisfy \(S\subset S_*\) and \(\overline{S_*}\subset D_{1/2}\) directly; no choice by the sufficient crude ball estimate is depicted. The cup/trace signs are DL.2, DL.4 and DL.9. The genuine auxiliary lift is DL.19–DL.22. The matrices and nonzero Koszul composition are exactly DL.27; the valid Hom contraction is DL.26. The figure source and exact coordinate data reproduce this section. Human comparison: [Micro-hyperbolic systems, §§3.1–3.2.5](https://perso.imj-prg.fr/pierre-schapira/wp-content/uploads/schapira-pub/Microhyp.pdf) and [HolIII, IV.3–IV.4](https://www.kurims.kyoto-u.ac.jp/~kenkyubu/kashiwara/HolIII.pdf); the complete proof is DL.1–DL.8 and the analytic appendix.*

#### The analytic input for DL.5

The following proof shows that on a domain with a smooth strict plurisubharmonic exhaustion every closed locally \(L^2\) form of positive degree has a locally \(L^2\) \(\bar\partial\) primitive. It supplies the global vanishing needed on the exact cone-topology basis domains. No boundary-growth or boundary-regularity hypothesis is imposed.

#### DLH.1. Hilbert existence

A closed Hilbert subspace has an orthogonal projection: a distance-minimizing sequence is Cauchy by the parallelogram identity applied to its midpoints, completeness gives a minimizer, and variation by real and imaginary multiples shows its residual is orthogonal. A bounded linear functional is represented by inner product with a vector: its closed kernel has one-dimensional orthogonal complement when the functional is nonzero, and scaling any vector in that complement gives the representative and its norm. The zero functional uses zero.

The \(L^2\) spaces here are complete. From a Cauchy sequence select a subsequence with summable successive \(L^2\) distances. The sums of absolute successive differences have bounded \(L^2\) norm by the triangle inequality; monotone convergence shows their pointwise limit is in \(L^2\). The subsequence therefore converges almost everywhere and in \(L^2\), and so does the original sequence. Multiplication by the square root of a positive weight gives weighted completeness.

Let \(T: H_1\dashrightarrow H_2\), \(S: H_2\dashrightarrow H_3\) be densely defined, with \(\ker S\) closed, \(ST=0\), and
\[
\begin{gathered}
\|T^*f\|^2+\|Sf\|^2\ge\|f\|^2\\
\quad(f\in\operatorname{Dom}T^*\\
\cap\operatorname{Dom}S).
\end{gathered}
\tag{DLH.1}
\]
For \(g\in\ker S\), take \(f\in\operatorname{Dom}T^*\) and decompose \(f=f_1+f_2\) orthogonally to \(\ker S\). Since \(\operatorname{im}T\subset\ker S\), \(T^*f_2=0\); hence \(f_1\) is in the same adjoint domain and \(T^*f_1=T^*f\). Applying (DLH.1) to \(f_1\) gives
\[
 |\langle g,f\rangle|\le\|g\|\|T^*f\|.
 \tag{DLH.2}
\]
Extend the bounded functional \(T^*f\mapsto\langle f,g\rangle\) to the closure of its image, and by zero to its orthogonal complement. Hilbert representation gives \(u\), \(\|u\|\le\|g\|\), with \(\langle u,T^*f\rangle=\langle g,f\rangle\). On tests this is the required weak equation.

#### DLH.2. Weights and the actual graph domain

Take compact exhaustion sets \(K_\nu\subset\operatorname{int}K_{\nu+1}\) and smooth cutoffs \(\eta_\nu=1\) near \(K_\nu\), supported in \(\operatorname{int}K_{\nu+1}\). Their derivative supports leave every compact; their supremum squared derivative norm is locally bounded. An ordinary locally finite smooth partition gives a smooth majorant \(\psi\) with
\[
 |\bar\partial\eta_\nu|^2\le e^\psi .
 \tag{DLH.3}
\]
The majorant is obtained by summing constants bounding the locally bounded function on each relatively compact partition support. This needs local finiteness, not a claim that consecutive closed exhaustion shells are disjoint. Ordinary cutoffs and partitions are the constructions proved in R.5/E.2.

Set \(\varphi_1=\varphi-2\psi\), \(\varphi_2=\varphi-\psi\), \(\varphi_3=\varphi\). Use maximal distributional \(\bar\partial\) from weighted \(L^2\) degree \(q-1\) to \(q\) as \(T\), and degree \(q\) to \(q+1\) as \(S\). Their domains are dense because compact tests are dense by exhaustion and convolution. The kernel of \(S\) is closed since convergence implies distributional convergence on compacts. Also \(ST=0\).

Let the domain have complex dimension \(n\). For a degree-\(q\) form \(f=\sum'_{|J|=q}f_J\,d\bar z_J\), extend the coefficients alternatingly to all ordered indices. Then \(f_{jK}\) is the coefficient of the contraction by the \(j\)-th basis vector, and \(\sum'_K\sum_j|f_{jK}|^2=q\sum'_J|f_J|^2\). For compact smooth \(u\) of degree \(q-1\) and \(f\) of degree \(q\), the actual coefficient calculation is
\[
\begin{gathered}
\langle\bar\partial u,f\rangle_{\varphi_2}\\
=\sum_K'\sum_j\int\\
\partial_{\bar z_j}u_K\,\overline{f_{jK}}\,e^{-\varphi_2}\\
=\sum_K'\int u_K\,\\
\overline{\begin{gathered}-\sum_j e^{\varphi_1}\\
\partial_{z_j}(e^{-\varphi_2}f_{jK})\end{gathered}}\\
\,e^{-\varphi_1}.
\end{gathered}
\tag{DLH.4a}
\]
The second equality is ordinary compact integration by parts in each coordinate. Thus the adjoint expression is
\[
\begin{gathered}
T^*f=-\sum_K'\sum_j\\
e^{\varphi_1}\partial_{z_j}(e^{-\varphi_2}f_{jK})\,d\bar z_K\\
=:\vartheta f .
\end{gathered}
\tag{DLH.4}
\]
It holds on tests and distributionally on the adjoint domain; the expansion is (DLH.4a).

Compact tests are dense in \(\operatorname{Dom}T^*\cap\operatorname{Dom}S\) for its graph norm. First multiply by \(\eta_\nu\). The \(S\)-error is \(\bar\partial\eta_\nu\wedge f\), with squared norm bounded by \(\int_{\Omega\setminus K_\nu}|f|^2e^{-\varphi_2}\). Testing \(T(\eta_\nu u)\) proves the product remains in the adjoint domain with error
\(-e^{-\psi}\sum_{K,j}(\partial_{z_j}\eta_\nu)f_{jK}d\bar z_K\).
Its squared \(\varphi_1\)-norm is at most \(q\) times that same tail integral, since \(\sum_{K,j}|f_{jK}|^2=q|f|^2\). Both errors vanish.

Then mollify the compact form. Weights are equivalent to constants on its fixed compact. For a smooth first-order operator \(a\partial_{x_j}\), its mollification commutator is
\[
\begin{gathered}
\int\left[\begin{gathered}
\frac{\begin{gathered}a(x)\\-a(x-\varepsilon y)\end{gathered}}{\varepsilon}\partial_j\rho(y)\\
+\partial_ja(x-\varepsilon y)\rho(y)
\end{gathered}\right]\\
f(x-\varepsilon y)\,dy .
\end{gathered}
\tag{DLH.5}
\]
The mean value bound gives a uniform \(L^2\) operator bound using
\(\int(|y||\partial_j\rho|+|\rho|)\). For smooth compact \(f\) the commutator tends to zero directly; approximate a compact \(L^2\) coefficient by smooth ones to conclude. Zero-order terms follow from ordinary mollification convergence. Applying this to \(\vartheta\) and to constant-coefficient \(\bar\partial\) proves graph density. No identification of an adjoint domain with a formal maximal domain is made.

#### DLH.3. The a priori estimate

Write \(\delta_j=\partial_{z_j}-\varphi_{z_j}\). We derive the complete weighted identity on compact forms. Let \(e_j\) mean exterior multiplication by \(d\bar z_j\), and let \(i_j=e_j^*\) be its contraction for the pointwise Euclidean form inner product. On an increasing basis wedge, \(i_je_k+e_ki_j=\delta_{jk}\): when \(j=k\), exactly one of the two terms returns the wedge, according as the index is present or absent; when \(j\ne k\), their insertion/removal permutations have opposite signs. Put \(A_j=\partial_{\bar z_j}f\). Expanding \(\|\sum_j e_jA_j\|^2\) and substituting this relation gives
\[
\begin{gathered}
|\bar\partial f|^2\\
=\sum_j|A_j|^2\\
-\sum_{j,k}\langle i_kA_j,i_jA_k\rangle\\
=\sum_J'\sum_j|\partial_{\bar z_j}f_J|^2\\
-\sum_K'\sum_{j,k}\\
\partial_{\bar z_j}f_{kK}\,\\
\overline{\partial_{\bar z_k}f_{jK}}.
\end{gathered}
\tag{DLH.6a}
\]
The formula includes repeated-index cancellations and every cross term. In the weighted inner product, compact integration by parts says that the adjoint of \(\partial_{\bar z_k}\) is \(-\delta_k\). Since \(\delta_k\partial_{\bar z_j}-\partial_{\bar z_j}\delta_k=\varphi_{k\bar j}\), two integrations by parts give, for each \(K,j,k\),
\[
\begin{gathered}
\int \partial_{\bar z_k}f_{jK}\,\\
\overline{\partial_{\bar z_j}f_{kK}}\,e^{-\varphi}\\
=-\int f_{jK}\,\\
\overline{\begin{gathered}\partial_{\bar z_j}(\delta_k f_{kK})\\
+\varphi_{k\bar j}f_{kK}\end{gathered}}\,e^{-\varphi}\\
=\int \delta_jf_{jK}\,\\
\overline{\delta_kf_{kK}}\,e^{-\varphi}\\
-\int\varphi_{j\bar k}f_{jK}\overline{f_{kK}}\,e^{-\varphi}.
\end{gathered}
\tag{DLH.6b}
\]
Here \(\overline{\varphi_{k\bar j}}=\varphi_{j\bar k}\) because \(\varphi\) is real. The sum of the left side is the cross sum in (DLH.6a), after exchanging the names \(j,k\). Substitution into its integral gives
\[
\begin{gathered}
\sum_K'\int\left|\sum_j\delta_jf_{jK}\right|^2e^{-\varphi}\\
+\int|\bar\partial f|^2e^{-\varphi}\\
=\sum_J'\sum_j\int\\
|\partial_{\bar z_j}f_J|^2e^{-\varphi}\\
+\sum_K'\sum_{j,k}\int\\
\varphi_{j\bar k}f_{jK}\overline{f_{kK}}e^{-\varphi}.
\end{gathered}
\tag{DLH.6}
\]
The only commutator is
\(\delta_k\partial_{\bar z_j}-\partial_{\bar z_j}\delta_k
=\varphi_{k\bar j}\). Equations (DLH.6a)–(DLH.6b) give the alternating expansion and both integrations explicitly.

Since
\(\vartheta_\varphi f=e^\psi T^*f+\sum_{K,j}\psi_{z_j}f_{jK}d\bar z_K\),
Cauchy–Schwarz bounds its squared norm by
\(2\|T^*f\|_{\varphi_1}^2+
2q\int|\partial\psi|^2|f|^2e^{-\varphi}\).
The last term of (DLH.6) is at least
\(q\int\lambda_\varphi|f|^2e^{-\varphi}\). If
\[
 \lambda_\varphi\ge2|\partial\psi|^2+2e^\psi,
 \tag{DLH.7}
\]
then \(2\|T^*f\|^2+\|Sf\|^2\ge2q\|f\|_{\varphi_2}^2\), which implies (DLH.1), as \(q\ge1\). Graph density extends this to the actual domains. DLH.1 gives
\[
\begin{gathered}
\bar\partial u=g,\\
\int|u|^2e^{-\varphi+2\psi}\\
\le\int|g|^2e^{-\varphi+\psi}.
\end{gathered}
\tag{DLH.8}
\]

#### DLH.4. Weights for unrestricted local data

Let \(\psi_0\) be the smooth strict exhaustion, translated to be nonnegative. On each compact sublevel its minimum Levi eigenvalue \(\mu(t)\) is positive and the maximum \(M(t)\) of \(2|\partial\psi|^2+2e^\psi\) is finite. Choose a smooth increasing convex \(\chi\) whose derivative dominates \(M(t)/\mu(t)\). For example choose increasing constants \(B_k\) dominating the bounds through level \(k+2\), and set
\(\chi'(t)=B_0+\sum_{k\ge0}(B_{k+1}-B_k)\sigma(t-k)\)
for a fixed increasing smooth step \(\sigma\), zero below zero and one above one. This is locally finite and nondecreasing. Its additive constant and the constants \(B_k\) can be increased as needed. The Hessian chain rule gives
\[
 \lambda_{\chi\circ\psi_0}
       \ge\chi'(\psi_0)\lambda_{\psi_0},
 \tag{DLH.9}
\]
so \(\varphi=\chi\circ\psi_0\) satisfies (DLH.7).

For locally \(L^2\) input \(g\), write \(M_k\) for the nonnegative right side of the following inequality, and choose the additive constant at least \(M_1\) and \(B_j\ge M_k\) for every \(k\le j+2\). Then
\[
\begin{gathered}
\chi(k-1)\ge\\
k+\log\left(\begin{gathered}1+\int_{\{\psi_0\le k\}}\\
|g|^2e^\psi\end{gathered}\right)\\
\quad(k\ge1).
\end{gathered}
\tag{DLH.10}
\]
Each integral is finite on that compact. For \(k\ge2\), throughout the unit interval \([k-2,k-1]\), all steps through index \(k-3\) are complete, so \(\chi'\ge B_{k-2}\ge M_k\). Its integral over that interval, with the nonnegative earlier value, proves (DLH.10); the additive constant handles \(k=1\). At every real \(t\in[m,m+1]\), similarly \(\chi'(t)\ge B_m\), which was chosen to dominate the Hessian requirement through level \(m+2\). The construction is therefore explicit and simultaneous. The integral of \(|g|^2e^{-\varphi+\psi}\) over \(k-1\le\psi_0<k\) is at most \(e^{-k}\). Hence the right side of (DLH.8) is finite, while its primitive is locally \(L^2\) because all weights are bounded above and below on compacts.

This proves the global existence used on the exact domains (DL.15). It provides the cone-topology concentration input and no broader infinite-order or propagation theorem.


The figure has a full-resolution PNG, SVG, editable plotting source and exact coordinate data.


### 5.44. Full homogeneous coefficients and ordinary infinite-order kernels

We construct the common-domain full homogeneous coefficient ring, its complete infinite products and derivative tails, and a canonical linear pole/log kernel map. Supplied strict angular data admit an explicit normalization with all overlap and missing-face primitives. The ordinary diagonal infinite-order subring has a canonical residue product and arbitrary ordinary holomorphic coordinate overlaps.

The full geometric identification for arbitrary projectivized microlocal sections and the mixed pole/log canonical product remain separate assertions, denoted IK.G and IK.M. The strict angular hypothesis is retained throughout. The ordinary calculation supplies neither assertion, and no infinite-order module-derived instantiation is made. The two separate geometric assertions are proved in [§5.45, IG.1–IG.5 and IM.1–IM.6](#5-45-the-full-geometric-infinite-order-kernel-ring-and-its-canonical-product). The conditional input and ordinary conclusions in this section retain their stated scope.

Free human specifications are [Kashiwara–Schapira, Micro-hyperbolic systems, §§1.3 and 3.1](https://perso.imj-prg.fr/pierre-schapira/wp-content/uploads/schapira-pub/Microhyp.pdf), and [Kashiwara–Kawai, Holonomic systems III, III.2, printed pp.883–884](https://www.kurims.kyoto-u.ac.jp/~kenkyubu/kashiwara/HolIII.pdf). The distinct real-conic symbol problem is treated in the freely accessible [Aoki–Honda–Yamazaki, Foundation of symbol theory for analytic pseudodifferential operators, I](https://www.jstage.jst.go.jp/article/jmath/69/4/69_1715/_pdf), and [Komori, The equivalence of pseudodifferential operators and their symbols via Čech–Dolbeault cohomology](https://content.ems.press/assets/public/full-texts/serials/prims/61/3/14299041/online/10.4171-prims-61-3-3.pdf). No symbol theorem from these references replaces a proof below.

The earlier full resolutions and controlled excision are [§5.34, R.1–R.7](#5-34-full-current-resolutions-and-controlled-proper-image-comparison); the finite relative cube, normalized trace, tensor signs and normal-line calibration are [§5.35, V.1–V.6](#5-35-relative-cones-unshifted-cup-and-the-normalized-trace). Finite-subring calibration uses [§5.41, CP.2](#5-41-canonical-products-for-full-convergent-spatial-symbols) and [§5.39, NL.11–NL.12 and NL.22–NL.27](#5-39-canonical-normal-localization-of-convergent-spatial-symbols). Unrestricted locally \(L^2\) existence on domains with smooth strictly plurisubharmonic exhaustions is proved in [§5.43, DLH.1–DLH.4](#5-43-the-canonical-all-degree-action-and-its-derived-module-structure). We first verify that this analytic hypothesis holds on every ordinary puncture intersection.

#### OC.1. Explicit exhaustions on every ordinary puncture intersection

Let \(M=U\times\prod_{i=1}^N\Delta_{R_i}\), with \(U\) an actual bounded base polydisc. Write all its base and normal coordinates as \(z_j\), centered at zero, with respective radii \(R_j\). A translated coordinate has the same argument after subtracting its center. For \(J\subset\{1,\ldots,N\}\), put

\[
\begin{gathered}
M_J=\left\{\begin{gathered}
(u,w)\in M:\\
w_i\ne0\text{ for }i\in J
\end{gathered}\right\}.
\end{gathered}
\tag{OC.1}
\]

The empty subset gives \(M\). On \(M_J\) define

\[
\begin{gathered}
\psi_J=\\
\sum_j\left(\begin{gathered}
|z_j|^2\\
-\log(R_j^2-|z_j|^2)
\end{gathered}\right)\\
+\sum_{i\in J}|w_i|^{-2}.
\end{gathered}
\tag{OC.2}
\]

This is smooth on exactly the stated domain. In one coordinate direct differentiation gives

\[
\begin{gathered}
\partial_z\partial_{\bar z}\bigl[-\log(R^2-|z|^2)\bigr]\\
={R^2\over(R^2-|z|^2)^2}>0,\\
\partial_w\partial_{\bar w}|w|^{-2}=|w|^{-4}>0.
\end{gathered}
\tag{OC.3}
\]

The \(|z_j|^2\) terms contribute the identity Levi matrix. Thus the complete Levi form is positive definite. Every term is bounded below on the bounded product, and approaching any outer coordinate boundary makes its logarithmic term tend to \(+\infty\). Approaching a puncture \(w_i=0\), \(i\in J\), makes \(|w_i|^{-2}\) tend to \(+\infty\). Consequently each sublevel stays a positive distance from every excluded boundary and puncture. It is closed in a bounded ambient product, hence compact there and contained in \(M_J\). This proves a true smooth strictly plurisubharmonic exhaustion on every finite cover intersection, including the whole product.

#### OC.2. Actual holomorphic cohomology vanishing

Let \(\mathcal L^q\) be the sheaf of forms with locally \(L^2\) coefficients and locally \(L^2\) distributional Dolbeault differential. The differential maps this sheaf into the next term, since its second differential is zero. Multiplication by a smooth function preserves both conditions on compacts by the distributional Leibniz rule. Thus each term is a \(C^\infty\)-module sheaf.

The augmented complex \(0\to\mathcal O\to\mathcal L^0\to\mathcal L^1\to\cdots\) is exact. For local positive-degree closed forms, apply DLH.1–DLH.4 on a small polydisc; OC.2 with an empty puncture set gives its smooth strictly plurisubharmonic exhaustion, with the exact Levi coefficients OC.3. The resulting locally \(L^2\) primitive has differential the given locally \(L^2\) form, so it belongs to \(\mathcal L^{q-1}\). In degree zero, §5.34, R.3's distributional holomorphicity proof identifies the kernel with ordinary holomorphic functions.

Each term is ordinary-section acyclic by §5.34, R.5's complete \(C^\infty\)-module injective-resolution and partition argument. §5.34, R.6's finite double-complex elimination therefore identifies global \(\mathcal L\) cohomology with the actual right-derived holomorphic sections. On \(M_J\), DLH.1–DLH.4 applied to OC.2 provides a global locally \(L^2\) primitive for every positive-degree global closed form. Again that primitive lies in the stated maximal graph sheaf. Hence

\[
\begin{gathered}
H^q(M_J,\mathcal O)=0\\
(q>0),
 \\
 H^0(M_J,\mathcal O)=\mathcal O(M_J).
\end{gathered}
\tag{OC.4}
\]

The input top-form coefficient line is holomorphically trivial on the local product. Apply the same proof in that frame; it changes no estimate or differential. The ordinary smooth Dolbeault section complexes on these opens have the same cohomology by §5.34, R.6's exact comparison, so their positive-degree closed sections also have actual smooth primitives. No distributional extension at a puncture and no bound on boundary growth has been used.

#### OC.3. Holomorphic-row reduction of the actual finite support cube

The diagonal is \(\Delta=\{w_1=\cdots=w_N=0\}\), and its complement is covered by \(U_i=\{w_i\ne0\}\). V.15's full cube has, for each subset \(J\), the section column \(\mathcal A^*(M_J)\), with total differential the alternating face restriction plus \((-1)^{|J|}\bar\partial\). R.7 and V.6 identify it with the actual relative supported object, rather than assuming the holomorphic cover is Leray.

OC.4 now proves that the holomorphic column augmentation is a quasi-isomorphism on every one of these finitely many actual opens. Here is the total-complex elimination. In a closed total cochain, begin with its smallest nonzero cube cardinality. Its component in positive Dolbeault degree is vertically closed, because no smaller-cardinality component contributes. Solve its vertical equation on that very \(M_J\), subtract its total boundary with the prescribed vertical sign, and remove it. Only a component of larger cube cardinality can be introduced. There are finitely many subsets and the Dolbeault degree is bounded. Continue until every surviving component has Dolbeault degree zero; its vertical closure makes it holomorphic. This constructs a representative in the holomorphic cube row. For injectivity, apply the same elimination to a boundary primitive whose total boundary has only holomorphic components. Its remaining degree-zero components are holomorphic and provide the boundary in that row. This proves the actual quasi-isomorphism, including boundaries.

It follows that the ordinary degree-\(N\) diagonal class is computed, on these cofinal actual products, by the quotient of holomorphic functions on \(M_{\{1,\ldots,N\}}\) by restrictions from the actual omitted-coordinate faces. The ordered face restriction signs are those of V.15. No row is discarded by a mere fine-sheaf assertion.

#### OC.4. Laurent representative, all pairings and zero detection

On the full coordinate-punctured product, one-variable Cauchy integrals split a holomorphic function into positive and negative Laurent powers. On a smaller common product, the positive projection in a coordinate uses a fixed outer circle and extends holomorphically through that coordinate's zero. Negative projections in distinct coordinates commute by finite compact integration. The identity

\[
\begin{gathered}
f-\Pi_{1,-}\cdots\Pi_{N,-}f\\
=\sum_{i=1}^N\Pi_{1,-}\cdots\Pi_{i-1,-}\Pi_{i,+}f
\end{gathered}
\tag{OC.5}
\]

therefore writes the difference as actual omitted-coordinate face functions. Their alternating deletion signs provide literal Čech primitives. The entire fully negative Laurent subseries converges normally on every punctured compact, by iterated Cauchy estimates on smaller inner circles. The domain is chosen before summing any index.

For any multiindex \(\alpha\in\mathbf N^N\), the compact product-torus pairing with \(w^\alpha\) annihilates every omitted face by the one-coordinate Cauchy theorem and selects the fully negative coefficient of \(w^{-\alpha-\mathbf1}\). Thus if all these pairings vanish, every fully negative coefficient vanishes, and OC.5 gives an actual face boundary. Conversely every face boundary has all pairings zero. This proves separation in this **completely computed ordinary diagonal local cohomology**, for all base parameters on the common product.

Arbitrarily small inner circles give the ordinary every-ε coefficient bounds after dividing the selected coefficient by \(\alpha!\); conversely those bounds give normal convergence of the entire negative subseries. Compact enlargement gives every fixed derivative tail. The canonical meaning and positive scalar of these product-torus pairings are supplied by the full two-support trace construction TC.1–TC.3; the direct Laurent calculation proves their detection without any solution-module faithfulness premise.

Under an ordinary local biholomorphism, the diagonal supported object and the input top-form line transform by their actual pullback. R.1/R.7 and V.4 give the coordinate invariance of this object and its proper trace, including the Jacobian. In the new coordinates OC.1–OC.4 apply on a cofinal local product and detect the transformed class by all its ordinary normal monomial pairings. Thus the transformed class is the class determined by OR.13's actual coefficient pairings, proving the ordinary geometric coordinate correspondence used in OR.4.

This supplement is confined to ordinary coordinate-punctured products and their diagonal local cohomology. It does not produce strict angular data from an arbitrary projectivized microlocal section, prove the mixed pole/log trace, establish support-enlargement injectivity, or infer any matrix-flatness, faithfulness, propagation or wider programme result.

![An explicit strictly plurisubharmonic exhaustion on a punctured disc](assets/puncture-exhaustion.png)

The exact \(N=1\) example of (OC.2), with unit base radii and base coordinates zero, is
\(\psi(w)=|w|^2-\log(1-|w|^2)+|w|^{-2}\) on \(0<|w|<1\).
Its Levi coefficient is \(1+(1-|w|^2)^{-2}+|w|^{-4}>0\).
The left panel shows its divergence at both excluded boundaries; the right shows
the compact annular sublevels for \(c=4,6,9\). The displayed level radii are
numerical bisection samples. The general product exhaustion and the cohomology
proof are OC.1–OC.4. A higher-dimensional slice fixes any additional punctured
normal coordinates at nonzero constants and shifts the exhaustion by a constant.
The plotting source,
exact function and numerical data,
full PNG and
editable SVG reproduce the illustration.
Free human background for the weighted estimate and Hilbert existence method:
[McNeal–Varolin, *L² estimates for the ∂̄ operator*, §§1.1 and 3.1](https://arxiv.org/abs/1502.08047).
The complete existence proof is [§5.43, DLH.1–DLH.4](#5-43-the-canonical-all-degree-action-and-its-derived-module-structure).

#### IK.1. Common-domain coefficients and exact projective chart restrictions

Fix base coordinates \(u=(t,x_1,\ldots,x_d)\), \(N=d+1\), and a cotangent chart with \(\tau\ne0\), \(z=\xi/\tau\). After an affine linear base change the central normalized covector is \(z=0\). Let \(U\times Z\) be one product of open polydiscs. A coefficient sequence is a collection \(p_j\in\mathcal O(U\times Z)\), \(j\in\mathbf Z\), with, on each compact \(K\Subset U\times Z\),

\[
\begin{gathered}
|p_{-n}|_K\le B_K C_K^n n!,\\

 |p_j|_K\le B_{K,\varepsilon}\frac{\varepsilon^j}{j!}
 \\
(j\ge0,\ \varepsilon>0).
\end{gathered}
\tag{IK.1}
\]

The same actual product supports every index. There is no upper index bound. Write \(P_j(u;\tau,\xi)=\tau^j p_j(u,\xi/\tau)\); the sequence \(\sum_jP_j\) is notation for these homogeneous coefficients. Its negative part is not an ordinary Laurent sum.

Fixed finite base or \(z\) derivatives preserve (IK.1) on a smaller product: the iterated Cauchy formula costs \(\nu!\rho^{-\lvert\nu\rvert}\) for a fixed multiorder \(\nu\) and positive coordinate margins \(\rho\). This constant is independent of \(j\). Restrictions are literal restrictions of every holomorphic coefficient. Germ equality means equality on a single smaller product, rather than independently at each degree. Local coefficient sequences glue: each coefficient glues holomorphically, and a compact set is covered by finitely many smaller compact coefficient charts; take their largest negative constant and sum their positive constants for each fixed \(\varepsilon\). Thus the class is a sheaf on the chosen base coordinate domain and its normalized projective charts.

For another homogeneous normalization \(\tau'=L(\tau,\xi)\ne0\), with the base coordinates still fixed, the coefficient is

\[
 p'_j=(\tau/\tau')^j p_j(u,z). \tag{IK.2}
\]

On a compact overlap both \(\lvert\tau/\tau'\rvert\) and its inverse are bounded. Multiplying by their \(j\)-th powers replaces \(C\) and rescales the input \(\varepsilon\); the every-\(\varepsilon\) quantifier is preserved. The transition identities are exact because the homogeneous functions \(P_j\) are the same functions. This proves projective covariable chart restrictions in fixed base coordinates. It does not prove nonlinear base-coordinate transformation of the microdifferential symbol or its geometric sheaf identification.

#### IK.2. One factorial majorant for binary and triple contractions

We first prove the numerical estimate used for all infinite contractions. For \((E,D,L>0)\), \(a=E(D+L)<1\), define

\[
\begin{gathered}
S_\ell(E,D,L)\\
=\sum_{\substack{J,M,m\ge0\\J-M-m=\ell}}
 E^J D^M L^m\frac{M!m!}{J!}.
\end{gathered}
\tag{IK.3}
\]

If \(\ell\ge0\), \(J=M+m+\ell\), and

\[
\begin{gathered}
\frac{M!m!}{(M+m+\ell)!}\le\frac1{\ell!},\\

 S_\ell\le \frac{E^\ell}{\ell!(1-ED)(1-EL)}.
\end{gathered}
\tag{IK.4}
\]

The factorial inequality follows because \((M+m+\ell)!/(M!m!\ell!)\) is a positive integer. The last sum is the product of two convergent geometric series.

For \(\ell=-n<0\), put \(q=M+m\ge n\). Since \(M!m!\le q!\),

\[
\begin{gathered}
\frac{M!m!}{(q-n)!}\le n!\binom qn,
 \\
 \sum_{M+m=q}D^ML^m\le(D+L)^q.
\end{gathered}
\]

The second inequality is the binomial expansion with its positive integer coefficients removed. Differentiating the geometric series \(n\) times proves

\[
 S_{-n}\le n!\frac{(D+L)^n}{(1-a)^{n+1}}. \tag{IK.5}
\]

All differentiations take place inside its radius of absolute convergence. In particular \(a\le1/2\) gives \(S_{-n}\le2[2(D+L)]^n n!\).

These estimates have literal tails. For \(\ell\ge0\), the part with \(M+m\ge R\) is at most

\[
 \frac{E^\ell}{\ell!}\frac{a^R}{1-a}. \tag{IK.6}
\]

For \(\ell=-n\), \(R\ge n\), the corresponding bound is

\[
\begin{gathered}
n! E^{-n}\sum_{q\ge R}\binom qn a^q\\
=E^{-n}a^n\frac{d^n}{da^n}\frac{a^R}{1-a}.
\end{gathered}
\tag{IK.7}
\]

The right side tends to zero as \(R\to\infty\): differentiating the displayed rational expression produces a fixed finite sum of terms bounded by a polynomial in \(R\) times \((a^{R-n})\). A cutoff in any one of \((J,M,m)\), or in any component used to form them, is controlled by these tails, because \(J=M+m+\ell\) at fixed output degree. These are actual numerical tails, not a topology on a formal completion.

Here is the reduction from coefficients to (IK.3). Fix a compact smaller base/projective product. Enlarge it slightly, still inside the common coefficient domain. There are uniform base Cauchy margins \(\rho>0\) and a uniform full-covariable polydisc margin \(\delta>0\) around each point \(((1,z))\). Every point of the latter polydisc has \(\tau\ne0\) and \(\xi/\tau\) in the enlarged projective product. On it \(\lvert\tau\rvert^j\le A^{\lvert j\rvert}\), for one \(A\ge1\). Thus the Cauchy bounds for \(P_j\), including a covariable multiorder \(\alpha\), are

\[
\begin{gathered}
|\partial_{(\tau,\xi)}^\alpha P_j|
\le\alpha!\delta^{-\lvert\alpha\rvert}\\
\begin{cases}B (AC)^{-j}(-j)!,&j<0,\\
B_{\varepsilon}(A\varepsilon)^j/j!,&j\ge0.
\end{cases}
\end{gathered}
\tag{IK.8}
\]

The analogous base Cauchy estimate costs \(\alpha!\rho^{-\lvert\alpha\rvert}\). For finitely many input operators choose common margins, a largest negative constant, and a common positive \(\varepsilon\); their positive prefactors are then finite. The domains and margins are chosen before any index summation.

For a binary contraction of length \(m=\lvert\alpha\rvert\), the factor left after division by \(\alpha!\) is bounded by \(m!(\delta\rho)^{-m}\). There are at most \((N^m)\) multiorders of length \(m\): words of length \(m\) in \(N\) letters map onto them by taking multiplicities. Absorb that count into \((L^m)\).

For a triple contraction, use multiorders \(a,b,c\in\mathbf N^N\), total length \(m=|a|+|b|+|c|\). The derivatives are

\[
\begin{gathered}
\frac{1}{a!b!c!}
(\partial_\zeta^{a+b}P_j)\\
(\partial_u^a\partial_\zeta^c Q_k)
(\partial_u^{b+c}R_h),\\
\zeta=(\tau,\xi).
\end{gathered}
\tag{IK.9}
\]

Cauchy leaves the factorial factor \(((a+b)!(b+c)!/b!)\). Coordinate by coordinate,

\[
 (a_i+b_i)!(b_i+c_i)!/b_i!\le(a_i+b_i+c_i)!.
\]

Indeed the \(a_i\) consecutive factors beginning at \((b_i+1)\) are at most the corresponding factors beginning at \((b_i+c_i+1)\). Multiplying and using the multinomial inequality bounds this factor by \(m!\). Counts of the \((3N)\) derivative components cost at most \(((3N)^m)\), again absorbed into \((L^m)\). Mixed base/covariable Cauchy bounds use the same enlarged product; they introduce no new index-dependent domain.

Group the nonnegative homogeneous input indices into total \(J\), and the absolute values of the negative indices into total \(M\). There are at most three inputs. Products of negative factorials are at most \(M!\). For \(r\ge1\) nonnegative indices their factorial product is at least \((J!/r^J)\), by the multinomial expansion. Counting their possible compositions costs at most another \((r^J)\); counting the at most three negative compositions costs at most \((3^M)\). Enlarge \(E\) to \(9A\varepsilon\), \(D\) to \((3AC)\), and \(L\) to include the derivative counts and margins. The finitely many sign choices cost a constant, at most eight times the product of the input prefactors. Each sign-choice sum is bounded by a subseries of (IK.3). If there are no nonnegative inputs, \(J=0\); the same enlarged positive majorant still bounds that finite-degree subseries. Consequently (IK.4)–(IK.7) apply to every binary and every triple contraction.

The input \(\varepsilon\) is arbitrarily small. To obtain a specified output positive bound \(\eta^\ell/\ell!\), choose \(E\le\eta\) and \(E(D+L)\le1/2\). The output prefactor may depend on \(\eta\). For negative output choose one such fixed \(E\), obtaining one factorial constant from (IK.5). This proves the exact quantifiers of (IK.1) for all output coefficients.

#### IK.3. The actual coordinate ring, unit, associativity, and derivative tails

Define the product by the full contraction

\[
\begin{gathered}
(P\circ Q)_\ell(u,z)=\\
\left.\sum_{j+k-|\alpha|=\ell}
\frac{\begin{gathered}\partial_\zeta^\alpha P_j\,\\
\partial_u^\alpha Q_k\end{gathered}}{\alpha!}\right|_{\tau=1,\xi=z}.
\end{gathered}
\tag{IK.10}
\]

IK.2 proves absolute normal convergence on every compact of the fixed smaller product, including the infinitely many summands at a fixed \(\ell\). Each summand is holomorphic and homogeneous of degree \(\ell\), so its sum is holomorphic and homogeneous. The output has both exact growth conditions by (IK.4)–(IK.5). IK.6–IK.7 give the actual tails of the grouped full contraction.

Each fixed finite derivative of the tails converges normally as well. Choose, once, a further compact enlargement within the smaller product used for IK.10. Apply (IK.6) or (IK.7) uniformly on that enlargement, and apply the iterated Cauchy integral for the chosen fixed output derivative. Its constant is independent of every summation index and of the cutoff. Thus the differentiated tail is at most that Cauchy constant times the displayed tail. This proves all fixed base and normalized-covariable derivative tails without differentiating a conditionally convergent series.

The unit is the sequence \(P_0=1\), \(P_j=0\) for \(j\ne0\): derivatives of 1 annihilate every nonzero contraction, and the remaining term is the original coefficient. Addition, scalar multiplication, and restrictions are literal, and distributivity follows from absolute convergence. The product is compatible with the fixed-base projective normalizations IK.2 because IK.10 differentiates the same homogeneous functions before making either normalization.

For associativity, expand \((P\circ Q)\circ R\). In the outer covariable derivative \((b+c)\), distribute \(b\) derivatives to \(P\) and \(c\) to \(Q\). The binomial coefficient cancels the outer \(((b+c)!)\), giving exactly IK.9. Expanding \(P\circ(Q\circ R)\) instead distributes its outer base derivative \((a+b)\) between \(Q\) and \(R\), producing the same IK.9, with the same coefficient \((1/(a!b!c!))\). Each term has output degree

\[
\begin{gathered}
\ell=j+k+h\\
-|a|-|b|-|c|.
\end{gathered}
\tag{IK.11}
\]

The triple estimate of IK.2 proves absolute normal convergence of this entire series and all its fixed output derivatives. Tonelli for the absolute scalar majorant therefore justifies both derivative Leibniz expansions and every regrouping. The two products are equal on the single smaller product. This proves associativity as an identity of actual holomorphic coefficient germs, rather than by formal polynomial continuity.

For any finite diagram of coefficient operators take the intersection of their finitely many actual products and the least of the finitely many margins. Apply this argument to each of the finitely many products appearing in its identities. All coefficients, maps, idempotents, and algebraic identities then live on one smaller product. This statement concerns the actual coordinate ring only; it does not manufacture a geometric ring action or a module-derived lift.

#### IK.4. The ordinary zero-section diagonal coefficient calculation

This is a separate, complete calculation of the source's ordinary diagonal local cohomology. Let \(v=u+w\) be a normal diagonal coordinate on \(U\times\Delta^N\), with input frame \(dv_1\wedge\cdots\wedge dv_N\). The local cohomology \(\mathcal H^N_\Delta(\Omega^{(0,N)})\) is computed by the finite coordinate puncture cube, using the exact finite relative-Čech augmentation of the earlier programme §5.35, V.6, with the full resolution proof §5.34, R.1–R.7. Those proofs apply to arbitrary holomorphic coefficients and impose no distribution growth condition. OC.1–OC.3 proves positive holomorphic cohomology vanishing on every actual puncture intersection and the finite elimination reducing the full Dolbeault cube to its holomorphic row. OC.4 proves that all normal monomial pairings detect precisely the actual face boundaries. Thus the holomorphic-row calculation below is justified before any coefficient is extracted.

For a holomorphic top-cube function on \((0<|w_i|<r_i)\), iterated Laurent expansion is normally convergent on smaller product annuli. This can be proved directly: integrate on two nested circles for each coordinate, expand the geometric Cauchy denominators, and estimate each resulting positive/negative geometric series on a smaller annulus. Perform the finitely many coordinate integrations successively; all circles are compact and Cauchy kernels are uniformly bounded there. The terms with a nonnegative power in some \(w_i\) extend across \(w_i=0\); they are precisely boundaries from the cube face omitting that coordinate. The fully negative part is unique, since integrating against every normal monomial annihilates all such faces and selects its Laurent coefficient. The splitting is realized by those actual Cauchy integrals, so it is a splitting of representatives and boundaries, not merely an equality of dimensions.

The resulting unique normal expression is

\[
\begin{gathered}
K_0(u,w)=\frac1{(2\pi i)^N}\\
\sum_{\alpha\in\mathbf N^N}
a_\alpha(u)\frac{\alpha!}{w^{\alpha+\mathbf1}}.
\end{gathered}
\tag{IK.12}
\]

On any compact base set, the Cauchy coefficient estimate on a normal product circle of arbitrarily small positive radii \(\rho_i\) gives

\[
 |a_\alpha|\le M_\rho\frac{\prod_i\rho_i^{\alpha_i+1}}{\alpha!}.
 \tag{IK.13}
\]

Taking all \(\rho_i\le\varepsilon\) proves the ordinary every-\(\varepsilon\) coefficient condition on the same base domain. Conversely that condition makes IK.12 normally convergent on every coordinate-punctured compact: choose \(\varepsilon<\min_i|w_i|\), and sum the product of \(N\) geometric series. The same argument after compact enlargement supplies every fixed derivative tail. Thus the ordinary diagonal geometric sheaf is exactly the ordinary infinite-order coefficient class, including actual germs.

It acts on every holomorphic germ: on a common input polydisc with margins \(\rho_i\ge\rho>0\), Cauchy gives \( |\partial^\alpha f|\le\alpha!\rho^{-|\alpha|}|f|\). Choose \(\varepsilon<\rho\), and sum \(B_\varepsilon|f|(1-\varepsilon/\rho)^{-N}\). Coefficient germs are separated by these actions: apply them to \(e^{\langle u,\lambda\rangle}\). The quotient of the output by this nowhere-zero function is the entire symbol \(\sum_\alpha a_\alpha(u)\lambda^\alpha\); its Taylor coefficients recover every \(a_\alpha\). This is ordinary operator separation, with no claim concerning arbitrary module sections.

Grouping \(a_\alpha\zeta^\alpha\) by total degree gives a subclass of IK.1 whose intrinsic positive coefficient \(P_j(u;\tau,\xi)\) is a homogeneous polynomial of total covariable degree \(j\ge0\), with no negative coefficients. Its normalized polynomial \(p_j(u,z)=P_j(u;1,z)\) has spatial degree **at most** \(j\), retaining every time-power index \(m=j-|\beta|\ge0\). For example \(P_3=\tau^3+\tau\xi^2\) gives \(p_3=1+z^2\). OR.1–OR.4 retain every \(\alpha=(\alpha_t,\alpha_x)\), and AN.18 includes all \(m+|\beta|=j\). Conversely every such subclass with the IK.1 bounds gives the ordinary class: apply the polynomial Cauchy coefficient estimate on a full unit covariable polydisc after finite homogeneous rescaling; \(j!\ge\alpha!\), and a fixed exponential cost is absorbed by rescaling \(\varepsilon\). In IK.10, covariable differentiation of a polynomial of degree \(j\) vanishes for \(|\alpha|>j\), so no negative output degree occurs. The ring product agrees with the ordinary Leibniz action by the same absolutely convergent Cauchy bounds. This identifies the ordinary ring calculation and its inclusion into the coordinate ring. It does not yet prove that the off-zero geometric \(E^\infty\) sheaf restricts compatibly to that ordinary class across all cotangent neighborhoods; that requires IK.G below.

#### IK.5. Full spatial-index holomorphic pole/log kernels

Choose one slightly larger normalized polydisc \((|z_i|<R_i)\) within the common coefficient domain. Expand

\[
\begin{gathered}
p_j(u,z)=\sum_{\beta\in\mathbf N^d}p_{j\beta}(u)z^\beta,
 \\
 a_{m\beta}=p_{m+|\beta|,\beta}.
\end{gathered}
\tag{IK.14}
\]

On a fixed compact base set, the spatial Cauchy coefficient estimate is

\[
\begin{gathered}
|p_{-a,\beta}|\le B C^a a! R^{-\beta},\\

 |p_{j\beta}|\le B_\varepsilon\varepsilon^j R^{-\beta}/j!,
 \\
 R^{-\beta}=\prod_iR_i^{-\beta_i}.
\end{gathered}
\tag{IK.15}
\]

Use \(w=s-t\), \(\zeta_i=y_i-x_i\). Define the cut-plane one-variable functions for integers \(m\ge0\), \(n\ge0\),

\[
\begin{gathered}
\Phi_{m+1}(-w)=\frac{m!}{w^{m+1}},\\
\Phi_{-n}(-w)=-\frac{(-w)^n}{n!}\\
(\log w+\gamma-H_n),\\
H_0=0.
\end{gathered}
\tag{IK.16}
\]

Here (log w) is principal, cut on the negative real ray. The constant \(\gamma\) can be defined by the convergent sequence \(H_n-\log n\): its successive differences \(1/(n+1)-\log(1+1/n)\) have absolute value at most \((1/n^2)\), by integrating \((1/(1+x))\) on \(([0,1/n])\); hence the sequence is Cauchy. Only its finite constant and the explicitly displayed normalization are used. Differentiation proves \(\partial_T\Phi_\lambda(T)=\Phi_{\lambda+1}(T)\) for these integer indices, including \(n=0\), directly from \(H_n-H_{n-1}=1/n\). This fixes the source's signs and factorials.

The full kernel series is

\[
\begin{gathered}
K_P=\frac1{(2\pi i)^N}\\
\left\{\begin{gathered}
\sum_{m\ge0,\beta\ge0}p_{m+|\beta|,\beta}\\
\frac{m!\beta!}{w^{m+1}\zeta^{\beta+\mathbf1}}\\
-\sum_{n\ge0,\beta\ge0}p_{|\beta|-n-1,\beta}\\
\frac{\beta!(-w)^n}{n!\zeta^{\beta+\mathbf1}}\\
(\log w+\gamma-H_n)
\end{gathered}\right\}.
\end{gathered}
\tag{IK.17}
\]

Every spatial index is retained. The first sum is \(K_0\), the coefficient of \(-\log w\) in the second is \(G_P\), and its remaining harmonic-number part is \(K_2\). Thus \(K_P=K_0-\log w\,G_P+K_2\). The first sum is holomorphic wherever all coordinate differences are nonzero, on one small base neighborhood. Indeed

\[
 \frac{m!\beta!}{(m+|\beta|)!}\le1
\]

by \(\beta!\le|\beta|!\) and the binomial inequality. On a compact coordinate-punctured set choose \(\varepsilon\) smaller than every \((|w|)\) and every \(R_i|\zeta_i|\). The absolute sum, with the harmless common factor \(|w\prod_i\zeta_i|^{-1}\), is bounded by

\[
\begin{gathered}
B_\varepsilon(1-\varepsilon/|w|)^{-1}\\
\prod_i(1-\varepsilon/(R_i|\zeta_i|))^{-1}.
\end{gathered}
\tag{IK.18}
\]

For \(G_P\), split according to its homogeneous input index \(j=|\beta|-n-1\). Let \(b=|\beta|\). On a compact set of the thick complement with \((|w|)\) small and \(|\zeta_i|>A_i|w|\), choose \((A_i>R_i^{-1})\). Choose a positive number \(Y>\max_i(R_i|\zeta_i|)^{-1}\), locally uniformly on that compact, such that \(q=|w|Y<1\). The complete homogeneous-polynomial sum obeys

\[
\begin{gathered}
\sum_{|\beta|=b}\prod_i(R_i|\zeta_i|)^{-\beta_i}\\
\le C_Y Y^b,\\
C_Y=\\
\prod_i(1-(R_i|\zeta_i|Y)^{-1})^{-1}.
\end{gathered}
\tag{IK.19}
\]

To prove it, divide by \((Y^b)\) and enlarge the sum from degree \(b\) to every spatial multiindex; the latter is exactly the product of geometric series. Compact strict margins make \(C_Y\) uniformly finite.

If \(j=-a<0\), \(n=a+b-1\), and

\[
 \frac{a!\beta!}{(a+b-1)!}\le a+b.
\]

Put \(r=C|w|<1\). The absolute negative-homogeneous part of \(G_P\), apart from \((2\pi)^{-N}\prod_i|\zeta_i|^{-1}\), is at most

\[
\begin{gathered}
\frac{BC_Y}{|w|}\\
\left\{\begin{gathered}
\frac{r}{(1-r)^2(1-q)}\\
+\frac{rq}{(1-r)(1-q)^2}
\end{gathered}\right\}.
\end{gathered}
\tag{IK.20}
\]

This follows by summing \(((a+b)r^a q^b)\), \(a\ge1,b\ge0\). At \(w=0\) the displayed removable expression has a finite bound when the spatial differences stay nonzero. More directly each original term is an ordinary nonnegative integer power \((w^n)\); normal convergence on a slightly larger \(w\)-disc proves holomorphic extension there.

If \(j\ge0\), \(b=n+j+1\). Use \(\beta!\le b!\), (IK.19), and set \(x=\varepsilon Y\), choosing the input \(\varepsilon\) so that \((x+q<1)\). Its absolute sum is bounded by

\[
\begin{gathered}
B_\varepsilon C_Y Y\\
\sum_{n,j\ge0}(n+1)\binom{n+j+1}{j}\\
q^n x^j\\
=\frac{B_\varepsilon C_Y Y}{(1-x-q)^2}.
\end{gathered}
\tag{IK.21}
\]

The identity first sums \(j\) using the \(((n+1))\)-st derivative of the geometric series, then sums \(((n+1)(q/(1-x))^n)\). Both sums are absolute and positive. This is the mixed positive-homogeneous/negative-normal part absent from a merely negative-homogeneous estimate.

Equations IK.20–IK.21 prove that \(G_P\) is holomorphic on one thick-complement neighborhood, including \(w=0\) where every spatial difference is nonzero. \(H_n\le n\) for \(n\ge1\); inserting its factor differentiates the negative-homogeneous generating series by \(r\partial_r+q\partial_q-1\), since \(n=a+b-1\), and the mixed series by \(q\partial_q\). These are finite derivatives of convergent rational majorants and only increase their denominator powers. Thus \(K_2\) has the same holomorphic domain. Multiplying \(G_P\) by the locally bounded chosen logarithm on a cut-plane compact proves normal convergence of IK.17. All fixed derivative tails follow by compact enlargement and Cauchy, as in IK.3.

For completeness these series have quantitative index tails, not just pointwise convergence. Replace each strict ratio in IK.18–IK.21 by a slightly larger one, keeping \((r'<1)\) and \((q'+x'<1)\). Termwise comparison bounds a tail \(a\ge A\) by ((r/r')^A) times the larger negative majorant, and a tail \(j\ge J\) of IK.21 by ((x/x')^J) times its larger majorant. A spatial-degree tail is controlled by choosing \((Y')\) strictly larger than \(Y\), with the same strict inequalities, and extracting \(((Y/Y')^b)\). A normal-degree tail is controlled by \((q/q')\). Finite powers from \(H_n\) and fixed derivatives are absorbed by these larger majorants or by Cauchy on a larger compact. The finite product of chosen margins is common to the complete coefficient sequence. No coefficient receives its own domain.

The thick complement here has the source III.2 inequality \(>\). This construction makes no assertion about the opposite \(<\) complement used in III.3's special fibration class, or about the distinct \(\widetilde E^\infty,R^\infty,A^\infty\) tensors.

#### IK.6. Literal holomorphic relative-cone classes, branch corrections and finite calibration

Choose one closed thick ordering cone

\[
\begin{gathered}
G_A=\left\{\begin{gathered}
w\in\mathbf R_{\le0},\\
|\zeta_i|\le A_i(-\Re w)
\end{gathered}\right\},\\
A_i>R_i^{-1},
\end{gathered}
\tag{IK.22}
\]

on a sufficiently small convex base neighborhood. This is the initial line-time/spatial-thick support: \(\Im w=0\), so its ordinary real interior is empty. AN.5 below gives the literal enlargement to a full-dimensional pointed cone and explicit proper source margins; no equality of the two support topologies or injectivity of support enlargement is assumed. The ordered complement cover is \(U_0=\{w\notin\mathbf R_{\le0}\}\), \(U_i=\{|\zeta_i|>A_i|w|\}\), \(1\le i\le d\). IK.17 is holomorphic on its top intersection and its two non-pole parts extend holomorphically to the spatial intersection \(U_1\cap\cdots\cap U_d\) after the logarithm is removed. Let \(r_N=(-1)^{N(N-1)/2}\). Use the earlier exact full relative cone and finite cube of programme §5.35, V.1/V.6, with the ungraded input top-form frame. The complement top-cochain representative is

\[
 (0,-r_NK_P), \tag{IK.23}
\]

equivalently the ordinary top-cube coefficient \(r_N(-1)^N K_P\). These signs are the exact V.15–V.17 calibration, rather than an action-based identification. The full cone has \(d(a,b)=(\bar\partial a,ra-Db)\), where \(D=d_C+(-1)^r\bar\partial\) on complement Čech degree \(r\).

The \(K_2\) term is a **literal** cone boundary. For \(d\ge1\), put \(c_{1\cdots d}=r_N K_2\) on the spatial face of complement Čech degree \(d-1\), with every other ordered entry zero. Its coefficients are holomorphic, so \(Dc\) is only the Čech differential. On top indices \(0,1,\ldots,d\), the only nonzero summand deletes index 0 and has sign (+1); hence \(d(0,c)=(0,-r_NK_2)\). For \(d=0\), \(K_2\) is holomorphic on the whole normal disc and \(d(-K_2,0)=(0,-K_2)\). Thus no harmonic-number holomorphic term has been silently discarded.

Changing the branch to \(\log w+2\pi i k\) changes \(K_P\) by \(-2\pi i kG_P\). The same construction, with \(K_2\) replaced by that holomorphic difference, is its exact relative-cone primitive. This proves branch independence of the class. Restrictions are literal restrictions of coefficients and cochains. Enlarging \(A_i\) restricts to a smaller complement cover and the same series on the common top intersection; the finite cover refinement/augmentation of V.6 gives the actual restriction map between support cones. Hence this is a canonical linear germ map from the coordinate coefficient class into the directed holomorphic support-class system, with a common thick cone for any finite operator list.

For the finite-order subring, choose \(k\) with \(P=\partial_t^k\circ P_0\), \(P_0\) of order at most zero, on the common actual chart. When \(k=0\), IK.17 modulo the explicit \(K_2\) boundary is precisely the order-zero kernel (§5.41, CP.15): its only nonnegative normal term is \(p_{0,0}/(w\prod_i\zeta_i)\), and \(n=a+|\beta|-1\) in its negative part. The complete earlier proof [§5.41, CP.2](#5-41-canonical-products-for-full-convergent-spatial-symbols) gives the literal canonical class, including the full finite complement homotopy.

Differentiating IK.17 at a fixed finite order is permitted by its derivative tails. The integer identity in IK.16 and the derivatives of the coefficients give \(K_{\partial_t\circ P}=\partial_tK_P\) as a complement-cochain identity. The complete normal-localization proof [§5.39, NL.11–NL.12 and NL.22–NL.27](#5-39-canonical-normal-localization-of-convergent-spatial-symbols) proves the supported principal-log correction and the canonical normal-generator product, including its origin term. Therefore induction on this **finite** \(k\) identifies IK.23 with the finite-order canonical class. This establishes finite-subring calibration with the source branch, all factorials and every normal sign. It does not take a current limit as \(k\to\infty\).

Indeed \(P=\sum_{j\ge0}\partial_t^j/(j!)^2\) gives \(K=e^{1/w}/(2\pi i w)\). The complete local proof IK.9 below proves that its finite partial diagonal currents have no ordinary distribution limit. IK.23 is instead a holomorphic complement cochain with actual compact convergence there; it requires no extension of its essential singularity to an ordinary point-supported distribution.

#### IK.7. Proper geometry supplied independently of the missing infinite product comparison

The cone \(G_A\) is closed, convex and stable under addition: its normal ray parameter \(\sigma=-\Re w\ge0\) adds, and each spatial inequality follows from the triangle inequality. If two cone differences compose between fixed endpoints, the intermediate normal real coordinate is in the closed segment between the endpoints; the spatial intermediate coordinates lie in closed discs bounded by \(A_i\sigma_{\mathrm{total}}\). Thus the intermediate fiber is closed and bounded. On a fixed compact set of endpoints lying in an inner base polydisc, choose once an outer base polydisc containing all these discs with a positive margin. The intermediate fibers are then compact inside the outer source domain. This is the exact proper triple-to-endpoint geometry of the §5.41, CP.3 construction and applies to any finite list of the present coefficient kernels. It depends only on the common cone, not on differential order.

This establishes the geometric properness premise for a future infinite cup/excision/trace comparison. It does **not** prove convergence of its chain representatives across the positive-head singularity or show that IK.23 intertwines that canonical product with IK.10. The existing finite current primitive cannot be summed in the ordinary current topology. A proof of the full holomorphic relative-cone comparison is still required.

#### IK.8. Exact inverse estimates for supplied pole/log normal-form data

This theorem has an explicit input: a pole/log normal-form pair on one common neighborhood. It does not assert that an arbitrary geometric projectivized microfunction already has this form. Put aside the factor \((2\pi i)^{-N}\). Let \(F_0(u,w,\zeta)\) be holomorphic where every coordinate difference is nonzero, with only fully negative Laurent powers in all normal coordinates. Let \(G(u,w,\zeta)\) be holomorphic on one thick complement \(|\zeta_i|>a_i|w|\), including \(w=0\) when every spatial difference is nonzero, and with only fully negative spatial Laurent powers. Terms missing a negative power in some spatial coordinate have the explicit missing-face boundary interpretation of IK.4; here the supplied normal-form data are already so normalized.

Take fixed numbers \(A_i>a_i\). For every sufficiently small \(\rho>0\), the joint cycle

\[
 |w|=\rho,\qquad |\zeta_i|=A_i\rho \tag{IK.24}
\]

lies in that same actual thick complement. On a compact base set let \(M_\rho\) be the supremum of \(|G|\) on this compact cycle. The Taylor-in-\(w\), Laurent-in-\(\zeta\) coefficient \(g_{n\beta}\), \(n\ge0,\beta\ge0\), is extracted by its finite iterated Cauchy integral. Its value does not depend on the particular permissible circle radii: deform one circle at a time within the common domain, using the one-variable Cauchy integral on the intervening annulus. Existence of the expansion follows on each fixed spatial product annulus by the same geometric Cauchy-denominator expansion as IK.4, with the \(w\)-disc chosen smaller than \(\min_i|\zeta_i|/a_i\). The pieces agree on overlaps by uniqueness of their coefficients.

Thus

\[
\begin{gathered}
|g_{n\beta}|\\
\le M_\rho\,\rho^{-n}\prod_i(A_i\rho)^{\beta_i+1}\\
=B_\rho\,\rho^{|\beta|-n}A^\beta,\\
B_\rho=M_\rho\rho^d\prod_iA_i.
\end{gathered}
\tag{IK.25}
\]

The kernel signs in IK.17 give

\[
\begin{gathered}
a_{-n-1,\beta}=(-1)^n\frac{n!}{\beta!}g_{n\beta},
 \\
 p_{j\beta}=a_{j-|\beta|,\beta}.
\end{gathered}
\tag{IK.26}
\]

Fix the single normalized covariable neighborhood
\(\sum_iA_i|z_i|<1\), and a compact subset on which
\(q=\sum_iA_i|z_i|\le q_*<1\).
For \(j=-s<0\), \(n=|\beta|+s-1\); the complete spatial sum is bounded by

\[
\begin{gathered}
\sum_\beta |p_{-s,\beta}z^\beta|\\
\le B_\rho\rho^{1-s}\\
\sum_{b\ge0}\frac{(b+s-1)!}{b!}q^b\\
=B_\rho\rho^{1-s}(s-1)!\\
(1-q)^{-s}.
\end{gathered}
\tag{IK.27}
\]

The first equality of spatial summation uses the multinomial formula
\(\sum_{|\beta|=b}(A|z|)^\beta/\beta!=q^b/b!\).
The last equality is the \((s-1)\)-st derivative of the geometric series. One fixed \(\rho\) gives the required \(s!\) negative bound with
\(C=[\rho(1-q_*)]^{-1}\). Every spatial index and every singular base parameter allowed by the supplied holomorphic data is included.

For \(j\ge0\), the log part has \(b=|\beta|\ge j+1\) and \(n=b-j-1\). Its complete spatial sum satisfies

\[
\begin{gathered}
\sum_{|\beta|\ge j+1}|p_{j\beta}z^\beta|\\
\le B_\rho\rho^{j+1}\\
\sum_{b\ge j+1}\frac{(b-j-1)!}{b!}q^b\\
\le\frac{B_\rho\rho^{j+1}q^{j+1}}{(j+1)!(1-q)} .
\end{gathered}
\tag{IK.28}
\]

The factorial denominator contains \(j+1\) consecutive positive integers, whose product is at least \((j+1)!\). For any prescribed \(\varepsilon>0\), choose \(\rho\) sufficiently small that \(\rho q_*\le\varepsilon\). Its prefactor \(B_\rho\) can depend on this choice, but the base/covariable neighborhood remains the same. Hence IK.28 has the every-\(\varepsilon\) positive bound. Both IK.27–IK.28 converge normally on all compact subsets of this one neighborhood, so they define holomorphic \(p_j\) there.

The fully negative Laurent coefficients of \(F_0\) give
\(a_{m\beta}\), \(m\ge0\), by dividing its coefficient of
\(w^{-m-1}\zeta^{-\beta-\mathbf1}\) by \(m!\beta!\).
The ordinary diagonal calculation IK.4 gives the every-\(\varepsilon\) ordinary differential coefficient bound. For each homogeneous \(j=m+|\beta|\), their normalized \(p_j\) is a finite polynomial and has IK.1 after rescaling \(\varepsilon\), by the multinomial formula. Adding that pole contribution to the log contribution IK.28 gives the full positive \(p_j\); negative \(p_j\) comes from IK.27.

Conversely the coordinate sequence reconstructed this way recovers every coefficient of the supplied \(F_0,G\). The uniqueness of their actual Cauchy expansions and the normal convergence in IK.5 prove equality of the functions on the common domains. Its canonical harmonic-number \(K_2\) is an explicit cone boundary by IK.6. We have therefore proved a bijection, with explicit common-domain estimates and inverse signs, between the full coefficient class and these **supplied normalized pole/log pairs**, modulo the stated missing-face and normal-holomorphic boundaries. The geometric task IK.G is now specifically the reduction of arbitrary sections of \(\gamma^{-1}\gamma_*C^R_\Delta\) to these normal-form data, with exact angular descent/overlap cochains. That reduction cannot be replaced by assuming HolIII III.2's recalled construction.

As a check retaining a genuinely infinite positive head, let
\(P=\sum_{j\ge0}\partial_t^j/(j!)^2\) and \(T\) be multiplication by \(t\).
The only nonzero derivatives of \(t\) have order 0 or 1, so IK.10 gives

\[
\begin{gathered}
{}[P,T]=\sum_{r\ge0}\frac{\partial_t^r}{r!(r+1)!},
 \\
 K_P=\frac{e^{1/w}}{2\pi i w},
 \\
 K_{[P,T]}=\frac{e^{1/w}-1}{2\pi i}.
\end{gathered}
\tag{IK.29}
\]

The input multiplication kernel \(sK_P\) minus the output multiplication kernel \(tK_P\) is \(e^{1/w}/(2\pi i)\). Its difference from \(K_{[P,T]}\) is exactly the entire function \(1/(2\pi i)\), with the literal one-dimensional normal-holomorphic cone boundary of IK.6. This is a full positive-head example of a needed cochain correction, not a general infinite product theorem or a distribution limit.

![Exact full contraction indices, rational tail bounds and proper intermediate geometry](assets/infinite-kernel-geometry.png)

**Figure IK.1.** Left: a finite displayed window of the exact degree-zero index relation \(m=j+k\ge0\); both mixed signs and the unbounded positive head occur. The theorem retains every index outside the window. Center: exact rational samples of IK.6 at \(\ell=0\), and IK.7 at \(\ell=-3\) divided by its prefactor \(6E^{-3}\), for \(E=1/64,D=2,L=4\). Right: the exact one-spatial-coordinate real slice of section IK.7's intermediate fiber between endpoints \((0,0)\) and \((-1,0)\), at \(A=2\). The full proof uses closed complex spatial discs in every coordinate. The drawing does not replace the complex-domain proof or infer any current limit. The plotting source and exact data reproduce the illustration. Human geometric specifications are Micro-hyperbolic systems §3.1 and HolIII III.2.


Plotting source, exact data, full PNG, and editable SVG.

#### IK.9. Why the infinite positive head has no ordinary distribution limit

Consider the one-variable actual ordinary infinite-order expression

\[
 P(\partial_t)=\sum_{j\ge0}\frac{\partial_t^j}{(j!)^2}.        \tag{IK.30}
\]

It satisfies the positive class (IK.1) and (OR.1): \(1/(\varepsilon^j j!)\le\exp(1/\varepsilon)\), since it is a term of the exponential series. Thus \(1/(j!)^2\le\exp(1/\varepsilon)\varepsilon^j/j!\) for every \(\varepsilon>0\). For a holomorphic germ with Cauchy radius \(\rho\), its action is absolutely bounded by \(\|f\|\exp(1/\rho)\) and hence is a genuine holomorphic-germ operator.

In the Cauchy orientation \(w=s-t\), its positive-head kernel is exactly

\[
\begin{gathered}
K(w)=\frac1{2\pi i}\sum_{j\ge0}\frac1{j!w^{j+1}}\\
=\frac{e^{1/w}}{2\pi i\,w}\\
(w\ne0).
\end{gathered}
\tag{IK.31}
\]

Indeed \(\partial_t^j(s-t)^{-1}=j!(s-t)^{-j-1}\). The series converges normally on every punctured compact: \(|w|\ge\rho>0\) bounds it by \((2\pi\rho)^{-1}\exp(1/\rho)\). It has an essential singularity at zero. For an input germ \(f\), Cauchy pairing on a sufficiently small circle gives exactly (IK.30), because the coefficient of \(s^{-1}\) in \(e^{1/s}f(s)/s\) is \(\sum_j f^{(j)}(0)/(j!)^2\).

There is no ordinary distribution supported at the origin inducing this germ functional. Here is a local proof of the required finite-jet fact. On a fixed compact neighborhood, continuity of a distribution \(T\) gives an estimate by smooth derivative seminorms through some finite order \(M\). If all derivatives through \(M\) of a test function \(f\) vanish at zero, take \(\chi_r(w)=\chi(w/r)\) with \(\chi=1\) near zero. Support at zero gives \(T(f)=T(\chi_rf)\). Taylor's formula and Leibniz show, for \(|\alpha|\le M\),

\[
\begin{gathered}
\sup|\partial^\alpha(\chi_rf)|\\
=O(r^{M+1-|\alpha|})=O(r).
\end{gathered}
\tag{IK.32}
\]

The support of \(\chi_rf\) lies in the fixed compact for small \(r\). The distribution estimate therefore makes \(T(f)=0\). Subtracting the Taylor polynomial times a fixed cutoff proves that a point-supported distribution depends on finitely many jets. This argument uses the real two-dimensional smooth jets when \(t\) is complex, so it does not assume holomorphic distribution theory.

For \(f(t)=t^m\) near zero and \(m>M\), every smooth jet through \(M\) vanishes, while (IK.30) gives

\[
\begin{gathered}
(Pf)(0)\\
=\frac{m!}{(m!)^2}=\frac1{m!}\ne0.
\end{gathered}
\tag{IK.33}
\]

This is the contradiction. The finite partial operator functionals are supported at zero. If they converged in the ordinary distribution topology, their limit would be supported there (tests away from zero are annihilated), and on every cutoff monomial their evaluations eventually stabilize to (IK.33). The same contradiction applies. Thus the finite diagonal-current partial sums have no distribution limit, even though their holomorphic complement kernels converge normally.

Thus full infinite-order kernels require actual holomorphic relative-Čech classes or analytic functionals. The all-degree relative-kernel action of §5.43 applies **after** a genuine full \(E^\infty\)-to-class-ring map has been established and its cone-open, support and common-collar hypotheses have been verified. An abstract map to derived endomorphisms alone supplies no such geometric comparison.


![The full positive head converges on punctured complements but has no finite-jet distribution representative](assets/infinite-order-obstruction.png)

**Figure IK.2.** Exact coefficient and functional data for (IK.30)–(IK.33). Left: \(j!a_j=1/j!\), which gives the holomorphic Cauchy estimate. Center: the exact real sample \(w=1/k\), \(2\le k\le40\), of \(\log_{10}|2\pi K(w)|=(k+\log k)/\log10\), together with finite partial kernels; these are numerical samples on the positive real slice of the punctured complex domain, not a distribution boundary-value assertion. Right: exact rational monomial evaluations of the full operator and the order-three partial operator. For \(m\ge4\) the partial operator vanishes and the full operator does not. The complete proof is IK.9; source normalization is HolIII III.2, printed p.883. The reproducible source and exact data describe the numerical samples; the proof is IK.9.

Plotting source, exact data, full PNG, and editable SVG.

#### AN.1. The exact strict angular input

Write the base variable as \(u=(t,x)\), the normal diagonal difference as \((w,\zeta_1,\ldots,\zeta_d)\), and \(N=d+1\). Fix one base polydisc \(U\), positive \(r,r_i,a_i\), and the actual Reinhardt thick complement

\[
\begin{gathered}
T_a=\left\{\begin{gathered}
0<|w|<r,\\
a_i|w|<|\zeta_i|<r_i\\
(1\le i\le d)
\end{gathered}\right\}.
\end{gathered}
\tag{AN.1}
\]

The universal normal-angle cover of this domain has coordinate \(L\), with \(w=e^L\), \(\Re L<\log r\), and deck transformation \(L\mapsto L+2\pi i\). A strict angular datum is a holomorphic function \(F(u,L,\zeta)\) on that cover and a holomorphic function \(H(u,w,\zeta)\) on

\[
\begin{gathered}
T_a^0=\left\{\begin{gathered}
|w|<r,\\
a_i|w|<|\zeta_i|<r_i
\end{gathered}\right\},
\end{gathered}
\tag{AN.2}
\]

such that the literal equality

\[
\begin{gathered}
F(u,L+2\pi i,\zeta)-F(u,L,\zeta)\\
=H(u,e^L,\zeta)
\end{gathered}
\tag{AN.3}
\]

holds on the **same** domain. In particular, the defect extends holomorphically to \(w=0\) wherever the spatial differences are nonzero. An equivalent finite presentation may be used only when it supplies finitely many lifted normal-angle intervals, actual holomorphic representatives on their corresponding thick domains, and literal overlap corrections that each extend holomorphically to the entire **common** domain AN.2. After an actual face-resolved choice of representatives has removed spatial-face discrepancies, these are equalities of functions on the overlaps. Subtract the cumulative extended corrections along the ordered interval chain and glue equal functions; the final correction after one circuit is \(H\). This finite construction therefore requires common-domain extensions of its corrections as part of the input. Overlap germs alone, or equality in a stalkwise cohomology quotient, are not sufficient.

Set

\[
\begin{gathered}
G=-\frac{H}{2\pi i},\\
 F_0=F+LG.
\end{gathered}
\tag{AN.4}
\]

Equation AN.3 makes \(F_0\) deck-invariant, so it is a single-valued holomorphic function on \(U\times T_a\). This is a literal function equality on the cover. It preserves every base parameter and introduces no coefficient-dependent domain.

#### AN.2. Cauchy splitting, actual missing faces, and common domains

Shrink once so that \(r'<r\), \(r_i'<r_i/2\), and \(a_i r'<r_i/4\). The finite shrink depends only on the original domains. In a spatial annulus, split a holomorphic function into its negative and nonnegative Laurent parts. The nonnegative projection in the \(i\)-th variable is given, for a locally fixed \(R_i\) with \(\max(a_i|w|,|\zeta_i|)<R_i<r_i\), by

\[
\begin{gathered}
(\Pi_{i,+}f)(\zeta_i)\\
=\frac{1}{2\pi i}\int_{|\eta_i|=R_i}
\frac{f(\eta_i)}{\eta_i-\zeta_i}\,d\eta_i,\\
\Pi_{i,-}=1-\Pi_{i,+}.
\end{gathered}
\tag{AN.5}
\]

For the positive part one may use the fixed contour \(R_i=r_i/2\) on the shrunken domain. It therefore extends holomorphically to \(|\zeta_i|<r_i/2\), including \(\zeta_i=0\), while the other thick-complement inequalities are retained. Contour deformation makes different locally fixed contour choices equal. Derivatives in all remaining complex variables commute with a locally fixed compact contour integral, so the projections are jointly holomorphic. Projections in distinct spatial variables commute by repeated integration over the product of finitely many compact circles.

The negative projection also has a direct expansion by an inner Cauchy circle. The usual geometric denominator expansion shows normal convergence on any smaller annulus; applying it in finitely many variables gives normal convergence on compact subsets of AN.1. To see why no uniform inner radius in \(w\) is required for holomorphicity, cover a compact subset by finitely many small products on each of which the circles can be held fixed. Their values agree on overlaps by one-variable contour deformation. This is a finite compact cover, not a different germ domain for each Laurent index.

Write \(\Pi_-=\Pi_{1,-}\cdots\Pi_{d,-}\). The exact finite telescoping identity is

\[
\begin{gathered}
f-\Pi_- f\\
=\sum_{i=1}^d
     \Pi_{1,-}\cdots\Pi_{i-1,-}\Pi_{i,+}f.
\end{gathered}
\tag{AN.6}
\]

Each summand extends across the spatial coordinate \(\zeta_i=0\). On a chosen normal-log cut its coefficient is thus defined on the actual Čech face omitting spatial index \(i\). Apply AN.6 to both \(F_0\) and \(G\). The terms arising from \(LG\) are still holomorphic on those faces because the normal-log cut remains among their defining opens. The fully spatial-negative part \(\Pi_-G\) retains its extension to \(w=0\) on AN.2: on a compact neighborhood with fixed nonzero spatial differences, choose the spatial contour radii first and the normal disc small enough that all contour points stay in AN.2. AN.5 then gives a holomorphic function at \(w=0\). These locally defined extensions agree by uniqueness of the negative Laurent coefficients.

Next split \(\Pi_-F_0\) in \(w\). Its nonnegative normal Laurent part \(J\) extends to \(w=0\) on a thick complement, so it is an actual face omitting the normal-cut index. Its fully negative normal part \(K_0\) has fully negative powers in every normal coordinate. It extends to the full coordinate-punctured product

\[
\begin{gathered}
0<|w|<r',\\
 0<|\zeta_i|<r_i'.
\end{gathered}
\tag{AN.7}
\]

Here is the convergence argument for this extension. Given a compact subset of AN.7, choose circles with \(0<\rho<\min|w|\), \(0<\rho_i<\min|\zeta_i|\), and \(a_i\rho<\rho_i\). They lie in the original thick domain if \(\rho\) is sufficiently small. Iterated Cauchy integrals on that fixed product cycle give, for the coefficient \(c_{m\beta}\) of \(w^{-m-1}\zeta^{-\beta-\mathbf1}\),

\[
 |c_{m\beta}|\le M_{\rho,\rho_i}
           \rho^{m+1}\prod_i\rho_i^{\beta_i+1}.
 \tag{AN.8}
\]

Divide by the target negative powers and sum the resulting finite product of strictly convergent geometric series. This proves normal convergence of the **entire** fully negative subseries on that compact. The cycle and domain were chosen before summing any index. Fixed derivatives and index tails follow by taking a compact enlargement and applying Cauchy's formula there. The coefficient values from different permissible cycles agree: the radii region \(\rho_i>a_i\rho\) is connected in logarithmic radii, so a deformation through overlapping products of permissible annuli, one coordinate at a time, connects any two cycles. Thus this is one actual holomorphic function, not separately chosen coefficient germs.

Consequently, on each log cut, the original datum has the literal form

\[
\begin{gathered}
F=K_0-LG_-+J\\
+\sum_{i=1}^d S_i,\\
G_-=\Pi_-G,
\end{gathered}
\tag{AN.9}
\]

where \(J\) extends to the normal face and \(S_i\) extends to spatial face \(i\). All terms share the one shrunken base neighborhood; \(K_0\) has AN.7 and \(G_-\) has AN.2, with possibly enlarged constants \(a_i\). No finite spatial-degree cutoff has been made.

#### AN.3. Literal relative-cone primitives and branch gluing

Use ordered complement indices \(0,1,\ldots,d\), with index 0 for the normal-log cut and index \(i\) for the \(i\)-th thick spatial complement. The calibrated top representative is \((0,-r_NF)\), where \(r_N=(-1)^{N(N-1)/2}\). Its full-cone differential is

\[
\begin{gathered}
d(a,b)=(\bar\partial a,ra-Db),\\
D=d_C+(-1)^q\bar\partial\\
\text{in complement Čech}\\
\text{ degree }q.
\end{gathered}
\tag{AN.10}
\]

Put \(S_0=J\). For \(d\ge1\), define a degree-\((N-1)\) relative-cone primitive by its holomorphic complement entries

\[
\begin{gathered}
c_{0\cdots\widehat i\cdots d}
     =r_N(-1)^i S_i\\
(0\le i\le d),
 \\
 a=0.
\end{gathered}
\tag{AN.11}
\]

These are complement Čech degree \(N-2\). The sign of deleting index \(i\) in \(d_C\) is \((-1)^i\); its product with AN.11's sign is +1. Therefore

\[
\begin{gathered}
d(0,c)\\
=(0,-r_N(J+\textstyle\sum_iS_i))
\end{gathered}
\tag{AN.12}
\]

as an actual cochain identity. All Dolbeault differentials of the coefficients vanish. There is only one top ordered index set, so no other top entry has been suppressed. When \(d=0\), the holomorphic normal-face term is removed instead by \(d(-J,0)=(0,-J)\). This is the appropriate whole-domain primitive for a one-open complement cover.

Changing the lifted logarithm by \(2\pi i k\) changes \(K_0-LG_-\) by \(-2\pi i kG_-\). That difference extends to the normal face. Formula AN.11 with only \(i=0\) nonzero is its literal primitive; for \(d=0\) use the preceding whole-domain version. These primitives add exactly when the integers \(k\) add. They hence satisfy every triple-overlap identity, rather than only showing pairwise equality of abstract classes. Restrictions, base parameter changes preserving the fixed coordinates, and finite angular refinements are restrictions of these same functions and primitives.

This proves angular normalization and class gluing **for the strict datum AN.1–AN.3**. To conclude IK.G one must still prove that every section of \(\gamma_*C^R_\Delta\) has such a finite common-domain, face-resolved angular presentation, and that zero classes have exactly the displayed holomorphic face primitives. The microlocal support cones appearing in the definition of \(C^R\) do not automatically have ray support. The change from those cones to the cuts and thick spatial faces used here is a substantive support/descent theorem. It remains unproved in this unit.

#### AN.4. Complete coefficient bounds extracted from these data

We derive the complete coefficient estimates directly from the supplied strict data. Temporarily omit the common factor \((2\pi i)^{-N}\). Choose fixed \(A_i>a_i\). For every sufficiently small \(\rho>0\), the joint cycle \(|w|=\rho\), \(|\zeta_i|=A_i\rho\) lies in AN.2. On a compact base set Cauchy extraction from

\[
\begin{gathered}
G_-\\
=
 \sum_{n\ge0,\beta\ge0}g_{n\beta}(u)
                  w^n\zeta^{-\beta-\mathbf1}
\end{gathered}
\tag{AN.13}
\]

gives

\[
\begin{gathered}
|g_{n\beta}|\le B_\rho\rho^{|\beta|-n}A^\beta,
 \\
 B_\rho=M_\rho\rho^d\prod_iA_i.
\end{gathered}
\tag{AN.14}
\]

The expansion exists and is unique on thick domains by AN.2's Cauchy argument. The integer kernel convention is

\[
\begin{gathered}
a_{-n-1,\beta}=(-1)^n\frac{n!}{\beta!}g_{n\beta},
 \\
 p_{j\beta}=a_{j-|\beta|,\beta}.
\end{gathered}
\tag{AN.15}
\]

Thus \(K_0-LG_-\) has exactly the source pole/log signs. On the single normalized covariable neighborhood \(\sum_iA_i|z_i|<1\), put \(q=\sum_iA_i|z_i|\). The multinomial formula yields, for \(s\ge1\),

\[
\begin{gathered}
\sum_\beta|p_{-s,\beta}z^\beta|\\
\le B_\rho\rho^{1-s}(s-1)!\\
(1-q)^{-s}.
\end{gathered}
\tag{AN.16}
\]

Indeed \(n=|\beta|+s-1\), and summing at spatial degree \(b\) gives \((b+s-1)!q^b/b!\). The sum is \((s-1)!(1-q)^{-s}\), obtained by differentiating a geometric series \(s-1\) times. One fixed \(\rho\) gives the negative factorial estimate on each compact with \(q\le q_*<1\).

For \(j\ge0\), the log part has \(b\ge j+1\) and \(n=b-j-1\). It satisfies

\[
\begin{gathered}
\sum_{|\beta|\ge j+1}|p_{j\beta}z^\beta|\\
\le \frac{B_\rho\rho^{j+1}q^{j+1}}
             {(j+1)!(1-q)}.
\end{gathered}
\tag{AN.17}
\]

The ratio \((b-j-1)!/b!\) is at most \(1/(j+1)!\); sum the remaining geometric tail. Given **any** \(\varepsilon>0\), choose \(\rho\) with \(\rho q_*\le\varepsilon\). Only \(B_\rho\) changes; the base and covariable neighborhoods remain fixed. AN.17 gives the required every-\(\varepsilon\) positive estimate.

For the pole coefficients \(c_{m\beta}\) of AN.8, set \(a_{m\beta}=c_{m\beta}/(m!\beta!)\). Taking the same scaled circles gives \(|a_{m\beta}|\le B'_{\rho}\rho^{m+|\beta|}A^\beta/(m!\beta!)\). Grouping \(m+|\beta|=j\) yields

\[
\begin{gathered}
\sum_{m+|\beta|=j}|a_{m\beta}z^\beta|\\
\le B'_{\rho}\frac{[\rho(1+q)]^j}{j!}.
\end{gathered}
\tag{AN.18}
\]

Choose \(\rho(1+q_*)\le\varepsilon\). This proves the full positive pole estimate on that same neighborhood. Every coefficient \(p_j\) is holomorphic there by normal convergence. Every fixed finite base or normalized-covariable derivative satisfies the corresponding bounds and index tails: take one compact enlargement, apply the undifferentiated bounds there, and multiply by the fixed Cauchy derivative constant. No index changes the domain. The estimates preserve all spatial indices and all positive homogeneous contributions to the negative-normal log part.

Reconstruction recovers \(K_0\) and \(G_-\) by their unique Cauchy coefficients. The harmonic-number normalization of the source, \(-(-w)^n(\gamma-H_n)/n!\), adds a holomorphic normal-face function with the same thick domain. The bound \(H_n\le n\) and a slightly larger geometric ratio prove convergence; AN.11 supplies its exact boundary. Hence strict angular data reduce to the supplied pole/log normal form with all signs, actual inverse coefficients and full growth bounds. This remains a conditional reduction from AN.3, not the missing projectivized-sheaf identification.

#### AN.5. Full-dimensional support enlargement and proper source margins

The closed support \(G_A\) in IK.22 has \(\Im w=0\) and ordinary real interior empty. It is an initial line-time/spatial-thick support. It is not the full-dimensional cone required by DL/DLH/DLG, and its quotient/open-set topology is not identified with that of a full-dimensional cone. For every \(\epsilon>0\), support forgetting sends its class into the closed convex full-dimensional cone

\[
\begin{gathered}
G_{A,\epsilon}=\\
\left\{\begin{gathered}
\Re w\le0,\\
|\Im w|\le\epsilon(-\Re w),\\
|\zeta_i|\le A_i(-\Re w)
\end{gathered}\right\}.
\end{gathered}
\tag{AN.19}
\]

The inclusion is literal. This cone is an intersection of the closed halfspaces \(\Re w\le0\), \(\pm\Im w\le\epsilon(-\Re w)\), and the convex norm epigraphs \(|\zeta_i|\le A_i(-\Re w)\); hence it is closed and convex. It has interior because at \(w=-1,\zeta=0\) all inequalities except the redundant \(\Re w\le0\) are strict, and \(\Re w<0\) is strict too. It is stable under addition by the triangle inequality. It is pointed: belonging to the cone and its negative forces \(\Re w=0\), then every other coordinate is zero. Its complementary support map is the restriction of the full-cone complex from the larger complement \(M\setminus G_A\) to the smaller complement \(M\setminus G_{A,\epsilon}\), with the identity on the total component. This is a chain map by AN.10. A class for \(G_A\) therefore has a canonical support-enlargement image; no injectivity of that map is asserted.

For its proper triple-to-endpoint geometry, write \(\sigma_i=-\Re w_i\ge0\). Their sum is the final \(\sigma\). Thus \(0\le\sigma_i\le\sigma\), \(|\Im w_i|\le\epsilon\sigma\), and \(|\zeta_{i,k}|\le A_k\sigma\). Over a compact endpoint set every intermediate point is in a closed bounded family with these fixed bounds. Here is an explicit source-margin choice. Translate the base center to zero and let the outer coefficient polydisc have normal radius \(R\) and spatial radii \(R_k\). Choose inner endpoint radii \(\delta_t,\delta_k>0\) with

\[
\begin{gathered}
\delta_t<\frac{R}{2(1+2\sqrt{1+\epsilon^2})},\\

 \delta_k<R_k/4,\\

 \delta_t<R_k/(8A_k)\\
\text{for every }k.
\end{gathered}
\tag{AN.20}
\]

Then \(\sigma\le2\delta_t\). Every intermediate normal coordinate has modulus at most \(\delta_t+2\sqrt{1+\epsilon^2}\delta_t<R/2\), and every intermediate spatial coordinate has modulus at most \(\delta_k+2A_k\delta_t<R_k/2\). The entire closed bounded fiber family thus has a positive margin inside the actual outer source. It is compact there, giving the proper support map. For any finite operator list use the largest of finitely many \(A_k\), one positive \(\epsilon\), and this one margin choice. These facts supply support enlargement and properness only. An actual DL/DLG instantiation also requires the cone-open, round-domain, output-support and common-collar hypotheses, together with the separate ring-map and full mixed-product arguments. These support bounds alone do not give a derived instantiation.

![The exact line-time support enlargement and bounded intermediate fibers](assets/full-cone-enlargement.png)

**Figure AN.2.** Exact slices of AN.19 with \(\epsilon=1/2\), one spatial variable and \(A=2\). Left: the initial normal ray \(\Im w=0\) lies in the full-dimensional wedge. Center: between endpoints \(w=0\) and \(w=-1\), the intermediate normal difference is \(-\sigma+i\eta\), with \(0\le\sigma\le1\) and \(|\eta|\le\tfrac12\min(\sigma,1-\sigma)\). Right: at spatial endpoint difference zero, the intermediate spatial coordinate is in a **closed complex disc** of radius \(2\min(\sigma,1-\sigma)\); points are exact rational samples with \(\sigma=k/20\). The full properness proof applies in every spatial dimension, inside an outer actual source domain with a positive margin. The support map's injectivity and the derived cone-open/common-collar hypotheses are not inferred. The plotting source and exact data reproduce the illustration.


Plotting source, exact data, full PNG, and editable SVG.

#### OR.1. The complete ordinary class and its binary product

Let \(U\subset\mathbf C^N\) be one base polydisc. An ordinary infinite-order operator is

\[
\begin{gathered}
A=\sum_{\alpha\in\mathbf N^N}a_\alpha(u)\partial_u^\alpha,
 \\

 |a_\alpha|_K\le B_{K,\varepsilon}
                 \frac{\varepsilon^{|\alpha|}}{\alpha!}
 \\
\text{for every }\varepsilon>0.
\end{gathered}
\tag{OR.1}
\]

All coefficients are holomorphic on \(U\). Given a second operator \(B\) on the same domain and a fixed smaller coefficient compact, choose a larger compact and coordinate Cauchy margin \(R>0\). On it

\[
 |\partial_u^\gamma b_\beta|
 \le \gamma!R^{-|\gamma|}
      B_{\varepsilon_B}\frac{\varepsilon_B^{|\beta|}}{\beta!}.
 \tag{OR.2}
\]

The ordinary Leibniz coefficient at \(\eta\) is the entire infinite sum

\[
\begin{gathered}
c_\eta\\
=
 \sum_{\substack{\delta+\beta=\eta\\\gamma\ge0}}
 \binom{\delta+\gamma}{\gamma}
 a_{\delta+\gamma}\partial_u^\gamma b_\beta.
\end{gathered}
\tag{OR.3}
\]

Here \(\delta=\alpha-\gamma\). The fixed output index does **not** bound \(\gamma\). Its full absolute majorant is

\[
\begin{gathered}
|c_\eta|
\le B_{\varepsilon_A}B_{\varepsilon_B}
(1-\varepsilon_A/R)^{-N}\\
\frac{(\varepsilon_A+\varepsilon_B)^{|\eta|}}{\eta!},\\
\varepsilon_A<R.
\end{gathered}
\tag{OR.4}
\]

To verify it, cancel the three factorials in OR.1–OR.3. The remaining summand is \((\varepsilon_A/R)^{|\gamma|}\varepsilon_A^{|\delta|}\varepsilon_B^{|\beta|}/(\delta!\beta!)\). The \(\gamma\)-sum is the product of \(N\) geometric series, and the finite \(\delta+\beta=\eta\) sum is the coordinatewise binomial expansion divided by \(\eta!\). For any specified output \(\varepsilon\), choose \(\varepsilon_A+\varepsilon_B\le\varepsilon\) and \(\varepsilon_A<R/2\). This proves the exact every-\(\varepsilon\) condition, including every spatial index.

At a fixed \(\eta\), the tail with \(|\gamma|\ge M\) tends to zero geometrically. For example choose \(q=\varepsilon_A/R<q'<1\); it is bounded by \((q/q')^M\) times the same majorant with \(q'\). Every fixed output derivative tail has this bound times a fixed Cauchy constant on a compact enlargement. The domain and margins are fixed before all index summations. OR.3 is therefore an actual holomorphic coefficient product.

The action on a holomorphic germ \(f\), with common Cauchy margin \(\rho\), is normally convergent with bound \(B_\varepsilon|f|(1-\varepsilon/\rho)^{-N}\), for \(\varepsilon<\rho\). Applying that estimate on successive compact enlargements justifies differentiating and regrouping the composition. Its coefficient formula is OR.3. The exponential germs \(e^{\langle u,\lambda\rangle}\) separate operators: the output divided by the exponential is the entire series \(\sum_\alpha a_\alpha(u)\lambda^\alpha\), whose Taylor coefficients recover \(a_\alpha\). Hence operator composition proves unit and associativity for this ordinary class. This is ordinary operator separation, not separation of any extended module's sections.

#### OR.2. The holomorphic diagonal kernel and residue-trace map

Its fully negative diagonal kernel is

\[
\begin{gathered}
K_A(u,v)=\frac1{(2\pi i)^N}\\
\sum_\alpha a_\alpha(u)
\frac{\alpha!}{(v-u)^{\alpha+\mathbf1}}.
\end{gathered}
\tag{OR.5}
\]

On a compact where each difference is nonzero, choose \(\varepsilon<\min_i|v_i-u_i|\) and sum a product of geometric series. Thus OR.5 is normally convergent on the coordinate-punctured domain, with every fixed derivative and index tail by a compact enlargement. Conversely Laurent coefficient extraction on arbitrarily small normal circles gives OR.1; the fully negative Laurent splitting removes all actual missing-face boundaries. The actual ordinary geometric calculation is OC.1–OC.4 and IK.4.

For the intermediate holomorphic trace put \(a=s-u\), \(b=v-s\), \(c=v-u=a+b\). A holomorphic top cochain \(T(u,a,b)\) of the two diagonal puncture cubes is defined wherever each \(a_i,b_i\) is nonzero. On an output compact with each \(c_i\ne0\), select fixed positive \(\rho_i<\frac13\min|c_i|\), small enough that \(|a_i|=\rho_i\) and \(b_i=c_i-a_i\) remain inside the actual source. Define

\[
\begin{gathered}
(\mathcal R T)(u,c)=\\
\int_{|a_1|=\rho_1}\cdots\\
\int_{|a_N|=\rho_N}
T\left(\begin{gathered}u,a,\\c-a\end{gathered}\right)\\
\,da_1\cdots da_N.
\end{gathered}
\tag{OR.6}
\]

Each circle has positive complex orientation. All integrals are over one compact product torus separated from both singularity lists. Locally fixed circles prove holomorphic dependence. Deforming within \(0<\rho_i<|c_i|\) proves independence of these circles, one coordinate at a time. Local choices therefore agree and give an actual top cochain on the output puncture cube.

There are two kinds of missing source faces. If \(T\) extends across \(a_i=0\), its \(a_i\) integral vanishes by Cauchy's theorem; \(b_i=0\) is outside that circle. If \(T\) extends across \(b_i=0\), deform its \(a_i\) circle through \(a_i=c_i\) to a fixed larger small circle. That crossing is permissible precisely because this source face is holomorphic at \(b_i=0\). Holding the larger circle fixed for \(|c_i|\) smaller than half its radius gives an output function holomorphic at \(c_i=0\). Thus this source boundary maps to the actual output face omitting \(c_i\). There is no argument from convergence in a quotient. For the output face function \(h_i\), its calibrated relative-cone primitive is the entry \(r_N(-1)^{i-1}h_i\) on that face, by the deletion sign, just as AN.11. This explicitly proves compatibility of OR.6 with source boundaries and the finite cube augmentation.

The canonical meaning of OR.6 is proved by the full two-support cube, proper collar and retained relative trace in TC.1–TC.3 below. Every lower source and target complement entry participates in that comparison, with the signs in TC.15. OC.1–OC.4 supplies the actual holomorphic-row reduction and all-monomial class detection used in this ordinary calculation.

All fixed derivatives and limits of these collar calculations commute with their finite compact integrals. This continuity statement concerns holomorphic functions on the specified separated collars; it makes no claim about distribution convergence at the normal origin or about the mixed thick-cone ray endpoints.

#### TC.1. One coordinate: all four source entries

Use variables \((u,a,c)\), with \(s=u+a\), \(v=u+c\), and \(b=c-a=v-s\). The two source supports are \(A=\{a=0\}\) and \(B=\{b=0\}\); their intersection is \(a=c=0\). Write

\[
\begin{gathered}
U_A=M\setminus A,\\
U_B=M\setminus B,\\
U=U_A\cup U_B\\
=M\setminus(A\cap B).
\end{gathered}
\tag{TC.1}
\]

On an actual source polydisc, a total-degree-\(q\) two-support cube has four entries

\[
\begin{gathered}
(g,h_A,h_B,f)\in\\
I^q(M)\oplus I^{q-1}(U_A)\\
\oplus I^{q-1}(U_B)\\
\oplus I^{q-2}(U_A\cap U_B).
\end{gathered}
\tag{TC.2}
\]

Here \(I\) is the full Dolbeault resolution. In this order its differential is

\[
\begin{gathered}
d_K(g,h_A,h_B,f)=\\
\left(\begin{gathered}
\delta g,\ r_Ag-\delta h_A,\\
r_Bg-\delta h_B,\\
h_B-h_A+\delta f
\end{gathered}\right).
\end{gathered}
\tag{TC.3}
\]

This is V.15 with the ordered subset \((A,B)\), including the sign on the top deletion. Two one-dimensional ordinary kernels have top-cube entries \(-K_A\) and \(-K_B\). Their unshifted source cup consequently has the top function \(f=K_AK_B\), so its complete degree-two cube is \((0,0,0,f)\). The coefficient \(f\) is defined only on \(U_A\cap U_B\). All holomorphic parameters and every normally convergent coefficient index may be retained in this function.

Choose \(0<\kappa_1<\kappa_2<1\), for example \(\kappa_1=1/8\), \(\kappa_2=1/4\). Let \(\theta\) be a fixed smooth function equal to zero on \(( -\infty,\kappa_1^2]\), equal to one on \([\kappa_2^2,\infty)\), and taking values in \([0,1]\). On \(U\) put

\[
\begin{gathered}
\rho_A(a,c)=\\
\begin{cases}\theta(|a|^2/|c|^2),&\!\!c\ne0,\\
1,&\!\!c=0,\ a\ne0,
\end{cases}\\
\rho_B=1-\rho_A.
\end{gathered}
\tag{TC.4}
\]

This is smooth on the **whole union complement** \(U\). At a point with \(c=0,a\ne0\), a neighborhood has \(|a|>\kappa_2|c|\), so \(\rho_A\) is identically one there. It vanishes near \(A\setminus B\), where \(c\ne0\); \(\rho_B\) vanishes near \(B\setminus A\), because there \(a=c\) and the ratio is 1. Thus these are an actual smooth partition subordinate to \(U_A,U_B\). No value or smooth extension is required at the excluded intersection.

The explicit finite augmentation of the full cube to the single relative cone is

\[
\begin{gathered}
\Phi(g,h_A,h_B,f)=\\
\left(\begin{gathered}
g,\ \rho_A h_A+\rho_B h_B\\
+\delta\rho_A\wedge f
\end{gathered}\right).
\end{gathered}
\tag{TC.5}
\]

Each product is extended by zero only where its multiplying cutoff vanishes on a neighborhood. The derivative \(\delta\rho_A\) is supported where both \(a\) and \(b\) are nonzero. Indeed its support for \(c\ne0\) has

\[
\begin{gathered}
\kappa_1|c|\le |a|\le\kappa_2|c|,\\

 |b|\ge(1-\kappa_2)|c|.
\end{gathered}
\tag{TC.6}
\]

For \(c=0\) it vanishes everywhere on \(U\). Formula TC.5 is therefore an actual smooth cochain on \(U\), including neighborhoods of either individual support face away from their intersection.

To check its differential, apply \(\delta\) to the second component, using \(\delta\rho_B=-\delta\rho_A\) and \(\delta(\delta\rho_A\wedge f)=-\delta\rho_A\wedge\delta f\). The result is

\[
\begin{gathered}
r g-\delta\left(\begin{gathered}
\rho_Ah_A+\rho_Bh_B\\
+\delta\rho_A\wedge f
\end{gathered}\right)\\
=\rho_A(r_Ag-\delta h_A)\\
+\rho_B(r_Bg-\delta h_B)\\
+\delta\rho_A\wedge(h_B-h_A+\delta f).
\end{gathered}
\tag{TC.7}
\]

This is exactly the complement component of \(\Phi d_K\). The total components are both \(\delta g\). Thus TC.5 is a chain map in every degree, with every base-antiholomorphic term retained. The general finite Čech row contraction in V.6 proves that this augmentation is the actual relative-support comparison; TC.7 also verifies it directly for this two-support cube.

For the top holomorphic function it gives

\[
 \Phi(0,0,0,f)=(0,\delta\rho_A\,f).
 \tag{TC.8}
\]

There is no whole-domain \(\rho_A f\) cochain in this equation. Only its derivative times \(f\) occurs on the actual union complement.

#### TC.2. Proper collar, parameter boundary, and retained relative trace

Choose an outer coefficient source with normal radius \(R\), and an inner endpoint domain \(|c|<r\), \(r<R/4\). Choose one smooth source collar \(\eta(a)\), equal to one on \(|a|\le R/3\), supported on \(|a|<R/2\), with all its source points inside the original domain. Shrink \(r\), if needed, so that \(\kappa_2r<R/3\). Then \(b=c-a\) has modulus less than \(3R/4\) on the collar. The source collar is properly supported over the inner endpoint domain; over every compact endpoint set its support is in a fixed compact subset of the actual outer coefficient source. Compact base parameters are handled by the same finite margin choice.

The controlled relative localization is

\[
\begin{gathered}
L_\eta(g,h)\\
=(\eta g+\delta\eta\wedge h,\ \eta h).
\end{gathered}
\tag{TC.9}
\]

Its total first component is extended across the support only on a neighborhood where \(\delta\eta=0\), exactly as in R.7. For TC.8, TC.6 and the choice of collar give

\[
\begin{gathered}
\delta\eta\wedge\delta\rho_A f=0,\\

 \eta\delta\rho_A f=\delta\rho_A f.
\end{gathered}
\tag{TC.10}
\]

Thus its localized relative pair is still \((0,\delta\rho_A f)\). The first component is the literal zero whole-domain form. The complement component remains defined on \(U\); it has not been extended at \(a=c=0\). On a compact subset of the output complement \(c\ne0\), TC.6 puts its entire fiber support in a compact annulus separated from both poles and from the outer source boundary. Its relative trace there is consequently an ordinary compact smooth integration. Nothing is asserted about pushing this complement coefficient across \(c=0\).

In source degree 2 the one-dimensional normalized relative trace has the sign

\[
\begin{gathered}
\operatorname{Tr}^{\rm rel}(g,h)\\
=(\operatorname{Tr}_2g,-\operatorname{Tr}_1h).
\end{gathered}
\tag{TC.11}
\]

It is the full-cone version of the \([1]\)-shifted trace; the minus on the complement entry is essential. The ordinary coefficient-frame trace \(\operatorname{Tr}_1\) integrates the fiber \(d\bar a\wedge da\) coefficient with scalar +1, as proved in V.4. Terms in \(\delta\rho_A\) containing only a base-antiholomorphic differential have insufficient real fiber degree and vanish under this trace. They remain part of the source cochain in TC.8–TC.10, so no differential or parameter term has been deleted before tracing.

For fixed \(c\ne0\), use an outer circle \(|a|=\rho\) with \(\kappa_2|c|<\rho<|c|\) and an inner circle of radius less than \(\kappa_1|c|\). On the outer circle \(\rho_A=1\); on the inner circle it is zero. The only singularity inside the outer circle is \(a=0\); the second singularity \(a=c\) lies outside. Stokes on the intervening annulus gives

\[
\begin{gathered}
\int \delta\rho_A\,f\wedge da\\
=\int_{|a|=\rho}f\,da.
\end{gathered}
\tag{TC.12}
\]

The equality uses only compact coefficients on that annulus and its boundary. Combining TC.8–TC.12 gives the actual retained target pair

\[
 (0,-\textstyle\int_{|a|=\rho} f\,da).
 \tag{TC.13}
\]

The parameter-dependent partition is smooth on all of the source union complement, including \(c=0,a\ne0\); the single proper collar is fixed over the whole inner endpoint domain. The final coefficient is holomorphic on the output complement by a locally fixed permissible circle and one-variable Cauchy deformation. This is the complete ordinary two-support cup/excision/relative-trace roof. It requires neither a distribution extension of an essential singularity nor a uniform distribution-order bound.

#### TC.3. Every normal coordinate and every complement entry

For \(N\) coordinates retain the entire ordered \(2N\)-support cube for \(A_i=\{a_i=0\}\), \(B_i=\{c_i-a_i=0\}\). It has one entry on every actual intersection of the corresponding complements. Apply TC.5 in coordinate \(i\) with every remaining cube entry, all other variables, and its ordered holomorphic coefficient frame retained. A cutoff in coordinate \(i\) is TC.4 with \(a_i,c_i\), and its collar is TC.9 with one fixed outer source radius. These maps commute with restriction in every unprocessed coordinate. Equation TC.7 verifies their complete differential, including parameter-antiholomorphic derivatives.

The finite tensor-to-cube sign is V.16,

\[
 (-1)^{\sum_{i<j}q_i r_j},
 \tag{TC.14}
\]

for antiholomorphic degrees \(q_i\) and complement degrees \(r_i\). Therefore one may apply TC.5/TC.9/TC.11 successively without replacing the remaining cube by its top holomorphic coefficient. At intermediate stages all its lower Čech/Dolbeault entries are still present. This is an iteration of actual chain maps and controlled localization roofs; the shift on each trace uses V.14 and TC.11. The final holomorphic residue is a **top coordinate of the target cube**, not a degree-\((N-1)\) Dolbeault complement function.

Here are the exact signs in that final coordinate. Each one-coordinate relative trace contributes the positive connecting class \((0,-\mathcal R_i f)\). Its ordinary one-support cube top coordinate is \(-\mathcal R_i f\), giving \((-1)^N\) for the ordered normal list. Regrouping the \(N\) one-coordinate holomorphic lines into the single ungraded input top-form frame uses V.14/V.17 and contributes
\(r_N=(-1)^{1+\cdots+(N-1)}\). The between-kernel unshifted cup sign \((-1)^{N^2}\) and the source-degree-\(2N\) trace sign \((-1)^{N^2}\) cancel by V.13. No further double suspension is introduced. Hence the final ordinary top cube coefficient and the corresponding full-cone Čech top coefficient are respectively

\[
\begin{gathered}
r_N(-1)^N\mathcal R f,\\
-r_N\mathcal R f,\\
\mathcal R f=\\
\int_{|a_1|=\rho_1}\cdots\\
\int_{|a_N|=\rho_N}
f\left(\begin{gathered}u,a,\\c-a\end{gathered}\right)\\
\,da_1\cdots da_N.
\end{gathered}
\tag{TC.15}
\]

The cone-to-cube multiplier is \((-1)^{N-1}\), so the two coordinates in TC.15 agree exactly. For \(N=1,2,3\), the ordinary top factors are \(-1,-1,+1\); the full-cone top factors are \(-1,+1,+1\). This check detects both the missing connecting sign and the misplaced line suspension in dimension two.

All source collars fit on one common outer polydisc: take \(|c_i|<r_i<R_i/4\), collar support \(|a_i|<R_i/2\), and \(\kappa_2r_i<R_i/3\). A finite operator/matrix list permits taking the smallest of its finitely many positive margins. On every output compact where the traced coordinates are nonzero, all transition products are compact and separated from both source singularity lists. Arbitrary joint holomorphic coefficients and their entire normally convergent index families may be multiplied there. Repeated integration is justified by bounded compact coefficients and Fubini; each fixed derivative commutes with these finite integrations. Where some output coordinate is zero, keep its corresponding target cube face and the other entries until that coordinate is processed. No singular coefficient is extended across that face. This verifies the domains and complete complement roof in every normal dimension.

Missing source faces behave as required: a function extending across \(a_i=0\) has zero residue in that coordinate; a function extending across \(b_i=0\) permits deforming the \(a_i\) circle through \(a_i=c_i\) to a fixed small circle and gives a function extending across \(c_i=0\). Its target missing-face coefficient supplies the literal Čech boundary with the deletion sign of V.15. This statement is about the complete target cube after TC.15; it does not drop the source or target complement terms during the trace.

#### TC.4. Canonical ordinary conclusions and remaining scope

The complete ordinary product estimates OR.1/OR.3 and the ordinary coordinate-change coefficient estimates OR.4 have canonical-class interpretations through TC.1–TC.3 and the actual ordinary cohomology calculation OC.1–OC.4. Specifically, for the complete ordinary all-spatial class, normal convergence on the fixed separated tori gives

\[
 \mathcal R(K_AK_B)=K_{A\circ B},
 \tag{TC.16}
\]

with all fixed-output infinite contractions and every derivative tail controlled by OR.4/OR.11. TC.15 identifies this equality with the full ordinary canonical cup/excision/normalized trace, including every target complement coordinate and every sign. OC.4’s complete finite torus/Laurent separation then gives the ordinary coordinate-class correspondence in OR.4. These are the complete ordinary coefficient formulas and ordinary canonical scope.

Full IK.G and IK.M are proved in [§5.45, IG.1–IG.5 and IM.1–IM.6](#5-45-the-full-geometric-infinite-order-kernel-ring-and-its-canonical-product). Ordinary trace and coordinate conclusions alone supply neither assertion. No derived instantiation or later matrix-flatness, faithfulness, reconstruction, propagation, D-type/C1/analytic proper/GL/BB/KL conclusion is asserted in this ordinary unit.

#### OR.3. Complete ordinary canonical multiplication

Apply OR.6 to \(T=K_A(u,s)K_B(s,v)\). On the selected torus \(K_B(s,v)\) is holomorphic in \(s\) throughout the inside polydisc: its only coordinate kernel singularities are at \(s_i=v_i\), outside the selected circles. Choose once a slightly larger inside polydisc still separated from those singularities and from the coefficient-domain boundary. OR.1 for \(A\) and Cauchy's derivative estimate for \(K_B\) give an absolute sum bounded by a product of geometric series there. Termwise Cauchy integration is therefore justified and gives

\[
\begin{gathered}
\int K_A(u,s)K_B(s,v)\,ds\\
=\sum_\alpha a_\alpha(u)
                    \partial_s^\alpha K_B(s,v)|_{s=u}.
\end{gathered}
\tag{OR.9}
\]

For a fixed \(\alpha\), differentiate OR.5 for \(B\) on the common separated domain and apply the finite derivative Leibniz rule. Its term indexed by \(\beta\) and \(\gamma\le\alpha\) is

\[
\begin{gathered}
\frac1{(2\pi i)^N}
\binom\alpha\gamma a_\alpha\partial^\gamma b_\beta\\
\frac{(\alpha-\gamma+\beta)!}
{c^{\alpha-\gamma+\beta+\mathbf1}}.
\end{gathered}
\tag{OR.10}
\]

The **entire** sum is absolute. Use OR.4 with \(\varepsilon_A+\varepsilon_B<\min_i|c_i|\), as well as \(\varepsilon_A<R/2\). After grouping \(\eta=\alpha-\gamma+\beta\), the sum is bounded by

\[
\begin{gathered}
\frac{B_{\varepsilon_A}B_{\varepsilon_B}}
{(2\pi)^N\prod_i|c_i|}\\
(1-\varepsilon_A/R)^{-N}\\
\prod_i\!\!\bigl(1-(\varepsilon_A+\varepsilon_B)/|c_i|\bigr)^{-1}.
\end{gathered}
\tag{OR.11}
\]

Thus every regrouping in OR.9–OR.10 is justified on the fixed output compact, and OR.3 proves

\[
\begin{gathered}
\mathcal R\bigl(K_A(u,s)K_B(s,v)\bigr)\\
=K_{A\circ B}(u,v).
\end{gathered}
\tag{OR.12}
\]

It is an equality of holomorphic top cochains, in the canonical residue trace calculated in OR.2. The \((2\pi i)^{-N}\) normalization is exact: two kernels supply its square and the \(N\) positive Cauchy residues supply \((2\pi i)^N\). The trace/cup calibration has no further scalar. The diagonal unit is \(\prod_i[2\pi i(v_i-u_i)]^{-1}\), by the same positive Cauchy residue, on both sides.

Triple products may be computed on two nested product tori, with radii chosen once from the finite list of coefficient domains and endpoint separation margins. All coefficients and Cauchy factors are bounded on the resulting compact product of tori, and OR.11's stronger interior margins give absolute convergence of every operator series there. Repeated compact integration hence satisfies Fubini. Alternatively OR.12 and the already proved ordinary associativity in OR.1 give the same result. Neither route passes to ordinary distributions. Every finite ordinary matrix, idempotent, differential, relation and homotopy uses the finite intersection of these domains, so the same ordinary unital class map applies on one neighborhood to all its entries. An infinite-order module-derived instantiation additionally requires the full IK.G/IK.M comparison and the cone-open, support and common-collar hypotheses of §5.43.

#### OR.4. Arbitrary ordinary holomorphic coordinate overlaps

The ordinary statement is also intrinsic under arbitrary local biholomorphic base-coordinate changes; no linearity hypothesis is needed here. Let \(v=\phi(u)\) be a biholomorphism and let \(A\) have OR.1 on \(U\). Work on one smaller relatively compact base domain \(U'\Subset U\). For \(\alpha\ge0\) define

\[
\begin{gathered}
a'_\alpha(\phi(u))=\\
\frac1{\alpha!}\\
\left[A_{u'}(\phi(u')-\phi(u))^\alpha\right]_{u'=u}.
\end{gathered}
\tag{OR.13}
\]

Here \(u\) is held fixed as a holomorphic parameter while \(A\) differentiates the variable \(u'\). The notation does not differentiate the parameter along with the input variable. The series defining OR.13 is normally convergent on every compact of \(U'\): choose a fixed input tube about that compact with positive margin inside \(U\), use the derivative Cauchy bound for the holomorphic function \((\phi(u')-\phi(u))^\alpha\), and choose the input coefficient \(\varepsilon\) smaller than that margin. Thus \(a'_\alpha\) is holomorphic on the one domain \(\phi(U')\), independent of \(\alpha\).

For every \(\delta>0\), continuity of the finitely many coordinate functions of \(\phi\) on a compact enlargement gives a uniform input radius \(r_\delta>0\) on which
\(\max_i|\phi_i(u')-\phi_i(u)|<\delta\). OR.1's action bound, with \(\varepsilon<r_\delta/2\), then gives

\[
 |a'_\alpha|_K\le C_{K,\delta}
                \frac{\delta^{|\alpha|}}{\alpha!}.
 \tag{OR.14}
\]

The radius used to estimate the action may depend on \(\delta\); the common coefficient domain \(\phi(U')\) does not. Hence OR.14 has the exact every-\(\delta\) quantifier. No Faà di Bruno expansion or formal coefficient transformation has been assumed.

Let \(f\) be holomorphic on an actual input neighborhood of \(v=\phi(u)\). Choose a smaller radius \(R>0\) for its Taylor series, then choose the preceding input tube with coordinate variation less than \(\delta<R\). On that tube,

\[
\begin{gathered}
f(\phi(u'))=\\
\sum_\alpha\frac{\partial_v^\alpha f(\phi(u))}{\alpha!}\\
(\phi(u')-\phi(u))^\alpha
\end{gathered}
\tag{OR.15}
\]

converges normally on a slightly larger tube than the one used for the action. OR.1's sup-norm action estimate permits applying \(A_{u'}\) termwise. Equations OR.13–OR.15 give

\[
\begin{gathered}
{}[A(f\circ\phi)](u)\\
=\sum_\alpha a'_\alpha(\phi(u))\\
\partial_v^\alpha f(\phi(u)).
\end{gathered}
\tag{OR.16}
\]

The right side is absolutely bounded by
\(C_{K,\delta}|f|(1-\delta/R)^{-N}\). Therefore the ordinary coordinate-change operator is actual and has the complete ordinary class. Applying the same argument to \(\phi^{-1}\) gives its inverse. The action separation proved in OR.1 makes compositions of coordinate changes agree exactly on triple overlaps and makes the product invariant.

This also gives the ordinary diagonal geometric overlap. In any base coordinates, OC.1–OC.4 and the finite puncture cube/Laurent argument of OR.2 prove that a diagonal local-cohomology class is uniquely determined by its pairings with all holomorphic normal monomials; those pairings are its fully negative Laurent coefficients. The pairing on the compact normal torus is the canonical residue trace just proved in OR.2. Changing coordinates pulls back the input holomorphic top form with its Jacobian and applies the ordinary change of variables on that same compact integration cycle; the earlier V.4 determinant/orientation proof gives its coordinate invariance. Hence the naturally transformed diagonal class has the monomial pairings OR.13 and, by the proved Laurent separation, is precisely the class of the kernel of \(A'\). This use of separation is confined to the completely calculated ordinary diagonal local cohomology. It does not identify the off-zero projectivized microlocal sheaf or establish separation of arbitrary extended-module sections.

![Strict angular defect, exact Čech deletion signs, and nested ordinary residue tori](assets/angular-residue-geometry.png)

**Figure AN/OR.1.** Left: exact four-interval cover of the normal-angle circle, with interval length \(3\pi/4\) and centers \(0,\pi/2,\pi,3\pi/2\); the jump \(H\) is an explicit AN.3 input, not a theorem about arbitrary microfunctions. Center: the actual deletion signs of AN.11 for \(N=3\), where \(r_3=-1\); each product of the Čech deletion sign with the primitive sign is \(r_3\). Right: the one-coordinate complex slice of OR.6 with \(u=0\), \(v=1\), intermediate residue circle radius \(1/4\), and an outer input circle radius \(3/2\). The two kernel singularities are separated on the entire intermediate circle by exact lower bounds \(1/4\) and \(3/4\). The general theorem uses products of these circles in every normal coordinate, actual positive margins in the coefficient domain, and complete infinite index sums. The plotting source and exact rational data reproduce the illustration. Human geometric specifications are the freely accessible sources linked above; proofs are AN.1–AN.4 and OR.1–OR.4.


Plotting source, exact data, full PNG, and editable SVG.

#### Mathematical conclusions and remaining scope

| Assertion | Proof locations and mathematical hypotheses |
|---|---|
| Common-domain full coefficient sheaf, every spatial index | IK.1/IK.14–IK.15 The base coordinates are fixed. |
| Full infinite binary/triple contractions and every fixed derivative tail | IK.2–IK.3, especially IK.3–IK.11 |
| Unit, associativity, finite common-domain coefficient diagrams | IK.3 |
| Fixed-base projective covariable chart overlaps | IK.2 and homogeneous IK.10 |
| Ordinary diagonal \(D^\infty\) coefficient/local-cohomology identity | OC.1–OC.4 and IK.4; §5.34 R.1–R.7 and §5.35 V.6 for the exact finite cube |
| Full pole/log/holomorphic kernel convergence, including mixed positive-homogeneous negative-normal terms | IK.5 |
| Actual holomorphic forward class, branch/harmonic-term boundaries, finite calibration | IK.6; §5.41 CP.2 and §5.39 NL.11–NL.12/NL.22–NL.27 The asserted map is linear. |
| Common proper triple geometry | IK.7 |
| Exact inverse full coefficient extraction from supplied pole/log normal forms | IK.8, especially IK.24-IK.28 |
| IK.G: full \(\gamma^{-1}\gamma_*C^R_{\Delta}\) identification with the full coefficient class; nonlinear coordinate transitions and zero-section/off-zero compatibility | Complete cofinal-support, angular-witness, zero-detection, nonlinear and ordinary compatibility proof: [§5.45, IG.1–IG.5 and IM.1–IM.6](#5-45-the-full-geometric-infinite-order-kernel-ring-and-its-canonical-product) |
| IK.M: canonical full holomorphic relative-cone cup/excision/normalized-trace comparison with IK.10 | Complete bounded-normalization, full cup/excision/trace and final-support primitive proof: [§5.45, IG.1–IG.5 and IM.1–IM.6](#5-45-the-full-geometric-infinite-order-kernel-ring-and-its-canonical-product) |
| Whole-ring degree-one/all-degree action, support preservation and finite full-infinite-order derived diagrams | Complete actual finite-diagram realization and module-derived action: [§5.46, ID.1–ID.6](#5-46-finite-infinite-order-diagrams-on-actual-derived-support-modules); exact final-cone, cone-open, round-domain, restricted action-source and common-collar hypotheses retained The hypotheses are ID.1–ID.4. |
| All finite matrix relations, right flatness, proper-ideal detection and unit injectivity | Matrix proof: [Section 5.47](#5-47-arbitrary-finite-matrices-and-right-flatness-of-the-full-infinite-order-ring). Ideal/unit/faithful-extension proof: [Section 5.48](#5-48-proper-ideals-and-faithful-extension-to-the-full-infinite-order-ring). Other scalar changes are separate assertions. The matrix and ideal/unit extension hypotheses apply; other scalar changes are separate assertions. |
| Ordinary/mixed scalar changes, infinite Späth division and balanced reconstruction | The genuine microlocal/ordinary extension is proved in §§5.47–5.48. Full infinite Späth sum division: [Section 5.49](#5-49-full-infinite-spath-division-and-balanced-reconstruction), ISD.1–ISD.5 and ISB.1–ISB.5. Actual special fibration class: ISF.1–ISF.5. Both algebraic balanced reconstruction maps, whole singular germ, off-ray vanishing and full contact transport: ISR.1–ISR.7. The stated division, special fibration and reconstruction hypotheses apply. |
| Propagation, purity, arbitrary-section separation, monodromy stability/full infinite-order D-type linearity | Separate module and solution statements. Not asserted by these local kernel arguments. |
| Faithful finite D-type embedding, finite poles, intrinsic order/half order, full C1, initial generators, arbitrary proper analytic regularity | Separate module and proper-image statements. Not asserted by these local kernel arguments. |
| General BB/KL, GL14/GL16, affine, critical and factorization assertions | Outside the local kernel and support statements. Outside the stated local kernel and support results. |

| Assertion | Proof locations and mathematical hypotheses |
|---|---|
| Normalize supplied strict angular universal-cover data, retaining every spatial Laurent index | AN.1–AN.3, actual projections AN.5, primitives AN.11–AN.12 AN.3 is an explicit hypothesis. |
| Recover full homogeneous coefficients on one common product, both factorial and every-\(\varepsilon\) bounds, actual tails | AN.4, AN.14–AN.18 The hypotheses are the stated strict angular data. |
| Complete ordinary all-spatial infinite product and fixed-index infinite contractions | OR.1, OR.3–OR.4 |
| Ordinary diagonal holomorphic cup/excision/normalized residue trace and canonical ordinary kernel multiplication | OR.2–OR.3, TC.1–TC.4 and OC.1–OC.4; finite-cube V.6 and compact smooth collar trace V.4–V.5 |
| Arbitrary ordinary holomorphic coordinate transitions, actual class correspondence and triple overlaps | OR.4, OR.13–OR.16 and the proved ordinary Laurent pairing separation |
| Full IK.G identification of arbitrary \(\gamma^{-1}\gamma_*C^R_\Delta\) sections with the coefficient sheaf | Complete cofinal-support, angular-witness, zero-detection, nonlinear and ordinary compatibility proof: [§5.45, IG.1–IG.5 and IM.1–IM.6](#5-45-the-full-geometric-infinite-order-kernel-ring-and-its-canonical-product) |
| Full IK.M product of arbitrary mixed pole/log thick-cone kernels | Complete bounded-normalization, full cup/excision/trace and final-support primitive proof: [§5.45, IG.1–IG.5 and IM.1–IM.6](#5-45-the-full-geometric-infinite-order-kernel-ring-and-its-canonical-product) |
| Whole infinite-order support-preserving degree-one/all-degree action and module-derived finite diagrams | Complete actual finite-diagram realization and module-derived action: [§5.46, ID.1–ID.6](#5-46-finite-infinite-order-diagrams-on-actual-derived-support-modules); exact final-cone, cone-open, round-domain, restricted action-source and common-collar hypotheses retained The hypotheses are ID.1–ID.4. |
| Finite matrix relations, right flatness, proper-ideal detection/unit injectivity, ordinary and mixed scalar changes | Full geometric and genuine ordinary matrix/right-flatness proof: [Section 5.47](#5-47-arbitrary-finite-matrices-and-right-flatness-of-the-full-infinite-order-ring). Proper-ideal/unit/faithful-extension proof: [Section 5.48](#5-48-proper-ideals-and-faithful-extension-to-the-full-infinite-order-ring). Other ordinary/mixed scalar changes are separate assertions. The geometric and genuine ordinary extension described here is the stated scope. |
| Infinite Späth division and balanced reconstruction | [Section 5.49](#5-49-full-infinite-spath-division-and-balanced-reconstruction), ISD.1–ISD.5, ISF.1–ISF.5, ISR.1–ISR.7 and complete ISB.1–ISB.5: actual x-dependent spatial relations, both growth regimes, all spatial/derivative/common-domain controls, both balanced inverses, singular-inclusive whole germ, off-ray vanishing and full contact transport. The stated fibration and full coefficient hypotheses apply. |
| Moving-sector propagation, singular-support purity, arbitrary-section separation, special-action/monodromy stability, infinite-order D-type linearity | Separate module and solution statements. Not asserted by these local kernel arguments. |
| Faithful finite D-type embedding, finite poles/intrinsic order/half-order/full C1/initial generators, arbitrary analytic proper regularity, general BB/KL/GL14/GL16/affine/critical/factorization assertions | These global assertions require their own mathematical hypotheses and proofs. Outside the stated local kernel and support results. |

The full infinite-order geometric ring theorem is proved in [§5.45, IG.1–IG.5 and IM.1–IM.6](#5-45-the-full-geometric-infinite-order-kernel-ring-and-its-canonical-product). Supplied strict angular data and the ordinary diagonal subring retain their explicit scope. Section 5.47 proves the arbitrary finite-matrix relation criterion and right flatness of the full infinite-order microlocal ring over the finite-order ring for all algebraic left modules, including the genuine ordinary zero-section version. Section 5.48 proves proper-ideal detection and arbitrary-module unit injectivity and faithful exact extension for the full geometric and genuine ordinary zero-section rings. Section 5.49 proves full infinite Späth sum division and both algebraic balanced reconstruction maps, the distinct special fibration class, whole-germ singular scope, off-ray vanishing and full contact transport. Other scalar changes, propagation, analytic proper regularity and every other unresolved original mathematical programme assertion remain open.


### 5.45. The full geometric infinite-order kernel ring and its canonical product

We identify the full homogeneous coefficient class with the source-defined complex-projectivized diagonal sheaf. We derive strict angular data from arbitrary sections over the complete normal-phase fiber, detect zero classes, and prove actual nonlinear coordinate and zero-section compatibility. The full mixed canonical product is then proved in IM.1–IM.6. Every spatial index, both growth regimes, fixed derivative tails and one common actual coefficient domain are retained.

The free human definitions are [Kashiwara–Schapira, Micro-hyperbolic systems](https://perso.imj-prg.fr/pierre-schapira/wp-content/uploads/schapira-pub/Microhyp.pdf), §1.2, printed/PDF p.5, and §1.3, p.6. The first page defines the stalks by supported cohomology on the comonoidal topology and specifies the antipodal convention; the second defines \(C^\infty=\gamma^{-1}\gamma_*C^R\) off zero. The same local-support specification is written explicitly in [Aoki–Honda–Yamazaki, Foundation of symbol theory for analytic pseudodifferential operators, I](https://www.jstage.jst.go.jp/article/jmath/69/4/69_1715/_pdf), §2, printed p.1721, PDF p.7. Their recalled symbol theorem is not a premise.

The earlier full Dolbeault/current resolutions and controlled excision are [§5.34, R.1–R.7](#5-34-full-current-resolutions-and-controlled-proper-image-comparison); the finite relative cube, normalized trace, tensor signs and normal-line calibration are [§5.35, V.1–V.6](#5-35-relative-cones-unshifted-cup-and-the-normalized-trace). Global locally \(L^2\) existence and the actual graph-domain estimates are proved in [§5.43, DLH.1–DLH.4](#5-43-the-canonical-all-degree-action-and-its-derived-module-structure). The complete homogeneous coefficients, forward kernels and inverse estimates are [§5.44, IK.1–IK.8](#5-44-full-homogeneous-coefficients-and-ordinary-infinite-order-kernels); its [AN.1–AN.4 normalization](#an-1-the-exact-strict-angular-input) is applied below only after we derive the required angular data from arbitrary geometric sections. The ordinary puncture resolution is [OC.1–OC.4](#oc-1-explicit-exhaustions-on-every-ordinary-puncture-intersection), and the complete ordinary two-support trace is [TC.1–TC.4](#tc-1-one-coordinate-all-four-source-entries). The ordinary canonical products and arbitrary ordinary coordinate changes are [OR.3–OR.4](#or-3-complete-ordinary-canonical-multiplication).

#### IG.1. Unfold the actual microlocal-support stalk and prove the cofinal cones

Let \(u=(t,x)\), \(v=u+(w,\zeta)\), \(d=N-1\). The diagonal has complex codimension \(N\). The source's comonoidal topology identifies all positive real multiples of a covector and gives, with its antipodal convention in the chosen \(w=s-t\) orientation,

\[
\begin{gathered}
C^R_{\Delta,p}=
\underset{D,H}{\mathop{\rm colim}}\ H_H^N(D;\mathcal O),\\
C_\Delta(H)_{u_0}\setminus\{0\}\\
\subset\{-\Re\langle(w,\zeta),p\rangle>0\}.
\end{gathered}
\tag{IG.1}
\]

Here \(D\) is an actual neighborhood of \((u_0,0)\), \(H\) is closed, and arrows are restriction in \(D\) and support forgetting on inclusion of supports. The same formula at nearby covectors with a strict polar margin defines the local section represented by that supported class. Formula IG.1 is the stated sheaf definition: a finite cocycle uses one neighborhood, and two such cocycles have the same germ exactly when their difference is a boundary after one later restriction/support map. This follows directly from filtered colimits of vector-space complexes: a cocycle, a primitive and every equality have finitely many entries and occur at one later index. No completion, coefficient-wise neighborhood choice or omitted symbol construction is involved.

Normalize the central covector to \(p=(1,0)\), using a fixed linear base-coordinate change. For a representative support \(H\), its normal cone intersected with the unit sphere is compact. The function \(-\Re w\) is strictly positive on that compact set by IG.1, so it has a positive minimum \(\mu\). If the set is empty, a sequence of nonzero normal displacements approaching the diagonal would produce a unit normal-cone vector, a contradiction; thus \(H\) is contained in the diagonal locally and there is no extra off-diagonal support to bound. Otherwise the compact minimum gives

\[
\begin{gathered}
-\Re w\ge\mu\sqrt{|w|^2+\sum_i|\zeta_i|^2}\\
\text{on the unit-direction}\\
\text{normal cone.}
\end{gathered}
\tag{IG.2}
\]

Choose \(0<\lambda<\pi/2\) and \(B_i>0\), with strict margins large enough to contain that cone in

\[
\begin{gathered}
T_{\lambda,B,\phi}=\\
\left\{\begin{gathered}
|\arg(-e^{i\phi}w)|\le\lambda,\\
|\zeta_i|\le B_i|w|
\end{gathered}\right\},
\end{gathered}
\tag{IG.3}
\]

where the zero normal vector is included. At \(\phi=0\), one may take \(\lambda>\arccos\mu\) and \(B_i>\mu^{-1}\). After one shrink of \(D\), the **actual** support \(H\) is contained in IG.3. If no shrink worked, choose points of \(H\) with base tending to \(u_0\) and normal norm tending to zero, normalize their normal displacements and pass to a sphere subsequence. Its limit is in \(C_\Delta(H)\) by the definition of that normal cone and violates the strict containment margin, a contradiction. This also proves uniform containment on a smaller base neighborhood; otherwise the failing base points provide the same sequence.

These standard cones are themselves admissible on an actual covector neighborhood. For \(p'=e^{i(\phi+\eta)}(1,z)\),

\[
\begin{gathered}
-\Re\langle(w,\zeta),p'\rangle\\
\ge\left(\begin{gathered}\cos(\lambda+|\eta|)\\-\sum_iB_i|z_i|\end{gathered}\right)|w|.
\end{gathered}
\tag{IG.4}
\]

For \(|\eta|\le\eta_*\), \(\lambda+\eta_*<\pi/2\), and
\(\sum_iB_i|z_i|<\frac12\cos(\lambda+\eta_*)\), this is strictly positive on every nonzero vector of IG.3. Thus replacing \(H\) by IG.3 is a legitimate support map on a common base/covariable product and a phase interval. This proves cofinality in the actual source-defined stalks; it is not an assertion that an arbitrary real-conic class has ray support.

For later proper geometry the standard support lies in the closed convex, full-dimensional cone

\[
\begin{gathered}
\mathcal G_\phi=\\
\left\{\begin{gathered}
-\Re(e^{i\phi}w)\ge0,\\
|\Im(e^{i\phi}w)|\\
\le\tan\lambda[-\Re(e^{i\phi}w)],\\
|\zeta_i|\\
\le\tfrac{B_i}{\cos\lambda}[-\Re(e^{i\phi}w)]
\end{gathered}\right\}.
\end{gathered}
\tag{IG.5}
\]

Its interior contains \(e^{i\phi}w=-1,\zeta=0\). The inclusion is a literal support-enlargement chain map, and the explicit margins/properness proof AN.19–AN.20 applies. No injectivity of support enlargement or equality of cone quotient topologies is asserted.

#### IG.2. Top holomorphic cochains and exact zero witnesses

Take one actual product source \(M=U\times\Delta_w\times\prod_i\Delta_{\zeta_i}\), restricted from \(D\). The complement of IG.3 has the exact ordered cover

\[
\begin{gathered}
V_{0,\phi}=\\
\{w\notin e^{-i\phi}(-\text{sector}_\lambda)\},\\
V_i=\{|\zeta_i|>B_i|w|\},\\
1\le i\le d.
\end{gathered}
\tag{IG.6}
\]

The normal open \(V_{0,\phi}\) is a connected normal sector of aperture \(2\pi-2\lambda\), strictly between \(\pi\) and \(2\pi\). Its branch of \(L=\log w\) has imaginary part in one interval \((\alpha,\beta)\) of that length. A normal-face intersection omitting index 0 includes \(w=0\) whenever its spatial differences are nonzero.

We prove that every finite cover intersection is \(\mathcal O\)-acyclic on this **whole** domain. Let \(I\) be its set of spatial indices. Start with the squared norm of all base and normal coordinates and add the usual outer-disc barriers \(-\log(R_j^2-|q_j|^2)\). For each \(i\in I\) add

\[
 |\zeta_i^{-1}|^2-\log(1-|B_iw/\zeta_i|^2).
 \tag{IG.7}
\]

If index 0 is present, add

\[
\begin{gathered}
|L|^2-\log(\Im L-\alpha)\\
-\log(\beta-\Im L).
\end{gathered}
\tag{IG.8}
\]

Every term is smooth and plurisubharmonic on the specified intersection. For a holomorphic function \(h\), \(|h|^2\) has Levi form \(|dh|^2\). Direct differentiation gives the nonnegative Levi form \(|dh|^2/(1-|h|^2)^2\) of \(-\log(1-|h|^2)\). The last two terms of IG.8 are convex functions of positive harmonic functions, so their Levi forms are the nonnegative squares of the corresponding derivatives divided by the squares of those harmonic functions. The squared coordinate norm makes the sum strictly plurisubharmonic. It is an exhaustion: outer boundaries, a required spatial puncture, a forbidden normal angle and \(w\to0\) when index 0 is present each force a displayed term to tend to infinity. Add a constant to make it nonnegative.

The complete DLH.1–DLH.4 analytic existence proof now solves every closed positive-degree locally \(L^2\) Dolbeault form on each intersection. To identify this with actual holomorphic cohomology, use the maximal graph sheaf \(\mathcal L^q\) of locally \(L^2\) forms whose distributional Dolbeault differential is locally \(L^2\). Multiplication by a smooth function preserves both conditions, so it is a \(C^\infty\)-module sheaf. Its augmentation by \(\mathcal O\) is locally exact: DLH.1–DLH.4 on a small actual polydisc solves positive degree, and R.3's distributional holomorphicity identifies degree zero. The primitive has the given locally \(L^2\) differential and hence lies in the graph sheaf. R.5 proves ordinary-section acyclicity of these module sheaves, and R.6's finite acyclic-resolution comparison computes actual derived holomorphic sections by their global complex. DLH.1–DLH.4 on IG.7–IG.8 now makes that global complex exact in positive degree. Thus positive-degree holomorphic cohomology vanishes on every stated intersection. This is an explicit graph-sheaf/exhaustion/existence argument, rather than an invocation of a named Stein theorem. The source product \(M\) has the same property using its outer-disc barriers.

The finite V.6 augmentation therefore reduces the actual supported complex to the holomorphic cube for IG.6. For the diagonal input top-form twist, use the ungraded frame and the V.17 calibration: an unsigned physical kernel function has ordinary top cube coordinate \(r_N(-1)^N\) times that function, or full-cone Čech top coordinate \(-r_N\) times it. The functions below are expressed in that physical kernel coordinate; multiplying every representative and face primitive by the same fixed nonzero calibration changes no equality or zero test. Its top degree \(N\) has one holomorphic function on

\[
 \Omega_\phi\times
 \{|\zeta_i|>B_i|w|\text{ for all }i\},
 \tag{IG.9}
\]

with every outer source bound and base variable retained. Its top boundaries are exactly

\[
 f=\sum_{i=0}^d(-1)^i h_{0\cdots\widehat i\cdots d},
 \tag{IG.10}
\]

where each \(h\) is an actual holomorphic function on that omitted-coordinate face. This conclusion includes zero detection at the level of representatives: if the supported class is zero, it has a finite holomorphic face primitive on a common actual smaller product. The cover acyclicity and the finite cube comparison prove this statement; it is not assumed from an equality of quotient classes.

For a support equal to the union of two neighboring standard cones with the same \(\lambda,B\), its normal closed sector is their connected union and its spatial inequality is unchanged. If the phase-center gap \(h\) satisfies \(h<2\lambda\) and \(2\lambda+h<\pi\), the normal complement is again a connected sector of aperture between \(\pi\) and \(2\pi\). The same exhaustion and face-primitive proof applies verbatim. This is the exact overlap domain used next.

#### IG.3. Arbitrary projectivized sections give strict angular data

Let \(s\) be an arbitrary germ of \(\gamma_*C^R_\Delta\) at \((u_0,[1:0])\). By definition it is a section of \(C^R\) on the inverse image of a projective neighborhood, hence over every normal phase. The positive real radial variable carries no additional data: the source's topology is the inverse image of the spherical topology, so restrictions on each positive real orbit are the same support germs. The relevant fiber is the compact phase circle.

Choose finitely many phase neighborhoods with supported representatives by IG.1–IG.2. Restrict the finitely many base/covariable products to one common smaller product. On their phase overlaps the representatives are equal as sections of \(C^R\). A zero germ in the filtered support complex has one later neighborhood/support primitive by IG.1. On compact subarcs of the finitely many overlaps choose finitely many such witnesses, and restrict to one common source. This finite selection occurs **before** any coefficient index appears.

For these finitely many supports and witnesses choose strict polar margins on their compact phase subarcs. Compact unit normal-direction sets give one positive minimum margin. Refine the circle to a sufficiently fine cyclic grid \(\phi_i\), retaining the already chosen witnesses on the subarcs assigned to the grid. Choose one \(\lambda<\pi/2\), common \(B_i\), and grid spacing \(h\) so that all assigned supports/witnesses are contained, with a strict margin, in the corresponding standard cones, and

\[
 0<h<\min(2\lambda,\pi-2\lambda).
 \tag{IG.11}
\]

This choice is finite and possible: first take a uniform angular margin from the selected compact subarcs and supports; then take the grid step smaller than that margin and enlarge the finitely bounded spatial constants. No new support witness is required by this refinement. A representative/witness assigned to a grid point or neighboring overlap remains admissible throughout its compact subarc by the same strict margin. Enlarging it to the standard cone uses IG.2's sequential containment proof.

We obtain top functions \(f_i\) on IG.9. The difference of neighboring functions, on their common connected normal complement, is an actual top boundary for the union support. By IG.10 it has one normal-face term \(h_i\), extending holomorphically to \(w=0\) on the **whole** common thick spatial complement, and finitely many spatial-face terms. Apply the full negative spatial Laurent projection from AN.5–AN.6. It kills each spatial-face term: a function holomorphic across \(\zeta_k=0\) has no negative Laurent coefficient in that variable. Let \(F_i=\Pi_-f_i\) and \(H_i=\Pi_-h_i\), including the deletion sign in the latter. Then the literal equality is

\[
 F_i-F_{i+1}=H_i
 \tag{IG.12}
\]

on their common normal complement. Each \(H_i\) is holomorphic on one common domain
\(\{|w|<r,\ B_k|w|<|\zeta_k|<r_k\}\), including the normal zero face. The actual primitive, not an analytic-continuation assumption, supplies that common-domain extension. All negative spatial projections are defined by the same locally fixed Cauchy circles and agree on overlaps as proved in AN.2.

Lift the normal complements to intervals on the real normal-angle line, with equal length \(2\pi-2\lambda\) and consecutive center spacing \(h\). They cover that line. Put \(C_0=0\), \(C_{i+1}=C_i+H_i\), extending the indices periodically, and glue the functions \(F_i+C_i\) on these lifted intervals. Equation IG.12 makes adjacent pieces equal. If two more distant lifted intervals overlap, their intersection lies in every intervening interval because the intervals have equal length and ordered centers; hence the adjacent equalities telescope there as well. This proves actual gluing on all overlaps, including the complete lifted phase cover.

After one circuit the glued function satisfies

\[
\begin{gathered}
F(L+2\pi i)-F(L)\\
=\sum_i H_i=:H.
\end{gathered}
\tag{IG.13}
\]

The finite sum \(H\) is holomorphic on the entire common normal-zero thick complement. Thus **arbitrary** source-defined projectivized sections have the strict datum AN.1–AN.3. IG.10–IG.12 gives the common-domain overlap extension with explicit bounds; compactness of the phase circle alone is insufficient.

AN.2–AN.4 applies to this datum. It gives \(K_0-\log w\,G\), modulo literal normal and spatial face primitives, with \(K_0\) fully negative in every normal Laurent variable, \(G\) holomorphic at normal zero on the thick complement and fully negative in the spatial Laurent variables. Cauchy extraction gives all \(p_j(u,z)\) on one normalized covariable neighborhood, the factorial negative bound and every-\(\varepsilon\) positive bound, including all spatial indices and the positive-homogeneous/negative-normal mixed terms. These are the proved estimates AN.14–AN.18, with every fixed derivative and tail obtained by one compact enlargement. This proves surjectivity from the full coefficient class onto the actual projectivized geometric germs.

#### IG.4. Zero-class detection and the inverse sheaf map

Conversely, a full coefficient sequence has its normally convergent holomorphic pole/log kernel, with the fixed source factorial signs, as in IK.5–IK.6. Remove its explicit holomorphic harmonic-number face boundary. For every phase choose the corresponding branch of \(\log w\) on IG.9. Branch changes are multiples of the normal-holomorphic function \(G\); AN.11–AN.12 gives their literal common face primitives and their triple-overlap identity. Thus these representatives define a section over the whole phase fiber, with one common base/covariable product. This is an actual map to \(\gamma_*C^R\), not merely to one real-conic stalk.

Suppose its geometric germ is zero. At a central phase, IG.1 and IG.10 give, after one actual later support/domain choice, a holomorphic missing-face expression for \(K_0-LG\). Project fully negatively in the spatial variables. The kernel is already so normalized, and the spatial-face terms vanish; hence

\[
 K_0-LG=J,
 \tag{IG.14}
\]

where \(J\) is holomorphic on a thick normal-zero complement. Fix a compact spatial product annulus and choose a small normal punctured disc entirely in that complement. The three functions in IG.14 are holomorphic there on a normal angular sector. Continue this literal identity around its normal circle. The functions \(K_0,G,J\) are single-valued; the lifted logarithm changes by \(2\pi i\). Subtracting the two continuations gives \(2\pi iG=0\). The identity theorem gives \(G=0\) on the actual common thick neighborhood. Now \(K_0=J\) extends across normal zero. Its fully negative normal Laurent coefficients are therefore zero, so \(K_0=0\). The unique Cauchy coefficient extraction AN.15/AN.18 makes every reconstructed coefficient germ zero.

This proves injectivity. More generally equal geometric germs give equal coefficient germs by applying the same argument to their difference, on the finite intersection of their domains. All changes of initial phase cover, representatives, face witnesses, Cauchy radii, cone constants and branches therefore give the same actual coefficient germ. Surjectivity and injectivity identify sheaves, since all constructions commute with restriction and equality is on one smaller product, not separately in each degree. Finite operator diagrams use the finite intersection of these source and coefficient products and one common admissible full-dimensional cone enlargement IG.5.

#### IG.5. Actual nonlinear coordinates and zero-section compatibility

Let \(\widehat u=\phi(u)\) be any local biholomorphism. Its map on diagonal pairs is an actual biholomorphism preserving the diagonal. It pulls back the input top-form frame with the holomorphic Jacobian determinant, and carries the cotangent direction by the dual derivative. It preserves the source comonoidal-support germs. Here is the support estimate needed to verify that statement rather than assume it. On one compact enlargement Taylor's integral formula gives

\[
\begin{gathered}
\phi(u+q)-\phi(u)\\
=d\phi_u(q)+E(u,q),\\
|E(u,q)|\le C|q|^2.
\end{gathered}
\tag{IG.15}
\]

The inverse derivative and its norm are uniformly bounded there. A cofinal support with strict polar margin \(\mu|q|\) for the corresponding transformed covectors retains at least half that margin for \(|q|<\mu/(2C')\), after absorbing the bounded derivative/covector constants into \(C'\). Finitely many phase supports and witnesses permit the minimum of their positive margins and this single normal-radius shrink. The transformed supports can consequently be enlarged to the standard cones of IG.3 in the new coordinates by the same cofinal argument IG.1. The Jacobian multiplier and all its fixed derivatives are holomorphic on one common pair domain and introduce no index-dependent neighborhood.

Apply IG.2–IG.3 and AN.14–AN.18 in the transformed chart. The resulting full coefficients are holomorphic on one actual common new base/projective product and have exactly both growth conditions, all fixed derivative tails and every spatial index. This is the nonlinear transition map; it is obtained from an actual geometric germ and explicit normal-form bounds, not a formal transformation series. Its inverse comes from \(\phi^{-1}\). Composition on triple overlaps is exact: pullbacks of cochains compose, the input top-form determinants obey the ordinary chain rule, and IG.4 makes the coefficient representative unique. Hence these transition maps identify the coordinate coefficient sheaf with the intrinsic projectivized kernel sheaf. Multiplicativity of the full transition maps uses IK.M; coordinate-ring naturality therefore also uses its complete mixed-product comparison.

At the zero section the source definition is \(H^N_\Delta(\Omega^{(0,N)})\). For the ordinary coordinate-puncture cover, every intersection has the explicit proper strictly plurisubharmonic exhaustion consisting of the squared coordinate norm, all outer-disc barriers, and \(\sum_{i\in I}|q_i^{-1}|^2\) for its required normal punctures. The same DLH.1–DLH.4 existence and R/V finite-column elimination used in IG.2 therefore reduce this cover to its holomorphic row; V.6 alone would not supply that analytic acyclicity. Its complete coordinate puncture cube has the unique fully negative Laurent representative of OR.5; Cauchy extraction on arbitrarily small product circles proves exactly the ordinary every-\(\varepsilon\) class OR.1. Its off-zero map is the actual support-forgetting map \(\Delta\subset T_{\lambda,B,\phi}\), phase by phase. The kernel has no log part, so the just constructed identification sends it to

\[
\begin{gathered}
P_j(u;\tau,\xi)=\\
\sum_{m+|\beta|=j}a_{m\beta}(u)\tau^m\xi^\beta,\\
j\ge0;\\
P_j=0\quad(j<0).
\end{gathered}
\tag{IG.16}
\]

The full \(P_j\) is a homogeneous polynomial of total degree \(j\); its normalized \(p_j(u,z)\) has spatial degree at most \(j\). Every time/spatial multiindex is retained. These are literal restrictions of the same diagonal class, so they agree on a common neighborhood and are compatible with the source's zero/off-zero restriction maps. Not every off-zero germ extends to zero; no such extension is asserted. For an actual germ near zero, its zero-section representative is the ordinary class just calculated. Under nonlinear coordinates the support maps commute with pair pullback, and the corrected ordinary two-support trace TC.1–TC.4 supplies the ordinary top-form pairing correspondence. Thus the entire zero-section compatibility, including arbitrary ordinary coordinate overlaps, is actual.

![Actual cofinal support slices and connected angular descent](assets/infinite-projective-descent.png)

**Figure IG.1.** Left: exact one-spatial-coordinate real slice of IG.2 with initial strict polar margin \(\mu=2/3\), whose spatial bound is \(|\Re\zeta|\le\sqrt5\sigma/2\), \(\sigma=-\Re w\); it lies strictly inside the standard bound \(2\sigma\). The full proof uses complex spatial norms. Center: \(\lambda=\pi/3\), phase spacing \(h=\pi/4\); neighboring support sectors have union aperture \(11\pi/12<\pi\), and their common connected normal complement has aperture \(13\pi/12>\pi\). A normalized covariable disc \(|z|<1/16\), with phase half-neighborhood \(\pi/24\), has the exact positive polar margin \(\cos(3\pi/8)-1/8\). Right: IG.10–IG.14's actual face-witness projection, normal-zero overlap extension, monodromy datum and zero test. The editable figure, exact data and reproducible source are linked below. The outer normal/spatial source radii are 3.

Editable SVG · Exact data · Figure source


#### Mathematical identification and its product interface

| Assertion | Complete proof |
|---|---|
| Source-defined support stalks and cofinal actual pointed cones | IG.1, compact normal-direction argument IG.2–IG.4; definition on inspected free source p.5 |
| Whole-domain holomorphic top representatives and actual missing-face zero witnesses | IG.2, explicit strict PSH exhaustions IG.7–IG.8, exact DLH.1–DLH.4 existence and R/V resolutions |
| Arbitrary whole-phase/projectivized section gives strict common-domain angular datum | IG.3, IG.10–IG.13, including finite actual overlap witnesses |
| Full homogeneous coefficient bounds and inverse signs, every spatial index and fixed derivative tail | AN.2–AN.4 applied to the angular datum derived in IG.3 |
| Geometric zero detection and actual germ/sheaf identification | IG.4, monodromy followed by negative Laurent uniqueness |
| Arbitrary nonlinear geometric coordinate overlaps and triple identities | IG.5, actual Taylor remainder, source pullback and zero detection |
| Ordinary zero-section and actual off-zero restriction compatibility | IG.16, the complete ordinary Laurent calculation, and [§5.44, OC.1–OC.4 and TC.1–TC.4](#oc-1-explicit-exhaustions-on-every-ordinary-puncture-intersection) |
| Full canonical mixed pole/log proper cup/excision/trace and ring compatibility | IM.1–IM.6 below |
| Full infinite-order support-preserving degree-one/all-degree and derived diagrams | Actual finite full-infinite diagrams and all-degree strict module-derived action are proved in [§5.46, ID.1–ID.6](#5-46-finite-infinite-order-diagrams-on-actual-derived-support-modules), with every geometric and common-domain premise checked |
| Every matrix relation/flatness/faithfulness, ordinary/mixed scalar change, Späth/reconstruction, propagation/purity/separation, special/monodromy/D-type, finite poles/order/half-order/C1/initial generator/arbitrary analytic proper, BB/KL/GL14/GL16/affine/critical/factorization and sourcing/proof obligation | Matrix relations/right flatness are proved in §5.47; proper-ideal detection/unit injectivity/faithful exact extension in §5.48; full infinite Späth sum division, actual special fibration class and balanced reconstruction in [Section 5.49](#5-49-full-infinite-spath-division-and-balanced-reconstruction), ISD/ISF/ISR with the complete ISB appendix. Other ordinary/mixed scalar changes and every other listed obligation retain their separate scope after IM.6 |

The geometric identification is multiplicative by IM.1–IM.6 below. A particular module-derived application still requires the exact cone-open, round-domain, support-to-output and common-collar hypotheses of §5.43; no such application is inferred here.


**The full mixed holomorphic relative-cone product.**

We work in the actual holomorphic relative-cochain complexes, with fixed proper collars and a finite, bounded Dolbeault/Čech contraction chosen for the entire input family. In particular, the essential kernel \(e^{1/w}/(2\pi iw)\) remains a holomorphic complement coefficient throughout.

The complete coefficient/forward-kernel construction is [§5.44, IK.1–IK.8](#5-44-full-homogeneous-coefficients-and-ordinary-infinite-order-kernels), strict angular normalization is [AN.1–AN.4](#an-1-the-exact-strict-angular-input), and the full ordinary trace is [TC.1–TC.4](#tc-1-one-coordinate-all-four-source-entries). IG.1–IG.5 above supply the arbitrary-section geometric identification and its zero detector.

Finite-order equality of the actual canonical classes, including all principal-log and normal-localization corrections, is [§5.41, CP.1–CP.6](#5-41-canonical-products-for-full-convergent-spatial-symbols) together with [§5.39, NL.1–NL.7](#5-39-canonical-normal-localization-of-convergent-spatial-symbols). The full resolution, cube, excision and normalized-trace proofs are [§§5.34–5.35, R.1–R.7 and V.1–V.6](#5-34-full-current-resolutions-and-controlled-proper-image-comparison). The weighted existence, Hilbert representation, graph approximation and reweighting arguments used here are [§5.43, DLH.1–DLH.4](#5-43-the-canonical-all-degree-action-and-its-derived-module-structure).

The freely accessible human specifications are [Micro-hyperbolic systems](https://perso.imj-prg.fr/pierre-schapira/wp-content/uploads/schapira-pub/Microhyp.pdf), §3.1, and [Holonomic systems III](https://www.kurims.kyoto-u.ac.jp/~kenkyubu/kashiwara/HolIII.pdf), III.2. Their recalled symbol calculus is not used to discharge a premise. All new analytic contraction and limit arguments are proved below.

#### IM.1. A bounded finite-column contraction on one actual family

Consider any one of the finite relative cubes in IG.2, including its whole-domain component. Every face domain has the explicit smooth proper strictly plurisubharmonic exhaustion IG.7–IG.8; the whole domain has the ordinary disc exhaustion. Let \(m\) be its complex dimension. Work with a specified family of cochains and their differentials that is locally bounded and convergent on each compact face domain. The family may include every positive-head truncation of a finite operator list and their actual output/overlap cochains. There is no coefficient-dependent domain.

For each face choose the graph-domain cutoffs and the smooth derivative majorant \(\psi\) of DLH.2. Define degree weights

\[
\begin{gathered}
\varphi_q=\Phi+(q-m)\psi,\\
 0\le q\le m+1.
\end{gathered}
\tag{IM.1}
\]

Choose \(\Phi=\chi\circ\psi_0\), with \(\psi_0\) the displayed strict exhaustion, so large that every \(\varphi_{q+1}\), \(1\le q\le m\), satisfies DLH.7. It is enough that the smallest Levi eigenvalue of \(\Phi\) dominate
\((m+1)\|\partial\bar\partial\psi\|+2|\partial\psi|^2+2e^\psi\). DLH.4's explicit increasing convex step construction supplies this domination on successive compact exhaustion levels. The same construction makes the locally bounded envelope of **all** specified input cochains and their differentials square-integrable with each of the finitely many weights. To see this directly, on exhaustion shell \(k-1\le\psi_0<k\) increase \(\chi(k-1)\) above \(k+\log(1+\int_{\psi_0\le k}E^2e^{(m+1)|\psi|})\). The shell contribution is then at most \(e^{-k}\). The one envelope \(E\) is locally finite because the entire family converges on compacts.

Choose face weights in increasing intersection cardinality. For every restriction from a parent face to a smaller child intersection, enlarge the child's \(\Phi\) further so that

\[
\begin{gathered}
\varphi_{q,\mathrm{child}}\\
\ge\varphi_{q,\mathrm{parent}}\big|_{\mathrm{child}}\\
\text{for every }q.
\end{gathered}
\tag{IM.2}
\]

There are finitely many parents and degrees; their required values and Hessian bounds are locally bounded on the child's exhaustion sublevels, so the same convex construction supplies them simultaneously. Restrictions are then bounded with norm at most one. Equal face domains use the same weights. Include all neighboring-phase intersection domains in this finite selection; restrictions between the finitely many covers obey the same condition.

Let \(H_q=L^2_{(0,q)}(W,\varphi_q)\), and let \(\delta_q\) be maximal distributional \(\bar\partial\). It is closed: convergence in the source and target norms gives distributional convergence on compacts and preserves the distributional equation. Its kernel is closed. DLH.1–DLH.4, with the consistent consecutive three weights IM.1, gives
\(\operatorname{im}\delta_{q-1}=\ker\delta_q\), and a solution of norm at most the norm of the right side. Choose the unique solution orthogonal to \(\ker\delta_{q-1}\); it is linear and has the same norm bound, since orthogonal projection removes a closed-kernel vector without changing its differential. Denote this bounded right inverse on closed forms by \(R_q\). Put

\[
\begin{gathered}
h_q=R_q\Pi_{\ker\delta_q},\\

 \pi_0=\Pi_{\ker\delta_0},\\
 \pi_q=0\ (q>0).
\end{gathered}
\tag{IM.3}
\]

The projections exist by the Hilbert proof DLH.1. Weak degree-zero closed forms are holomorphic by R.3. Directly, on the actual maximal graph domains,

\[
\begin{gathered}
\delta h+h\delta=1-i\pi,\\

 h^2=0,\\
 h i=0,\\
 \pi h=0,\\
\pi i=1.
\end{gathered}
\tag{IM.4}
\]

For \(q>0\), \(\delta h_q=\Pi_{\ker\delta_q}\), while \(h_{q+1}\delta f\) is the unique solution orthogonal to that kernel, namely \(f-\Pi_{\ker\delta_q}f\). The same argument in degree zero gives the first identity with the holomorphic projection. The output of \(R_q\) is orthogonal to \(\ker\delta_{q-1}\), proving the three side identities. This proves an actual bounded contraction, not a choice of an unproved continuous inverse.

Let \(d_v=(-1)^r\delta\) in cube cardinality \(r\), and let \(d_h=d_C\) be the finite restriction differential. Set \(h_v=(-1)^rh\), so IM.4 holds with \(d_v\). If the cube has at most \(L\) opens, \(d_hh_v\) increases cardinality and its \((L+1)\)-st power is zero. Define the finite operators

\[
\begin{gathered}
A=\sum_{k=0}^{L}(-d_hh_v)^k,\\

 \mathsf H=h_vA,\\
 \mathsf P=\pi A.
\end{gathered}
\tag{IM.5}
\]

With \(D=d_v+d_h\), they satisfy

\[
\begin{gathered}
D\mathsf H+\mathsf H D=1-i\mathsf P,
 \\
 \mathsf P D=d_C\mathsf P,
 \\
 \mathsf P i=1.
\end{gathered}
\tag{IM.6}
\]

Here is the algebraic verification. The finite sum is \((1+d_hh_v)^{-1}\). Use \(d_vh_v+h_vd_v=1-i\pi\), \(d_vd_h+d_hd_v=0\), and \(d_h^2=0\). Expansion gives
\([D,1+d_hh_v]=-d_h+d_hi\pi-d_hh_vd_h\). Substitute this into the commutator with its finite inverse, use \(h_vi=0\) and \(d_hi=i d_C\), and telescope to obtain the first identity. Applying \(D\) on both sides gives \(i(\mathsf P D-d_C\mathsf P)=0\), so the second holds because \(i\) is injective. The third follows from \(h_vi=0\). No infinite operator series appears in this verification.

All operators in IM.5 are bounded on the finite sum of face Hilbert spaces, and preserve the actual graph domains used by these identities. Restrictions preserve graph domains because they commute with distributional differentiation and satisfy IM.2 in both adjacent degrees. A convergent locally bounded input family fits these one-time weights; dominated convergence makes it converge in each chosen graph norm. IM.5 therefore gives convergence of its primitives and holomorphic projections. Holomorphic weighted-\(L^2\) convergence implies normal convergence on each compact: the iterated disc mean-value inequality bounds its square by a fixed local \(L^2\) norm, weights are comparable to constants on a compact enlargement, and Cauchy bounds every fixed derivative. This is the precise continuity used below.

#### IM.2. The full holomorphic cup/excision/trace cochains

Fix \(P,Q\) in the full coefficient class on one actual common base/projective product. Let \(R=P\circ Q\) be the complete infinite contraction of IK.2–IK.3. Choose a common slightly larger coefficient compact, all base/covariable Cauchy margins, the negative factorial constant and the positive every-\(\varepsilon\) prefactors before any summation. IK.5 gives normally convergent pole/log/holomorphic kernels on the coordinate-punctured and source thick-complement domains. Its inequalities are the source III.2 inequalities \(>\), not the distinct III.3 special-fibration inequalities \(<\).

Choose finitely many normal phases with the source domains/cones of IG.2–IG.3, with adjacent spacing \(h<2\lambda\) and \(2\lambda+h<\pi\). A common \(\lambda,B_i\) is allowed for the finite symbol/product list, but every cone is **phase-local**: at phase \(\phi_i\) use its rotated convex cone \(\mathcal G_i=\mathcal G_{\phi_i}(\lambda,B)\) of IG.5. At a neighboring pair use the separate cone centered at \(\psi_i=(\phi_i+\phi_{i+1})/2\), with half-angle \(\alpha=\lambda+h/2\) and spatial slope \(B_i/\cos\alpha\). It contains both phase cones: on their union the rotated normal angle has absolute value at most \(\alpha\), and

\[
\begin{gathered}
-\Re(e^{i\phi_i}w)\\
\le\frac{\cos\lambda}{\cos\alpha}[-\Re(e^{i\psi_i}w)]
\end{gathered}
\tag{IM.18}
\]

holds at the appropriate endpoint phase, by \(\cos(h/2)+\tan\alpha\sin(h/2)=\cos\lambda/\cos\alpha\). Thus its spatial slope contains the endpoint slopes as well. Since \(\alpha<\pi/2\), this neighboring cone is pointed. No single pointed cone is chosen to contain the whole circle of rotated phase cones. The domains of all input symbols and finitely many intermediate products are intersected once; every subsequent Cauchy margin, cone constant and collar is fixed on a smaller common product.

Choose once

\[
\begin{gathered}
\lambda<\lambda'<(\pi-h)/2,\\
B'_k>\frac{B_k}{\cos(\lambda+h/2)}\\
>\frac{B_k}{\cos\lambda}.
\end{gathered}
\tag{IM.19}
\]

The stronger spatial choice also handles the neighboring convex cones on the same product. It retains \(h<2\lambda'\) and \(2\lambda'+h<\pi\). Define the standard normalization supports
\(T'_i=T_{\lambda',B',\phi_i}\), and on a neighboring pair
\(T'_{i,i+1}=T_{\lambda'+h/2,B',\psi_i}\).
They respectively contain \(\mathcal G_i\) and the neighboring convex cone just constructed. Their complements have exactly the normal/spatial open covers IG.6, with the new angle and spatial constants; every intersection has the literal IG.7–IG.8 exhaustion. Thus these supports, not an undeclared cover of a convex cone, are the normalization complexes to which IM.1 will be applied.

For the final full-dimensional convex supports take
\(\mathcal G'_i=\mathcal G_{\phi_i}(\lambda',B')\), and on each neighbor
\(\mathcal G'_{i,i+1}=\mathcal G_{\psi_i}(\lambda'+h/2,B')\).
Then \(T'_i\subset\mathcal G'_i\) and
\(T'_{i,i+1}\subset\mathcal G'_{i,i+1}\) by IG.5. Each is pointed and stable under addition. All of this geometry is selected before any truncation or weight construction. Put \(\alpha'=\lambda'+h/2<\pi/2\) and
\(K_k=B'_k/\cos\alpha'\). Choose a neighboring-overlap half-width \(\eta_*>0\) with \(\alpha'+\eta_*<\pi/2\), and shrink one covariable polydisc so that

\[
\begin{gathered}
\sum_k K_k|z_k|\\
<\tfrac12\cos(\alpha'+\eta_*).
\end{gathered}
\tag{IM.20}
\]

Give the actual covector-phase neighborhood of \(\phi_i\) half-width \(h/2+\eta_*\). Its neighboring intersection is centered at \(\psi_i\) and has half-width \(\eta_*\), including the last/first overlap on the lifted phase line. These phase intervals cover the entire circle since their half-width exceeds \(h/2\). For a phase cone, the sum of its normal half-angle and this covector half-width is \(\lambda'+h/2+\eta_*=\alpha'+\eta_*\); for its neighboring cone, the corresponding sum is also \(\alpha'+\eta_*\). Thus the very same IM.20 polar margin covers both kinds of supports on their actual phase neighborhoods, rather than only at the finite phase centers.

IG.4's polar inequality, applied to the radial bound \(|\zeta_k|\le K_k|w|\), proves admissibility for every support in this finite phase/neighbor system. The covariable shrink is one-time and common to all homogeneous and spatial indices.

![Phase-local cones and the forward normalization support maps](assets/infinite-phase-local-support-bridge.png)

**Figure IM.2.** The exact example has \(\lambda=\pi/6\), \(\lambda'=\pi/4\), \(h=\pi/12\), \(B=1\), \(B'=3/2\). Its final neighboring normal half-angle is \(7\pi/24\), so the support union aperture is \(7\pi/12<\pi\). Covector-phase intervals have half-width \(\pi/16=h/2+\eta_*\); neighboring overlaps have half-width \(\pi/48=\eta_*\). The central panel is the real slice \(\Im(e^{i\phi}w)=\Im\zeta=0\); the proof uses complex spatial norms. Endpoints have coordinate radii \(1/64\), the collar is one on radius \(1/4\) and supported in radius \(1/2\), and the outer source radii are 3. The arrows are the literal support maps IM.21–IM.23: the boundary is on \(T'\), then mapped forward to final \(\mathcal G'\).

Editable SVG · Exact data · Figure source


In triple difference coordinates \(a=s-u\), \(b=v-s\), the singularity lists are independent. Put the actual two kernel relative cubes on the source \((u,a,b)\) and form their entire tensor-to-cube map V.16. Holomorphic joint coefficients may be multiplied after that map. All lower complement entries and all Dolbeault components are retained; this is the same full operation corrected in TC.1–TC.3, with more general cone complement opens.

Augment this finite source cube to the full cone for the intersection of the two actual kernel supports using V.6's smooth Čech row contraction. Choose a fixed smooth partition on that union complement subordinate to its finite complement opens. Such a partition exists by the explicit locally finite ball/closure construction R.5: group the locally finite small-ball functions by the finitely many opens and normalize their sum. On each compact only finitely many balls occur, and their closures lie inside the respective opens. Multiplying a holomorphic face coefficient by the corresponding cutoff or its derivative consequently extends by zero only on a neighborhood where the cutoff is zero. It never extends an essential singularity across its support. This addresses individual pole/log/spatial faces as well as the normal origin.

The source intersection has a proper intermediate fiber. In the rotated normal coordinate write \(\sigma_a=-\Re a_t\ge0\), \(\sigma_b=-\Re b_t\ge0\); their sum is the final \(\sigma\). All imaginary and spatial intermediate coordinates are bounded by the common constants times \(\sigma\). Choose the actual outer source first, and an inner endpoint product satisfying AN.20 for the largest constants of the finite final phase/neighbor cones just selected. These have normal half-angle at most α′ and spatial slope at most K_k. For endpoint radii δ one may require δ+2δ/cosα′<R_t/3 and δ+2K_kδ<R_k/3 in the actual source polydisc. A fixed collar equal to one on these inner intermediate polydiscs and supported inside the source half-radii then works for every phase and neighboring pair. Only this compact collar and the domain margins are uniform across the finite phase system; its entire circle of supports is not put in one pointed cone. Thus the entire intersection over the inner endpoint domain is inside a fixed compact intermediate collar with a positive source-boundary margin. Use one collar \(\eta\), equal to one near this entire closed intersection and compactly supported in that actual source. The controlled localization is

\[
\begin{gathered}
L_\eta(a_0,b_0)=\\
\left(\begin{gathered}
\eta a_0+\delta\eta\wedge b_0,\\
\eta b_0
\end{gathered}\right).
\end{gathered}
\tag{IM.7}
\]

The first coefficient is a genuine whole-domain cochain: \(\delta\eta\) is supported away from the closed intersection, where \(b_0\) is smooth. The second remains a complement coefficient. On the output complement the source intersection is empty, since the cone is stable under addition. There \(\eta b_0\) is properly supported over the output and may be integrated on a compact smooth intermediate fiber. At an individual kernel support face the augmented coefficient is already smooth by the subordinate partition; there is no undefined pole evaluation at a ray endpoint.

Apply the actual normalized relative trace

\[
\begin{gathered}
(a_0,b_0)\longmapsto\\
\left(\begin{gathered}
\operatorname{Tr}_qa_0,\\
(-1)^N\operatorname{Tr}_{q-1}b_0
\end{gathered}\right),\\
\operatorname{Tr}_q=(-1)^{N(q-N)}I_q.
\end{gathered}
\tag{IM.8}
\]

These are the V.4/W.3 chain-map signs; TC.11 is their one-dimensional case. The base-antiholomorphic components remain in IM.7–IM.8; only insufficient fiber-degree terms vanish upon integration. Compact integration of the first component is on the collar-derivative region separated from the source intersection. Compact integration of the second is only on the actual output complement. No complement coefficient is extended across the output support.

![The complete ordinary two-support trace on its actual union complement](assets/infinite-two-support-trace.png)

**Figure IM.3.** The two-pole calibration is [§5.44, TC.1–TC.4](#tc-1-one-coordinate-all-four-source-entries). At \(c=1/8\), the cutoff transition is \(1/64\le|a|\le1/32\), so \(|b|\ge3/32\). The full cube is augmented to \((0,\bar\partial\rho_A f)\) on the union complement and traced to \((0,-\oint f\,da)\) on \(c\ne0\). The central plot is a real parameter slice; the marked joint intersection is excluded. This figure illustrates the earlier complete trace proof used by IM.7–IM.8; it does not introduce a whole-domain pole extension.

Editable SVG · Exact data · Figure source


Denote the resulting output relative cycle by \(\mathcal C_i(P,Q)\) at phase \(i\), on the convex support \(\mathcal G_i\). After this output is formed, **literally forget its support** to \(T'_i\) before weights, positive truncations or holomorphic-row normalization. If \(H\subset K\), write \(\mathsf F_{H,K}\) for that chain map. In the full relative cone it is

\[
\begin{gathered}
\mathsf F_{H,K}(a,b)=(a,b|_{M\setminus K}),\\
\mathsf F_{K,L}\mathsf F_{H,K}=\mathsf F_{H,L}.
\end{gathered}
\tag{IM.21}
\]

It keeps the whole-domain entry, restricts every complement entry and commutes with the total differential. When covers differ, use the actual V.6 refinement/augmentation roof with all entries, and then the IG.6 standard cover of \(T'_i\). On a neighboring pair first use its pointed common convex support and then \(T'_{i,i+1}\). The same restrictions supply the comparison maps from either phase cube to that neighboring standard cube. This is a map of actual complexes; no support-forgetting injectivity is claimed. The normalization bridge is exactly

\[
\begin{gathered}
\mathcal G_i\longrightarrow T'_i
\longrightarrow\mathcal G'_i,\\
\mathcal G_{i,i+1}\longrightarrow T'_{i,i+1}\\
\longrightarrow\mathcal G'_{i,i+1}.
\end{gathered}
\tag{IM.22}
\]

All face exhaustion and bounded-normalization arguments below take place on the standard \(T'\) cubes. The output cycle represents exactly the canonical unshifted cup, controlled support/excision and shifted normalized trace. This is a construction by the actual R/V augmentation roofs, not an operator evaluation. The ordinary subring calibration is TC.15; the earlier finite pole/log calibration is CP.1–CP.6/NL. The same source collar and finite partitions work for the entire input family.

All fixed output derivatives of these cochains depend continuously on the kernel coefficients on compact subsets of their **actual** holomorphic face domains. To verify this, on any compact source union-complement set a subordinate partition has finitely many terms supported on compact subsets of the relevant face opens. Kernel convergence and all fixed derivatives hold there by IK.5. The source total coefficient in IM.7 uses only the compact collar-derivative region. On a compact output-complement set, properness gives a compact union-complement fiber family, so differentiation under the fixed compact smooth integration gives the same bounds. These statements cover all individual ray/spatial endpoints by the full augmentation, rather than integrating a singular raw product across them.

#### IM.3. Actual angular overlap primitives for the product

A branch change of the kernel \(P\) is its explicit normal-face boundary \(d\beta_P\), where \(\beta_P\) is AN.11 with the full normally convergent \(G_P\). The harmonic-number holomorphic part is another explicit normal-face boundary. The same holds for \(Q,R\), with their exact source factorial signs. These are actual common-domain cochains; no supported logarithmic term is discarded.

For a bilinear unshifted cup \(\mu\), closed degree-\(N\) inputs satisfy

\[
\begin{gathered}
\mu\left(\begin{gathered}P+d\beta_P,\\Q+d\beta_Q\end{gathered}\right)\\
-\mu(P,Q)\\
=d\left\{\begin{gathered}
\mu(\beta_P,Q)\\
+\!(-1)^N\mu\left(\begin{gathered}P\\+d\beta_P,\\\beta_Q\end{gathered}\right)
\end{gathered}\right\}.
\end{gathered}
\tag{IM.9}
\]

This is the ordinary differential Leibniz rule with \(|\beta_P|=N-1\). A trace of degree \(-N\) obeys \(d\operatorname{Tr}=(-1)^N\operatorname{Tr}d\), so the corresponding output primitive is \((-1)^N\) times the trace of the braces. Append the \(-\beta_R\) term when subtracting the output kernel.

All covers and support enlargements may be refined to the finite neighboring-phase common cover of IG.3. If two augmentation partitions differ, apply V.6's finite Čech row contraction to their difference: kill its lowest nonzero row, subtract its total boundary, and continue through the finite Dolbeault degrees. This explicitly constructs the comparison homotopy. Each step is multiplication by the fixed partition, restriction and a fixed derivative. Collar changes are the exact IM.7 homotopy: for \(\eta'-\eta\), the primitive is \(((\eta'-\eta)b_0,0)\), extended where that difference is supported away from the intersection. Its differential is the difference of the localized cochains. A common properly supported collar may instead be used throughout, eliminating that change. The resulting finite comparison homotopies include every complement component and all shift signs IM.8.

Consequently the phase cycles

\[
\begin{gathered}
D_i=\mathsf F_{\mathcal G_i,T'_i}\mathcal C_i(P,Q)\\
-\kappa'_i(R)
\end{gathered}
\tag{IM.10}
\]

have actual neighboring overlap primitives \(T_i\) on the standard common support \(T'_{i,i+1}\) and common domain. Here \(\kappa'_i(R)\) is the forward kernel supported on \(T'_i\), or equivalently its original forward class mapped through \(T_{\lambda,B,\phi_i}\subset\mathcal G_i\subset T'_i\). Restrictions to the neighboring support are understood in every difference. The degree-zero chain maps IM.21 add no trace, shift, or coefficient-frame sign. They are the finite cochains just constructed from IM.9, the branch/harmonic face primitives, and the cover/collar homotopies. Their construction consists of fixed finite operations and compact traces on actual common domains, so the continuity statement of IM.2 applies to them too. This supplies the strict chain-level angular data needed for normalization; it is not an assertion of equality of abstract phase classes without witnesses.

#### IM.4. Full positive truncations, all infinite contractions and actual tails

Let \(P^{[J]}\) retain every negative homogeneous coefficient of \(P\) and every nonnegative coefficient through \(J\); define \(Q^{[J]}\) similarly. They are actual finite-order symbols, retaining **all** spatial Taylor indices, on the same coefficient product. Their input bounds have the same constants as the full symbols. Let \(R^{[J]}=P^{[J]}\circ Q^{[J]}\).

The complete fixed-degree contraction and its derivative tails are the IK.2–IK.3 estimates. To make their uniform use explicit, the absolute scalar majorant is

\[
\begin{gathered}
S_\ell(E,D,L)=\\
\sum_{J_+-M_--m=\ell}
\begin{gathered}
E^{J_+}D^{M_-}L^m\\
\frac{M_-!m!}{J_+!},
\end{gathered}\\
E(D+L)<1.
\end{gathered}
\tag{IM.11}
\]

All base/covariable margins and constants are chosen once. For \(\ell\ge0\) its bound is
\(E^\ell/[\ell!(1-ED)(1-EL)]\); for \(\ell=-n<0\) it is
\(n!(D+L)^n/[1-E(D+L)]^{n+1}\). Its positive-degree cutoff tails are geometric, and its negative-degree cutoff tails are the finite derivatives of \(a^K/(1-a)\), \(a=E(D+L)<1\). Thus truncating one positive input index gives a tail tending to zero at every fixed output degree, uniformly on the common coefficient compact. Cauchy on one enlargement gives every fixed base and normalized-covariable derivative tail. The arbitrary input every-\(\varepsilon\) choice makes \(E\) as small as required for any output positive bound. Negative output uses one fixed such choice. None of these constants depends on the truncation index.

It follows that \(R^{[J]}_\ell\to R_\ell\) normally for each \(\ell\), while **all** \(R^{[J]}\) have one uniform factorial negative bound and the uniform every-\(\varepsilon\) positive bounds. Spatial coefficient extraction on the common enlarged \(z\)-polydisc has one Cauchy cost \(r_z^{-|\beta|}\). The full forward-kernel bounds IK.18–IK.21 therefore dominate every \(K_{R^{[J]}}\) by the same summable series on each permitted compact complement. This includes the mixed positive-homogeneous/negative-normal terms \(j=|\beta|-n-1\ge0\); their majorant is the exact rational series \((1-x-q)^{-2}\), with one fixed strict choice \(x+q<1\). Termwise coefficient convergence and the common dominator give

\[
\begin{gathered}
K_{P^{[J]}}\to K_P,\\

 K_{Q^{[J]}}\to K_Q,\\

 K_{R^{[J]}}\to K_R
\end{gathered}
\tag{IM.12}
\]

normally on every actual holomorphic face domain, with every fixed derivative. Fixed derivative tails follow by compact enlargement. The logarithmic branch factors are bounded on the relevant cut compacts; the harmonic-number factors are absorbed by slightly larger strict ratios. Every spatial, homogeneous and contraction index is included. This is convergence of holomorphic complement cochains, not of ordinary diagonal currents.

IM.2–IM.3 now gives convergence of the entire output/overlap cochain family
\(D_i^{[J]},T_i^{[J]}\) to \(D_i,T_i\) on every compact actual domain, including their differentials. These cycles and their overlap primitives have already been mapped by IM.21 into the fixed standard T′ phase/neighbor cubes. Fit this **whole** family to the one-time weights and operators IM.1–IM.6 on exactly those IG.7–IG.8 face domains, with λ′ and λ′+h/2 in place of the initial angles. No acyclicity for an unspecified convex-cone cover is being used. Their projections and their actual primitives then converge in those fixed weighted graph models. This is why an arbitrary sequence of high-order finite current primitives is unnecessary and is not summed.

#### IM.5. A literal product-difference primitive, without a closed-image assumption

Normalize each phase cycle on its standard T′_i cube using IM.6; every support/domain restriction was fixed in IM.19–IM.22 before this step. Since its degree is \(N\), its holomorphic cube projection has only the top holomorphic row entry. Write it as \(f_i\) in the physical kernel coordinate of IG.2. The primitive \(\mathsf H_iD_i\) is an actual relative-cube primitive for the difference between \(D_i\) and that top entry.

On a neighboring overlap, first restrict both phase cubes and their cochains to the fixed standard T′_{i,i+1} cube, with its IG.7–IG.8 exhaustions and common weights. There
\(D_i-D_{i+1}=dT_i\). Subtract the two normalization primitives and project on the common holomorphic overlap cube. Because that projection is a chain map and is the identity on holomorphic rows, we obtain the actual holomorphic face primitive

\[
\begin{gathered}
t_i=\mathsf P_{i,i+1}
\left\{\begin{gathered}
T_i-\mathsf H_iD_i\\
+\mathsf H_{i+1}D_{i+1}
\end{gathered}\right\},\\
f_i-f_{i+1}=d_Ct_i.
\end{gathered}
\tag{IM.13}
\]

Apply the finite negative spatial Cauchy projections to these top and face entries. They kill every spatial-face coefficient and produce the normal-zero correction \(H_i\) on the whole common thick domain. Glue their lifted functions with cumulative corrections exactly as IG.12–IG.13. Finally AN.2–AN.4 gives a normally convergent pole/log pair \(\nu(D)=(K_0,G)\), with every coefficient bound on one product. All these steps are finite sums, the bounded operators IM.5 and actual compact Cauchy integrals. Thus

\[
 \nu(D^{[J]})\longrightarrow\nu(D)
 \tag{IM.14}
\]

normally on all actual compact pole/log domains, including the normal-zero face domains for \(G\). No unspecified inverse or quotient topology is used.

For each finite-order pair the earlier CP.1–CP.6/NL proof says that \(D^{[J]}\) is zero as a canonical kernel germ, with its full principal-log correction. IG.4's geometric zero test therefore gives \(\nu(D^{[J]})=0\) as a germ. These normalized functions are holomorphic on the same connected actual domains chosen above; vanishing on a smaller nonempty neighborhood implies vanishing throughout by the identity theorem. Hence every term on the left of IM.14 is the literal zero pair. It follows that \(\nu(D)=0\). This passage uses a proved bounded normalization and zero detector, rather than assuming that the image of a boundary operator is closed or that cohomology commutes with an unspecified limit.

Here is the resulting literal boundary. For each phase, decompose its projected top entry by the actual AN.6 spatial Cauchy projections. Let \(S_{i,k}\) be its spatial-face terms, and let \(C_i\) be the finite cumulative normal-zero correction in IG.3. Since its normalized pole/log pair is zero, its glued fully spatial-negative function has only the nonnegative normal part \(J\). Thus its fully negative spatial top entry is \(J-C_i\), holomorphic on the normal face. With \(r_N=(-1)^{N(N-1)/2}\), define the actual face primitive in **full-cone Čech coordinates** by

\[
\begin{gathered}
c_{i,\widehat0}=r_N(J-C_i),\\

 c_{i,\widehat k}=r_N(-1)^kS_{i,k}\\
(1\le k\le d).
\end{gathered}
\tag{IM.15}
\]

For \(N\ge2\), these are Čech degree \(N-2\) complement coordinates. Their V.15 conversion to the ordinary relative cube contributes \((-1)^{N-2}\); the physical top function \(f_i\) has full-cone top coordinate \(-r_Nf_i\), whose Čech degree \(N-1\) conversion contributes \((-1)^{N-1}\) and yields the calibrated ordinary top \(r_N(-1)^Nf_i\). Thus IM.15 and its differential use the same exact frame/shift convention as IG.2/V.17. For \(N=1\), there is no degree-minus-one complement face; the normal-face term instead uses the whole-domain primitive of AN.3. This exception retains the TC one-dimensional roof and its negative relative trace. These functions are obtained by the displayed Cauchy integral projections, on their actual omitted-coordinate faces; all fixed derivatives and tails were controlled in IM.12–IM.14. Add this face primitive to the actual \(\mathsf H_iD_i\), with the cone-to-cube identification V.15. The result \(B_{P,Q,i}\) satisfies

\[
\begin{gathered}
dB_{P,Q,i}\\
=D_i=\mathsf F_{\mathcal G_i,T'_i}\mathcal C_i(P,Q)\\
-\kappa'_i(P\circ Q).
\end{gathered}
\tag{IM.16}
\]

This is a literal relative-cone identity on \(T'_i\), with all total and complement entries retained. Its Dolbeault primitives are locally \(L^2\) on exactly those standard-cube face domains, and its holomorphic face primitives are normally convergent there. None is an ordinary distribution extension of the positive-head singularity. The boundary is **not** asserted on the initial \(\mathcal G_i\). Map it forward once more to the specified full-dimensional convex support \(\mathcal G'_i\):

\[
\begin{gathered}
d\mathsf F_{T'_i,\mathcal G'_i}B_{P,Q,i}\\
=\mathsf F_{\mathcal G_i,\mathcal G'_i}\mathcal C_i(P,Q)\\
-\mathsf F_{T'_i,\mathcal G'_i}\kappa'_i(P\circ Q).
\end{gathered}
\tag{IM.23}
\]

The analogous identity holds on \(\mathcal G'_{i,i+1}\) for the neighboring comparisons. Every final support remains admissible by IM.20. The collar was selected for these final cones as well. Naturality of the entire cup/support/excision/trace roof identifies the first class with the canonical product of the input classes mapped to \(\mathcal G'_i\): the restriction maps commute with cup, their full augmentation roofs commute after the finite partition homotopies of IM.3, and compact trace commutes with restriction. If a source-support enlargement is used before cup, the same collar is one near its enlarged intersection; its excision comparison is therefore valid on the same proper family. Consequently IM.23 gives equality of the full canonical thick-cone classes on this exact common **final** support:

\[
 \kappa(P)*\kappa(Q)=\kappa(P\circ Q).
 \tag{IM.17}
\]

The original holomorphic harmonic-number normalization and every branch change are among the explicit normal-face primitives used in IM.3/IM.15. Consequently IM.16 does not remove a supported principal-log correction by formal differentiation or by evaluation on solutions. Finite-subring calibration is the earlier CP/NL proof; TC supplies the corrected ordinary all-dimensional trace and line signs. The infinite correction is the full-cochain primitive IM.16.

#### IM.6. Unital ring, triple products, coordinates, and finite diagrams

The diagonal unit is the exact finite calibrated unit. IM.17 and the actual coefficient associativity IK.3 make \(\kappa\) a unital ring map for the full class. IG.3–IG.4 makes it the geometric sheaf identification; hence the ring product is the intrinsic canonical product on the source-defined projectivized diagonal sheaf. Arbitrary nonlinear transition maps IG.5 are now multiplicative because the actual pair pullback, unshifted cup and normalized proper trace commute with that coordinate change. V.4's determinant/real-orientation computation applies on the same compact collar integrations, and IG.15 supplies the common support margins.

For a triple product use one pointed full-dimensional cone at each fixed phase for all three kernels and their finitely many intermediate products; use a separate pointed midpoint cone on each neighboring overlap. Choose λ,λ′,B,B′ once for that finite list as in IM.18–IM.22. Repeat the literal normalization bridge and final-support map there. Only the compact intermediate collar and outer-source/inner-endpoint margins are common across the finite phase system; there is no pointed cone containing the full phase circle. Nonnegative normal lengths add, bounding both intermediate points in one compact family inside the actual coefficient domain. The complete triple contraction majorant is the IK.2 estimate with three homogeneous inputs and all three multiorders; its factorial factor is bounded by \(M_-!m!/J_+!\), as proved there. It gives absolute convergence, the factorial/every-\(\varepsilon\) output bounds and every fixed derivative tail. The corresponding finite full-cube cups and two compact proper traces have the same signs by V.12–V.14. Their order is immaterial by compact Fubini after the specified shift/line reordering. Alternatively both are the same actual coefficient triple product under the injective identification IG.4. No formal positive-degree completion occurs in either argument.

For any finite list of full symbols, matrices, relations, idempotents, maps, inverses and homotopies, take the finite intersection of their actual coefficient domains and the minimum of their finitely many Cauchy margins. Choose common negative and spatial constants and a finite phase cover. At each phase choose one common cone for that finite symbol/product list, and on each neighboring overlap choose its separate pointed cone, retaining 2λ′+h<π. Choose the standard T′ normalization cubes, the final convex G′ supports and one compact proper collar for the resulting finite system before performing any infinite contraction. The proof above fits all their finitely many product/overlap cochains and their entire positive-truncation families into one fixed finite weighted model. Every relation then holds on a single smaller actual neighborhood. There is no shrinking for each positive degree, spatial Taylor index or relation coefficient.

![Controlled holomorphic tails and the finite full-cochain product primitive](assets/infinite-holomorphic-product.png)

**Figure IM.1.** Left: exact samples \(w=1/k\), \(2\le k\le12\), of \(2\pi|K(w)|=k e^k\) for \(P=\sum_{j\ge0}\partial_t^j/(j!)^2\), and finite positive cutoffs \(J=1,3,8\), whose scaled values are the exact rational numbers \(k\sum_{j=0}^Jk^j/j!\). These are punctured-complement samples, not current boundary values. Center: for \(N=2\), the output cube has cardinalities \(r=0,1,2\), all ambient Dolbeault degrees \(q=0,\ldots,4\), and the finite inverse \(A=1-d_hh_v+(d_hh_v)^2\); no infinite homotopy series is drawn or used. Right: IM.13–IM.16 retains the full output and overlap cochains, proves the zero pair by bounded normalization, and constructs its actual relative-cone primitive. The editable figure, exact data and reproducible source are linked below. The full proof includes every normal dimension, every spatial Taylor index, both growth regimes and every fixed derivative tail.

The right panel displays IM.16 on the standard support \(T'_i\), with the actual \(\mathsf F_{\mathcal G_i,T'_i}\) support map and \(\kappa'_i\); its next arrow is IM.23 on final \(\mathcal G'_i\).

Editable SVG · Exact data · Figure source


This proves the full coefficient/geometric-kernel ring, its canonical mixed product, unit, associativity, nonlinear coordinate compatibility and ordinary zero-section restriction. A particular all-degree module-derived application still requires the exact cone-open, round-domain, common-collar and output-support hypotheses of §5.43. Section 5.47 proves the full arbitrary finite-matrix relation theorem and right flatness for all algebraic left modules, including the genuine ordinary zero-section restriction. Section 5.48 proves extension faithfulness and unit injectivity for every algebraic left module and sheaf in the full geometric and genuine ordinary zero-section rings. Section 5.49, ISD.1–ISD.5, ISF.1–ISF.5 and ISR.1–ISR.7 proves full infinite Späth sum division, the distinct special fibration class and both balanced reconstruction maps for the whole singular-inclusive germ, including off-ray vanishing and the explicit full contact shear. Other ordinary and mixed scalar changes remain open. Sections5.50–5.86 prove moving-sector propagation, purity, arbitrary-section separation, the special action and monodromy stability, full infinite-order D-type linearity and the faithful D-type embedding. Sections5.87–5.89 prove finite poles, intrinsic orders, the half-order criterion and coherent characteristic-stable cutoffs. Section5.90, GG.A–GG.G proves initial ordinary generators near compact sets. Arbitrary analytic proper regularity requires its further argument. The kernel-ring statement has no representation-theoretic assertion about generalized or nonactual central characters, other regular complex characters, singular or parabolic blocks, IC/Hecke/KL, stack or ind-scheme categories, restricted local systems, or affine, critical and factorization categories.


### 5.46. Finite infinite-order diagrams on actual derived support modules

We realize any specified finite full-symbol diagram, including its product, zero, idempotent, differential, map and contraction identities, in one canonical ring on one newly chosen actual domain. Under the checked final-cone and support hypotheses, that ring acts on a strict module-derived support object in every degree. Supplied bounded finite-projective contractions transfer by the explicit tensor and Hom formulas below.

The complete geometric ring and mixed product are [§5.45, IG.1–IG.5 and IM.1–IM.6](#5-45-the-full-geometric-infinite-order-kernel-ring-and-its-canonical-product). The entire earlier [§5.43, DL.1–DL.9, DLG.1–DLG.7 and DLH.1–DLH.4](#5-43-the-canonical-all-degree-action-and-its-derived-module-structure) supplies the full relative-kernel action, analytic concentration and genuine module-derived construction; the [§5.44 coefficient and trace proofs](#5-44-full-homogeneous-coefficients-and-ordinary-infinite-order-kernels) retain every homogeneous/spatial index and exact sign. A finite diagram uses one common domain; the whole symbol stalk need not be represented on a preassigned domain. The all-module right-flatness proof is Section 5.47; proper-ideal detection, arbitrary-module unit injectivity and faithful exact extension are proved in Section 5.48. Other scalar changes require separate proofs.

The freely accessible human target is [Micro-hyperbolic systems, Corollary3.2.5](https://perso.imj-prg.fr/pierre-schapira/wp-content/uploads/schapira-pub/Microhyp.pdf). Every mathematical premise is proved here or in the complete earlier lesson proofs linked above.

#### ID.1. The final cone is precisely an admitted full thick cone

Fix any finite list of full infinite-order symbol germs, including all entries, products, idempotents, differentials, maps, inverses and homotopies in a specified finite diagram. Fix a nonzero real cotangent phase over its normalized projective chart. Translate the base point to zero and rotate the complex time coordinate so that this phase is zero. The pair pullback and input top-form determinant in IG.5/IM.6 transport the ring and its products, so this coordinate normalization preserves the canonical coefficient-frame conventions.

IG/IM supplies one actual product of coefficient domains, one common phase and neighboring support system, and literal finite product/zero witnesses before any infinite index is summed. At the chosen phase its final support is

\[
\begin{gathered}
G=\left\{\begin{gathered}-\Re w=l\ge0,\\
|\Im w|\le\epsilon l,\\
|\zeta_j|\le A_jl\end{gathered}\right\},\\
\epsilon=\tan\lambda'>0,\\
A_j=B'_j/\cos\lambda'>0.
\end{gathered}
\tag{ID.1}
\]

This is exactly the full-dimensional cone of §5.43, (DL.11). In particular, no line support with empty ordinary interior is used. The cone is closed and convex by the triangle inequality in each constraint. If both a vector and its negative belong, their real-time lengths are zero, so every coordinate is zero; it is pointed. The strict inequalities hold at \(w=-1,\zeta=0\), giving ordinary interior. Put

\[
\begin{gathered}
K_G=\left(1+\epsilon^2+\sum_jA_j^2\right)^{1/2}.\\
g\in G\Longrightarrow |g|\le K_G[-\Re g_t].
\end{gathered}
\tag{ID.2}
\]

The assertion follows by squaring each displayed coordinate bound and adding. The empty spatial list, hence normal dimension one, is allowed. For a common neighboring cone use its own midpoint coordinates and its own constants \(\epsilon=\tan(\lambda'+h/2)\), \(A_j=B'_j/\cos(\lambda'+h/2)\); these are positive and finite by IM.19. A finite comparison uses the finite maximum of these constants. It does not put all rotated phases in one pointed cone.

#### ID.2. One actual round coefficient domain and one ring for the finite diagram

In these coordinates take

\[
\begin{gathered}
D_R=(R e_t+\operatorname{int}G)\\
\cap(-R e_t-\operatorname{int}G),\\ R>0.
\end{gathered}
\tag{ID.3}
\]

Here \(e_t\) is the positive real time unit vector. Explicitly \(|\Re t|<R\), and the transverse bounds are \(|\Im t|<\epsilon(R-|\Re t|)\), \(|x_j|<A_j(R-|\Re t|)\). Thus it contains zero and is bounded by \(|u|<K_GR\). If \(u,z\in D_R\) and \(v-u,z-v\in G\), then \(v\) lies in its first open cone by addition to \(u\), and in its second by subtraction from \(z\). Hence \(v\in D_R\): this is the required roundness, not an assertion about ordinary Euclidean balls.

Choose the outer base/normal coefficient product first. A sufficiently small \(R\) puts \(\overline{D_R}\) inside the base coefficient domain and makes every difference of two points of \(D_R\) fit strictly inside every selected normal source radius: its time modulus is at most \(2R\sqrt{1+\epsilon^2}\) and its spatial moduli at most \(2A_jR\). Take the minimum of finitely many strict upper bounds on \(R\). The choice depends on this finite symbol/witness list, not on any homogeneous or spatial coefficient index.

Let \(\mathcal R=E(G;D_R)\) be the whole canonical degree-\(N\) kernel cohomology ring of §5.43. Each full symbol's forward holomorphic relative representative first forgets support from its standard \(T'\) to this final \(G\), then restricts to \(D_R\times D_R\). These are literal relative-complex maps. All complement entries remain on their actual complements; no infinite positive head is extended as an ordinary current.

The full product primitive IM.16 is on \(T'\); IM.23 already maps it forward to the final \(G\). Its restriction to \(D_R\times D_R\) therefore remains a literal boundary there. The resulting class is the product in \(\mathcal R\), not a different outer-domain product: for compact endpoint sets, the kernel-supported intermediate points belong to \(D_R\) by roundness. Nonnegative lengths add, so (ID.2) bounds those points. The closed intersection is consequently compact inside the actual intermediate domain. The controlled excision/collar comparison of DLG.3–DLG.5 restricts the outer cup/trace roof to this one proper family. Its finite partition homotopies keep every complement entry and its normalized trace sign. Thus the restricted IM.23 compares exactly the inner-domain canonical product.

IG.4 supplies actual finite zero witnesses for germ identities. IM.6 places their finite union and every selected product comparison on one later support/domain system. Reducing \(R\) there, as above, realizes every specified relation in the **same** \(\mathcal R\). The diagonal class is its genuine unit by the exact finite-subring calibration and DLG.5. Finite idempotents, complexes and their maps therefore become actual finite \(\mathcal R\)-diagrams with all their equations holding. This neither requires one fixed domain for every possible germ nor promotes coefficient-wise vanishing to an unwitnessed common-domain relation.

#### ID.3. Verify the input cone-open premise, rather than inherit it silently

Choose input opens \(\Omega_0\subset\Omega_1\) that are \(G\)-open for the **final** (ID.1) cone, and put \(S=\Omega_1\setminus\Omega_0\). Require its ordinary closure to lie in a sufficiently small output neighborhood \(U\Subset D_R\). Being open for an earlier smaller cone alone is not this premise.

There is a concrete test for the usual directional inputs. Write \(y=(\Im t,x)\), let \(g(y)\) have Lipschitz constant \(L\) in the actual transverse Euclidean metric, and put

\[
\begin{gathered}
M_G=\left(\epsilon^2+\sum_jA_j^2\right)^{1/2},\\
\Omega_g=\{-\Re t>g(y)\},\\ LM_G<1.
\end{gathered}
\tag{ID.4}
\]

For \(a\in G\), write \(\sigma=-\Re a_t\ge0\). Its transverse displacement has norm at most \(M_G\sigma\). Then

\[
\begin{gathered}
-\Re(t+a_t)-g(y+a_y)\\
\ge -\Re t-g(y)\\
+(1-LM_G)\sigma>0.
\end{gathered}
\tag{ID.5}
\]

So \(\Omega_g+G\subset\Omega_g\). Finite intersections/unions of such invariant opens are invariant, as is every strict lower real-time halfspace. Alternatively the admitted basis \(x+a e_t+\operatorname{int}G\) is invariant by convex-cone addition. These furnish actual input opens whenever their relative support satisfies the stated compact output condition. This calculation does not assert that every preexisting directional input passes the enlarged constants.

Choose \(U\subset B(0,r)\) and \(a>r\) so small that

\[
 \overline{(U+G)\cap\{\Re t\ge-a\}}\Subset D_R.
 \tag{ID.6}
\]

This is an actual choice. If \(v=u+g\), \(|u|<r\), and \(\Re v_t\ge-a\), then \(-\Re g_t\le a+r\) and \(|v|\le r+K_G(a+r)\). An ordinary ball about zero lies compactly inside \(D_R\); first fix its radius, then choose \(r,a\) with that last bound smaller. All finitely many cones in an admissible comparison can use their largest \(K_G\). Intersecting both input opens with \(U+G\) leaves \(S\) unchanged and gives the exact supported object by the proved open-excision map of DL.7/DLG.7. From now on \(\Omega=\Omega_1\subset U+G\).

#### ID.4. Actual properness and the support-to-output inclusions

Set \(\omega=\{\Re t<-a\}\), \(S_*=\Omega\setminus\omega\). Both \(S\) and \(S_*\) are closed in \(\Omega_G\), and \(S\subset S_*\) because \(\omega\cap U=\varnothing\). Condition (ID.6) puts the ordinary closure of \(S_*\) compactly inside \(D_R\).

Let \(K\Subset\Omega\) be a compact output family. Its relevant auxiliary inputs belong to

\[
 (K+G)\cap\{\Re t\ge-a\}.
 \tag{ID.7}
\]

This set is bounded by (ID.2). It is closed: a convergent sequence has a subsequence of its \(K\)-coordinates converging in \(K\), and the corresponding differences converge in the closed cone. It lies in \(\Omega\) because \(\Omega+G\subset\Omega\), and it lies inside the coefficient domain by (ID.6). Thus it is compact **inside** the actual source. Intersecting it with the relative closed support \(S_*\), or \(S\), preserves that compactness. This proves properness for the canonical action, including the source boundary condition.

The support-to-output premise is checked on the actual action source. Define \(A_G=\left\{\begin{gathered}(u,v)\in(\Omega\cap D_R)\\\times(\Omega\cap D_R):v-u\in G\end{gathered}\right\}\), with \(p_1\) and \(p_2\) the restrictions of the output and input projections. If an output \(u\in\Omega_0\), every supported input \(v\in u+G\) lies in \(\Omega_0\) and cannot lie in \(S\). Every output in this relation already lies in \(\Omega\); hence an output paired with an input in \(S\) lies in \(\Omega\setminus\Omega_0=S\). Therefore

\[
 p_1(A_G\cap p_2^{-1}S)\subset S.
 \tag{ID.8}
\]

The same argument with the invariant halfspace \(\omega\) gives the corresponding inclusion for \(S_*\). The collar construction of DLG.4, applied to these compact source families, supplies actual properly supported cutoffs equal to one near the whole intersections. Its finite-output partition permits one collar for all entries of the specified finite diagram. Derivative-collar terms are away from the closed intersection; their complement coefficients are therefore defined there. No unrestricted complement function is extended over a kernel support.

If one performs several consecutive products, their nonnegative lengths add to the endpoint length. All intermediate points stay in \(D_R\) by roundness and satisfy the same bounds. One final collar can be chosen for the whole finite intersection. DLG.5's projection formula and eight-entry triple comparison then apply literally. The essential positive head has remained a holomorphic complement coefficient throughout.

#### ID.5. The genuine module-derived object and every-degree action

All hypotheses of the earlier complete DL/DLG construction are now checked for the actual full infinite symbols realized in \(\mathcal R\). Its exact analytic DLH existence and cone-topology concentration give

\[
\begin{gathered}
\mathcal F=H^1_{S_*}(q_{G*}\mathcal O_\Omega),\\
R\Gamma_{S_*}(q_{G*}\mathcal O_\Omega)\\
\simeq\mathcal F[-1].
\end{gathered}
\tag{ID.9}
\]

DLG.6–DLG.7 constructs a **strict** sheaf of left \(\mathcal R\)-modules \(\mathcal F\) using the full arbitrary-relative-kernel cup, complete cube augmentation, controlled proper excision and normalized trace. Its unit and multiplication identities are ordinary sheaf identities. These earlier proofs act on the whole kernel ring; the present full symbols require no supported-current shortcut or new growth premise.

Use its actual module-injective resolution \(J^\bullet_\mathcal R(\mathcal F)\), or its common natural vector-injective model endowed with this strict action. Then

\[
\begin{gathered}
\mathcal Q_\mathcal R=\\
\Gamma_SJ^\bullet_\mathcal R(\mathcal F)[-1]\\
\in D(\mathcal R\operatorname{-Mod}(\Omega_G)).
\end{gathered}
\tag{ID.10}
\]

The complete DL.6/DLG.7 concentration roof identifies its underlying vector-derived object with \(R\Gamma_S(q_{G*}\mathcal O_\Omega)\), and identifies its operation by each \(r\in\mathcal R\) with the original canonical whole-relative-kernel operation **in every degree**. The proof uses the literal equality of support functors \(\Gamma_S\Gamma_{S_*}=\Gamma_S\) on injectives, the full finite support cube and the same proper collar. Thus (ID.10) is an actual object in the module-derived category, not an arbitrarily chosen collection of chain lifts of vector-derived endomorphisms.

Apply the actual selected full-symbol classes of ID.2 to this strict module object. Every finite operator relation, matrix identity and composition in the diagram holds there because it holds in the one ring \(\mathcal R\). For a coefficient-domain, cone or auxiliary-halfspace change, use DL.7/DLG.7's actual finite module-derived comparison roofs. A common cone change is admitted only after rechecking ID.3–ID.4 for the common enlarged cone. Under those premises the comparisons commute with the full ring maps and the canonical all-degree action; their successive maps compose because they originate from the same ordinary injective resolution. This supplies the actual finite full-infinite-order germ diagrams.

#### ID.6. Finite projectives, idempotents and contractions

For a bounded exact complex of finite projectives over the full symbol-germ ring, start with its highest nonzero differential onto the last term. Projectivity splits that surjection. Its kernel is a direct summand of the preceding projective term, hence projective; descend finitely. The resulting contraction \(h\) satisfies \(dh+hd=1\). This is the complete algebraic splitting of DL.8; it uses neither Noetherianity of the infinite ring nor its flatness over the finite ring.

Write each projective term as a split idempotent of a finite free term. Its differentials, maps, idempotents and the contraction have finitely many full-symbol entries. Include all their equations in the finite list of ID.1–ID.2. Their images in \(\mathcal R\) form the actual realized complex \(P^\bullet_\mathcal R\), with the realized idempotents, maps and contraction. This is a finite-diagram realization; it is not a claim that arbitrary infinite-ring modules extend underived or that one preassigned domain receives the entire symbol stalk.

For right projective terms, the actual tensor totalization with (ID.10) has

\[
\begin{gathered}
d(p\otimes q)=\\
d_Pp\otimes q+(-1)^i p\otimes d_Qq,\\
H=h\otimes1,\\dH+Hd=1\qquad(p\in P^i).
\end{gathered}
\tag{ID.11}
\]

The two mixed terms cancel with signs \((-1)^{i-1}+(-1)^i=0\). For left projective terms and a degree-\(k\) Hom element,

\[
\begin{gathered}
d\phi=d_Q\phi-(-1)^k\phi d_P,\\
H\phi=(-1)^k\phi h,\\ dH+Hd=1.
\end{gathered}
\tag{ID.12}
\]

Expanding gives cancellation of \(d_Q\phi h\) with signs \((-1)^k+(-1)^{k+1}\), leaving \(\phi(hd_P+d_Ph)=\phi\). Restrict these formulas through the split idempotents. All sums in the projective direction are finite even if the support resolution is unbounded above. The exact finite-column filtration of DL.8 proves that these projective tensor/Hom complexes compute the corresponding derived models. The contraction therefore transfers in the actual module-derived category with all degree, idempotent and map identities intact.

This proves finite full-infinite-order instantiation under the checked geometric hypotheses. [Section 5.47](#5-47-arbitrary-finite-matrices-and-right-flatness-of-the-full-infinite-order-ring) proves the full finite-matrix relation criterion and genuine microlocal/ordinary all-module right flatness; [Section 5.48](#5-48-proper-ideals-and-faithful-extension-to-the-full-infinite-order-ring) proves proper-ideal detection, arbitrary-module unit injectivity and faithful exact extension. Section 5.49 supplies complete full infinite Späth sum division and balanced reconstruction, the actual special fibration source class and the explicit full contact transport. Sections5.50–5.89 prove the specified scalar comparisons, propagation, purity, separation, D-type realization and coherent characteristic cutoffs; Section5.90, GG.A–GG.G proves the initial ordinary generators near compact sets. The arbitrary analytic proper-image theorem requires its further argument.

![One actual round-domain margin and the strict module-derived mechanism](assets/full-infinite-derived-instantiation.png)

**Figure ID.1.** Exact normal-dimension-two example at one real phase: \(\lambda'=\pi/4\), \(B'=3/2\), so \(\epsilon=1\), \(A=3/\sqrt2\), \(K_G=\sqrt{13/2}\). The left panel is only the real slice \(\Im t=\Im x=0\); the full domain has the complex-norm inequalities of ID.2. Here \(R=1/32\), \(r=1/1024\), \(a=1/512\), and the auxiliary closure has norm at most \((1+3\sqrt{13/2})/1024<1/(32\sqrt2)\). The open complex ball of the latter radius lies in \(D_R\) because \(|\Re t|+|\Im t|\le\sqrt2|u|\) and \(|\Re t|+|x|/A\le\sqrt{1+A^{-2}}|u|<\sqrt2|u|\); its dashed boundary is excluded. The difference-time bound \(\sqrt2/16\), spatial bound \(3/(16\sqrt2)\), and base bound \(\sqrt{13/2}/32\) are all strictly below the illustrative common outer coefficient radius \(1/4\). With \(L=1/8\), the actual directional condition is \(LM_G=\sqrt{11/2}/8<1\). The right panel constructs a strict whole-\(\mathcal R\) action before the module-derived support object, retaining its shift and finite contraction maps. Exact general proof: ID.1–ID.6 and the named earlier DL/DLG/DLH proofs. Free human target: [Micro-hyperbolic systems, Corollary3.2.5](https://perso.imj-prg.fr/pierre-schapira/wp-content/uploads/schapira-pub/Microhyp.pdf).

Editable SVG, exact data, and reproducible plotting source.


### 5.47. Arbitrary finite matrices and right flatness of the full infinite-order ring

We prove the finite-matrix relation criterion for the full geometric infinite-order ring and right flatness for all algebraic left modules. The ordinary zero-section version retains the original polynomial basis and relation shifts. Every relation coefficient has one actual domain, every positive-epsilon bound, one factorial negative bound and every spatial index. Section 5.48 proves proper-ideal detection, arbitrary-module unit injectivity and faithful exact extension. Section 5.49 proves the distinct special fibration class and full infinite Späth sum division and balanced reconstruction. Other scalar changes and the broader programme require separate proofs.

The earlier actual finite-order input is [§5.19, complete homogeneous products and ordered jets](#5-19-actual-factorial-growth-symbols-and-ordered-jets), [§5.20, the one-norm product estimates](#5-20-one-norm-multiplication-units-and-dummy-division), [§5.21, actual simultaneous division and finite stalk relations](#5-21-actual-simultaneous-division-and-finite-stalk-relations), [§5.22.1, the actual normal element](#5-22-1-the-actual-normal-element), and [§5.24.1, chart normalization](#5-24-1-the-filtration-used-here) and [§5.24.2, strict image generation](#5-24-2-an-additional-consequence-of-the-actual-division). The full coefficient class and contractions are [§5.44, IK.1–IK.5](#5-44-full-homogeneous-coefficients-and-ordinary-infinite-order-kernels); the actual geometric ring, zero detection, nonlinear coordinates, canonical mixed product and final-support witnesses are [§5.45, IG.1–IG.5 and IM.1–IM.6](#5-45-the-full-geometric-infinite-order-kernel-ring-and-its-canonical-product). The coefficient division and all-index convergence arguments used below are proved in MFA.1–MFA.4 and MFB.1–MFB.5 before the matrix proof MF.1–MF.9.

**Analytic coefficient division with uniform radii.**

This is a local proof used by the matrix argument. All coefficient operators below are complex linear. They need not be linear over the coefficient ring or over an operator ring. That distinction is essential for a nonprojective relation module.

#### MFA.1. Norms, initial exponents and finite heads

Let \(w=(w_1,\ldots,w_q)\) be finitely many complex variables centered at the point. For polyradius \(R\), put
\[
\begin{gathered}
|f|_R=\sum_\alpha |f_\alpha|R^\alpha,\\
|v|_{R,c}=\sum_i c_i|v_i|_R.
\end{gathered}
\tag{MFA.1}
\]
The first norm is finite on every strictly smaller polydisc than a polydisc of holomorphy: the iterated Cauchy bound gives \( |f|_R\le M\prod_a(1-R_a/S_a)^{-1}\) for \(R<S\). Coefficient convolution proves submultiplicativity. These coefficient spaces are weighted \(\ell^1\) Banach spaces. A restriction to any set of Taylor exponents has norm at most one.

For a submodule \(I\subset\mathcal O_{w,0}^{b}\), order a nonzero vector's Taylor exponents by total degree, then lexicographic order, then component. Its initial exponent is translated by multiplication by a Taylor monomial. Thus its initial diagram is upward closed in each component of \(\mathbf N^q\). An upward closed subset has finitely many minimal elements: every infinite sequence in \(\mathbf N^q\) has an earlier term componentwise at most a later term. Induct on \(q\), taking a subsequence with constant or increasing first coordinate, and then apply induction to the remaining coordinates. An infinite set of incomparable minima is impossible. Each exponent dominates a minimum, by minimizing in the finite box below it.

Choose finitely many actual \(g_i\in I\) realizing the minima \((\alpha_i,j_i)\), with leading coefficient one. Their initial orthants cover the diagram. Choose positive rational weights \(v_a\) close enough to one that every different Taylor exponent in each \(g_i\) has weight strictly larger than its head. Here is the finite justification. If \(L=\max_i|\alpha_i|\), a small hierarchical perturbation \(v_a=1+\delta^a\) realizes lexicographic comparison among exponents of degree at most \(L\). Make it smaller still so \((L+1)\min v_a>L\max v_a\). All higher total degrees then have larger weight than every head. At the same exponent, other nonzero components occur only after \(j_i\). Put \(c_j=\kappa^j\) and choose \(\kappa>0\) so that the weighted sum of those finitely many same-exponent coefficients is less than \(c_{j_i}/16\).

Set \(R_a(r)=r^{v_a}\). For small enough \(r_0>0\), all divisors are holomorphic on a larger polydisc, and, simultaneously for every \(r\in[r_0/2,r_0]\),
\[
\begin{gathered}
\max_i\frac{|g_i-w^{\alpha_i}e_{j_i}|_{R(r),c}}{c_{j_i}R(r)^{\alpha_i}}\\
<\theta<\tfrac12.
\end{gathered}
\tag{MFA.2}
\]
Indeed every different exponent has a positive weight gap. Its divided radius contribution decreases as \(r\) decreases. Dominate their sum by the absolute Taylor series at one fixed larger radius and apply dominated convergence as \(r_0\to0\). The already fixed same-exponent component tolerance supplies the remaining bound. This argument controls the entire analytic series, not a finite Taylor approximation.

#### MFA.2. A literal bounded splitting

Order the heads, assign each exponent in their orthants to the first containing head, and write \(\Delta_i\) for those disjoint assigned sets and \(\Delta_0\) for their complement. Let \(V_i\) be the coefficient-support space obtained by taking component \(j_i\), restricting to \(\Delta_i\), and dividing by \(w^{\alpha_i}\). Give \(V=\bigoplus_iV_i\) the norm
\[
 |a|_V=\sum_i c_{j_i}R^{\alpha_i}|a_i|_R.              \tag{MFA.3}
\]
Let \(d\) be the restriction-and-division operator, \(T_h a=\sum_i a_iw^{\alpha_i}e_{j_i}\), and \(T_ga=\sum_i a_ig_i\). Disjoint supports give \( |df|_V\le |f|_{R,c}\), \(dT_h=1_V\), and \(\ker d\) is exactly the coefficient-support space \(\Delta_0\). Submultiplicativity and \(MFA.2\) give \(\|d(T_g-T_h)\|\le\theta\). Hence
\[
\begin{gathered}
h=\\
(1+d(T_g-T_h))^{-1}d,\\
p=1-T_gh.
\end{gathered}
\tag{MFA.4}
\]
The inverse is its absolutely convergent Neumann series on the stated Banach spaces. The identities \(hT_g|_V=1_V\), \(hp=0\), and \(dp=0\) follow by multiplying \(MFA.4\); in particular \(p^2=p\). The input, every quotient, and the remainder have one common actual polydisc.

For \(f\in I\), its remainder \(pf=f-\sum_i(hf)_ig_i\) belongs to \(I\): this uses a finite sum of actual holomorphic coefficient multiples, and does not require closure of \(I\). If nonzero, its initial exponent belongs to the diagram of \(I\), whereas all its exponents lie in \(\Delta_0\). This contradiction gives \(pf=0\). Thus \(\operatorname{im}T_g=I\), \(\ker p=I\), and these finitely many heads generate \(I\). This also proves the analytic coefficient Noetherianity used here.

Viewed in ordinary vector coefficient norms, \(h,p,T_g\) are bounded uniformly for all \(r\in[r_0/2,r_0]\). The head division constants are bounded by the fixed minima \(R(r_0/2)^{-\alpha_i}\); the inverse norm is at most \((1-\theta)^{-1}\); and the finite divisor norms are bounded on the largest radius. They are the same operators at every radius. The coefficient Neumann identities and uniqueness of the support-normalized quotient prove this assertion, independently of the numerical norm used to establish convergence.

#### MFA.3. The image operator and zero detection

Suppose a holomorphic matrix \(W_0:\mathcal O^r\to\mathcal O^s\) has coefficient kernel \(\mu\). Apply MFA.1–MFA.2 to \(\mu\), obtaining \(T_g,h,p\). Apply them also to \(\operatorname{im}W_0\). Each of that image's finitely many standard divisors is a finite holomorphic combination of the rows of \(W_0\). Compose its quotient map with those finite combinations to obtain a bounded coefficient operator \(s:\mathcal O^s\to\mathcal O^r\), with \(W_0s f=f\) for \(f\in\operatorname{im}W_0\). Put \(k=ps\). Then
\[
                     kW_0=p.                        \tag{MFA.5}
\]
For \(v-sW_0v\in\mu=\ker p\), applying \(p\) proves \(MFA.5\). All three coefficient operators \(h,p,k\) are bounded on the same radius interval, after taking one smaller \(r_0\) for their finite data. This is a coefficient splitting of the image and its relations. No splitting over \(\mathcal O\), \(A\), or \(B\) has been asserted.

#### MFA.4. Homogeneous polynomial coefficients at the zero section

For ordinary operators, use base variables \(u\) and polynomial covariables \(\rho\), retaining the integer shifts of each free basis vector. Every component of a homogeneous vector has one prescribed covariable degree. The initial diagram is upward closed under base Taylor and covariable polynomial monomials; the same Dickson proof supplies finitely many homogeneous standard divisors. The radius argument above applies to their Taylor series in \(u\) and their finite polynomials in \(\rho\).

The standard divisors can be chosen homogeneous: take the homogeneous summand containing a selected initial monomial of a vector in the graded submodule. It has the same head, since extracting that summand only removes monomials. When a homogeneous image divisor is expressed in the rows of a graded matrix, project each coefficient to the homogeneous degree appropriate for that row; the homogeneous part of the equality remains the whole divisor. Thus the finite image combinations used in MFA.3 preserve the shifts too.

Every support projection, head division and full-divisor multiplication preserves total covariable degree with the indicated free-basis shifts. The Neumann series in \(MFA.4\) therefore stays in that graded component. For a fixed component degree, only finitely many covariable exponents can occur, so its limit is a polynomial in \(\rho\) whose base coefficients are holomorphic on the common domain. Quotients with negative prescribed polynomial degree are zero. The same norm bounds hold uniformly in the total degree: their proof uses fixed head monomials, fixed divisor norms, and the same \(\ell^1\) radius norms, with fixed component radius factors to accommodate the finite shifts. For arbitrary holomorphic base input, one repeats the radius choice on its intersection with the finite divisor domain. This proves the precise polynomial-preserving version needed at the zero section.

Every mathematical result in this coefficient-division argument has been proved above. The complete earlier [§5.21.1–§5.21.5](#5-21-actual-simultaneous-division-and-finite-stalk-relations) gives the corresponding actual finite-order operator division, used in MF.2 below.

**Full infinite-order order-lowering estimate.**

This convergence argument proves convergence of the series used for arbitrary matrix relations. It includes every normalized spatial derivative, arbitrary many correction steps, the unbounded positive head, the factorial negative tail, and a single domain chosen before the summation indices.

#### MFB.1. Operators and one domain

In a cotangent chart \(\tau\ne0\), write \(F_j=\tau^jp_j(u,z)\), \(z=\xi/\tau\). The actual class has one holomorphic coefficient product, with, on each compact,
\[
\begin{gathered}
|p_j|\le M_\varepsilon\varepsilon^j/j!\\
(j\ge0, \varepsilon>0),\\
|p_{-n}|\le MD^nn! (n\ge1).
\end{gathered}
\tag{MFB.1}
\]
We use the absolute Taylor norms \(MFA.1\) on one family \(R(r)\), \(r\in[r_0/2,r_0]\). A Cauchy estimate on a fixed enlargement converts the input compact bounds to these norms with constants independent of \(j\). Coefficient operators \(h,p,k\), or a finite composition of these, have uniform norm \(H\) on that interval by MFA.2–MFA.3. Every right-hand actual matrix is order zero, with coefficients \(Q_{-q}\) satisfying \( |Q_{-q}|\le AC^q q!\) on a fixed enlargement, including \(q=0\). The finite list has one such bound and domain.

The right Leibniz term acting on a left coefficient has indices
\[
\begin{gathered}
\alpha\in\mathbf N^N,\quad q\ge0,\\
K=q+|\alpha|,\\
\frac1{\alpha!}(\partial_\rho^\alpha F_j)(\partial_u^\alpha Q_{-q}).
\end{gathered}
\tag{MFB.2}
\]
It lowers homogeneous degree by \(K\). Subtracting its principal ordinary multiplication removes precisely \(q=0,\alpha=0\); every remaining term lowers degree by at least one. The coefficient operators constructed in MFA.1–MFA.4 preserve degree. A word of correction terms interspersed with such coefficient operators therefore has total loss \(K\ge\) its number of correction terms. All matrix and component sums are finite.

Choose the same fixed outer margin for the right factors throughout. Its Cauchy estimate costs \(\alpha!\delta^{-|\alpha|}\); the factorial in \(MFB.2\) cancels it. If a total-order version is used, \(|\alpha|!/\alpha!\le N^{|\alpha|}\), so the difference is absorbed in an exponential constant. Across a word with tail indices \(q_i\), let \(Q=\sum_iq_i\) and \(L=\sum_i|\alpha_i|\), thus \(K=Q+L\). Their tail factorial product is at most \(Q!\).

#### MFB.2. Derivatives interspersed with coefficient divisions

At normalized \(\tau=1\), one covariable differentiation is exactly
\[
\begin{gathered}
\partial_{\xi_a}F_j=\tau^{j-1}\partial_{z_a}p_j,\\
\partial_\tau F_j=\tau^{j-1}(j-E_z)p_j,\\
E_z=\sum_a z_a\partial_{z_a}.
\end{gathered}
\tag{MFB.3}
\]
These equations retain every spatial index. They also specify all signs. At an intermediate step starting from input degree \(m\), the current degree has absolute value at most \(|m|+K\). A coefficient operator preserves that degree, although it need not commute with \(E_z\) or differentiation.

To bound a word, insert intermediate radii decreasing from \(r_0\) to \(r_0/2\), allocating equal scalar radius loss \(r_0/(2L)\) to each of its \(L\) derivatives; when \(L=0\) no derivative margin is consumed. All \(R_a(r)=r^{v_a}\) have a fixed positive lower derivative on this compact interval, so each coordinate margin is at least \(c/L\), with fixed \(c>0\). Cauchy's formula, the bounded normalized \(z_a\), and \(MFB.3\) bound each one-derivative step by
\[
\begin{gathered}
C_0(|m|+K+L)\\
\le 2C_0(|m|+K)\quad\\
(K>0).
\end{gathered}
\tag{MFB.4}
\]
The \(H\) bound for every inserted coefficient operator holds at its own intermediate radius. Thus no commutation of a division operator with a derivative has been used, and no new radius is lost for a division operator. The count of those operators is bounded by a fixed multiple of the number of correction factors, hence by a fixed multiple of \(K\).

For \(M=|m|\ge0\), \(K=Q+L>0\),
\[
\begin{gathered}
(M+K)^LQ!\\
\le e^K\frac{(M+K)!}{M!}.
\end{gathered}
\tag{MFB.5}
\]
To prove it, \(Q!\le(M+K)^Q\). Also
\(\prod_{a=1}^K(M+a)/(M+K)\ge K!/K^K\ge e^{-K}\): each factor increases with \(M\), and the last inequality follows from
\(\sum_{a=1}^K\log a\ge\int_1^K\log t\,dt=K\log K-K+1\).
Equations \(MFB.4\)–\(MFB.5\), submultiplicativity and the right-factor Cauchy estimates bound each word, before coefficient summation, by an exponential constant times that one factorial ratio.

There are at most exponentially many words of total loss \(K\). Ordered compositions of a positive integer \(K\) number \(2^{K-1}\); their subdivisions into \(q_i\) and the \(N\) derivative components number at most \((N+2)^{2K}\), a deliberately loose bound obtained by placing separators among \(K\) units and coloring the units. Finite matrix entries and coefficient-operator choices cost another exponential factor. Absorb all these constants, all fixed input bounds, and \(H\) into \(C_1\ge1\). The sum of the absolute norms of **all** terms of loss \(K\), for any one of the Neumann or iterated correction expressions below, is bounded by
\[
                  C_1^K\frac{(|m|+K)!}{|m|!}|p_m|_{R(r_0)}.
                                                               \tag{MFB.6}
\]
The case \(K=0\) is the single bounded coefficient operation and is covered by a fixed prefactor. This argument also applies to truncated words uniformly in the truncation; the bound includes each contraction and every tail coefficient rather than replacing them by a formal product.

#### MFB.3. Positive output, negative output and explicit tails

For positive output \(j\ge0\), input degree is \(m=j+K\ge0\). The integer inequality
\[
\begin{gathered}
\frac{(j+2K)!j!}{(j+K)!^2}\le\binom{2K}{K}\\
\le4^K
\end{gathered}
\tag{MFB.7}
\]
follows by writing the first ratio as \(\prod_{a=1}^K(j+K+a)/(j+a)\), whose factors decrease with \(j\), and then using the binomial expansion. From \(MFB.1\), \(MFB.6\),
\[
\begin{gathered}
|\text{output}_j|\le\\
C_2M_\varepsilon\frac{\varepsilon^j}{j!}\\
\sum_{K\ge0}(4C_1\varepsilon)^K.
\end{gathered}
\tag{MFB.8}
\]
Given any desired \(\eta>0\), choose \(0<\varepsilon\le\eta\) with \(9C_1\varepsilon<1/2\). The coefficient domain and \(C_1\) do not depend on this choice. This proves the positive estimate for **every** \(\eta\). Terms with \(K\ge R\) are bounded by the right side of \(MFB.8\) times \((4C_1\varepsilon)^R\), with the geometric denominator retained.

For negative output \(-n\), the input is \(m=K-n\). When \(K<n\), \(MFB.6\) gives
\[
\begin{gathered}
C_1^K\frac{n!}{(n-K)!}\\
M D^{n-K}(n-K)!\\
=M n!C_1^KD^{n-K}.
\end{gathered}
\tag{MFB.9}
\]
This part is finite; its sum is at most \(Mn!(D+C_1)^n\), since the binomial coefficients dominate the corresponding \(n\) summands. For \(K\ge n\), write \(m=K-n\ge0\). The multinomial expansion proves
\((n+2m)!/(n!m!m!)\le3^{n+2m}\). Consequently the full positive-input part is bounded by
\[
\begin{gathered}
C_2M_\varepsilon n!(3C_1)^n\\
\sum_{m\ge0}(9C_1\varepsilon)^m.
\end{gathered}
\tag{MFB.10}
\]
Choose one fixed \(\varepsilon<1/(18C_1)\). Equations \(MFB.9\)–\(MFB.10\) prove one factorial negative constant for every \(n\), on the same domain. The tail \(K\ge R>n\) in \(MFB.10\) is bounded by the displayed prefactor times
\((9C_1\varepsilon)^{R-n}/(1-9C_1\varepsilon)\).

These bounds prove absolute normal convergence of the whole expressions, including infinitely many terms at a fixed output degree. They justify rearrangement, finite matrix sums and multiplication identities by the triangle inequality and absolute scalar convergence. On a fixed further inner product, the Cauchy formula applies to each chosen finite output derivative. Its constant is independent of all cutoffs and degrees. Applying it to the displayed tails proves normal convergence of every fixed base and normalized-spatial derivative. All spatial Taylor exponents remain in the norm; none is truncated or bounded in advance.

#### MFB.4. Actual inverses and vanishing iteration

Let \(L\) be one of the above order-lowering correction operators. Define
\[
\begin{gathered}
(1+L)^{-1}f=\\
\sum_{a\ge0}(-L)^a f.
\end{gathered}
\tag{MFB.11}
\]
Equations \(MFB.6\)–\(MFB.10\) prove that this expression is an actual coefficient sequence in the complete class, on one domain. Reserve an intermediate radius, for example \(3r_0/4\): the same proof first gives normal convergence there. Apply one further correction on the inner radius \(r_0/2\). The Cauchy formula permits differentiation of the convergent series, and the direct word bound MFB.6, with the additional correction included, permits its reordering there. The finite geometric identity gives left and right inverses, because its remainder \(L^{a+1}f\) has total loss at least \(a+1\) and tends to zero coefficientwise with the explicit tails above. This is a limit of normally convergent holomorphic coefficient functions. It is not an ordinary supported-distribution limit, a formal completion, or a limit in a cohomology quotient.

In particular, if an actual sequence satisfies \(s=-Ls\), then \(s=(-L)^a s\) for every finite \(a\). If \(s\) is the output of a previous division, select one new, fixed smaller outer radius inside its holomorphic domain and the same coefficient-divisor domain. MFA.2 makes the identical coefficient operators bounded on its radius interval; the support-normalized analytic operator is unchanged. This is a single additional shrinking made before any iteration index, not shrinking at each iteration. For positive output use \(MFB.8\) with \(K\ge a\). For fixed negative output \(-n\), take \(a>n\) and use \(MFB.10\) with \(m\ge a-n\). Both tend to zero, so every actual coefficient of \(s\) vanishes on the same inner product. Hence \(s=0\). This is the zero-remainder detection needed for full matrix relations with no highest positive homogeneous coefficient.

#### MFB.5. Ordinary zero-section version

In the ordinary polynomial coefficient spaces of MFA.4, the covariable derivative in \(MFB.2\) is an ordinary derivative of a homogeneous polynomial. Its Cauchy bound on the same radius interval is at most \(C_0L\) per derivative; including it in \(MFB.4\) gives the same bound. Every finite-order right matrix has only finitely many tail offsets \(q\) relative to its principal shifted degree. A homogeneous space of negative polynomial degree is zero. All intermediate coefficient operations preserve the prescribed shifted polynomial degrees by MFA.4, and the finite ordinary Leibniz product preserves the corresponding graded spaces.

Thus \(MFB.6\)–\(MFB.8\) apply with finite free-basis shifts. A fixed shift changes the positive factorial bound by a fixed polynomial in the degree; this is absorbed by choosing the input \(\varepsilon\) smaller than the requested output value, since \(j^a\theta^j\) is bounded for every fixed \(a\) and \(0<\theta<1\). The finitely many degrees below the shifts satisfy the same convergent tail bounds individually. Every output component remains a homogeneous polynomial of its actual nonnegative degree. This proves \(MFB.11\) and zero detection in the genuine ordinary \(D^\infty\) class at the zero section; it does not substitute an ordinary ring for the microlocal argument MFB.1–MFB.4.

**Arbitrary finite matrices in the genuine infinite-order geometric ring.**

The theorem proved in MF.1–MF.8 is the full finite-matrix relation criterion and right flatness of \(B=\mathcal E^\infty_p\) over \(A=\mathcal E_p\), including the ordinary zero-section restriction. The tensor product is the ordinary algebraic tensor product.

The earlier complete [geometric identification IG and product IM](#5-45-the-full-geometric-infinite-order-kernel-ring-and-its-canonical-product) are exact premises. They identify the actual common-domain coefficient class with the source-defined complex-projectivized diagonal kernel ring, with its input top-form twist, nonlinear coordinate changes, actual pole/log kernels, full cone enlargement and ordinary zero-section restriction. This proof establishes the new relation theorem; it does not infer it from geometric convergence, finite module-derived instantiation, a finite-projective contraction, or completion flatness.

Free human context is Kashiwara–Schapira, [Micro-hyperbolic systems](https://perso.imj-prg.fr/pierre-schapira/wp-content/uploads/schapira-pub/Microhyp.pdf), §8, printed/PDF p.43. That page recalls flatness before its noncharacteristic theorem and supplies no proof of the relation criterion. Every argument used below is proved in MFA, MFB and MF, or in the complete earlier programme proofs linked above. Complete earlier programme dependencies are linked above.

#### MF.1. Sides, statement and germs

All modules being extended are left \(A\)-modules; \(B\) is a right \(A\)-module through the proved unital geometric ring inclusion. Write a free module as rows. For every finite \(r\times s\) matrix \(D\) over \(A\), its left-linear map is \(v\mapsto vD\), where every matrix multiplication is the actual operator product. Put \(K_D=\{v\in A^r:vD=0\}\). We prove
\[
\begin{gathered}
\ker(B^r\xrightarrow{\,D\,}B^s)\\
=\operatorname{im}(B\otimes_AK_D\longrightarrow B^r).
\end{gathered}
\tag{MF.1}
\]
Specifically, a relation \(bD=0\) has an expression \(b=\sum_{\nu=1}^lc_\nu v_\nu\) with finitely many actual \(v_\nu\in K_D\), actual \(c_\nu\in B\), and one common coefficient product and the full bounds MFB.1 for every \(c_\nu\). An equality of germs is represented by all coefficients on one smaller product, as in the proved ring theorem. No degree-dependent shrinking is permitted.

If \(D=0\), use the ordinary standard basis rows of \(A^r\) and the components of \(b\); empty matrix sizes are included in this convention. If the ambient complex dimension is zero, \(A=B=\mathbf C\) at a point and the assertion is the usual exact scalar relation statement. These cases need no cotangent chart or derivative estimate.

For a point off the zero section choose a chart \(\tau\ne0\). Translate the base point and normalize its central \(z=\xi/\tau\) to zero by the fixed affine cotangent coordinate change of the exact earlier [§5.24.1](#5-24-1-the-filtration-used-here). This preserves the full contraction pairing. Let \(A_0\) denote the actual order-zero factorial ring and \(h=\tau^{-1}\). The exact earlier [§5.21](#5-21-actual-simultaneous-division-and-finite-stalk-relations) proves that every submodule of \(A_0^s\) has finite standard generators, on an actual common coefficient domain. The complete [§5.24.2](#5-24-2-an-additional-consequence-of-the-actual-division) proves strictness with the homogeneous-index filtration. The actual normal-element localization [§5.22.1](#5-22-1-the-actual-normal-element) identifies \(A=A_0[h^{-1}]\); it is not a formal localization of an unnamed coefficient ring.

The equality with that actual localization also follows directly at coefficient level. If \(P\) has finite upper homogeneous bound \(m\ge0\), then \(Ph^m\) is the literal right coefficient shift and is order zero. A fixed shift preserves factorial bounds, by \((n+m)!\le2^{n+m}n!m!\). Hence \(P=(Ph^m)h^{-m}\), with both endpoints actual. Right multiplication by \(h\) is injective because it shifts the whole coefficient sequence, and its actual normal automorphism is the complete finite formulas (5.22a)–(5.22b) in [§5.22.1](#5-22-1-the-actual-normal-element). The inverse \(h^{-1}=\tau\) is the proved finite geometric ring inverse.

#### MF.2. Strict normalization of an arbitrary matrix

First right-multiply each target column by a suitable fixed power of \(h\) so that all entries of \(D\) have order at most zero. This is an invertible right target change over both \(A\) and \(B\), and leaves the source relation module unchanged. Denote the resulting matrix again by \(D\).

Let \(J=\operatorname{im}(A^rD)\subset A^s\), \(J_0=J\cap A_0^s\). Apply the full actual finite division theorem to the left \(A_0\)-submodule \(J_0\). Its finite standard generators \(P_i\) have homogeneous orders \(n_i\ge0\). For \(F\in J_0\cap H^aA_0^s\), the exact strictness proof gives
\[
\begin{gathered}
F=\sum_i Q_iP_i,\\
Q_i\in H^{\max(0,a-n_i)}A_0.
\end{gathered}
\tag{MF.2}
\]
This is precisely the actual Neumann division of [§5.21](#5-21-actual-simultaneous-division-and-finite-stalk-relations), with its quotient support constraints; no infinite formal degree reduction is used.

Normalize the rows by left multiplication:
\[
\begin{gathered}
W_i=h^{-n_i}P_i,\\
P_i=h^{n_i}W_i.
\end{gathered}
\tag{MF.3}
\]
The earlier programme's explicit formula (5.21i) in [§5.21.2](#5-21-2-initial-exponents-and-the-finite-diagram) proves that \(W_i\) is actual order zero: multiplication by \(\tau^{n_i}\) has only finitely many base derivatives, and a fixed homogeneous shift preserves factorial growth on one smaller domain. Every \(P_i\) belongs to \(J\), so there is a finite matrix \(G\) over \(A\) with \(W=GD\). Conversely the rows of \(D\) belong to \(J_0\); (MF.2) expresses them in \(P_i\), hence in \(W_i\), giving a finite \(A\)-matrix \(J_1\) with
\[
                  W=GD,\qquad D=J_1W.                \tag{MF.4}
\]
All entries and these finite exact identities are represented on one neighborhood.

There is a stronger strictness consequence needed below. If \(F\in J_0\) has order at most \(-a\), \(a\ge0\), its coefficients in \(W\) may all have order at most \(-a\). In (MF.2), write \(Q_iP_i=(Q_ih^{n_i})W_i\); the coefficient has homogeneous index at least \(\max(0,a-n_i)+n_i\ge a\). Thus \(W\) is a strict order-zero image presentation. Its principal coefficient matrix is \(W_0:\mathcal O^l\to\mathcal O^s\).

#### MF.3. Actual lifts of every principal relation

Let \(\mu=\ker W_0\subset\mathcal O^l\). Each \(v\in\mu\) has an actual order-zero lift: initially use its holomorphic coefficient vector \(v\) as an order-zero symbol. Then \(vW\) has homogeneous index at least one. Strictness in MF.2 expresses \(vW=aW\) with every component of \(a\) of index at least one. Therefore \(v-a\) is an actual relation, of principal vector \(v\).

Apply MFA.1 to \(\mu\). Choose its finitely many standard holomorphic divisors \(g_\nu\) realizing the whole initial diagram. The preceding paragraph lifts each to an actual order-zero row \(R_\nu\) with
\[
           RW=0,\qquad R_0=(g_\nu)_\nu.              \tag{MF.5}
\]
There are finitely many such rows, with one common domain and one factorial tail constant. The finite generating proof for \(\mu\) is given in MFA.1–MFA.2, so it requires no citation to a missing coefficient-kernel theorem. These rows need not generate an arbitrary module by an \(A\)-linear contraction; their principal analytic coefficient division will do the work.

Let \(T_0c=cR_0\), restricted to the quotient support space \(V\) of MFA.2, and let \(T c=cR\) be actual right multiplication. MFA.2 gives \(hT_0|_V=1\), \(p=1-T_0h\), \(hp=0\), \(\ker p=\mu\). MFA.3 supplies \(kW_0=p\). Their norms are uniform on one common radius interval. \(E=T-T_0\) and \(W_-=({}\cdot W)-({}\cdot W_0)\) lower homogeneous order by at least one and have exactly the finite-matrix form covered by MFB.1–MFB.4.

Here and through MF.4, \(h\) is the complex-linear coefficient division operator of MFA.2. The normal generator \(h=\tau^{-1}\) used in MF.1–MF.2 is a different, explicitly scoped object. Juxtaposition of coefficient operators means composition: \(kW_0\) sends \(v\) to \(k(vW_0)\). Actual matrix juxtaposition elsewhere means the stated row product in the operator ring.

#### MF.4. Complete infinite division of a relation

For an arbitrary actual \(b\in B^l\) with \(bW=0\), set
\[
\begin{gathered}
c=(1+hE)^{-1}hb\\
=\sum_{a\ge0}(-hE)^ahb,\\
s=b-Tc.
\end{gathered}
\tag{MF.6}
\]
MFB proves the absolute normal convergence of this whole sum on one coefficient product, including every contraction and spatial index. It proves the positive estimate for every requested \(\varepsilon\), one negative factorial constant, and all fixed derivative tails. Thus all finitely many components of \(c\) and \(s\) are actual \(B\)-sections. The support space \(V\) is preserved by \(hE\) because \(h\) has values in \(V\); all its coefficient-support restrictions remain literal closed restrictions in MFA.2. Formula (MF.6) and \(hT_0|_V=1\) give
\[
\begin{gathered}
hs=hb-(1+hE)c=0,\\
sW=0.
\end{gathered}
\tag{MF.7}
\]
The second equality uses \(RW=0\) and the proved actual associativity; absolute convergence in MFB permits the displayed rearrangement. Since \(hs=0\), \(ps=s\). The exact coefficient identity \(kW_0=p\) and \(sW=0\) imply
\[
                       s=-kW_-s.                    \tag{MF.8}
\]
MFB.4 applies to the order-lowering operator \(kW_-\). Iterating (MF.8) gives zero in every output coefficient with the explicit geometric tails MFB.8/MFB.10, including negative outputs after the finite part is passed. Hence \(s=0\). This is actual zero detection for a relation with no largest positive homogeneous degree. Therefore \(b=cR\).

The same construction for a finite-order input has no positive coefficients above its finite upper bound. Every lowering word in (MF.6) preserves that bound, so \(c\) also has a finite upper bound. Its negative estimates are already proved in MFB.9–MFB.10. Thus \(c\in A\). Consequently the finitely many rows \(R_\nu\) generate \(\ker(A^l\to A^s)\) as well.

#### MF.5. Transfer back to the original finite matrix

For the original relation \(bD=0\), (MF.4) gives \((bJ_1)W=0\). By MF.4 there is an actual finite coefficient row \(c\) with \(bJ_1=cR\). Therefore
\[
 b=b(1-J_1G)+cRG.                                    \tag{MF.9}
\]
Every row of \(1-J_1G\) is an actual \(A\)-relation for \(D\), because \((1-J_1G)D=D-J_1W=0\). Every row of \(RG\) is such a relation because \(RGD=RW=0\). They form one finite list \(v_\nu\). The coefficients in (MF.9) are the components of \(b\) and \(c\), all actual \(B\)-sections with the proved complete common-domain bounds. This proves the difficult inclusion in (MF.1). The reverse inclusion follows from the finite equations \(v_\nu D=0\) and associativity. This step fixes the row sides explicitly; there is no reversal to right-module relations.

#### MF.6. The ordinary zero section, with full positive heads

At a zero-section point the proved geometric ring restricts to \(B=D^\infty_u\), with ordinary homogeneous polynomial coefficients of degree \(j\), and \(A=D_u\). The full polynomial degree statement is about the homogeneous \(P_j(u,\rho)\); a normalized \(p_j(u,z)\) has degree at most \(j\), because powers of the normal covariable give lower spatial degrees.

Here is the strict finite setup rather than an appeal to microlocal division. Give \(D_u\) the nonnegative order filtration, whose multiplication has ordinary principal product in \(\mathcal O_u[\rho]\). MFA.4 proves finite homogeneous standard generation of every graded submodule of a finite shifted free coefficient module. Choose finitely many actual image rows \(P_i\) lifting its principal generators. Subtracting a combination of them lowers the finite order by at least one, so ordinary degree reduction terminates. It expresses every actual image row with the expected shifted order bounds. This proves strictness and finite generation directly. For a principal relation, its initial actual lift has image of lower order; strictness expresses that image with lower source order. Subtracting those coefficients produces an actual relation with the prescribed principal vector. Apply MFA.4 to the resulting graded principal kernel and lift its finitely many homogeneous standard divisors.

Retain the row shifts \(m_i=\operatorname{ord}P_i\) and the shifted relation orders \(r_\nu=\max_i(\operatorname{ord}(R_{\nu i})+m_i)\), omitting zero entries. The principal component \(R_{\nu i,0}\) has covariable degree \(r_\nu-m_i\). In total degree \(j\), component \(i\) has ordinary polynomial covariable degree \(j-m_i\), and coefficient \(\nu\) has degree \(j-r_\nu\). A negative polynomial degree is the zero space. The operators \(h_j,p_j,k_j\) of MFA.4 respect exactly those spaces and are uniformly bounded in \(j\) on the common radius family; their identities are the shifted version of MFA.4–MFA.5. The finite ordinary Leibniz corrections lower total degree and preserve these graded spaces.

Now use (MF.6)–(MF.8) in these spaces, with MFB.5. Every output is a polynomial of its actual covariable degree and has the positive estimate for every \(\varepsilon\) on one base domain and one fixed covariable ball; finite shifts are absorbed by the explicit epsilon slack in MFB.5. Homogeneity extends this estimate to every fixed covariable compact: scaling from the fixed ball costs a fixed exponential in the degree, absorbed by selecting the input epsilon smaller, while the base domain stays fixed. The full infinite tail converges normally, and the same iteration proves zero remainder. The transfer (MF.9) is ordinary finite matrix multiplication over \(D\). This proves (MF.1) at every zero-section point for genuine ordinary \(D^\infty\), with the entire unbounded positive head. It proves no unsupported equality between microlocal and ordinary module tensor products away from their actual ring maps.

#### MF.7. From the criterion to right flatness for all modules

For completeness the algebraic passage includes arbitrary modules. The finite microlocal stalk \(A\) is left Noetherian: its actual \(A_0\) is left Noetherian by the complete finite division proof, and normal localization preserves that property. To see the latter directly, for an \(A\)-submodule of a finite \(A\)-free module, multiply each individual row on the left by a sufficiently large power of \(h\) to put it in \(A_0^r\). Its intersection with \(A_0^r\) is finitely generated over \(A_0\); localizing those finitely many generators recovers every row. For ordinary \(A=D_u\), the finite homogeneous standard-generation and terminating order argument in MF.6 prove the same assertion.

Let \(L\subset A^s\) be finite and choose a finite free surjection \(A^r\to L\), with kernel \(K\). Right exactness of algebraic tensor says that \(\ker(B^r\to B\otimes_A L)\) is the image of \(B\otimes_AK\). Criterion (MF.1), applied to the composite \(A^r\to A^s\), identifies this image with \(\ker(B^r\to B^s)\). Thus \(B\otimes_A L\to B^s\) is injective.

For an injection \(N\subset M\) of finite left modules, choose a finite free surjection \(F\to M\), kernel \(K\), and inverse image \(N'\subset F\) of \(N\). Noetherianity makes \(K,N'\) finite. The preceding paragraph gives injectivity of both \(B\otimes_AK\) and \(B\otimes_AN'\) into \(B\otimes_AF\). An element of \(B\otimes_AN\) lifts to \(B\otimes_AN'\). If it becomes zero in \(B\otimes_AM\), right exactness expresses its image in \(B\otimes_AF\) by an element of \(B\otimes_AK\). Injectivity for \(N'\subset F\) makes these lifts equal; their image in \(B\otimes_AN\) is zero. Therefore tensor preserves finite-module injections.

For arbitrary \(N\subset M\), an element of \(B\otimes_AN\) and its claimed zero image use finitely many tensor summands. A zero equality in an algebraic tensor quotient is a finite sum of its defining additivity and balancing relations. Take the finite submodule \(N_0\subset N\) generated by its summand entries and a finite submodule \(M_0\subset M\) containing \(N_0\) and all entries of those finite relation witnesses. The zero equality already holds in \(B\otimes_AM_0\). The finite injection \(N_0\subset M_0\) kills the element. Hence tensor preserves every injection. Together with right exactness, this proves right \(A\)-flatness of \(B\).

The same argument pointwise proves sheaf flatness: exactness of a sequence of sheaves is exactness on every stalk, algebraic sheaf tensor has the stated stalk tensor, and the pointwise result applies. No assumption of coherence on the arbitrary modules in this conclusion remains.

#### MF.8. Actual geometric support, finite diagrams and precise boundary

Choose the input \(b\), the finite rows \(D,W,R\), the finite matrices \(G,J_1\), all coefficient divisors and the common domain once for each finite relation problem. MFA and MFB then give all output coefficients on one smaller actual product independent of homogeneous, Taylor, contraction and iteration indices. The negative constant and each positive-epsilon prefactor may change; the domain does not. For any finite family of relations or module maps, intersect this finite list of domains and choose the maxima of its finitely many negative constants and the input epsilon small enough for all its constants \(C_1\). Thus every finite diagram and every finite homotopy identity used in its tensor extension is an equality of actual sections on one common domain. The proof is local at a point; it does not assert one universal domain for every germ of an arbitrary module.

Here are concrete common kernel margins for that finite list. Choose fixed Taylor covariable radii \(R_i>0\) strictly inside its final holomorphic coefficient domain, a compact base enlargement there, and one largest negative factorial constant \(C_-\ge1\) for all inputs, relation coefficients and the finitely many products in MF.9. Choose spatial slopes \(A_i>2/R_i\), and one endpoint neighborhood with \(|w|<1/(4C_-)\), also contained in the common base endpoint domain. The source thick complement is \(|\zeta_i|>A_i|w|\). Locally on a compact of that complement with \(w\ne0\), choose \(Y>\max_i(R_i|\zeta_i|)^{-1}\) with \(|w|Y<3/4\); the strict slopes allow this. Choose the positive input epsilon so \(\varepsilon Y<1/8\). Thus the literal mixed pole/log majorants IK.18–IK.21 have \(r=C_-|w|<1/4\), \(q=|w|Y<3/4\), \(x+q=\varepsilon Y+|w|Y<7/8\). At \(w=0\) with spatial differences nonzero, use a fixed finite local \(Y\) and the removable extension proved in IK.5. A compact set has finitely many such neighborhoods; the minimum of their epsilon choices works for its finite family. These choices concern whole coefficients and all spatial Taylor indices. They do not assign a separate domain to any degree.

Carry these coefficient sections to the proved source-defined geometric ring by IG and IM. Choose a finite phase grid with \(h<2\lambda\) and \(2\lambda+h<\pi\), and \(B_i\) large enough for the preceding fixed spatial slopes. At each phase and each neighbor take its separate convex cone. Choose once \(\lambda<\lambda'<(\pi-h)/2\) and \(B'_i>B_i/\cos(\lambda+h/2)\). Literally map support to the larger standard \(T'\), use its IG.7–IG.8 cube and exhaustions, and then map any boundary forward to the final convex \(G'\) containing \(T'\). Reserve the positive overlap width \(\eta_*\) with \(\lambda'+h/2+\eta_*<\pi/2\), and restrict to the smaller admissible covariable product of IM.2. This latter restriction does not replace the larger Taylor radii \(R_i\) on which the already constructed coefficients and the preceding kernel bounds remain holomorphic. Choose the one common compact proper collar of IM.2 for this finite phase family and the fixed smaller endpoint domain; it is uniform across phases, while the pointed cones remain phase-local. All complement-cube terms and trace signs are literal premises; no support-forgetting map is presumed injective.

Equality here is first the actual coefficient equality (MF.9), and the proved injective geometric identification maps it to the actual ring equality on these common final supports. This route retains the full mixed pole/log canonical product. The signs are the proved full-cone Čech signs, with ordinary top cube coordinate \(r_N(-1)^N f\); the \(N=1\) whole-domain exception remains. No new cochain trace, infinite ordinary current limit, or arbitrary distribution completion is introduced by the coefficient division.

The constructed relation coefficients need not be canonical under a coordinate change. Their existence and the kernel equality are intrinsic: the proved nonlinear coordinate-overlap ring maps transport the finite identity and its finite \(A\)-relation rows on a common overlap. At the zero section MF.6 supplies the actual polynomial version rather than extending a punctured-chart quotient across a singular face.

The theorem here is the finite-matrix relation criterion and its right-flatness consequence for the actual full geometric \(E^\infty/E\), together with the actual \(D^\infty/D\) zero-section right-flatness just proved. It does not establish proper-ideal detection, extension-faithfulness and arbitrary-module unit injectivity, flatness of \(E^R/E\), the special fibration-dependent class, mixed \(E^\infty/D^\infty\) scalar change, infinite Späth division or balanced reconstruction, moving-sector propagation, purity, singular-inclusive separation, finite D-type/C1, half-order/contact transport, the initial analytic generator or arbitrary analytic proper regularity. Section5.90, GG.A–GG.G supplies the initial generator near compact sets. The arbitrary analytic proper-image theorem requires its further argument. Finite module-derived instantiation has its separate geometric proof in Section 5.46; the flatness proof here neither uses it as a premise nor proves its other geometric statements.

#### MF.9. Decisive calibrations

For \(D=(x,\partial_x)^{\mathsf t}\), the actual row \(G=(\partial_x,-x)\) has \(GD=1\). Then \(P=1-DG\) has \(PD=0\), and every \(bD=0\) satisfies \(b=bP\). The commutator is \(\partial_xx-x\partial_x=1\). This checks the matrix sides in MF.9; it is a split calibration, not a proof for arbitrary relations.

A full positive/negative calibration has \(u=(x,y)\), covariables \((\tau,\xi)\), and
\[
\begin{gathered}
C(\tau)=\sum_{j\ge0}\frac{\tau^j}{(j!)^2}\\
+\sum_{n\ge1}n!\tau^{-n},\\
R=(-y,x),\\
b=C\circ R=\\
(-yC,\ xC+\partial_\tau C).
\end{gathered}
\tag{MF.10}
\]
The positive estimate holds for every epsilon, since \((\varepsilon^{-j}/j!)\) is bounded; the negative coefficients have factorial constant one. The negative symbolic tail is not claimed to converge as an ordinary Laurent function. For \(D=(x,y)^{\mathsf t}\), \(RD=0\), and
\[
\begin{gathered}
b_1\circ x=-xyC-y\partial_\tau C,\\
b_2\circ y=xyC+y\partial_\tau C,\\
bD=0.
\end{gathered}
\tag{MF.11}
\]
Only the indicated first derivatives occur because the right factors are linear. Thus the full actual relation includes a positive-head and negative-tail derivative correction with the specified sign. Dropping that correction gives a false ordinary-product calculation. The coefficient identities and bounds are exact, not a truncation argument for flatness.

Explicitly the positive prefactor for \(C\) may be \(e^{1/\varepsilon}\), because each term \(\varepsilon^{-j}/j!\) is at most its exponential sum. The positive coefficient of \(\partial_\tau C\) is \(1/(j!(j+1)!)\), with the same bound. Its coefficient at degree \(-n\) is \(-(n-1)(n-1)!\) for \(n\ge2\), and zero for \(n=1\), bounded in absolute value by \(n!\). On a compact base product \(|x|\le X,|y|\le Y\), the negative constants for \(b_1,b_2\) can be \(Y,X+1\), with factorial constant one. Every index in this mixed calibration is therefore in the complete actual class.

![One loss parameter controls both growth regimes](assets/infinite-matrix-flatness-loss-and-relations.png)

**Figure MF.1.** The exponent plane has input degree \(m=K-n\) for negative output \(-n\), with the exact division line \(K=n\). The finite negative-input region uses MFB.9, and the infinite positive-input region uses MFB.10. The positive-output formula is MFB.8 with input \(m=j+K\). The middle panel shows the exact coefficient splitting and actual relation formula MF.6–MF.9, with row multiplication on the right. The right panel is the exact two-variable calibration MF.10–MF.11, retaining \(\partial_\tau C\). Coordinates and all displayed constants are in the reproducible figure source and data. These are proof mechanisms; no plotted finite cutoff substitutes for a bound. Free human target context is Micro-hyperbolic systems §8 p.43; the proofs are MFA.1–MFA.4, MFB.1–MFB.5 and MF.1–MF.8 above.

Editable SVG, exact data, and reproducible plotting source.

![Fixed coefficient domains and derivative-margin allocation](assets/infinite-matrix-flatness-common-domain.png)

**Figure MF.2.** Exact real slice of the coefficient radius family \(R_u(r)=r^{3/2}\), \(R_z(r)=r\), suitable for the illustrative principal relation \((-z,u)\). The normalized sample radii are \(r/r_0=1,7/8,3/4,5/8,1/2\), corresponding to \(L=4\) equal scalar margins \(r_0/8\). The full domains are complex polydiscs; this is not a geometric support cone. For the general matrix the finite coefficient heads determine its own weights \(v_a\), and the derivative-margin lower bound uses their fixed minima on the same compact radius interval. Bounded coefficient divisions consume no radius margin, including when inserted between derivatives. MFB.4 reserves one additional fixed interval for the whole zero-remainder iteration. Proof: MFA.1–MFA.3 and MFB.2/MFB.4. Coordinates and the distinction between this explicit sample and the arbitrary-matrix proof are retained in the source and data.

Editable SVG, exact data, and reproducible plotting source.



### 5.48. Proper ideals and faithful extension to the full infinite-order ring

We prove that every proper finite-order left ideal remains proper in the full geometric infinite-order ring, including the genuine ordinary zero section. Combined with the right-flatness theorem of [§5.47, proof sections MF.1–MF.7](#5-47-arbitrary-finite-matrices-and-right-flatness-of-the-full-infinite-order-ring), this gives unit injectivity and faithful exact, conservative scalar extension for every algebraic left module and every sheaf of left modules. The precise tensor-cohomology and support consequences are included below.

The complete analytic coefficient and convergence arguments are given here as proof sections FEA.1–FEA.4 and FEB.1–FEB.5 before FEP.1–FEP.10. Earlier actual finite division and strictness are [§5.21](#5-21-actual-simultaneous-division-and-finite-stalk-relations) and [§5.24.2](#5-24-2-an-additional-consequence-of-the-actual-division). The full coefficient and kernel bounds are [§5.44, proof sections IK.1–IK.5](#5-44-full-homogeneous-coefficients-and-ordinary-infinite-order-kernels), and the source-defined ring, mixed product, coordinate maps and final-support witnesses are [§5.45, proof sections IG.1–IG.5 and IM.1–IM.6](#5-45-the-full-geometric-infinite-order-kernel-ring-and-its-canonical-product). Section 5.49 proves full infinite Späth sum division and balanced reconstruction with the distinct special fibration source class. Other scalar changes, propagation and the broader programme require separate proofs.

The complete coefficient argument below is used for proper-ideal detection without invoking right flatness. Every coefficient operator is bounded complex linear on the stated common domain. The later unit-injectivity consequences use the right-flatness theorem of [§5.47, proof section MF.7](#5-47-arbitrary-finite-matrices-and-right-flatness-of-the-full-infinite-order-ring).

**Analytic coefficient division with uniform radii.**

This is a local proof used by the matrix argument. All coefficient operators below are complex linear. They need not be linear over the coefficient ring or over an operator ring. That distinction is essential for a nonprojective relation module.

#### FEA.1. Norms, initial exponents and finite heads

Let \(w=(w_1,\ldots,w_q)\) be finitely many complex variables centered at the point. For polyradius \(R\), put
\[
\begin{gathered}
|f|_R=\sum_\alpha |f_\alpha|R^\alpha,\\
|v|_{R,c}=\sum_i c_i|v_i|_R.
\end{gathered}
\tag{FEA.1}
\]
The first norm is finite on every strictly smaller polydisc than a polydisc of holomorphy: the iterated Cauchy bound gives \( |f|_R\le M\prod_a(1-R_a/S_a)^{-1}\) for \(R<S\). Coefficient convolution proves submultiplicativity. These coefficient spaces are weighted \(\ell^1\) Banach spaces. A restriction to any set of Taylor exponents has norm at most one.

For a submodule \(I\subset\mathcal O_{w,0}^{b}\), order a nonzero vector's Taylor exponents by total degree, then lexicographic order, then component. Its initial exponent is translated by multiplication by a Taylor monomial. Thus its initial diagram is upward closed in each component of \(\mathbf N^q\). An upward closed subset has finitely many minimal elements: every infinite sequence in \(\mathbf N^q\) has an earlier term componentwise at most a later term. Induct on \(q\), taking a subsequence with constant or increasing first coordinate, and then apply induction to the remaining coordinates. An infinite set of incomparable minima is impossible. Each exponent dominates a minimum, by minimizing in the finite box below it.

Choose finitely many actual \(g_i\in I\) realizing the minima \((\alpha_i,j_i)\), with leading coefficient one. Their initial orthants cover the diagram. Choose positive rational weights \(v_a\) close enough to one that every different Taylor exponent in each \(g_i\) has weight strictly larger than its head. Here is the finite justification. If \(L=\max_i|\alpha_i|\), a small hierarchical perturbation \(v_a=1+\delta^a\) realizes lexicographic comparison among exponents of degree at most \(L\). Make it smaller still so \((L+1)\min v_a>L\max v_a\). All higher total degrees then have larger weight than every head. At the same exponent, other nonzero components occur only after \(j_i\). Put \(c_j=\kappa^j\) and choose \(\kappa>0\) so that the weighted sum of those finitely many same-exponent coefficients is less than \(c_{j_i}/16\).

Set \(R_a(r)=r^{v_a}\). For small enough \(r_0>0\), all divisors are holomorphic on a larger polydisc, and, simultaneously for every \(r\in[r_0/2,r_0]\),
\[
\begin{gathered}
\max_i\frac{|g_i-w^{\alpha_i}e_{j_i}|_{R(r),c}}{c_{j_i}R(r)^{\alpha_i}}\\
<\theta<\tfrac12.
\end{gathered}
\tag{FEA.2}
\]
Indeed every different exponent has a positive weight gap. Its divided radius contribution decreases as \(r\) decreases. Dominate their sum by the absolute Taylor series at one fixed larger radius and apply dominated convergence as \(r_0\to0\). The already fixed same-exponent component tolerance supplies the remaining bound. This argument controls the entire analytic series, not a finite Taylor approximation.

#### FEA.2. A literal bounded splitting

Order the heads, assign each exponent in their orthants to the first containing head, and write \(\Delta_i\) for those disjoint assigned sets and \(\Delta_0\) for their complement. Let \(V_i\) be the coefficient-support space obtained by taking component \(j_i\), restricting to \(\Delta_i\), and dividing by \(w^{\alpha_i}\). Give \(V=\bigoplus_iV_i\) the norm
\[
 |a|_V=\sum_i c_{j_i}R^{\alpha_i}|a_i|_R.              \tag{FEA.3}
\]
Let \(d\) be the restriction-and-division operator, \(T_h a=\sum_i a_iw^{\alpha_i}e_{j_i}\), and \(T_ga=\sum_i a_ig_i\). Disjoint supports give \( |df|_V\le |f|_{R,c}\), \(dT_h=1_V\), and \(\ker d\) is exactly the coefficient-support space \(\Delta_0\). Submultiplicativity and equation \(FEA.2\) give \(\|d(T_g-T_h)\|\le\theta\). Hence
\[
\begin{gathered}
h=\\
(1+d(T_g-T_h))^{-1}d,\\
p=1-T_gh.
\end{gathered}
\tag{FEA.4}
\]
The inverse is its absolutely convergent Neumann series on the stated Banach spaces. The identities \(hT_g|_V=1_V\), \(hp=0\), and \(dp=0\) follow by multiplying equation \(FEA.4\); in particular \(p^2=p\). The input, every quotient, and the remainder have one common actual polydisc.

For \(f\in I\), its remainder \(pf=f-\sum_i(hf)_ig_i\) belongs to \(I\): this uses a finite sum of actual holomorphic coefficient multiples, and does not require closure of \(I\). If nonzero, its initial exponent belongs to the diagram of \(I\), whereas all its exponents lie in \(\Delta_0\). This contradiction gives \(pf=0\). Thus \(\operatorname{im}T_g=I\), \(\ker p=I\), and these finitely many heads generate \(I\). This also proves the analytic coefficient Noetherianity used here.

Viewed in ordinary vector coefficient norms, \(h,p,T_g\) are bounded uniformly for all \(r\in[r_0/2,r_0]\). The head division constants are bounded by the fixed minima \(R(r_0/2)^{-\alpha_i}\); the inverse norm is at most \((1-\theta)^{-1}\); and the finite divisor norms are bounded on the largest radius. They are the same operators at every radius. The coefficient Neumann identities and uniqueness of the support-normalized quotient prove this assertion, independently of the numerical norm used to establish convergence.

#### FEA.3. The image operator and zero detection

Suppose a holomorphic matrix \(W_0:\mathcal O^r\to\mathcal O^s\) has coefficient kernel \(\mu\). Apply proof sections FEA.1–FEA.2 to \(\mu\), obtaining \(T_g,h,p\). Apply them also to \(\operatorname{im}W_0\). Each of that image's finitely many standard divisors is a finite holomorphic combination of the rows of \(W_0\). Compose its quotient map with those finite combinations to obtain a bounded coefficient operator \(s:\mathcal O^s\to\mathcal O^r\), with \(W_0s f=f\) for \(f\in\operatorname{im}W_0\). Put \(k=ps\). Then
\[
                     kW_0=p.                        \tag{FEA.5}
\]
For \(v-sW_0v\in\mu=\ker p\), applying \(p\) proves equation \(FEA.5\). All three coefficient operators \(h,p,k\) are bounded on the same radius interval, after taking one smaller \(r_0\) for their finite data. This is a coefficient splitting of the image and its relations. No splitting over \(\mathcal O\), \(A\), or \(B\) has been asserted.

#### FEA.4. Homogeneous polynomial coefficients at the zero section

For ordinary operators, use base variables \(u\) and polynomial covariables \(\rho\), retaining the integer shifts of each free basis vector. Every component of a homogeneous vector has one prescribed covariable degree. The initial diagram is upward closed under base Taylor and covariable polynomial monomials; the same Dickson proof supplies finitely many homogeneous standard divisors. The radius argument above applies to their Taylor series in \(u\) and their finite polynomials in \(\rho\).

The standard divisors can be chosen homogeneous: take the homogeneous summand containing a selected initial monomial of a vector in the graded submodule. It has the same head, since extracting that summand only removes monomials. When a homogeneous image divisor is expressed in the rows of a graded matrix, project each coefficient to the homogeneous degree appropriate for that row; the homogeneous part of the equality remains the whole divisor. Thus the finite image combinations used in proof section FEA.3 preserve the shifts too.

Every support projection, head division and full-divisor multiplication preserves total covariable degree with the indicated free-basis shifts. The Neumann series in equation \(FEA.4\) therefore stays in that graded component. For a fixed component degree, only finitely many covariable exponents can occur, so its limit is a polynomial in \(\rho\) whose base coefficients are holomorphic on the common domain. Quotients with negative prescribed polynomial degree are zero. The same norm bounds hold uniformly in the total degree: their proof uses fixed head monomials, fixed divisor norms, and the same \(\ell^1\) radius norms, with fixed component radius factors to accommodate the finite shifts. For arbitrary holomorphic base input, one repeats the radius choice on its intersection with the finite divisor domain. This proves the precise polynomial-preserving version needed at the zero section.

Every mathematical result in this coefficient argument has been proved above. The complete earlier [§5.21.1–§5.21.4](#5-21-actual-simultaneous-division-and-finite-stalk-relations) gives the corresponding actual finite-order operator division, used in proof sections FEP.2–FEP.3 below.

The complete coefficient argument below is used for proper-ideal detection without invoking right flatness. Every coefficient operator is bounded complex linear on the stated common domain. The later unit-injectivity consequences use the right-flatness theorem of [§5.47, proof section MF.7](#5-47-arbitrary-finite-matrices-and-right-flatness-of-the-full-infinite-order-ring).

**Full infinite-order order-lowering estimate.**

This appendix proves convergence of the series used for arbitrary matrix relations. It includes every normalized spatial derivative, arbitrary many correction steps, the unbounded positive head, the factorial negative tail, and a single domain chosen before the summation indices.

#### FEB.1. Operators and one domain

In a cotangent chart \(\tau\ne0\), write \(F_j=\tau^jp_j(u,z)\), \(z=\xi/\tau\). The actual class has one holomorphic coefficient product, with, on each compact,
\[
\begin{gathered}
|p_j|\le M_\varepsilon\varepsilon^j/j!\\
(j\ge0, \varepsilon>0),\\
|p_{-n}|\le MD^nn! (n\ge1).
\end{gathered}
\tag{FEB.1}
\]
We use the absolute Taylor norms of equation \(FEA.1\) on one family \(R(r)\), \(r\in[r_0/2,r_0]\). A Cauchy estimate on a fixed enlargement converts the input compact bounds to these norms with constants independent of \(j\). Coefficient operators \(h,p,k\), or a finite composition of these, have uniform norm \(H\) on that interval by proof sections FEA.2–FEA.3. Every right-hand actual matrix is order zero, with coefficients \(Q_{-q}\) satisfying \( |Q_{-q}|\le AC^q q!\) on a fixed enlargement, including \(q=0\). The finite list has one such bound and domain.

The right Leibniz term acting on a left coefficient has indices
\[
\begin{gathered}
\alpha\in\mathbf N^N,\quad q\ge0,\\
K=q+|\alpha|,\\
\frac1{\alpha!}(\partial_\rho^\alpha F_j)(\partial_u^\alpha Q_{-q}).
\end{gathered}
\tag{FEB.2}
\]
It lowers homogeneous degree by \(K\). Subtracting its principal ordinary multiplication removes precisely \(q=0,\alpha=0\); every remaining term lowers degree by at least one. The coefficient operators from FEA.1–FEA.4 preserve degree. A word of correction terms interspersed with such coefficient operators therefore has total loss \(K\ge\) its number of correction terms. All matrix and component sums are finite.

Choose the same fixed outer margin for the right factors throughout. Its Cauchy estimate costs \(\alpha!\delta^{-|\alpha|}\); the factorial in equation \(FEB.2\) cancels it. If a total-order version is used, \(|\alpha|!/\alpha!\le N^{|\alpha|}\), so the difference is absorbed in an exponential constant. Across a word with tail indices \(q_i\), let \(Q=\sum_iq_i\) and \(L=\sum_i|\alpha_i|\), thus \(K=Q+L\). Their tail factorial product is at most \(Q!\).

#### FEB.2. Derivatives interspersed with coefficient divisions

At normalized \(\tau=1\), one covariable differentiation is exactly
\[
\begin{gathered}
\partial_{\xi_a}F_j=\tau^{j-1}\partial_{z_a}p_j,\\
\partial_\tau F_j=\tau^{j-1}(j-E_z)p_j,\\
E_z=\sum_a z_a\partial_{z_a}.
\end{gathered}
\tag{FEB.3}
\]
These equations retain every spatial index. They also specify all signs. At an intermediate step starting from input degree \(m\), the current degree has absolute value at most \(|m|+K\). A coefficient operator preserves that degree, although it need not commute with \(E_z\) or differentiation.

To bound a word, insert intermediate radii decreasing from \(r_0\) to \(r_0/2\), allocating equal scalar radius loss \(r_0/(2L)\) to each of its \(L\) derivatives; when \(L=0\) no derivative margin is consumed. All \(R_a(r)=r^{v_a}\) have a fixed positive lower derivative on this compact interval, so each coordinate margin is at least \(c/L\), with fixed \(c>0\). Cauchy's formula, the bounded normalized \(z_a\), and equation \(FEB.3\) bound each one-derivative step by
\[
\begin{gathered}
C_0(|m|+K+L)\\
\le 2C_0(|m|+K)\quad\\
(K>0).
\end{gathered}
\tag{FEB.4}
\]
The \(H\) bound for every inserted coefficient operator holds at its own intermediate radius. Thus no commutation of a division operator with a derivative has been used, and no new radius is lost for a division operator. The count of those operators is bounded by a fixed multiple of the number of correction factors, hence by a fixed multiple of \(K\).

For \(M=|m|\ge0\), \(K=Q+L>0\),
\[
\begin{gathered}
(M+K)^LQ!\\
\le e^K\frac{(M+K)!}{M!}.
\end{gathered}
\tag{FEB.5}
\]
To prove it, \(Q!\le(M+K)^Q\). Also
\(\prod_{a=1}^K(M+a)/(M+K)\ge K!/K^K\ge e^{-K}\): each factor increases with \(M\), and the last inequality follows from
\(\sum_{a=1}^K\log a\ge\int_1^K\log t\,dt=K\log K-K+1\).
Equations \(FEB.4\)–\(FEB.5\), submultiplicativity and the right-factor Cauchy estimates bound each word, before coefficient summation, by an exponential constant times that one factorial ratio.

There are at most exponentially many words of total loss \(K\). Ordered compositions of a positive integer \(K\) number \(2^{K-1}\); their subdivisions into \(q_i\) and the \(N\) derivative components number at most \((N+2)^{2K}\), a deliberately loose bound obtained by placing separators among \(K\) units and coloring the units. Finite matrix entries and coefficient-operator choices cost another exponential factor. Absorb all these constants, all fixed input bounds, and \(H\) into \(C_1\ge1\). The sum of the absolute norms of **all** terms of loss \(K\), for any one of the Neumann or iterated correction expressions below, is bounded by
\[
                  C_1^K\frac{(|m|+K)!}{|m|!}|p_m|_{R(r_0)}.
                                                               \tag{FEB.6}
\]
The case \(K=0\) is the single bounded coefficient operation and is covered by a fixed prefactor. This argument also applies to truncated words uniformly in the truncation; the bound includes each contraction and every tail coefficient rather than replacing them by a formal product.

#### FEB.3. Positive output, negative output and explicit tails

For positive output \(j\ge0\), input degree is \(m=j+K\ge0\). The integer inequality
\[
\begin{gathered}
\frac{(j+2K)!j!}{(j+K)!^2}\le\binom{2K}{K}\\
\le4^K
\end{gathered}
\tag{FEB.7}
\]
follows by writing the first ratio as \(\prod_{a=1}^K(j+K+a)/(j+a)\), whose factors decrease with \(j\), and then using the binomial expansion. From equation \(FEB.1\) and equation \(FEB.6\),
\[
\begin{gathered}
|\text{output}_j|\le\\
C_2M_\varepsilon\frac{\varepsilon^j}{j!}\\
\sum_{K\ge0}(4C_1\varepsilon)^K.
\end{gathered}
\tag{FEB.8}
\]
Given any desired \(\eta>0\), choose \(0<\varepsilon\le\eta\) with \(9C_1\varepsilon<1/2\). The coefficient domain and \(C_1\) do not depend on this choice. This proves the positive estimate for **every** \(\eta\). Terms with \(K\ge R\) are bounded by the right side of equation \(FEB.8\) times \((4C_1\varepsilon)^R\), with the geometric denominator retained.

For negative output \(-n\), the input is \(m=K-n\). When \(K<n\), equation \(FEB.6\) gives
\[
\begin{gathered}
C_1^K\frac{n!}{(n-K)!}\\
M D^{n-K}(n-K)!\\
=M n!C_1^KD^{n-K}.
\end{gathered}
\tag{FEB.9}
\]
This part is finite; its sum is at most \(Mn!(D+C_1)^n\), since the binomial coefficients dominate the corresponding \(n\) summands. For \(K\ge n\), write \(m=K-n\ge0\). The multinomial expansion proves
\((n+2m)!/(n!m!m!)\le3^{n+2m}\). Consequently the full positive-input part is bounded by
\[
\begin{gathered}
C_2M_\varepsilon n!(3C_1)^n\\
\sum_{m\ge0}(9C_1\varepsilon)^m.
\end{gathered}
\tag{FEB.10}
\]
Choose one fixed \(\varepsilon<1/(18C_1)\). Equations \(FEB.9\)–\(FEB.10\) prove one factorial negative constant for every \(n\), on the same domain. The tail \(K\ge R>n\) in equation \(FEB.10\) is bounded by the displayed prefactor times
\((9C_1\varepsilon)^{R-n}/(1-9C_1\varepsilon)\).

These bounds prove absolute normal convergence of the whole expressions, including infinitely many terms at a fixed output degree. They justify rearrangement, finite matrix sums and multiplication identities by the triangle inequality and absolute scalar convergence. On a fixed further inner product, the Cauchy formula applies to each chosen finite output derivative. Its constant is independent of all cutoffs and degrees. Applying it to the displayed tails proves normal convergence of every fixed base and normalized-spatial derivative. All spatial Taylor exponents remain in the norm; none is truncated or bounded in advance.

#### FEB.4. Actual inverses and vanishing iteration

Let \(L\) be one of the above order-lowering correction operators. Define
\[
\begin{gathered}
(1+L)^{-1}f=\\
\sum_{a\ge0}(-L)^a f.
\end{gathered}
\tag{FEB.11}
\]
Equations \(FEB.6\)–\(FEB.10\) prove that this expression is an actual coefficient sequence in the complete class, on one domain. Reserve an intermediate radius, for example \(3r_0/4\): the same proof first gives normal convergence there. Apply one further correction on the inner radius \(r_0/2\). The Cauchy formula permits differentiation of the convergent series, and the direct word bound in equation (FEB.6), with the additional correction included, permits its reordering there. The finite geometric identity gives left and right inverses, because its remainder \(L^{a+1}f\) has total loss at least \(a+1\) and tends to zero coefficientwise with the explicit tails above. This is a limit of normally convergent holomorphic coefficient functions. It is not an ordinary supported-distribution limit, a formal completion, or a limit in a cohomology quotient.

In particular, if an actual sequence satisfies \(s=-Ls\), then \(s=(-L)^a s\) for every finite \(a\). If \(s\) is the output of a previous division, select one new, fixed smaller outer radius inside its holomorphic domain and the same coefficient-divisor domain. Proof section FEA.2 makes the identical coefficient operators bounded on its radius interval; the support-normalized analytic operator is unchanged. This is a single additional shrinking made before any iteration index, not shrinking at each iteration. For positive output use equation \(FEB.8\) with \(K\ge a\). For fixed negative output \(-n\), take \(a>n\) and use equation \(FEB.10\) with \(m\ge a-n\). Both tend to zero, so every actual coefficient of \(s\) vanishes on the same inner product. Hence \(s=0\). This is the zero-remainder detection needed for full matrix relations with no highest positive homogeneous coefficient.

#### FEB.5. Ordinary zero-section version

In the ordinary polynomial coefficient spaces of proof section FEA.4, the covariable derivative in equation \(FEB.2\) is an ordinary derivative of a homogeneous polynomial. Its Cauchy bound on the same radius interval is at most \(C_0L\) per derivative; including it in equation \(FEB.4\) gives the same bound. Every finite-order right matrix has only finitely many tail offsets \(q\) relative to its principal shifted degree. A homogeneous space of negative polynomial degree is zero. All intermediate coefficient operations preserve the prescribed shifted polynomial degrees by proof section FEA.4, and the finite ordinary Leibniz product preserves the corresponding graded spaces.

Thus equations \(FEB.6\)–\(FEB.8\) apply with finite free-basis shifts. A fixed shift changes the positive factorial bound by a fixed polynomial in the degree; this is absorbed by choosing the input \(\varepsilon\) smaller than the requested output value, since \(j^a\theta^j\) is bounded for every fixed \(a\) and \(0<\theta<1\). The finitely many degrees below the shifts satisfy the same convergent tail bounds individually. Every output component remains a homogeneous polynomial of its actual nonnegative degree. This proves equation \(FEB.11\) and zero detection in the genuine ordinary \(D^\infty\) class at the zero section; it does not substitute an ordinary ring for the microlocal argument FEB.1–FEB.4.

**Full proper-ideal detection and faithful right scalar extension.**

The proper-left-ideal proof in sections FEP.1–FEP.6 uses convergent coefficient quotients without assuming flatness. The complete right-flatness theorem in [§5.47, proof sections MF.1–MF.7](#5-47-arbitrary-finite-matrices-and-right-flatness-of-the-full-infinite-order-ring) is used in FEP.8–FEP.9 to prove unit injectivity and faithful exact extension for all algebraic left modules and sheaves.

The source-defined geometric ring and its full coefficient/product identification are proved in [§5.45, IG.1–IG.5 and IM.1–IM.6](#5-45-the-full-geometric-infinite-order-kernel-ring-and-its-canonical-product). The complete coefficient and convergence proofs FEA/FEB are supplied above, without using matrix flatness for proper-ideal detection. The earlier actual finite operator division and strictness proofs are [§5.21](#5-21-actual-simultaneous-division-and-finite-stalk-relations) and [§5.24.2](#5-24-2-an-additional-consequence-of-the-actual-division).

Free human context is Kashiwara–Schapira, [Micro-hyperbolic systems](https://perso.imj-prg.fr/pierre-schapira/wp-content/uploads/schapira-pub/Microhyp.pdf), §1.3 printed/PDF p.6 and the complete §8, printed/PDF pp.42–46, including its flatness recall on p.43. Page 6 specifies the complex-projectivized ring, input-form twist and ordinary zero-section restrictions; page 43 recalls flatness before the noncharacteristic theorem. Neither page supplies the proper-ideal argument or unit injectivity used here. No recalled external symbol, flatness, separation or reconstruction theorem is imported.

#### FEP.1. Exact target, sides and two coefficient regimes

Let \(X\) be any complex manifold of finite dimension, with the genuine source-defined rings in the proved geometric construction. At any cotangent stalk write \(A=\mathcal E_p\), \(B=\mathcal E^\infty_p\), with its proved unital geometric ring map. Modules are left \(A\)-modules and \(B\) is a right \(A\)-module. For every proper left ideal \(I\subset A\), the ideal extension is the algebraic left \(B\)-ideal
\[
\begin{gathered}
BI=\left\{\begin{gathered}
\sum_{\nu=1}^l b_\nu a_\nu:\\
l<\infty,\\
b_\nu\in B,\\
a_\nu\in I
\end{gathered}\right\}.\\
\text{We prove }1\notin BI,\\
B/BI\ne0.
\end{gathered}
\tag{FEP.1}
\]
The product is the full actual operator product. No closure of this ideal and no completed tensor product is used.

Off the zero section choose \(\tau\ne0\), normalized \(z=\xi/\tau\), base variables \(u\), and translate the normalized point to \(w=(u,z)=0\) by the exact finite coordinate change. Every symbol has coefficients on one holomorphic product, with
\[
\begin{gathered}
P=\sum_{j\in\mathbf Z}\tau^jp_j(u,z),\\
|p_{-n}|_K\le MC^nn!,\\
|p_j|_K\le M_\varepsilon\varepsilon^j/j!\\
(j\ge0,\ \varepsilon>0).
\end{gathered}
\tag{FEP.2}
\]
Every compact \(K\) has its constants, and the domain is independent of \(j,\varepsilon\). This is the complete geometric ring proved in [§5.45](#5-45-the-full-geometric-infinite-order-kernel-ring-and-its-canonical-product), including all spatial Taylor indices and the entire mixed pole/log product. At the zero section \(A=D_u\), \(B=D^\infty_u\), and \(P_j(u,\rho)\) is a homogeneous polynomial of degree \(j\ge0\) with the corresponding every-epsilon bound. A normalized \(p_j(u,z)\) then has degree at most \(j\), not necessarily degree exactly \(j\).

The ring \(B\) is nonzero: its unit has coefficient \(1\) at degree \(0\), all other coefficients zero, and the proved geometric identification detects this actual nonzero coefficient sequence. If \(I=0\), equation (FEP.1) follows immediately. In ambient dimension \(0\), \(A=B=\mathbf C\) at a point and this is the only proper ideal. We henceforth treat a nonzero proper ideal in positive dimension.

#### FEP.2. Strict finite presentation of an arbitrary proper microlocal ideal

Let \(A_0\) be the actual order-zero factorial ring, \(h=\tau^{-1}\), and \(I_0=I\cap A_0\). The complete earlier [§5.21](#5-21-actual-simultaneous-division-and-finite-stalk-relations) and [§5.24.1–§5.24.2](#5-24-2-an-additional-consequence-of-the-actual-division) prove actual standard generation and strictness for every \(A_0\)-submodule, by its common-domain Banach division and support-normalized Neumann series. This is a complete earlier proof premise, not a completion-flatness claim.

The initial diagram of \(I_0\) is saturated at homogeneous index \(0\). Indeed if an element has first index \(n\ge0\), left multiplication by \(\tau^n\) normalizes it to actual order zero, still in \(I\). The exact equation (5.21i) in [§5.21.2](#5-21-2-initial-exponents-and-the-finite-diagram) has only finitely many base derivatives and preserves the factorial class on one common smaller domain. Its principal Taylor head is unchanged. Thus an initial exponent \((n,\alpha)\) dominates the realized exponent \((0,\alpha)\), and every minimal exponent has \(n=0\).

Select the finite actual standard rows \(W_i\in I_0\) realizing these minima. All are order zero, and their principal holomorphic functions \(g_i\) realize the whole initial diagram of the coefficient ideal
\[
\begin{gathered}
\mu=\left\{\begin{gathered}
\text{principal}\\
\text{ coefficient of}\\
\text{ an order-zero}\\
\text{ element of }I_0
\end{gathered}\right\}\\
\subset\mathcal O_{w,0}.
\end{gathered}
\tag{FEP.3}
\]
This set is an ideal: addition and principal multiplication by a holomorphic order-zero coefficient are ordinary coefficient operations. Its diagram is the index \(0\) slice of the diagram of \(I_0\). Proof sections FEA.1–FEA.2 give its literal bounded analytic standard division. The complete strict finite division proof gives
\[
\begin{gathered}
F\in I_0\cap H^aA_0\quad\Longrightarrow\\
F=\sum_i q_iW_i,\\
q_i\in H^aA_0.
\end{gathered}
\tag{FEP.4}
\]
An arbitrary element of \(I\) has a finite upper order bound; multiplying on the left by a sufficiently large power of \(h\) puts it in \(I_0\). The fixed shift and actual normal automorphism preserve the class. Equation (FEP.4) then implies \(I=AW\), with \(W\) the finite column of its rows. Therefore \(BI=BW\) as algebraic ideals. The equality uses only finite actual expressions, and their common domains are intersected once for each finite witness.

Every \(g_i\) vanishes at \(w=0\). If \(g_i(0)\ne0\), the complete actual elliptic inversion proof [§5.20.3](#5-20-3-actual-elliptic-inverses-and-the-stalk-radical) gives an inverse for \(W_i\) in \(A_0\) on a common domain; since \(W_i\in I\), left multiplying by that inverse would give \(1\in I\). This contradicts properness. Consequently
\[
                       \mu\subset\mathfrak m_{w,0}.   \tag{FEP.5}
\]
In particular every coefficient standard head of \(\mu\) has strictly positive Taylor degree.

#### FEP.3. Coefficient splittings and actual relation lifts

All coefficient operators below are bounded complex linear. Juxtaposition of these operators means composition; actual row multiplication remains explicitly \(bW\) or \(cR\). In particular none of the splittings is assumed \(A\)-linear or \(B\)-linear.

Let \(T_0b=bW_0=\sum_i b_ig_i\) be ordinary principal multiplication. Apply the complete analytic division in proof sections FEA.1–FEA.2 to the \(g_i\), with the assigned disjoint head orthants. It supplies the quotient-support space \(V\subset\mathcal O^l\), division operator \(\mathsf d\), and remainder projection \(\mathsf p\), on one radius interval:
\[
\begin{gathered}
\mathsf d T_0|_V=1,\\
T_0\mathsf d=1-\mathsf p,\\
\mathsf d\mathsf p=0,\\
\ker\mathsf p=\mu,\qquad\mathsf p^2=\mathsf p.
\end{gathered}
\tag{FEP.6}
\]
Their norms are uniform at every intermediate radius. Since the constant monomial is in none of the positive-degree head orthants of equation (FEP.5), the literal support restriction in proof section FEA.2 gives
\[
                  \mathsf d1=0,\qquad\mathsf p1=1.    \tag{FEP.7}
\]
These identities concern the whole analytic coefficient functions, not their values at one point alone.

Set \(\mathsf v=\mathsf d T_0\) on the source. Then \(\mathsf v^2=\mathsf v\), \(\operatorname{im}\mathsf v=V\), and \(\ker\mathsf v=\ker T_0=:\mu_s\). To check the kernel equality, \(T_0b\in\mu\), so \((1-\mathsf p)T_0b=T_0b\), whence \(T_0\mathsf v b=T_0b\). Its vanishing is equivalent to \(\mathsf v b=0\). Thus \(1-\mathsf v\) maps onto \(\mu_s\).

Every principal relation \(r\in\mu_s\) has an actual order-zero relation lift. Regard its holomorphic vector as an actual order-zero symbol. Then \(rW\) has index at least \(1\). It belongs to \(I_0\), so equation (FEP.4) expresses it as \(aW\) with \(a\) of index at least \(1\). The row \(r-a\) is an actual relation of principal vector \(r\). Choose the finitely many analytic standard divisors of \(\mu_s\) from proof section FEA.1 and lift them in this way. They give a finite actual matrix \(R\) with
\[
\begin{gathered}
RW=0,\\
\operatorname{im}R_0=\mu_s.
\end{gathered}
\tag{FEP.8}
\]
All its negative tails are actual factorial tails on one domain. No scalar-extension flatness has entered this lifting.

Proof section FEA.2 applied to \(R_0\) supplies a bounded coefficient quotient \(\mathsf d_R\), with values in its quotient-support space \(V_R\), such that
\[
 R_0\mathsf d_R f=f\ (f\in\mu_s),\qquad
 \mathsf d_R R_0|_{V_R}=1.
\]
Here \(R_0\) denotes the map \(c\mapsto cR_0\). Define
\[
\begin{gathered}
\mathsf q=\mathsf d_R(1-\mathsf v).\\
R_0\mathsf q=1-\mathsf v,\\
\mathsf q R_0|_{V_R}=1,\\
\ker\mathsf q=V.
\end{gathered}
\tag{FEP.9}
\]
The first equality follows since \(1-\mathsf v\) has image \(\mu_s\). The second uses \(\mathsf vR_0=0\). For the third, \(\mathsf q b=0\) implies \((1-\mathsf v)b=R_0\mathsf q b=0\); conversely it vanishes on \(V\). Thus the source splitting is chosen to use precisely the image-quotient complement \(V\). An unrelated coefficient complement would not prove the following annihilation identity.

#### FEP.4. Actual infinite quotient and exact annihilation of the ideal

Let \(T b=bW\) be actual multiplication, \(E=T-T_0\), \(T_Rc=cR\), and \(E_R=T_R-R_0\). Both \(E,E_R\) lower homogeneous order by at least \(1\). They consist of the complete right-symbol tails and contractions, not finite-order truncations.

The complete convergence proof in sections FEB.1–FEB.5 proves that any word of these corrections interspersed with the bounded coefficient operators of equations (FEP.6)–(FEP.9), with total loss \(K\) and input degree \(m\), has its complete absolute sum bounded on one inner coefficient product by
\[
                 C_1^K\frac{(|m|+K)!}{|m|!}|f_m|.     \tag{FEP.10}
\]
The constant depends on the finite operators and fixed margins, not \(K,m\). For positive output \(j\), this is bounded by a prefactor times \(\varepsilon^j/j!\sum_K(4C_1\varepsilon)^K\), using the input degree \(m=j+K\). For negative output \(-n\), its \(K<n\) part is bounded by \(Mn!\sum_{K<n}C_1^KD^{n-K}\); its \(K\ge n\) part is bounded by \(M_\varepsilon n!(3C_1)^n\sum_{m\ge0}(9C_1\varepsilon)^m\).
These are the full bounds in equations (FEB.8)–(FEB.10). One can choose epsilon arbitrarily small on the same domain. Thus every inverse below is its actual absolutely normally convergent Neumann series, with every positive-epsilon bound, one factorial negative bound, every spatial index and all fixed derivative tails. Proof section FEB.4 reserves an intermediate radius for composition and one additional fixed interval if needed, before any iteration index. For all the finitely many composites written in this section, one may reserve in advance twelve scalar-radius subintervals between \(r_0\) and \(r_0/2\). All coefficient operators are uniformly bounded on that whole interval; each derivative word uses the fixed margin of its assigned subinterval. Taking the largest of their finite constants gives the same equation (FEP.10) form. This accounts for source inversion, its actual product, target inversion and the finite identity comparisons on one final common product. There is no shrinking at each iteration or at each homogeneous degree.

Define the target operator
\[
                    \mathcal P=\mathsf p(1+E\mathsf d)^{-1}.
                                                               \tag{FEP.11}
\]
Since \(\mathsf d\mathsf p=0\), the series fixes the image of \(\mathsf p\), so \(\mathcal P\mathsf p=\mathsf p\). It also satisfies \(\mathcal P(1+E\mathsf d)=\mathsf p\). From \(T_0\mathsf d=1-\mathsf p\) we obtain the literal identity
\[
\begin{gathered}
\mathcal P T\mathsf d\\
=\mathcal P(1-\mathsf p)+\mathcal P E\mathsf d\\
=\mathcal P-\mathsf p+\mathsf p-\mathcal P=0.
\end{gathered}
\tag{FEP.12}
\]
Every reordering is justified by equation (FEP.10), including all contractions and every principal relation coefficient.

Equation (FEP.12) alone annihilates multiplication on \(V\), not on the entire free source. To handle an arbitrary actual \(b\in B^l\), use the lifted relations:
\[
\begin{gathered}
c=(1+\mathsf q E_R)^{-1}\mathsf q b,\\
v=b-T_Rc.
\end{gathered}
\tag{FEP.13}
\]
The series converges in the full actual class by equation (FEP.10). Its quotient values remain in \(V_R\). Equation (FEP.9) gives \(\mathsf qv=\mathsf qb-(1+\mathsf qE_R)c=0\), hence \(v=\mathsf vv\in V\), coefficientwise on the same domain. The actual equation \(RW=0\) and associativity give \(Tb=Tv\). Put \(f=T_0v\); then \(v=\mathsf df\). Therefore equation (FEP.12) proves
\[
\begin{gathered}
\mathcal P Tb=0\\
\text{for every }b\in B^l.
\end{gathered}
\tag{FEP.14}
\]
This is the full complement/source correction. It uses actual relation lifts and a bounded coefficient source splitting, rather than an assumed module contraction or flatness.

For completeness this operator gives a full complex-linear quotient normal form. For any \(f\in B\), put \(a=(1+\mathsf dE)^{-1}\mathsf df\), \(s=f-Ta\). Its equation gives \(\mathsf ds=0\), so \(s=\mathsf ps\). The complete word series identity
\[
 (1+E\mathsf d)^{-1}=1-E(1+\mathsf dE)^{-1}\mathsf d
\]
is proved by expanding both absolutely convergent sides, or by the finite geometric identity and the tails in proof section FEB.4. Applying \(\mathsf p\) yields \(\mathcal Pf=s\), because \(\mathsf d(f-Ea)=a\). Thus \(\ker\mathcal P=BW\) by equation (FEP.14) and \(f=Ta+\mathcal Pf\), and \(\operatorname{im}\mathcal P=\mathsf pB\), with \(\mathcal P^2=\mathcal P\). The quotient \(B/BW\) is thereby identified with \(\mathsf pB\) as complex vector spaces of actual germs. No \(B\)-linear splitting, canonical coordinate-independent projection, or special fibration/Späth reconstruction is asserted.

#### FEP.5. Proper-ideal detection without flatness

By equation (FEP.7), \(E\mathsf d1=0\). Therefore equation (FEP.11) is literally
\[
                             \mathcal P1=1.          \tag{FEP.15}
\]
If \(1\in BI=BW\), it would be a finite expression \(Tb=1\) in the actual ring on a common coefficient neighborhood. Applying equation (FEP.14) would give \(\mathcal P1=0\), contrary to equation (FEP.15) and the nonzero geometric unit. Hence \(1\notin BI\) and \(B/BI\ne0\) for every proper microlocal left ideal \(I\). All actual germs involved in a proposed finite witness have their coefficient domains intersected once; the complete series are then formed on one smaller product. A cohomological closure or coefficientwise domains cannot alter this contradiction.

#### FEP.6. The genuine ordinary zero section

At a zero-section point repeat the argument in the graded polynomial coefficient spaces of proof sections FEA.4 and FEB.5. Here are its exact finite premises. The principal ring of \(D_u\), with its nonnegative order filtration, is \(\mathcal O_u[\rho]\). For the graded ideal \(\operatorname{gr}I\), proof section FEA.4 supplies finitely many homogeneous coefficient standard divisors \(g_i\). Choose actual \(W_i\in I\) with those principal symbols. Subtracting their finite principal combinations lowers ordinary order by at least \(1\), so finite descending order reduction terminates and proves \(I=DW\) and strictness. For a homogeneous principal relation, its finite actual initial lift has image of smaller order; strictness removes that image with smaller source order. This gives exact ordinary relation lifts. Proof section FEA.4 supplies the finite homogeneous standard relation list.

Retain the free source shifts \(m_i=\operatorname{ord}W_i\), and relation shifts \(r_\nu=\max_i(\operatorname{ord}R_{\nu i}+m_i)\), omitting zero entries. In total degree \(j\), the source component has polynomial covariable degree \(j-m_i\), the relation quotient has degree \(j-r_\nu\), and the principal relation component has degree \(r_\nu-m_i\). Negative polynomial degrees mean zero spaces. Every coefficient quotient, projection and relation correction in equations (FEP.6)–(FEP.13) respects those shifts; their bounds are uniform in total degree. Every finite ordinary Leibniz correction lowers total degree. Proof section FEB.5 therefore proves the complete positive every-epsilon convergence and all derivative tails, with one common base domain; no negative ordinary covariable degree is introduced.

Properness again excludes a constant Taylor head. A homogeneous \(g_i\) of positive covariable degree vanishes at \(\rho=0\). If it has degree \(0\) and nonzero value at the base point, its actual \(W_i\) is multiplication by a nonvanishing holomorphic function, hence invertible in \(D_u\) by its literal holomorphic reciprocal. That would put \(1\) in \(I\). Thus the graded principal ideal is contained in the maximal Taylor ideal at \((u,\rho)=(u_0,0)\). Its support division gives \(\mathsf d1=0\), \(\mathsf p1=1\), now in total degree \(0\). The identical actual equations (FEP.11)–(FEP.15) prove \(1\notin D^\infty I\).

Each output stays a homogeneous polynomial of its actual covariable degree and has every positive-epsilon bound on a fixed covariable ball. Homogeneity extends the bound to every other fixed covariable compact by a fixed exponential rescaling absorbed in epsilon slack; the base domain is unchanged. This proves the full ordinary proper-ideal criterion with the unbounded positive head. The ordinary argument has been supplied in addition to the complete nonzero-chart microlocal proof; neither is substituted for the other.

#### FEP.7. Canonical algebraic tensor quotient and proved right flatness

For any left ideal \(I\), ordinary algebraic tensor has inverse maps
\[
\begin{gathered}
B\otimes_A(A/I)\\
\longrightarrow B/BI,\\
b\otimes(a+I)\\
\longmapsto ba+BI,\\
B/BI\\
\longrightarrow B\otimes_A(A/I),\\
b+BI\\
\longmapsto b\otimes(1+I).
\end{gathered}
\tag{FEP.16}
\]
Balancing proves the first well defined; \(bj\otimes1=b\otimes j=0\) for \(j\in I\) proves the second. Their compositions are identities. Neither map uses flatness or completion.

From this point use the complete theorem of [§5.47, proof sections MF.1–MF.7](#5-47-arbitrary-finite-matrices-and-right-flatness-of-the-full-infinite-order-ring): \(B\) is right \(A\)-flat for all algebraic left modules at every nonzero cotangent stalk and at the genuine ordinary zero section. Proper-ideal detection in FEP.1–FEP.6 was proved independently of flatness; FEP.8–FEP.9 now use that exact earlier right-flatness proof for the unit and faithful-extension consequences.

#### FEP.8. Unit injectivity for arbitrary left modules

Let \(M\) be any algebraic left \(A\)-module, with no finiteness assumption, and \(0\ne m\in M\). Its left annihilator \(I=\{a:am=0\}\) is proper, and \(A/I\to M\), \(a+I\mapsto am\), is an injection onto its cyclic submodule. By the proved right-flatness theorem, tensor keeps that injection. In equation (FEP.16) the element \(1\otimes(1+I)\) is the nonzero class \(1\) by the proper-ideal theorem. Its image in \(B\otimes_AM\) is \(1\otimes m\), so
\[
\begin{gathered}
M\longrightarrow B\otimes_AM,\\
m\longmapsto1\otimes m\\
\text{ is injective}\\
\text{ for every }M.
\end{gathered}
\tag{FEP.17}
\]
The map is \(A\)-linear after restriction of scalars, since \(a\otimes m=1\otimes am\). This proves the unit theorem with the entire arbitrary-module quantifier. It also proves \(B\otimes_AM=0\Rightarrow M=0\). No claim about a solution object, degree-one operator faithfulness, separation by sectorial solutions, or the infinite reconstruction theorem is used.

The functor is faithful on morphisms. If \(f:M\to N\) has \(B\otimes f=0\), then for every \(m\), \(1\otimes f(m)=0\); unit injectivity for \(N\) gives \(f(m)=0\), hence \(f=0\). Algebraic tensor is right exact: for \(L\subset M\), the inverse generator-and-balancing maps give \((B\otimes_AM)/\operatorname{im}(B\otimes_AL)=B\otimes_A(M/L)\), exactly as in equation (FEP.16), with \(m+L\) in place of \(a+I\). A surjection is surjective after tensor since each simple tensor lifts. Right flatness supplies injectivity and thus exactness. If extension of a map is an isomorphism, its extended kernel and cokernel vanish, so object detection makes both zero and the original map is an isomorphism. Thus the functor is faithful, exact and conservative for all algebraic left modules. Fullness, extension-group injectivity and effective descent are not conclusions of this argument.

Conversely, unit injectivity for every module implies equation (FEP.1) by applying it to the nonzero class \(1\) in \(A/I\) and using equation (FEP.16). Under flatness, detection of nonzero objects alone also implies \(B/BI\ne0\) for every proper ideal, since \(A/I\ne0\). Thus the proper-ideal test, unit injectivity and faithful scalar-extension criterion have the exact stated relation; no weaker operator-action faithfulness is substituted.

#### FEP.9. Sheaves, finite common domains and precise remaining boundary

For sheaves of left \(\mathcal E\)-modules, the algebraic sheaf tensor has stalk \(B\otimes_AM_p\), and the unit's stalk is equation (FEP.17). Here is the elementary stalk calculation. Each stalk tensor has finitely many section entries, which can be represented on one common neighborhood; each claimed equality is a finite sum of additivity and balancing relations, whose section entries and identities can again be represented on one common smaller neighborhood. These operations give mutually inverse maps between the stalk of the tensor presheaf and the tensor of stalks. Sheafification does not change stalks, since a germ of locally compatible representatives is represented near its point by one of them. An injective stalk at every point means that the sheaf kernel is zero: a section with zero germ everywhere vanishes on a neighborhood of each point, and hence vanishes by the sheaf gluing axiom. The same argument for kernels and locally lifted cokernel germs tests exactness and isomorphisms. Consequently the unit theorem, object detection, exactness and faithful/conservative scalar-extension functor hold for arbitrary sheaves of left modules as well. At the zero section these are the actual \(D^\infty/D\) statements just proved, using the complete ordinary right-flatness proof in [§5.47, proof sections MF.6–MF.7](#5-47-arbitrary-finite-matrices-and-right-flatness-of-the-full-infinite-order-ring).

There is a tensor-theoretic corollary for complexes. In each degree, tensor the two exact sequences defining cycles, boundaries and their quotient cohomology. Exactness gives \(H^j(B\otimes_AC^\bullet)=B\otimes_AH^j(C^\bullet)\). Object detection therefore reflects acyclicity, and applied to the cone of a chain map reflects quasi-isomorphisms. This argument is degreewise and uses no bounded-projective contraction. It does not construct the separately commissioned canonical module-derived geometric action or imply a full embedding on higher Ext groups.

For sheaf support defined by nonzero stalks, \(\operatorname{Supp}(\mathcal E^\infty\otimes_{\mathcal E}M)=\operatorname{Supp}M\): a zero input stalk has zero tensor, and a nonzero input stalk is detected by the unit theorem. The same statement uses \(D^\infty/D\) at the ordinary zero section. This is a statement about actual sheaf supports; the later coherence, characteristic-cycle and geometric purity obligations remain separate.

Each proposed finite ideal witness or finite diagram uses only finitely many actual ring elements, finite ideal/relation lifts, coefficient operators and ring identities. Intersect their coefficient products once and select common radii, all base/covariable margins, the largest finite right-tail constants, and epsilon small enough for all constants in equation (FEP.10). The domain is independent of every homogeneous, spatial, contraction and iteration index. This is enough for each cyclic element of an arbitrary module; no one universal neighborhood is asserted for infinitely many unrelated module germs.

The coefficient outputs are actual geometric ring sections by the geometric identification IG/IM in [§5.45](#5-45-the-full-geometric-infinite-order-kernel-ring-and-its-canonical-product). In this kernel paragraph write \(\omega=s-t\), \(\zeta_i=y_i-x_i\); the proved kernel normal variable \(w\) is here \(\omega\), distinguished from the earlier coefficient Taylor vector \(w=(u,z)\). Explicitly, choose Taylor covariable radii \(R_i\) inside the final common coefficient domain, one largest negative constant \(C_-\ge1\), slopes \(A_i>2/R_i\), and a common endpoint domain \(|\omega|<1/(4C_-)\). On the thick complement \(|\zeta_i|>A_i|\omega|\), choose local \(Y>\max_i(R_i|\zeta_i|)^{-1}\) with \(|\omega|Y<3/4\) and then epsilon with \(\varepsilon Y<1/8\). The full mixed kernel majorants in equations (IK.18)–(IK.21) have \(r<1/4\), \(q<3/4\), \(q+x<7/8\); at \(\omega=0\) use their proved removable extension. A finite compact cover supplies the joint epsilon. This retains every spatial index and the positive-homogeneous/negative-normal mixed terms.

Choose phase-cone spatial slopes \(B_i>A_i\). Use the proved finite phase grid \(h<2\lambda\), \(2\lambda+h<\pi\), separate phase/neighbor pointed cones, and the one-time bridge \(\lambda<\lambda'<(\pi-h)/2\), \(B'_i>B_i/\cos(\lambda+h/2)\). Forget support literally to the larger standard \(T'\), use its actual cube and exhaustions, and map any resulting boundary forward to a final convex \(G'\) containing \(T'\). Reserve its overlap width and common proper compact collar and a smaller admissible covariable product. The larger Taylor radii used in the kernel bounds remain a holomorphic coefficient enlargement. All complement terms and full-cone Čech/ordinary conversion signs are the proved ones, including ordinary top entry \(r_N(-1)^N f\) and the \(N=1\) exception. No support-forgetting injectivity or ordinary supported-distribution limit is inferred.

The auxiliary coefficient quotient \(\mathcal P\) depends on coordinates and standard heads. Its existence proves an intrinsic nonvanishing statement: proved nonlinear coordinate ring maps transport the finite ideal identity and unit to every overlap. No canonical coordinate-independent module projection is needed. Ordinary compatibility is supplied directly in FEP.6.

The precise proved scope is proper-left-ideal detection in the full geometric \(E^\infty/E\) and ordinary \(D^\infty/D\) stalks, with all-domain/growth/support controls. Unit injectivity and faithful exact extension for arbitrary algebraic modules and sheaves follow using the complete earlier right-flatness theorem identified in FEP.7. The other scalar changes, \(E^R\), the special fibration class, coordinate-commuting \(A^\infty\), mixed \(E^\infty/D^\infty\) flatness, infinite Späth division/balanced reconstruction, propagation, purity, singular-inclusive separation, D-type/C1, half-order/contact transport and arbitrary analytic proper regularity require their matching arguments. Section5.90, GG.A–GG.G proves the initial ordinary generators near compact sets. No such theorem is implied by this coefficient quotient. Section 5.49 separately proves the special fibration class, full infinite Späth sum division and algebraic balanced reconstruction, with the explicit full contact shear; the other listed obligations remain open.

#### FEP.10. A decisive joint-domain calibration

Take one base coordinate \(x\) and its covariable \(\tau\), at \(x=0,\tau\ne0\), with any remaining base/covariable parameters passive. For a nonzero finite-order \(b\), the principal coefficient of \(b\circ x\) is the nonzero product of its leading coefficient with \(x\). Hence its order is the order of \(b\), and \(Ax\) has a strict principal image \((x)\). In the principal division, \(\mathsf d\) deletes the constant Taylor term and divides by \(x\), while \(\mathsf p\) takes the constant coefficient in \(x\). The actual right product is
\[
\begin{gathered}
b\circ x=xb+\partial_\tau b,\\
\mathcal P\\
=\mathsf p\sum_{a\ge0}(-\partial_\tau\mathsf d)^a,\\
\mathcal P1=1,\\
\mathcal P(b\circ x)=0.
\end{gathered}
\tag{FEP.18}
\]
For this calibration \(\partial_\tau\) and the Taylor division in \(x\) commute. That special commutation is not assumed in the general proof. The finite upper-bounded/negative-tail version of the same normally convergent quotient annihilates \(Ax\) and fixes \(1\), proving this finite ideal proper directly, without using flatness or assuming the target. The same formula then applies to the full \(B\) class by the complete convergence proof FEB.1–FEB.5.

The exact calibration retains arbitrary normal and spatial powers:
\[
\begin{gathered}
\mathcal P(x^a\tau^m\xi^\beta)\\
=(-1)^a m^{\underline a}\tau^{m-a}\xi^\beta,\\
a\ge0,\ m\in\mathbf Z,\ \beta\in\mathbf N^d.
\end{gathered}
\tag{FEP.22}
\]
Only Taylor degree \(a\) contributes to the projection series, so this follows by the ordinary finite derivative formula. The identity
\(m^{\underline{a+1}}=m(m-1)^{\underline a}\) then proves cancellation for
\((x^a\tau^m\xi^\beta)\circ x=x^{a+1}\tau^m\xi^\beta+
m x^a\tau^{m-1}\xi^\beta\), for every integer \(m\), including negative \(m\). Its homogeneous input degree is \(m+|\beta|\), so this includes negative-normal powers with positive homogeneous degree. At the ordinary zero section the polynomial calibration uses \(m\ge0\). The full local FEB.1–FEB.5 estimates, rather than coefficientwise formal continuation of this calibration, control arbitrary infinite sums over all these indices.

The tempting entire expression
\[
\begin{gathered}
\widetilde b(x,\tau)=\frac{1-e^{-x\tau}}x\\
=\sum_{j\ge1}(-1)^{j+1}\frac{x^{j-1}\tau^j}{j!},\\
\widetilde b\circ x=1
\end{gathered}
\tag{FEP.19}
\]
is holomorphic at \(x=0\) by its displayed power series. The differential identity follows directly from \(x\widetilde b=1-e^{-x\tau}\), \(\partial_\tau\widetilde b=e^{-x\tau}\). It is not in the actual class: on any fixed base compact \(|x|\le r_x\), \(r_x>0\), its degree-\(j\) norm is \(r_x^{j-1}/j!\), and the every-epsilon inequality would require
\[
\begin{gathered}
r_x^{-1}(r_x/\varepsilon)^j\le M_\varepsilon\\
(j\ge1).
\end{gathered}
\tag{FEP.20}
\]
For \(0<\varepsilon<r_x\), the left side is unbounded. Restricting to \(x=0\) alone gives the finite expression \(\tau\), but does not give a section on any common neighborhood. The failure is exactly the uniform joint domain and every-epsilon quantifier.

Its finite truncation \(b_L\) obeys
\[
\begin{gathered}
b_L\circ x=1+r_L,\\
r_L=(-1)^{L+1}\frac{x^L\tau^L}{L!},\\
\mathcal P r_L=-1.
\end{gathered}
\tag{FEP.21}
\]
The first equality follows by cancellation of successive positive powers. The last follows from the exact finite quotient formula: only Taylor degree \(a=L\) contributes, with \((-1)^L\partial_\tau^L[(-1)^{L+1}\tau^L/L!]=-1\).
At \(x=1/2,\tau=1\) the ordinary entire-function residual magnitude \(2^{-L}/L!\) tends to zero. Its required positive-symbol weighted norm on \(|x|\le1/2\) with \(\varepsilon=1/4\) is exactly \(2^L\), and its quotient value remains \(-1\). Hence that pointwise limit cannot be used to put \(1\) in the algebraic ideal \(Bx\). The general proof uses the actual tails in equation (FEP.10), which do tend to zero in every fixed requested epsilon estimate after choosing a smaller input epsilon.

Strict generation before taking the coefficient quotient is essential. At the ordinary zero section, \(Dx+D\partial_x=D\), because the exact finite right products give
\[
          \tau\circ x-x\circ\tau=1.                  \tag{FEP.23}
\]
Although the original two principal symbols generate the proper commutative ideal \((x,\tau)\), the graded ideal of the actual left image contains \(1\). The principal source relation \((-\tau,x)\) of that original presentation has actual image \(-1\), so it is not an actual relation. Our FEP.2/FEP.6 construction first uses the strict standard generators of the whole actual ideal, and FEP.3 lifts principal relations only after that step. It therefore never mistakes this improper ideal for a proper one.

![Actual ideal detection and the excluded exponential candidate](assets/infinite-faithful-extension-joint-domain.png)

**Figure FEP.1.** The left panel shows the exact analytic source correction, target quotient and unit test in equations (FEP.6)–(FEP.15), with actual row products and complex-linear coefficient operators distinguished. The right panel displays exact samples \(L=1,\ldots,20\), \(r_x=1/2,\varepsilon=1/4,x=1/2,\tau=1\): required weighted residual \(2^L\), entire-function residual \(2^{-L}/L!\), and \(\mathcal Pr_L=-1\). Their general identities, not these finite samples, are equations (FEP.20)–(FEP.21). The illustration records the complete joint-domain obstruction and does not claim convergence in the actual ring from ordinary entire-function values. Reproducible coordinates, sources and data are retained; free human geometric context is Micro-hyperbolic systems §1.3/§8, with the local proof sections FEP.1–FEP.10 and FEA/FEB above.

Editable SVG, exact data, and reproducible source.



### 5.49. Full infinite Spath division and balanced reconstruction

We prove the full infinite Späth sum decomposition and both algebraic balanced reconstruction maps for the entire generic holonomic germ. The proof includes singular branches, nonreduced fibres, every spatial Taylor index, both homogeneous growth regimes, fixed derivative and iteration tails, off-ray vanishing and the actual generic-position contact shear. The coordinate-commuting ring, uniform finite spatial-polynomial ring, decorated special fibration sheaf and full spatial-convergent ring remain distinct.

The division is proved in ISD.1–ISD.5, the source realization and full product of the decorated special sheaf in ISF.1–ISF.5, and reconstruction and contact transport in ISR.1–ISR.7. The complete convergence argument ISB.1–ISB.5 is included after them; its earlier coefficient-norm premise is the complete MFA.1–MFA.4 proof in [§5.47](#5-47-arbitrary-finite-matrices-and-right-flatness-of-the-full-infinite-order-ring). The arbitrary-module faithful and support theorem used for whole-support detection is [§5.48, FEP.8–FEP.9](#5-48-proper-ideals-and-faithful-extension-to-the-full-infinite-order-ring).

The complete actual finite product, norm, units, Nakayama and division arguments are [§5.19–§5.21](#5-19-actual-factorial-growth-symbols-and-ordered-jets), the actual normal shift is [§5.22.1](#5-22-1-the-actual-normal-element), and chart normalization and strictness are [§5.24.1–§5.24.2](#5-24-1-the-filtration-used-here).

The complete whole-germ finite input is [§5.28, Q.1–Q.7](#5-28-convergent-contact-quantization-and-whole-germ-generic-position), [§5.29, G.1–G.5](#5-29-actual-finite-generation-over-the-coordinate-commuting-ring), [§5.30, F.1–F.7](#5-30-actual-polynomial-flatness-and-reconstruction-after-finite-realization), [§5.31, L.1–L.8](#5-31-a-bounded-actual-generic-position-complex-with-polynomial-operator-matrices), and [§5.32, J.1–J.3](#5-32-coherent-actual-lattices-and-principal-support-for-the-whole-germ). The full coefficient/kernel arguments are [§5.44, IK.1–IK.5, AN.1–AN.4 and TC.1–TC.4](#5-44-full-homogeneous-coefficients-and-ordinary-infinite-order-kernels) and [§5.45, IG.1–IG.5 and IM.1–IM.6](#5-45-the-full-geometric-infinite-order-kernel-ring-and-its-canonical-product). The strong spatial-coefficient source construction uses the complete [§5.43, DLH.1–DLH.4](#5-43-the-canonical-all-degree-action-and-its-derived-module-structure) existence argument and the finite comparisons in [§5.34, R.5–R.6](#5-34-full-current-resolutions-and-controlled-proper-image-comparison) and [§5.35, V.6](#5-35-relative-cones-unshifted-cup-and-the-normalized-trace).

Free human context: Kashiwara and Kawai, [On the holonomic systems of microdifferential equations III, author-hosted full PDF](https://www.kurims.kyoto-u.ac.jp/~kenkyubu/kashiwara/HolIII.pdf), III.3–III.5, printed pp.887–905. The complete III.5 continuation includes Corollary 3.5.7 and Theorem 3.5.8 through printed p.905. The printed commutator inconsistency on p.902 is discussed in ISD.1, where the correct coordinate commutator is derived directly. The source specifies targets and conventions; every mathematical premise is proved here or at the exact complete earlier locators above.

**Actual full infinite Späth division.**

The full source-defined coefficient/ring premise is [§5.45, IG.1–IG.5 and IM.1–IM.6](#5-45-the-full-geometric-infinite-order-kernel-ring-and-its-canonical-product). The complete lowering proof ISB.1–ISB.5 is included below. The complete actual finite input is [§5.28](#5-28-convergent-contact-quantization-and-whole-germ-generic-position)–[§5.32](#5-32-coherent-actual-lattices-and-principal-support-for-the-whole-germ). The faithful-extension theorem [§5.48, FEP.8–FEP.9](#5-48-proper-ideals-and-faithful-extension-to-the-full-infinite-order-ring) is used for proof section ISR.5 support detection, not for division.

#### ISD.1. Exact rings, sides and the finite whole-germ input

Use \(u=(t,x_1,\ldots,x_d)\), \(\rho=(\tau,\xi)\), \(z=\xi/\tau\), \(h=\tau^{-1}\), and \(p_0=(0;1,0)\). Write \(E\) for the actual finite-order ring and \(B=E^\infty_{p_0}\) for the proved source-defined full geometric stalk. Its normal symbols are

\[
\begin{gathered}
P=\sum_{j\in\mathbf Z}\tau^jp_j(u,z),\\
|p_{-n}|_K\le M_K C_K^n n!,\\
|p_j|_K\le M_{K,\epsilon}\epsilon^j/j!\\(j\ge0)
\end{gathered}
\tag{ISD.1}
\]

for every \(\epsilon>0\), with every coefficient holomorphic on one product independent of \(j,\epsilon\). All spatial Taylor indices are retained. All modules are left modules, free-module maps use row vectors and multiplication on the right, and every tensor below is algebraic.

Define \(A^\infty\subset B\) by \([P,x_i]=0\) for all \(i\), and \(R^\infty\subset B\) by one uniform finite bound on the spatial polynomial degree. Precisely,

\[
\begin{gathered}
A^\infty=\left\{\begin{gathered}p_j(u,z)\\\text{ independent }\\\text{of }z,\\\ \text{all }j\end{gathered}\right\},\\
R^\infty=\bigoplus_{\beta\in\mathbf N^d}A^\infty\partial_x^\beta .
\end{gathered}
\tag{ISD.2}
\]

The direct sum means finite support in \(\beta\) for an entire operator, not a bound varying with \(j\). To verify it, the actual Leibniz formula gives \([P,x_i]=\partial_{\xi_i}P\). A common power of each \(\operatorname{ad}x_i\) vanishes exactly when each normalized coefficient is polynomial with a common coordinatewise degree bound; finitely many such bounds give a uniform total degree. Taylor extraction on a smaller fixed \(z\)-polydisc and the fixed shift \(j=m+|\beta|\) put each coefficient of \(\partial_x^\beta\) in \(A^\infty\). Conversely each finite displayed sum has a uniform polynomial bound. The product of two such sums remains finite in \(\beta\): the Leibniz rule in \(x\) is finite for their spatial powers, while all \(\tau,t\) contractions are the actual proved product in \(A^\infty\). Thus both are genuine subrings, with the original unit. They are not the decorated special sheaf of proof section ISF.1 below.

Let \(M\) be any whole coherent actual holonomic \(E\)-germ in generic position, without a regularity or smooth-branch restriction. [§5.32, J.1–J.3](#5-32-coherent-actual-lattices-and-principal-support-for-the-whole-germ) constructs an actual coherent lattice \(M_0\subset M\), with \(EM_0=M\), injective \(h\), and finite nilpotent principal fibre. [§5.28, Q.1–Q.7](#5-28-convergent-contact-quantization-and-whole-germ-generic-position) puts every whole reduced Lagrangian germ in that position and transports its actual lattice. [§5.29, G.1–G.5](#5-29-actual-finite-generation-over-the-coordinate-commuting-ring) makes \(M_0\) finite over the coordinate-commuting \(A_0\); these are the complete earlier proofs of this finite input.

The map \(A_0\to\mathbf C\) sending an operator to its index-zero coefficient at \(u=0\) is a ring map. Its kernel \(\mathfrak m\) is two-sided. Every element outside \(\mathfrak m\) is an actual unit by the proved elliptic inverse in [§5.20.3](#5-20-3-actual-elliptic-inverses-and-the-stalk-radical), which preserves the \(z\)-independent subring. Thus \(\mathfrak m=\operatorname{Jac}(A_0)\): a proper left ideal contains no element outside it, and \(A_0/\mathfrak m=\mathbf C\). The finite-generator Nakayama proof applies. Choose generators \(U=(u_1,\ldots,u_r)^{\mathsf T}\) of \(M_0\) whose residues form a basis of \(M_0/\mathfrak mM_0\). The case \(M=0\) has \(r=0\) and every assertion is immediate.

Since \(z_\nu M_0\subset M_0\), there are actual matrices \(C_\nu\in M_r(A_0)\) such that

\[
\begin{gathered}
z_\nu U=C_\nu U,\\
B_\nu=\tau\circ C_\nu\in M_r(A(1)),\\
(D_\nu I-B_\nu)U=0,\\
L_\nu=h\circ(D_\nu I-B_\nu)\\
=z_\nu I-C_\nu .
\end{gathered}
\tag{ISD.3}
\]

Here \(D_\nu=\partial_{x_\nu}\), \(h\tau=1\), and \(hz_\nu=z_\nu h\); hence the last equality is an exact actual product identity. Each \(B_\nu\) commutes with every \(x_i\), retains arbitrary holomorphic \(x\)-dependence, and has order at most one. Put \(A_\nu(u)=\sigma_0(C_\nu)\). The matrices \(A_\nu(0)\) represent the commuting nilpotent \(z_\nu\)-actions on the finite principal fibre. In particular they commute with each other at \(u=0\). No commutation of the lifts \(C_\nu\), or of the matrices \(A_\nu(u)\) on the whole free frame, is assumed.

The inspected free source HolIII printed p.902 / PDF90 prints “free from \(D_1,\ldots,D_d\), i.e. \([D_\mu,B_\nu]=0\)” in (3.5.11). Its earlier definition of \(A\), and its principal \(A_\nu(t,x)\tau\), require the spatial-derivative-free condition \([x_\mu,B_\nu]=0\). The displayed \(D_\mu\) commutator is an internal printed inconsistency: \([D_\mu,B_\nu]\) would differentiate the allowed \(x\)-dependent coefficients. Equation ISD.3 proves the precise condition used here directly, so no unproved interpretation of that printed commutator is a premise. The actual source page is in the linked free PDF.

#### ISD.2. One fixed family of principal division operators

Commuting nilpotent matrices admit a common strict triangular basis. Here is the elementary argument. The kernel of the first nilpotent is nonzero and invariant under all the others. Successively intersect with the next kernel; its restriction is nilpotent, so a nonzero invariant space has nonzero kernel. This yields a common vector killed by all matrices. Apply the same argument to their induced commuting nilpotent matrices on the quotient by that vector and induct on dimension. The resulting invariant flag has zero diagonal actions. A constant change of the generators \(U\) puts all \(A_\nu(0)\) in this strict triangular form.

For any finite input family, choose its common coefficient product and positive outer spatial radii \(R_\nu^*\). A weighted component norm makes all these fixed strict triangular matrices as small as desired: for an upper triangular matrix, take component weights \(c_j=\delta^j\), so right row multiplication has factors \(c_j/c_i=\delta^{j-i}\) for \(j>i\); use the reversed exponents for the lower triangular convention. Decrease \(\delta>0\), then shrink the base radii while holding the spatial radii fixed. The absolute Taylor norm of every nonconstant base term tends to zero by domination by its convergent Taylor series. Therefore one weighted row/matrix norm satisfies

\[
\begin{gathered}
\|A_\nu(u)\|_r\le a_\nu<R_\nu^*/8
\\\text{for every }1/2\le r\le1,\\
R_\nu(r)=rR_\nu^* .
\end{gathered}
\tag{ISD.4}
\]

The base polyradii also scale over this fixed interval. The finite matrices remain fixed; the component weights and base radius may depend on the common input family, as an actual stalk argument permits. This does not choose a radius for each coefficient or each epsilon.

For a holomorphic row \(f=\sum_{k\ge0}f_k(u,z_{\ne\nu})z_\nu^k\), define coefficientwise, using ordinary matrix multiplication,

\[
\begin{gathered}
p_\nu f=\sum_{k\ge0} f_k A_\nu^k,\\
d_\nu f=\sum_{k\ge1}\sum_{v=0}^{k-1}\\
f_k z_\nu^{k-1-v} A_\nu^v,\\
T_{\nu,0}q=q(z_\nu I-A_\nu).
\end{gathered}
\tag{ISD.5}
\]

These series are holomorphic on the fixed products of equation ISD.4. In the absolute Taylor norm, summing the geometric factors gives

\[
\begin{gathered}
\|p_\nu\|_r\le(1-a_\nu/R_\nu(r))^{-1}\\
\le4/3,\\
\|d_\nu\|_r\le(R_\nu(r)-a_\nu)^{-1}\\
\le 8/(3R_\nu^*) .
\end{gathered}
\tag{ISD.6}
\]

Indeed a \(z_\nu^k\)-term of norm \(\|f_k\|R_\nu^k\) contributes at most \(\|f_k\|a_\nu^k\) to \(p_\nu\), and at most \(\|f_k\|R_\nu^{k-1}\sum_{v<k}(a_\nu/R_\nu)^v\) to \(d_\nu\). Absolute convergence proves the same estimates with all other spatial and base Taylor indices intact. The scalar telescoping identity for \(z^kI-A^k\), with coefficient order unchanged, yields

\[
\begin{gathered}
T_{\nu,0}d_\nu=1-p_\nu,\\
d_\nu T_{\nu,0}=1,\\
d_\nu p_\nu=0,\\ p_\nu^2=p_\nu .
\end{gathered}
\tag{ISD.7}
\]

For \(d_\nu T_{\nu,0}=1\), truncate a holomorphic quotient at spatial degree \(K\); the terminal matrix power is bounded by \((a_\nu/R_\nu)^K\), hence tends to zero. This also proves the identity for every holomorphic germ on that product. The image of \(p_\nu\) and the kernel of \(d_\nu\) consist exactly of rows independent of \(z_\nu\). The maps are bounded complex linear coefficient maps, not \(E\)-linear or \(A^\infty\)-linear projections.

These constructions commute with restriction and preserve independence of any other spatial variable already eliminated, since \(A_\nu(u)\) has no spatial covariable. Their formulas on different allowable radii give the same germs by equation ISD.7 and the uniqueness of the displayed principal division.

#### ISD.3. Actual infinite correction, all tails and common domains

Let \(T_\nu q=q\circ L_\nu\) and \(E_\nu=T_\nu-T_{\nu,0}\). Every term of \(E_\nu\) loses at least one homogeneous degree. The right tail \(C_\nu-A_\nu\) has order at most minus one and factorial bounds. All contractions with the principal \(A_\nu(u)\), including its \(t\) and every \(x\) derivative, have positive contraction order. The right term \(z_\nu I\) is base independent, so its product is ordinary multiplication and has no derivative correction.

Choose a finite radius ladder from the outer product to a fixed final product, reserving a further intermediate product for each finite composition below and for their verification. Since there are only \(d\) divisions and finitely many matrices, this requires finitely many shrinkings. No shrinking is indexed by a homogeneous degree, a word length or epsilon. On that ladder equation ISD.6 gives one uniform bound \(H\) for every \(d_\nu,p_\nu\).

The complete local lowering proof ISB.1–ISB.5 now applies to \(d_\nu E_\nu\). Its hypotheses are exactly equations ISD.4–ISD.6 and the finite actual right order-zero matrix \(L_\nu\). For total loss \(K\), input homogeneous degree \(m\), and one fixed coefficient compact, its complete word majorant is

\[
\begin{gathered}
C_1^K\frac{(|m|+K)!}{|m|!}\,\|f_m\|_{\rm outer}.
\end{gathered}
\tag{ISD.8}
\]

The complete proof in ISB.1–ISB.5 retains every right-tail index, every spatial/base contraction, all \(\tau\)-Euler derivatives and every inserted bounded coefficient map. It allocates the entire fixed Cauchy margin over a word before estimating it. In particular it does not assume that \(d_\nu\) commutes with base differentiation or with actual operator multiplication.

Thus the actual inverse and quotient are

\[
\begin{gathered}
q_\nu=(1+d_\nu E_\nu)^{-1}d_\nu F\\
=\sum_{\ell\ge0}(-d_\nu E_\nu)^\ell d_\nu F,\\
s_\nu=F-T_\nu q_\nu .
\end{gathered}
\tag{ISD.9}
\]

These are actual \(B\)-rows, not formal completions. For clarity the complete two-sided summation from equation ISD.8 gives the following bounds. For positive output degree \(j\), input \(m=j+K\) has
\[
\begin{gathered}
\frac{(j+2K)!j!}{(j+K)!^2}\le4^K,\\
\|(q_\nu)_j\|\le H M_\epsilon\frac{\epsilon^j}{j!}\\
\sum_{K\ge0}(4C_1\epsilon)^K .
\end{gathered}
\tag{ISD.10}
\]

Given any desired output epsilon, choose an input epsilon no larger than it and with \(4C_1\epsilon<1/2\); the coefficient product does not change with that choice. For negative output \(-n\), losses \(K<n\) give at most \(HM n!(C+C_1)^n\), while \(K\ge n\), with input \(m=K-n\ge0\), give
\[
\begin{gathered}
\|(q_\nu)_{-n}\|\\\le HM n!(C+C_1)^n\\
+HM_\epsilon n!(3C_1)^n\\
\sum_{m\ge0}(9C_1\epsilon)^m .
\end{gathered}
\tag{ISD.11}
\]

Here \((n+2m)!/(n!m!^2)\le3^{n+2m}\). One fixed small epsilon gives a genuine factorial negative bound. Tails in \(K\), or in inverse word length \(\ell\) since \(K\ge\ell\), have the corresponding geometric tails in equations ISD.10–ISD.11. In equation ISD.11 a tail beginning beyond \(K=n\) is bounded by the tail beginning at \(m=K-n\). Every fixed base or spatial derivative has the same tails on the reserved smaller product by Cauchy. This proves joint normal convergence, the every-epsilon positive class and the negative factorial class for all spatial indices on the same domain.

The actual product theorem places \(s_\nu\) in \(B\) as well. The convergent inverse identity, justified on the intermediate product before passing to the final one, gives \(d_\nu s_\nu=0\). Therefore \(s_\nu=p_\nu s_\nu\) is independent of \(z_\nu\). Conversely if \(T_\nu q+s=0\), with \(s\) independent of \(z_\nu\), applying \(d_\nu\) and its actual inverse gives \(q=0\), then \(s=0\). This is uniqueness for one normalized one-variable division; it is not a directness assertion for the simultaneous sum.

#### ISD.4. All spatial relations and the printed sum

Apply equation ISD.9 successively to \(F\), then its first remainder, and so on in the fixed order \(\nu=1,\ldots,d\). At stage \(\nu\) work in the coefficient subspace independent of \(z_1,\ldots,z_{\nu-1}\). It is preserved by \(d_\nu,p_\nu,E_\nu\): the right coefficients are independent of all \(z\), spatial derivatives in already eliminated variables vanish, and the normal Euler derivative preserves that independence. The final remainder is independent of every \(z_\nu\). Setting \(a_\nu=q_\nu\circ h\) gives the literal actual decomposition

\[
\begin{gathered}
F=\sum_{\nu=1}^d a_\nu\circ(D_\nu I-B_\nu)\\
+s,\\
a_\nu\in B^r,\quad s\in(A^\infty)^r .
\end{gathered}
\tag{ISD.12}
\]

Indeed \(L_\nu=h\circ(D_\nu I-B_\nu)\), so the normalization is on the left; using \((D_\nu-B_\nu)h\) here would require an additional conjugation and would not give equation ISD.12 with the stated matrices. Right multiplication by \(h\) in \(a_\nu=q_\nu h\) is the fixed homogeneous shift \(j\mapsto j-1\), preserving both full bounds on the same product. The finite spatial list and all inverse series have a common final domain and the derivative/tail controls of equations ISD.10–ISD.11. Thus
\[
\begin{gathered}
B^r=\sum_{\nu=1}^d B^r(D_\nu I-B_\nu)\\
+(A^\infty)^r .
\end{gathered}
\tag{ISD.13}
\]

This proves the full spatial sum decomposition for the actual matrices supplied by the whole-germ lattice. No uniform finite spatial degree is imposed on \(F\) or \(a_\nu\). The remainder belongs to the coordinate-commuting ring only after every spatial variable has been eliminated.

The printed decomposition is a sum. For \(d=2,r=1,B_1=B_2=0\), the coefficient rows \(a_1=D_2,a_2=-D_1\) give \(a_1D_1+a_2D_2=0\) although neither is zero. Thus arbitrary coefficient tuples are not unique, even in the finite subring. The ordered division above selects one tuple; no uniqueness of all simultaneous representations, no directness, and no identification of the entire module quotient with a free \(A^\infty\)-module is claimed. Other actual \(A^\infty\)-relations of \(M\), including torsion and nilpotents, remain.

For \(d=0\), equation ISD.13 reads \(B^r=(A^\infty)^r\) and requires no division. For a finite family of inputs, maps or relations, all data can be represented first on one common product and the finite construction above then supplies one common final product and all the stated bounds. It does not assert one universal representative neighbourhood for every element of an unrestricted stalk.

#### ISD.5. Exact calibrations and retained limits

For one spatial variable and the constant nilpotent matrix \(J=\left(\begin{smallmatrix}0&1\\0&0\end{smallmatrix}\right)\), the normalized divisor \(L=zI-J\) has no correction \(E_\nu\). The exact principal and full division is
\[
\begin{gathered}
z^k v=\left(\sum_{a=0}^{k-1}z^{k-1-a}vJ^a\right)\\
(zI-J)+vJ^k .
\end{gathered}
\tag{ISD.14}
\]

It retains the nilpotent remainder \(vJ\) for \(k=1\); replacing \(J\) by its eigenvalue zero would change the actual module. For \(F_j=\tau^j/(1-z/r)\), \(|z|<r\), the quotient and remainder are
\[
\begin{gathered}
s_j=\tau^jv(I-J/r)^{-1},\\
q_j=\tau^jv\,r^{-1}(1-z/r)^{-1}\\
(I-J/r)^{-1}.
\end{gathered}
\tag{ISD.15}
\]

The finite inverse \(I+J/r\) verifies the formulas exactly, while the input has infinitely many spatial Taylor indices. Multiply each already degree-\(j\) equation ISD.15 by \(1/(j!)^2\), then sum over \(j\ge0\). Its single normal prefactor is therefore \(\tau^j/(j!)^2\), with no repeated \(\tau^j\). This gives the actual full positive-head calibration and every epsilon bound, since \(1/(j!)^2\le e^{1/\epsilon}\epsilon^j/j!\). All of its spatial Taylor indices remain.

The nonreduced complete earlier example [§5.31, L.8](#5-31-a-bounded-actual-generic-position-complex-with-polynomial-operator-matrices) also retains the independent \(A\)-relation \(T_*=tI-h\operatorname{diag}(0,1)\), with the exact second differential \(T_*-hI\). Equation ISD.13 eliminates spatial derivatives without removing that relation. Its nilpotent \(zI-J\) remainder is therefore not a claim that \(M^\infty\) is free over \(A^\infty\).

The construction uses holomorphic complement kernels through the proved geometric identification, never an ordinary supported-distribution limit of the positive head. Its full finite-family support, overlap and trace bridge is stated in proof section ISR.6 below. The faithful-extension theorem [§5.48, FEP.8–FEP.9](#5-48-proper-ideals-and-faithful-extension-to-the-full-infinite-order-ring) detects the nonzero finite module inside its extension in proof section ISR.5; it is not needed for equation ISD.13 itself. All propagation, purity, separation, \(E^R\)-module injection, D-type, C1, half-order and analytic-proper obligations remain separate.

**The distinct decorated special fibration sheaf.**

This local proof retains precisely the special fibration \(F(t,x)=x\) and the decorated sheaf \(\widetilde E^\infty\) of the freely author-hosted HolIII III.3, printed pp.888–889. It is distinct from full \(E^\infty\), \(A^\infty\) and \(R^\infty\). The complete coefficient/kernel and angular proofs are at [§5.44, IK/AN/TC](#5-44-full-homogeneous-coefficients-and-ordinary-infinite-order-kernels) and [§5.45, IG/IM](#5-45-the-full-geometric-infinite-order-kernel-ring-and-its-canonical-product). The complete free source context is III.3–III.5, printed pp.887–905.

#### ISF.1. Joint coefficient specification and signs

Put \(\omega=s-t\). A special germ has a pair \((P_0,K)\), with
\[
\begin{gathered}
P_0=\sum_{m\ge0,\beta\ge0}p_{m,\beta}(t,x)\\
D_t^mD_x^\beta\in D^\infty,\\
K(t,s,x,D_x)\\
=\sum_{\beta\ge0}k_\beta(t,s,x)D_x^\beta .
\end{gathered}
\tag{ISF.1}
\]
Every coefficient of \(P_0\) is holomorphic on one \(U\); on each compact and for every epsilon,
\[
\begin{gathered}
|p_{m,\beta}|\le M_\epsilon
\frac{\epsilon^{m+|\beta|}}{m!\beta!}.
\end{gathered}
\tag{ISF.2}
\]
Every coefficient of \(K\) is holomorphic on one joint product in \((t,s,x)\), including \(t=s\), and on each compact
\[
\begin{gathered}
\text{for every }\delta>0,\\
|k_\beta(t,s,x)|\le M_\delta\,\delta^{|\beta|}/\beta! .
\end{gathered}
\tag{ISF.3}
\]
The product is independent of indices and epsilon. Separate normal and spatial epsilons are equivalent to equation ISF.2 by using their minimum; this reserves normal margins independently of a desired spatial output bound.

Write \(J=-K\), and expand on a fixed smaller joint product as
\[
\begin{gathered}
J(t,s,x,D_x)=\sum_{q\ge1,\beta\ge0}\\
a_{-q,\beta}(t,x)\frac{(t-s)^{q-1}}{(q-1)!}D_x^\beta .
\end{gathered}
\tag{ISF.4}
\]
The associated full symbol is \(P_0+\sum_{q,\beta}a_{-q,\beta}\tau^{-q}\xi^\beta\). Its normal relative-kernel expression is
\[
\begin{gathered}
\frac1{2\pi i}\\\left\{\begin{gathered}
P_0(t,x,D_t,D_x)\frac1{\omega}\\
-J(t,s,x,D_x)\log\omega
\end{gathered}\right\}.
\end{gathered}
\tag{ISF.5}
\]
Thus equation ISF.5 has the source's \(+K\log(s-t)\) sign. Full spatial Cauchy factors and the input top form are included on embedding in the diagonal cube. The positive residue scalar and all-dimensional top-cube sign are the proved TC/IG/IM signs of [§5.44, TC.1–TC.4](#5-44-full-homogeneous-coefficients-and-ordinary-infinite-order-kernels) and [§5.45, IG/IM](#5-45-the-full-geometric-infinite-order-kernel-ring-and-its-canonical-product).

For a fixed coefficient compact choose an outer \(s-t\) circle of radius \(\eta>0\) in the joint product. Cauchy and equation ISF.3 give
\[
\begin{gathered}
|a_{-q,\beta}|\le M_\delta C^q(q-1)!
\frac{\delta^{|\beta|}}{\beta!},\\ q\ge1,
\end{gathered}
\tag{ISF.6}
\]
where one \(C\ge1\) is independent of \(\delta\). Conversely equation ISF.6 with that independence makes equation ISF.4 normally convergent for \(|t-s|<1/C\), with equation ISF.3 on every smaller compact, for every delta. Allowing a new \(C\) for every spatial epsilon without a common normal radius would not prove the special class.

#### ISF.2. Actual geometric map on every normal fibre

On a compact normalized covariable set put \(Z=\sum_i\sup|z_i|\), \(y=C\delta Z<1\). In positive homogeneous degree \(j\), the negative-normal part has \(q=|\beta|-j\ge1\). The multinomial identity and \(j!(b-j-1)!\le b!\) give
\[
\begin{gathered}
|p_j^{\rm neg}|
\le\frac{M_\delta C^{-j}}{j!}\sum_{b\ge j+1}y^b\\
=M_\delta\frac{(\delta Z)^j}{j!}\frac{y}{1-y}.
\end{gathered}
\tag{ISF.7}
\]
In degree \(-n\), \(q=b+n\), and
\[
\begin{gathered}
|p_{-n}^{\rm neg}|\\
\le M_\delta C^n n!\sum_{b\ge0}\binom{n+b}{b}y^b\\
=M_\delta C^n n!(1-y)^{-n-1}.
\end{gathered}
\tag{ISF.8}
\]
The scalar sum identity follows by differentiating the absolutely convergent geometric series \(n\) times. For positive output choose delta arbitrarily small relative to the desired epsilon and \(Z\); for the negative bound choose one delta with \(y<1/2\). No domain changes. The differential part has normalized homogeneous polynomials of degree at most \(j\), bounded by \(M_\epsilon(\epsilon(1+Z))^j/j!\). These are the full factorial/every-epsilon bounds. At \(Z=0\) the estimates have their literal constant-coefficient interpretation (the positive negative-normal part is zero).

All sums converge normally on one product, with geometric tails in spatial degree from equations ISF.7–ISF.8 and every derivative tail on a reserved smaller product. The proved geometric identification therefore gives maps \(j_p:\widetilde E^\infty_{q_0}\to E^\infty_p\) for all \(p\) above \(q_0\) with \(\tau\ne0\). Multiplicativity is proved in proof section ISF.4. Bounds may depend on the covariable compact; the base domain is fixed.

At \(p_0\) the map is injective. Proof section IG.4's geometric zero test makes every homogeneous coefficient zero. Taylor extraction in \(z\), with \(j=m+|\beta|\), makes all \(p_{m,\beta}\) and \(a_{-q,\beta}\) zero. All Taylor coefficients of joint holomorphic \(J\) vanish at \(s=t\), so \(J=0\) on a smaller connected product.

This is the coordinate realization of the source-defined decorated sheaf. The following uniform-coefficient resolution argument ensures that arbitrary source sections, and zero witnesses, really supply the strict angular input; applying the supplied-data AN proof alone would not establish that fact.

For a finite family of relative cocycles with spatial coefficients indexed by beta, use a countable compact exhaustion of its finitely many whole/sector domains. Partition-of-unity and Dolbeault graph representatives have all their fixed smooth derivative bounds in the same every-spatial-epsilon class: multiplication by each of the finitely many smooth cutoff functions changes only its compact constants. There are weights \(c_\beta>0\) such that
\[
\begin{gathered}
\frac{c_\beta}{\beta!\,a^{|\beta|}}\longrightarrow\infty\\
\quad\text{as }|\beta|\longrightarrow\infty\\
\quad\text{for every }a>0,\\
\sum_\beta c_\beta^2\|g_\beta\|_{L^2(K)}^2<\infty
\end{gathered}
\tag{ISF.18}
\]
on every compact \(K\), for every entry of this finite family and its finitely needed graph derivatives. Here is the full choice. For each integer \(k\), the input bounds at spatial epsilon \(1/(4k)\) control all first \(k\) compact sets, all finite entries and derivatives of order at most \(k\). Choose increasing thresholds \(n_k\) so that for total degree \(b\ge n_k\) all their constants times \(4^{-b}\) are at most \(2^{-b}\). Put \(k(b)=\max\{k:n_k\le b,\ k\le b\}\), with \(k(b)=1\) for an initial finite range, and \(c_\beta=\beta!k(|\beta|)^{|\beta|}\). Then \(k(b)\to\infty\), giving the first property. On each fixed compact, for sufficiently large \(b\), every weighted coefficient is at most \(2^{-b}\) by the choice at \(k(b)\); the number of degree-\(b\) multiindices is at most \((b+1)^d\), so its square sum converges. The finite remaining coefficients are harmless. Thus the weights are chosen once for the full family and all spatial epsilons, not separately for each beta.

The complete weighted existence proof [§5.43, DLH.1–DLH.4](#5-43-the-canonical-all-degree-action-and-its-derived-module-structure) extends to the Hilbert sum of these weighted coefficients with the same scalar normal/base weight. This extension follows directly from that proof: the integration-by-parts/Levi inequality is first summed over finitely many beta and then passed to the increasing sums of squares. A Cauchy sequence in the coefficient Hilbert sum converges coordinatewise, and its squared norm is bounded by the limit inferior of the finite sums, proving completeness. Finite component vectors are dense. The scalar graph core approximation applies to each of a finite number of components and then to their dense union, so all adjoint/graph identities persist. The proof's bounded functional and representing Hilbert vector therefore solve the joint closed equation with the same estimate. Choose the scalar convex exhaustion weight to dominate the locally finite total norms in equation ISF.18, exactly by [§5.43, DLH.4](#5-43-the-canonical-all-degree-action-and-its-derived-module-structure)'s explicit exhaustion construction; the sector exhaustions in equations (IG.7)–(IG.8) of [§5.45, IG.2](#5-45-the-full-geometric-infinite-order-kernel-ring-and-its-canonical-product) with no spatial normal faces are strictly plurisubharmonic and proper on the actual normal-sector/product domains. This choice depends on the whole finite cocycle family, not on beta, and no coefficient domain is changed.

Consequently the Dolbeault graph resolution is locally exact and fine in this strong coefficient family. The complete finite-resolution comparisons [§5.34, R.5–R.6](#5-34-full-current-resolutions-and-controlled-proper-image-comparison) and [§5.35, V.6](#5-35-relative-cones-unshifted-cup-and-the-normalized-trace) eliminate its finitely many positive Dolbeault columns on the whole and sector domains. The actual primitives stay locally in the same weighted Hilbert family: smooth compact multipliers and finite restrictions preserve its local square sums, so each next finite equation has the same weights from equation ISF.18. A holomorphic output coefficient on a compact polydisc satisfies the ordinary mean-value bound
\(\sup_{K'}|f_\beta|\le C_{K',K}\|f_\beta\|_{L^2(K)}\);
this follows by the Cauchy/mean-value formula in each coordinate and Cauchy–Schwarz. The Hilbert bound is \(C/c_\beta\), which by equation ISF.18 implies every bound in equation ISF.3 on that compact; the finite low degrees are bounded individually. A further fixed Cauchy margin controls each derivative. Thus the resulting holomorphic relative-cube representatives and every holomorphic zero primitive are actual ordinary infinite-order spatial operator sections on the same product. Solving each beta with an unrelated weight or shrinking would not prove this conclusion, and is not used.

Now unfold the source comonoidal-support stalk in its single normal coordinate as in the complete [§5.45, IG.1](#5-45-the-full-geometric-infinite-order-kernel-ring-and-its-canonical-product) proof. The compact unit-direction argument gives cofinal closed normal sectors with a strict polar margin; there is no spatial normal displacement in \(X\times_Y X\). A section over the compact normal phase circle has a finite angular cover, and each representative, transition equality and primitive occurs on a common later restriction/support index because each relative cube has finitely many entries. Use the preceding strong-coefficient elimination on the union of two neighbouring sectors as well. It gives literal holomorphic angular representatives whose discrepancies are whole-normal-face primitives on one smaller joint product, with equation ISF.3. Subtract their cumulative whole-face corrections around the finite ordered cover. After one circuit the remaining defect is one joint holomorphic spatial operator \(H\). This is exactly the strict angular datum of proof sections AN.1–AN.3, now derived from arbitrary decorated source sections with its all-index domain and zero-witness properties. No support-forgetting injection is assumed.

Apply the complete proof sections AN.2–AN.3 Cauchy/Laurent normalization on fixed normal contours. Its scalar contour bounds are independent of beta and preserve equation ISF.3. The normal polar terms give \(P_0\); for each normal epsilon choose a sufficiently small positive Laurent contour before extraction. Its bound is \(M_{r,\delta}r^m/m!\), simultaneously with \(\delta^{|\beta|}/\beta!\). Choosing \(r,\delta\) below a desired epsilon proves equation ISF.2. The angular mismatch gives the joint holomorphic \(K\), with equation ISF.3. The remaining whole-normal-domain entries are the explicit proof section AN.3 coboundaries. Conversely equations ISF.1–ISF.5 produce those angular entries and mismatches by normally convergent series. The AN zero test and the just-proved actual holomorphic zero witnesses establish both directions and zero detection. This is the source-defined decorated class, with neither coefficient-dependent domains nor a supported-distribution limit.

#### ISF.3. Distinct rings and a decisive spatial example

For \(R^\infty\), fixed shifts of proof section ISD.1 put its positive-normal part in equation ISF.2. Its finite spatial set and factorial negative bound give equation ISF.6 with one \(C\): for fixed beta and large \(q\), \((q-|\beta|)!\le q!\), and the extra factor \(q\) relative to \((q-1)!\) is absorbed by \(2^q\). Bound the finitely many other terms separately. On that finite spatial set delta changes only a finite constant. Consequently
\[
\begin{gathered}
A^\infty\subset R^\infty\subset\widetilde E^\infty_{q_0}
\xrightarrow{j_p}E^\infty_p .
\end{gathered}
\tag{ISF.9}
\]
The ordinary operator \(\sum_{b\ge0}D_{x_1}^b/(b!)^2\) is special by equation ISF.2 and has no uniform finite spatial degree, so is outside \(R^\infty\). On \(|z_1|<1\),
\[
\begin{gathered}
P=\tau^{-1}(1-z_1)^{-1}\\
=\sum_{b\ge0}\tau^{-b-1}\xi_1^b
\end{gathered}
\tag{ISF.10}
\]
is an actual finite-order \(E\)-operator, hence full \(E^\infty\). Its \(J\) coefficient at \(D_{x_1}^b\) is \((t-s)^b/b!\). For fixed nonzero \(t-s\), equation ISF.3 fails at \(\delta<|t-s|\). Its differential part is zero, and pair uniqueness excludes another special representation. This spatial translation kernel cannot be called local in \(x\) as an ordinary infinite-order germ operator.

The rings remain distinct before tensoring. Proof sections ISR.2–ISR.4 prove the particular module comparisons without identifying rings or exchanging algebraic balancing relations.

#### ISF.4. Full product and the retained endpoint cochains

Here is a direct spatial product bound, uniform in all additional holomorphic parameters. On nested base polydiscs with coordinate margin \(\rho>0\), let \(A=\sum a_\beta D_x^\beta\), \(H=\sum b_\gamma D_x^\gamma\) satisfy equation ISF.3. The full coefficient sum is
\[
\begin{gathered}
c_\delta=\sum_{\gamma\le\delta,\alpha\ge0}\\
\binom{\delta-\gamma+\alpha}{\alpha}\\
a_{\delta-\gamma+\alpha}\partial_x^\alpha b_\gamma .
\end{gathered}
\tag{ISF.11}
\]
Cauchy gives the absolute bound
\[
\begin{gathered}
|c_\delta|\le M_A M_H\,\frac{(2\epsilon)^{|\delta|}}{\delta!}\\
(1-\epsilon/\rho)^{-d},\\ \epsilon<\rho .
\end{gathered}
\tag{ISF.12}
\]
Indeed each term is bounded by
\(\epsilon^{|\delta|}(\epsilon/\rho)^{|\alpha|}/((\delta-\gamma)!\gamma!)\)
times \(M_AM_H\). Summing \(\gamma\le\delta\) gives \(2^{|\delta|}/\delta!\), and summing alpha is geometric. For any output epsilon choose an input epsilon at most half of it and below \(\rho/2\). The tail \(|\alpha|\ge L\) has an additional factor
\((\epsilon/\rho)^{L/2}(1-\sqrt{\epsilon/\rho})^{-d}\).
For fixed delta, truncating either input spatial degree gives the same tail with \(L\) reduced by \(|\delta|\). One extra fixed Cauchy margin controls every fixed derivative. The same proof in \(d+1\) variables gives the ordinary \(D^\infty\) product, including all normal indices. Thus no spatial cutoff is imposed.

For joint holomorphic \(J\), temporarily use its ordinary Volterra expression
\[
\begin{gathered}
V_Jf(t,x)=\\\int_a^t J(t,s,x,D_x)\\
f(s,x)\,ds
\end{gathered}
\tag{ISF.13}
\]
on a larger convex normal disc. The anchor \(a\) is fixed independently of \(x\), outside the smaller endpoint discs. Equation ISF.3 and Cauchy for \(f\) justify the full spatial sum. This computes a local kernel; it does not make \(D_t^{-1}\) a two-sided inverse on holomorphic functions. There are four product cases.

For \(P_0Q_0\), the full ordinary product was just proved. For a positive normal derivative times a negative kernel, repeated differentiation gives the exact identity
\[
\begin{gathered}
D_t^m V_J=V_{\partial_t^mJ}\\
+\sum_{k=0}^{m-1}D_t^{m-1-k}\\
\bigl((\partial_t^kJ)(t,t)\,\cdot\,\bigr).
\end{gathered}
\tag{ISF.14}
\]
Every spatial derivative is spread by equation ISF.11, with its actual coefficient order. In the full \(P_0\) sum, the normal bound \(\epsilon^m/m!\) cancels the integral derivative factor \(m!\eta^{-m}\), where \(\eta>0\) is a fixed joint normal Cauchy margin. The sum is bounded by \(\sum_m(\epsilon/\eta)^m\).

For its ordinary boundary coefficient at \(D_t^l\), \(m\ge l+1\), a valid bound is
\(4^m m!\eta^{-(m-l)}/l!\) times a joint kernel norm, enlarged by \(\max(1,\eta)\).
To verify it, expand \(D_t^{m-1-k}\) in equation ISF.14. The \(l\)-coefficient has binomial factor \(\binom{m-1-k}{l}\). Its remaining diagonal derivative is a sum of at most \(2^{m-1-k-l}\) mixed \(t,s\) derivatives of total order \(m-1-l\), bounded by \((m-1-l)!\eta^{-(m-1-l)}\). The binomial factor is at most \(m!/(l!(m-l)!)\), and the number of summands is at most \(m\le2^m\). The claimed bound follows. After multiplication by \(\epsilon^m/m!\), the output is bounded by
\((4\epsilon)^l/l!\) times the geometric sum
\(\sum_{m-l\ge1}(4\epsilon/\eta)^{m-l}\).
Combining equation ISF.12, with independent spatial epsilons chosen first, proves equation ISF.2 and equation ISF.3 for both resulting parts on one product.

For a negative kernel times \(P_0\), integrate by parts after spreading all spatial derivatives:
\[
\begin{gathered}
\int_a^t H(t,s)D_s^m f(s)\,ds\\
=\sum_{k=0}^{m-1}(-1)^k\\
\bigl[\begin{gathered}
(\partial_s^kH)(t,s)\\D_s^{m-1-k}f(s)
\end{gathered}\bigr]_{s=a}^{s=t}\\
+(-1)^m\int_a^t(\partial_s^mH)(t,s)\\
f(s)\,ds .
\end{gathered}
\tag{ISF.15}
\]
Here \(H\) includes the actual ordered spatial product of \(J(t,s)\) and the coefficient of the \(m\)-th source normal derivative. The upper endpoint gives the ordinary \(D^\infty\)-part; the integral gives the joint holomorphic \(J\)-part. On an intermediate product, the \(s\)-Cauchy estimate for \(H\) and the preceding diagonal/Leibniz bound give the same geometric normal tails and every-epsilon positive bounds. All spatial sums are controlled by equation ISF.12.

The lower endpoint is retained. Represent evaluation at the separated anchor by the positive Cauchy kernel \((s'-a)^{-1}/(2\pi i)\). The exact kernel contributed by the lower endpoints in equation ISF.15, with the source normal top-form \(ds'\), is
\[
\begin{gathered}
H_a(t,s')=
-\sum_{m\ge1}\sum_{k=0}^{m-1}\\
(-1)^k(\partial_s^kH_m)(t,a)\,\\
\frac{(m-1-k)!}{2\pi i\,(s'-a)^{m-k}} .
\end{gathered}
\tag{ISF.17}
\]
Here \(H_m\) is the actual ordered spatial kernel product for the \(m\)-th positive source derivative. The minus sign is the lower end of \([\,\cdot\,]_a^t\), and the derivative factorial is the Cauchy formula for \(f^{(m-1-k)}(a)\). On the endpoint product \(|a-s'|\) has one positive lower margin. The input \(\epsilon^m/m!\) cancels the mixed \(k!\,(m-1-k)!\) derivative/evaluation bound, and \(m\le2^m\) absorbs its finite boundary sum. Choose epsilon below both fixed margins. This makes equation ISF.17 normally holomorphic in \((t,s')\), including \(t=s'\); equation ISF.12 supplies the spatial operator bounds for every epsilon. It is a literal whole-normal-face cochain and its relative-cube coboundary. The complete [§5.44, AN.3](#5-44-full-homogeneous-coefficients-and-ordinary-infinite-order-kernels) and [§5.45, IG.4](#5-45-the-full-geometric-infinite-order-kernel-ring-and-its-canonical-product) give that normal-face formula; equation (IM.15) in [§5.45, IM.5](#5-45-the-full-geometric-infinite-order-kernel-ring-and-its-canonical-product)'s \(N=1\) whole-domain exception retains precisely such an entry with the calibrated trace sign. This is not a discarded distribution or a zero inferred after evaluating a solution.

For two negative kernels the output Volterra kernel is
\[
\begin{gathered}
L(t,s,D_x)=\\\int_s^t J(t,v,D_x)\\
H(v,s,D_x)\,dv .
\end{gathered}
\tag{ISF.16}
\]
Parametrize \(v=s+\theta(t-s)\), \(0\le\theta\le1\). One small convex normal disc puts the entire compact segment family in the larger joint product. Equation ISF.12 gives every spatial epsilon bound uniformly there, and integration multiplies it by a bounded length. Thus \(L\) is jointly holomorphic with equation ISF.3. Normal Taylor expansion gives the exact beta integral and
\(\binom{-q}{k}=(-1)^k\binom{q+k-1}{k}\)
of the complete earlier [§5.33, K.3](#5-33-a-convergent-negative-order-polynomial-kernel-complex). Equation ISF.12 now justifies all its spatial infinite sums; the joint normal Taylor radius gives all normal tails. Therefore it is the actual full negative-normal symbol product.

All outer, intermediate, endpoint and anchor margins are fixed before any index or epsilon. The estimates prove absolute normal convergence of every normal/spatial contraction, every fixed derivative tail and all joint domains. Finite normal/spatial truncations have the proved canonical mixed pole/log trace with literal complement terms. The estimates pass their actual holomorphic formulas and face primitives to normally convergent limits on the same domains. The bounded AN/IG/IM normalization and zero detector identify this product with the canonical full geometric product by the full argument in [§5.45, IM.4–IM.5](#5-45-the-full-geometric-infinite-order-kernel-ring-and-its-canonical-product), with no closed-image or cohomology-limit assumption. One may equivalently compare the actual coefficient products above and apply injective proof section IG.4; every whole-normal-face endpoint and log/harmonic correction has been represented by its explicit face coboundary.

In equation ISF.16, \(J_{\rm out}=L\), so the special log coefficient is \(K_{\rm out}=-L\), consistent with equation ISF.5. The same \(J=-K\) convention applies to equations ISF.14–ISF.15. The unit is \((1,0)\). Actual associativity follows from the full proved ring product and the injective pair map; finite triple families have common products/collars by proof section ISR.6. This proves the decorated special sheaf of rings and all \(j_p\) homomorphisms. Spatial locality follows from equation ISF.3's holomorphic-germ estimate; it is not transferred to all full \(E^\infty\)-operators.

#### ISF.5. Retained boundaries

The chosen-fibration special sheaf has the stated exact joint bounds, full product and maps to the full ring. The full special tensor for arbitrary monodromy ideals and the prescribed-complement D-type realization are proved in §5.79, FB.D–FB.F. Complex relative propagation for the whole generic-position actual complex is proved in Section5.80, CP.F--CP.G. Common continued solutions, section separation and faithful realization of an arbitrary holonomic module remain separate mathematical obligations.

**Full infinite balanced reconstruction for the entire generic holonomic germ.**

The full division is proved in ISD.1–ISD.5, the distinct decorated special source sheaf in ISF.1–ISF.5, and the complete tail estimates in ISB.1–ISB.5. The complete earlier right-flatness theorem is [§5.47, MF.1–MF.7](#5-47-arbitrary-finite-matrices-and-right-flatness-of-the-full-infinite-order-ring). Faithful extension is [§5.48, FEP.1–FEP.9](#5-48-proper-ideals-and-faithful-extension-to-the-full-infinite-order-ring); it is used for proof section ISR.5, not for division or the balanced inverse maps.

#### ISR.1. Definitions and full finite input

Let \(q_0=0\), \(p_0=(0;1,0)\), \(E_p\) the finite-order stalk, \(B_p=E^\infty_p\), and
\[
\begin{gathered}
N=M^\infty_{p_0}=B_{p_0}\otimes_{E_{p_0}}M_{p_0}.
\end{gathered}
\tag{ISR.1}
\]
This is an ordinary algebraic tensor of a right finite-ring module with a left module. The whole finite holonomic germ, including singular branches and nonreduced principal fibres, has the lattice, actual generic-position reduction and full-chart finite polynomial complex of complete earlier [§5.28, Q.1–Q.7](#5-28-convergent-contact-quantization-and-whole-germ-generic-position), [§5.29, G.1–G.5](#5-29-actual-finite-generation-over-the-coordinate-commuting-ring), [§5.30, F.1–F.7](#5-30-actual-polynomial-flatness-and-reconstruction-after-finite-realization), [§5.31, L.1–L.6](#5-31-a-bounded-actual-generic-position-complex-with-polynomial-operator-matrices) and [§5.32, J.1–J.3](#5-32-coherent-actual-lattices-and-principal-support-for-the-whole-germ). No generic smooth stratum or quotient of supported sections replaces it. Keep the actual generators \(U\) and \(B_\nu\) of equation ISD.3.

The finite module \(M\) is generated by \(U\) over \(E\); hence \(N\) is generated by their images over \(B_{p_0}\). Equation ISD.12 applies to every finite row of coefficients and kills the spatial relations in \(N\). Therefore
\[
\begin{gathered}
N=\sum_{i=1}^r A^\infty u_i .
\end{gathered}
\tag{ISR.2}
\]
This is generation, not a free-module identification or a finite presentation over \(A^\infty\). All other actual relations of \(M\) remain after tensoring. In particular its normal torsion, nilpotent spatial actions and singular supported sections have not been removed. This conclusion does not require extension-faithfulness.

\(R^\infty\) is a single base-germ ring: finite spatial polynomials with coefficients in \(A^\infty\) extend to every finite normalized covariable product above \(q_0\), with their original base coefficient domain. Proof section ISF.2 gives the maps \(j_p:\widetilde E^\infty_{q_0}\to B_p\), with \(R^\infty\subset\widetilde E^\infty_{q_0}\). Thus both tensors
\[
\begin{gathered}
T_R(p)=B_p\otimes_{R^\infty}N,\\
T_S(p)=B_p\otimes_{\widetilde E^\infty_{q_0}}N
\end{gathered}
\tag{ISR.3}
\]
have literal right actions on their first factor and left actions on \(N\) from the indicated maps. The middle tensor of the fibre reconstruction formula is \(T_S\), not a tensor over \(A^\infty\). No tensor is completed.

#### ISR.2. The first balanced isomorphism along the ray

On the complex ray \(p\in\mathbf C^\times p_0\), the proved projectivized coefficient/geometric construction canonically identifies the stalks \(B_p\) and \(B_{p_0}\): the homogeneous coefficient sequence is the same, with the ambient nonzero \(\tau\)-coordinate and its actual angular branch transitions. Proof sections IG.2–IG.5 supply their intrinsic class overlap, including log-face corrections; it is not an identification of real-conic rings. The finite module and its presentation have the same conic identification.

Define
\[
\begin{gathered}
\mu_R:T_R(p)\longrightarrow N,\\
b\otimes n\longmapsto bn,\\
\iota_R:N\longrightarrow T_R(p),\\
n\longmapsto1\otimes n .
\end{gathered}
\tag{ISR.4}
\]
The multiplication is balanced over \(R^\infty\): \((ba)n=b(an)\) is the actual ring/module associativity. Its unit gives \(\mu_R\iota_R=1\). To prove the other identity, let a tensor element involve finitely many \(b\otimes n\). By equation ISR.2 write \(n=\sum_i c_i u_i\) with actual \(c_i\in A^\infty\), on a common product for that finite family. Balancing reduces it to a finite coefficient row \(F_i=b\circ c_i\) tensored with \(U\). Apply the full equation ISD.12 decomposition to \(F\). For each spatial relation, algebraic balancing gives
\[
\begin{gathered}
\sum_i(a_\nu(D_\nu I-B_\nu))_i\\
\otimes u_i\\
=\sum_j(a_\nu)_j\otimes\\
((D_\nu I-B_\nu)U)_j=0 .
\end{gathered}
\tag{ISR.5}
\]
Every entry of \(D_\nu I-B_\nu\) is in \(R\subset R^\infty\). For the actual remainder \(s\in(A^\infty)^r\),
\[
\begin{gathered}
\sum_i s_i\otimes u_i
=1\otimes\sum_i s_i u_i .
\end{gathered}
\tag{ISR.6}
\]
Thus every tensor equals \(1\otimes\mu_R\) of itself. This proves \(\iota_R\mu_R=1\), with finite algebraic identities and actual full coefficients. It does not choose a unique simultaneous quotient.

The maps are \(B_p\)-linear. Multiplication already is; the inverse's linearity follows either from its two inverse identities or directly from
\(\iota_R(bn)=b\iota_R(n)\), proved by the same equation ISD.12 balancing for that finite row. Consequently
\[
\begin{gathered}
B_p\otimes_{R^\infty}N\xrightarrow[\;\iota_R\;]{\;\mu_R\;}N\\
\quad\text{is an actual }B_p\\
\text{-module }\text{isomorphism}.
\end{gathered}
\tag{ISR.7}
\]
The displayed arrows denote mutually inverse maps with the formulas in equation ISR.4.

These maps are natural for every actual \(B_{p_0}\)-linear morphism of such extensions: multiplication commutes with that morphism and the unit formula does too. Their finite diagrams commute because both composites are their literal formulas, not because a particular chosen division is natural. All entries and finite balancing witnesses have a common representative product by proof sections ISD.3–ISD.4. No unrestricted infinite family of germs is assigned one universal representative neighbourhood.

#### ISR.3. The distinct special balancing, with both inverse maps

For \(p\in\mathbf C^\times p_0\), define
\[
\begin{gathered}
\mu_S:T_S(p)\longrightarrow N,\\
b\otimes n\longmapsto bn,\\
\iota_S:N\longrightarrow T_S(p),\\
n\longmapsto1\otimes n .
\end{gathered}
\tag{ISR.8}
\]
They are balanced by the genuine special ring map in proof section ISF.4 and its restriction action on \(N\). Let \(q:T_R(p)\to T_S(p)\) be the quotient adding the special balancing relations; its elementary tensors have the same formula. It is surjective because algebraic tensor generators are still the same \(b\otimes n\). Clearly \(\mu_Sq=\mu_R\), \(q\iota_R=\iota_S\), and \(\mu_S\iota_S=1\). For every element of \(T_S\), choose a finite tensor lift \(v\in T_R\). Then
\[
\begin{gathered}
\iota_S\mu_S(qv)
=q\iota_R\mu_R(v)=qv .
\end{gathered}
\tag{ISR.9}
\]
This proves both inverse identities and \(B_p\)-linearity. It also proves \(q\) is an isomorphism: its inverse is \(\iota_R\mu_S\). Each additional special balancing relation was already zero under the inverse multiplication in \(T_R\); no identification of the two coefficient rings is inferred.

Thus the first and middle reconstruction tensors agree through actual balanced maps:
\[
\begin{gathered}
B_p\otimes_{R^\infty}N\\
\xrightarrow{\ q\ }B_p\otimes_{\widetilde E^\infty_{q_0}}N\\
\xrightarrow{\ \mu_S\ }N ,
\end{gathered}
\tag{ISR.10}
\]
with inverses \(\iota_R\mu_S\) and \(\iota_S\), respectively. This is not an isomorphism involving \(B_p\otimes_{A^\infty}N\); no such tensor is substituted.

#### ISR.4. Exact off-ray vanishing

Let \(p=(q_0;\tau,\xi)\), \(\tau\ne0\), with \(z=\xi/\tau\ne0\). Choose \(\nu\) with \(z_\nu(p)\ne0\). Since \(A_\nu(0)\) is nilpotent, of index at most \(r\),
\[
\begin{gathered}
(z_\nu(p)I-A_\nu(0))^{-1}\\
=\sum_{k=0}^{r-1}z_\nu(p)^{-k-1}A_\nu(0)^k .
\end{gathered}
\tag{ISR.11}
\]
On one small product about \(p\), its holomorphic principal matrix inverse is therefore defined with a positive nonvanishing determinant margin. The actual normalized \(L_\nu=z_\nu I-C_\nu\) has a finite-order inverse there. For completeness, let \(H\) be its principal holomorphic inverse, as an actual order-zero matrix. Both \(H\circ L_\nu-I\) and \(L_\nu\circ H-I\) have order at most minus one with factorial bounds on the product. The complete lowering argument in proof sections ISB.1–ISB.4 gives the normally controlled actual negative-order inverses of \(I+(H L_\nu-I)\) and \(I+(L_\nu H-I)\), with finite upper order and one smaller product. They produce a left and a right inverse of \(L_\nu\); associativity makes them equal. This is an actual finite \(E_p\) inverse. Because \(D_\nu I-B_\nu=\tau\circ L_\nu\), its inverse is \(L_\nu^{-1}\circ h\).

In \(T_R(p)\), the images \(1\otimes u_i\) obey the matrix relation \(D_\nu I-B_\nu\) by its literal entries in \(R^\infty\). Multiplication by the displayed actual inverse makes every image zero. By equation ISR.2 the images of \(U\) generate the tensor, so \(T_R(p)=0\). The quotient \(T_R(p)\to T_S(p)\) gives \(T_S(p)=0\). The balanced inverse comparison between the two zero modules is their unique zero map; the target of the fibre reconstruction formula is zero there, not \(N\).

We have proved the full source fibre formula
\[
\begin{gathered}
B_p\otimes_{R^\infty}N\\
\simeq B_p\otimes_{\widetilde E^\infty_{q_0}}N\\
\simeq\begin{cases}
N,&p\in\mathbf C^\times p_0,\\
0,&\begin{gathered}p\in\{\tau\ne0\}\\\cap\pi^{-1}(q_0),\\p\notin\mathbf C^\times p_0 .\end{gathered}
\end{cases}
\end{gathered}
\tag{ISR.12}
\]
Each off-ray inverse has its own open nonvanishing principal margin; it is not extended across the excluded ray. A finite off-ray diagram uses a finite common neighbourhood at its points and a finite compact covering if necessary. This is compatible with the fixed base-germ special/polynomial coefficients; it does not assert a holomorphic inverse on a product containing its zero determinant.

#### ISR.5. Whole support, singular branches and actual contact transport

The complete faithful-extension theorem [§5.48, FEP.1–FEP.9](#5-48-proper-ideals-and-faithful-extension-to-the-full-infinite-order-ring) gives unit injectivity and exact faithful extension for every algebraic left module and every sheaf. Thus the full extension of the whole finite actual sheaf has
\[
\begin{gathered}
M\hookrightarrow E^\infty\otimes_E M,\\
\operatorname{Supp}(E^\infty\otimes_E M)\\
=\operatorname{Supp}M .
\end{gathered}
\tag{ISR.13}
\]
The complete [§5.32, J.2](#5-32-coherent-actual-lattices-and-principal-support-for-the-whole-germ) proves that the right side is the reduced principal support of the actual lattice, including every branch. Thus the convention defining support by this full \(E^\infty\)-extension now agrees with that finite support. This does not assert purity of individual infinite-order sections or any equality involving \(E^R\).

The finite bounded polynomial complex of earlier [§5.31, L.1–L.6](#5-31-a-bounded-actual-generic-position-complex-with-polynomial-operator-matrices) remains an exact complex resolving the extension by the complete right-flatness theorem [§5.47, MF.1–MF.7](#5-47-arbitrary-finite-matrices-and-right-flatness-of-the-full-infinite-order-ring). Every differential and its finite maps use one full base neighbourhood and length at most \(2d+1\). The terminal exactness is the proved scalar extension of that actual finite complex; no exactness over \(R^\infty\) or Noetherianity of \(A^\infty,E^\infty\) is inferred.

For clarity the particular generic-position contact shear also extends to the full class, so it transports the original full extension rather than only its finite module. Keep the exact earlier \(a_C(\rho)=\frac12\xi^{\mathsf T}C\xi/\tau\), \(b_C=\partial_\rho a_C\), \(R_a\) and coefficient formula in equations (Q.9)–(Q.10) of [§5.28, Q.4](#5-28-convergent-contact-quantization-and-whole-germ-generic-position). For an arbitrary input degree \(m\in\mathbf Z\), a resummation word of total loss \(K\) has total base derivative order \(K+\ell\le2K\). The complete [§5.28, Q.5](#5-28-convergent-contact-quantization-and-whole-germ-generic-position) counting and \( (K+\ell)!/\ell!\le4^K K!\) give
\[
\begin{gathered}
\|\begin{gathered}
\text{all transform terms }\\
\text{of loss }K
\end{gathered}\|\\
\le L^K K!\,\|p_m\|_{\rm outer}.
\end{gathered}
\tag{ISR.14}
\]
The constants and shifted base/covariable products are fixed before all \(m,K\); the estimate has no finite-upper-order assumption because \(\tau^m\) is an ordinary normal prefactor and \(R_a\) differentiates only the base coefficient.

For positive output \(j\), input \(m=j+K\) gives
\[
\begin{gathered}
\|q_j\|\le M_\epsilon\frac{\epsilon^j}{j!}
\sum_{K\ge0}(L\epsilon)^K ,
\end{gathered}
\tag{ISR.15}
\]
using \(K!j!\le(j+K)!\). Given an output epsilon use a smaller input epsilon with \(L\epsilon<1/2\). For negative output \(-n\), the negative inputs \(K<n\) give \(M n!(C+L)^n\); positive inputs \(m=K-n\ge0\) give
\[
\begin{gathered}
\|q_{-n}\|\le M n!(C+L)^n\\
+M_\epsilon n!L^n(1-L\epsilon)^{-n-1}.
\end{gathered}
\tag{ISR.16}
\]
The last identity is the convergent binomial sum
\(\sum_{m\ge0}\binom{n+m}{n}(L\epsilon)^m\).
Every spatial index is in the holomorphic coefficient norm. Positive loss tails are geometric; the positive-input tail in equation ISR.16 starting at \(m>J\) is bounded by
\[
\begin{gathered}
M_\epsilon n!(2L)^n
\frac{(2L\epsilon)^{J+1}}{1-2L\epsilon}
\\(2L\epsilon<1).
\end{gathered}
\tag{ISR.17}
\]
Fixed derivatives use one further common Cauchy margin. These estimates give actual full-class continuity of the transformation and of its compositions.

Truncate only the positive homogeneous head at \(J\); the negative actual tail stays intact, so these are finite-order inputs. The complete earlier [§5.28, Q.6](#5-28-convergent-contact-quantization-and-whole-germ-generic-position) proves product preservation and \(T_{-C}T_C=T_CT_{-C}=1\) on each such input. Equations ISR.15–ISR.17 and the proved full product bounds pass their coefficients and every derivative tail to normally convergent limits on common intermediate products. They prove the same identities for every full input. Therefore \(T_C\) is an actual full sheaf ring isomorphism extending the finite contact quantization, with inverse \(T_{-C}\). Its coordinate values and contact signs are exactly equation (Q.18) of [§5.28, Q.6](#5-28-convergent-contact-quantization-and-whole-germ-generic-position); no new formal exponential operator is admitted.

Restrict the module action along this actual isomorphism: \(a\cdot_{\rm new}m=T_C(a)\cdot_{\rm old}m\), with \(T_C:B_{\rm new}\to B_{\rm old}\) extending the corresponding finite-ring isomorphism. Its tensor-extension comparison is the explicit balanced map
\[
\begin{gathered}
B_{\rm new}\otimes_{E_{\rm new}}M_{\rm new}\\
\longrightarrow B_{\rm old}\otimes_{E_{\rm old}}M_{\rm old},\\
b\otimes m\longmapsto T_C(b)\otimes m
\end{gathered}
\tag{ISR.18}
\]
with the finite action transported by the same restriction; the inverse uses \(T_{-C}\). Balancing follows from multiplicativity on the finite subring, and both compositions are identities. Thus every original whole holonomic germ and its full extension can be made generic by the proved shear, including its singular points, and the conclusions of equations ISR.2–ISR.13 apply to that transported whole germ. The decorated special sheaf belongs to the resulting chosen fibration; it is not asserted independent of that choice.

#### ISR.6. Common products, support enlargement, overlaps and trace signs

All finite inputs, divisors, coefficient operations, identities and inverses are first represented on one common actual product. Equation ISD.4 then fixes weighted base/covariable radius profiles; proof section ISD.3 fixes its finite outer/intermediate/final ladder. Every homogeneous index, spatial Taylor index and every output epsilon uses that same final product. The special normal kernel equation ISF.4 has a joint normal product with one radius independent of spatial epsilon. Its product proof fixes a single convex normal disc and a separated anchor before summation.

To pass these actual operators and identities to the proved geometric kernels, choose common final Taylor radii \(R_i>0\) and a common negative factorial constant \(C_-\ge1\) for the finite list. The complete proved IG/IM/IK estimates in [§5.44](#5-44-full-homogeneous-coefficients-and-ordinary-infinite-order-kernels) and [§5.45](#5-45-the-full-geometric-infinite-order-kernel-ring-and-its-canonical-product) give thick complement margins
\[
\begin{gathered}
A_i>2/R_i,\\
|\omega|<1/(4C_-),\\
|\zeta_i|>A_i|\omega|,\\\zeta_i=y_i-x_i .
\end{gathered}
\tag{ISR.19}
\]
On each compact choose \(Y>\max_i(R_i|\zeta_i|)^{-1}\) with
\(|\omega|Y<3/4\), and a positive epsilon with \(\epsilon Y<1/8\). The mixed Taylor majorant has \(r<1/4\), \(q<3/4\), \(x+q<7/8\), including positive-homogeneous/negative-normal terms. This is the exact proved bound; choosing epsilon does not shrink the product. The removable \(\omega=0\) extension and a finite compact cover provide the same bounds on the actual complement.

Choose the finite phase grid \(h_{\rm grid}<2\lambda\), \(2\lambda+h_{\rm grid}<\pi\), and spatial cone constants \(B_i>A_i\). Use a pointed full-dimensional convex cone at each phase for its finite operator/product list and a separate pointed midpoint cone on each neighbouring overlap. Only the compact proper intermediate collar and endpoint/source margins are uniform over the finite phase family. No pointed cone contains the full phase circle.

The output on a convex cone \(G_\phi\) is literally mapped by support forgetting to one larger standard
\(T_{\lambda',B',\phi}\), with
\[
\begin{gathered}
\lambda<\lambda'<(\pi-h_{\rm grid})/2,\\
B'_i>\\B_i/\cos(\lambda+h_{\rm grid}/2).
\end{gathered}
\tag{ISR.20}
\]
This includes the required \(B'_i>B_i/\cos\lambda\). Choose the smaller admissible covariable product and the finite polar margins before any coefficient/tail normalization. On this standard cone use the full cube and exhaustions in equations (IG.7)–(IG.8) of [§5.45, IG.2](#5-45-the-full-geometric-infinite-order-kernel-ring-and-its-canonical-product). The resulting boundary is on \(T'\); map it forward to a further convex \(G'\) containing \(T'\), or keep that exact common final support. Support-forgetting injectivity is never inferred.

The literal cups, full two-support cubes, excision maps and traces use the proper common collar of proved TC/IM arguments in [§5.44, TC.1–TC.4](#5-44-full-homogeneous-coefficients-and-ordinary-infinite-order-kernels) and [§5.45, IM.1–IM.6](#5-45-the-full-geometric-infinite-order-kernel-ring-and-its-canonical-product). Their base-antiholomorphic components are retained before trace. The full-cone Čech top coordinate is \(-r_N f\); its conversion in degree \(N-1\) gives the ordinary top cube coordinate
\[
\begin{gathered}
r_N(-1)^N f,\\ r_N=(-1)^{N(N-1)/2}.
\end{gathered}
\tag{ISR.21}
\]
The omitted-coordinate complement faces in degree \(N-2\) acquire \((-1)^{N-2}\). At \(N=1\) there is a whole-domain normal-face primitive and the exact trace \((0,-\oint f\,da)\), with positive residue scalar; equation ISF.15 retains the lower endpoint, whose explicit cochain is equation ISF.17. These conventions are unchanged by the division or balanced tensors.

Actual nonlinear base overlaps use the proved [§5.45, IG.5](#5-45-the-full-geometric-infinite-order-kernel-ring-and-its-canonical-product) pair pullback, Jacobian/top-form conversion and coordinate zero test. The chosen division coefficients need not be coordinate invariant: its sum identity is an actual ring/module identity and pulls back as such. Multiplication and unit maps in equations ISR.4 and ISR.8 are intrinsic and hence agree on all overlaps; their inverse identities show that no choice of coefficient normal form changes them. Equations ISR.14–ISR.18 separately handle the generic-position contact shear, beyond nonlinear base-coordinate overlaps.

#### ISR.7. Calibrations, illustration locators and retained obligations

Equations ISD.14–ISD.15 give exact full-spatial and full-positive-head division with the nonzero nilpotent \(J\) retained. The scalar two-variable tuple \((D_2,-D_1)\) proves nonuniqueness of arbitrary simultaneous coefficient tuples. Equation ISF.10 exhibits an actual finite-order full operator excluded from the decorated special class. Equation ISR.11 is the exact finite nilpotent off-ray inverse, while equations ISR.5–ISR.9 give both balanced inverse maps. The complete earlier [§5.31, L.8](#5-31-a-bounded-actual-generic-position-complex-with-polynomial-operator-matrices) example retains its normal relation \(tI-h\operatorname{diag}(0,1)\) and the essential differential correction \(T_*-hI\).

The figures below show exact stated real slices and algebraic diagrams of these formulas. Their editable SVGs, exact rational data and reproducible Python source accompany their captions. Every coordinate, parameter and proof locator is specified; they illustrate the complete arguments.

The proof establishes full infinite sum division and both algebraic balanced reconstruction maps, the distinct special fibration class used by the middle tensor, every growth/domain/spatial/tail estimate, whole-germ singular and branch scope, off-ray vanishing and full contact transport. The full infinite coefficient class is used throughout.

The following remain separate: other scalar changes (including mixed \(E^\infty/D^\infty\), real-conic \(E^R\) and their module injections), moving-sector propagation and continuation, infinite section purity/separation, finite-dimensional solution families and monodromy enlargement, D-type/finite-pole reconstruction, intrinsic order comparison, C1, half-order/contact conventions beyond the explicit shear, arbitrary analytic proper/support-proper regularity. Section5.90, GG.A–GG.G supplies the initial ordinary generators near compact sets.

The complete lowering argument is included here. It controls the actual Spath quotient inverses; its coefficient maps are bounded complex linear and every shrinking is fixed before all homogeneous indices. The exact earlier coefficient-norm premise is [§5.47, MFA.1–MFA.4](#5-47-arbitrary-finite-matrices-and-right-flatness-of-the-full-infinite-order-ring).

**Full infinite-order order-lowering estimate.**

This appendix proves convergence of the series used for arbitrary matrix relations. It includes every normalized spatial derivative, arbitrary many correction steps, the unbounded positive head, the factorial negative tail, and a single domain chosen before the summation indices.

#### ISB.1. Operators and one domain

In a cotangent chart \(\tau\ne0\), write \(F_j=\tau^jp_j(u,z)\), \(z=\xi/\tau\). The actual class has one holomorphic coefficient product, with, on each compact,
\[
\begin{gathered}
|p_j|\le M_\varepsilon\varepsilon^j/j!\\
(j\ge0, \varepsilon>0),\\
|p_{-n}|\le MD^nn! (n\ge1).
\end{gathered}
\tag{ISB.1}
\]
We use the absolute Taylor norms \(MFA.1\) on one family \(R(r)\), \(r\in[r_0/2,r_0]\). A Cauchy estimate on a fixed enlargement converts the input compact bounds to these norms with constants independent of \(j\). Coefficient operators \(h,p,k\), or a finite composition of these, have uniform norm \(H\) on that interval by proof sections MFA.2–MFA.3. Every right-hand actual matrix is order zero, with coefficients \(Q_{-q}\) satisfying \( |Q_{-q}|\le AC^q q!\) on a fixed enlargement, including \(q=0\). The finite list has one such bound and domain.

The right Leibniz term acting on a left coefficient has indices
\[
\begin{gathered}
\alpha\in\mathbf N^N,\quad q\ge0,\\
K=q+|\alpha|,\\
\frac1{\alpha!}(\partial_\rho^\alpha F_j)(\partial_u^\alpha Q_{-q}).
\end{gathered}
\tag{ISB.2}
\]
It lowers homogeneous degree by \(K\). Subtracting its principal ordinary multiplication removes precisely \(q=0,\alpha=0\); every remaining term lowers degree by at least one. The coefficient operators constructed in [§5.47, MFA.1–MFA.4](#5-47-arbitrary-finite-matrices-and-right-flatness-of-the-full-infinite-order-ring) preserve degree. A word of correction terms interspersed with such coefficient operators therefore has total loss \(K\ge\) its number of correction terms. All matrix and component sums are finite.

Choose the same fixed outer margin for the right factors throughout. Its Cauchy estimate costs \(\alpha!\delta^{-|\alpha|}\); the factorial in \(ISB.2\) cancels it. If a total-order version is used, \(|\alpha|!/\alpha!\le N^{|\alpha|}\), so the difference is absorbed in an exponential constant. Across a word with tail indices \(q_i\), let \(Q=\sum_iq_i\) and \(L=\sum_i|\alpha_i|\), thus \(K=Q+L\). Their tail factorial product is at most \(Q!\).

#### ISB.2. Derivatives interspersed with coefficient divisions

At normalized \(\tau=1\), one covariable differentiation is exactly
\[
\begin{gathered}
\partial_{\xi_a}F_j=\tau^{j-1}\partial_{z_a}p_j,\\
\partial_\tau F_j=\tau^{j-1}(j-E_z)p_j,\\
E_z=\sum_a z_a\partial_{z_a}.
\end{gathered}
\tag{ISB.3}
\]
These equations retain every spatial index. They also specify all signs. At an intermediate step starting from input degree \(m\), the current degree has absolute value at most \(|m|+K\). A coefficient operator preserves that degree, although it need not commute with \(E_z\) or differentiation.

To bound a word, insert intermediate radii decreasing from \(r_0\) to \(r_0/2\), allocating equal scalar radius loss \(r_0/(2L)\) to each of its \(L\) derivatives; when \(L=0\) no derivative margin is consumed. All \(R_a(r)=r^{v_a}\) have a fixed positive lower derivative on this compact interval, so each coordinate margin is at least \(c/L\), with fixed \(c>0\). Cauchy's formula, the bounded normalized \(z_a\), and \(ISB.3\) bound each one-derivative step by
\[
\begin{gathered}
C_0(|m|+K+L)\\
\le 2C_0(|m|+K)\quad\\
(K>0).
\end{gathered}
\tag{ISB.4}
\]
The \(H\) bound for every inserted coefficient operator holds at its own intermediate radius. Thus no commutation of a division operator with a derivative has been used, and no new radius is lost for a division operator. The count of those operators is bounded by a fixed multiple of the number of correction factors, hence by a fixed multiple of \(K\).

For \(M=|m|\ge0\), \(K=Q+L>0\),
\[
\begin{gathered}
(M+K)^LQ!\\
\le e^K\frac{(M+K)!}{M!}.
\end{gathered}
\tag{ISB.5}
\]
To prove it, \(Q!\le(M+K)^Q\). Also
\(\prod_{a=1}^K(M+a)/(M+K)\ge K!/K^K\ge e^{-K}\): each factor increases with \(M\), and the last inequality follows from
\(\sum_{a=1}^K\log a\ge\int_1^K\log t\,dt=K\log K-K+1\).
Equations \(ISB.4\)–\(ISB.5\), submultiplicativity and the right-factor Cauchy estimates bound each word, before coefficient summation, by an exponential constant times that one factorial ratio.

There are at most exponentially many words of total loss \(K\). Ordered compositions of a positive integer \(K\) number \(2^{K-1}\); their subdivisions into \(q_i\) and the \(N\) derivative components number at most \((N+2)^{2K}\), a deliberately loose bound obtained by placing separators among \(K\) units and coloring the units. Finite matrix entries and coefficient-operator choices cost another exponential factor. Absorb all these constants, all fixed input bounds, and \(H\) into \(C_1\ge1\). The sum of the absolute norms of **all** terms of loss \(K\), for any one of the Neumann or iterated correction expressions below, is bounded by
\[
\begin{gathered}
C_1^K\frac{(|m|+K)!}{|m|!}|p_m|_{R(r_0)}.
\end{gathered}
\tag{ISB.6}
\]
The case \(K=0\) is the single bounded coefficient operation and is covered by a fixed prefactor. This argument also applies to truncated words uniformly in the truncation; the bound includes each contraction and every tail coefficient rather than replacing them by a formal product.

#### ISB.3. Positive output, negative output and explicit tails

For positive output \(j\ge0\), input degree is \(m=j+K\ge0\). The integer inequality
\[
\begin{gathered}
\frac{(j+2K)!j!}{(j+K)!^2}\le\binom{2K}{K}\\
\le4^K
\end{gathered}
\tag{ISB.7}
\]
follows by writing the first ratio as \(\prod_{a=1}^K(j+K+a)/(j+a)\), whose factors decrease with \(j\), and then using the binomial expansion. From \(ISB.1\), \(ISB.6\),
\[
\begin{gathered}
|\text{output}_j|\le\\
C_2M_\varepsilon\frac{\varepsilon^j}{j!}\\
\sum_{K\ge0}(4C_1\varepsilon)^K.
\end{gathered}
\tag{ISB.8}
\]
Given any desired \(\eta>0\), choose \(0<\varepsilon\le\eta\) with \(9C_1\varepsilon<1/2\). The coefficient domain and \(C_1\) do not depend on this choice. This proves the positive estimate for **every** \(\eta\). Terms with \(K\ge R\) are bounded by the right side of \(ISB.8\) times \((4C_1\varepsilon)^R\), with the geometric denominator retained.

For negative output \(-n\), the input is \(m=K-n\). When \(K<n\), \(ISB.6\) gives
\[
\begin{gathered}
C_1^K\frac{n!}{(n-K)!}\\
M D^{n-K}(n-K)!\\
=M n!C_1^KD^{n-K}.
\end{gathered}
\tag{ISB.9}
\]
This part is finite; its sum is at most \(Mn!(D+C_1)^n\), since the binomial coefficients dominate the corresponding \(n\) summands. For \(K\ge n\), write \(m=K-n\ge0\). The multinomial expansion proves
\((n+2m)!/(n!m!m!)\le3^{n+2m}\). Consequently the full positive-input part is bounded by
\[
\begin{gathered}
C_2M_\varepsilon n!(3C_1)^n\\
\sum_{m\ge0}(9C_1\varepsilon)^m.
\end{gathered}
\tag{ISB.10}
\]
Choose one fixed \(\varepsilon<1/(18C_1)\). Equations \(ISB.9\)–\(ISB.10\) prove one factorial negative constant for every \(n\), on the same domain. The tail \(K\ge R>n\) in \(ISB.10\) is bounded by the displayed prefactor times
\((9C_1\varepsilon)^{R-n}/(1-9C_1\varepsilon)\).

These bounds prove absolute normal convergence of the whole expressions, including infinitely many terms at a fixed output degree. They justify rearrangement, finite matrix sums and multiplication identities by the triangle inequality and absolute scalar convergence. On a fixed further inner product, the Cauchy formula applies to each chosen finite output derivative. Its constant is independent of all cutoffs and degrees. Applying it to the displayed tails proves normal convergence of every fixed base and normalized-spatial derivative. All spatial Taylor exponents remain in the norm; none is truncated or bounded in advance.

#### ISB.4. Actual inverses and vanishing iteration

Let \(L\) be one of the above order-lowering correction operators. Define
\[
\begin{gathered}
(1+L)^{-1}f=\\
\sum_{a\ge0}(-L)^a f.
\end{gathered}
\tag{ISB.11}
\]
Equations \(ISB.6\)–\(ISB.10\) prove that this expression is an actual coefficient sequence in the complete class, on one domain. Reserve an intermediate radius, for example \(3r_0/4\): the same proof first gives normal convergence there. Apply one further correction on the inner radius \(r_0/2\). The Cauchy formula permits differentiation of the convergent series, and the direct word bound in equation ISB.6, with the additional correction included, permits its reordering there. The finite geometric identity gives left and right inverses, because its remainder \(L^{a+1}f\) has total loss at least \(a+1\) and tends to zero coefficientwise with the explicit tails above. This is a limit of normally convergent holomorphic coefficient functions. It is not an ordinary supported-distribution limit, a formal completion, or a limit in a cohomology quotient.

In particular, if an actual sequence satisfies \(s=-Ls\), then \(s=(-L)^a s\) for every finite \(a\). If \(s\) is the output of a previous division, select one new, fixed smaller outer radius inside its holomorphic domain and the same coefficient-divisor domain. Proof section MFA.2 makes the identical coefficient operators bounded on its radius interval; the support-normalized analytic operator is unchanged. This is a single additional shrinking made before any iteration index, not shrinking at each iteration. For positive output use \(ISB.8\) with \(K\ge a\). For fixed negative output \(-n\), take \(a>n\) and use \(ISB.10\) with \(m\ge a-n\). Both tend to zero, so every actual coefficient of \(s\) vanishes on the same inner product. Hence \(s=0\). This is the zero-remainder detection needed for full matrix relations with no highest positive homogeneous coefficient.

#### ISB.5. Ordinary zero-section version

In the ordinary polynomial coefficient spaces of proof section MFA.4, the covariable derivative in \(ISB.2\) is an ordinary derivative of a homogeneous polynomial. Its Cauchy bound on the same radius interval is at most \(C_0L\) per derivative; including it in \(ISB.4\) gives the same bound. Every finite-order right matrix has only finitely many tail offsets \(q\) relative to its principal shifted degree. A homogeneous space of negative polynomial degree is zero. All intermediate coefficient operations preserve the prescribed shifted polynomial degrees by proof section MFA.4, and the finite ordinary Leibniz product preserves the corresponding graded spaces.

Thus \(ISB.6\)–\(ISB.8\) apply with finite free-basis shifts. A fixed shift changes the positive factorial bound by a fixed polynomial in the degree; this is absorbed by choosing the input \(\varepsilon\) smaller than the requested output value, since \(j^a\theta^j\) is bounded for every fixed \(a\) and \(0<\theta<1\). The finitely many degrees below the shifts satisfy the same convergent tail bounds individually. Every output component remains a homogeneous polynomial of its actual nonnegative degree. This proves \(ISB.11\) and zero detection in the genuine ordinary \(D^\infty\) class at the zero section; it does not substitute an ordinary ring for the microlocal argument in proof sections ISB.1–ISB.4.

#### Reproducible figures for full division and reconstruction

![Full spatial division, nilpotent remainder and special-class distinction](assets/infinite-spath-division-and-special-class.png)

**Figure ISD.1.** Left: the exact real slice of equation ISD.15 retains its nilpotent matrix and row, radius one and real spatial coordinate from minus three-fifths to three-fifths. Both quotient coordinates are the reciprocal of one minus that coordinate; both remainder coordinates equal one. The nilpotent second coordinate survives. Center: equation ISF.10 uses normal difference one-quarter and spatial epsilon one-eighth; its coefficient-bound ratio is exactly two to the spatial degree, illustrating the proved failure of equation ISF.3. Right: proof sections ISD.2–ISD.4 and ISB.1–ISB.4 keep all spatial indices and both growth regimes on one common final domain. The first two panels are exact stated real/rational samples; the third is an algebraic and estimate diagram with no geometric scale. Free human context: [Kashiwara–Kawai, HolIII](https://www.kurims.kyoto-u.ac.jp/~kenkyubu/kashiwara/HolIII.pdf), III.3/III.5, printed pp.888–889/900–903.

Editable SVG, exact data, and reproducible source.

![Both balanced inverses and the nilpotent off-ray inverse](assets/infinite-balanced-reconstruction-and-off-ray.png)

**Figure ISR.1.** Left: the exact diagram of proof sections ISR.2–ISR.3 uses two distinct coefficient rings, ordinary algebraic tensors, multiplication maps and unit inverses. The sum decomposition removes relation tensors and balances the coordinate-commuting remainder without a free-module identification. Right: the exact real samples of equation ISR.11 use reciprocal integer spatial coordinate, unit normal covariable and zero base coordinates; the inverse's row-sum norm is the integer plus its square. Proof section ISR.4 proves off-ray vanishing, and the inverse has no extension through the excluded ray. The diagram has no geometric scale. The whole-support conclusion uses the complete [§5.48, FEP.8–FEP.9](#5-48-proper-ideals-and-faithful-extension-to-the-full-infinite-order-ring) theorem. Free human context: [Kashiwara–Kawai, HolIII](https://www.kurims.kyoto-u.ac.jp/~kenkyubu/kashiwara/HolIII.pdf), III.5, equations (3.5.8)/(3.5.16), printed pp.901/903.

Editable SVG, exact data, and reproducible source.


### 5.50. Full infinite coefficient support depth and finite duality

Put \(A=\mathcal E\) and \(B=\mathcal E^\infty\), with the actual geometric product on a complex projective cotangent chart \(V\). If every local component of a closed analytic subset \(Z\subset V\) has complex codimension at least \(c\), then
\[
\mathcal H_Z^q(B)=0\quad(q<c).
\tag{SD.1}
\]
This includes singular points, components of different dimensions and nonreduced defining ideals: the support functor depends on the underlying closed subset. For a bounded complex \(Q\) of finite locally projective right \(A\)-modules there is a natural comparison
\[
\begin{gathered}
R\Gamma_Z\mathcal Hom_{A^{\mathrm{op}}}(Q,B)
\\ \simeq\mathcal Hom_{A^{\mathrm{op}}}(Q,R\Gamma_ZB).
\end{gathered}
\tag{SD.2}
\]
Its right-hand supported representative has the actual commuting left \(B\) and right \(A\) actions. The proof retains both homogeneous growth regimes, every spatial index and fixed derivative on common coefficient domains. The final finite-dual argument gives the support bound for modules under its stated Ext-concentration hypothesis.

The full coefficient identification and product are [§5.44, IK.1–IK.3 and AN.1–AN.5](#5-44-full-homogeneous-coefficients-and-ordinary-infinite-order-kernels) and [§5.45, IG.1–IG.5 and IM.1–IM.6](#5-45-the-full-geometric-infinite-order-kernel-ring-and-its-canonical-product). The scalar existence theorem used below is the complete [§5.43, DLH.1–DLH.4](#5-43-the-canonical-all-degree-action-and-its-derived-module-structure), including its graph-domain and exhaustion-weight proof. Distributional holomorphicity, partitions and acyclic resolutions are [§5.34, R.1–R.7](#5-34-full-current-resolutions-and-controlled-proper-image-comparison); the supported complement cube with its signs is [§5.35, V.6](#5-35-relative-cones-unshifted-cup-and-the-normalized-trace). These are proof dependencies. The freely accessible [Kashiwara–Kawai, Holonomic systems III, I.2.2, printed pp.841–842](https://www.kurims.kyoto-u.ac.jp/~kenkyubu/kashiwara/HolIII.pdf) is additional reading; none of its recalled results substitutes for the argument.

#### SD.A. An exact auxiliary-variable realization of the entire coefficient sheaf

Use fixed base coordinates \(u=(t,x)\), and the normalization \(\tau=1\),
\(z=\xi/\tau\). Write \(v=(u,z)\), of complex dimension m. The earlier proved sheaf
identification IG.1--IG.5 identifies \(B|_V\) with sequences \(p_j\in\mathcal O(V)\),
\(j\in\mathbf Z\), with, on each compact \(K\) contained in \(V\),
\[
\begin{gathered}
\sup_K|p_{-n}|\le B_K C_K^n n!
\\ (n\ge1),
\\ \sup_K|p_j|\le B_{K,\epsilon}\epsilon^j/j!
\\ (j\ge0,\ \epsilon>0).
\end{gathered}
\tag{SD.3}
\]
One coefficient domain supports every index. Both restrictions and
equalities are actual sheaf restrictions/equalities on a common smaller
domain. Split the sequence into its nonnegative and negative parts,
writing \(B=B_+\oplus B_-\) as sheaves of complex vector spaces. This
splitting is used for analytic depth, not asserted to be a ring or
right-A splitting.

Let \(\pi:V\times\mathbf C_s\to V\) be projection and \(i:V\to V\times\mathbf C_s\) the zero
section. Define
\[
\begin{gathered}
F_+(v,s)=\sum_{j\ge0}j!p_j(v)s^j,
\\ F_-(v,s)=\sum_{n\ge1}\frac{p_{-n}(v)}{n!}s^{n-1}.
\end{gathered}
\tag{SD.4}
\]
These give exact sheaf isomorphisms
\[
\begin{gathered}
B_+=\pi_*\mathcal O_{V\times\mathbb C},
\\ B_-=i^{-1}\mathcal O_{V\times\mathbb C}.
\end{gathered}
\tag{SD.5}
\]
Here inverse image denotes germs on neighbourhoods of the zero section;
it is not restriction modulo the auxiliary coordinate. In particular,
\(B_-\) is not \(\mathcal O_V\) and not a formal power-series sheaf.

For the first assertion fix a compact \(K\) in \(V\) and \(R>0\). Choose \(\epsilon\)
with \(\epsilon R<1\). SD.3 bounds the entire positive sum and its tail by
\[
\begin{gathered}
\sup_{K\times\{|s|\le R\}}|F_+|
\le\frac{B_{K,\epsilon}}{1-\epsilon R},
\\ \sup|\text{tail}_{j\ge J}|
\le\frac{B_{K,\epsilon}(\epsilon R)^J}{1-\epsilon R}.
\end{gathered}
\tag{SD.6}
\]
Thus \(F_+\) is jointly holomorphic on the whole \(V\times\mathbf C\). Conversely, for
\(F\) holomorphic on that product, the fixed auxiliary contour \(|s|=R\) gives
\[
\begin{gathered}
p_j(v)=\frac1{j!}\frac1{2\pi i}
\\ \int_{|s|=R}\frac{F(v,s)}{s^{j+1}}\,ds,
\\ |p_j|_K\le M_{K,R}R^{-j}/j!.
\end{gathered}
\tag{SD.7}
\]
Given \(\epsilon\), take \(R=1/\epsilon\). The \(v\)-domain is unchanged as \(\epsilon\)
varies. Taylor uniqueness proves both inverse identities.

For the negative assertion SD.3 makes \(F_-\) normally convergent for
\(|s|<1/C_K\) near \(K\), with the actual tail bound
\[
\begin{gathered}
\sup_{K,\ |s|\le r}|\text{tail}_{n\ge L}|
\\ \le\frac{B_K C_K(C_Kr)^{L-1}}{1-C_Kr},
\\ C_Kr<1.
\end{gathered}
\tag{SD.8}
\]
These local holomorphic functions agree by their coefficient expansions,
and hence define a holomorphic germ along \(V\times\{0\}\). Conversely such
a germ has, near each compact \(K\), one radius \(r>0\) on which it has a
representative bounded by \(M\) on \(|s|=r\). Cauchy gives
\[
\begin{gathered}
p_{-n}=n![s^{n-1}]F_-,
\\ |p_{-n}|_K\le M r^{1-n}n!,
\end{gathered}
\tag{SD.9}
\]
which is SD.3 with, for example, \(C=\max(1,r^{-1})\) and an enlarged finite
constant. A compact is covered by finitely many representative products,
so one positive radius works for that compact. The radius may decrease
towards the boundary of \(V\); the coefficient domain \(V\) itself does not.
Germ equality of \(F_-\) gives all coefficient equalities on one smaller
\(v\)-domain, and the converse follows by the same expansions. This proves
SD.5 as an actual sheaf assertion, not only as stalkwise vector spaces.

Every fixed \(v\)-derivative preserves these constructions and tail bounds:
first reserve a compact enlargement in the one \(v\)-domain, then apply its
Cauchy integral to SD.6 or SD.8. The derivative cost is independent of
\(j\), \(n\) and the cutoff. If \(p_j\) is Taylor-expanded in any spatial covariable
polydisc, its complete coefficients \(p_{j,\beta}\) are extracted on fixed
circles with bounds
\[
\begin{gathered}
|p_{j,\beta}|\le B_\epsilon\epsilon^j R^{-\beta}/j!
\\ (j\ge0),
\\ |p_{-n,\beta}|\le B C^n n!R^{-\beta}.
\end{gathered}
\tag{SD.10}
\]
All multiindices \(\beta\) are retained. No finite spatial-degree truncation
occurs anywhere below. The earlier proved IK/AN normal kernel extraction then
retains its positive-homogeneous/negative-normal mixed terms as well.

For analytic depth we may compose every \(p_j\) with any biholomorphic change
of the \(v\)-coordinates. Compact preimages prove SD.3 with unchanged
quantifiers. This is an auxiliary presentation of the underlying
complex-vector sheaf to flatten a smooth analytic support. It does not
assert that an arbitrary biholomorphism mixing \(u\) and \(z\) is a quantized
contact transformation or a ring transition. The actual ring transition
and right action remain those of the earlier proved geometric identification.

#### SD.B. Actual fine resolutions and acyclicity of the auxiliary families

The exact earlier programme analytic inputs are DLH.1--DLH.4, including its
graph-core proof and explicit simultaneous convex-exhaustion weight,
and R.1--R.7, including distributional holomorphicity, actual injective
resolutions and their chain comparison. The precise earlier proof locators are linked above. We use their proved result: on a domain
having a smooth strictly plurisubharmonic exhaustion, every closed
positive-degree locally \(L^2\) Dolbeault form has a locally \(L^2\) primitive.
We do not use a named Stein or vector-valued existence theorem as an
unproved premise.

Let \(L^q\) be the maximal locally \(L^2\) graph sheaf of (0,\(q\))-forms on the
(\(m\)+1) dimensional auxiliary product: both a form and its distributional
Dolbeault differential are locally \(L^2\). Its differential is the full
Dolbeault differential in \(v\) and s. The complex
\[
\begin{gathered}
0\longrightarrow\mathcal O\longrightarrow L^0
\\ \xrightarrow{\bar\partial}\cdots\longrightarrow L^{m+1}
\longrightarrow0
\end{gathered}
\tag{SD.11}
\]
is exact. Positive-degree local exactness is DLH.1–DLH.4 on a small polydisc;
a primitive has the given \(L^2\) differential, so lies in the graph sheaf.
Degree zero is exactly O by the complete R.3 distributional
holomorphicity proof. Multiplication by a smooth function preserves both
graph conditions by the product rule.

First consider \(B_+\). The sheaves \(\pi_*L^q\) are modules over \(\mathcal C^\infty_V\):
multiply by the pullback of a smooth base function. They are therefore
ordinary-section acyclic by the explicit partition/injective proof
R.5. Their augmented complex resolves \(\pi_*\mathcal O\) locally. Indeed, a closed
germ is represented on \(U\times\mathbf C\); after shrinking \(U\) to a polydisc,
that product has the strict proper exhaustion
\[
\begin{gathered}
|v|^2+|s|^2
\\ {}+\sum_a-\log(R_a^2-|v_a|^2),
\end{gathered}
\tag{SD.12}
\]
translated by a constant if necessary. DLH.1–DLH.4 solve its closed form on the
whole product, so the image germ is exact. Thus \(\pi_*L^*\) is a finite
ordinary-section-acyclic resolution of \(B_+\). R.6'\(s\) finite double-complex
comparison applies to it, and computes actual derived sections.

Next consider \(B_-\). Inverse image is exact on sheaves (stalks are filtered
germ colimits, whose finite equations/witnesses occur at a common later
index). Hence \(i^{-1}L^*\) resolves \(i^{-1}\mathcal O\). It is a
\(\mathcal C^\infty_V\)-module termwise, again using smooth multiplication in \(v\),
and is ordinary-section acyclic by R.5. We need an actual global
primitive for any closed section over a punctured product, not a
separately chosen primitive for each n.

Let \(U\) be a product of discs and punctured discs in v. It has the smooth
nonnegative strict exhaustion
\[
\begin{gathered}
\psi_0(v)=|v|^2
\\ {}+\sum_a-\log(R_a^2-|v_a|^2)
\\ {}+\sum_{a\text{ punctured}}|v_a^{-1}|^2+K.
\end{gathered}
\tag{SD.13}
\]
Each barrier is psh; differentiation of \(-\log(R^2-|w|^2)\) gives positive
Levi form \(R^2/(R^2-|w|^2)^2\), and \(|w^{-1}|^2\) is the squared modulus of
a holomorphic function. Boundary or puncture approach forces divergence.
The coordinate norm makes the sum strict.

A section of \(i^{-1}L^q\) on \(U\) is represented by an \(L^q\) form on some open
neighbourhood of \(U\times\{0\}\). Here is the precise neighbourhood argument.
Choose local representatives on products above a locally finite,
relatively compact cover of U. Refining it by open sets whose closures
lie inside the original base sets leaves only finitely many relevant
representatives near each base point. Equality of the germ sections
gives equality on a still smaller auxiliary neighbourhood of that point
for each of these finitely many pairs. Their finite minimum is positive
locally. A positive continuous minorant is obtained from a subordinate
partition of unity and sufficiently small positive constants on each
refinement member. Shrink the auxiliary radius below that minorant.
The representatives now agree on all overlaps and glue. If the section
is closed as a germ, apply the same construction once more to its zero
Dolbeault differential. It is then represented by a literally closed
form \(g\) on a neighbourhood \(T\) of the entire zero section. No coefficient
index has been chosen in this construction.

Every such \(T\) contains a tube
\[
\begin{gathered}
\Omega_\chi=\{(v,s):v\in U,
\\ |s|<\exp[-\chi(\psi_0(v))]\},
\end{gathered}
\tag{SD.14}
\]
for a smooth nonnegative increasing convex \(\chi\). To choose it, on each
compact sublevel \(\{\psi_0\le k\}\) take a positive lower bound for the tube
radii inside T. Choose increasing constants bounding the negative
logarithms of these lower bounds through level \(k\)+2. The explicit smooth
step-function construction of DLH.4 gives an increasing convex \(\chi\) above
these constants on their levels. Increasing its initial constant
handles the first compact. Thus SD.14 is cofinal among the actual
neighbourhoods used by these germ sections.

This tube has a smooth strictly psh proper exhaustion. Put
\[
\begin{gathered}
t(v,s)=e^{2\chi(\psi_0(v))}|s|^2,
\\ \rho(v,s)=\psi_0(v)+|s|^2
\\ {}-\log(1-t(v,s)).
\end{gathered}
\tag{SD.15}
\]
For \(s\) nonzero, \(\log t=2\chi(\psi_0)+\log|s|^2\) is psh and exponentiation is
convex increasing, so \(t\) is psh there. It is smooth also at \(s=0\); its
Levi form there is the nonnegative limit of the Levi forms away from
zero. The function \(-\log(1-t)\) is convex increasing for \(t<1\), with
nonnegative first and second derivatives. The Hessian chain rule makes
it psh. The term \(\psi_0\) is strict in \(v\) and \(|s|^2\) is strict in \(s\), hence
\(\rho\) is strict in all \(m\)+1 coordinates. A \(\rho\)-sublevel bounds \(\psi_0\) and
\(t\) away from 1. The base then lies in a compact in \(U\), and the auxiliary
coordinate stays uniformly inside the tube boundary, so that sublevel
is compact in \(\Omega_\chi\). This proves properness, including at \(s=0\).

Apply the complete DLH.1–DLH.4 proof on \(\Omega_\chi\) to g. It gives one actual
primitive on that tube for every positive degree. Its inverse-image
germ solves the original section equation on U. This proves
\[
\begin{gathered}
H^q(U,B_-)=0\quad(q>0),
\\ H^q(U,B_+)=0\quad(q>0).
\end{gathered}
\tag{SD.16}
\]
For \(B_+\), use the product \(U\times\mathbf C\) and the exhaustion \(\psi_0+|s|^2\).
For \(B_-\), use \(i^{-1}L^*\), its ordinary-section acyclicity and R.6, just
as for \(B_+\). No neighbourhood-cohomology continuity theorem is assumed.

SD.16 proves acyclicity for the full family at once. Intermediate graph
forms are auxiliary forms on one domain/tube; every holomorphic output
is recovered by the Cauchy maps SD.7/SD.9 with exactly SD.3 and all fixed
derivatives SD.10. Scalar existence alone without SD.5, these actual resolutions,
cofinal tubes and inverse Cauchy estimates would not prove this claim.
The possible datum-adapted variable Hilbert-weight route is unnecessary:
no assertion about uniform graph cores for degree-dependent weights is
used here. The present argument neither certifies nor needs that route.

#### SD.C. Smooth support: a full-family Laurent cube with literal primitives

Let \(S\) be a smooth complex submanifold of codimension c. Flatten it by
ordinary holomorphic \(v\)-coordinates, permissible for the underlying
coefficient sheaf by SD.A. Near a chosen point take one polydisc
\(D=\Delta^c_w\times\Delta^{m-c}_y\) with \(S=(w_1=\cdots=w_c=0)\). For
\(I\) contained in \(\{1,\ldots,c\}\), let \(D_I\) puncture precisely the \(w_i\) with \(i\) in \(I\);
\(D_\varnothing=D\). The complement \(D\setminus S\) has the finite ordered cover \(D_{i}\).
The exact support/injective comparison of R.7 and finite complement-cube
proof V.6 apply to the graph resolutions of SD.B. The partition homotopy
uses functions of \(v\), so is valid for both \(\pi_*L\) and \(i^{-1}L\).
Every \(D_I\) is acyclic for \(B\) by SD.16. Eliminating the finitely many
positive graph columns therefore reduces actual derived supported
sections to the holomorphic-family cube
\[
\begin{gathered}
K^p(D,B)=\bigoplus_{|I|=p}B(D_I),
\\ 0\le p\le c,
\\ (dc)_I={}
\\ \sum_{r=1}^{|I|}(-1)^{r-1}
c_{I\setminus\{i_r\}}|_{D_I}.
\end{gathered}
\tag{SD.17}
\]
The empty term has degree zero, so this is the shifted support cone,
not the unshifted complement Cech complex. The actual map from the
injective supported resolution to this cube is the R.7/V.6 roof,
with their declared cone signs. SD.16 supplies its needed family
acyclicity; it is not a coefficientwise formal use of V.6.

We now prove exactness below degree \(c\), including actual family
primitives. In a punctured coordinate \(w\) use its Laurent expansion.
Write \(\Pi_i^-\) for the sum of strictly negative \(w_i\) powers and \(\Pi_i^+\)
for the sum of nonnegative powers. Its coefficients are the actual
Cauchy integrals on circles lying within the common input domain. The
positive sum extends across \(w_i=0\) on a smaller disc; the negative sum
is holomorphic on the punctured disc. Contour independence follows
from Cauchy'\(s\) theorem on a compact annulus. The expansions converge
normally on smaller product annuli by the geometric series bounds
from those circles.

Here those are family projections, not isolated scalar projections.
For \(B_+\), apply them to \(F_+\) on \(D_I\times\mathbf C\). Fixed normal circles preserve
holomorphicity for every auxiliary \(s\), so the output is still an entire
auxiliary family on its one base domain. For \(B_-\), represent \(F_-\) on a
tube above \(D_I\). On each compact in the remaining variables, fixed
normal circles are a compact subset of \(D_I\) and have a uniform positive
auxiliary radius in the tube. For \(\Pi^+\) choose circles outside the
smaller output \(w\)-disc; its positive geometric sum then extends on
that whole disc and that auxiliary radius. For \(\Pi^-\) choose circles
inside the output punctured annulus, giving its negative geometric sum.
Contours on overlapping compact annuli give the same auxiliary germs
by Cauchy'\(s\) theorem after taking the finite minimum of their auxiliary
radii. Thus the projections glue to an actual holomorphic germ along
the output zero section. Its radius can vary on punctured base compacts,
as allowed in SD.3; there is no index-dependent base domain. SD.7/SD.9
recover all original homogeneous indices, with the exact every-\(\epsilon\)
and factorial bounds. Fixed derivatives use one more common margin.

Here are the explicit uniform estimates, including their tails. For
one normal coordinate, on an output annulus \(a\le|w|\le b\) choose contours
\(\rho_-<a\le b<\rho_+\) inside the common input normal annulus. If the function
on those contours is bounded by \(M_-\) and \(M_+\), then Cauchy gives
\[
\begin{gathered}
|c_k|\le M_+\rho_+^{-k}
\\ (k\ge0),
\\ |c_{-h}|\le M_-\rho_-^h
\\ (h\ge1).
\end{gathered}
\tag{SD.17a}
\]
Thus the positive and negative sums are bounded, respectively, by
\(M_+/(1-b/\rho_+)\) and \(M_-(\rho_-/a)/(1-\rho_-/a)\). Their tails starting at
normal degrees \(L\) and -\(L\) are bounded by the same denominators times
\((b/\rho_+)^L\) and \((\rho_-/a)^L\). For several projected coordinates use the
product contour; its coefficient bound is the product of the respective
contour powers times one bound \(M\) for the full function, and summing
gives the product of the displayed geometric denominators. A cutoff
in any coordinate is controlled by its corresponding geometric tail
times the other denominators. All contours for the finitely many cube
entries are chosen on one common product before any index summation.

For \(F_+\), on every auxiliary circle \(|s|=L_0\) the full product normal
contour bound is finite. If \(G\) is any projected family and \(A_K\) is that
product of geometric denominators, auxiliary Cauchy gives
\[
\begin{gathered}
|p_j^G|_K
\\ \le A_K M_{K,L_0}L_0^{-j}/j!,
\\ |p_{-n}^G|_K
\\ \le A_K M_{K,r_0}r_0^{1-n}n!.
\end{gathered}
\tag{SD.17b}
\]
The first inequality pertains to \(F_+\); choose \(L_0=1/\epsilon\), possible
for every \(\epsilon\) on the same projected base product. The second
pertains to \(F_-\), with one \(r_0>0\) smaller than the tube radii on the
compact product of all required normal contours. Choose \(r_0\) once
before \(n\) is varied. These inequalities prove the actual output
quantifiers. For a fixed derivative enlarge its coefficient compact
within the projected domain and apply its Cauchy derivative cost to
these bounds and tails. Spatial Taylor extraction then applies SD.10
on fixed additional circles for every \(\beta\). This supplies explicit
full-family primitive bounds as well as the sheaf existence assertion.

Projections in distinct coordinates commute by the finite product
contour integral and its normally convergent geometric sums. Consequently
each cube entry has the direct decomposition
\[
\begin{gathered}
B(D_I)=\bigoplus_{J\subset I}T_J,
\\ T_J=\{\text{families strictly}
\\ \text{negative in every }w_j,
\\ j\in J,\text{ holomorphic}
\\ \text{across every}
\\ \text{other normal coordinate}\}.
\end{gathered}
\tag{SD.18}
\]
This equality is used after a single common polydisc shrink for the
finitely many cube entries. Each \(T_J\) is represented on \(D_J\); extending
the nonnegative variables is exactly the construction just proved.
The decomposition is unique by Laurent uniqueness. In any overlap or
restriction the projections agree, so it is a decomposition of the
local cube germs. The shrinking is chosen before recovering any
homogeneous coefficient or spatial Taylor index.

The cube now splits into finitely many blocks indexed by J. In block \(J\)
the entries are \(T_J\) for each \(I\) containing \(J\); the differential adds
indices in \(\{1,\ldots,c\}\setminus J\) with SD.17'\(s\) alternating signs. It is an identity
map on that same actual \(T_J\). If \(J\) is not the whole index set, choose
a in its complement. The homotopy removes a from \(I\) when a is present,
with sign (-1) to the number of elements of \(I\) before a, and is zero
when a is absent. Checking the two cases shows
\[
 dh+hd=1.                                                         \tag{SD.19}
\]
The term that adds/removes a supplies the identity. Every other added
index \(b\) occurs twice with opposite signs, since moving past a changes
its ordered position by one. Removing a never removes an element of \(J\),
so the homotopy remains inside the block. If \(J\) is the whole index set,
there is only \(I=J\) in degree c. Therefore the entire cube has zero
cohomology in degrees below c. Its actual lower-degree primitive is
the sum of the finitely many block homotopies of its actual projected
families. The estimates, domains and fixed derivatives of these
primitives were established before their inverse coefficient recovery.

Taking the filtered limit over smaller polydiscs is exact: every finite
equation, and every witness to its being a boundary, already holds at
one common later neighbourhood. Thus this proves
\[
 \mathcal H_S^q(B)=0\quad(q<c)                                     \tag{SD.20}
\]
as a sheaf assertion for every smooth \(S\), with all coefficients retained.
For \(c=0\) the below-zero conclusion is automatic; empty support has zero
supported complex. No globally uniform polydisc on a whole chart is
claimed for a local sheaf theorem, but each datum/primitive uses one
actual domain for every infinite index and fixed derivative.

#### SD.D. Singular analytic support by closed exceptional-set induction

The earlier programme lesson Analytic germs, local parametrization and the Nullstellensatz supplies the following results with their complete proofs. Proposition1.2 gives the finite irreducible germ
decomposition; Theorem4.1 gives for every positive-dimensional
irreducible component a finite projection and a nonzero discriminant,
with a dense smooth covering outside its inverse image; Theorem5.4 and
Proposition5.5 give its dimension and strict dimension drop of every
proper analytic subset. These are exact earlier programme proof
dependencies, not a stratification theorem imported from HolIII.

At any point choose a small representative with finitely many
irreducible components \(Z_a\). For each positive-dimensional component
of maximum dimension \(d\) choose the closed analytic exceptional set
\(E_a\) cut out on it by its finite-projection discriminant. It is proper,
and has dimension less than d. Put \(W\) equal to the union of those \(E_a\),
all intersections of distinct maximum-dimensional components, and
all lower-dimensional components. These are finitely many closed
analytic sets on a common neighbourhood; their union is analytic
(multiply their finite local vanishing ideals). Every intersection
is proper in each distinct component, by irredundancy and
Proposition5.5, hence has dimension below d. Thus \(\dim W<d\), and outside
\(W\) the set \(Z\) is a disjoint smooth submanifold, of codimension \(m\)-d.
Zero-dimensional reduced components are isolated points on a small
representative and already satisfy the smooth computation. Nothing
about a Whitney condition or a globally finite stratification is used.

For completeness the exact localization step, including the derived
bound, is proved on the actual injective model. For a complex-vector
sheaf \(F\) use R.5'\(s\) functor
\[
\begin{gathered}
G(F)(U)=\prod_{x\in U}F_x,
\\ I^q(F)=G(Q^qF),
\\ QF=G(F)/F.
\end{gathered}
\tag{SD.21}
\]
It is an injective resolution. The proof is the stalk-map adjunction:
maps to \(i_{x*}T\) are maps of stalks to the vector space \(T\), and such maps
extend from subspaces; products of these skyscraper injectives are
injective. Its exact quotient iteration and chain comparison are
the complete R.5--R.6 proofs. If \(T\) is closed,
\[
 \Gamma_TG(F)(U)=\prod_{x\in U\cap T}F_x                            \tag{SD.22}
\]
with zero components elsewhere. Indeed vanishing on \(U\setminus T\) is precisely
that condition. The right side is itself a product of skyscraper
injectives. Open restriction commutes with \(G\) and the quotient iteration.

For \(W\subset Z\) and \(j:V\setminus W\to V\), SD.22 gives the degreewise exact
sequence of injective complex-vector sheaves
\[
\begin{gathered}
0\longrightarrow\Gamma_W I^*(F)
\\ \longrightarrow\Gamma_Z I^*(F)
\\ \longrightarrow j_*\Gamma_{Z\setminus W}
(I^*(F)|_{V\setminus W})
\\ \longrightarrow0.
\end{gathered}
\tag{SD.23}
\]
The last arrow forgets components indexed by \(W\), and is surjective by
zero extension of the other components. Its last complex represents
\(Rj_*R\Gamma_{Z\setminus W}(F|_{V\setminus W})\), since its source terms
are injective by SD.22 and \(j_*\) sends injectives to injectives
(it is right adjoint to exact open restriction).

We also spell out why \(Rj_*\) cannot introduce lower cohomology. Suppose
a bounded-below complex \(C\) has \(H^q(C)=0\) for \(q<c\). Its good truncation
\(T\) has zero terms below \(c\), \(T^c=C^c/\operatorname{im}(d^{c-1})\), and the unchanged terms
above c. The natural \(C\to T\) is a quasi-isomorphism by direct computation
of its kernels/images. Apply the exact Godement functors \(I^r\) to \(T\),
\(r\ge0\), and totalize with differential \(d_T+(-1)^q d_I\) on \(T\)-degree q.
Every diagonal has finitely many entries: \(c\le q\le\text{total degree}\). The
vertical-resolution elimination of R.6 proves its augmentation is a
quasi-isomorphism, and its total terms are injective. It has no terms
below c. For an already injective representative of \(C\), compare it to
this total through its Godement double complex. The R.6 exact-cone
contraction for bounded-below injective complexes makes both comparisons
homotopy equivalences; applying \(j_*\) preserves their actual homotopies.
Thus \(Rj_*C\) can be represented with no terms below \(c\), proving the bound
without assuming \(j_*\) exact on arbitrary sheaves.

Now induct on the maximum dimension \(d\) of Z. The zero-dimensional
case is SD.C. Let all components of \(Z\) have codimension at least \(c\),
so \(d\le m-c\). The constructed \(W\) has dimension at most \(d\)-1; by induction,
\(\mathcal H_W^q(B)=0\) for \(q<m-d+1\), in particular for \(q<c+1\).
On \(V\setminus W\) the support \(Z\setminus W\) is smooth of codimension \(m-d\ge c\), so SD.C
makes its support complex concentrated in degrees at least c.
The proved \(Rj_*\) bound preserves that lower bound. The long exact
cohomology sequence of SD.23 then gives
\(\mathcal H_Z^q(B)=0\) for \(q<c\). This includes points in \(W\) itself:
the conclusion comes from the supported complexes at those stalks,
not extension by continuity from regular points. The induction
terminates after finitely many dimension drops near every point.
This proves SD.1 for arbitrary singular and non-pure-dimensional Z.

All chart passages are legitimate. The geometric identification gives
the actual \(B\) sheaf on each normalized complex projective chart. The
coefficient presentation used above proves its underlying support
cohomology, and these local zero assertions glue. For a homogeneous
analytic set upstairs, its projective chart support is analytic;
the local nonzero radial product does not change its complex
codimension. The requested homogeneous supports are therefore included.

#### SD.E. The actual support-preserving right \(E\) action

The earlier proved IK.1--IK.3 and IM.1--IM.6 identify the full canonical
geometric product with the normally convergent coefficient contraction
\[
\begin{gathered}
(b\circ a)_\ell(v)
\\ =\left.\begin{gathered}
\sum_{j+k-|\alpha|=\ell}\frac1{\alpha!}
\\ {}\cdot\partial_\zeta^\alpha(\tau^jp_j(u,z))
\\ {}\cdot\partial_u^\alpha(\tau^ka_k(u,z))
\end{gathered}\right|_{\tau=1,\xi=z},
\\ a\in A,\ b\in B.
\end{gathered}
\tag{SD.24}
\]
The earlier exact numerical factorial estimates bound every fixed
degree'\(s\) full infinite contraction, retain all spatial indices, give
actual tails and every fixed derivative tail on common enlarged/smaller
products, and recover the exact SD.3 quantifiers. Earlier proved \(M\) proves
that this is the source-defined product, with actual geometric
common supports/trace maps. Hence \(b\cdot a=b\circ a\) defines the actual
right-A sheaf action, not a formal module action.

If \(b\) vanishes on an open set, all \(p_j\) vanish there and every derivative
in SD.24 vanishes there; its normally convergent sum is zero. Equivalently
the earlier proved sheaf product restricts to the product of restrictions.
Therefore
\[
 \operatorname{supp}(b\cdot a)\subset\operatorname{supp}(b).          \tag{SD.25}
\]
No smooth multiplier or fixed-order truncation of \(B\) is used to assert
this support property. For a finite family of a and \(b\), the earlier proved
finite-diagram common-domain property takes the finite intersection of
their coefficient domains and reserves the Cauchy margins before all
infinite sums. This is sufficient for every matrix and homotopy below.

On SD.21 put, for a in A(\(U\)),
\[
 ((t_x)_{x\in U}\cdot a)_x=t_x\cdot a_x.                            \tag{SD.26}
\]
Restrictions commute with this formula. The augmentation \(F\to G(F)\),
quotient \(Q\), its iterations and all differentials are right-A linear
whenever \(F\) is a right-A sheaf. For \(F=B\) this is an actual right-A
injective resolution in the category of underlying complex-vector
sheaves; injectivity as right-A sheaves is neither needed nor claimed.
SD.22 is a right-A subsheaf because its zero components remain zero.
Thus
\[
 C_Z=\Gamma_Z I^*(B)                                               \tag{SD.27}
\]
is an actual right-A representative of \(R\Gamma_Z B\), with SD.1 as its
cohomological lower bound. The geometric/auxiliary computation and this
module representative identify the same derived support object through
the actual injective augmentation. We do not demand that the auxiliary
Dolbeault differential admit a coefficientwise microdifferential action.
An acyclic vector-space model was used to prove the bound; the actual
right action is present on the canonical model SD.27.

The same construction retains the actual left-\(B\) action. Left product
with a local \(B\) section restricts to zero wherever its target section
is zero, since the earlier proved product is a sheaf product. On stalk products
use left multiplication by its germ. Earlier proved associativity makes this
commute with the right-A action. Both actions descend to every quotient
\(Q^qB\), commute with every Godement differential, and preserve SD.22.
Hence \(C_Z\) is in fact a (\(B\),A) bimodule complex, and the evaluation and
finite-Hom maps below are left-\(B\) linear as well as the required
right-A target maps. This retains the operator action in the actual
module reconstruction.

This model also represents derived support in the category of right-A
sheaves, if that category is used to define it. Here is the verification.
Let \(U_0\) be the exact forgetful functor to complex-vector sheaves. Its
left adjoint sends \(F\) to \(F\otimes_{\mathbf C}A\) with right multiplication in A;
it is exact since stalk tensor products over the field \(C\) are exact.
Consequently \(U_0\) sends injective right-A sheaves to injective
complex-vector sheaves: extend a map through that exact left adjoint
and use its adjunction. There are enough injective right-A sheaves.
For a complex-vector injective \(J\), the internal sheaf
\(\mathcal Hom_{\mathbf C}(A,J)\), with right action
\( (\phi\cdot a)(b)=\phi(ab)\), is right-A injective. Its adjunction
with \(U_0\) sends a right-A map to evaluation at 1, and sends a
complex-vector map \(f:U_0F\to J\) to the right-A map taking \(t\) to
\(b\mapsto f(tb)\). Thus a map from a right-A subobject extends by
complex-vector injectivity of J. Choose an embedding \(U_0F\to J\);
the resulting coinduced map \(F\to\mathcal Hom_{\mathbf C}(A,J)\) is injective because
evaluation at 1 is that embedding. Iterate its quotient to obtain
a right-A injective resolution. Its underlying complex is a
complex-vector injective resolution, by the just-proved forgetful
property. \(\Gamma_Z\) commutes with \(U_0\) on sections. The underlying
right-A derived support therefore is the ordinary complex-vector
derived support, with its actual right action. In particular every
underlying complex-vector injective right-A sheaf is \(\Gamma_Z\)-acyclic
in the right-A category, even if it is not right-A injective.
Applying the acyclic-resolution comparison to SD.21 proves that
SD.27 represents that module-derived support as well. This closes
the category comparison instead of inferring it from vector-space
quasi-isomorphisms without a right action.

#### SD.F. Finite-Hom/support comparison on sheaves and common resolutions

For any right-A sheaf \(Q\) and right-A sheaf \(F\) the literal equality is
\[
\begin{gathered}
\Gamma_Z\mathcal Hom_{A^{op}}(Q,F)
\\ =\mathcal Hom_{A^{op}}(Q,\Gamma_ZF).
\end{gathered}
\tag{SD.28}
\]
Over \(U\) the left side consists of right-A sheaf morphisms \(Q|_U\to F|_U\)
whose restriction on \(U\setminus Z\) is zero. Such a morphism sends every local
section of \(Q\) to a section vanishing off \(Z\), hence factors uniquely through
the kernel subsheaf \(\Gamma_ZF\). Conversely a morphism into that kernel
restricts to zero off Z. This proves equality on sections, compatible
with restrictions and natural in all right-linear maps. It uses sheaf
morphisms on every subopen, not only module maps between global sections.

If \(Q\) is finite locally projective, locally choose a finite free right
A-module \(A^h\), an idempotent e, and \(Q=eA^h\), with inclusion/projection
whose composition is the identity on Q. Then Hom(\(Q\),\(F\)) is the image of
precomposition with e on \(F^h\). It is a direct summand of \(F^h\) as a
complex-vector sheaf. Consequently Hom(\(Q\),-) is exact: finite products
preserve an exact stalk sequence, and an idempotent summand preserves
surjectivity (apply the idempotent to a lift) and kernels. All matrices
and equations are actual sheaf matrices on one neighbourhood.

For the same local splitting,
\(\mathcal Hom_{A^{op}}(Q,I^q(B))\) is a direct summand of
\(I^q(B)^h\). Its higher supported cohomology is zero. Indeed the latter
is injective as a complex-vector sheaf by SD.21, so its derived supported
complex has only degree zero; naturality carries the idempotent through
that complex, and the direct summand has zero higher cohomology too.
This is a local assertion and suffices for the sheaf functor \(R\Gamma_Z\).
Exactness of finite Hom makes \(\mathcal Hom(Q,I^*(B))\) a resolution of \(\mathcal Hom(Q,B)\).
The finite acyclic-resolution comparison R.6 (with \(\Gamma_Z\) in place of
its left exact functor) therefore gives the actual comparison
\[
\begin{gathered}
R\Gamma_Z\mathcal Hom_{A^{op}}(Q,B)
\\ \simeq\Gamma_Z\mathcal Hom_{A^{op}}(Q,I^*(B))
\\ =\mathcal Hom_{A^{op}}(Q,C_Z).
\end{gathered}
\tag{SD.29}
\]
The equality is SD.28 in each degree; this identifies the natural map.
It is not a chosen stalk vector-space isomorphism. Left multiplication
on the target commutes with all maps, so this is left-\(B\) linear on the
actual bimodule model just constructed.

For a bounded finite-projective complex \(Q\), use the total Hom differential
\[
\begin{gathered}
(dF)^n={}
\\ d_I F^n-(-1)^n F^n d_Q,
\\ \mathcal Hom(Q,I)^n
\\ =\bigoplus_i\mathcal Hom(Q^i,I^{i+n}).
\end{gathered}
\tag{SD.30}
\]
There are only finitely many i. Each total term is a finite sum of the
proved supported-acyclic summands. Applying the exact finite Hom
functors to the augmented resolution, finite-diagonal column
elimination proves the quasi-isomorphism
\(\mathcal Hom(Q,B)\to\operatorname{Tot}\mathcal Hom(Q,I)\). The same acyclic-resolution comparison then
computes its derived support. SD.28 commutes with both source
precomposition and target differentials; it therefore gives a literal
identity of the supported total complexes. This proves SD.2 with the
precise differential SD.30 and actual right-A target \(C_Z\).

The result is natural under actual chain maps \(Q\to Q'\). It is also
independent of the chosen finite resolution, with the required sheaf
and domain qualifications. For two bounded finite locally projective
resolutions \(R\),\(S\) of a sheaf \(N\), a stalk lifting is constructed downwards:
lift \(R^0\to N\) through \(S^0\to N\); then lift \(f^{i+1}d_R^i\) through
\(S^i\to\ker d_S^i\). Projectivity provides these lifts. A difference \(k\)
of two lifts is similarly lifted to \(h^i\), giving
\[
 k^i=d_S^{i-1}h^i+h^{i+1}d_R^i.                                   \tag{SD.31}
\]
These are finitely many maps between finite summands. Their entries
are actual \(E\) germs; represent all entries, splitting idempotents and
equalities on one common neighbourhood, and shrink once for the
finitely many relations. Stalk surjectivity supplies a neighbourhood
for each required lift of finitely many generating sections; the finite
intersection supplies the single comparison domain. No assertion of
global projectivity of a locally projective sheaf is used. Maps in
both directions lift the identity, so their compositions are homotopic
to the identities by SD.31.

For a target complex \(C\) the induced Hom homotopy on degree \(n\) is
\(K(F)=(-1)^n Fh\). Expanding SD.30 gives
\[
\begin{gathered}
dK(F)+K(dF)
\\ =F(d_Sh+hd_R)=Fk.
\end{gathered}
\tag{SD.32}
\]
The two target-differential terms cancel. The remaining two source
terms have sign plus. Equality SD.28 commutes with these homotopies
because every right action and every map restricts off Z. Hence
different finite resolution choices induce the same actual derived
finite-Hom/support morphism on their common sheaf neighbourhoods.
All target sections are full \(B\) sections; their infinite coefficients
already have common domains, and there are finitely many source
operator entries. This is the sheaf comparison needed by the finite-dual reduction.

#### SD.G. Coefficient families, the support cube and the auxiliary tube

![Full coefficient realization and the singular support depth mechanism](assets/full-coefficient-depth-mechanism.png)

The figure is an exact mechanism diagram, not a drawing of an analytic
set's embedding. Its top arrows are the actual SD.4/SD.7/SD.9 inverse
maps; the auxiliary disc in the negative box represents a variable-radius
germ neighbourhood, not a globally uniform radius. Its middle \(c=2\)
cube shows all four Laurent-sign blocks and SD.19's contraction for
every proper \(J\); only \(J=\{1,2\}\) remains, in degree two. Its bottom exact
sequence shows SD.23 with \(\dim W<\dim Z\) and the proved lower bound for
\(Rj_*\). It explicitly includes singular points. The reproducible source, exact data, PNG and editable SVG accompany the diagram. The free human source is HolIII
I.2.2; the complete proof locators are SD.A--SD.F.

![Exact variable-radius auxiliary tube](assets/auxiliary-variable-tube.png)

This second figure plots the real \(v,s\) slice of the actual complex tube
with \(U\) the unit complex disc, \(\psi_0=-\log(1-|v|^2)\), \(\chi(t)=1+t\), and
\(|s|<\exp(-1)(1-|v|^2)\). The displayed sample coordinates have \(v\)-spacing
1/250 and the exact analytic radius formula; open boundary points are
excluded. The complex domain and all signs/constants are stated in the
right panel, and SD.15 proves its smooth strict proper exhaustion.
For general data SD.14 chooses \(\chi\) to fit the actual representative;
the figure is an exact special example of that construction, not an
assertion of a fixed tube radius for all data. Reproducible source, exact samples, full PNG and editable SVG give this slice.


#### FD.1. Finite duality and a support-degree bound

Let \(A\) be an associative unital ring, let \(A\to B\) be a unital homomorphism, and let \(M\) have a resolution \(P\) of finite projective left \(A\)-modules in cochain degrees \(-m,\ldots,0\). Assume that \(B\) is flat as a right \(A\)-module. Put \(Q=\operatorname{Hom}_A(P,A)\), a complex of right modules. Suppose
\[
\begin{gathered}
H^j(Q)=0\quad(j\ne r),
\\ N=H^r(Q),
\\ 0\le r\le m.
\end{gathered}
\tag{FD.1}
\]
Our conventions are \(K[s]^i=K^{i+s}\), \(d_{K[s]}=(-1)^s d_K\), and
\[
\begin{gathered}
d(f)={}
\\ d_{\mathrm{target}}f-(-1)^i f d_{\mathrm{source}}
\\ \text{for a Hom element}
\\ \text{of degree }i.
\end{gathered}
\tag{FD.2}
\]

**Proposition FD.1.** The right module \(N\) has a resolution \(R\) of finite projective right modules in degrees \(-r,\ldots,0\), and there is a natural derived isomorphism of left \(B\)-modules
\[
\begin{gathered}
B\otimes_A M
\\ \simeq R\operatorname{Hom}_{A^{\mathrm{op}}}(N,B)[r].
\end{gathered}
\tag{FD.3}
\]
If a bounded-below right-module complex \(C\) has \(H^q(C)=0\) for \(q<c\), then
\[
\begin{gathered}
H^j\!\left(\operatorname{Hom}_{A^{\mathrm{op}}}(R,C)[r]\right)=0
\\ (j<c-r).
\end{gathered}
\tag{FD.4}
\]
All tensors and Homs here are algebraic. No completion or growth comparison is implicit.

**Finite projective duals.** A finite generating set of a projective left module \(L\) gives a split surjection \(A^h\to L\). Its dual makes \(L^\vee=\operatorname{Hom}_A(L,A)\) a finite projective right module. When \(L\) is free with basis \(e_a\), evaluation is the isomorphism
\[
\begin{gathered}
B\otimes_A L\longrightarrow
\operatorname{Hom}_{A^{\mathrm{op}}}(L^\vee,B),
\\ b\otimes l\longmapsto\bigl(f\longmapsto bf(l)\bigr),
\\ g\longmapsto\sum_a g(e_a^\vee)\otimes e_a
\\ \text{in the inverse direction}.
\end{gathered}
\tag{FD.5}
\]
Balancing is the identity \(ba f(l)=b f(al)\). Passing to complementary summands proves the same isomorphism for finite projective \(L\). It commutes with maps between such modules.

**Removing the exact tail.** Every \(Q^j\) is finite projective. If its largest nonzero degree \(t\) exceeds \(r\), exactness makes \(Q^{t-1}\to Q^t\) surjective. Split this map using projectivity of \(Q^t\). The decomposition of \(Q^{t-1}\) into its kernel and the chosen copy of \(Q^t\) separates a contractible identity complex \(Q^t\to Q^t\); the preceding differential lands in the kernel because its composite is zero. Removing this summand leaves finite projectives. Repeating finitely gives a complex \(Q'\) in degrees \(0,\ldots,r\) with the same cohomology and explicit homotopy equivalence \(Q\simeq Q'\). Set \(R=Q'[r]\) and augment its degree-zero term to \(N\) by the quotient of cycles by boundaries. FD.1 gives exactness in negative degrees and the required degree-zero cohomology. Thus \(R\) is the asserted resolution and
\[
Q\simeq Q'=R[-r].
\tag{FD.6}
\]
No projectivity of \(N\) is assumed.

**Evaluation and its signs.** For \(p\in P^i\), define
\[
\begin{gathered}
J^i(b\otimes p)(f)=(-1)^i bf(p),
\\ f\in Q^{-i}.
\end{gathered}
\tag{FD.7}
\]
By FD.5 this is balanced, right linear in \(f\), and an isomorphism in each degree. The dual differential is \(d_Q^q(f)=(-1)^{q+1}f d_P^{-q-1}\). Consequently, evaluated on \(f\in Q^{-i-1}\),
\[
\begin{gathered}
d_{\operatorname{Hom}}J^i(b\otimes p)(f)
\\ =(-1)^{i+1}bf(d_Pp)
\\ =J^{i+1}(b\otimes d_Pp)(f).
\end{gathered}
\tag{FD.8}
\]
Therefore \(J:B\otimes_A P\to\operatorname{Hom}_{A^{\mathrm{op}}}(Q,B)\) is a chain isomorphism and is left \(B\)-linear. Right flatness of \(B\) preserves every kernel/image sequence of the augmented resolution \(P\), so its tensor complex has cohomology \(B\otimes_A M\) in degree zero and zero elsewhere. Hom of each removed identity complex is contractible. FD.6 thus replaces the right-hand complex by \(\operatorname{Hom}(R[-r],B)\). The isomorphism
\[
\begin{gathered}
\operatorname{Hom}(R[-r],B)
\\ \longrightarrow\operatorname{Hom}(R,B)[r]
\end{gathered}
\tag{FD.9}
\]
multiplies a map of degree \(i\) by \((-1)^{ri}\). Indeed, the source differential is \((-1)^{i+r+1}f d_R\), while the target differential is \((-1)^{i+1}f d_R\). The exponents \(ri+i+1\) and \(r(i+1)+i+r+1\) differ by \(2r\), verifying the chain identity. This proves FD.3 once the derived-Hom and resolution comparisons are justified.

#### FD.2. Cone, target and resolution comparisons

For a chain map \(u:C\to C'\), its cone has
\[
\begin{gathered}
T^n=C'^n\oplus C^{n+1},
\\ d_T(v,w)={}
\\ (d_{C'}v+uw,-d_Cw).
\end{gathered}
\tag{FD.10}
\]
The square is zero because \(u\) commutes with differentials. We prove directly that the cone is acyclic exactly when \(u\) is a quasi-isomorphism.

Suppose first that \(u\) is a quasi-isomorphism and \((v,w)\) is a cone cycle. Then \(w\) is a cycle and \(uw=-d_{C'}v\). Injectivity on cohomology gives \(w=d_Ca\), with \(a\in C^n\). Subtracting the boundary \(d_T(0,-a)=(-ua,w)\) leaves the cycle \((v+ua,0)\). Surjectivity on cohomology gives \(v+ua=ub+d_{C'}t\), where \(b\in C^n\) is a cycle and \(t\in C'^{n-1}\). The original cycle is therefore \(d_T(t,b-a)\).

Conversely, suppose \(T\) is acyclic. If \(w\in C^n\) is a cycle and \(uw=d_{C'}v\), then \((-v,w)\) is a cone cycle in degree \(n-1\). Writing it as a boundary shows \(w=-d_Ca\), proving injectivity of \(H^n(u)\). For any cycle \(v\in C'^n\), acyclicity writes \((v,0)=(d_{C'}t+ua,-d_Ca)\). Thus \(a\) is a cycle and \([v]=H^n(u)[a]\), proving surjectivity.

If \(L\) is projective and \(T\) acyclic, a cycle \(f:L\to T^n\) factors through \(Z^n(T)\). The surjection \(T^{n-1}\to Z^n(T)\) admits a lift of \(f\) by projectivity, so \(f=dg\). Hence \(\operatorname{Hom}(L,T)\) is acyclic.

To pass to the whole bounded source \(R\), let \(R_{\ge k}\) contain degrees \(k,\ldots,0\). These are subcomplexes, and
\[
\begin{gathered}
0\longrightarrow R_{\ge k+1}\longrightarrow R_{\ge k}
\\ \longrightarrow R^k[-k]\longrightarrow0
\end{gathered}
\tag{FD.11}
\]
is degreewise split. Applying \(\operatorname{Hom}(-,T)\) gives a reversed degreewise split exact sequence. Its left term is a shift of \(\operatorname{Hom}(R^k,T)\), and its right term is \(\operatorname{Hom}(R_{\ge k+1},T)\). In an exact sequence of complexes with acyclic ends, a middle cycle maps to a quotient boundary \(dv\). Lift \(v\) to \(w\); subtracting \(dw\) leaves a cycle coming from the subcomplex. That cycle is a boundary there, so the original middle cycle is a boundary. Starting at \(k=0\) and proceeding finitely to \(-r\) proves acyclicity of \(\operatorname{Hom}(R,T)\).

Under the component identification \((f,g)\),
\[
\begin{gathered}
\operatorname{Hom}(R,\operatorname{cone}u)
\\ =\operatorname{cone}\bigl(\operatorname{Hom}(R,u)\bigr).
\end{gathered}
\tag{FD.12}
\]
Here \(f\) has Hom degree \(n\), \(g\) has degree \(n+1\), and the differential is \((d_{\operatorname{Hom}}f+ug,-d_{\operatorname{Hom}}g)\). Its second component is \(-d_Cg-(-1)^n g d_R\), exactly the degree-\(n+1\) Hom sign. The cone criterion therefore shows that Hom from \(R\) preserves target quasi-isomorphisms.

For resolution independence, let \(R,S\) be finite projective resolutions of the same right module \(N\), extended by zero outside their bounded nonpositive ranges. Lift \(R^0\to N\) through \(S^0\to N\). Once \(f^{i+1}\) is defined, \(f^{i+1}d_R^i\) lands in the kernel of the next differential of \(S\), using the augmentation at degree zero. Exactness makes \(S^i\) surject onto that kernel, and projectivity of \(R^i\) supplies \(f^i\). This constructs \(f:R\to S\) lifting the identity. Interchanging \(R,S\) gives \(g\).

Any difference \(k\) of two lifts is null-homotopic. First lift \(k^0\), whose image is in \(\ker(S^0\to N)\), to \(h^0:R^0\to S^{-1}\). Given \(h^{i+1}\), applying \(d_S^i\) to \(k^i-h^{i+1}d_R^i\) gives zero by the previous-degree identity. Exactness and projectivity lift this map to \(h^i\). Thus
\[
k^i=d_S^{i-1}h^i+h^{i+1}d_R^i.
\tag{FD.13}
\]
In particular \(gf\) and \(fg\) are homotopic to the identities. For a target complex \(C\), the induced Hom homotopy on a degree-\(n\) map is
\[
\begin{gathered}
K(F)=(-1)^n Fh,
\\ dK(F)+K(dF)
\\ =F(d_Sh+hd_R)=Fk.
\end{gathered}
\tag{FD.14}
\]
The two target-differential terms cancel and both source terms have positive sign. Hom therefore turns the resolution comparisons into homotopy equivalences. Together with FD.12 this proves that the bounded projective complex computes the resolution-independent derived Hom in FD.3. Each total degree uses only finitely many source terms.

#### FD.3. The lower bound and the actual support interface

Replace \(C\) by its good truncation \(C_{\ge c}\): it has zero terms below \(c\), term \(C^c/\operatorname{im}d_C^{c-1}\) at \(c\), and the original terms above it. The natural quotient map is a quasi-isomorphism because \(H^q(C)=0\) for \(q<c\). At degree \(c\) both cohomologies are \(\ker d_C^c/\operatorname{im}d_C^{c-1}\); above it they agree and below it they vanish. FD.2 proves that \(\operatorname{Hom}(R,-)\) preserves this map.

A term of \(\operatorname{Hom}(R,C_{\ge c})\) has total degree \(p+q\), where \(0\le p\le r\) and \(q\ge c\). Each total degree is a finite sum. The complex has no terms below \(c\); shifting by \([r]\) gives
\[
p+q-r\ge c-r.
\tag{FD.15}
\]
This proves FD.4 for the entire finite projective resolution.

![Finite dual degree bounds](assets/finite-dual-degree-bound.png)

*Figure FD.1. For \(r=2\), \(c=3\), all resolution columns \(p=0,1,2\) are shown, with the finite window \(q=3,4,5\) and an upward arrow for the remaining rows. Each cell has degree \(p+q\ge3\); the shift \([2]\) lowers the minimum to one. The general bound is FD.15. This diagram proves no analytic coefficient depth by itself. Reproducible source, exact data, PNG and editable SVG. Free human context: [Holonomic systems III, I.2.2 and I.3.8](https://www.kurims.kyoto-u.ac.jp/~kenkyubu/kashiwara/HolIII.pdf); the complete algebraic argument is FD.1–FD.3 above.*

For sheaves the same argument is applied locally. Splittings, lifts and their identities involve finitely many operator entries; represent their germs and shrink once to make all identities literal. SD.F proves the natural sheaf Hom/support comparison and its compatibility with these finite-resolution maps. Consequently the following implication retains the actual actions and common domains. Suppose an actual finite \(E\)-module \(M\) has a finite locally projective resolution, its dual Ext is concentrated in degree \(r\), and \(B=E^\infty\) is right flat over \(E\). If \(Z\) has complex codimension at least \(c\), then
\[
\begin{gathered}
\mathcal H_Z^j(B\otimes_E M)=0
\\ (j<c-r).
\end{gathered}
\tag{FD.16}
\]
Indeed FD.3 identifies the module with the shifted finite Hom complex. SD.2 takes its derived support to \(\operatorname{Hom}_{E^{\mathrm{op}}}(R,C_Z)[r]\), with the actual bimodule representative \(C_Z\) of SD.27. SD.1 supplies its lower bound, so FD.4 gives FD.16. In particular a local section supported on such a \(Z\) is zero when \(c>r\). Right flatness here is the complete [§5.47 finite-matrix theorem](#5-47-arbitrary-finite-matrices-and-right-flatness-of-the-full-infinite-order-ring). The concentrated-Ext condition remains an explicit hypothesis of FD.16. Neither this implication nor coefficient depth alone establishes the real-module injection, arbitrary-section separation or the full moving-sector comparison.



### 5.51. Actual analytic microlocal Ext and support purity

For every coherent holonomic left module over the actual finite-order analytic microdifferential sheaf, the dual Ext vanishes outside degree \(n=\dim_{\mathbf C}X\). The shifted dual \(R\mathcal Hom_E(M,E)[n]\) is a right holonomic module in degree zero. This statement includes singular, reducible and nonreduced holonomic germs. The proof below supplies the full commutative analytic estimate, actual principal-Hom injection, right-side estimates and contact-rank argument. Its support-purity consequence concerns every section of the full extension \(E^\infty\otimes_E M\).

#### EC.0. Conventions and earlier actual operator proofs

Let the base dimension be \(n=d+1\), and use a nonzero-covector chart

\[
\begin{gathered}
u=(t,x_1,\ldots,x_d),
\\ \rho=(\tau,\xi_1,\ldots,\xi_d),
\\ h=\tau^{-1},\qquad z_i=\xi_i/\tau.
\end{gathered}
\tag{EC.0}
\]

The normalized contact chart has dimension \(m=2d+1=2n-1\). Its holomorphic
germ ring is \(R=\mathbf C\{u,z\}\). Write \(A=E_0=\mathscr E(0)\) and

\[
\begin{gathered}
E=A[h^{-1}],
\\ A/(h)=R,
\\ ha=\sigma(a)h,
\\ \sigma(a)\equiv a\pmod h.
\end{gathered}
\tag{EC.1}
\]

All localizations here are actual finite-order normal localizations.
They consist of finite-order shifts of actual factorial-growth symbols.
The target statement uses the right module

\[
\begin{gathered}
\mathcal Ext_E^j(M,E),
\\ \mathbb D_E^r(M)
\\ =R\mathcal Hom_E(M,E)[r].
\end{gathered}
\tag{EC.2}
\]

No left side change is implicit. With cohomological shift

\[
 K[r]^i=K^{i+r},
 \tag{EC.3}
\]

concentration in Ext degree \(n\) means that \(\mathbb D_E^n(M)\) is a
right module in degree zero. Thus the integer called \(r\) in the support
reduction and in HolIII I.3.8 is exactly \(n=\dim_{\mathbf C}X\) in this
holonomic application.

The earlier actual operator proofs used here are:

1. [Sections 5.19–5.20, S.1–S.4 and B.1–B.3.1](#5-19-actual-factorial-growth-symbols-and-ordered-jets): homogeneous normal symbols, their actual Leibniz product, common-domain factorial estimates and elliptic inversion.
2. [Sections 5.21 and 5.24, Theorems 5.21.1 and 5.24.2](#5-21-actual-simultaneous-division-and-finite-stalk-relations): finite actual left relations and common-neighbourhood coherence.
3. [Section 5.22.1](#5-22-exact-faithful-positive-degree-completion-at-the-dummy-zero-section), equations 5.22a–5.22b: both actual normal automorphisms and the trivial reduction of their normal action modulo \(h\).
4. [Section 5.32, J.1–J.3](#5-32-coherent-actual-lattices-and-principal-support-for-the-whole-germ): actual coherent \(h\)-injective lattices, exact reduced principal/actual support, and whole-germ contact reduction with a finite principal fibre. The actual contact isomorphism is [§5.28, Q.1–Q.7](#5-28-convergent-contact-quantization-and-whole-germ-generic-position).
5. [Section 5.31, L.3–L.6](#5-31-a-bounded-actual-generic-position-complex-with-polynomial-operator-matrices): a finite free actual lattice resolution with exact principal reduction on one neighbourhood, and its actual normal localization. Its bound is \(q=2d+1\).

The first-order trace input is the complete subargument of Good filtrations and the characteristic variety, Theorem 7.0, from “Localization of first-order modules” through “Return to the radical ideal”, equations 7.1d–7.1s. It applies to a rational first-order ring with central square-zero \(\epsilon\), commutative Noetherian quotient, and \(\ker\epsilon=\epsilon B\), together with a finite module satisfying \(\ker\epsilon=\epsilon V\). It proves that each minimal support prime of \(V/\epsilon V\) is involutive for the commutator biderivation. Its denominator localization, coefficient field in the finite Artin quotient and finite-length trace are proved there. It assumes neither finite type over a field nor Jacobi for that biderivation. EC.D below checks all these hypotheses on the actual quotients.

Analytic component, parametrization, vanishing-ideal and dimension assertions are the complete Analytic germs, Theorems 4.1, 5.1, 5.4 and Proposition 5.5. The analytic Ext proof below identifies the other exact earlier local-algebra inputs.

#### EC.A. The commutative analytic Ext estimate

The required statement is for the analytic ring itself:

\[
\begin{gathered}
R=\mathbf C\{w_1,\ldots,w_m\},
\\ N\text{ finite over }R,
\\ c=m-\dim\operatorname{Supp}N,
\\ \operatorname{Ext}_R^j(N,R)=0\quad(j<c),
\\ \operatorname{Supp}\operatorname{Ext}_R^j(N,R)
\\ \subseteq\operatorname{Supp}N,
\\ \dim\operatorname{Supp}\operatorname{Ext}_R^j(N,R)
\\ \le m-j.
\end{gathered}
\tag{EC.A}
\]

The zero module has zero Ext in every degree. Here is the complete analytic proof of the displayed estimate.

##### AC.A. The ambient local hypotheses

Let
\[
R=\mathbf C\{w_1,\ldots,w_m\},\qquad m\geq0.
\]
The following facts genuinely apply to this ring and every algebraic prime localization of it:

1. It is Noetherian local, with maximal ideal \(\mathfrak m=(w_1,\ldots,w_m)\); every finite module and every submodule of a finite module is Noetherian and finite.
2. It is regular local of dimension \(m\), and each \(R_{\mathfrak p}\) is regular local of dimension \(\operatorname{ht}_R\mathfrak p\).
3. Each such local ring is Cohen–Macaulay; a regular element in the maximal ideal of a nonzero finite module drops its depth exactly by one.
4. For every prime \(\mathfrak p\),
   \[
\begin{gathered}
\dim(R/\mathfrak p)
\\ =m-\operatorname{ht}_R\mathfrak p.
\end{gathered}
\tag{AC.1}
\]
5. Every finite \(R_{\mathfrak p}\)-module has a finite free resolution of length at most \(\operatorname{ht}_R\mathfrak p\).

The exact earlier proofs appear in AC.B. Their hypotheses apply to convergent germs and every prime localization. In particular AC.1 is the general Cohen–Macaulay local-ring dimension formula. The arguments below establish the analytic Ext bounds directly over this ring.

##### AC.B. Exact earlier programme proofs

The following are earlier programme proofs at their stated Noetherian, local, regular or Cohen–Macaulay hypotheses. No finite-type hypothesis over a field is imposed on the convergent ring or its prime localizations.

| Result used | Complete earlier programme proof |
|---|---|
| Convergent units, Weierstrass preparation and division, analytic Noetherianity | Analytification, Section 2 and Lemma 2.1 |
| Finite modules and submodules; localization of Noetherian rings | Noetherian and Artinian rings, Proposition 1.2 and Theorem 2.1 |
| Exact localization, prime correspondence, Nakayama and support detection | Localization, Theorems 2.1, 3.1, 4.2, 5.1 and Proposition 6.1 |
| General local dimension bound and regular-element dimension drop | Local dimension, Theorem 3.2 and Proposition 4.2 |
| Associated primes, zero divisors, localization, finite prime avoidance and prime filtrations | Associated primes, Lemma 1.1, Theorems 1.2, 2.2, 3.1–3.2 and Solution 8.5 |
| Depth zero and exact depth drop | Depth, Lemmas 2.1–2.2 and Theorem 2.3 |
| Cohen–Macaulay associated components, localization and the general local dimension formula | Depth, Theorems 3.1, 4.1, 5.1 and Lemma 5.4 |
| Regular local rings are Cohen–Macaulay; regular quotients remain Cohen–Macaulay | Depth, Theorem 6.1 and Corollary 6.2 |
| Minimal resolutions, Auslander–Buchsbaum and the finite free-resolution bound | Projective dimension, Theorem 2.2 and Corollary 4.3, with complete Sections 1–4 |
| Regularity at every prime localization | Regular local rings, Lemma 3.1 and Theorem 3.2, with Lemma 2.1 and Theorem 2.2 |
| Resolution comparison, homotopies and the second-variable Ext exact sequence | Resolutions, Tor and Ext, Theorem 1.1 and Sections 2 and 4 |
| Finite analytic parametrization, Nullstellensatz and analytic dimension | Analytic germs, Theorems 4.1, 5.1, 5.4 and Proposition 5.5 |
| Exact actual lattice and principal support for the whole coherent germ | [Section 5.32, J.0–J.2](#5-32-coherent-actual-lattices-and-principal-support-for-the-whole-germ) |


##### AC.C. The convergent ambient ring

**AC.2 (the analytic ambient ring).** The ring \(R\) is Noetherian regular local of dimension \(m\).

**Proof.** The constant-term unit test makes \(R\) local. A germ with zero constant term is a finite sum \(\sum_iw_if_i\) with convergent \(f_i\): assign each nonconstant Taylor monomial to the first coordinate dividing it. The assigned subseries converge absolutely on a smaller polydisc, and division by the assigned coordinate is only a fixed monomial shift. Thus the maximal ideal is exactly \((w_1,\ldots,w_m)\). Taylor coefficients show that the classes of the \(w_i\) give a basis of \(\mathfrak m/\mathfrak m^2\): there are no nonzero linear terms in \(\mathfrak m^2\), and every term of degree at least two is a finite sum of degree-two monomials times convergent germs by the same assignment. Hence the embedding dimension is \(m\).

Here is the earlier analytic Noetherian argument with its analytic hypothesis retained. Induct on \(m\), starting from \(\mathbf C\). If \(I\ne0\), choose a nonzero \(f\in I\). A complex linear coordinate change makes its first nonzero homogeneous part nonzero on the last coordinate axis. Analytic Weierstrass preparation gives a monic polynomial \(P\in I\), of degree \(r\) in that variable, whose lower coefficients belong to \(R_{m-1}=\mathbf C\{w_1,\ldots,w_{m-1}\}\). Analytic division identifies \(R/(P)\) as the finite free \(R_{m-1}\)-module of polynomial remainders of degree less than \(r\). By induction and Noetherian Proposition 1.2 it is a Noetherian module. The image of \(I\) in it has finitely many generators. Their lifts, together with \(P\), generate \(I\) over \(R\). The zero ideal needs no generators. This is the earlier analytification Lemma 2.1, with its complete Section 2 preparation/division proof; it concerns convergent germs throughout.

The coordinate ideals
\[
\begin{gathered}
(0)\subsetneq(w_1)\subsetneq\cdots
\\ \subsetneq(w_1,\ldots,w_m)
\end{gathered}
\tag{AC.3}
\]
are prime. Indeed their quotients are the remaining convergent power-series rings; these are domains because the lowest nonzero total-degree homogeneous pieces of two nonzero series have nonzero product in a polynomial ring. The containments are strict by Taylor coefficients. Thus \(\dim R\geq m\). The general Noetherian local Proposition 4.2 in the earlier local-dimension lesson gives \(\dim R\leq\operatorname{embdim}R=m\). Equality proves regularity. At \(m=0\) this is the field case, with empty coordinate list. \(\square\)

Consequently the earlier regular-local lesson Theorem 3.2 applies to **every algebraic prime** \(\mathfrak p\) of \(R\), not just primes representing smooth analytic components. Its proof localizes arbitrary projective resolutions, uses finite global dimension and the proved regularity characterization. The height equality is the exact prime-chain correspondence. By the earlier depth lesson Theorem 6.1 all these localizations are CM. By the earlier projective-dimension lesson Corollary 4.3, or the earlier regular-local lesson Theorem 2.2, their finite modules have finite free resolutions bounded by their dimensions.

For clarity, the dimension formula used here is also the following exact CM argument. Write \(h=\dim R_{\mathfrak p}\). The proof of the earlier depth lesson Theorem 5.1 constructs an \(R\)-regular list \(a_1,\ldots,a_h\) inside \(\mathfrak p\). Its quotient \(Q\) is CM of dimension \(m-h\), by the exact dimension/depth drops. Its localization at \(\mathfrak p\) is nonzero by Nakayama and has dimension zero by the same drops in the CM ring \(R_{\mathfrak p}\). Thus \(\mathfrak p\) is minimal over the list: a smaller prime over the list would give a positive-length chain in that localized quotient. The CM associated-component Theorem 4.1 then gives
\[
\dim(R/\mathfrak p)=\dim Q=m-h.
\]
This is exactly earlier Lemma 5.4. It uses neither algebraic Noether normalization nor finite type over \(\mathbf C\). Together with \(h=\operatorname{ht}\mathfrak p\), it proves AC.1.

##### AC.D. Regular sequences inside annihilator ideals

**AC.4.** Let \(A\) be a Noetherian regular local ring, and let \(I\subsetneq A\). If every prime containing \(I\) has height at least \(r\), then \(I\) contains an \(A\)-regular sequence of length \(r\).

**Proof.** Suppose \(a_1,\ldots,a_j\in I\) have been chosen regularly, with \(j<r\), and set \(Q_j=A/(a_1,\ldots,a_j)\). It is a nonzero finite module. For \(\mathfrak q\in\operatorname{Ass}_A Q_j\), an associated element survives at \(\mathfrak q\): a denominator outside \(\mathfrak q\) could not annihilate it. It is annihilated there by the maximal ideal. The depth-zero test therefore gives \(\operatorname{depth}_{A_{\mathfrak q}}(Q_j)_{\mathfrak q}=0\). Each earlier sequence entry belongs to \(\mathfrak q\), since it annihilates \(Q_j\). Exact localization preserves its injectivity, and all intermediate localized quotients are nonzero since the final quotient is nonzero. The localized sequence is consequently regular. The localized ambient ring is regular and CM, so the exact depth drops give
\[
\begin{gathered}
0=\operatorname{depth}_{A_{\mathfrak q}}(Q_j)_{\mathfrak q}
\\ =\operatorname{ht}_A\mathfrak q-j.
\end{gathered}
\tag{AC.5}
\]
Every associated prime of \(Q_j\) therefore has height \(j<r\), so none contains \(I\). Its associated-prime set is finite. Finite prime avoidance selects \(a_{j+1}\in I\) outside the union of these primes. The zero-divisor theorem makes multiplication by it injective on \(Q_j\). Since \(I\) is proper in a local ring, it lies in the maximal ideal, and Nakayama keeps the next quotient nonzero. Induction constructs the desired list. For \(r=0\), use the empty sequence. \(\square\)

The quotients \(Q_j\) are not asserted to be regular. The only regular rings in this argument are \(A\) and its prime localizations.

##### AC.E. Direct lower Ext vanishing

**AC.6.** Let \(A\) be a commutative ring, \(N,T\) modules, and suppose \(a_1,\ldots,a_r\) annihilate \(N\) and act as a regular sequence on \(T\). Then
\[
\begin{gathered}
\operatorname{Ext}_A^j(N,T)=0
\\ (0\leq j<r).
\end{gathered}
\tag{AC.7}
\]
No finiteness or injective resolution is needed for this assertion.

**Proof.** Choose a free resolution \(F_\bullet\to N\). If \(aN=0\), multiplication by \(a\) on \(F_\bullet\) is homotopic to zero. Here is the actual construction. Put \(s_{-1}=0\). Since the image of \(a:F_0\to F_0\) lies in the kernel of the augmentation, freeness lifts it across \(d_1:F_1\twoheadrightarrow\ker\epsilon\), giving \(s_0\) with \(d_1s_0=a\operatorname{id}_{F_0}\). Once the lower \(s_i\)'s have been chosen, the map
\[
a\operatorname{id}_{F_i}-s_{i-1}d_i
\]
has image in \(\ker d_i\), because
\[
\begin{gathered}
d_i(a\operatorname{id}_{F_i}-s_{i-1}d_i)
\\ =a d_i-(a\operatorname{id}_{F_{i-1}}-s_{i-2}d_{i-1})d_i
\\ =0.
\end{gathered}
\]
Lift it across \(d_{i+1}\) to \(s_i\). Thus
\[
a\operatorname{id}_{F_i}=d_{i+1}s_i+s_{i-1}d_i
\tag{AC.8}
\]
in every degree. This is also the comparison proof in the earlier resolution lesson Theorem 1.1 specialized to the zero endomorphism of \(N\).

In \(\operatorname{Hom}_A(F_\bullet,T)\), precomposition with multiplication by \(a\) equals coefficient multiplication by \(a\), since \(A\) is commutative. Precomposition with the \(s_i\)'s gives a cochain homotopy from that map to zero by AC.8. Thus every \(a\in\operatorname{ann}_A N\) kills \(\operatorname{Ext}_A^j(N,T)\) for all \(j\) and every target \(T\).

Induct on \(r\). The assertion is vacuous for \(r=0\). For \(r\geq1\), every map \(N\to T\) has image annihilated by \(a_1\). Injectivity of \(a_1\) on \(T\) makes this image zero. This proves degree zero. Put \(T'=T/a_1T\). The remaining \(a_2,\ldots,a_r\) are regular on \(T'\) and still annihilate \(N\). The degreewise exact Hom complexes for
\[
0\longrightarrow T\xrightarrow{a_1}T\longrightarrow T'\longrightarrow0
\]
give, for \(1\leq j<r\), the exact segment
\[
\begin{gathered}
\operatorname{Ext}_A^{j-1}(N,T')
\\ \longrightarrow\operatorname{Ext}_A^j(N,T)
\\ \xrightarrow{a_1}\operatorname{Ext}_A^j(N,T).
\end{gathered}
\tag{AC.9}
\]
The first term is zero by induction, since \(j-1<r-1\). The last arrow is therefore injective. It is also zero by AC.8, since \(a_1N=0\). Hence its source is zero. This proves every degree in AC.7. The connecting-homomorphism construction is the explicit one in the earlier resolution lesson Section 2. \(\square\)

This argument works directly over the actual convergent ring and uses no change-of-rings premise or finite-type hypothesis.

##### AC.F. The analytic Ext theorem

**AC.10.** For every finite \(R\)-module \(N\) and every integer \(j\geq0\), \(\operatorname{Ext}_R^j(N,R)\) is finite and
\[
\begin{gathered}
\operatorname{Supp}_R\operatorname{Ext}_R^j(N,R)
\\ \subseteq\operatorname{Supp}_R N
\\ {}\cap\{\mathfrak p:\operatorname{ht}_R\mathfrak p\geq j\}.
\end{gathered}
\tag{AC.11}
\]
Consequently
\[
\begin{gathered}
\dim\operatorname{Supp}_R\operatorname{Ext}_R^j(N,R)
\\ \leq m-j.
\end{gathered}
\tag{AC.12}
\]
For \(N\ne0\), put \(c=m-\dim\operatorname{Supp}_R N\). Then
\[
\begin{gathered}
\operatorname{Ext}_R^j(N,R)=0
\\ (0\leq j<c).
\end{gathered}
\tag{AC.13}
\]
For \(N=0\), all groups vanish. Empty support has dimension \(-\infty\).

**Proof.** Noetherianity gives a resolution by finite free modules: choose finite generators, then finite generators of the finite kernel, and repeat. Applying Hom into \(R\) gives a cochain complex of finite free modules, so every kernel, image and cohomology module is finite. Exact localization and Hom from finite free modules give
\[
\begin{gathered}
\operatorname{Ext}_R^j(N,R)_{\mathfrak p}
\\ \cong\operatorname{Ext}_{R_{\mathfrak p}}^j
(N_{\mathfrak p},R_{\mathfrak p}).
\end{gathered}
\tag{AC.14}
\]
This is zero if \(N_{\mathfrak p}=0\). It is also zero for \(j>\operatorname{ht}\mathfrak p\), because AC.2 and the exact regular-local theorem give a finite free resolution of \(N_{\mathfrak p}\) of length at most \(\dim R_{\mathfrak p}=\operatorname{ht}\mathfrak p\); resolution comparison identifies its Ext with AC.14. This proves AC.11.

Let \(Q=\operatorname{Ext}_R^j(N,R)\). If \(Q\ne0\), every prime of its support has height at least \(j\), so AC.1 gives \(\dim(R/\mathfrak p)\leq m-j\) for each such prime. More explicitly, the finite prime-filtration theorem makes \(\operatorname{Supp}Q\) a finite union of the closures \(V(\mathfrak p_i)\) of its filtration primes. Each filtration factor \(R/\mathfrak p_i\) is supported inside \(\operatorname{Supp}Q\), by exact localization, so its height is at least \(j\). A finite closed union has dimension the maximum of the component dimensions. Hence \(\dim\operatorname{Supp}Q\leq m-j\). If \(j>m\), AC.11 already makes the support empty, hence \(Q=0\) by local detection. The zero case satisfies AC.12 under the empty-support convention.

For the lower vanishing assume \(N\ne0\) and put \(I=\operatorname{ann}_R N\). Finiteness proves \(\operatorname{Supp}N=V(I)\): if \(N_{\mathfrak p}=0\), each member of a finite generating set is killed by a denominator outside \(\mathfrak p\); their product is in \(I\setminus\mathfrak p\). Conversely an element of \(I\setminus\mathfrak p\) becomes an invertible annihilator, so kills the whole localization. The ideal \(I\) is proper.

The support dimension is the maximum of \(\dim(R/\mathfrak q)\) over its finitely many minimal support primes. To see the finite-union dimension statement directly, these primes give a finite closed cover of \(V(I)\). Any chain in a finite union of closed sets lies entirely in the member containing its generic point: closedness contains all the specializations in the chain. Thus its dimension is the maximum of their dimensions. AC.1 now gives
\[
\begin{gathered}
c=m-\dim\operatorname{Supp}N
\\ =\min_{\mathfrak q\in\operatorname{Supp}N}
\operatorname{ht}_R\mathfrak q.
\end{gathered}
\tag{AC.15}
\]
In particular every prime containing \(I\) has height at least \(c\). AC.4 constructs an \(R\)-regular sequence of length \(c\) lying **in the annihilator** \(I\), so every entry kills \(N\) itself. Apply AC.6 with target \(T=R\). This proves AC.13. No CM, reducedness, smoothness, equidimensionality or generic freeness was imposed on \(N\) or on \(R/I\). \(\square\)

##### AC.G. Analytic support and the actual principal module

The support dimensions in AC.10 are the analytic germ dimensions needed in the microdifferential application. An ideal and its radical have the same prime spectrum and Krull dimension. The analytic Nullstellensatz and analytic-dimension Theorem 5.4 in the earlier analytic-germ lesson identify \(\dim(R/I)\) with the dimension of its reduced analytic zero germ, component by component. Here is the finite-module support interface without an additional coherence premise. Present \(N\) as the cokernel of a finite matrix \(R^a\to R^b\), and represent all its entries on one common neighbourhood. Its finite-type analytic cokernel has zero stalk at a point exactly when that matrix is surjective over the local holomorphic ring, which finite Nakayama makes equivalent to rank \(b\) over its residue field. Its support is therefore the zero germ of the \(b\times b\) minors. The same criterion at an algebraic prime \(\mathfrak p\) of \(R\), over the residue field of \(R_{\mathfrak p}\), says that the prime zero set of those minors is \(\operatorname{Supp}_R N=V(\operatorname{ann}N)\). Thus their radical equals \(\sqrt{\operatorname{ann}N}\), and the analytic Nullstellensatz identifies the analytic support germ with the zero germ of that annihilator. When there are fewer than \(b\) columns, the minor ideal is zero; when \(b=0\), the module is zero and the support is empty. The same construction applies to the finite Ext modules. All nilpotent/embedded relations remain in the modules used by the proof. Only the dimension measurement uses the reduced zero set.

For the microlocal application with \(R=\mathbf C\{t,x_1,\ldots,x_d,z_1,\ldots,z_d\}\), the precise ambient dimension is \(m=2d+1\). For a finite principal module of support dimension \(s\), AC.13 gives vanishing below \(2d+1-s\). AC.12 gives the upper support estimate \(\dim\operatorname{Supp}\operatorname{Ext}_R^j(N,R)\leq2d+1-j\). Section 5.32, J.1–J.2 is the earlier **actual finite-order** lattice/principal support input; it is not replaced by a formal support argument. A homogeneous analytic Lagrangian support in the full nonzero cotangent space has its dimension lowered by one on the transverse \(\tau=1\) slice. In the usual base dimension \(d+1\), this slice has dimension \(d\), so the principal codimension is \(d+1\).


#### EC.B. The actual opposite ring and right finiteness

For clarity we supply the actual-domain side check needed by the dual
complex. This also permits use of the left support construction for a
right module without changing its normalized contact support.

##### EC.B.1. The adjoint remains an actual symbol

Write an order-zero normal symbol as

\[
\begin{gathered}
P(u,\rho)=\sum_{a\ge0}p_{-a}(u,\rho),
\\ p_{-a}=a_a(u,z)\tau^{-a}.
\end{gathered}
\tag{EC.B1}
\]

Its coefficients on every fixed smaller common polydisc satisfy
\(\lvert a_a\rvert\le B C^a a!\). Define

\[
\begin{gathered}
T(P)(u,\rho)={}
\\ \sum_{\alpha\in\mathbf N^n}\frac{1}{\alpha!}
\partial_\rho^\alpha\partial_u^\alpha P(u,-\rho).
\end{gathered}
\tag{EC.B2}
\]

The homogeneous coefficient of order \(-N\) uses exactly
\(a+|\alpha|=N\). It is a finite sum. The antipodal change has the same
normalized coordinates \(z\), and only multiplies the coefficient
of order \(-a\) by \((-1)^a\).

Choose a compact target polydisc in \((u,z)\). Choose two positive fixed
Cauchy radii \(\delta_u,\delta_\rho\) such that its complex \(u\)- and
\(\rho\)-polydiscs remain in one common original coefficient domain,
and \(\lvert\tau\rvert\ge1-r>0\) on the latter. This is possible by
continuity of \((\tau,\xi)\mapsto\xi/\tau\). Homogeneity gives

\[
\begin{gathered}
\lvert p_{-a}(u,-\rho)\rvert\le B C_1^a a!,
\\ C_1=C/(1-r).
\end{gathered}
\tag{EC.B3}
\]

The two independent Cauchy estimates, for \(k=|\alpha|\), give

\[
\begin{gathered}
\left|\frac{1}{\alpha!}
\partial_\rho^\alpha\partial_u^\alpha p_{-a}(u,-\rho)\right|
\\ \le B C_1^a a!\,\alpha! D^k
\\ \le B C_1^a D^k N!,
\\ D=(\delta_u\delta_\rho)^{-1}.
\end{gathered}
\tag{EC.B4}
\]

Here \(\alpha!\le k!\) and \(a!k!\le(a+k)!=N!\).
There are \(\binom{k+n-1}{n-1}\le2^{k+n-1}\) such multi-indices.
Summing over \(a+k=N\), using that every binomial coefficient is at
least one, yields the explicit actual-class bound

\[
\begin{gathered}
\lvert (TP)_{-N}\rvert
\\ \le B2^{n-1}(C_1+2D)^N N!.
\end{gathered}
\tag{EC.B5}
\]

Thus (EC.B2) defines an actual order-zero symbol on the one chosen
smaller common domain. It is not a passage from an arbitrary formal
series to an analytic operator. The same estimate after a finite order
shift handles every finite-order symbol.

##### EC.B.2. It is an anti-involution

Put \(J(P)(u,\rho)=P(u,-\rho)\) and
\(D_*=\sum_i\partial_{u_i}\partial_{\rho_i}\). Then \(T=e^{D_*}J\).
All exponential expressions in this paragraph mean finite expressions
at each homogeneous coefficient. Since \(JD_*=-D_*J\),

\[
\begin{gathered}
T^2=e^{D_*}Je^{D_*}J
\\ =e^{D_*}e^{-D_*}=1.
\end{gathered}
\tag{EC.B6}
\]

This equality can be read coefficientwise: at every positive derivative
order its coefficient is the binomial sum
\(\sum_{a=0}^k\binom{k}{a}(-1)^a=0\).

For the product, let \(\mu\) denote ordinary multiplication and put

\[
 B_* =\sum_i\partial_{\rho_i}\otimes\partial_{u_i},\qquad
 C_* =\sum_i\partial_{u_i}\otimes\partial_{\rho_i}.
\]

The actual homogeneous Leibniz product is
\(P\circ Q=\mu e^{B_*}(P\otimes Q)\).
The ordinary product rule gives

\[
 D_*\mu=\mu(D_*\otimes1+1\otimes D_*+B_*+C_*).
\]

All the constant coefficient derivatives commute. Antipodal change
converts \(B_*\) to \(-B_*\). Consequently

\[
\begin{gathered}
T(P\circ Q)
\\ =\mu e^{D_*\otimes1+1\otimes D_*+B_*+C_*}
\\ {}e^{-B_*}(JP\otimes JQ)
\\ =\mu e^{C_*}(TP\otimes TQ)
\\ =TQ\circ TP.
\end{gathered}
\tag{EC.B7}
\]

Each coefficient here has finitely many summands; B.1 and the earlier
actual product estimate ensure that both endpoints are actual symbols
on a common smaller domain. Equations (EC.B6)--(EC.B7) prove an actual
anti-isomorphism \(A\simeq A^{\mathrm{op}}\). Its principal action on
\(R=A/(h)\) is the identity, and \(T(h)=-h\).

The same formula on sheaf germs commutes with restriction, and the
bounded domain check just proved lets every finite list of entries be
represented on one common smaller domain. Left actual Noetherianity
and finite-matrix coherence therefore imply their right versions:
apply \(T\) entrywise to a right relation matrix, reverse products, use
the earlier finite left kernel list, and transpose back. This operation
preserves the normalized principal support. It is an auxiliary local
side check; the dual in (EC.2) itself remains on the right.

##### EC.B.3. Actual right Nakayama

The earlier actual unit theorem makes every \(1-ah\) invertible in
\(A\), because its principal symbol is one. If a finite right module
\(V\) satisfies \(V=Vh\), choose a generating list of least length.
For a last generator \(v_s\), write

\[
 v_s=\sum_{i=1}^s v_i a_i h.
\]

Then
\(v_s(1-a_sh)=\sum_{i<s}v_i a_i h\). Multiplying on the right by
the actual inverse of \(1-a_sh\) eliminates \(v_s\), contradicting
minimality unless \(V=0\). This proves precisely the finite right
Nakayama consequence needed below. No completeness assertion is used.

#### EC.C. A finite actual principal-Hom injection

Let \(M_0\subset M\) be the actual \(h\)-injective lattice and let

\[
\begin{gathered}
0\longrightarrow L_q\xrightarrow{d_q}L_{q-1}
\\ \longrightarrow\cdots\xrightarrow{d_1}L_0
\\ \longrightarrow M_0\longrightarrow0
\end{gathered}
\tag{EC.C1}
\]

be the bounded actual free \(A\)-resolution supplied by Section 5.31,
L.4--L.6. Its quotient modulo \(h\) is an exact finite free
\(R\)-resolution of \(\bar M_0=M_0/hM_0\). This exact principal
resolution, proved there for the whole lattice, is retained; it is not
the principal matrix of an arbitrary uncorrected generating list.

Set

\[
\begin{gathered}
C^j=\operatorname{Hom}_A(L_j,A),
\\ \delta^j(f)=f\,d_{j+1}.
\end{gathered}
\tag{EC.C2}
\]

These are finite free right \(A\)-modules, and
\(\delta^{j+1}\delta^j=0\). Their cohomology is the actual
\(H^j=\operatorname{Ext}_A^j(M_0,A)\). It is a finite coherent right
module by B.2 and the finite kernel/image constructions. Evaluation on
the finite bases gives the exact identity of finite complexes

\[
\begin{gathered}
C^\bullet/C^\bullet h
\\ =\operatorname{Hom}_R(L_\bullet/hL_\bullet,R).
\end{gathered}
\tag{EC.C3}
\]

In particular its degree-\(j\) cohomology is
\(F^j=\operatorname{Ext}_R^j(\bar M_0,R)\).

**Finite principal-Hom injection.** There is an injective map of
right \(R\)-modules

\[
 H^j/H^j h\ \hookrightarrow\ F^j.
 \tag{EC.C4}
\]

**Proof.** Reduce an actual cycle \(z\in C^j\) modulo \(h\) and take
its class in the cohomology of (EC.C3). A boundary reduces to a boundary,
and a multiple of \(h\) reduces to zero, so the map is well defined.
If its value is zero, lift its finite principal boundary to
\(y\in C^{j-1}\). There is an actual \(w\in C^j\) with

\[
 z-\delta y=w h.
 \tag{EC.C5}
\]

Right \(A\)-linearity gives \((\delta w)h=0\). Multiplication by
\(h\) is injective on the free right term \(C^{j+1}\): in the actual
normal symbol a right factor \(h\) is an exact coefficient shift, and
has zero product only on the zero symbol. Thus \(\delta w=0\).
Equation (EC.C5) says \([z]=[w]h\), proving that the kernel before
quotienting is exactly \(H^jh\). Right \(R\)-linearity follows by reducing
the actual right action, whose differential commutes with every right
coefficient. This proves (EC.C4). \(\square\)

At a contact-chart point where \(F^j\) has zero stalk, (EC.C4) and
B.3 force \(H^j=0\). Consequently

\[
 \operatorname{Supp}H^j\subseteq\operatorname{Supp}F^j.
 \tag{EC.C6}
\]

This is a statement about the actual coherent right Ext module. The
injection (EC.C4) is stronger than the local vanishing implication. It
retains all nilpotents and all actual sections of the original module.
It uses one principal reduction and one actual boundary lift per tested
cycle. There is no infinite descending-order reduction, formal
completion, filtration-page stabilization or analytic/algebraic
comparison in this argument.

##### EC.C.1. Finite-order localization and degree signs

Normal localization of (EC.C1) gives the actual free \(E\)-resolution
of \(M\) already proved in L.4--L.6. Evaluation on each finite free
basis identifies its dual complex with \(C^\bullet[h^{-1}]\).
For a right module, an element of the normal localization is represented
by a finite fraction \(v h^{-k}\), and it is zero exactly when
\(vh^a=0\) for some finite \(a\ge0\). Normality moves coefficients
across denominators using \(\sigma\), so finitely many fractions have
a common denominator. The fraction criterion proves exactness: for a
fraction killed in a localized quotient, a finite further power of
\(h\) sends its numerator to the original kernel; its shifted fraction
then comes from that kernel. Surjections lift numerators. Thus the
localization commutes with the finite cycle and boundary sequences,
and

\[
\begin{gathered}
\operatorname{Ext}_E^j(M,E)=H^j[h^{-1}],
\\ \operatorname{Supp}\operatorname{Ext}_E^j(M,E)
\\ \subseteq\operatorname{Supp}F^j.
\end{gathered}
\tag{EC.C7}
\]

The right Ext is coherent by B.2 and the corresponding finite localized
kernel construction. Its support in (EC.C7) is normalized contact
support. Passing back to the conic cotangent chart adds precisely the
one nonzero scaling coordinate.

For explicit cohomological signs, put \(P^{-j}=L_j\) and use
\(d_P^{-j}=d_j\). The standard Hom differential is

\[
 d_{\mathrm{Hom}}^j(f)=(-1)^{j+1}f\,d_{j+1}.
 \tag{EC.C8}
\]

Multiplication in degree \(j\) by
\(\kappa_j=(-1)^{j(j+1)/2}\) is a chain isomorphism from (EC.C2)
to that standard Hom complex, since
\(\kappa_{j+1}=(-1)^{j+1}\kappa_j\). Therefore no Ext degree or
shift changes when using (EC.C2). Every degree uses the finite number
of terms of (EC.C1); there is no infinite totalization.

#### EC.D. Actual analytic microlocal Bernstein bound

We prove the minimum support dimension for every nonzero coherent
actual \(E\)-module, without importing the algebraic \(D\)-module
inequality and without assuming a smooth holonomic normal form.

Let \(V\ne0\) be such a left module. Section 5.32, J.1--J.2 constructs
a finite actual \(h\)-injective lattice \(V_0\subset V\), and proves

\[
 \operatorname{Supp}V=
 \operatorname{Supp}(V_0/hV_0)
 \tag{EC.D1}
\]

on the normalized chart. This part of J.1--J.2 applies to every coherent
module, independently of holonomicity or the finite-fibre argument J.3.

Form the actual finite first-order quotients

\[
\begin{gathered}
B=A/h^2A,
\\ W=V_0/h^2V_0,
\\ \epsilon=h.
\end{gathered}
\tag{EC.D2}
\]

Equation (EC.1) makes \(\epsilon\) central in \(B\), because
\([h,a]=(\sigma(a)-a)h\in h^2A\). Its square is zero. Cancellation
of the injective normal \(h\) action gives

\[
\begin{gathered}
\ker(\epsilon:B\to B)=\epsilon B,
\\ \ker(\epsilon:W\to W)=\epsilon W.
\end{gathered}
\tag{EC.D3}
\]

For example, \(hv\in h^2V_0\) means \(hv=h^2w\), hence \(v=hw\).
The same cancellation holds in \(A\). The quotient ring is \(R\), and
\(W/\epsilon W=\bar V_0\) is finite over \(R\). Also \(W\) is finite
over \(B\) by any finite generating set of \(V_0\). The two kernel/image
identities prove flatness over the dual numbers exactly as in the
earlier first-order trace proof: choose a rational vector-space basis
of the quotient, lift it, and use reduction and one multiplication by
\(\epsilon\) to prove generation and independence. No infinite-order
coefficient tensor product is present.

The exact earlier first-order trace lemma consequently proves that
every minimal support prime of \(\bar V_0\) is involutive for the
commutator biderivation of (EC.D2). We now compute that biderivation
and the precise resulting dimension bound.

For principal functions \(a(u,z),b(u,z)\), the first contraction in
the actual normal product gives

\[
\begin{gathered}
\{a,b\}_c=\sum_{i=1}^d\Bigl[
\\ (\partial_{z_i}a)(\partial_{x_i}-z_i\partial_t)b
\\ {}-(\partial_{z_i}b)(\partial_{x_i}-z_i\partial_t)a
\Bigr],
\\ \{z_i,x_j\}_c=\delta_{ij},
\\ \{z_i,t\}_c=-z_i.
\end{gathered}
\tag{EC.D4}
\]

Indeed \(\partial_{\xi_i}a=h\partial_{z_i}a\) and
\(\partial_\tau a=-h\sum_i z_i\partial_{z_i}a\); all second and
higher contractions, or contractions involving a positive \(h\)-order
coefficient, vanish after reduction of the commutator modulo \(h^2\).
This is the antisymmetric biderivation

\[
\begin{gathered}
\Pi=\sum_{i=1}^d
\\ \partial_{z_i}\wedge(\partial_{x_i}-z_i\partial_t).
\end{gathered}
\tag{EC.D5}
\]

Its rank is exactly \(2d\). Its kernel on covectors is the one-dimensional
span of

\[
 \alpha=dt+\sum_i z_i\,dx_i.
 \tag{EC.D6}
\]

The \(2d\) vectors in (EC.D5) are independent and span \(\ker\alpha\),
and their paired coefficients in \(\Pi\) are nondegenerate. These
observations prove both rank and kernel. No assertion of Jacobi for
(EC.D4) is made or required: this contact-chart biderivation is the
one supplied by the actual first-order normal quotient.

Here is the finite-witness passage from prime involutivity to conormal
isotropy. Fix one minimal analytic support prime \(\mathfrak p\), and
write \(s=\dim R/\mathfrak p\). If \(s=m\), its conormal is zero
and the required dimension bound is immediate. Otherwise the exact
earlier analytic-germs Lemma 3.4 and Theorem 4.1 produce adapted linear
coordinates \(w=(w',w'')\), a primitive coordinate \(a=w_{s+1}\),
a monic minimal polynomial \(W_a(w',T)\), a nonzero discriminant
\(\delta(w')\), and the explicit finite equation list

\[
\begin{gathered}
g_0=W_a(w',a),
\\ g_k=\delta(w')w_k
\\ {}-B_k(w',a)
\\ (s+2\le k\le m),
\\ (g_0,g_{s+2},\ldots,g_m)\subseteq\mathfrak p.
\end{gathered}
\tag{EC.D6a}
\]

Their zero set on \(\delta\ne0\) is the generic-sheet support of that
component. At each of its points their normal derivative determinant is
\(\partial_TW_a(w',a)\,\delta^{m-s-1}\ne0\), by the simple-root
part of Theorem 4.1. Thus their differentials span its entire
\((m-s)\)-dimensional conormal space. This uses actual finite equations,
rather than arbitrary newly chosen graph germs.

The trace lemma gives \(\{g_a,g_b\}_c\in\mathfrak p\) for every
pair from this finite list. Choose finite generators of
\(\mathfrak p\) and represent these finitely many ideal-membership
identities on one smaller common neighbourhood. Their brackets then
vanish everywhere on its component, by the analytic Nullstellensatz.
Choose a point of its dense generic-sheet locus away from the other
components. The exact earlier proper-subset dimension drop ensures
that such points exist. At this point the bracket identities and
(EC.D4)--(EC.D5) give \(\Pi|_{U\times U}=0\) for its full conormal
space \(U\). No new reduced-ideal sheaf coherence theorem is required
for this finite list and its finite identities.

On covectors modulo the one-dimensional kernel of \(\Pi\), this is
an isotropic subspace of a \(2d\)-dimensional symplectic vector space.
Its dimension is at most \(d\): for an isotropic subspace \(S\), the
isomorphism of a nondegenerate alternating form with the vector-space
dual gives \(\dim S^\perp=2d-\dim S\), and \(S\subseteq S^\perp\).
Thus

\[
\begin{gathered}
\dim U\le d+1,
\\ \dim\operatorname{Supp}V
\\ \ge(2d+1)
\\ {}-(d+1)=d.
\end{gathered}
\tag{EC.D7}
\]

The analytic component dimension equals the dimension of its dense
smooth pieces, by the exact earlier analytic dimension proof. Hence
every component has dimension at least \(d\); in particular a nonzero
coherent actual \(E\)-module has normalized support dimension at least
\(d=n-1\), or conic cotangent support dimension at least \(n\).

For a right module apply B.2 to make it a left module. Its principal
action is the identity on \(R\), so the normalized support is the same.
Alternatively the opposite first-order ring reverses the sign of
(EC.D4), which preserves isotropic subspaces and their dimension bound.
This proves (EC.D7) on both sides. The zero module is excluded only
from the minimum-dimension assertion. For \(d=0\), (EC.D7) is the
correct trivial bound zero.

#### EC.E. Full holonomic concentration

Let \(M\) be an arbitrary coherent holonomic actual \(E\)-module germ.
If \(M=0\), all its Ext groups are zero. Otherwise its conic support is
Lagrangian of pure dimension \(n\); its normalized support therefore has
pure dimension \(d=n-1\). Section 5.32, J.1--J.3 supplies the actual
coherent lattice, whole-germ contact reduction and bounded resolution
used in C, without removing any branch or nilpotent part. Its principal
module \(\bar M_0\) has exactly this reduced support by J.2. Thus

\[
\begin{gathered}
\operatorname{codim}_R\operatorname{Supp}\bar M_0
\\ =(2d+1)-d
\\ =d+1=n.
\end{gathered}
\tag{EC.E1}
\]

Apply (EC.A). For \(j<n\), \(F^j=0\); (EC.C4), actual finite right
Nakayama and (EC.C7) give

\[
\begin{gathered}
\operatorname{Ext}_E^j(M,E)=0
\\ (j<n).
\end{gathered}
\tag{EC.E2}
\]

For every \(j\), the actual support estimate is

\[
\begin{gathered}
\operatorname{Supp}\operatorname{Ext}_E^j(M,E)
\\ \subseteq\operatorname{Supp}F^j
\subseteq\operatorname{Supp}M,
\\ \dim\operatorname{Supp}\operatorname{Ext}_E^j(M,E)
\\ \le m-j=2n-1-j.
\end{gathered}
\tag{EC.E3}
\]

If \(j>n\), the last bound is strictly less than \(n-1=d\). A nonzero
right coherent actual \(E\)-module would violate the independently proved
actual bound (EC.D7). Hence

\[
\begin{gathered}
\operatorname{Ext}_E^j(M,E)=0
\\ (j>n).
\end{gathered}
\tag{EC.E4}
\]

For \(j=n\), (EC.E3) gives contained support of dimension at most \(d\).
Any nonzero component has dimension at least \(d\) by D. At its smooth
points away from the other components it lies in the original
Lagrangian component of the same dimension, so its tangent space agrees
there with that Lagrangian tangent space. The exact earlier
proper-analytic-subset dimension drop makes such a contained component
an original component. Thus the right Ext is holonomic, or is zero.

Consequently the exact result proved by this complete local argument is

\[
\begin{gathered}
\boxed{\begin{gathered}
\mathcal Ext_{\mathscr E_X}^j(M,\mathscr E_X)=0
\\ (j\ne n),
\\ n=\dim_{\mathbf C}X.
\end{gathered}}
\end{gathered}
\tag{EC.E5}
\]

All arguments are local on actual common domains and commute with
restriction. Under the earlier actual contact ring isomorphism, transport
the whole finite resolution and apply that isomorphism to every finite
matrix entry and every Hom value. This identifies both dual complexes,
their right action and their cohomology; the contact biholomorphism
preserves the support dimensions. Thus the theorem transports back to
the original germ after the generic-position construction. It applies
at every germ in every nonzero-covector cotangent chart of the actual
\(E\) sheaf defined by Sections 5.19--5.33. Right Ext and the shift \([n]\) in
(EC.2)--(EC.3) have precisely the convention required in the finite dual
support reduction.


![The actual principal-Hom injection and exact contact support degree bounds](assets/actual-microlocal-ext-mechanism.png)

*Figure EC.1. The left panels show the actual injection (EC.C4), its one
finite boundary lift (EC.C5), and the rank/conormal estimate
(EC.D5)--(EC.D7). The right panel uses the exact illustrative dimensions
\(d=2\), \(n=3\), \(m=q=5\): commutative codimension is three, and
degree \(j=4\) would have support dimension at most one, below the actual
nonzero minimum two. The general integer is \(r=n\). It illustrates
the full mechanism without imposing those illustrative dimensions on
the theorem. Free human reading: the first-order trace discussion in
Ginzburg I.2 and HolIII I.3.8, linked below. The complete reproducible source, SVG, full PNG and exact data accompany the figure. No infinite-coefficient depth
or I23 section-separation statement is inferred.*


#### EC.F. Actual domains and the remaining comparison scope

The explicit finite injection EC.C4 supplies the principal comparison. The proof needs neither negative-order completeness nor an interchange of analytic microlocalization and formal completion. Its actual bounded lattice resolution and exact principal reduction are L.4–L.6; the actual lattice for the arbitrary coherent Ext module is J.1–J.2. The first-order trace theorem has exactly the ring and module hypotheses checked in EC.D2–EC.D3.

Free human reading context is [Kashiwara–Kawai, Holonomic systems III, I.3.8, printed p.847](https://www.kurims.kyoto-u.ac.jp/~kenkyubu/kashiwara/HolIII.pdf), and [Ginzburg, Lectures on D-modules, I.2](https://math.berkeley.edu/~nadler/ginzburg.dmodules.pdf). These freely accessible texts provide additional reading. Every result used in the argument is proved above or at the precise earlier programme proofs, and a citation supplies no omitted proof step.

#### PG.1. Support purity for the full infinite-order extension

Let \(n=\dim_{\mathbf C}X\ge1\), let \(V\) be a normalized nonzero-covector chart of dimension \(2n-1\), and let \(M\) be any coherent holonomic left \(E\)-module on a sufficiently small common neighbourhood. Put \(B=E^\infty\) and \(M^\infty=B\otimes_E M\), with the ordinary algebraic tensor product. For every closed complex analytic subset \(Z\subset V\) whose local components have codimension at least \(c\),
\[
\mathcal H_Z^j(M^\infty)=0
\quad(j<c-n).
\tag{PG.1}
\]
In particular, if \(c>n\), restriction to the complement is injective:
\[
\begin{gathered}
M^\infty\longrightarrow j_*\bigl(M^\infty|_{V\setminus Z}\bigr),
\\ j:V\setminus Z\hookrightarrow V.
\end{gathered}
\tag{PG.2}
\]
The assertion includes singular and nonreduced holonomic germs and every section of the full coefficient extension. It assumes containment of the section's support in this analytic \(Z\); it makes no claim for arbitrary closed subsets of a smaller topological dimension.

**Proof.** The zero module is immediate. Otherwise EC.E proves that
\(N=\mathcal Ext_E^n(M,E)\) is a coherent right module and that all other dual cohomology vanishes. The actual bounded finite resolution is the one used in EC.C. Proposition FD.1 of [§5.50](#5-50-full-infinite-coefficient-support-depth-and-finite-duality) removes the exact positive dual tail by finite projective splittings. It supplies a finite locally projective right resolution \(R\) of \(N\), in degrees \(-n,\ldots,0\), and the left \(B\)-linear identification
\[
M^\infty\simeq
\mathcal Hom_{E^{\mathrm{op}}}(R,B)[n].
\tag{PG.3}
\]
Here the right flatness is precisely [§5.47, MF.1–MF.7](#5-47-arbitrary-finite-matrices-and-right-flatness-of-the-full-infinite-order-ring). The evaluation sign and the shift sign are FD.7–FD.9. This identifies the actual tensor module, with degree-zero cohomology, rather than a formal completion.

Let \(C_Z\) be the actual \((B,E)\)-bimodule representative of \(R\Gamma_ZB\) constructed in SD.E. SD.1 gives \(H^q(C_Z)=0\) for \(q<c\). The natural finite-Hom/support map SD.2 and its signed total differential give
\[
\begin{gathered}
R\Gamma_Z M^\infty
\\ \simeq\mathcal Hom_{E^{\mathrm{op}}}(R,C_Z)[n].
\end{gathered}
\tag{PG.4}
\]
Both maps are the actual sheaf maps of SD.F and FD.1–FD.3 and are left \(B\)-linear. To see the degree bound explicitly, replace \(C_Z\) by its good truncation at \(c\), with quotient by the degree-\(c\) boundaries and zero terms below \(c\). The two actions commute with the differential and preserve its images, so this truncation retains the bimodule structure. It is quasi-isomorphic to \(C_Z\). FD.2 proves that Hom from the bounded projective resolution preserves this quasi-isomorphism. A surviving Hom term then has source degree \(-p\), target degree \(q\), and shifted total degree
\[
\begin{gathered}
p+q-n\ge c-n,
\\ 0\le p\le n,\quad q\ge c.
\end{gathered}
\tag{PG.5}
\]
Only finitely many \(p\)'s occur in each total degree. Hence the right side of PG.4 has no cohomology below \(c-n\), proving PG.1. Splittings, lifts, idempotents and chain identities involve finitely many actual operator entries; SD.F represents all of them on one common neighbourhood before any infinite coefficient sum. All positive every-epsilon bounds, negative factorial bounds, spatial indices and fixed derivatives therefore remain those of the full sheaf \(B\).

For a sheaf in degree zero, \(\mathcal H_Z^0\) is the kernel of restriction off \(Z\): directly, its sections are the local sections zero on that complement. If \(c>n\), PG.1 kills this kernel and proves PG.2. Thus a full \(M^\infty\)-section vanishing off \(Z\) is zero. No coherence of that section's own support is needed. \(\square\)

#### PG.2. Extension across the singular exceptional set

The reduced normalized support \(Y\) of a nonzero holonomic \(M\) is pure of dimension \(n-1\). Choose the closed analytic exceptional set \(W\subset Y\) by the finite discriminant/component construction of SD.D: include the exceptional loci of its irreducible components and their pairwise intersections. Outside \(W\) the components are disjoint smooth submanifolds. The earlier analytic dimension-drop proof gives
\[
\begin{gathered}
\dim W\le n-2,
\\ \operatorname{codim}_V W\ge n+1.
\end{gathered}
\tag{PG.6}
\]
For \(n=1\), the support consists locally of isolated smooth points and \(W\) can be empty. Applying PG.2 in all other cases proves that restriction off \(W\) detects every section of \(M^\infty\). In particular a section zero on the complement of this exceptional set cannot survive on its singular points or component intersections. The proof uses supported complexes at those points, not continuity from regular points.

![Contact support, its exceptional point and the finite-dual degree bound](assets/holonomic-support-purity.png)

*Figure PG.1. The geometric panel is the real slice \(t=0\), \(x,z\in\mathbf R\), of the exact complex analytic set \(Y=\{t=0,\ xz=0\}\subset\mathbf C^3_{t,x,z}\), with exceptional set \(W=\{0\}\). The two complex branches are Legendrian for \(\alpha=dt+z\,dx\): on the \(x\)-branch \(z=dt=0\), and on the \(z\)-branch \(dx=dt=0\). On either nonzero radial conification the tautological form \(\theta=\tau\alpha\) consequently pulls back to zero. Its differential pulls back to zero as well, and the smooth cone has dimension two, half the ambient symplectic dimension four; it is therefore Lagrangian. This illustrates the support geometry for base dimension \(n=2\); it assumes no classification or selected presentation of \(M\). Here \(\operatorname{codim}W=3\), so \(c-n=1\). The other panels give the actual general maps PG.3–PG.4 and inequality PG.5, which leave no degree-zero section supported on \(W\). Hollow branch points depict the complement of \(W\); the node itself remains part of \(Y\). Reproducible source, exact data, full PNG and editable SVG. Free human context: [Holonomic systems III, I.3.8 and IV.6.2](https://www.kurims.kyoto-u.ac.jp/~kenkyubu/kashiwara/HolIII.pdf); the complete argument is PG.1–PG.2 above.*

This support-purity theorem supplies the extension step once vanishing off the analytic exceptional set is known. It does not establish that vanishing from solution pairings, construct a common solution family, or solve the finite-matrix relation lifting from \(E^\infty\) to \(E^{\mathrm R}\). The real-module injection, moving-sector comparison and full I23/I24 separation assertions retain those distinct requirements.



### 5.52. Full Euler-matrix division and comparison reductions

Every constant complex Euler matrix admits a unique full infinite-order coefficient remainder. The division below retains arbitrary matrix size, every nilpotent entry, all spatial covariables, every positive-epsilon estimate and the negative factorial regime on one coefficient domain. Two general comparison reductions follow: injectivity can be detected through an actual embedding into another coefficient module, and generic injectivity extends across the singular analytic exceptional set by §5.51.

#### AM.0. The actual symbols and the full matrix statement

Use coordinates \((t,x_1,\ldots,x_d)\) near \(t=0\), with nonzero normalized covector \(\tau=1\), \(\xi=0\). Put \(E=\mathscr E_p\), \(B=\mathscr E^\infty_p\), \(h=\tau^{-1}\), and \(z_i=\xi_i/\tau\). The actual normal composition, its normal element and all finite shifts are the complete proofs in [§§5.19–5.22](#5-19-actual-factorial-growth-symbols-and-ordered-jets). The full geometric infinite-order coefficient class and its product are [§5.45, IG.1–IG.5 and IM.1–IM.6](#5-45-the-full-geometric-infinite-order-kernel-ring-and-its-canonical-product). The all-index convergence proof [§5.49, ISB.1–ISB.5](#5-49-full-infinite-spath-division-and-balanced-reconstruction) gives a general lowering argument; the Euler case has the explicit quotient series proved here.

On one fixed smaller holomorphic coefficient product, the full class is
\[
\begin{gathered}
P=\sum_{j\in\mathbb Z}p_j(t,x,z)\tau^j,
\\ |p_j|\le M_\epsilon\epsilon^j/j!\quad(j\ge0),
\\ |p_{-n}|\le M C_0^n n!\quad(n\ge1).
\end{gathered}
\tag{AM.2}
\]
The positive estimate holds for every \(\epsilon>0\). Constants depend on a chosen compact smaller product, while that product is fixed before choosing epsilon. Every fixed derivative has the corresponding estimate on a fixed further smaller product by the coefficient Cauchy formula.

For any constant \(q\times q\) complex matrix \(A\), let \(N_A\) have its finite left \(E\)-presentation
\[
\begin{gathered}
(tD_tI-A)e=0,
\\ D_{x_i}e=0\quad(1\le i\le d),
\\ e=(e_1,\ldots,e_q)^{\mathsf T}.
\end{gathered}
\tag{AM.3}
\]
Coefficient rows multiply the displayed relation matrices on the right, with actual operator composition in the written order. There is no transposed action. Tensoring this presentation with \(B\) means the ordinary algebraic tensor product. Its quotient is \(B^q\) modulo the rows generated by the displayed relations: the balancing relations identify \(b\otimes(Pe_i)\) with \((bP)\otimes e_i\), so the tensor of a quotient is exactly that cokernel. No completion or tensor-flatness assertion is needed for this quotient description. For \(q=0\) all rows are empty and the statement is immediate; assume \(q>0\) below.

#### AM.A. A complete coefficient normal form

**Proposition AM.A.** Every row of \(B^q\), modulo the left row span of AM.3, has a unique representative
\[
 R=\sum_{j\in\mathbb Z}r_j(x)\tau^j
 \tag{AM.4}
\]
with the complete bounds AM.2 on one common domain. The division coefficients also belong to the full class \(B\). Thus this is a normal form in the actual module \(B\otimes_E N_A\), not a coefficientwise formal quotient.

**Proof: spatial division.** For a holomorphic coefficient on a polydisc,
\[
\begin{gathered}
f(t,x,z)-f(t,x,0)
\\ =\sum_{i=1}^d z_i g_i(t,x,z).
\end{gathered}
\tag{AM.5}
\]
One explicit choice successively sets the first \(i-1\) coordinates to zero and divides the difference in the \(i\)-th coordinate by \(z_i\). The quotient extends holomorphically at zero by its Taylor series. On a fixed smaller polydisc, the one-variable Cauchy formula bounds all these finitely many maps by a constant independent of the homogeneous index. Since right composition with \(D_{x_i}\) is exactly multiplication of the normal symbol by \(\xi_i=\tau z_i\), AM.5 writes the original row as its \(z=0\) row plus \(\sum_i H_iD_{x_i}\). Each \(H_i\) is the corresponding coefficient sequence shifted by one order. Fixed shifts preserve AM.2: in the positive range choose the input epsilon smaller than the requested output epsilon; fixed degree polynomials are absorbed by that choice. In the negative range \((n+a)!\le C_a^n n!\) for fixed \(a\), after increasing a prefactor for finitely many small indices. This proves actual spatial division on one common domain.

The spatial ideal is stable under right composition with the normal Euler relation, since \(D_{x_i}\) commutes with it. The principal evaluation at \(z=0\) also commutes with that composition: the normal derivative \(\partial_\tau\) is \(\tau^{-1}(j-\sum z_i\partial_{z_i})\) on homogeneous degree \(j\), whose second term vanishes at \(z=0\). Thus it remains to do division for \(f_j(t,x)\).

**Proof: normal division.** Put \(a=A+I\). Multiplication of AM.3 on the left by the invertible \(hI\) gives the equivalent relation
\[
\begin{gathered}
L=tI-ah,
\\ h(tD_tI-A)=L.
\end{gathered}
\tag{AM.6}
\]
The sign and the additional identity matrix follow from \(ht=t h-h^2\). If \(Q=\sum q_j(t,x)\tau^j\), the exact normal product is
\[
\begin{gathered}
(QL)_j=tq_j
\\ {}+q_{j+1}((j+1)I-a).
\end{gathered}
\tag{AM.7}
\]
There are no omitted higher contractions: \(t\) has only its first nonzero derivative, and \(a\) is constant.

Write \(Tg=g(0,x)\) and \(Dg=(g-Tg)/t\), so \(g=tDg+Tg\). On a coefficient compact with normal radius \(r>0\), the maximum principle applied to the holomorphic quotient gives
\[
\begin{gathered}
\|Tg\|\le\|g\|,
\\ \|Dg\|\le(2/r)\|g\|.
\end{gathered}
\tag{AM.8}
\]
Indeed the second bound holds on the radius-\(r\) circle, hence inside it; the other variables are parameters. Successive quotients use the same radius, not a sequence of shrinking radii. Put \(b=2/r\), and let \(Kq\) denote the second term in AM.7. Then
\[
\begin{gathered}
q=(1+DK)^{-1}Df,
\\ R=T(f-Kq).
\end{gathered}
\tag{AM.9}
\]
The inverse in this displayed expression is now constructed, rather than assumed. Its degree-\(j\) entry is
\[
\begin{gathered}
q_j=\sum_{k\ge0}(-1)^k D^{k+1}f_{j+k}
\\ {}\prod_{s=1}^k((j+s)I-a).
\end{gathered}
\tag{AM.10}
\]
The order of the factors is the written order on the right of the coefficient row. They commute with one another because all are polynomials in \(a\); no commutation of \(a\) with a coefficient row is used. The empty product is the identity.

Choose an integer \(S\ge1\) with \(S\ge\|a\|\), using a submultiplicative matrix norm. For \(j\ge0\), the norm of the product in AM.10 is at most
\[
 \frac{(j+k+S)!}{(j+S)!}.
 \tag{AM.11}
\]
Consequently the summand is bounded by
\[
\begin{gathered}
bM_\epsilon\epsilon^j(b\epsilon)^k
\\ {}\frac{(j+k+S)!}{(j+S)!(j+k)!}
\\ \le \frac{C_S bM_\epsilon}{j!}
\\ {}\epsilon^j(j+1)^S(b\epsilon)^k(k+1)^S.
\end{gathered}
\tag{AM.12}
\]
The elementary factorial inequality follows by bounding the \(S\) factors in \((j+k+S)!/(j+k)!\) by \(C_S(j+1)^S(k+1)^S\), and using \((j+S)!\ge j!\). Given an output \(\eta>0\), take \(\epsilon\le\eta/2\) and \(b\epsilon<1/2\). The sum in \(k\) converges; \((j+1)^S(\epsilon/\eta)^j\) is bounded, since the ratio of successive terms tends to \(\epsilon/\eta<1\). This gives \(\|q_j\|\le M'_\eta\eta^j/j!\) for every \(\eta\), on the same domain.

For \(j=-n<0\), split AM.10 into \(k<n\) and \(k\ge n\). In the first part,
\[
\begin{gathered}
\prod_{s=1}^k(|-n+s|+S)
\\ =\frac{(n-1+S)!}{(n-k-1+S)!}.
\end{gathered}
\tag{AM.13}
\]
After multiplication by the bound \(M C_0^{n-k}(n-k)!\) for the input coefficient, this is at most
\(M C_0^{n-k}(n-1+S)!\), since \(S\ge1\). The sum of the first \(n\) terms is therefore bounded by \(M' C_1^n n!\): the remaining factors are \(b^{k+1}\), at most \(n\) summands, and a fixed polynomial in \(n\). Each is absorbed in one exponential constant.

In the second part put \(k=n+m\), \(m\ge0\). The product bound is exactly
\[
\begin{gathered}
\prod_{s=1}^{n+m}(|-n+s|+S)
\\ =\frac{S(n-1+S)!}{(S!)^2}
\\ {}\cdot(m+S)!.
\end{gathered}
\tag{AM.14}
\]
Using the positive input bound with one fixed \(\epsilon\) satisfying \(b\epsilon<1/2\), its sum is at most
\[
\begin{gathered}
C M_\epsilon b^{n+1}(n-1+S)!
\\ {}\sum_{m\ge0}(b\epsilon)^m(m+1)^S
\\ \le M'' C_2^n n!.
\end{gathered}
\tag{AM.15}
\]
This supplies one negative factorial constant for the entire sequence. For each fixed \(j\), the tail of AM.10 is bounded by a geometric tail times a fixed polynomial. Hence it converges normally in the coefficient variables. Reserving one additional fixed coefficient margin and applying the Cauchy formula gives the same statement for every fixed derivative.

The finite geometric identity for AM.9 passes to this normal limit. Both \(D\) and the fixed-index map \(K\) are bounded on the coefficient compacts used above; its remainder is a tail of AM.10. Thus \(q=Df-DKq\) and \(f=tq+Kq+R=QL+R\). Equation AM.7 and the just-proved estimates show that \(R=T(f-Kq)\) again satisfies both parts of AM.2.

For uniqueness, if \(QL+R=0\) with \(R\) independent of \(t\), applying \(D\) gives \(q=-DKq\). Iterating this finite identity \(k\) times gives the degree-\(j\) term \((-1)^kD^kq_{j+k}\prod_{s=1}^k((j+s)I-a)\). For fixed \(j\) and large \(k\), the positive input estimate for \(q_{j+k}\), with \(b\epsilon<1/2\), bounds it by a geometric sequence times a fixed polynomial, exactly as in AM.12 or AM.14. It tends to zero. Therefore every \(q_j\) is zero on the same coefficient domain, and then \(R=0\). Together with the spatial quotient compatibility, this proves uniqueness modulo all the relations AM.3. \(\square\)

#### AM.R. Injectivity can be detected through an embedding

Let \(R\to B\to C\) be unital ring maps, and let \(M,N\) be left \(R\)-modules. Write \(\iota_M:B\otimes_RM\to C\otimes_RM\) for the natural map, and similarly for \(N\). If there is an injective left \(B\)-linear map \(T:B\otimes_RM\to B\otimes_RN\) and \(\iota_N\) is injective, then \(\iota_M\) is injective.

**Proof.** The explicit map
\[
\begin{gathered}
C\otimes_B(B\otimes_R M)
\\ \longrightarrow C\otimes_R M,
\\ c\otimes(b\otimes m)\longmapsto cb\otimes m
\end{gathered}
\tag{AM.R1}
\]
has inverse \(c\otimes m\mapsto c\otimes(1\otimes m)\). Each map respects additivity and the two tensor balancing relations. For instance \(cb'\otimes(b\otimes m)=c\otimes(b'b\otimes m)\) has the same image, and \(br\otimes m=b\otimes rm\) has the same image after the remaining \(R\)-balancing. In the reverse composite, \(c\otimes(b\otimes m)=cb\otimes(1\otimes m)\). These identities prove both inverse identities. The same construction holds for \(N\).

They identify \(C\otimes_B T\) with a map \(\widetilde T:C\otimes_RM\to C\otimes_RN\), and on a generator \(b\otimes m\) give the commuting relation
\[
 \widetilde T\,\iota_M=\iota_N T.
 \tag{AM.R2}
\]
If \(\iota_M(s)=0\), this makes \(\iota_N(Ts)=0\). Injectivity of \(\iota_N\) gives \(Ts=0\), and injectivity of \(T\) gives \(s=0\). No injectivity or flatness of \(C\otimes_B T\) was assumed. In particular a finite family of maps into modules with injective comparison suffices when their combined map is injective: tensor products commute with a finite direct sum by distributing each generator, and the comparison is then the direct sum of its injective component maps. \(\square\)

This lemma requires an actual \(B\)-linear embedding and the proved target comparisons. It does not supply those embeddings or infer them from a finite-order contact transformation. A ring inclusion by itself remains insufficient after tensoring with an arbitrary module.

#### AM.S. Extending generic injectivity across the analytic exceptional set

Let \(M\) be any coherent holonomic actual \(E\)-module on a normalized nonzero-covector chart \(V\), with base dimension \(n\ge1\), and put \(M^\infty=E^\infty\otimes_E M\). Let \(\phi:M^\infty\to\mathcal F\) be a morphism to any sheaf of abelian groups. Construct the closed complex analytic exceptional set \(W\subset\operatorname{Supp}M\) by [§5.51, PG.2](#pg-2-extension-across-the-singular-exceptional-set): it includes the finite discriminant exceptional loci and all component intersections. The components outside it are disjoint and smooth, and
\[
 \operatorname{codim}_V W\ge n+1.
 \tag{AM.S1}
\]
For \(n=1\), take the empty exceptional set. If every stalk map of \(\phi\) off \(W\) is injective, then \(\phi\) is injective everywhere.

**Proof.** If \(M=0\), there is nothing to prove. For a local section \(s\) in the kernel, every germ off \(W\) is zero by the assumed injectivity. The sheaf axiom therefore gives \(s|_{V\setminus W}=0\) on the local open in question. The complete actual support-purity theorem [§5.51, PG.1–PG.2](#pg-1-support-purity-for-the-full-infinite-order-extension) applies with \(c=n+1>n\), including after restriction to that open. It makes \(s=0\). The kernel's own support need not be analytic or coherent; containment in the fixed analytic set \(W\) is enough. The empty exceptional set case follows directly from the stalkwise assertion. This proves the result for every section, including singular and nonreduced holonomic germs. \(\square\)

The coefficient normal form AM.A and reductions AM.R–AM.S do not prove an arbitrary holonomic real-module comparison. Such an application still requires actual evaluation or embeddings on each smooth characteristic branch, the appropriate compatible contact transformations, and the moving-sector/common-solution comparison for section separation. These are distinct proof requirements.

![Full coefficient division and the retained nilpotent remainder](assets/euler-model-full-coefficient-division.png)

*Figure AM.1. The quotient formula and both complete growth regimes are AM.8–AM.15 on the same coefficient domain. The matrix sample is exactly \(A=J=\left(\begin{smallmatrix}0&1\\0&0\end{smallmatrix}\right)\), \(L=tI-(J+I)h\). Normal composition gives \(\tau L=t\tau I-J\); its first coefficient row proves \((t\tau,0)=(\tau,0)L+(0,1)\). The second component retains the nilpotent relation \(tD_te_1=e_2\). This is an equation/estimate diagram, with no sampled geometry. The complete mathematical argument is AM.A. Editable plotting source, exact data, full PNG and SVG. Free human context: [Kashiwara–Kawai, Holonomic systems III](https://www.kurims.kyoto-u.ac.jp/~kenkyubu/kashiwara/HolIII.pdf), I.3 and IV.6. Every result used above is proved here or in the precise earlier programme proofs; the free paper supplies additional reading.*


### 5.53. Real-kernel comparison for all constant Euler matrices

For every finite constant complex Euler matrix, the full infinite-order
coefficient module embeds into its actual real-kernel extension. This
includes every integer resonance, arbitrary nonsemisimple and nilpotent
blocks, both infinite coefficient tails and every finite free presentation
of the same model. The proof constructs a module for the whole real
kernel stalk, identifies the positive diagonal and negative Volterra
actions, and detects the complete normal remainder by power and logarithm
functions.

#### AM.T. The model, its sides and exact earlier proofs

Use the same nonzero covector and coordinates as
[§5.52, AM.0–AM.A](#5-52-full-euler-matrix-division-and-comparison-reductions):
\(t=0,\) \(\tau=1,\) \(\xi=0,\)
\(E=\mathscr E_p,\) \(B=\mathscr E^\infty_p,\)
\(C=\mathscr E^{\mathbb R}_p,\) \(h=D_t^{-1}.\)
Let \(A\) be any constant \(q\times q\) complex matrix and let \(N_A\)
be the entire left module with the finite relations AM.3,
\((tD_tI-A)e=0\) and \(D_{x_i}e=0\).
The ordinary algebraic tensor module \(B\otimes_E N_A\) has the
unique normal row \(R=\sum_jr_j(x)\tau^j\) of AM.4. Its coefficients
satisfy the full bounds AM.2, on one fixed coefficient domain.
The quotient is the explicit all-index division AM.5–AM.15.
The result proved below is the injectivity in AM.29 and the actual
finite relation row AM.31; no completed tensor product is used.

The actual coefficient rings and finite shifts are
[§§5.19–5.22](#5-19-actual-factorial-growth-symbols-and-ordered-jets).
The smooth/current/injective supported resolutions, ordinary sections
and controlled proper excision are
[§5.34, R.5–R.7](#5-34-full-current-resolutions-and-controlled-proper-image-comparison).
The ordered support cube, input top-form calibration and all-degree
trace are
[§5.35, V.1–V.6](#5-35-relative-cones-unshifted-cup-and-the-normalized-trace).
The complete finite-order action, with the full ambient primitive, is
[§5.42, CA.1–CA.7](#5-42-the-canonical-degree-one-directional-action).
The full-relative-cochain action and its finite naturality are
[§5.43, DLG.1–DLG.7](#5-43-the-canonical-all-degree-action-and-its-derived-module-structure);
its DLH.1–DLH.4 proves the analytic existence input.
Actual ordinary diagonal Čech cohomology and the canonical residue
comparison are
[§5.44, OC.1–OC.4, OR.2 and TC.1–TC.3](#5-44-full-homogeneous-coefficients-and-ordinary-infinite-order-kernels).
The source-defined geometric coefficient sheaves, their maps and
products are
[§5.45, IG.1–IG.5 and IM.1–IM.6](#5-45-the-full-geometric-infinite-order-kernel-ring-and-its-canonical-product).
The additional real-stalk/domain calculation is proved in AM.H below.

#### AM.B. Actual real kernels on a halfspace and the two coefficient tails

##### AM.B.1. One target for the whole real stalk

Put \(N=d+1\), \(H=\{\Re t\ge0\}\times\mathbf C^d\), and
\[
\begin{gathered}
\mathcal S=\underset{D\ni0}{\mathop{\rm colim}}\,
H_H^1(D;\mathcal O),
\\ U_D=D\cap\{\Re t<0\}.
\end{gathered}
\tag{AM.16}
\]
Here \(D\) ranges over product polydiscs about zero; support is always
intersected with its actual domain. The input difference is \(w=s-t\).
This choice agrees with the negative displacement in IG.1, rather than
silently reversing the real phase.

Both \(D\) and \(U_D\) are acyclic for \(\mathcal O\). To verify this with
the proved analytic existence theorem, take OC.2's sum of the coordinate
disc barriers, and on \(U_D\) add \(-\log(-\Re t)\). If \(t=u+iv\), the
last term has \(t,\bar t\) Levi coefficient \(1/(4u^2)>0\). It is
bounded below on the bounded domain and diverges at \(u=0\). The other
barriers diverge at every outer boundary. Their sum is therefore a
smooth strictly plurisubharmonic exhaustion. DLH.1–DLH.4 and the maximal
graph resolution/augmentation argument OC.2 prove all the claimed
positive-degree vanishing, for unrestricted holomorphic data. Hence the
full derived supported object is computed by the actual two-term
holomorphic complex
\[
\begin{gathered}
\left[\mathcal O(D)\longrightarrow\mathcal O(U_D)\right],
\\ \text{in degrees }0,1.
\end{gathered}
\tag{AM.16a}
\]
The arrow is restriction. Its kernel is zero by holomorphic uniqueness.
Thus \(\mathcal S=\operatorname{colim}_D\mathcal O(U_D)/\mathcal O(D)\).
We identify a function class \([f]\) with the full relative class
\((0,-f)\), consistently for input and output; this specifies the
connecting sign in that quotient identification.
A class is zero exactly when its representative agrees, on a later
left half-polydisc, with a holomorphic function on the whole polydisc.
No topology on this quotient is being used.

We give the whole-ring geometry. AM.H below reproduces the actual real
stalk calculation from IG.1, so that a finite set of arbitrary real
classes and equality witnesses has one actual kernel cone
\[
\begin{gathered}
G=
\\ \left\{\begin{gathered}
l=-\Re(s-t)\ge0,\\
|\Im(s-t)|\le\epsilon l,\\
|y_i-x_i|\le A_i l
\end{gathered}\right\},
\\ \epsilon,A_i>0.
\end{gathered}
\tag{AM.17}
\]
There is no upper bound imposed on these finitely many cone constants.
If the input belongs to \(H\) and the difference belongs to \(G\), then
\[
\begin{gathered}
0\le\Re s\le\Re t,\quad l\le\Re t,
\\ |\Im s-\Im t|\le\epsilon\Re t,
\\ |y_i-x_i|\le A_i\Re t.
\end{gathered}
\tag{AM.18}
\]
In particular the output belongs to \(H\). Choose an outer source box
inside both the chosen input domain and the actual kernel domain. By first making the output box small,
AM.18 puts the entire closed intersection above a compact output set
in a smaller compact source box, strictly inside the outer one. Its
projection to the output is proper. This also works for a finite
product: its nonnegative displacement lengths add to
\(\Re t-\Re s\le\Re t\), so every intermediate point satisfies the same
bounds. The complement is \(G\)-open in the exact required direction:
if \(\Re t<0\) and \(s-t\in G\), then \(\Re s=\Re t-l<0\).

Apply DLG.1–DLG.5 on these actual domains. Use the full kernel pair
\((a,b)\), its four-entry cup with an input pair, the finite-cube
augmentation, proper collar and normalized trace. The support-to-image
condition is AM.18; the degree is \(N+1-N=1\). DLG.4 sends boundaries
to boundaries with their full primitives retained. DLG.5 gives
associativity and the positive diagonal unit. DLG.3 and DLG.7 compare
restrictions, support enlargements and collars. A new, larger finite
cone list is accommodated by a further output shrink, while \(H\)
itself does not change. AM.H makes representatives and finite equality
witnesses simultaneous. Consequently this construction gives a
unital left action of the whole source-defined stalk \(C\) on
\(\mathcal S\). It does not assert that every real kernel is a symbol
or is represented by a supported current.

The unit and its derivatives act as the usual multiplication and
derivatives, by U.5–U.8 and E.13. On \(\mathcal S\), \(D_t\) is an
isomorphism: primitives exist on the convex left half-disc, with one
fixed negative anchor, and depend holomorphically on \(x\). If \(D_tf\)
extends across zero, subtract a full-disc primitive; the remaining
function is constant in \(t\) and holomorphic in \(x\), so extends too.
This proves both surjectivity and zero kernel. Since \(hD_t=D_th=1\)
in \(C\), its actual action is this inverse.

##### AM.B.2. Positive infinite order by a complex-diagonal calculation

Let \(R=\sum_{j\in\mathbf Z}r_j(x)D_t^j\) be a normal row from AM.A.
The argument applies componentwise, so first take one scalar entry.
Split \(R=R_++R_-\), with \(j\ge0\) in \(R_+\) and \(j<0\) in \(R_-\).
The first is an ordinary infinite differential operator: its
multiindices are \((j,0,\ldots,0)\), and AM.2 gives exactly OR.1's
every-epsilon ordinary coefficient bound. Its physical holomorphic
diagonal kernel, with the input form \(ds\wedge dy_1\wedge\cdots\wedge dy_d\),
is
\[
\begin{gathered}
K_+(t,x;s,y)=
\\ \frac1{(2\pi i)^N}
\\ {}\left(\sum_{j\ge0}\frac{j!r_j(x)}{(s-t)^{j+1}}\right)
\\ {}\prod_{i=1}^d\frac1{y_i-x_i}.
\end{gathered}
\tag{AM.19a}
\]
All products with an empty spatial list are one. On a compact where
\(|s-t|\ge\rho>0\), choose \(\varepsilon<\rho\). The time sum is bounded by
\(M_\varepsilon\rho^{-1}\sum_{j\ge0}(\varepsilon/\rho)^j\).
On any actual coordinate-punctured compact the spatial factors are
bounded too. Compact enlargement and the holomorphic Cauchy estimates
prove the same convergence for each fixed derivative. Thus AM.19a is
an actual holomorphic cochain, not a distributional sum at the origin.
OC.1–OC.4 identifies its augmented finite normal Čech cube with the
actual \(\Delta\)-supported object; IG.1 and IG.5 identify support
forgetting \(\Delta\subset\{s-t\in G\}\) with the actual map to \(C\).

We record the supported-action comparison, including the input
complement. On an ordinary output open \(V\) take a small normal
polytube inside \(p_2^{-1}V\), locally about each point of the diagonal.
For a holomorphic input \(f\), the physical top normal coefficient is
\(K_+f(s,y)\); its literal ordinary cube entry is
\(r_N(-1)^N K_+f(s,y)\). Define the top trace to be
\([r_N(-1)^N]^{-1}\) times the integral of that literal entry.
Thus the resulting formula integrates the physical coefficient over the positively ordered circles
\(|s-t|=\rho_0,\ |y_i-x_i|=\rho_i\), inside that polytube. The scalar
\((2\pi i)^{-N}\) is already present in \(K_+\), so it is not inserted
a second time. Integration annihilates every normal face boundary:
the face omitting coordinate \(i\) is holomorphic across that
coordinate, and its circle integral is zero by Cauchy's theorem.
Set the trace to zero in smaller normal degrees. This is a chain map
from the augmented normal cube to \(\mathcal O_V[-N]\). Radii can be
shrunk on overlaps; their integrals agree by Cauchy's theorem on the
annuli. The maps commute with every base restriction and every base
Čech face.

The shifts also specify the base-complement sign. With normal cube
degree \(p\), the total differential on normal and base cubes is
\(d_{\rm normal}+(-1)^p d_{\rm base}\).
The residue map takes values in \(\mathcal O[-N]\).
Fixing a closed kernel of normal degree \(N\) maps the base complex
to the total complex shifted by \([N]\). At that degree the base
differential has combined factor
\((-1)^N(-1)^N=1\).
The shifted residue therefore gives the same positive ordinary map
on base degrees zero and one, including the restriction arrow.
These are the tensor/shift conventions of V.14–V.17; no second
normalizing scalar or additional suspension is inserted.

This is the normalized canonical trace on that cube. In one coordinate
the connecting class for \(1/(2\pi i(s-t))\) has trace \(f(t)\), by U.5.
In the ordered \(N\)-coordinate frame, V.15–V.17 gives physical top
cube coordinate \(r_N(-1)^N K_+\), where
\(r_N=(-1)^{N(N-1)/2}\). The ordered connecting maps and trace supply
the inverse factor, exactly as the all-coordinate comparison
TC.1–TC.3. Equivalently the spatial reduction has
\(r_N(-1)^d r_d=1\), as proved in CA.5. The finite maps in V.6,
R.6–R.7 and TC.1–TC.3 commute with the base restrictions just used.
Their cochain homotopies retain every remaining complement entry.
They therefore identify this holomorphic normal-cube trace with the
canonical cup/excision/trace before, as well as after, passing to
cohomology. This comparison concerns the complete normally convergent
coefficient on separated circles; it does not take a limit of
finite-order currents.

Here is why the same identification computes the action with support
\(H\). It is a support calculation, not merely equality of two maps on
ordinary holomorphic functions. For a sheaf injective \(I\) on the
pair space the actual support subfunctors satisfy
\[
\begin{gathered}
\Gamma_\Delta\Gamma_{p_2^{-1}H}I
\\ =\Gamma_{\Delta\cap p_2^{-1}H}I
\\ =\Gamma_{\Delta\cap p_1^{-1}H}I
\\ =\Gamma_{p_1^{-1}H}\Gamma_\Delta I.
\end{gathered}
\tag{AM.19b}
\]
These are literal equalities of subsheaves: sections have support in
both closed sets, and \(p_1=p_2\) on \(\Delta\). For a closed inclusion
\(i\), \(\Gamma_Z=i_*i^!\) preserves injectives: \(i_*\) is exact, so
its right adjoint \(i^!\) preserves injectives; \(i_*\) also preserves
injectives since its left adjoint \(i^{-1}\) is exact. Thus applying
AM.19b to an injective resolution proves the corresponding derived
support identities. Similarly
\(\Gamma_Hp_{1*}I=p_{1*}\Gamma_{p_1^{-1}H}I\), and \(p_{1*}\) preserves
injectives because \(p_1^{-1}\) is exact. This proves the derived
support/direct-image comparison. On diagonal-supported objects the
projection is proper; R.7's controlled zero-extension roof implements
the same comparison with the actual proper trace and its collars.

We prove the needed comparison by a support adjunction and an explicit
cube homotopy. All derived categories in this paragraph are categories
of complex-vector sheaves on the actual base or pair domain. They are
not categories of the vector spaces obtained after global sections.

First let \(F\) be a bounded-below complex, let \(Z\) be any closed
subset of a base domain, and let
\(j_Z:R\Gamma_Z F\to F\) be support forgetting. If a bounded-below
complex \(K\) has all its cohomology sheaves supported in \(Z\), then
\[
\begin{gathered}
j_Z\circ-:
\\ \operatorname{Hom}(K,R\Gamma_Z F)
\\ \xrightarrow{\ \sim\ }\operatorname{Hom}(K,F).
\end{gathered}
\tag{AM.19d}
\]
Here is a proof, including the model used for this assertion.
For the closed inclusion \(i:Z\hookrightarrow D\), both \(i^{-1}\)
and \(i_*\) are exact, as checked on stalks. The unit
\(K\to i_*i^{-1}K\) is a quasi-isomorphism: it is the identity on the
cohomology stalks in \(Z\), and both sides have zero cohomology outside
\(Z\). Thus \(K\) may be represented by a complex whose every term is
supported in \(Z\). Take a bounded-below injective target resolution
\(I\) of \(F\). As proved after AM.19b, \(\Gamma_Z I\) is a complex
of injectives and represents \(R\Gamma_Z F\). Every sheaf map from a
term supported in \(Z\) to \(I^r\) has image in \(\Gamma_Z I^r\):
it vanishes off \(Z\). This statement holds term by term for chain
maps and for chain homotopies. Consequently the two Hom complexes
are literally equal. The injective comparison of R.6 identifies their
homotopy classes with the derived morphisms, proving AM.19d.
The bounded-below injective Hom assertion used here has the following
direct lifting proof. For an acyclic complex \(L\) and a chain map
\(u:L\to I\), start in the bottom nonzero target degree \(r\).
The map \(u^r\) kills \(\ker d_L^r=\operatorname{im}d_L^{r-1}\);
factor it through \(\operatorname{im}d_L^r\) and extend to
\(L^{r+1}\to I^r\) by injectivity. This is the first homotopy term.
Subtract its homotopy boundary. The next residual component again
kills the kernel of the source differential, by the chain-map
identity. Repeating upward makes \(u\) null-homotopic.
The same construction in each shifted degree says that the Hom
complex from an acyclic complex to \(I\) is acyclic. It follows,
by applying it to the cone of a quasi-isomorphism, that Hom into
\(I\), and likewise into \(\Gamma_ZI\), computes derived Hom.
In particular, two maps \(K\to R\Gamma_Z F\) are equal if their
composites with \(j_Z\) agree. This statement is used on sheaf
complexes before taking ordinary sections; no injectivity claim for
maps out of the half-disc quotient is being made.

Now retain the complex-diagonal support \(A=\Delta\) and put
\(B=p_2^{-1}H\) in the pair space. The intersection \(A\cap B\)
projects into \(H\). Both \(A\) itself and \(A\cap B\) are proper
over a smaller output domain: \(p_1|_\Delta\) is the identity.
The R.7/DLG.4 construction therefore defines the ordinary diagonal
operation \(\mathcal T:\mathcal O\to\mathcal O\) and its supported
operation
\[
 a_H^\Delta:R\Gamma_H\mathcal O\longrightarrow R\Gamma_H\mathcal O.
\]
The former is a degree-zero derived endomorphism of one degree-zero
sheaf. A nonnegative injective target resolution shows that it is
uniquely its sheaf map on \(\mathcal O\): degree-zero cycles land in
the degree-zero kernel and no degree-minus-one boundary exists.
Its holomorphic normal-cube calculation above consequently
identifies this whole morphism with \(T_+\) of AM.19c.

We check the support-forgetting square for these two operations on
the actual four-entry cube, rather than inferring it from equality
on holomorphic functions. Write the full relative kernel
\(k=(a,b)\) and input \(e=(c,f)\), with kernel degree \(N\).
DLG.1 gives a four-entry cup
\[
 F(k,e)=
 (ac,\ bc,\ (-1)^Naf,\ (-1)^{N-1}bf).
\]
Let \(\pi_A\) discard the input-complement column:
\[
\begin{gathered}
\pi_A(x,x_A,x_B,z)
\\ =(x,x_A),
\\ \pi_AF(k,e)=(ac,bc)
\\ =k\mathbin{\cup}c.
\end{gathered}
\tag{AM.19e}
\]
The differential DLG.2 makes \(\pi_A\) a literal chain map into
the single \(A\)-support cone. Projection \(e\mapsto c\) is input
support forgetting by R.7. Thus AM.19e is the ordinary kernel cup
after forgetting input support, on every full cochain and in every
degree, with \(b\) retained.

For clarity the augmentation homotopy also commutes with this
projection. In DLG.5–DLG.6 let \(\Phi_\rho\) be the cube-to-intersection
cone map, let \(\iota\) be its displayed inverse, and let \(H_\rho\)
be its displayed homotopy. If \(j_{A\cap B,A}\) forgets source
support from \(A\cap B\) to \(A\), then
\(j_{A\cap B,A}=\pi_A\iota\). Therefore
\[
\begin{gathered}
\pi_A-j_{A\cap B,A}\Phi_\rho
\\ =d(\pi_AH_\rho)+(\pi_AH_\rho)d,
\\ \pi_AH_\rho(x,x_A,x_B,z)
\\ =(0,-\rho_Bz).
\end{gathered}
\tag{AM.19f}
\]
The second coefficient is defined on \(U_A\): the cutoff
\(\rho_B\) is subordinate to \(U_B\), so its product with \(z\)
extends by zero within \(U_A\). It is not extended across \(A\).
This verifies the domain of the entire homotopy.
In fact the complement entry of either side is
\(\rho_B(x_A-x_B)-\bar\partial\rho_A\wedge z\):
the two \(\rho_B\bar\partial z\) terms cancel and
\(\bar\partial\rho_B=-\bar\partial\rho_A\).
The whole-domain entries are zero.

Choose one proper collar \(\eta=1\) near all of \(A\), over a
precompact output open. Then it is one near \(A\cap B\) as well.
It exists by the explicit bump construction DLG.4 and can be
chosen in the original source domain after shrinking the output.
The same \(L_\eta\) commutes with the source support-forgetting
map: its two entries are
\((\eta x+\bar\partial\eta\wedge y,\eta y)\), and
\(\bar\partial\eta=0\) near \(A\). Its ambient correction is legal
for both support sets. Every traced homotopy term now has proper
support. Applying the normalized all-degree trace to AM.19f and
the cup gives the homotopy
\((-1)^N Q_N L_\eta\pi_AH_\rho F(k,-)\), by the two
degree-\(N\) sign identities DLG.4/DLG.16.
Forgetting the target \(H\)-support means retaining the trace's
ambient first component, exactly as R.7 and DL.4 prescribe.
Thus the following square commutes as a morphism of the
sheaf-derived objects:
\[
 \begin{gathered}
 j_H a_H^\Delta=T_+j_H,\\
 j_H:R\Gamma_H\mathcal O\longrightarrow\mathcal O.
 \end{gathered}
 \tag{AM.19g}
\]
R.6–R.7 identifies the same finite smooth/cube roof with the
injective supported model. No top-entry-only or global
cohomology argument has replaced that comparison.

The cohomology of \(R\Gamma_H\mathcal O\) is supported in \(H\),
by the off-support contraction R.22. Apply AM.19d to this source,
with \(F=\mathcal O\). Naturality gives
\(j_H(R\Gamma_H T_+)=T_+j_H\); AM.19g and the uniqueness in
AM.19d force
\[
 a_H^\Delta=R\Gamma_H T_+.
 \tag{AM.19h}
\]
This proves the canonical supported-action comparison.

Finally the actual \(C\)-class of \(R_+\) is obtained by support
forgetting from \(\Delta\) to the real cone \(G\), as shown above.
The intersections of both supports with \(p_2^{-1}H\) are proper;
for \(G\) this is AM.18. A common collar may be chosen near their
union and the full diagonal, which is still proper over a small
output domain. DLG.3–DLG.4's support-forgetting homotopy therefore
identifies the two actions on \(R\Gamma_H\mathcal O\).
It follows that the action of this whole \(C\)-class is the
operation in AM.19h, not only an operation of its finite
truncations or a separate formal diagonal symbol.

Apply derived ordinary sections to AM.19h on \(D\). Both entries
of AM.16a are acyclic by AM.B.1, and \(T_+\) is a local sheaf
map commuting with restriction. R.6's natural augmentation
comparison therefore represents the resulting map on AM.16a by
\((T_+,T_+)\), with its actual restriction arrow. On \(U_D\),
choose each input circle smaller than the distance to its
boundary. A full-disc input can be restricted and its circle
shrunk to this one; Cauchy's annulus theorem makes that square
commute. A function defined only on \(U_D\) is never integrated
on a circle crossing its input boundary.

Repeated Cauchy integration, with the geometric bound established
before integration, now gives the strict local sheaf map
\[
\begin{gathered}
T_+f(t,x)
\\ =\sum_{j\ge0}r_j(x)\partial_t^j f(t,x).
\end{gathered}
\tag{AM.19c}
\]
The same formula holds on \(D\) and on \(U_D\), locally on each open.
Its series converges normally on every compact there: choose one
input Cauchy radius and then \(\varepsilon\) smaller than that radius.
Thus the canonical action of \(R_+\) on AM.16a is represented by
\((T_+,T_+)\). In the full relative convention an input \((0,-f)\)
maps to \((0,-T_+f)\), after the proved augmentation comparisons.
This establishes the supported action of the complete positive
infinite part without an estimate for a purported infinite ambient
distribution or an unspecified sequence of primitives.

##### AM.B.3. Negative order by the complete finite-order Volterra proof

The whole \(R_-=\sum_{n\ge1}r_{-n}(x)D_t^{-n}\) belongs to the actual
finite-order ring \(E\): its upper order is \(-1\) and AM.2 supplies
its one negative factorial bound. It need not be a finite Laurent
polynomial. CA.1 admits \(g=0\); its input is exactly \(\Re t<0\).
The complete CA.3–CA.6 construction, including CA.33's full smooth
ambient primitive and all tangential indices, therefore applies to
this whole operator. Its actual action is the convergent Volterra
series
\[
\begin{gathered}
T_-f(t,x)=
\\ \sum_{n\ge1}\frac{r_{-n}(x)}{(n-1)!}
\\ {}\int_a^t(t-s)^{n-1}f(s,x)\,ds,
\\ a<0.
\end{gathered}
\tag{AM.20}
\]
Choose the anchor and a fixed small half-disc before summing.
Straight segments from \(a\) to a compact output family lie in a
compact subset of the larger left half-disc; let their lengths and
\(|t-s|\) be at most \(L\). The \(n\)-th anchored primitive is bounded
by \(M_KL^n/(n-1)!\). Its product with \(r_{-n}\) is at most
\(MM_K n(C_0L)^n\). Shrink the whole configuration to make \(C_0L<1\).
This proves normal convergence; fixed derivatives follow by one
additional compact Cauchy margin in \(t,x\). CA.33 identifies this
series with the actual class, with its full ambient primitive,
instead of identifying separate finite classes and taking a quotient
limit.

Changing \(a\) to \(b\) subtracts the fixed-path integral from \(a\)
to \(b\), with integrand \((t-s)^{n-1}f(s,x)/(n-1)!\).
That term is a polynomial in \(t\), holomorphic in \(x\).
On a small full output disc its sum has the bound
\(C n[C_0(r+L_0)]^{n-1}\), where \(L_0\) bounds the fixed path and its
distance from zero. Choose both anchors and the full disc small
enough for this ratio to be less than one. The sum is holomorphic on
that full disc. Thus anchored choices give the same \(\mathcal S\)
class. These holomorphic anchor differences are distinct from CA.33's
smooth ambient comparison primitive.

Additivity of the actual \(C\)-action and the two established
identifications give, for every holomorphic left-half-disc input,
\[
\begin{gathered}
Rf=T_+f+T_-f
\\ \pmod{\mathcal O}.
\end{gathered}
\tag{AM.21}
\]
Writing \(D_t^{-n}\) for these anchored iterates gives exactly
\(\sum_{j\in\mathbf Z}r_j(x)D_t^jf\) as a normally convergent
holomorphic representative on that half-disc, up to the proved
full-disc holomorphic gauge. The fixed domain comes before all
coefficient indices and all positive epsilon choices.

#### AM.C. Power/log functions detect every normal coefficient

First take a scalar exponent \(\lambda\notin\mathbb Z\). On the left half-disc choose a branch and let \(f_\lambda=t^\lambda\). Its iterated derivatives and primitives, modulo holomorphic functions, are
\[
\begin{gathered}
D_t^j f_\lambda=c_j(\lambda)t^{\lambda-j},
\\ c_j=\lambda(\lambda-1)\cdots
\\ {}(\lambda-j+1)\ (j\ge0),
\\ c_{-n}=\prod_{s=1}^n(\lambda+s)^{-1}.
\end{gathered}
\tag{AM.22}
\]
Every coefficient is nonzero. For fixed \(\lambda\), finite-product estimates give
\( |c_j|\le K^{j+1}j!\) and \(|c_{-n}|\le K^{n+1}/n!\). For example, beyond \(2|\lambda|+2\), each denominator has modulus at least half its positive integer index; the finitely many initial factors are nonzero and are absorbed in a prefactor. The derivative bound follows in the same way, or directly by Cauchy's formula on a circle whose radius is a fixed proportion of \(|t|\). These estimates and AM.2 make
\[
\begin{gathered}
Rf_\lambda=t^\lambda V(t,x),
\\ V=\sum_{j\in\mathbb Z}r_j(x)c_j(\lambda)t^{-j}
\end{gathered}
\tag{AM.23}
\]
normally convergent on a punctured disc after one fixed normal shrink.

We prove the full-disc primitive-gauge assertion directly. Let \(Q_0=f\)
and \(Q_n'=Q_{n-1}\) for \(n\ge1\), with all branches on the left
half-disc. For the noninteger power use
\(Q_n=c_{-n}t^{\lambda+n}\). At a fixed negative anchor \(a\),
\[
\begin{gathered}
|Q_l(a)|\le
\\ M_a K^l |a|^l/l!\quad(l\ge1),
\end{gathered}
\tag{AM.23a}
\]
where \(K\) depends on \(\lambda\) but not on how small \(a\) is;
\(M_a\) may depend on \(a\). This follows from the finite-product
bound for \(c_{-l}\), with \(|a^\lambda|\) in \(M_a\).
Taylor subtraction at the anchor gives the exact identity
\[
\begin{gathered}
Q_n(t)
\\ {}-D_{t,a}^{-n}f(t)
\\ =\sum_{k=0}^{n-1}
\\ {}\frac{Q_{n-k}(a)(t-a)^k}{k!}.
\end{gathered}
\tag{AM.23b}
\]
Both sides have the same \(n\)-th derivative and the displayed
polynomial matches the first \(n\) derivatives at \(a\), which proves
the identity by repeated integration. On the whole disc \(|t|\le r\),
the binomial theorem and AM.23a bound this polynomial by
\[
 \frac{M_a}{n!}\bigl(K|a|+r+|a|\bigr)^n.
 \tag{AM.23c}
\]
Choose the small disc and then, for example, \(a=-r/2\), so that
\(C_0[K|a|+r+|a|]<1\). Multiplication by
\(|r_{-n}|\le MC_0^n n!\) now gives a convergent geometric series of
polynomials on the full disc. This is a holomorphic full-disc gauge,
uniformly on each fixed tangential compact. Fixed derivatives follow
by one further Cauchy margin. Consequently AM.21 with the anchored
primitives is exactly the class of the explicit representative AM.23;
no quotient-space limit or multivalued full-circle bound is needed.

If AM.23 is zero in \(\mathcal S\), it agrees on a smaller left half-disc with a holomorphic germ at zero. Analytic continuation around the circle changes it by
\((e^{2\pi i\lambda}-1)t^\lambda V\). The germ has no monodromy, so \(V=0\). Integrating its normally convergent Laurent series over circles gives \(r_jc_j=0\) for every \(j\), hence \(r_j=0\).

For an integer exponent \(m\), use
\[
\begin{gathered}
E_k(t)=
\\ \begin{cases}
\begin{gathered}t^k\log t/k!,\\k\ge0,\end{gathered}\\
\begin{gathered}(-1)^{-k-1}(-k-1)!t^k,\\k<0.\end{gathered}
\end{cases}
\end{gathered}
\tag{AM.24}
\]
The identities \(D_tE_k=E_{k-1}\) hold modulo holomorphic functions, including \(k=0\). They follow by one differentiation, with the additional polynomial derivative term in the nonnegative case discarded only because it is holomorphic on a full disc. Thus
\[
\begin{gathered}
RE_m=
\\ U(t,x)\log t+V_-(t,x),
\\ U=
\\ \sum_{j\le m}\frac{r_j(x)t^{m-j}}{(m-j)!},
\\ V_-=
\\ \sum_{j>m}r_j(x)(-1)^{j-m-1}
\\ {}(j-m-1)!t^{m-j}.
\end{gathered}
\tag{AM.25}
\]
The first series is holomorphic at zero, and the second is a
normally convergent strictly negative Laurent series. Indeed the
factorials in AM.25 differ from AM.2's factorials by the fixed shift
\(m\). Their ratios cost a fixed polynomial in the index; a smaller
negative radius or the arbitrary positive epsilon absorbs it.

Here are exact primitive gauges in this integer case. Write
\(H_l=\sum_{v=1}^l1/v\), with \(H_0=0\). If \(m\ge0\), set
\[
\begin{gathered}
Q_n=E_{m+n}
\\ {}-(H_{m+n}-H_m)
\\ {}\frac{t^{m+n}}{(m+n)!}.
\end{gathered}
\tag{AM.25a}
\]
If \(m<0\), put \(k=m+n\); set \(Q_n=E_k\) for \(k<0\), and
\[
\begin{gathered}
Q_n=E_k-H_k\,t^k/k!
\\ (k\ge0).
\end{gathered}
\tag{AM.25b}
\]
In both cases \(Q_0=E_m\) and \(Q_n'=Q_{n-1}\).
For positive \(k\), this follows by
\(H_k-H_{k-1}=1/k\); at \(k=0\), it is
\((\log t)'=1/t\), with exactly the sign in AM.24.
At a fixed negative \(a\), these \(Q_l\) satisfy AM.23a:
the factorial shift is bounded by a polynomial in \(l+1\),
\(H_{l+|m|}\le C_m+\log(l+1)\), and a fixed exponential \(K^l\)
absorbs both; powers of \(a\), its logarithm and all finitely many
initial terms are in \(M_a\). Here \(K\) can again be chosen
independently of the small anchor. Thus AM.23b–AM.23c prove normal
convergence of the full-disc anchored Taylor gauge.

The difference \(Q_n-E_{m+n}\) in AM.25a–AM.25b is a polynomial.
After multiplication by \(r_{-n}\), its supremum on a full small
disc is bounded by a constant times
\((C_0r)^n(n+1)^{|m|}(1+\log(n+1))\).
The finitely many terms with \(m+n<0\) have zero difference.
The polynomial series therefore also converges on a full disc when
\(C_0r<1\). Positive derivatives of \(E_m\) have a polynomial
discrepancy from \(E_{m-j}\) only for the finitely many
\(0\le j\le m\) when \(m\ge0\); after that the extra polynomial has
been differentiated to zero. For \(m<0\) there is no such discrepancy.
This proves AM.25 as an equality of actual classes, with every
infinite polynomial gauge accounted for.

If the class is zero, AM.25 equals a holomorphic germ on the left
half-disc. Uniqueness of analytic continuation around the puncture
then forces its jump \(2\pi iU\) to be the zero **function**.
This is not an inference from its quotient class: a nonzero
holomorphic function would itself have zero class. Uniqueness of
Taylor coefficients kills all \(r_j\) for \(j\le m\). The remaining
strictly negative Laurent series is holomorphic at zero only if
every negative Laurent coefficient is zero, by circle integration.
This kills the other \(r_j\) as well, for every integer \(m\).

For a Jordan block \(A=\lambda I+J\) of size \(q\), with \(J^q=0\) and its ones immediately above the diagonal, put
\[
 F_A=\sum_{\ell=0}^{q-1}J^\ell f_{\lambda,\ell}(t).
 \tag{AM.26}
\]
For noninteger \(\lambda\), take \(f_{\lambda,\ell}=t^\lambda(\log t)^\ell/\ell!\). For \(m\ge0\), take \(f_{m,\ell}=t^m(\log t)^{\ell+1}/[m!(\ell+1)!]\). For \(m<0\), take \(f_{m,\ell}=(-1)^{-m-1}(-m-1)!t^m(\log t)^\ell/\ell!\). Direct differentiation gives
\[
\begin{gathered}
(tD_tI-A)F_A=0
\\ \pmod{\mathcal O},
\\ D_{x_i}F_A=0.
\end{gathered}
\tag{AM.27}
\]
Indeed \((tD_t-\lambda)f_{\lambda,\ell}=f_{\lambda,\ell-1}\) for \(\ell\ge1\); the degree-zero term is zero modulo holomorphic functions. The powers of \(J\) then give precisely \(JF_A\).

All entries admit the full action AM.21, which was proved for every holomorphic input on the left half-disc. Positive derivatives satisfy the Cauchy estimate on a fixed sub-sector compact. For negative iterates, repeatedly integrate \(t^b(\log t)^\ell\): when \(b\ne-1\), integration gives
\[
\begin{gathered}
t^{b+1}\sum_{v=0}^\ell
\\ {}\frac{(-1)^v\ell!}{(\ell-v)!(b+1)^{v+1}}
\\ {}(\log t)^{\ell-v};
\end{gathered}
\tag{AM.28}
\]
when \(b=-1\), it gives \((\log t)^{\ell+1}/(\ell+1)\).
Differentiation verifies both formulas.

For an explicit common estimate write an iterated primitive as
\(Q_n=t^{b+n}P_n(\log t)\), where \(P_0(z)=z^\ell\).
If \(b+n\ne0\), use the finite polynomial inverse
\[
\begin{gathered}
P_n=
\\ (b+n+\partial_z)^{-1}P_{n-1}
\\ =\sum_{v=0}^{L}\frac{(-\partial_z)^vP_{n-1}}{(b+n)^{v+1}}.
\end{gathered}
\tag{AM.28a}
\]
If \(b+n=0\), put \(P_n(z)=\int_0^zP_{n-1}(u)\,du\).
This resonance occurs at most once, so one fixed
\(L=\ell+1\) bounds every polynomial degree. In both cases
\((b+n)P_n+P_n'=P_{n-1}\), and hence \(Q_n'=Q_{n-1}\).
For \(P=\sum p_vz^v\) put \(\|P\|_*=\sum v!|p_v|\).
Directly \(\|\partial_zP\|_*\le\|P\|_*\), and polynomial integration
with zero constant also has norm at most one. For sufficiently large
\(n\), \(|b+n|\ge n/2\ge2\); the finite geometric sum in AM.28a
therefore has operator norm at most \(2/|b+n|\le4/n\).
All earlier nonzero denominators and the possible resonant step form
one finite list. Taking its maximum constant gives
\(\|P_n\|_*\le C\,4^n/n!\) for every \(n\).
Evaluation at a fixed nonzero anchor gives
\[
\begin{gathered}
|Q_n(a)|\le
\\ |a^b|\,|a|^n\|P_n\|_*
\\ {}\sum_{v=0}^{L}|\log a|^v/v!
\\ \le M_a\,4^n|a|^n/n!.
\end{gathered}
\tag{AM.28b}
\]
Thus AM.23a–c applies to these complete primitive chains as well.
There is one common exponential constant for the finitely many entries
of \(F_A\), with full-disc normally convergent anchor gauges and all
fixed tangential derivatives. This proves the primitive estimates
for every Jordan entry; the actual action itself was already proved
for arbitrary holomorphic inputs in AM.B.

The matrix \(F_A\) is upper triangular, with the scalar detector just proved on every diagonal entry. If a normal row \(R=(R_1,\ldots,R_q)\) satisfies \(RF_A=0\) in \(\mathcal S^q\), its first column says \(R_1f_{\lambda,0}=0\), hence \(R_1=0\). Its second then says \(R_2f_{\lambda,0}=0\), and so on. All coefficients vanish. Finite direct sums preserve this detection. A constant similarity of \(A\) gives an invertible constant change of generators and of the detector matrix, so the same conclusion holds for every constant matrix with its split Jordan decomposition. No diagonalizability or semisimplicity is assumed.

For completeness, the required finite matrix reduction is elementary. A nonconstant complex polynomial has a root: its modulus tends to infinity at infinity, hence has a minimum; if a nonzero minimum occurred at \(z_0\), write its first nonconstant Taylor term as \(c(z-z_0)^k\) and choose the argument of \(z-z_0\) so that this term is a negative real multiple of the value at \(z_0\). For sufficiently small modulus the higher terms are smaller than the strict decrease, a contradiction. Repeated division splits the characteristic polynomial into linear factors. The identity \((uI-A)\operatorname{adj}(uI-A)=\det(uI-A)I\), compared coefficient by coefficient and then multiplied by powers of \(A\), gives the Cayley–Hamilton relation by telescoping. Bezout identities between its relatively prime factors produce the direct sum of the generalized eigenspaces: if \(p_i(u)=(u-\lambda_i)^{m_i}\) and \(P=\prod_i p_i\), polynomials \(b_i\) satisfy \(\sum_i b_i(P/p_i)=1\); applying these to \(A\) gives components annihilated by \(p_i(A)\), whose intersections are zero by the same coprime identities. Bezout itself is the terminating polynomial division algorithm, followed backwards.

On one generalized eigenspace let \(J=A-\lambda I\), so \(J^r=0\). Set \(K_i=\ker J^i\), \(K_0=0\), and \(K_{r+1}=K_r\). For each \(i\) choose representatives \(V_i\subset K_i\) of a basis of
\(K_i/(K_{i-1}+JK_{i+1})\). Their chains \(v,Jv,\ldots,J^{i-1}v\), \(v\in V_i\), span the space. In fact in \(K_i/K_{i-1}\), the defining quotient says that these representatives together with the image of \(J:K_{i+1}/K_i\to K_i/K_{i-1}\) span; descend from \(i=r\), then use the filtration by the \(K_i\)'s. This map induced by \(J\) is injective, because \(Jv\in K_{i-1}\) implies \(J^iv=0\). If \(a_i=\dim K_i-\dim K_{i-1}\), it follows that \(|V_i|=a_i-a_{i+1}\). The number of vectors in all the spanning chains is \(\sum_i i(a_i-a_{i+1})=\sum_i a_i=\dim K_r\). They therefore form a basis. Order each chain by \(J^{i-1}v,\ldots,Jv,v\); the resulting matrix has its ones immediately above the diagonal. This proves the exact Jordan convention used in AM.26, including all nilpotent blocks.

#### AM.D. Real-module injection and explicit matrix saturation for the models

For the model AM.3, the columns of \(F_A\) define \(E\)-linear maps to the actual left \(C\)-module \(\mathcal S\), by AM.27. They therefore extend to \(C\)-linear maps from \(C\otimes_E N_A\), by \(c\otimes v\mapsto c\phi(v)\). If an element of \(B\otimes_E N_A\) dies after extension to \(C\), take its unique normal row AM.4. Its value under all these maps is \(RF_A=0\). AM.C makes its normal row zero, and AM.A makes the original element zero. Thus
\[
\begin{gathered}
B\otimes_E N_A\longrightarrow C\otimes_E N_A
\\ \text{is injective}.
\end{gathered}
\tag{AM.29}
\]
For a zero-size matrix both tensor modules are zero, so this conclusion
also covers that case.

This also constructs the precise finite relation lift for its presentation, rather than only separating selected scalar symbols. Write its presentation matrix as the vertical stack of \(tD_tI-A\) and the \(D_{x_i}I\). Suppose a row \(c\) over \(C\) has product \(f=cP\in B^q\). Divide \(f\) as in AM.A:
\[
\begin{gathered}
f=QL+\sum_iH_iD_{x_i}
\\ {}+R.
\end{gathered}
\tag{AM.30}
\]
Every evaluation annihilates \(f\), because \(P F_A=0\) as an actual \(\mathcal S\)-class. Hence \(RF_A=0\), so \(R=0\). Using AM.6 gives the explicit \(B\)-row
\[
\begin{gathered}
b=(Qh,H_1,\ldots,H_d),
\\ bP=f=cP.
\end{gathered}
\tag{AM.31}
\]
AM.5 and AM.10 construct all its entries with the full positive every-epsilon bound, one negative factorial bound, all spatial indices and each fixed derivative on one common coefficient domain. The finite equality with the given real row has one later actual support/domain witness by AM.H. No tensor-flatness assertion about \(C\) or \(C/B\) is used.

For any other finite free presentation \(E^a\xrightarrow{P'}E^b\to N_A\to0\), ordinary tensoring identifies its cokernel with \(B\otimes_E N_A\). Explicitly, send a row in \(B^b\) to its balanced tensor combination of the given generators. Its inverse sends \(b\otimes \bar v\) to \(bv\) modulo the rows \(B^aP'\); replacing a lift \(v\) by a relation changes the row by such a row, and the formula respects additivity and the balancing relation. The same argument applies to \(C\). Hence if \(c'P'=f'\in B^b\), its tensor class dies over \(C\), AM.29 makes it zero over \(B\), and the displayed cokernel description supplies \(b'\in B^a\) with \(b'P'=f'\). All rows here have finitely many entries, and AM.H supplies a common actual witness for their equality. This proves presentation independence by the explicit balanced tensor quotient.

![Proper halfspace geometry, two coefficient tails and an exact detector](assets/euler-model-halfspace-and-detectors.png)

*Figure AM.2. Left: the exact complex-time slice of AM.18 has
\(\epsilon=1/4\), output \(t=3/5+i/10\), and tangential variables
suppressed. The input support is the whole right halfspace
\(\Re s\ge0\). Its intersection with the kernel cone is the closed
triangle with vertices \(-i/20\), \(3/5+i/10\), and \(i/4\).
Every edge and vertex is included. These points satisfy exactly
\(0\le\Re s\le3/5\) and
\(|\Im s-1/10|\le(3/5-\Re s)/4\); the bounded fibre is the
time section of the properness proof. Centre: the positive infinite
part is the complex-diagonal Čech/residue calculation AM.B.2, with
its derived input-support comparison. The entire negative tail is one
finite-order operator and uses the full Volterra calculation AM.B.3.
Right: the full sequence is \(r_j=1/(j!)^2\) for \(j\ge0\) and
\(r_{-n}=n!4^{-n}\) for \(n\ge1\).
It satisfies AM.2 since \(1/(j!\varepsilon^j)\le e^{1/\varepsilon}\)
and its negative factorial constant is \(1/4\). AM.25 gives
\(R\log t=U(t)\log t+V_-(t)\) modulo full-disc holomorphic germs,
where \(U(t)=1/(1-t/4)\) on \(|t|<4\) and
\(V_-=\sum_{j\ge1}(-1)^{j-1}/(j\,j!\,t^j)\) on the punctured disc.
The plotted curve samples this exact rational \(U\) on a negative
real interval; the marked values are \(U(-1)=4/5\) and
\(U(-1/2)=8/9\). The representative's jump is \(2\pi iU\) as a
function. It obstructs holomorphic extension of that representative;
its own quotient class is zero. The scalar full-disc gauge estimate
is AM.23a–c. All constant Jordan blocks are treated in AM.26–27,
and the relation lift is AM.30–31.*

Figure AM.2 has editable source,
exact data, a
full PNG and an
SVG.

![The full cube projection, proper homotopy and supported-action square](assets/euler-model-diagonal-support.png)

*Figure AM.3. In this diagram \(A=\Delta\), \(B=p_2^{-1}H\),
\(k=(a,b)\) has degree \(N\), and \(e=(c,f)\).
All four entries of the cup are displayed. Projection \(\pi_A\)
retains its whole-domain and kernel-complement entries, giving the
ordinary kernel cup after input support is forgotten (AM.19e).
The actual homotopy is
\(S=\pi_AH_\rho=(0,-\rho_Bz)\), on the kernel complement \(U_A\),
and \(\pi_A-j_{A\cap B,A}\Phi_\rho=dS+Sd\) (AM.19f).
Here \(z\) denotes the fourth cube coefficient, not a cotangent
coordinate. The single collar equals one near the entire diagonal;
the written traced homotopy is proper. The bottom square is the
equality AM.19g in the derived category of sheaves before taking
ordinary sections. The adjunction AM.19d makes its lift unique and
gives AM.19h. Every object, map and sign is proved in AM.B.2.
This is a diagram of exact cochain and derived maps, with no
sampled geometry.*

Figure AM.3 has editable source,
exact formula data
with a full PNG
and an SVG.

#### AM.H. The complete real-stalk representative lemma

This lemma identifies the entire source-defined real stalk and its finite equality witnesses. Its RM equation labels belong to this lemma.

Let \(X\) have complex dimension \(N\), let the diagonal in \(X\) times \(X\) be
\(\Delta\), and identify its conormal with \(T^*X\) by the ordered output and
input variables. The input holomorphic top form is ordered
\(dv_1\wedge\cdots\wedge dv_N\). The source definitions are
\[
\begin{gathered}
\mathcal C^{\mathbb R}_{Y|X}
\\ =\mathcal H^d_{T_Y^*X}(\pi^{-1}\mathcal O_X)^a,
\\ \mathcal E^{\mathbb R}_X
\\ =\mathcal C^{\mathbb R}_{\Delta|X\times X}
\\ {}\otimes_{p_2^{-1}\mathcal O_X}p_2^{-1}\Omega_X.
\end{gathered}
\tag{RM.1}
\]
Here \(d\) is the complex codimension of \(Y\), \(\pi\) is the comonoidal projection,
and a is the antipodal map. Off the zero section,
\[
\begin{gathered}
\mathcal C^\infty_{Y|X}
=\gamma^{-1}\gamma_*\mathcal C^{\mathbb R}_{Y|X},
\\ \gamma:T^*X\setminus T_X^*X
\\ \longrightarrow P^*X.
\end{gathered}
\tag{RM.2}
\]
The top-form twist gives the corresponding definition of
\(\mathcal E^\infty\). Thus a real stalk is a germ at one real phase;
a projectivized section supplies compatible germs around the whole
complex phase fiber. These are different sheaves.

Fix a nonzero \(p\), use a linear base change to normalize its central
covector to \((1,0)\), and write \(u=(t,x)\),
\(v=u+(w,\zeta)\), \(d=N-1\). The already proved IG.1 unfolds
(RM.1), with precisely its antipodal sign, as
\[
\begin{gathered}
C:=\mathcal E^{\mathbb R}_p
\\ =\mathop{\rm colim}_{D,H}
H_H^N(D;\mathcal O^{(0,N)}),
\\ C_\Delta(H)_{u_0}\setminus0
\\ \subset\{-\Re\langle(w,\zeta),p\rangle>0\}.
\end{gathered}
\tag{RM.3}
\]
The domains \(D\) are actual pair-neighborhoods; \(H\) is closed. Arrows are
restriction and support forgetting. The local top-form frame identifies
this with IG.1's untwisted coefficient calculation, with the fixed
V.17 calibration. Filtered colimits here commute with cohomology by a
direct argument: a closed finite cochain is closed at a later index, and
a zero cohomology class has its finite primitive at a later index.
The same argument makes finitely many equalities simultaneous at one
index. It makes no infinite collection of domains simultaneous.

We record why the ring models used below are cofinal, including their
actual domains. For the finitely many chosen supports, the compact
unit-normal-cone argument of IG.1–IG.2 gives one positive polar margin.
It supplies numbers \(0<\lambda<\pi/2\), \(B_j>0\), and an actual
smaller pair-neighborhood on which every support is contained in
\[
\begin{gathered}
T=\left\{\begin{gathered}
|\arg(-w)|\le\lambda,\\
|\zeta_j|\le B_j|w|
\end{gathered}\right\}.
\end{gathered}
\tag{RM.4}
\]
The normal zero vector is included. If containment failed arbitrarily
near the base and normal origin, normalize failing normal displacements
and extract a unit-sphere subsequence. Its limiting normal-cone vector
violates the strict margin used to choose T. This proves actual uniform
containment, not only tangent-cone containment at one point.

Enlarge T to the closed convex cone
\[
\begin{gathered}
G=\left\{\begin{gathered}
l=-\Re w\ge0,\\
|\Im w|\le\epsilon l,\\
|\zeta_j|\le A_jl
\end{gathered}\right\},
\\ \epsilon=\tan\lambda,
\\ A_j=B_j/\cos\lambda.
\end{gathered}
\tag{RM.5}
\]
Each support map is a literal relative-complex map. \(G\) is itself
admissible after a fixed phase/covariable shrink. For
\(p'=e^{i\eta}(1,z)\),
\[
\begin{gathered}
-\Re\langle(w,\zeta),p'\rangle
\\ \ge l\left(\begin{gathered}
\cos\eta-\epsilon|\sin\eta|
\\ {}-\sum_jA_j|z_j|
\end{gathered}\right).
\end{gathered}
\tag{RM.6}
\]
The parentheses is positive on one actual neighborhood of \((\eta,z)=0\).
On a nonzero vector of \(G\), l is positive. Thus the support enlargement
remains within the real stalk's defining polar class. Finite lists of
supports, cocycles, primitives and products use their common minimum
margin and common maximum cone constants.

Let \(e_t\) be the positive real time vector and put
\[
\begin{gathered}
D_R=(Re_t+\operatorname{int}G)
\\ {}\cap(-Re_t-\operatorname{int}G).
\end{gathered}
\tag{RM.7}
\]
Explicitly this is
\(|\Re t|<R\),
\(|\Im t|<\epsilon(R-|\Re t|)\), and
\(|x_j|<A_j(R-|\Re t|)\). It contains the base origin.
Choosing \(R\) sufficiently small puts its closure inside the selected
outer base domain and all pair differences inside the selected normal
domain: time differences have modulus at most
\(2R\sqrt{1+\epsilon^2}\), and spatial differences at most \(2A_jR\).
This one finite choice precedes every symbol index.

This domain is \(G\)-round. If \(u\) and \(v\) belong to \(D_R\) and
\(s-u,v-s\in G\), then addition of \(G\) preserves the first open-cone
inequality for \(s\), and subtraction of \(G\) preserves its second one.
Hence \(s\) belongs to \(D_R\). For compact endpoint families these inequalities
have positive uniform margins. The triangle inequality shows that every
intermediate point retains the corresponding endpoint margin.

For consecutive differences \(g_i\in G\), write their nonnegative
lengths as \(l_i=-\Re(g_i)_t\). Then
\[
\begin{gathered}
\sum_i l_i=-\Re(v_t-u_t),
\\ |g_i|\le K_Gl_i,
\\ K_G=\sqrt{1+\epsilon^2+\sum_jA_j^2}.
\end{gathered}
\tag{RM.8}
\]
Thus every intermediate point over compact endpoints is bounded and
stays a positive distance inside \(D_R\). The defining support intersection
is closed, so that family is compact inside the actual source. This
proves properness for any prescribed finite number of products.
Convexity gives \(G+G=G\), so projected product support is again \(G\).

The complete earlier DLG.1–DLG.5 proof now applies, without a symbol
normal-form assumption, to the whole ring
\[
\begin{gathered}
\mathcal R(G,D_R)
\\ =H^N_{\{v-u\in G\}}\left(\begin{gathered}
D_R\times D_R;\\
\mathcal O^{(0,N)}
\end{gathered}\right).
\end{gathered}
\tag{RM.9}
\]
Its product is the full relative cup, controlled proper excision and
normalized intermediate trace. It uses smooth relative representatives
and all complement entries; arbitrary singular currents are not
multiplied. DLG.5 proves associativity by the full finite cube, its
explicit homotopies, the projection formula and compact Fubini.
Its unit is the diagonal residue with the ordered positive Cauchy
normalization. DLG.3 and DLG.7 prove that a later domain, support or
collar choice intertwines these actual products. The new checks above
verify every required geometric hypothesis at the actual real stalk.

Every class of \(C\) has a representative in some (RM.9). Every finite
equality in \(C\) is represented at one later such ring: include its
primitive in the finite support list, perform (RM.4)–(RM.7), and map
that primitive by the same actual relative maps. Conversely each
(RM.9) is an admissible source stalk representative by (RM.6).
Consequently the colimit of these whole unital rings, with products
computed after common refinements, is precisely the source-defined
ring \(C.\) This statement includes arbitrary real germs, rather than the
subclass with homogeneous coefficient normal forms. Its multiplication
is the defining source cup/excision/trace multiplication because all
maps used are its actual cochain maps. No mixed flatness is used.


The actual target comparisons just proved may be used in
[§5.52, AM.R](#am-r-injectivity-can-be-detected-through-an-embedding)
when its same-source full-coefficient embedding is supplied.
For arbitrary coherent holonomic germs, the generic embeddings and
compatible contact transport on every smooth characteristic branch
remain additional requirements. The singular extension
[AM.S](#am-s-extending-generic-injectivity-across-the-analytic-exceptional-set)
applies only after its generic injectivity hypothesis has been proved.
The model theorem above does not assert those hypotheses.

Free additional reading:
[Kashiwara–Kawai, Holonomic systems III](https://www.kurims.kyoto-u.ac.jp/~kenkyubu/kashiwara/HolIII.pdf),
the definitions on printed pp.830–831 and the comparison discussed on
p.922. Every result used in the present model proof is proved here
or in the exact earlier programme proofs listed in AM.T.


### 5.54. Halfspace solutions and finite Euler extensions

Halfspace solution functionals separate every finite extension of the
full infinite-order constant Euler modules. The proof gives a finite
free resolution, contracts its solution complex, computes the integer
resonance obstructions and extends functionals across arbitrary finite
extensions. It also proves real-kernel comparison for every module
with an actual embedding into this extension category.

#### EE.0. Rings, modules and exact earlier proofs

Use the actual source stalk at \(p=(t,x;\tau,\xi)=(0,0;1,0)\), with
\(x\in\mathbf C^d\), and put
\[
\begin{gathered}
 E=\mathscr E_p,\\ B=\mathscr E^\infty_p,\\
 C=\mathscr E^{\mathbb R}_p.
\end{gathered}
\tag{EE.1}
\]
These are the rings actually constructed in [§5.19](#5-19-actual-factorial-growth-symbols-and-ordered-jets) and
[§§5.44–5.45](#5-44-full-homogeneous-coefficients-and-ordinary-infinite-order-kernels)
and [§5.53, AM.H](#am-h-the-complete-real-stalk-representative-lemma), with the unital inclusions \(E\to B\to C\) of
[§5.53, AM.T–AM.B](#5-53-real-kernel-comparison-for-all-constant-euler-matrices). The right \(E\)-flatness of \(B\) is the complete
finite-matrix theorem of [§5.47, MF.1–MF.7](#5-47-arbitrary-finite-matrices-and-right-flatness-of-the-full-infinite-order-ring). For a finite constant matrix
\(A\in M_q(\mathbf C)\), use the left module
\[
\begin{gathered}
 N_A=E^q/\mathcal R_A,\qquad\theta=tD_t,\\
 \mathcal R_A=\sum_{i=1}^d E^qD_{x_i}\\
 \hphantom{\mathcal R_A={}}+E^q(\theta I-A).
\end{gathered}
\tag{EE.2}
\]
Free left modules use coefficient rows; thus right multiplication by the
displayed matrices is a left-linear map. A homomorphism from a free module
to a left target is specified by a column of target elements.

The whole left \(C\)-module used here is exactly
\[
\begin{gathered}
 S=\underset{D}{\mathop{\rm colim}}S_D,\\
 S_D=\mathcal O(U_D)/\mathcal O(D),\\
 D=\{|t|<r\}\times P_x,\\
 U_D=D\cap\{\Re t<0\}.
\end{gathered}
\tag{EE.3}
\]
The polydiscs are centred at zero and shrink cofinally. This is not an
action defined only for a symbol subring: the complete relative-kernel
action and its common-domain refinements are proved in [§5.53, AM.B and AM.H](#5-53-real-kernel-comparison-for-all-constant-euler-matrices). In particular restrictions along both inclusions in EE.1 are actual
module actions. The equality in EE.3 with the supported cohomology stalk,
including the acyclicity and degree convention, is proved there.

The complete all-coefficient detector of [§5.53, AM.C–AM.D](#5-53-real-kernel-comparison-for-all-constant-euler-matrices) proves that
evaluation against the columns of \(F_A\) gives an injective map
\[
 B\otimes_E N_A\longrightarrow S^q.
 \tag{EE.4}
\]
Here \(F_A\) is precisely its power/logarithm matrix, with the integer
cases, negative-order primitive gauges, every positive-epsilon bound and
one full negative-factorial bound already proved there. No scalar
extension to \(C\) is presumed flat.

#### EE.1. A finite free resolution of every Euler module

Let \(K_k=E^q\otimes_{\mathbf C}\bigwedge^k\mathbf C^d\), with
the ordered basis \(e_1,\ldots,e_d\), and give it the homological Koszul
differential for right multiplication by \(D_{x_i}\). For an ordered
subset \(I=(i_1<\cdots<i_k)\), write \(e_I=e_{i_1}\wedge\cdots\wedge e_{i_k}\);
\(e_{I\setminus i_a}\) retains that order:
\[
\begin{gathered}
 d_K(Qe_I)=\sum_{a=1}^k(-1)^{a-1}\\
 \qquad(QD_{x_{i_a}})e_{I\setminus i_a}.
\end{gathered}
\tag{EE.5}
\]
We prove its exactness in positive degrees in the actual finite-order
symbol class, rather than invoking a regular-sequence criterion.

Write a normal symbol as \(\sum_{j\le m}\tau^jp_j(t,x,z)\),
\(z=\xi/\tau\). Right multiplication by \(D_{x_i}\) is exactly
multiplication by \(\tau z_i\). On exterior-valued coefficient functions
let \(d_z=\sum_i e_i\wedge\partial_{z_i}\), and let
\(\iota_z\) be contraction with the coordinate vector \(z\). If a
coefficient is homogeneous of degree \(\ell\) in \(z\) and has
exterior degree \(k\), direct differentiation and the exterior
anticommutation identities give
\[
 \iota_zd_z+d_z\iota_z=(\ell+k)I.
 \tag{EE.6}
\]
The same identity for a convergent holomorphic Taylor series is obtained
termwise on a smaller polydisc, by its uniform Cauchy convergence. Define
\[
\begin{gathered}
 h_K(\tau^jp_j)=\\
 \tau^{j-1}\int_0^1s^k(d_wp_j)(t,x,sz)\,ds.
\end{gathered}
\tag{EE.7}
\]
Formula EE.7 acts in exterior degree \(k\). The derivative \(d_w\) is taken in the coefficient's argument before
evaluation at \(sz\). A Taylor monomial of positive total
\(\ell+k\) is multiplied by \(1/(\ell+k)\) before the exterior
derivative; the constant scalar term has derivative zero. Consequently
EE.6 proves
\[
 d_Kh_K+h_Kd_K=I-\epsilon_0,
 \tag{EE.8}
\]
where \(\epsilon_0\) is evaluation at \(z=0\) in degree zero and
is zero in higher degrees.

All maps just used are defined on actual coefficient germs. Choose fixed
nested polydiscs, the inner one centred at \(z=0\), and reserve one
outer Cauchy margin \(\delta>0\). The whole radial segment \(sz\)
stays in the inner disc. If a coefficient has outer norm \(M_j\),
each derivative in EE.7 has norm at most \(\delta^{-1}M_j\), and
the integral and the finitely many exterior components add a constant
depending only on \(d,\delta\). The single shift \(j\mapsto j-1\)
preserves an upper finite order and the bound \(MC_0^n n!\) for the
negative coefficients: \((n-1)!\le n!\), with the finitely many
indices crossing zero absorbed in the prefactor. Thus \(h_K\) belongs
to the actual finite-order class on one fixed smaller domain. Formula
EE.8 is an identity of those actual symbols. It proves
\[
\begin{gathered}
 H_k(K)=0\quad(k>0),\\
 H_0(K)=E^q/\sum_iE^qD_{x_i}.
\end{gathered}
\tag{EE.9}
\]
It also supplies the unique representative in the latter quotient which
is independent of \(z\), as in [§5.52, AM.A](#am-a-a-complete-coefficient-normal-form)'s spatial division.

Right multiplication by \(\theta I-A\) commutes with EE.5. It
therefore gives a chain endomorphism \(f:K\to K\). Its induced map
on EE.9 is injective. Indeed write a nonzero representative
\(Q=\sum_{j\le m}q_j(t,x)\tau^j\), with \(q_m\ne0\).
Normal composition gives
\[
\begin{gathered}
 Q(\theta I-A)=\\
 \sum_{j\le m}tq_j\tau^{j+1}\\
 {}+\sum_{j\le m}(jq_j-q_jA)\tau^j.
\end{gathered}
\tag{EE.10}
\]
The top coefficient is \(tq_m\). Multiplication by \(t\) is
injective on each holomorphic coefficient germ: if \(tq_m=0\), it
vanishes on the nonempty open \(t\ne0\), hence everywhere by its
power series. This contradicts \(q_m\ne0\). A nonzero sequence
bounded above always has a largest nonzero index, so the argument
includes the entire infinite negative tail.

Form the homological cone
\[
\begin{gathered}
 F_k=K_k\oplus K_{k-1},\\
 d_F(a,b)=\\(d_Ka+f(b),-d_Kb).
\end{gathered}
\tag{EE.11}
\]
Use \(K_{-1}=0\), and augment \(F_0\) onto EE.2. The differential
squares to zero by the just-proved commutation. For completeness, the
short exact sequence of complexes
\(0\to K_\bullet\to F_\bullet\to K_{\bullet-1}\to0\), where the
last differential is \(-d_K\),
can be read without a long-exact-sequence citation: a cycle \((a,b)\)
has \(d_Kb=0\); subtracting a boundary kills \(b\) whenever its
positive-degree homology is zero, and the remaining \(a\) is then a
\(K\)-cycle. At degree one the possible remaining class of \(b\)
is killed by \(f\) in \(H_0(K)\), hence is zero by EE.10.
At degree zero the quotient is exactly the cokernel of \(f\) on
\(H_0(K)\), which is EE.2. Thus \(F\) is a finite free resolution
of \(N_A\), of length at most \(d+1\), with
\[
 \operatorname{rank}_E F_k=q\binom{d+1}{k}.
 \tag{EE.12}
\]
The binomial equality follows by grouping the subsets of
\(\{0,1,\ldots,d\}\) according to whether they contain zero.
The case \(d=0\) is the injective two-term presentation EE.10, and
the case \(q=0\) is the zero complex.

#### EE.2. Halfspace solutions have no positive obstruction degree

Apply \(\operatorname{Hom}_E(-,S)\) to EE.11. This is the cochain
Koszul complex on the commuting operators acting on columns in \(S^q\):
\[
\theta I-A,\ D_{x_1}I,\ldots,D_{x_d}I.
\tag{EE.13}
\]
The transpose is not inserted: a relation row \(P\) applied to a
column of images is \(Pv\). Reordering the first exterior generator
only changes the corresponding Koszul signs consistently.

The tangential cochain complex has an actual contraction on EE.3. For
a holomorphic tangential \(k\)-form \(\alpha\), \(k\ge1\), use
\(\iota_x\alpha(t,y)=\sum_i x_i\iota_{\partial_{y_i}}\alpha(t,y)\),
where the contraction vector is the fixed output \(x\). Set
\[
\begin{gathered}
 (H_x\alpha)(t,x)=\\
 \int_0^1s^{k-1}(\iota_x\alpha)(t,sx)\,ds.
\end{gathered}
\tag{EE.14}
\]
For degree zero put \(H_x=0\). Both the full polydisc \(D\) and
its left half \(U_D\) are preserved by \((t,x)\mapsto(t,sx)\).
Differentiation under the integral on compact subproducts is justified
by uniform Cauchy bounds. Differentiating the pullback along this radial
path, integrating its derivative from zero to one, and using the
exterior contraction identity proves
\[
 d_xH_x+H_xd_x=I-\pi_x,
 \tag{EE.15}
\]
where \(\pi_x\) evaluates a degree-zero function at \(x=0\) and
includes it as a function independent of \(x\); in positive degree it
is zero. Equivalently, for a coefficient monomial of \(x\)-degree
\(\ell\) and form degree \(k\), the radial derivative is
\((\ell+k)s^{\ell+k-1}\); its integral is one except for the
constant scalar term. This also proves the identity directly from
convergent Taylor series in \(x\), with \(t\) as parameter.
EE.14 sends forms holomorphic on \(D\) to forms holomorphic on
\(D\), so it descends to the quotient in EE.3 and then to its
cofinal germ. It commutes with \(\theta I-A\), because neither the
radial path nor its scalar factors depend on \(t\), and \(A\) is
constant. Explicitly write a total cochain as
\(\alpha+e_0\wedge\beta\), where \(e_0\) belongs to
\(T=\theta I-A\). Its differential is
\(d_x\alpha+e_0\wedge(T\alpha-d_x\beta)\).
Extend the homotopy by
\(\widetilde H(\alpha+e_0\wedge\beta)
=H_x\alpha-e_0\wedge H_x\beta\).
The two mixed terms cancel because \(TH_x=H_xT\), leaving EE.15
on each component. Projection and inclusion are evaluation at \(x=0\)
and extension as a function independent of \(x\), on both components.
Thus the entire complex EE.13 contracts onto
\[
\begin{gathered}
 S_0^q\xrightarrow{\theta I-A}S_0^q,\\
 S_0=\underset{r\downarrow0}{\mathop{\rm colim}}\,
       \mathcal O(V_r)/\mathcal O(\Delta_r),\\
 V_r=\Delta_r\cap\{\Re t<0\}.
\end{gathered}
\tag{EE.16}
\]
The first term has degree zero.

The last arrow is surjective. Given any representative
\(g\in\mathcal O(V_r)^q\), fix \(a\in V_r\), choose its
holomorphic logarithm branch, and put
\[
 v(t)=t^A\int_a^t s^{-A}g(s)\,\frac{ds}{s}.
 \tag{EE.17}
\]
The integral may be taken along the straight segment, which stays in
the convex domain \(V_r\). Its integrand is holomorphic there. If
\(h\) denotes that integrand, differentiation of
\(\int_0^1(t-a)h(a+u(t-a))\,du\) gives
\(\int_0^1\partial_u[u h(a+u(t-a))]du=h(t)\).
Thus the integral is a holomorphic primitive, with no continuation
assumption. The finite matrix exponential and its derivative give
\((\theta I-A)v=g\). Passing to its class proves the claimed
surjectivity; no bound near the excluded point zero is required for
a holomorphic representative of \(S_0\).

Consequently the resolution EE.11 proves
\[
\begin{gathered}
 \operatorname{Ext}_E^j(N_A,S)=0\\ (j>0).
\end{gathered}
\tag{EE.18}
\]
Here Ext is computed by free resolutions. Its independence from the
chosen resolution follows by the elementary lifting construction:
lift a map in degree zero through a free surjection, lift each next
map because the preceding difference is a cycle, and use exactness to
lift it to a boundary. The same induction constructs a homotopy
between any two lifts of the same augmented map. Applying Hom turns
that homotopy into a cochain homotopy. Thus EE.18 is an assertion about
the module, rather than only about the displayed complex.

We also compute the degree-zero space, without a classification
theorem. On \(W=\mathcal O(V_r)^q\), EE.17 is surjective and its
kernel is \(t^A\mathbf C^q\): differentiating \(t^{-A}v\)
makes it constant, and a holomorphic function with zero derivative
is constant on a convex connected domain. On
\(V=\mathcal O(\Delta_r)^q\), Taylor expansion identifies
\(\theta I-A\) with the finite matrix \(nI-A\) at coefficient
\(t^n\), \(n\ge0\). For \(n>2\|A\|\), its inverse is the
convergent geometric series
\(n^{-1}\sum_{k\ge0}(A/n)^k\), of norm at most \(2/n\).
The finitely many other invertible coefficients have bounded
inverses. Thus coefficientwise solutions converge on every smaller
disc. Only the finitely many nonnegative integral eigenvalues can
give obstructions. Write \(T_V=\theta I-A:V\to V\). The actual
convergent coefficient solutions prove
\[
\begin{gathered}
 \ker T_V\\=\bigoplus_{n\ge0}t^n\ker(nI-A),\\
 \operatorname{coker}T_V\\
 \simeq\bigoplus_{n\ge0}\operatorname{coker}(nI-A).
\end{gathered}
\tag{EE.19}
\]
Only the noninvertible summands in these finite-dimensional descriptions
are retained. The two dimensions are equal, say \(k\), by the finite
matrix rank formula. A class \([v]\in W/V\) killed by
\(\theta I-A\) has a well-defined obstruction
\([(\theta I-A)v]\) in the second space in EE.19. Changing its
representative adds an image from \(V\). This gives the exact sequence
\[
\begin{gathered}
 0\to\ker_V\to\ker_W\\
 \to\ker_{W/V}\\
 \to\operatorname{coker}_V\to0.
\end{gathered}
\tag{EE.20}
\]
Exactness at the third term follows by subtracting a preimage in
\(V\) when the obstruction vanishes; surjectivity at the last term
follows from EE.17. Therefore
\(\dim\ker_{W/V}=(q-k)+k=q\) on each fixed disc.

The columns of \(F_A\) of §5.53 are a basis for this space. Their
equations and the complete constant-matrix Jordan reduction are
proved in [§5.53, AM.C–AM.D](#5-53-real-kernel-comparison-for-all-constant-euler-matrices). For a Jordan block those columns form an
upper triangular matrix with diagonal entry \(f_{\lambda,0}\).
This entry has a nonzero class: a nonintegral power has nontrivial
function monodromy; a nonnegative integral power times \(\log t\)
has nonzero function jump; a negative integral power has a pole.
Each of these assertions, with its normalization, is proved in
[§5.53, AM.C](#5-53-real-kernel-comparison-for-all-constant-euler-matrices). If \(F_Ac\) is holomorphic across zero, its bottom
component first gives \(c_q=0\), and the preceding components in
turn give \(c_{q-1}=\cdots=c_1=0\). Apply this to each block and
the invertible constant change of basis. There are \(q\)
independent columns and the dimension is \(q\), so they span on
every such fixed disc. Use the common branch
\(\pi/2<\arg t<3\pi/2\) on all left half-discs. Restriction
carries these same columns to themselves. It is therefore an
isomorphism on their solution spaces. Since a finite set of relations
at the colimit germ vanishes on one common smaller disc, its solution
space is the colimit of these fixed-disc kernels. We have proved
\[
 \dim_{\mathbf C}\operatorname{Hom}_E(N_A,S)=q.
 \tag{EE.21}
\]

#### EE.3. Full infinite-order model modules and their finite extensions

Put \(T_A=B\otimes_E N_A\). Tensoring EE.11 with \(B\) remains
exact by the complete right-flatness theorem of [§5.47, MF.1–MF.7](#5-47-arbitrary-finite-matrices-and-right-flatness-of-the-full-infinite-order-ring);
its terms are finite free left \(B\)-modules. The elementary
adjunction sends a \(B\)-linear map to its values on
\(1\otimes N_A\), with inverse \(b\otimes m\mapsto b f(m)\).
It identifies the resulting Hom complex with EE.13, including all
differentials. Hence
\[
\begin{gathered}
 \operatorname{Ext}_B^j(T_A,S)=0\quad(j>0),\\
 \dim_{\mathbf C}\operatorname{Hom}_B(T_A,S)=q.
\end{gathered}
\tag{EE.22}
\]

Let \(\mathcal F\) be the following explicit class of left
\(B\)-modules. An object \(T\) is equipped with a finite filtration
\[
\begin{gathered}
 0=T_0\subset T_1\subset\cdots\\
 \subset T_m=T,\\
 T_a/T_{a-1}\simeq T_{A_a}.
\end{gathered}
\tag{EE.23}
\]
The length \(m\) and all block sizes \(q_a\) are arbitrary finite
integers; the extension classes are arbitrary. No splitting and no
descent of the extensions to \(E\) are required. Write
\(r(T)=\sum_aq_a\) for the rank attached to this supplied
filtration. We will in fact prove that it equals an intrinsic Hom
dimension, so it does not depend on the chosen filtration.

Here are the resolution details for arbitrary extensions over the
possibly non-Noetherian ring \(B\). For
\(0\to L\to M\to N\to0\), with finite free resolutions of
\(L,N\) of length at most \(d+1\), lift a basis of the degree-zero
free module for \(N\) to \(M\), and use the free module for \(L\)
together with these lifts to surject onto \(M\). Its kernel maps
onto the first kernel for \(N\), with kernel the first kernel for
\(L\): lift a relation for \(N\) to the direct sum, correct its
image in \(M\) by the free \(L\) part, and read the kernel from
the injection \(L\to M\). Repeat this construction on those
three kernels. Induction gives a free resolution of \(M\), with
degreewise split short exact sequence of resolutions for \(L,M,N\).
After the degree-\((d+1)\) free terms have been constructed, both
endpoint kernels are zero, so the middle kernel is zero too. The
middle terms are the finite direct sums of the
endpoint free terms. Thus every object of EE.23 has a finite free
resolution of the same length bound; no assertion about arbitrary
\(B\)-modules or Noetherianity has entered.

Apply Hom into \(S\) to that degreewise split sequence. This gives
a short exact sequence of cochain complexes in the reverse order.
Its connecting maps are obtained by lifting a cocycle to the middle
complex, taking its differential and reading it in the first
complex. The differential squared is zero, so this is a cocycle;
changing either lift changes it by a boundary. A class is zero
exactly when its lift can be corrected to a cocycle. This proves
the exact cohomology sequence directly. Induction from EE.22 yields
\[
\begin{gathered}
 \operatorname{Ext}_B^j(T,S)=0\quad(j>0),\\
 \dim_{\mathbf C}\operatorname{Hom}_B(T,S)=r(T)\\
 (T\in\mathcal F).
\end{gathered}
\tag{EE.24}
\]
In particular restriction of functionals is onto in every extension
whose quotient is in \(\mathcal F\).

The latter surjectivity has an explicit relation interpretation. For a
functional \(u:L\to S\), lift the degree-zero free module of a
resolution of \(N\) to \(M\). Its relation map has image in
\(L\); composing with \(u\) gives a degree-one cocycle. By
\(\operatorname{Ext}_B^1(N,S)=0\), it equals a coboundary from a
map on that degree-zero free module. More explicitly, let
\(z:F_0\to M\) be the chosen lift and
\(r:F_1\to L\) satisfy \(i r=z d_1\). Solve
\(\psi d_1=u r\), using the zero obstruction group.
The map \(L\oplus F_0\to M\), \((l,a)\mapsto i(l)+z(a)\),
has kernel consisting of \((-r(b),d_1b)\). The functional
\((l,a)\mapsto u(l)+\psi(a)\) kills those pairs and therefore
descends to the required \(M\to S\). All maps here are maps of
the actual full modules; no formal matrix limit or representativewise
scalar extension is used.

#### EE.4. Separation and real-kernel comparison on the whole extension class

**Theorem EE.1.** For every \(T\in\mathcal F\), evaluation is
injective. Put \(\mathcal H_T=\operatorname{Hom}_B(T,S)\); then
\[
\begin{gathered}
 \operatorname{ev}_T:T\\
 \longrightarrow\operatorname{Hom}_{\mathbf C}(\mathcal H_T,S),\\
 v\longmapsto(f\mapsto f(v)).
\end{gathered}
\tag{EE.25}
\]
Choose any complex basis of its \(r(T)\)-dimensional functional
space to identify the target with \(S^{r(T)}\). Then EE.25 is a
\(B\)-linear injection into that actual full \(C\)-module. Moreover
the natural comparison is injective:
\[
 T\longrightarrow C\otimes_BT.
 \tag{EE.26}
\]

**Proof.** For \(T_A\), EE.25 is the already proved full coefficient
detector EE.4, because EE.21–EE.22 identify its columns with a basis
of all functionals. Suppose
\(0\to L\xrightarrow{i}T\xrightarrow{\pi}N\to0\) is a step of
EE.23, and suppose the theorem's separation assertion is true for
\(L,N\). Let \(v\in T\) be killed by every functional to \(S\).
For each \(g:N\to S\), the functional \(g\pi\) kills \(v\),
so separation for \(N\) gives \(\pi v=0\). Write \(v=i(w)\).
Each functional \(u:L\to S\) extends to \(T\) by EE.24, hence
\(u(w)=0\). Separation for \(L\) gives \(w=0\), hence \(v=0\).
This proves EE.25 by induction for the entire arbitrary finite
filtration, including non-split extensions.

Every functional \(f:T\to S\) extends along scalar change to the
well-defined \(C\)-linear map
\[
\begin{gathered}
 C\otimes_BT\longrightarrow S,\\
 c\otimes v\longmapsto cf(v).
\end{gathered}
\tag{EE.27}
\]
Indeed \((cb)f(v)=c f(bv)\), which is exactly the balanced-tensor
relation. If \(1\otimes v=0\), EE.27 makes \(f(v)=0\) for
every functional. EE.25 gives \(v=0\). This proves EE.26. No
left-exactness of \(C\otimes_B-\) is being used. \(\square\)

**Corollary EE.2.** Let \(M\) be any left \(E\)-module for which
there is an actual \(B\)-linear injection
\(j:B\otimes_EM\hookrightarrow T\), with \(T\in\mathcal F\).
Then the following natural comparison is injective:
\[
B\otimes_EM\longrightarrow C\otimes_EM.
\tag{EE.28}
\]
The maps \(f\circ j\), for a basis of
\(\operatorname{Hom}_B(T,S)\), already separate the entire source.

**Proof.** EE.25 makes their joint evaluation injective on the
embedded module. Alternatively use the balanced isomorphism
\(C\otimes_B(B\otimes_EM)\simeq C\otimes_EM\),
\(c\otimes(b\otimes m)\mapsto cb\otimes m\), with inverse
\(c\otimes m\mapsto c\otimes(1\otimes m)\). The natural square
for \(j\) sends any element in the kernel of EE.28 to the kernel
of EE.26 for \(T\), hence to zero in \(T\); the injection \(j\)
then makes the original element zero. This is exactly the explicit
embedding reduction of [§5.52, AM.R](#am-r-injectivity-can-be-detected-through-an-embedding). \(\square\)

For example, an entire finite \(E\)-module filtration with quotients
\(N_{A_a}\) supplies EE.23 after tensoring with \(B\), by its
proved right flatness. The theorem also covers full \(B\)-extensions
which are not asserted to descend in this way. Neither assertion
claims membership for arbitrary holonomic germs.

#### EE.5. Applying the embedding criterion

For an application of Corollary EE.2, construct an actual embedding
of the entire full-coefficient module into a module with the finite
filtration EE.23. This is a hypothesis of the corollary. If coordinates
are changed, their finite-order, infinite-order and real-kernel actions
must be compatible with that embedding and with the comparison map.
The extension theorem alone does not supply such embeddings or contact
transport for arbitrary holonomic germs.

Once generic injectivity has been established on every required smooth
characteristic branch, [§5.52, AM.S](#am-s-extending-generic-injectivity-across-the-analytic-exceptional-set)
extends it across an analytic exceptional set of the stated
codimension. Its full support-purity hypothesis and the generic
injectivity hypothesis are both required.

![Exact resolutions, solution obstructions and extension separation](assets/euler-extension-separation.png)

*Figure EE.1. The left panel records the actual finite free resolution
EE.11–EE.12 and its Hom contraction EE.14–EE.18. The right panel
shows the exact functional sequence of EE.24 and the two deductions
in EE.25: \(\pi v=0\), then \(u(w)=0\) for every functional on
\(L\). All arrows have the displayed domains and codomains. The
bottom comparison is EE.26–EE.27 and uses the whole left
\(C\)-module \(S\) of EE.3. This is an exact diagram of proved
maps, with no sampled geometry or assertion of arbitrary holonomic
classification. Editable figure source,
exact diagram data,
full PNG and
SVG accompany the figure. Further free human
source context: [Kashiwara–Kawai, Holonomic systems III](https://www.kurims.kyoto-u.ac.jp/~kenkyubu/kashiwara/HolIII.pdf),
I.3; the proofs of the displayed maps are EE.1–EE.4 above.*


### 5.55. Actual Gevrey realization and Euler-stable lattices

On a smooth characteristic branch, restriction to the commutative ring of
actual factorial-growth coefficient symbols gives a finite module and,
on a dense analytic open, a free module with explicit flat connection
matrices. A coherent lattice stable under the Euler operator and tangential
derivatives gives an actual finite-order constant Euler normal form at
every point of a standard branch. All integer resonances and nilpotent
blocks are included.

#### DP.0. Actual rings and the precise input

Use the nonzero-covector coordinates of programme [§5.19](#5-19-actual-factorial-growth-symbols-and-ordered-jets):
\(u=(t,x_1,\ldots,x_d)\), \(z=\xi/\tau\), \(h=\tau^{-1}\).
Let \(E_0\) be the actual finite-order ring of order at most zero,
and \(E=E_0[h^{-1}]\) its proved normal localization. In particular
\[
 E_0/(h)=\mathbf C\{t,x,z\}.
 \tag{DP.1}
\]
Put \(v=(t,z_1,\ldots,z_d)\). The parameter ring used in this proof is
\[
 \begin{gathered}
 R_0=\\
 \left\{\begin{gathered}
  \sum_{n\ge0}a_n(x)h^n\\
  \text{with }|a_n|_K\le MC^n n!
 \end{gathered}\right\},\\
 R=R_0[h^{-1}].
 \end{gathered}
 \tag{DP.2}
\]
The coefficients have one common holomorphic \(x\)-domain; the bounds
hold on each smaller compact. These are actual symbol germs independent
of both \(t\) and \(z\). Their Leibniz product is ordinary coefficient
convolution: all \(t\)-derivatives of the right factor and all spatial
cotangent derivatives of the left factor vanish. Thus \(R_0\) and
\(R\) are commutative subrings. They are not rings of arbitrary formal
series. Their Gevrey series need not converge at a numerical \(h\ne0\).

Let \(M_0\) be a finite left \(E_0\)-module, and suppose
\[
 \begin{gathered}
 L=M_0/hM_0\\
 \text{is finite over }\mathbf C\{x\}.
 \end{gathered}
 \tag{DP.3}
\]
The image \(hM_0\) is a left submodule because \(h\) is normal.
No injectivity of \(h\) is needed until DP.4. For the smooth-support
application, [§5.32](#5-32-coherent-actual-lattices-and-principal-support-for-the-whole-germ), J.1–J.2 constructs an actual coherent lattice
\(M_0\subset M\), with \(EM_0=M\), for every coherent holonomic
\(E\)-germ. It proves equality of its reduced principal support with
the actual support. On a chart with support contained in
\(\Lambda=\{t=z=0\}\), DP.3 follows by the explicit analytic argument
in DP.3 below, including all principal nilpotents.

The full analytic norms, weighted coefficient completeness, support
projections and monomial division are [§5.21.1](#5-21-1-an-absolute-taylor-version-of-the-product-norm), equations 5.21a–5.21g.
Write \(\|\cdot\|_{P,\epsilon}\) for that norm at fixed polyradii
\(P\), with \(\lambda=2(d+1)\). We use its proved inequalities
\[
 \begin{gathered}
 \|A\circ B\|\le\|A\|\|B\|,\\
 \|AB\|\le\tfrac12\|A\|\|B\|,\\
 \|\partial_\rho^\alpha A\|
       \le(\epsilon/\lambda)^{|\alpha|}\|A\|.
 \end{gathered}
 \tag{DP.4}
\]
Here \(AB\) denotes ordinary multiplication of normal symbols, not
operator composition. Each derivative in the last line raises the
negative homogeneous index by one. All three inequalities retain the
actual \(\tau\)-derivatives. Division by a fixed normal-variable
monomial \(v^\gamma\), on its divisible coefficient support, has norm
at most \(P_v^{-\gamma}\). The head selection, hierarchical radii and
component weights are the complete Dickson and finite-head arguments
of [§5.21.2](#5-21-2-initial-exponents-and-the-finite-diagram)–[§5.21.3](#5-21-3-one-domain-and-radii-for-all-the-divisors) and [§5.29](#5-29-actual-finite-generation-over-the-coordinate-commuting-ring), G.2–G.3. The mixed time-head
correction needed for this restriction of scalars is proved next.

#### DP.1. Mixed time/cotangent heads with their full operator correction

Choose finite generators \(m_1,\ldots,m_b\) of \(M_0\) and let
\(I\subset E_0^b\) be their actual relation module. The principal
image of \(I\) is the kernel of
\(\mathbf C\{x,v\}^b\to L\). Indeed, if a vector maps into
\(hM_0\), lift its quotient by \(h\) to \(E_0^b\) and subtract
\(h\) times that lift. This is an actual relation with the same
principal vector. Specializing at \(x=0\) therefore gives a module
\(J\subset\mathbf C\{v\}^b\) with
\[
 \mathbf C\{v\}^b/J\simeq L/(x)L.
 \tag{DP.5}
\]
This vector space is finite dimensional by DP.3.

Order a head by least total \(v\)-degree, then least lexicographic
exponent, then component. The head diagram of \(J\) is upward
closed in each component. The complete finite-head proof cited in
DP.0 supplies finitely many minimal heads \((\gamma_i,j_i)\).
Its complement
\[
 \begin{gathered}
 \mathcal B=\{(\gamma,j):\\
 \gamma\notin\bigcup_{i:j_i=j}(\gamma_i+\mathbf N^{d+1})\}.
 \end{gathered}
 \tag{DP.6}
\]
is finite. To verify the required finite complement directly, each
coordinate \(v_l\) acts nilpotently on the finite-dimensional local
quotient in DP.5: for \(c\ne0\), \(v_l-c\) is an invertible
holomorphic germ, so its matrix has no nonzero eigenvalue. The
characteristic polynomial and its cofactor identity give
\(v_l^a=0\) on that quotient for \(a\) its dimension. A power of
the whole maximal ideal \((v)\) consequently kills it. Every
sufficiently large monomial vector belongs to \(J\), and hence to
its head diagram. The zero quotient case has the same conclusion
with an empty complement.

Lift each normalized selected head to an actual relation
\[
 \begin{gathered}
 P_i=v^{\gamma_i}e_{j_i}+A_i\in I,\\
 \gamma_i=(a_i,\beta_i).
 \end{gathered}
 \tag{DP.7}
\]
The lift exists by the principal-kernel assertion before DP.5 and by
lifting the finitely many specialized scalar coefficients as symbols
independent of \(x,h\). Its specialization at \(x=0\) has no term
before its selected head. The complete hierarchical-weight argument
of [§5.29](#5-29-actual-finite-generation-over-the-coordinate-commuting-ring), G.3, now applied to \(v=(t,z)\) and parameters \(x\),
has the following precise consequences. First choose hierarchical
positive normal-variable polyradii so that the other terms at
\(x=0,h=0\) are small relative to the head. Choose positive component
weights \(c_j\) to control later components at the same exponent.
Next shrink the \(x\)-radii to control all additional principal
terms, which vanish at \(x=0\). Finally shrink \(\epsilon\) to
control all positive-index coefficients and positive derivative
weights. Finite coefficient domination on an initially larger
polydisc proves these limits, as in G.3; no cancellation of norms
is assumed. The choices can be made so that
\[
 \begin{gathered}
 \|A_i\|_c<\tfrac18c_{j_i}P_v^{\gamma_i},\\
 \|F\|_c=\sum_jc_j\|F_j\|.
 \end{gathered}
 \tag{DP.8}
\]

Unlike a pure \(z\)-head, a time head has an operator correction.
The entire correction is finite and explicit:
\[
 \begin{aligned}
 &Q\circ(t^az^\beta)-Qt^az^\beta\\
 &\quad=\sum_{r=1}^a\binom ar
           (\partial_\tau^rQ)t^{a-r}z^\beta.
 \end{aligned}
 \tag{DP.9}
\]
This is the actual Leibniz formula: the right head depends on the
base only through \(t^a\), so the only internal derivatives are
\(\partial_\tau^rQ\) paired with \(\partial_t^r(t^a)\).
In particular the sign of the \(\tau\)-derivative is already in
that derivative; it is not replaced by a derivative with respect to
\(h\). Applying both remaining inequalities of DP.4 gives
\[
 \begin{gathered}
 \begin{aligned}
 &\|Q\circ(t^az^\beta)-Qt^az^\beta\|\\
 &\qquad\le\eta_{a,\beta}(\epsilon)\|Q\|,
 \end{aligned}\\
 \begin{aligned}
 \eta_{a,\beta}(\epsilon)&=\tfrac12\sum_{r=1}^a\binom ar\\
 &\quad\cdot(\epsilon/\lambda)^r\|t^{a-r}z^\beta\|.
 \end{aligned}
 \end{gathered}
 \tag{DP.10}
\]
For fixed polyradii each of the finitely many monomial norms is
bounded as \(\epsilon\downarrow0\): it is finite at some
\(\epsilon_0\), and its nonnegative weighted terms decrease.
Thus \(\eta_{a,\beta}(\epsilon)\to0\). Shrink the single
\(\epsilon\) once more so that
\[
 \begin{gathered}
 \eta_{a_i,\beta_i}(\epsilon)<\tfrac18P_v^{\gamma_i}\\
 \text{for every selected head}.
 \end{gathered}
 \tag{DP.11}
\]
Shrinking it preserves DP.8. There is no time-head correction when
\(a_i=0\).

#### DP.2. Actual finite generation over the commutative Gevrey ring

Partition the head diagram into disjoint assigned normal-variable
supports as in G.4. Allow every \(x\)-Taylor index and every
nonnegative homogeneous index in all coefficient sequences. The
remainder has normal indices in \(\mathcal B\). Define the ordinary
head map and its actual perturbation by
\[
 \begin{gathered}
 L_0(Q,S)\\
  =\sum_iQ_iv^{\gamma_i}e_{j_i}+S,\\
 \mathcal L(Q,S)\\
 =\sum_iQ_i\circ P_i+S,\\
 \|(Q,S)\|_*\\
  =\sum_ic_{j_i}P_v^{\gamma_i}\|Q_i\|\\
 \quad+\|S\|_c.
 \end{gathered}
 \tag{DP.12}
\]
The complete weighted coefficient norm and monomial-division proof
of [§5.21.1](#5-21-1-an-absolute-taylor-version-of-the-product-norm) gives a Banach-space inverse \(D=L_0^{-1}\), of norm
at most one with these norms. All its support projections have norm
at most one, and division by a head contributes exactly its inverse
radius bound. This remains true for the time variable: its removal
decreases the base falling factorials in equation 5.21f. Removing
the \(z\)-part decreases its cotangent falling factorials and rising
factorial initial values. No restriction to pure cotangent heads is
used by that monomial-division inequality.

Write \(\mathcal E=\mathcal L-L_0\). For each selected relation
its error is the sum of \(Q_i\circ A_i\) and DP.9 in its selected
component. Equations DP.4, DP.8 and DP.11 give
\[
 \|\mathcal E(Q,S)\|_c\le\tfrac14\|(Q,S)\|_*.
 \tag{DP.13}
\]
Consequently, for every input of finite norm, the actual series
\[
 (Q,S)=\sum_{k\ge0}(-D\mathcal E)^kDF
 \tag{DP.14}
\]
converges in the stated Banach space, with norm at most
\(\tfrac43\|F\|_c\), and solves
\(F=\sum_iQ_i\circ P_i+S\). The zero-derivative estimate 5.21c
supplies one common holomorphic coefficient polydisc and one
factorial constant for the entire output. This is not merely an
\(h\)-adic construction.

For an arbitrary actual input germ, first intersect its coefficient
domain with the finite divisor domains. Repeat the same hierarchical
normal-radius choices inside that domain, then shrink parameter
radii and \(\epsilon\). The head list and \(\mathcal B\) remain
fixed. The input has finite norm at sufficiently small such choices
by the complete norm-detection proof of §5.21.1. Thus DP.14 solves
division for every germ, with no requirement of one norm for every
possible germ.

The remainder is a finite ordinary-symbol sum
\[
 \begin{gathered}
 S=\sum_{(a,\beta,j)\in\mathcal B}
                 s_{a,\beta,j}t^az^\beta e_j,\\
 s_{a,\beta,j}=s_{a,\beta,j}(x,h)\in R_0.
 \end{gathered}
 \tag{DP.15}
\]
The finitely many coefficient extractions are bounded Cauchy maps on
one fixed smaller normal-variable product; the full factorial
bound and common \(x\)-domain are retained. Ordinary multiplication
in DP.15 cannot silently be replaced by operator composition.
The exact inverse ordering identity is
\[
 \begin{gathered}
 \begin{aligned}
 st^az^\beta&=\sum_{r=0}^a(-1)^r\binom ar\\
 &\quad\cdot(\partial_\tau^rs)\circ(t^{a-r}z^\beta),
 \end{aligned}\\
 s\in R_0.
 \end{gathered}
 \tag{DP.16}
\]
To prove it, expand each actual composition by DP.9 with its zero
term included. The coefficient of
\((\partial_\tau^ks)t^{a-k}z^\beta\) is
\(\binom ak\sum_{r=0}^k(-1)^r\binom kr\), hence is one for
\(k=0\) and zero otherwise. The identity
\(\sum_{r=0}^k(-1)^r\binom kr=(1-1)^k\) follows by multiplying
the finite polynomial \((1+X)^k\). Every derivative in DP.16 is
in \(R_0\), by DP.4 or by its explicit homogeneous coefficients.
The complement in DP.6 is downward closed in each component, so
\((a-r,\beta,j)\) is still in it.

Since the \(P_i\) are actual relations, DP.14–DP.16 prove the
whole-module identity
\[
 M_0=\sum_{(a,\beta,j)\in\mathcal B}
                R_0(t^az^\beta)m_j.
 \tag{DP.17}
\]
Here \(t^az^\beta\) is the indicated actual operator acting on
\(m_j\). This proves finiteness over \(R_0\) under DP.3. If
\(M=EM_0\), normal localization and the actual finite shift
formulas also give
\[
 M=\sum_{(a,\beta,j)\in\mathcal B}
                  R(t^az^\beta)m_j.
 \tag{DP.18}
\]
Restriction of scalars keeps the entire underlying module and all
its sections.

#### DP.3. Smooth support supplies the finite parameter input

Suppose \(M\) is a coherent holonomic actual \(E\)-germ, and its
support on a finite contact chart is contained in
\(\Lambda=\{t=z=0\}\). The coherent, \(h\)-injective lattice of
[§5.32](#5-32-coherent-actual-lattices-and-principal-support-for-the-whole-germ) has finite principal module \(L\) over
\(\mathbf C\{x,v\}\), with support contained in that same set.
The complete analytic Nullstellensatz, Theorem 5.1, puts a power of each \(v_i\) in
\(\operatorname{Ann}L\). There is therefore an integer \(k\ge1\)
such that
\[
 (v)^kL=0.
 \tag{DP.19}
\]
For example if \(v_i^{b_i}L=0\), take
\(k=1+\sum_i(b_i-1)\); every monomial of total degree \(k\)
contains some such power. These are actual annihilations in the
finite principal module, not an assumption that its ideal is reduced.

The quotient \(\mathbf C\{x,v\}/(v)^k\) is finite free over
\(\mathbf C\{x\}\), with basis the normal monomials of degree
less than \(k\): expand a convergent holomorphic function in \(v\)
and retain those finitely many Taylor coefficients. The remainder
lies in \((v)^k\) by grouping its absolutely convergent Taylor
series on a smaller polydisc. Thus \(L\) is finite over
\(\mathbf C\{x\}\), proving DP.3 and hence DP.17–DP.18 for
every such entire holonomic germ.

For this smooth-support application a uniform head list can also be
chosen directly. For every normal monomial \(v^\gamma\) of degree
\(k\) and every original component, DP.19 lifts it to an actual
relation whose entire principal symbol is that monomial. Its error
has positive homogeneous index, so its norm tends to zero with
\(\epsilon\). Use all these finitely many heads. Their complement
is exactly the normal monomials of degree less than \(k\), in
every component, on the same parameter neighbourhood. The only
additional head corrections are precisely DP.9. This gives DP.17
with one uniform finite generating list, without dropping any
principal nilpotents or parameter-supported sections.

This is also a coherent finite parameter sheaf on one smaller
\(x\)-neighbourhood. Take a finite holomorphic presentation of
\(L\) on the product. Tensor it with
\(\mathbf C\{x,v\}/(v)^k\); right exactness and DP.19 recover
the same module. Expanding the resulting finite matrix in the
finite normal-monomial basis gives a finite holomorphic matrix over
\(\mathbf C\{x\}\). All its coefficients and annihilation
identities are represented on a common smaller neighbourhood.

That finite parameter sheaf is locally free off a proper analytic
set. Here is an explicit local proof. Over the fraction field of
the domain \(\mathbf C\{x\}\), let its presentation matrix have
rank \(s\), and choose a nonzero \(s\)-minor \(g\). All larger
minors vanish as germs and hence identically on a smaller connected
neighbourhood. On \(g\ne0\), invert that submatrix and perform
finite row and column elimination. The remaining block is zero:
each of its entries multiplied by the determinant is a larger
minor, up to sign. Its cokernel is therefore free, of rank the
number of rows minus \(s\). If \(s=0\), all entries are zero
and take \(g=1\). If the cokernel is zero, it is free of rank
zero. This constructs the exceptional set \(g=0\) analytically,
without an infinite-module support assertion.

#### DP.4. Actual free lifting and the exact connection matrices

Work at any point of the parameter open where \(L\) is free of
rank \(r\), and choose lifts \(u_1,\ldots,u_r\) of its basis to
\(M_0\). Assume \(h\) acts injectively on \(M_0\), as it does
for the actual lattice inside \(M\). These lifts form a basis over
\(R_0\), with an actual finite argument.

Indeed choose the finite \(R_0\)-generating list \(G\) of DP.17.
Its elements modulo \(h\) are combinations of the lifted basis,
with coefficients holomorphic in \(x\). Lift these coefficients
as elements of \(R_0\). The finite error list belongs to
\(hM_0\); write its quotients by \(h\) as finite combinations
of \(G\). Thus, with the lists taken as columns and finite matrices
of the indicated sizes,
\[
 G=Fu+hCG.
 \tag{DP.20}
\]
The matrix \(I-hC\) is invertible in \(M_b(R_0)\). In fact all
its error coefficients have positive homogeneous index. Their full
norm tends to zero with \(\epsilon\), by domination by one finite
norm at an earlier \(\epsilon_0\). Choose it less than one in the
sum-of-entry matrix norm. The series \(\sum_{n\ge0}(hC)^n\)
converges in that Banach algebra, and has coefficients still
independent of \(t,z\). The norm-detection bound supplies actual
factorial coefficients on one common domain. Multiplication of
finite partial sums proves that its limit is the two-sided inverse.
Equation DP.20 therefore makes the \(u_i\) generate \(M_0\)
over \(R_0\). No formal Nakayama inference is used.

If \(\sum_i a_i u_i=0\), reduction modulo \(h\) makes every
\(a_i\) divisible by \(h\) in \(R_0\), since the chosen
residue classes are a basis. Divide the relation by the injective
action of \(h\), and repeat. Each \(a_i\) lies in every
\(h^nR_0\), so every coefficient of its actual symbol is zero.
Thus all \(a_i=0\). Division by one \(h\) preserves the actual
class: its factorial estimate changes by the factor \(n+1\),
absorbed by increasing its exponential constant. We have proved
\[
 M_0\simeq R_0^r,\qquad M\simeq R^r.
 \tag{DP.21}
\]
The second identity follows because normal localization expresses
every element as a finite sum of \(h^{-n}m\), \(m\in M_0\).
It is an identity for the entire original finite-order module after
restriction of scalars.

To state the matrices without transposition ambiguity, use coefficient
columns: \(v\mapsto\sum_jv_ju_j\). Define \(T,Z_i\) by the
columns which express \(tu_j,z_iu_j\) in this basis. All entries
are in \(R_0\), because these operators preserve the lattice.
The exact commutators with \(a\in R_0\) are
\[
 \begin{gathered}
 {}[t,a]=h^2\partial_ha,\\
 [z_i,a]=h\partial_{x_i}a.
 \end{gathered}
 \tag{DP.22}
\]
The first follows from \([t,h]=h^2\) by the finite monomial rule
and coefficient equality of actual symbols; equivalently
\([t,a]=-\partial_\tau a\). The second follows from
\(z_i=hD_{x_i}\) and \([D_{x_i},a]=\partial_{x_i}a\).
Every derivative is in the actual class: coefficient derivatives in
\(h\) add fixed polynomial factors in the homogeneous index,
absorbed in an exponential constant, and \(x\)-derivatives use
one fixed Cauchy margin. Thus the actions in DP.21 are exactly
\[
 \begin{gathered}
 t=h^2\partial_h+T,\\
 z_i=h\partial_{x_i}+Z_i,\\
 D_t=h^{-1}I,\\
 D_{x_i}=\partial_{x_i}+h^{-1}Z_i.
 \end{gathered}
 \tag{DP.23}
\]

The connection on this commutative actual Gevrey coefficient module is
\[
 \begin{gathered}
 \nabla_h=\partial_h+h^{-2}T,\\
 \nabla_i=\partial_{x_i}+h^{-1}Z_i.
 \end{gathered}
 \tag{DP.24}
\]
It is flat, with the exact equations
\[
 \begin{gathered}
 \begin{aligned}
 &h^2\partial_hZ_i-h\partial_{x_i}T\\
 &\qquad+[T,Z_i]=hZ_i,
 \end{aligned}\\
 \begin{aligned}
 &h(\partial_{x_i}Z_j-\partial_{x_j}Z_i)\\
 &\qquad+[Z_i,Z_j]=0.
 \end{aligned}
 \end{gathered}
 \tag{DP.25}
\]
Indeed the actual operator relations are
\([t,z_i]=hz_i\) and \([z_i,z_j]=0\). Substitute DP.23 and
cancel derivative terms, including
\([h^2\partial_h,h\partial_{x_i}]=h^2\partial_{x_i}\).
This gives DP.25 with the displayed signs. Direct expansion of the
curvatures of DP.24 gives respectively \(h^{-3}\) times the first
equation moved to the left, and \(h^{-2}\) times the second.
Thus both curvatures vanish.

The leading matrices \(T_0,Z_{i,0}\) commute and are jointly
nilpotent. More precisely DP.19 gives
\[
 \begin{gathered}
 T_0^a Z_{1,0}^{\beta_1}\cdots Z_{d,0}^{\beta_d}=0\\
 \text{if }a+|\beta|\ge k.
 \end{gathered}
 \tag{DP.26}
\]
They are exactly multiplication by \(t,z_i\) on \(L\) in the
chosen basis. This identifies their signs and their full principal
nilpotent structure. The commutation also follows by reducing DP.25
modulo \(h\). No reduced-symbol assumption has been inserted.

#### DP.5. Actual finite-order straightening by a holomorphic time shift

Let \(\varphi(x)\) be holomorphic, with
\(\varphi(0)=0\), \(\partial_x\varphi(0)=0\), and put
\[
 \begin{gathered}
 t'=t-\varphi(x),\quad x'=x,\\
 \tau'=\tau,\quad z'=z+\partial_x\varphi(x).
 \end{gathered}
 \tag{DP.27}
\]
These coordinates preserve \(dt+z\,dx\) exactly. We prove the
actual finite-order symbol isomorphism which implements their chain
rule; a formula only on coordinate generators would not suffice.
For \(|\gamma|\ge2\), write
\(\varphi_\gamma=\partial_x^\gamma\varphi/\gamma!\).
If \(P=\sum_{n\ge0}\tau^{-n}p_n(t,x,z)\), define its transformed
normal coefficient by the following finite expression at each \(N\).
Here \(w_\varphi=(t-\varphi(x),x,z+\partial\varphi(x))\).
Let \(\mathcal J_{K,\ell}\) be the finite set of ordered lists
\(\Gamma=(\gamma_1,\ldots,\gamma_\ell)\) with
\(|\gamma_i|\ge2\) and \(\sum_i(|\gamma_i|-1)=K\).
Write \(d_\Gamma=\sum_i\gamma_i\) and
\(c_\Gamma=\prod_i\varphi_{\gamma_i}(x)\).
Abbreviate \(p_{n,\Gamma}=(\partial_z^{d_\Gamma}p_n)(w_\varphi)\).
Then
\[
 \begin{gathered}
 q_N=\sum_{n+K=N}T_{n,K},\\
 \begin{aligned}
 T_{n,K}&=\sum_{\ell=0}^K\frac1{\ell!}\\
 &\quad\cdot\sum_{\Gamma\in\mathcal J_{K,\ell}}c_\Gamma p_{n,\Gamma}.
 \end{aligned}
 \end{gathered}
 \tag{DP.28}
\]
For \(K=0\), the only list is the empty list with \(\ell=0\),
so its term is the ordinary substitution into \(p_N\). For
\(K>0\), all lists have \(1\le\ell\le K\) and total
derivative length \(K+\ell\le2K\). Thus no infinite sum has
been hidden inside a homogeneous coefficient.

Here is its actual estimate. Fix nested coefficient products such
that all the shifted arguments in DP.28 stay in the inner target
product, with one additional covariable Cauchy margin \(\delta>0\).
This is possible since the shifts vanish at the origin. Choose
\(\eta>0\) such that the Taylor polydisc \(x+y\),
\(|y_i|\le\eta\), stays in a fixed holomorphy domain, and let
\(A_\varphi\) bound \(|\varphi|\) there. Then
\[
 \begin{gathered}
 |\varphi_\gamma|\le A_\varphi\eta^{-|\gamma|},\\
 |\partial_z^\alpha p_n|\le MC^n n!\,\alpha!\delta^{-|\alpha|}.
 \end{gathered}
 \tag{DP.29}
\]
For a list of loss \(K\), its total length is \(K+\ell\) and
\((K+\ell)!/\ell!\le4^K K!\). The number of ordered lists,
summed over \(1\le\ell\le K\), is at most \((4d^2)^K\):
there are \(\binom{K-1}{\ell-1}\) positive compositions of
\(K\), and at most \(d^{K+\ell}\) multiindex choices for the
corresponding length list. Sum over \(\ell\), and use
\(K2^K\le4^K\). The same finite count was proved in [§5.28](#5-28-convergent-contact-quantization-and-whole-germ-generic-position), Q.5.
Consequently one may take
\[
 \begin{gathered}
 L=32d^2\max(1,A_\varphi)\\
 \quad\cdot\max(1,(\delta\eta)^{-1})^2,\\
 d\ge1.
 \end{gathered}
 \tag{DP.30}
\]
so all terms at loss \(K\) have norm at most
\(MC^n n!L^K K!\). When \(d=0\), the map is the identity and
take \(L=0\). Summing the finitely many \(n+K=N\) and using
\(n!K!\le N!\) gives
\[
 |q_N|\le M(C+L)^N N!.
 \tag{DP.31}
\]
The homogeneous output is therefore an actual order-zero symbol on
one fixed smaller domain. This estimate uses no numerical value
of \(h\) and no formal-to-convergent substitution.

To identify the formula and prove multiplication and its inverse,
first take a normal symbol polynomial in \(t,x,z,h\). It represents
a finite sum of ordinary differential operators with coefficients
polynomial in \(t\) and holomorphic in \(x\), followed on the
right by finite powers of \(D_t^{-1}\). First the chain rule on
holomorphic functions gives the isomorphism of the ordinary operator
rings. It preserves \(D_t\), and extends algebraically to these
finite inverse powers. Explicitly, \([D_t,t]=1\) and iterated
\(t\)-derivatives of a polynomial vanish, so commuting a finite
inverse power past any such coefficient uses a finite expansion;
normal ordered monomials give the actual embedding in \(E\).
The inverse chain rule extends in the same way. No choice of a
primitive on holomorphic functions is being made. Its coordinate values are
\[
 \begin{gathered}
 t'\longmapsto t-\varphi(x),\\
 x'_i\longmapsto x_i,\quad D_{t'}\longmapsto D_t,\\
 D_{x'_i}\longmapsto D_{x_i}+\varphi_i D_t.
 \end{gathered}
 \tag{DP.32}
\]
The exact normal-symbol formula is DP.28. To verify it directly,
apply an ordinary differential operator to
\(\exp(\tau(t'+\varphi(x'))+\xi\cdot x')\).
Its gradient at the output is \((\tau,\xi+\tau\partial\varphi)\).
After removing that linear part, the remaining factor is
\[
 \exp\!\left(\tau\sum_{|\gamma|\ge2}
                             \varphi_\gamma y^\gamma\right).
 \tag{DP.33}
\]
The coefficient of \(y^\alpha\) is the finite sum
\(\sum_{\ell,\,\sum\gamma_i=\alpha}
   \tau^\ell\prod_i\varphi_{\gamma_i}/\ell!\).
Taylor's differential-operator formula contributes
\(\partial_{\xi}^\alpha P\) times precisely that coefficient:
the \(\alpha!\) from the derivative of the exponential cancels
the Taylor denominator. Collecting order loss
\(|\alpha|-\ell\) proves DP.28, including its sign. Equality of
these polynomials in the plane-wave parameters proves equality of
the finite differential operators. The right inverse powers of
\(D_t\) commute with the pullback, so the formula holds for every
polynomial symbol just specified.

For arbitrary actual order-zero symbols, each output coefficient
uses finitely many input coefficients and finitely many derivatives.
Approximate those coefficient functions by their Taylor polynomials
on a larger product. Cauchy convergence gives uniform convergence
of every required derivative on a fixed smaller product, also at
the shifted arguments. The polynomial identities for products and
the two opposite time shifts therefore pass to each output
coefficient. DP.31 puts their transforms and compositions in the
actual class. Equality of all their holomorphic coefficient germs
is equality of actual symbols on their common connected domain.
Thus DP.28 preserves the actual product and has inverse the formula
for \(-\varphi\). It fixes \(h\). Every finite order is obtained
by a finite right shift by \(D_t^m\), which is unchanged, so the
same proof gives an actual filtered isomorphism on \(E\). Its
principal map is exactly DP.27, since only the empty list contributes
at loss zero. This also proves its lattice and support transport.

For completeness the linear base changes used in the initial
normalization have the same actual property. If
\(u'=Au+b\), the exact normal-symbol substitution is
\((u',\rho')=(Au+b,A^{-\mathsf T}\rho)\). The contraction
\(\sum_j\partial_{\rho_j}^{(1)}\partial_{u_j}^{(2)}\)
is unchanged by these inverse linear matrices, so the complete
Leibniz product is preserved. There are no higher coordinate
derivatives. On a common nonzero-covector chart, write
\(\tau'=\tau a(z)\), with \(a\) holomorphic and bounded away
from zero. An input coefficient of index \(n\) acquires the
factor \(a(z)^{-n}\), bounded by \(C_1^n\); the fixed coordinate
substitution keeps a smaller product in the original domain. Its
factorial bound is therefore preserved. The inverse has the same
bound. Finite order shifts only add a fixed factor. These are
actual finite-order coordinate isomorphisms, not just checks on
coordinate generators.


#### DP.6. Generic characteristic geometry for the whole module

At a smooth homogeneous Lagrangian germ the finite quadratic contact
shear of [§5.28](#5-28-convergent-contact-quantization-and-whole-germ-generic-position) can make the projection to the \(x\)-coordinates
nonsingular. Its incidence proof applies to the linear tangent cone:
the kernel of \((x,z)\mapsto x+Cz\) misses that tangent space,
so the derivative from its \(d\)-dimensional Legendrian slice to
\(\mathbf C^d\) is an isomorphism. Here is the full local inverse
argument. Parametrize the smooth slice by \(w\in\mathbf C^d\).
After an invertible linear change its projected coordinate is
\(F(w)=w+R(w)\), with \(R(0)=0\), \(DR(0)=0\).
Choose a closed polydisc of radius \(r\) on which the operator norm
of \(DR\) is at most \(1/4\). For \(\|x\|<r/2\), the map
\(w\mapsto x-R(w)\) stays in that polydisc and has contraction
factor \(1/4\). Its iterates from \(w=x\) have a uniformly
geometric difference bound, so converge to a unique fixed point.
Each iterate is holomorphic in \(x\); the common Cauchy formula
on smaller polydiscs passes to the uniform limit, making the limit
holomorphic. The same derivative bound gives
\(\|F(w)-F(w')\|\ge3\|w-w'\|/4\), so it is the local inverse.
For \(d=0\) this step is empty. The smooth slice is consequently
given as \(t=\varphi(x),z=\psi(x)\). The contact equation gives
\(d\varphi+\sum_i\psi_i dx_i=0\), hence
\(\psi_i=-\partial_i\varphi\). The ordinary base change
\(t'=t-\varphi(x)\), \(x'=x\), with
\(z'=z+\partial_x\varphi\), puts its support into
\(t'=z'=0\). Its finite-order quantization is the chain-rule map:
\(D_t=D_{t'}\),
\(D_{x_i}=D_{x'_i}-(\partial_i\varphi)D_{t'}\).
The preceding DP.5 proves that this chain rule extends to the whole actual
finite-order class, with its inverse and factorial estimates. The finite lattice and actual
principal support transport of [§5.28](#5-28-convergent-contact-quantization-and-whole-germ-generic-position), Q.7 and [§5.32](#5-32-coherent-actual-lattices-and-principal-support-for-the-whole-germ) apply.

Thus DP.17–DP.26 apply to the whole coherent holonomic module on
each smooth characteristic branch, on the dense parameter open of
DP.3. The exceptional parameter set is a proper analytic subset,
so has dimension at most \(d-1\) by the earlier analytic-germs
Theorem 5.4 and Proposition 5.5 at the exact link in DP.3. Adding
the nonzero homogeneous covector scale gives
dimension at most \(d=n-1\) in \(T^*X\), hence ambient codimension
at least \(n+1\). In dimension \(d=0\), there is no exceptional
parameter set. This is a precise generic reduction; the original
module and the obligation at its exceptional and singular loci are
retained.

DP.17–DP.26 describe an actual Gevrey-coefficient flat connection.
Joint nilpotence of the leading matrices alone does not supply infinite-order
regularization or a separating Euler-extension target. These estimates do
not assert ordinary convergence in the numerical variable \(h\).
Arbitrary holonomic Euler membership and general real comparison additionally
require regularization and full real contact transport. The particular full
\(B\)-shear is proved in [§5.49](#5-49-full-infinite-spath-division-and-balanced-reconstruction), ISR.5. The generic-to-singular theorem
[§5.52](#5-52-full-euler-matrix-division-and-comparison-reductions), AM.S requires both its full support-purity hypothesis and generic
injectivity on the stated dense analytic open.


![Actual mixed normal division and whole dual-parameter realization](assets/actual-dual-parameter-realization.png)

*Figure DP.1. The input is the actual coherent lattice inside the whole
module, with standard smooth support. The time-head example is DP.9
with \(Q=h^2\), \(a=2\), \(\beta=1\): its first two
cotangent derivatives are \(-2h^3\) and \(6h^4\), giving
exactly the three displayed normal-symbol terms. The norm is the
weighted norm of DP.12, so the inverse bound is
\(\|(Q,S)\|_*\le4\|F\|_c/3\). The finite generators are DP.17;
the generic free coefficient module, actions, curvatures and joint
leading nilpotence are DP.21–DP.26. No numerical convergence in
\(h\) is used. Editable plotting source,
exact data,
full PNG and
SVG accompany the
figure. Free human context: [Kashiwara–Kawai, Holonomic systems III](https://www.kurims.kyoto-u.ac.jp/~kenkyubu/kashiwara/HolIII.pdf),
I.3 and III.5; the entire mixed-head and coefficient realization
proof is DP.1–DP.6 above.*

#### DP.7. An Euler-stable lattice gives an actual finite Euler normal form

**Theorem DP.R.** Suppose the whole left \(E\)-module \(M\) has
a coherent finite \(E_0\)-lattice \(M_0\subset M\), with \(EM_0=M\),
such that
\[
 \begin{gathered}
 \theta M_0\subset M_0,\\
 D_{x_i}M_0\subset M_0,\quad\theta=tD_t.
 \end{gathered}
 \tag{DP.34}
\]
Then at every point of this standard chart, \(M\) is isomorphic
over the actual finite-order ring \(E\) to a constant Euler module
\(N_A\), for a finite constant complex matrix \(A\). The proof
includes every integer resonance and every nilpotent block. It
does not require a formal or infinite-order scalar extension.
Here \(N_A\) is the module of §[§5.52](#5-52-full-euler-matrix-division-and-comparison-reductions)–5.53: a generator column
\(e\) with \((\theta I-A)e=0\) and \(D_{x_i}e=0\).

Condition DP.34 is exactly stability under the usual principal-
vanishing generators at \(\Lambda=\{t=z=0\}\). To check this
without a generation citation, an order-one principal symbol
vanishing on \(\Lambda\) is \(\tau f(t,x,z)\), with
\(f(0,x,0)=0\). Its convergent Taylor factorization
has the form \(f=ta+\sum_i z_i b_i\), with holomorphic
coefficients. Their order-zero symbols are actual. Subtract
\(a\circ\theta+\sum_i b_i\circ D_{x_i}\) from the given
operator. Its principal symbol is zero, so its order is at most
zero. Thus the left \(E_0\)-span of
\(E_0,\theta,D_{x_i}\) is the entire set of such order-one
operators. Stability under DP.34 is consequently stability under
the subring they generate. The finite stable-lattice condition is
condition (i) of the freely accessible HolIII Definition 1.1.11,
specialized to this smooth \(\Lambda\). None of the external
equivalences or classification theorems accompanying that definition
is used in the proof below.

##### DP.7.1. Freeness of the whole residue module at every parameter point

The exact identities \(t=h(\theta+I)\), \(z_i=hD_{x_i}\)
make \(tM_0,z_iM_0\subset hM_0\). Thus
\(L=M_0/hM_0\) is a finite module over \(O=\mathbf C\{x\}\),
with its ordinary flat connection induced by \(D_{x_i}\). That
operator descends because it commutes with \(h\), and satisfies
the usual Leibniz rule with \(O\). Unlike DP.3, this situation has
no exceptional parameter locus: we prove freeness at every point.

Choose a finite presentation \(O^a\xrightarrow{P}O^b\to L\to0\),
with columns as coefficient vectors. Here is the precise finiteness
argument. The actual left Noetherian theorem [§5.21](#5-21-actual-simultaneous-division-and-finite-stalk-relations) applies to \(E_0\),
and \(O=E_0/(h,t,z)\): first reduce modulo the normal \(h\),
then set the commuting \(t,z\) to zero in DP.1. Every ascending
chain of ideals of \(O\) lifts to one of left ideals of \(E_0\),
so it stabilizes. Every submodule of \(O^b\) is finite, by induction
on \(b\): its projection to the last coordinate is a finitely
generated ideal; lift those finitely many generators, and add the
finite generators of the projection kernel supplied by the induction.
The case \(b=1\) is exactly the ideal assertion. Apply this to the
kernel of a finite generator map onto \(L\). This gives the displayed
finite presentation. Lift each \(D_{x_i}\)-action on the finite
generators of \(L\) to a matrix \(B_i\) on \(O^b\). Its
kernel is preserved by \(\partial_i+B_i\), so there is a finite
matrix \(C_i\) with
\[
 \partial_iP+B_iP=PC_i.
 \tag{DP.35}
\]
Indeed each column on the left is a relation, and the columns of
\(P\) generate all relations. Let \(J_k\) be the ideal generated
by the \(k\)-minors of \(P\). Differentiate a minor using
DP.35. In its \(PC_i\) terms, one column is replaced by a
holomorphic linear combination of columns; in its \(B_iP\)
terms, one row is replaced by a linear combination of rows.
Multilinearity of the determinant expresses each term as a
holomorphic combination of \(k\)-minors of \(P\). Repeated rows
or columns give zero. Hence every \(J_k\) is stable under every
\(\partial_i\).

Any such ideal in \(O\) is zero or the whole ring. If it contains
a nonzero germ \(f\), choose a lowest-degree nonzero Taylor
coefficient \(c_\alpha\). Repeated differentiation gives an
element with value \(\alpha!c_\alpha\ne0\) at the point, which
is a unit. The reciprocal is a holomorphic germ by the convergent
geometric series around that nonzero value. The ideal is therefore
the whole ring.

Let \(s\) be the rank of \(P\) over the fraction field of
\(O\). If \(s>0\), \(J_s\ne0\), and thus \(J_s=O\);
some \(s\)-minor is a unit. All \((s+1)\)-minors are zero.
Finite row and column elimination around that unit minor makes
\(P\) an identity block plus a zero block, by the same Schur-
complement computation given in DP.3. Its cokernel is free of
rank \(r=b-s\). For \(s=0\), every matrix entry is zero and
the conclusion is immediate. This proves freeness at the given
point; the unit minor represents it on a neighbourhood. The zero
module and \(d=0\) are included.

DP.1–DP.2 and DP.4 now give the actual basis
\(M_0\simeq R_0^r\), \(M\simeq R^r\) at every point in
question. Its matrices have
\(T=hA(h,x)\), \(Z_i=hB_i(h,x)\), since their columns lie in
\(hM_0\). In these coefficient columns the flat connection is
\[
 \begin{gathered}
 \nabla_h=\partial_h+h^{-1}A(h,x),\\
 \nabla_i=\partial_{x_i}+B_i(h,x),\\
 A,B_i\in M_r(R_0).
 \end{gathered}
 \tag{DP.36}
\]
If \(r=0\), the theorem is already proved. Assume henceforth
\(r>0\).

##### DP.7.2. A tangential horizontal frame with full Gevrey bounds

Here is a convergent construction of the required parameter gauge.
For \(\epsilon>0\), let \(\mathcal G_\epsilon\) be the Banach
algebra of matrix series \(a=\sum_{n\ge0}a_nh^n\), with norm
\[
 \|a\|_\epsilon=\sum_{n\ge0}\|a_n\|\epsilon^n/n!.
 \tag{DP.37}
\]
Use an induced complex matrix norm, so \(\|I\|=1\). This is
a complete weighted coefficient space. Its product is bounded by
the product of the norms, since \(k!l!\le(k+l)!\). A uniform
actual estimate \(\|a_n(x)\|\le MC^n n!\) gives a bounded
\(\mathcal G_\epsilon\)-valued holomorphic function of \(x\)
on a smaller parameter product when \(C\epsilon<1\).
The coefficient series and its Cauchy integrals converge uniformly
in that Banach norm, proving the asserted holomorphy. Conversely
a bound in DP.37 gives the actual coefficient bound
\(\|a_n\|\le\|a\|_\epsilon\epsilon^{-n}n!\).

Solve first \(\partial_1U_1=-B_1U_1\), with
\(U_1(0,x_2,\ldots)=I\), by successive integrals on the
straight \(x_1\)-segment. The \(k\)-fold integral has norm
at most \(M_1^k|x_1|^k/k!\), if \(M_1\) bounds \(B_1\)
in DP.37 on the product. The series converges uniformly in that
Banach algebra, is holomorphic in all parameters and differentiates
to the stated equation. The same construction of
\(\partial_1V_1=V_1B_1\), \(V_1(0)=I\), gives its inverse:
\(\partial_1(V_1U_1)=0\), hence \(V_1U_1=I\).
Also \(U_1V_1\) and \(I\) solve the same initial value equation
\(W'=-B_1W+WB_1\); their difference is zero by iterating its
integral equation, whose bound is
\((2M_1|x_1|)^k/k!\) times an initial uniform bound.
Thus the inverse is two-sided. This proves existence and uniqueness
in the actual coefficient class, without evaluation of \(h\).

In this frame the first tangential connection is \(\partial_1\).
Expanding the product rule shows that gauge transformation sends
each curvature to its conjugate by \(U_1\). Tangential flatness
therefore makes every remaining transformed \(B_j\) independent
of \(x_1\). Solve successively in \(x_2,\ldots,x_d\), with
the same argument on the remaining variables. There are finitely
many steps and only fixed product shrinkings. Their product gives
\(U\in\operatorname{GL}_r(R_0)\) with all tangential
connections equal to \(\partial_i\). Its inverse has the same
actual common-domain property.

Under this coefficient gauge the new residue matrix is
\(U^{-1}AU+hU^{-1}\partial_hU\). It belongs to \(M_r(R_0)\):
\(h\)-differentiation multiplies the shifted coefficients by a
fixed polynomial in their index, which preserves factorial growth
after increasing its exponential constant. If necessary take a
single smaller \(\epsilon\) after all finitely many tangential
steps. Mixed flatness with \(\partial_i\) makes this matrix
independent of \(x\), coefficient by coefficient. Rename it
\(A(h)\). We have reduced the entire module to
\(\partial_h+A(h)/h\), with actual constant Gevrey coefficients,
and with all tangential actions ordinary derivatives.

##### DP.7.3. Finite integer shifts normalize every resonant residue

Integer changes of basis in \(R\), rather than a nonresonance
assumption, put all eigenvalues of \(A(0)\) in one real strip of
width one. Write \(b=\max\Re\lambda\) over its finite spectrum.
If some eigenvalues have \(\Re\lambda\le b-1\), take their
whole generalized eigenspaces \(W\), and let \(V\) be the sum
of the other generalized eigenspaces. The complete complex Jordan
and primary-decomposition proof is [§5.53](#5-53-real-kernel-comparison-for-all-constant-euler-matrices), AM.C–AM.D. Use the
constant decomposition \(W\oplus V\), and change the coefficient
frame by \(S=\operatorname{diag}(hI_W,I_V)\). The transformed
matrix is
\[
 A'=S^{-1}AS+hS^{-1}S'.
 \tag{DP.38}
\]
Both off-diagonal blocks of \(A(0)\) are zero. Thus the one
block divided by \(h\) in DP.38 still has nonnegative index;
all coefficients retain their actual factorial bounds by one
fixed shift. The new residue is block upper triangular, with
diagonal blocks \(A(0)|_W+I_W\), \(A(0)|_V\). Its eigenvalues
are the selected eigenvalues plus one, and the others unchanged,
with full algebraic multiplicities. The maximum real part remains
\(b\). Repeat until every real part is in \((b-1,b]\).
There are only finitely many steps: each individual eigenvalue
needs at most finitely many unit increases, and at every step at
least one of those remaining increases is made. Coincident
eigenvalues and any resulting nilpotent blocks are retained in
the next full generalized eigenspaces.

Every \(S\) and \(S^{-1}\) is an actual finite matrix over
\(R\); no new scalar extension or formal module is used.
These changes are independent of \(x\), so the tangential
actions stay ordinary derivatives. Call the final matrix again
\(A(h)\), with residue \(H=A(0)\). The difference of two of
its eigenvalues is never a nonzero integer, since their real
parts differ by strictly less than one.

##### DP.7.4. The Frobenius gauge lies in the actual factorial class

Seek \(V(h)=I+\sum_{n\ge1}V_nh^n\) such that
\[
 hV'=VH-A(h)V.
 \tag{DP.39}
\]
Writing \(A(h)=H+\sum_{k\ge1}A_kh^k\), its exact recursion is
\[
 \begin{gathered}
 \mathcal C_n(V_n)=-\sum_{k=1}^n A_kV_{n-k},\\
 \begin{aligned}
 \mathcal C_n(X)&=nX+HX\\
 &\quad-XH.
 \end{aligned}
 \end{gathered}
 \tag{DP.40}
\]
Each \(\mathcal C_n\), \(n\ge1\), is invertible. On a Hom
space between two generalized eigenblocks of \(H\), its
non-scalar part is the sum of two commuting nilpotent operators;
their sufficiently high power is zero by the finite binomial
formula. Its scalar part is \(n+\lambda_i-\lambda_j\), which
is nonzero by DP.7.3. Finite geometric inversion therefore proves
invertibility on each block, and the primary decomposition covers
the whole matrix space. In an induced norm, for
\(n>2\|\operatorname{ad}H\|\), the ordinary geometric inverse
of \(I+\operatorname{ad}H/n\) has norm at most two. Enlarge
one constant to cover the finitely many remaining inverses. Thus
there is a finite \(K\ge1\) such that
\[
 \|\mathcal C_n^{-1}\|\le K/n\quad(n\ge1).
 \tag{DP.41}
\]

Choose \(M,C\ge1\) with \(\|A_k\|\le MC^k k!\), and set
\(L=C(1+2KM)\). Induction from \(V_0=I\) gives
\[
 \begin{aligned}
 \|V_n\|&\le\frac{KM}{n}L^n n!\\
 &\quad\cdot\sum_{k=1}^n\frac{(C/L)^k}{\binom nk}\\
 &\le\frac{KM}{n}L^n n!\frac{C}{L-C}\\
 &\le L^n n!.
 \end{aligned}
 \tag{DP.42}
\]
All constants are fixed before \(n\). Hence this recursively
constructed gauge belongs to the actual \(R_0\). Its constant
term is \(I\); in DP.37 the norm of \(V-I\) is at most
\(L\epsilon/(1-L\epsilon)\), which is less than one at a
fixed sufficiently small \(\epsilon\). Its two-sided geometric
inverse is therefore also actual. Equation DP.39 holds as an
identity of actual symbols, because it holds in every coefficient.
The new connection is exactly \(\partial_h+H/h\). This proves
the analytic-class assertion needed for a module isomorphism,
including all earlier integer resonances, rather than only a
formal Frobenius calculation.

##### DP.7.5. The original finite-order module is the whole Euler model

In the resulting actual \(R\)-basis \(u_1,\ldots,u_r\), the
coefficient-column actions are
\(t=h^2\partial_h+hH\), \(D_{x_i}=\partial_i\).
Since \([t,h^{-1}]=-I\), their action on the basis is
\[
 \begin{gathered}
 D_{x_i}u_j=0,\\
 \theta u_j=\sum_k(H-I)_{kj}u_k.
 \end{gathered}
 \tag{DP.43}
\]
Thus the programme's generator-column convention gives an
\(E\)-linear surjection
\[
 N_{(H-I)^{\mathsf T}}\longrightarrow M.
 \tag{DP.44}
\]
It is surjective because the displayed vectors are a basis over
\(R\subset E\). It is also injective, by the actual division
proved in [§5.52](#5-52-full-euler-matrix-division-and-comparison-reductions), AM.A. Its division of a finite-order input remains
finite order: in the quotient formula AM.10, if the input
coefficients vanish above \(m\), then \(q_j=0\) for \(j>m\),
and the remainder has no coefficient above \(m\) either. The
spatial quotients use a single fixed order shift. All coefficients
therefore satisfy the finite-order actual bounds already proved
in AM.A. It follows that every element of the source in DP.44
has a unique normal row independent of \(t,z\), of finite upper
order. That row is precisely an element of \(R^r\). Its image
in \(M\) is its coefficient column in the actual basis just
constructed, so it is zero only if that row is zero. This proves
injectivity and the claimed finite-order isomorphism. The
transpose and the shift by \(-I\) in DP.44 are both forced by
the explicit coefficient-column convention and \(\theta=tD_t\).

Tensoring this actual isomorphism with the already constructed
\(B\) and \(C\) rings, and using the complete Euler comparison
[§5.53](#5-53-real-kernel-comparison-for-all-constant-euler-matrices), AM.29, proves comparison and finite halfspace separation
for every module with DP.34 on this same standard chart. No
flatness of \(C\) is required. After a contact change, the
comparison still requires compatible transport of those full
coefficient rings; DP.44 alone is not that transport theorem.

Free human source context: Kashiwara–Kawai,
[Holonomic systems III](https://www.kurims.kyoto-u.ac.jp/~kenkyubu/kashiwara/HolIII.pdf),
pp.833–834, Definition 1.1.11(i), and I.3. The actual division,
parameter freeness, Gevrey gauges and module isomorphism used here
are proved in DP.1–DP.7, using the exact earlier programme
estimates specified above. The recalled classification in that
paper is not a substitute for any of these proofs.


##### DP.7.6. An exact resonant matrix calculation

For the two-dimensional coefficient connection, let
\[
 \begin{gathered}
 A(h)=\begin{pmatrix}0&h^2\\0&2\end{pmatrix},\\
 S=\operatorname{diag}(h,1),\\
 \begin{aligned}
 A_1(h)&=S^{-1}AS\\
       &\quad+hS^{-1}S'\\
       &=\begin{pmatrix}1&h\\0&2\end{pmatrix},
 \end{aligned}\\
 \begin{aligned}
 H&=S^{-1}A_1S+hS^{-1}S'\\
  &=\begin{pmatrix}2&1\\0&2\end{pmatrix}.
 \end{aligned}
 \end{gathered}
 \tag{DP.45}
\]
Both equalities follow by multiplying the displayed diagonal
matrices; their added derivative term is \(\operatorname{diag}(1,0)\).
The two integer shifts retain the off-diagonal resonant term as
an actual nilpotent block. Formula DP.44 gives the Euler matrix
\((H-I)^{\mathsf T}=\left(\begin{smallmatrix}1&0\\1&1\end{smallmatrix}\right)\).
This is an illustration of the proved arbitrary-matrix theorem,
with no diagonalizability restriction.

![Actual Euler normal form from an Euler-stable lattice](assets/euler-stable-lattice-normal-form.png)

*Figure DP.2. All arrows concern the entire module under precisely
DP.34. The residue module is free at every point by the derivative-
stable presentation-minor proof DP.35, and its actual lift is
DP.21. The tangential gauge uses the Banach norm DP.37 and the
iterated-integral bound proved in DP.7.2. Integer shifts are DP.38;
the Sylvester inverses, factorial gauge bound and actual inverse
are DP.40–DP.42. The finite-order module isomorphism, including
its transpose and minus-one shift, is DP.43–DP.44. The exact
bottom example is the complete calculation DP.45. Editable
plotting source, exact data,
full PNG and
SVG are provided.
Free human context: [Kashiwara–Kawai, Holonomic systems III](https://www.kurims.kyoto-u.ac.jp/~kenkyubu/kashiwara/HolIII.pdf),
Definition 1.1.11(i) on pp.833–834 and I.3. The actual gauges and
normal-form proof used here are DP.7.1–DP.7.5 above.*


### 5.56. Whole real-kernel coordinate transport and balanced comparison

Ordinary holomorphic coordinates transport the entire real microlocal kernel ring, its projectivized infinite-order sections and their natural comparison. The proofs below retain the actual support systems, density and proper trace. Linear changes and holomorphic time shifts have compatible finite-order maps, explicit full-symbol formulas and balanced maps for every module.

#### BCT.0. Rings, coordinates and exact earlier proofs

Let \(X,X'\) have complex dimension \(N=d+1\), and let
\(F:X\to X'\) be a biholomorphism of actual neighborhoods. Put
\[
 \begin{gathered}
 u'=F(u),\\ p=(dF_u)^{\mathsf T}p',\\
 \widehat F(u,p)\\
 =\bigl(F(u),(dF_u)^{-\mathsf T}p\bigr).
 \end{gathered}
 \tag{BCT.1}
\]
The covectors are nonzero. Their real pairing is
\(-\Re\langle q,p\rangle\), with the sign of RM.3. Write
\[
 \begin{gathered}
 C_X=\mathcal E_X^{\mathbb R},\\
 B_X=\mathcal E_X^\infty,\\ E_X=\mathcal E_X.
 \end{gathered}
 \tag{BCT.2}
\]
The complete earlier real-stalk representative lemma
[§5.53, AM.H](#am-h-the-complete-real-stalk-representative-lemma)
proves the source definition and its cofinal whole ring representatives
RM.1–RM.9. In particular a stalk of \(C_X\) is the filtered colimit
of the actual supported groups
\[
 \begin{gathered}
 H_H^N(D;\mathcal O^{(0,N)}),\\
 C_\Delta(H)_{u_0}\setminus0\\
 \subset\{-\Re\langle q,p\rangle>0\}.
 \end{gathered}
 \tag{BCT.3}
\]
Here \(D\) is a pair domain, \(H\) is closed, and the coefficient
is the ordered holomorphic input top form. The product is the full
relative cup, controlled proper excision and the normalized intermediate
trace. Its definition includes arbitrary real germs, rather than only
those with homogeneous symbols.

The complete support/resolution and full trace proofs are
[§5.34, R.1–R.7](#5-34-full-current-resolutions-and-controlled-proper-image-comparison)
and [§5.35, V.1–V.6](#5-35-relative-cones-unshifted-cup-and-the-normalized-trace).
The compact-coordinate trace and its exact determinant/shift signs are
V.4, V.10–V.12. Cofinal actual round domains, positive lengths and every
finite product's properness are RM.4–RM.9. The complete projectivized
kernel identification and nonlinear base-coordinate transition are
[§5.45, IG.1–IG.5 and IM.1–IM.6](#5-45-the-full-geometric-infinite-order-kernel-ring-and-its-canonical-product).
Its actual coefficient and all-index tail estimates are
[§5.44, IK.1–IK.8](#5-44-full-homogeneous-coefficients-and-ordinary-infinite-order-kernels).
The finite-order time-shift formula is
[§5.55, DP.5](#dp-5-actual-finite-order-straightening-by-a-holomorphic-time-shift).
These are proved programme prerequisites, not external classification
assumptions.

#### BCT.1. An actual cofinal map of all real support germs

Let \(q=v-u\). On a fixed compact enlargement Taylor's integral
formula gives
\[
 \begin{gathered}
 F(u+q)-F(u)\\ =dF_uq+R(u,q),\\
 |R(u,q)|\le A|q|^2,\\
 |dF_uq|\ge m|q|,\\ |p'|\le P,\qquad m>0.
 \end{gathered}
 \tag{BCT.4}
\]
Indeed the integral of the derivative difference is bounded by one
second-derivative supremum times \(|q|^2\); invertibility of the
derivative on that compact gives a positive lower singular-value bound.
One may obtain it without a spectral theorem as
\(m=1/\sup\|(dF_u)^{-1}\|\). All norms here are fixed Euclidean
operator/vector norms.

Consider a target support with the actual strict margin
\[
 \begin{gathered}
 -\Re\langle q',p'\rangle\ge\mu|q'|,\\ \mu>0.
 \end{gathered}
 \tag{BCT.5}
\]
The unit-normal-cone compactness argument in RM.4–RM.6 supplies this
on a smaller domain and a fixed covector neighborhood for every finite
support list. If \(|q|\le m/(2A)\), BCT.4 gives
\(|q'|\ge m|q|/2\). Since the pulled-back covector is precisely
\(p=(dF_u)^{\mathsf T}p'\),
\[
 \begin{aligned}
 &-\Re\langle q,p\rangle\\
 &=-\Re\langle q',p'\rangle\\
 &\quad+\Re\langle R(u,q),p'\rangle\\
 &\ge\tfrac12\mu m|q|-PA|q|^2\\
 &\ge\tfrac14\mu m|q|.
 \end{aligned}
 \tag{BCT.6}
\]
when also \(|q|\le\mu m/(4PA)\). A zero Taylor-remainder
constant imposes no radius restriction. Thus the preimage support is
admissible at the pulled-back real phase on an actual common domain.
Apply the same argument to \(F^{-1}\) for the reverse map.

For completeness, its normal-cone assertion is exact. A convergent normal
sequence is given by \(u_k\to u_0\), \(q_k\to0\) and positive
real \(a_k\) with \(a_kq_k\to q\). BCT.4 gives
\(a_kR(u_k,q_k)\to0\). Therefore its transformed limit is
\(dF_{u_0}q\). Applying the inverse gives both inclusions in
\[
 \begin{gathered}
 C_{\Delta'}((F\times F)H)_{F(u_0)}\\
 =dF_{u_0}C_\Delta(H)_{u_0}.
 \end{gathered}
 \tag{BCT.7}
\]
This respects the antipodal sign in BCT.3. It also proves actual cofinality:
preimage and image pair neighborhoods and supports are inverse index
maps after the displayed restrictions. Restriction and support-forgetting
arrows commute with those maps. Any finite collection of cocycles,
primitives, products and equalities uses the common minimum of its
positive margins and the common maximum of its derivative constants.
No infinite collection of unrelated germ domains is made simultaneous.

#### BCT.2. The entire relative complex, including its input density

Put \(j_F(v)=\det dF_v\). The coefficient map is
\[
 \begin{gathered}
 f(u',v')\,dv'_1\wedge\cdots\wedge dv'_N\\
 \longmapsto f(F(u),F(v))\,j_F(v)\\
 \hphantom{\longmapsto{}}\cdot dv_1\wedge\cdots\wedge dv_N.
 \end{gathered}
 \tag{BCT.8}
\]
For smooth Dolbeault entries, take the full pullback of the
antiholomorphic form as well. Holomorphicity of \(F\) and its Jacobian
makes pullback commute with \(\bar\partial\). This follows directly
by the chain rule: the antiholomorphic differentials transform by the
conjugate derivative matrix, and the holomorphic multipliers have zero
Dolbeault differential. The same identity for the current resolution is
the transpose identity on compactly supported tests under a diffeomorphism;
R.4 identifies it with the smooth resolution. Arbitrary currents are not
multiplied.

On the relative cone, the map sends every entry by this same pullback.
For the convention of V.1,
\[
 \begin{gathered}
 d(a,b)=(\bar\partial a,ra-Db),\\
 (a,b)\longmapsto(F^*a,F^*b).
 \end{gathered}
 \tag{BCT.9}
\]
The restriction square commutes, so this is a chain map. On a full
ordered complement cube, pull back every face and preserve its index
order. Its differential \(d_C+(-1)^r\bar\partial\) commutes with
pullback with exactly the same signs. Thus no total or complement entry
is omitted. Finite cover refinements and their augmentation roofs are
transported as actual roofs. The same holds for the finite partition and
collar homotopies in R.7 and V.6: the cutoff functions and their
differentials are pulled back on their actual domains.

The inverse is the corresponding map for \(F^{-1}\). The coefficient
identity is
\(j_F(v)j_{F^{-1}}(F(v))=1\), by the finite determinant chain rule.
Hence BCT.8–BCT.9 give mutually inverse maps on every supported
cohomology group at the corresponding domains and supports. BCT.1 makes
them mutually inverse on the whole real stalk, with all finite equality
witnesses. Denote that additive stalk map by
\[
 \begin{gathered}
 T_C^F:C_{X',\widehat F(p)}\\ \longrightarrow C_{X,p}.
 \end{gathered}
 \tag{BCT.10}
\]
It is a map on all source-defined real germs. A coefficient symbol,
growth condition or solution-module faithfulness premise is not needed.

#### BCT.3. Proper collars, the product, its normalization and the unit

There is a concrete common collar for comparing the two product roofs.
After BCT.1, put the finitely many old supports inside a common convex
cone \(G\) of RM.5, with length \(l=-\Re q_t\) and
\(|q|\le K_Gl\). Select a ball \(B_\rho\) whose closed square lies inside both the
old pair source and its pulled-back new pair source. Such a ball exists
because both are actual neighborhoods of the diagonal base point.
Take endpoint norms less than \(\delta\), where
\[
 0<\delta<\frac{\rho}{2(1+2K_G)}.
 \tag{BCT.11}
\]
For any finite chain of consecutive supported differences,
\[
 \begin{gathered}
 \sum_i|q_i|\le K_G\sum_i l_i,\\
 \sum_i l_i=-\Re(v_t-u_t)\\ \le2\delta.
 \end{gathered}
 \tag{BCT.12}
\]
Every intermediate point therefore has norm at most
\((1+2K_G)\delta<\rho/2\). For compact endpoints, its support
family is closed and bounded in that fixed interior compact. This proves
properness inside both actual sources. A smooth cutoff equal to one near
that family and supported in \(B_\rho\) supplies the common collar.
This is the same finite compact cutoff construction used in R.7/V.4.
It works simultaneously for the finitely many maps and inverse maps.
The domains pulled back from the new round domains need not themselves
be round for a larger old linear cone; BCT.11–BCT.12 is the required
local product comparison instead.

On triple variables the diagram is literal:
\[
 \begin{gathered}
 (F\times F)\pi_{ij}\\
 =\pi'_{ij}(F\times F\times F).
 \end{gathered}
 \tag{BCT.13}
\]
The two pulled-back supports are exactly the preimages of the two new
supports. Full relative cup commutes with pullback because coefficient
multiplication and the ordered wedge product do. The full two-support
cube retains all four entries. Its support restrictions, collar excision
and the finite comparison homotopies commute on the collar just chosen.
The controlled excision theorem R.7 and the complete canonical product
comparison in IM.5 identify these roofs with the products in either
stalk. No injectivity of support enlargement is inferred.

The intermediate trace also commutes. In BCT.8 applied to two kernels,
the input densities contribute \(j_F(v_1)j_F(v_2)\). The
intermediate antiholomorphic block contributes
\(\overline{j_F(v_1)}\), so its full real fibre determinant is
\(|j_F(v_1)|^2\). V.4 proves the compact change-of-variable equality
and its current version, with the ordered complex orientation. The
remaining input density is exactly \(j_F(v_2)\). Holomorphic
base-dependent lift terms have fibre degree too small to survive the
trace, as proved there. The shifted trace sign is still
\((-1)^{N(q-N)}\); neither dimension nor degree has changed.
At \(q=N\) the sign is one. Thus there is no extra Cauchy residue
scalar or coordinate sign.

Consequently
\[
 \begin{gathered}
 T_C^F(P'Q')\\ =T_C^F(P')T_C^F(Q').
 \end{gathered}
 \tag{BCT.14}
\]
for every two elements of the entire real stalk. The inverse map for
\(F^{-1}\) is multiplicative by the same argument. Surjectivity now
proves the unit statement without a residue assumption: \(T_C^F(1)\)
acts on both sides as identity on every element of \(C_{X,p}\), so
it equals that ring's proved diagonal unit. Therefore BCT.10 is a unital
ring isomorphism.

For successive coordinate maps \(F:X\to X'\) and
\(G:X'\to X''\), actual pair pullbacks compose and
\(j_{G\circ F}(v)=j_G(F(v))j_F(v)\). Hence
\[
 \begin{gathered}
 T_C^{G\circ F}=T_C^F T_C^G,\\
 T_C^{\mathrm{id}}=\mathrm{id}.
 \end{gathered}
 \tag{BCT.15}
\]
Every finite composition is represented on one smaller common actual
neighborhood by BCT.1 and the same finite collar argument.


#### BCT.4. Projectivization and the exact full-coefficient square

The source definition RM.2 is
\[
 \begin{gathered}
 B_X=\gamma_X^{-1}\gamma_{X*}C_X,\\
 \gamma_X:\dot T^*X\longrightarrow P^*X.
 \end{gathered}
 \tag{BCT.16}
\]
Positive real scaling of a covector leaves the support condition in
BCT.3 unchanged, so the actual real cochain systems are real-conic.
The cotangent lift in BCT.1 commutes with every nonzero complex scaling,
by linearity of the dual derivative in the covector. It therefore induces
a map \(\overline F\) on the complex projectivization. Apply the
actual sheaf map BCT.10 to sections over \(\gamma\)-inverse images,
then take inverse-image germs. This gives
\[
 \begin{gathered}
 T_B^F:B_{X',\widehat F(p)}\\ \longrightarrow B_{X,p}.
 \end{gathered}
 \tag{BCT.17}
\]
The products and units of these sections are their pointwise real-stalk
products and units after common finite refinements. BCT.14–BCT.15 thus
prove that BCT.17 is a unital ring isomorphism with inverse
\(T_B^{F^{-1}}\), and prove its composition identities.

Concretely a projectivized section uses the finite angular cochain and
primitive system of IG.3. The phase circle is compact, and BCT.1 fixes
one set of derivative constants and finitely many positive margins for
it. Its pair pullback, input Jacobian, angular overlap primitives and
subsequent coefficient normalization are exactly those in IG.5. The
normal-form uniqueness in IG.4 therefore identifies BCT.17 with the
already proved nonlinear full-coefficient coordinate transition. No new
symbol or classification theorem is assumed.

Let \(i_X:B_X\to C_X\) be the source evaluation map. At a stalk it
sends a projectivized section to its germ at the chosen real phase.
Taking that germ and pulling back its actual representatives commute
literally. Thus the exact compatibility identity is
\[
 T_C^F i_{X'}=i_X T_B^F.
 \tag{BCT.18}
\]
It holds for every full infinite-order
coefficient germ and its source-defined real image, even though a general
real germ need not have a homogeneous normal form. No injectivity or
flatness of a restriction/support map is used to prove this identity.

#### BCT.5. The full time-shift formula and finite-order compatibility

Take
\[
 \begin{gathered}
 F_\varphi(t,x)\\ =(t-\varphi(x),x),\\
 \varphi(0)=0,\quad\partial\varphi(0)=0.
 \end{gathered}
 \tag{BCT.19}
\]
Its cotangent lift is
\(\tau'=\tau\), \(z'=z+\partial\varphi(x)\), as in DP.27.
Use the full coordinate class IK.1:
\[
 \begin{gathered}
 P=\\ \sum_{m\in\mathbf Z}\tau^m p_m(t,x,z),\\
 |p_{-n}|\le M C^n n!,\\
 |p_m|\le M_\epsilon\epsilon^m/m!,\\
 m\ge0,\qquad\epsilon>0.
 \end{gathered}
 \tag{BCT.20}
\]
Every coefficient has one common holomorphic product domain. Bounds are
on a chosen compact enlargement, and restrictions to any smaller compact
have the same type of estimates.

Use the ordered lists and functions of DP.28. In detail,
\(\mathcal J_{K,\ell}\) consists of the ordered lists
\(\Gamma=(\gamma_1,\ldots,\gamma_\ell)\) with
\(|\gamma_i|\ge2\) and \(\sum_i(|\gamma_i|-1)=K\).
Abbreviate \(f_\Gamma=(\partial_z^{d_\Gamma}f)(w_\varphi)\) in the
following display, where its derivative index and shifted argument are
defined in the same display:
\[
 \begin{gathered}
 d_\Gamma=\sum_i\gamma_i,\\
 c_\Gamma=\prod_i\frac{\partial^{\gamma_i}\varphi(x)}{\gamma_i!},\\
 w_\varphi=\left(\begin{gathered}
 t-\varphi(x),\ x,\\ z+\partial\varphi(x)
 \end{gathered}\right),\\
 \mathcal D_K^\varphi f=\sum_{\ell=0}^K\frac1{\ell!}\\
 \quad\cdot\sum_{\Gamma\in\mathcal J_{K,\ell}}c_\Gamma f_\Gamma.
 \end{gathered}
 \tag{BCT.21}
\]
The empty list gives \(\mathcal D_0^\varphi f=f(w_\varphi)\).
For each \(K\) the entire list is finite. The exact full formula is
\[
 \begin{gathered}
 q_j=\sum_{K\ge0}\mathcal D_K^\varphi p_{j+K},\\ j\in\mathbf Z.
 \end{gathered}
 \tag{BCT.22}
\]

##### BCT.5.1. All-index bounds on one product

The proof DP.29–DP.31 gives, on fixed shifted inner/outer products,
\[
 \begin{gathered}
 \|\mathcal D_K^\varphi f\|_{\rm inner}\\
 \le L^K K!\,\|f\|_{\rm outer}.
 \end{gathered}
 \tag{BCT.23}
\]
This bound is independent of the input homogeneous degree: \(\tau\)
is fixed and only the normalized \(z\)-coefficient is differentiated.
One explicit choice is DP.30's
\[
 \begin{gathered}
 L=32d^2\max(1,A_\varphi)\\
 \quad\cdot\max(1,(\delta\eta)^{-1})^2,\\ d\ge1.
 \end{gathered}
 \tag{BCT.24}
\]
Here \(\delta\) is the fixed covariable Cauchy margin,
\(\eta\) the fixed Taylor margin for \(\varphi\), and
\(A_\varphi\) its supremum there. The estimate counts every ordered
list, including all mixed spatial derivatives. When \(d=0\) or
\(\varphi\) is affine, all positive-loss terms vanish, and one can
use \(L=0\) for the resummation part.

For \(j\ge0\), \(K!j!\le(j+K)!\) gives
\[
 \begin{gathered}
 |q_j|\le\frac{M_\epsilon\epsilon^j}{j!}\\
 \quad\cdot\frac1{1-L\epsilon},\\ L\epsilon<1.
 \end{gathered}
 \tag{BCT.25}
\]
For any desired output epsilon choose the input epsilon at most that
number and with \(L\epsilon<1/2\). This proves the full
every-epsilon positive bound without changing the coefficient domain.

For \(j=-n<0\), split BCT.22 into \(K<n\) and \(K\ge n\).
The negative inputs are bounded by
\[
 \begin{gathered}
 M\sum_{K=0}^{n-1}C^{n-K}(n-K)!\\
 \quad\cdot L^K K!\\ \le M n!(C+L)^n.
 \end{gathered}
 \tag{BCT.26}
\]
Indeed \((n-K)!K!\le n!\), and the binomial expansion has every
remaining monomial with a positive integer coefficient. For the positive
inputs write \(K=n+m\), \(m\ge0\). Their sum is bounded by
\[
 \begin{gathered}
 M_\epsilon n!L^n\\
 \cdot\sum_{m\ge0}\binom{n+m}{n}(L\epsilon)^m\\
 =M_\epsilon n!L^n\\
 \quad\cdot(1-L\epsilon)^{-n-1}.
 \end{gathered}
 \tag{BCT.27}
\]
The last identity is the \(n\)-fold derivative of the geometric series
inside its radius. Choose one fixed such epsilon for all negative
outputs. BCT.26–BCT.27 gives one factorial negative constant for every
\(n\). Hence BCT.22 is an actual full symbol, not just a formal
transformation law.

There are also literal positive-input cutoff tails. For a positive output
\(j\), the omitted input degrees greater than \(J\) have bound
\[
 \frac{M_\epsilon\epsilon^j}{j!}
 \frac{(L\epsilon)^{\max(0,J-j+1)}}{1-L\epsilon}.
 \tag{BCT.28}
\]
For \(j=-n\), use \(\binom{n+m}{n}\le2^{n+m}\) in the omitted
part of BCT.27. If \(2L\epsilon<1\), its bound is
\[
 M_\epsilon n!(2L)^n
       \frac{(2L\epsilon)^{J+1}}{1-2L\epsilon}.
 \tag{BCT.29}
\]
These tails tend to zero at every fixed output degree. One fixed compact
enlargement and the Cauchy formula controls every specified derivative
as well. The product domains and margins precede all homogeneous and
spatial indices and every epsilon.

##### BCT.5.2. The geometric and finite-order maps are the same

We give the calibration argument, since an algebraic chain rule alone
would not identify a map of source-defined kernels. On the ordinary
coordinate generators, pair pullback BCT.8 gives the usual coefficient
and left-operator chain rule. The input density depends only on the
input variable, so it contributes no output derivative term. In
particular the geometric full ring map has
\[
 \begin{gathered}
 t'\longmapsto t-\varphi(x),\\ x'_i\longmapsto x_i,\\
 D_{t'}\longmapsto D_t,\\ D_{x'_i}\longmapsto\\ D_{x_i}+\varphi_iD_t.
 \end{gathered}
 \tag{BCT.30}
\]
It also sends \(D_{t'}^{-1}\) to \(D_t^{-1}\): these are actual
two-sided inverses in the unital kernel rings, so their images are
forced by multiplying both inverse identities. No primitive is made a
two-sided inverse on holomorphic functions.

Therefore the geometric map agrees with DP.28 on normal symbols
polynomial in \(t,x,z,h\). Their finite inverse normal powers are
the algebraic finite inverse powers used in the DP.5 proof. Equality
of their finite differential-operator chain rules and the actual normal
localization gives equality as kernel classes.

For an arbitrary finite-order input, keep its finite upper degree bound,
truncate its negative degrees and Taylor-expand its finitely many retained
coefficient functions on fixed smaller products. This gives polynomial
normal symbols \(P^{(s)}\) with coefficientwise holomorphic convergence,
one common finite upper order, and one uniform factorial negative bound.
Here is the uniform bound: the absolute Taylor sum on a smaller fixed
product is bounded by the product of the geometric Cauchy sums from an
outer product; that factor is independent of the homogeneous index and
the Taylor cutoff. Negative truncation only removes summands. The
finite-order transform DP.31, with a fixed normal shift if necessary,
gives the same uniform class for the transforms.

The forward pole/log kernels of these inputs and transforms converge
normally on common actual holomorphic complement face domains, with all
specified derivatives. This is precisely the all-spatial negative and
mixed Taylor majorant IK.18–IK.21 and its quantitative tails in IK.5;
finite upper degree gives the same positive bounds for the whole family.
Pullback by \(F\times F\) preserves compact convergence on its
preimage domains and multiplies by the fixed holomorphic Jacobian. BCT.1
fixes common cofinal phase supports for this entire uniformly bounded
family. The pulled-back angular overlap and whole-face entries are
retained.

At each of the finitely many phases, RM.5 contains these pulled-back
supports in a pointed cone with a strict angular margin. Choose the
standard normalization support by IM.19–IM.21: enlarge its normal
half-angle while retaining an angle less than a right angle, and enlarge
each spatial constant beyond the cone constant divided by the cosine of
its original half-angle. The neighboring phase overlaps use their own
pointed midpoint cones and their own standard supports, as in IM.18–IM.20.
The finite phase system admits one covariable shrink preserving every
strict polar margin. Thus every required support map goes forward into
its stated standard support; none is an inverse to support enlargement.

First pass from the pulled-back complement cover to those standard
IG.2 cubes by the finite smooth-resolution and partition-augmentation maps
of R.7/V.6. These are fixed finite cochain maps for the whole family.
Their restrictions and the finitely many derivatives of their fixed
cutoffs preserve compact convergence. Only the subsequent column solvers
are used on the known standard IG.2 face domains. Acyclicity of arbitrary
mixed or pulled-back cover intersections is not assumed.

To pass from these actual cochains to normalized projectivized
coefficients, use the complete bounded normalization construction IM.1
and IM.5 on the fixed standard face domains of IG.2. Its one-time weight
choice applies to this whole normally convergent family as follows.
On every compact exhaustion piece, the cochains and their differentials
have one finite supremum over \(s\). Choose the scalar exhaustion weight
to make the sum of these suprema's weighted square norms finite, exactly
as in DLH.4/IM.1. Compact convergence plus the resulting summable tail
gives convergence in that fixed graph norm. The finitely many bounded
column solvers and cube projections in IM.1 therefore give convergence
of the actual normalized top and face entries. Compact Cauchy
extraction, followed by AN.2–AN.4, preserves it coefficientwise. This
uses a proved bounded normalization and the IG.4 zero detector; it does
not assume a closed image of a cohomology boundary map.

For every \(s\) the two classes were equal. Each normalized
coefficient is holomorphic on the chosen common connected product. Its
zero germ is therefore the identically zero function by the identity
theorem, before taking any limit. Their normalized
coefficients are equal and converge to those of the two actual limiting
classes, so the limiting normalized coefficients are equal. IG.4 gives
equality of the projectivized kernel classes. It follows that the
geometric \(T_B^{F_\varphi}\) restricts to exactly the finite-order
DP.5 coordinate isomorphism.

Now let the input have the whole infinite positive head of BCT.20.
The finite-order truncations \(P^{[J]}\) retain that entire negative
tail and the positive head through \(J\). The input bounds are
uniform in \(J\). The actual cochain convergence of the whole angular
family is IK.18–IK.21 and IM.12, without any ordinary distribution
limit of high-order diagonal currents. Its fixed-coordinate pullback
has the same property. The just-described bounded normalization gives
coefficient convergence for the geometric maps. BCT.25–BCT.29 gives
coefficient and derivative convergence for BCT.22 with one uniform output
class. Both limits agree for every finite \(J\) by the finite-order
calibration, so they agree for the whole full input. This identifies
BCT.22 with the geometric map BCT.17. Its product, inverse and triple
overlap identities are consequently exact actual ring identities.

##### BCT.5.3. Linear maps and finite compositions

For \(F(u)=Au+b\), the whole normal-symbol map is exactly
\[
 \begin{gathered}
 u'=Au+b,\\ \rho'=A^{-\mathsf T}\rho.
 \end{gathered}
 \tag{BCT.31}
\]
The contraction \(\sum\partial_{\rho_i}^{(1)}\partial_{u_i}^{(2)}\)
is invariant under the two inverse matrices, proving the full ordered
product formula. There are no higher coordinate derivatives. On a
nonzero-covector chart let \(\tau'=\tau a(z)\), with \(a\)
and its reciprocal bounded on the compact overlap. Each homogeneous
coefficient acquires only a factor whose bound is exponential in its
degree, so the negative factorial and positive every-epsilon conditions
are preserved, with the same single product. The inverse has the same
bounds. This proves both the finite-order and full-class assertions.

Geometrically the generator chain rule is
\(D_{u'}=A^{-\mathsf T}D_u\). The image of an inverse linear
normal derivative is its actual elliptic inverse; its normal symbol has
no base derivatives and the two inverse products are literal. Polynomial
normal symbols therefore calibrate BCT.31. The same uniformly bounded
Taylor/negative truncation and actual kernel normalization argument of
BCT.5.2 proves equality with the geometric map on all finite inputs;
positive-head truncation proves it on the full class. The input top-form
factor is \(\det A\), and V.4 supplies the same positive real
\(|\det A|^2\) trace change, with no sign or extra scalar.

Finite compositions of these linear maps and holomorphic time shifts
therefore have compatible actual isomorphisms of all three rings
\(E,B,C\). Their composition and inverse identities hold on common
actual neighborhoods by BCT.15, BCT.18 and the finite-order calibration.


#### BCT.6. The whole module comparison is coordinate invariant

Let \(F\) be a finite composition of the linear and time-shift changes
of BCT.5.3. Abbreviate the source and target stalks by \(E,B,C\)
and \(E',B',C'\). The proved isomorphisms are compatible on the
entire finite subring and on the full coefficient image:
\[
 \begin{gathered}
 T_E:E'\longrightarrow E,\\ T_B:B'\longrightarrow B,\\ T_C:C'\longrightarrow C,\\
 T_B|_{E'}=T_E,\\ T_C i'=iT_B.
 \end{gathered}
 \tag{BCT.32}
\]
For any algebraic left \(E\)-module \(M\), give the same underlying
abelian group the \(E'\)-action \(a'\cdot m=T_E(a')m\), and
call it \(M'\). The maps
\[
 \begin{gathered}
 \Psi_B:B'\otimes_{E'}M'\\ \longrightarrow B\otimes_E M,\\
 b'\otimes m\longmapsto\\ T_B(b')\otimes m,\\
 \Psi_C:C'\otimes_{E'}M'\\ \longrightarrow C\otimes_E M,\\
 c'\otimes m\longmapsto\\ T_C(c')\otimes m.
 \end{gathered}
 \tag{BCT.33}
\]
are well defined on the actual algebraic tensors. For example the two
representatives of a \(B'\)-balancing relation have images
\(T_B(b')T_E(a')\otimes m\) and
\(T_B(b')\otimes T_E(a')m\), which are the same balancing
relation over \(E\). The identical calculation applies to \(C\).
The inverse maps use \(T_B^{-1},T_C^{-1}\) on coefficients and the
same underlying element of the module. Multiplication by the finite
subring proves their balancing, and both compositions are identities.
Thus these are whole-module isomorphisms, without a finite generation,
flatness, holonomicity or regularity hypothesis.

Let \(\eta_M:B\otimes_E M\to C\otimes_E M\) be the natural
coefficient map, and similarly define \(\eta_{M'}\). On every
simple tensor BCT.32 gives
\[
 \begin{gathered}
 \Psi_C\eta_{M'}(b'\otimes m)\\
 =\eta_M\Psi_B(b'\otimes m).
 \end{gathered}
 \tag{BCT.34}
\]
Every algebraic tensor is a finite sum of simple tensors, so the maps
agree everywhere. The vertical maps are isomorphisms; consequently
\[
 \begin{gathered}
 \eta_{M'}\text{ is injective}\\
 \Longleftrightarrow\quad\eta_M\text{ is injective}.
 \end{gathered}
 \tag{BCT.35}
\]
In particular the actual real comparison can be transported through each
ordinary coordinate change used to straighten a graph in DP.6. No
flatness assertion for \(C\) occurs in this proof.

The halfspace solution modules are transported as well. For an actual
new domain \(D'\) and \(U'=D'\cap\{\Re t'<0\}\), pullback
is an isomorphism
\[
 \begin{gathered}
 S'=\mathcal O(U')/\mathcal O(D'),\\
 \Theta_F:S'\longrightarrow S_F,\\ S_F=\\
 \mathcal O(F^{-1}U')/\mathcal O(F^{-1}D'),\\
 {}[f]\longmapsto[f\circ F].
 \end{gathered}
 \tag{BCT.36}
\]
Holomorphic extensions over \(D'\) pull back to holomorphic extensions
over \(F^{-1}D'\); the inverse proves both well-definedness and
bijectivity. The whole real-kernel action commutes with this map. Its
relative cup, support-to-output map and proper trace are the same actual
cochain operations as in BCT.3, now with a function cochain at the final
input. The input Jacobian in BCT.8 cancels the intermediate complex
change of variables in that trace, while the output function pulls back
with no density. BCT.11 fixes the common proper collar. Thus
\[
 \begin{gathered}
 \Theta_F(P'[f])\\ =(T_C P')\Theta_F([f]).
 \end{gathered}
 \tag{BCT.37}
\]
If a kernel action requires restriction, take all four terms on that
single smaller domain and transport its holomorphic-extension witnesses
by the same pullback. This is equality of actual quotient classes on
those common domains, and then of the actual germs. For the
time shift the old side has \(\Re(t-\varphi(x))<0\); it is not
silently replaced by an untransported flat halfspace. Every solution
functional and separating family therefore transports with its full
coefficient action.

#### BCT.7. A positive head with infinitely many negative transformed outputs

Here is an exact calibration which keeps the whole positive tail.
Take \(d=1\), \(\varphi(x)=x^2/2\), and define
\[
 \begin{gathered}
 H(\tau)=\sum_{m\ge0}\frac{\tau^m}{(m!)^2},\\
 P=H(\tau)(1-z)^{-1}.
 \end{gathered}
 \tag{BCT.38}
\]
This is an actual full symbol on \(|z|<r<1\). For every
\(\epsilon>0\),
\(1/(m!)^2\le e^{1/\epsilon}\epsilon^m/m!\), since the
positive exponential series contains its \(m\)-th term. This proves
the every-epsilon bound. There is no negative homogeneous input.

Put \(s=1-x-z\), on a product with \(s\ne0\), and
\[
 \begin{gathered}
 a_K=\frac{(2K)!}{2^K K!},\\
 Q=\sum_{K\ge0}\tau^{-K}\frac{a_K}{s^{2K+1}}.
 \end{gathered}
 \tag{BCT.39}
\]
This is actual finite order: \(a_K\le2^K K!\) follows from
\(\binom{2K}{K}\le4^K\), so one lower bound for \(|s|\)
gives its factorial constant for every \(K\). The exact operator
\(A=(1-x)D_t-D_x\) has
\[
 A\circ Q= D_t.
 \tag{BCT.40}
\]
Indeed its normal product is \(\tau sQ-\partial_xQ\). Its leading
coefficient is one, and the remaining coefficients cancel by
\(a_K=(2K-1)a_{K-1}\). The corresponding inverse
\(QD_t^{-1}\) is two sided. A right inverse has just been proved;
\(A((QD_t^{-1})A-1)=0\), and multiplication by the nonzero
leading symbol of \(A\) is injective on a finite-order series by
induction on its descending coefficients. Thus the parenthesized
finite-order series is zero. This does not use a highest degree for an
arbitrary full infinite-order series.

By the exact generator map, the old image of \((1-z')^{-1}\) is
this \(Q\). The actual \(H(D_t)\) commutes with its coefficients,
which are independent of \(t\), so the full product gives
\[
 \begin{gathered}
 b_{j,K}=\\ \frac{(2K)!}{2^K K!\,(j+K)!^2},\\
 q_j(0,0,0)=\\ \sum_{K\ge\max(0,-j)}b_{j,K},\\ j\in\mathbf Z.
 \end{gathered}
 \tag{BCT.41}
\]
These are the same coefficients as BCT.21–BCT.22: only the lists
with every \(\gamma_i=2\) occur, their length is \(K\), and
\(\partial_z^{2K}(1-z)^{-1}=(2K)!/(1-z)^{2K+1}\).
All terms of BCT.41 are strictly positive. In particular every negative
output coefficient is nonzero, despite the absent negative input.

For fixed \(j\), write its summand as \(b_{j,K}\). The exact
successive-term ratio is
\[
 \begin{gathered}
 \frac{b_{j,K+1}}{b_{j,K}}\\
 =\frac{2K+1}{(j+K+1)^2}.
 \end{gathered}
 \tag{BCT.42}
\]
For \(K\ge2|j|\) it is at most \(8/(K+2)\). For
\(K\ge\max(14,2|j|)\) it is therefore at most \(1/2\).
Consequently any cutoff \(R\) at least that large gives the exact
rational enclosure
\[
 \begin{gathered}
 S_{j,R}<q_j(0,0,0)\\ \le S_{j,R}+2b_{j,R+1},\\
 S_{j,R}=\sum_{K=\max(0,-j)}^R b_{j,K}.
 \end{gathered}
 \tag{BCT.43}
\]
The left inequality is strict because every tail is positive; the right
is the geometric bound just proved. This supplies convergent finite
calibrations for a genuinely infinite positive-to-negative transform.
No numerical convergence of the negative symbol series in \(h\) is
being asserted.

#### BCT.8. Scope of the coordinate theorem

BCT.10–BCT.18 proves whole real-stalk and full projectivized transport
for every ordinary local biholomorphism, with its actual density, support,
product, inverse and composition maps. BCT.19–BCT.37 identifies the
finite-order coordinate maps and balanced comparison for the linear and
holomorphic time-shift changes used in DP.6. Every statement concerns
the whole ring or module, with the explicitly stated hypotheses.

For a symmetric constant matrix \(C\), the quadratic contact shear
of §5.28 is
\[
 \begin{gathered}
 t'=t-z^{\mathsf T}Cz/2,\\ x'=x+Cz,\\ z'=z,\qquad\tau'=\tau.
 \end{gathered}
 \tag{BCT.44}
\]
For \(C\ne0\) its output base depends on the input covariable:
\(\partial(x+Cz)/\partial z=C\ne0\). It is therefore not the
cotangent lift of an ordinary base biholomorphism. The whole real-ring
map BCT.10 cannot be applied to it. Its actual full \(B\)-ring
quantization is already proved in §5.49, ISR.5, but a compatible real
contact kernel requires a separate construction. General holonomic
infinite-order regularization or an actual separating Euler-extension
embedding also requires its own proof; nilpotent leading matrices in
DP.26 alone do not supply that hypothesis.

Free human context for the source real microlocal and complex
projectivization definitions is [Kashiwara–Kawai, Holonomic systems III](https://www.kurims.kyoto-u.ac.jp/~kenkyubu/kashiwara/HolIII.pdf),
printed pp.830–831. The coordinate, tail, density and balanced-tensor
arguments used here are proved in BCT.1–BCT.7 and the exact earlier
programme proofs listed in BCT.0. No external contact-quantization or
classification theorem is used as a premise.


#### BCT.9. Exact support slices and coefficient enclosures

For the first figure take \(N=2\), \(\varphi(x)=x^2/2\), and
fix the output base point at \(u=0\). In normal variables
\(q=(w,\zeta)\), the actual pair map is
\[
 \begin{gathered}
 w'=w-\zeta^2/2,\\ \zeta'=\zeta.
 \end{gathered}
 \tag{BCT.45}
\]
A full complex target cone may be chosen as
\[
 \begin{gathered}
 l'=-\Re w'\ge0,\\ |\zeta|\le l',\\ |\Im w'|\le l'/4.
 \end{gathered}
 \tag{BCT.46}
\]
If \(|\zeta|\le1/2\), the inverse map has
\[
 \begin{gathered}
 l=-\Re w\ge l'-|\zeta|^2/2\\ \ge3l'/4,\\
 |\zeta|\le4l/3,\\ |\Im w|\le2l/3,\\
 -\Re w\\ \ge(3/\sqrt{29})|(w,\zeta)|.
 \end{gathered}
 \tag{BCT.47}
\]
Indeed \(|\zeta|^2/2\le|\zeta|/4\le l'/4\). The imaginary
part is bounded by \(l'/4+|\zeta|^2/2\le l'/2\le2l/3\).
The last inequality follows by squaring and using
\(|w|^2+|\zeta|^2\le(1+4/9+16/9)l^2\).
On the real slice it improves to \(-w\ge3|(w,\zeta)|/5\).
The curved boundary there is exactly
\(w=\zeta^2/2-|\zeta|\), with containing linear boundary
\(w=-3|\zeta|/4\). The base halfspace in the same real slice
is \(t<x^2/2\). These are fixed-point slices; the full variable-base
and all-dimensional support proof is BCT.1–BCT.3.

![Actual coordinate supports and the whole module comparison](assets/real-coordinate-supports-and-comparison.png)

*Figure BCT.1. The real base slice shows the exact pulled-back halfspace
of BCT.36. The normal slice is BCT.45–BCT.47 with
\(|\zeta|\le1/2\); it displays the curved preimage of BCT.46 and
its linear containing cone, with the exact real-slice margin \(3/5\).
The full complex slice has the proved margin \(3/\sqrt{29}\).
The bottom square is BCT.33–BCT.35 for the entire original module,
with the same source finite-ring action; its vertical arrows are actual
isomorphisms. Density, all complement entries, proper collars and trace
signs are proved in BCT.2–BCT.3. Editable plotting source,
exact data,
full PNG and
SVG accompany the
figure. Free human context is the real/projectivized definition in
[Kashiwara–Kawai, Holonomic systems III](https://www.kurims.kyoto-u.ac.jp/~kenkyubu/kashiwara/HolIII.pdf),
printed pp.830–831. The actual maps and estimates displayed here are
proved in BCT.1–BCT.6 and BCT.9.*

![Infinite positive-input contributions to negative homogeneous outputs](assets/full-positive-time-shift-calibration.png)

*Figure BCT.2. The input and coordinate change are exactly BCT.38.
For each integer \(-4\le j\le4\), the left marker is the rational
lower sum \(S_{j,24}\), and its error interval ends at the exact
rational upper bound \(S_{j,24}+2b_{j,25}\) of BCT.43. Only their
plotting coordinates are numerical; the data file retains each rational
numerator and denominator. The right panel displays exact successive-term
ratios BCT.42 for \(j=-4,0,4\); the region \(K\ge14\) has the
proved ratio bound \(1/2\) for these three cases. Every negative
output is strictly positive, although the input has no negative
homogeneous coefficient. The inverse-operator calibration and full
convergence argument are BCT.39–BCT.43. Editable plotting source,
exact rational data,
full PNG and
SVG are provided.
The full-class tail estimates are BCT.23–BCT.29; these coefficient
samples do not substitute for them. Free human context is the full
kernel definition in [Kashiwara–Kawai, Holonomic systems III](https://www.kurims.kyoto-u.ac.jp/~kenkyubu/kashiwara/HolIII.pdf),
printed pp.830–831; every displayed bound is proved above.*


### 5.57. Legendre incidence and the topological inverse of projective duality

A family of exact Legendre isomorphisms on localized polynomial operator algebras factors every symmetric quadratic contact shear. The proof retains the ordered inverse correction, gives holomorphic incidence identities and compact period bounds, and constructs both full topological open-incidence convolution inverses. These algebraic and topological results do not yet construct the compatible holomorphic microkernel or its action on the entire source-defined real operator stalk.

#### LI.0. Actual algebra and conventions

Let the base have dimension \(d+1\), with coordinates \((t,x)\),
normal derivative \(\tau=D_t\), spatial derivatives \(D_i=D_{x_i}\),
and \(h=\tau^{-1}\). Work first in the polynomial differential
operator ring localized at the actual normal derivative, denoted
\(A_d\). Concretely this is the subring of the actual finite-order
ring of §5.19 generated by \(t,x_i,D_i,\tau,\tau^{-1}\).
The complete product and actual normal inverse in that earlier section
give its associative multiplication. Its defining Weyl relations are

\[
 [\tau,t]=1,\qquad [D_i,x_j]=\delta_{ij},
 \tag{LI.1}
\]

with all other pairs of different coordinate/derivative generators
commuting. The order is the displayed left-operator order. In particular

\[
\begin{gathered}
{}[h,t]=-h^2,\\
z_i=D_i h,\\ E_x=\sum_i x_iD_i.
\end{gathered}
 \tag{LI.2}
\]

The first identity follows by multiplying \([\tau,t]=1\) on
both sides by the two-sided inverse. This inverse is an inverse in the
operator ring. It is not an inverse to differentiation on holomorphic
functions.

There are no additional polynomial relations in this concrete ring.
To see this, order an expression with base coordinates on the left and
derivatives on the right, using LI.1 and the inverse relation. This gives
a finite sum of \(t^a x^\beta D^\gamma\tau^k\), with
\(a\ge0\), \(\beta,\gamma\ge0\), \(k\in\mathbf Z\).
In the actual normal symbol these are distinct Laurent monomials in
\((t,x,\xi,\tau)\). A finite linear combination zero on a nonempty
open set has all coefficients zero: multiply by one power of \(\tau\)
to get a polynomial, and apply the one-variable polynomial identity
successively to each coordinate. Thus the ordered relations also give a
presentation of this subring. Checking LI.1 and invertibility of the
normal generator consequently defines a homomorphism of the whole
localized polynomial algebra.

Use \((s,y,\sigma,D_{y_i})\) for a second copy of this algebra.
The two normal variables will be identified by \(\sigma\mapsto\tau\).

#### LI.1. A family of exact localized Legendre maps

For any complex number \(\alpha\), define

\[
 \begin{gathered}
 L_\alpha(\sigma)=\tau,\qquad L_\alpha(\sigma^{-1})=h,\\
 L_\alpha(y_i)=D_i h,\\
 L_\alpha(D_{y_i})=-x_i\tau,\\
 L_\alpha(s)=t+(E_x+\alpha)h.
 \end{gathered}
 \tag{LI.3}
\]

**Proposition LI.A.** LI.3 is an isomorphism between the two whole
localized polynomial algebras. Its inverse is

\[
 \begin{gathered}
 L_\alpha^{-1}(\tau)=\sigma,\\
 L_\alpha^{-1}(x_i)=-D_{y_i}\sigma^{-1},\\
 L_\alpha^{-1}(D_i)=y_i\sigma,\\
L_\alpha^{-1}(t)\\ =s+(E_y+d-\alpha)\sigma^{-1},\\
 E_y=\sum_i y_iD_{y_i}.
 \end{gathered}
 \tag{LI.4}
\]

**Proof.** Put \(Y_i=D_i h\), \(V_i=-x_i\tau\), and
\(S=t+(E_x+\alpha)h\). Then
\([V_i,Y_j]=\delta_{ij}\), because
\(-x_iD_j+D_jx_i=\delta_{ij}\). All the \(Y_i\) commute,
all the \(V_i\) commute, and \(\tau\) commutes with both lists.
Also \([\tau,S]=1\).

The remaining two commutators include the normal-order terms. LI.2 gives
\([t,Y_i]=D_i h^2\), while
\([E_xh,Y_i]=-D_i h^2\). Therefore \([S,Y_i]=0\).
Similarly \([t,V_i]=x_i\), whereas
\([(E_x+\alpha)h,V_i]=-x_i\). One direct check of the latter is

\[
 \begin{gathered}
{}[(E_x+\alpha)h,x_i\tau]\\
=(E_x+\alpha)x_i\\ -x_i(E_x+\alpha)=x_i.
 \end{gathered}
 \tag{LI.5}
\]

Thus \([S,V_i]=0\). The images satisfy every defining relation,
and \(\tau\) remains invertible, so LI.3 defines the homomorphism.

The proposed inverse in LI.4 is also a homomorphism, before checking
its compositions. Indeed the same construction with parameter
\(d-\alpha\), with input and output names interchanged, is a
homomorphism by the calculation just completed. Follow it by spatial
reflection, which negates every spatial coordinate and spatial
derivative and fixes time and its derivative. Reflection preserves all
the defining relations, fixes the spatial Euler operator, and changes
the two spatial images to exactly LI.4. Its time image is already
the displayed time image in LI.4. This proves the required inverse-map
homomorphism assertion without assuming relations merely from a
proposed list of generator values.

Its spatial images satisfy the exact ordered identity

\[
 L_\alpha(E_y)=-E_x-d.
 \tag{LI.6}
\]

Indeed \((D_i h)(-x_i\tau)=-D_i x_i=-x_iD_i-1\).
Consequently the proposed image of \(t\) in LI.4 maps to
\(t+(E_x+\alpha)h+(-E_x-\alpha)h=t\).
The proposed images of \(x_i,D_i,\tau\) plainly map to the
original generators. Conversely the same computation gives
\(L_\alpha^{-1}(E_x)=-E_y-d\), so applying LI.4 to LI.3's
image of \(s\) gives \(s\); its images of \(y_i,D_{y_i}\)
and \(\sigma\) also return those generators. Thus both compositions
are identities on all generators and hence on the whole algebras. ∎

In particular, for the simple Cauchy normalization \(\alpha=0\),
the inverse time coordinate contains \(+d\sigma^{-1}\).
Omitting it makes the supposed inverse send the old \(t\) to
\(t-dh\), rather than \(t\).

Identify the names of the two polynomial algebras, and let \(J\) be
the ordinary spatial reflection: it sends \(x,D_x\) to
\(-x,-D_x\), and fixes \(t,\tau\). A second application of
LI.3 gives

\[
 \begin{gathered}
L_\alpha^2(x_i)=-x_i,\\ L_\alpha^2(D_i)=-D_i,\\
 L_\alpha^2(t)=t+(2\alpha-d)h,\\
 L_\alpha^2(\tau)=\tau.
 \end{gathered}
 \tag{LI.7}
\]

The spatial statements follow immediately from LI.3, and the time
statement follows by substituting LI.6 into
\(L_\alpha(t+(E_x+\alpha)h)\). For \(\alpha=0\),
LI.7 can be written

\[
 L_0^2=\operatorname{Ad}(\tau^{-d})\,J.
 \tag{LI.8}
\]

For an integer \(k\), induction on LI.1 gives
\(\tau^k t\tau^{-k}=t+kh\). The negative integers follow
from LI.2. All other generators commute with \(\tau\), proving
LI.8. For \(\alpha=d/2\), the square in LI.7 is exactly \(J\)
on the polynomial algebra. This is an algebraic statement for every
complex \(\alpha\); no fractional power of an operator was used.

#### LI.2. The exact contact factorization

On \(\tau\ne0\), use the contact form
\(dt+z^{\mathsf T}dx\). The contact map underlying LI.3 is

\[
 \begin{gathered}
\ell(t,x,z)\\ =(t+x^{\mathsf T}z,\ z,\ -x),\\
\ell^{-1}(s,y,w)\\ =(s+y^{\mathsf T}w,\ -w,\ y).
 \end{gathered}
 \tag{LI.9}
\]

These maps are literal inverses by substitution. Their contact property
is the identity

\[
\begin{gathered}
d(t+x^{\mathsf T}z)-x^{\mathsf T}dz\\
=dt+z^{\mathsf T}dx.
\end{gathered}
 \tag{LI.10}
\]

Let \(C=C^{\mathsf T}\) be any complex matrix, without an
invertibility or rank hypothesis, and put
\(\varphi(y)=-y^{\mathsf T}Cy/2\). Its ordinary time change has
the exact contact lift

\[
\begin{gathered}
F_\varphi(s,y,w)\\
=\left(\begin{gathered}s+y^{\mathsf T}Cy/2,\\ y,\ w-Cy\end{gathered}\right).
\end{gathered}
 \tag{LI.11}
\]

Applying LI.9, then LI.11, then the inverse in LI.9 gives

\[
 \begin{gathered}
 \ell^{-1}F_\varphi\ell(t,x,z)\\
 =(t-z^{\mathsf T}Cz/2,\ x+Cz,\ z)\\
 =\kappa_C(t,x,z).
 \end{gathered}
 \tag{LI.12}
\]

For example the intermediate new time is
\(t+x^{\mathsf T}z+z^{\mathsf T}Cz/2\), and the final
addition from the inverse map is
\(z^{\mathsf T}(-x-Cz)\). Their sum is the time in LI.12.
Thus the factorization covers every symmetric \(C\), including
every degenerate matrix and all dimensions. It is not a generic-rank
substitution for the required shear.

The time map in LI.11 already has compatible actual \(E,B,C\)
maps in §5.56. The additional map \(\ell\) depends on covariables;
it is the unresolved correspondence step, not an ordinary base change.

#### LI.3. Exact cancellation of the operator normalization

Write \(\Theta_C\) for the polynomial-algebra time map in LI.11:

\[
 \begin{gathered}
 \Theta_C(s)=s+y^{\mathsf T}Cy/2,\\
 \Theta_C(D_y)=D_y-Cy\sigma,\\
 \Theta_C(y)=y,\qquad\Theta_C(\sigma)=\sigma.
 \end{gathered}
 \tag{LI.13}
\]

The operator chain rule of §5.56 proves these identities on the actual
finite ring, so they also hold on its polynomial subring. Its inverse
is \(\Theta_{-C}\). Symmetry is needed for the derivative of the
quadratic coefficient and for the commuting transformed derivatives.

**Proposition LI.B.** For every \(\alpha\in\mathbf C\),

\[
 L_\alpha\Theta_C L_\alpha^{-1}
 \tag{LI.14}
\]

is independent of \(\alpha\), and its values are

\[
 \begin{gathered}
 x\longmapsto x+Cz,\\
 t\longmapsto t-z^{\mathsf T}Cz/2,\\
 D_x\longmapsto D_x,\qquad\tau\longmapsto\tau.
 \end{gathered}
 \tag{LI.15}
\]

**Proof.** For \(x\), the three steps in LI.14 are

\[
\begin{gathered}
-D_y\sigma^{-1}\\
\longmapsto -D_y\sigma^{-1}+Cy\\
\longmapsto x+Cz.
\end{gathered}
 \tag{LI.16}
\]

For \(D_x\), its inverse image \(y\sigma\) is unchanged by
\(\Theta_C\) and maps back to \(D_x\). The normal generator
is fixed in all three steps. For the time coordinate use the exact
inverse in LI.4. Since

\[
 \Theta_C(E_y)=E_y-y^{\mathsf T}Cy\sigma,
 \tag{LI.17}
\]

its middle image is

\[
\begin{gathered}
s+(E_y+d-\alpha)\sigma^{-1}\\ -y^{\mathsf T}Cy/2.
\end{gathered}
 \tag{LI.18}
\]

The first two terms map to \(t\) by Proposition LI.A. The last
maps to \(-z^{\mathsf T}Cz/2\), because all the \(z_i\)
commute. This proves every formula in LI.15. These elements generate
the algebra, so the maps agree everywhere and are independent of
\(\alpha\). ∎

LI.15 agrees with the exact generator values Q.18 of the finite-order shear. The established full finite and infinite shear
formulas in §§5.28 and 5.49 are retained. LI.B alone does not identify
an action on arbitrary real-kernel germs from its generator values.

#### LI.4. Holomorphic incidence functions and cleared operator relations

On the pair of bases define

\[
 f(t,x;s,y)=s-t-x^{\mathsf T}y.
 \tag{LI.19}
\]

It is a submersion since \(\partial_s f=1\). On an actual open set
where \(f\ne0\) and a holomorphic logarithm of \(f\) is chosen,
put \(K_\nu=\exp(-\nu\log f)\). For an integer \(\nu\)
this is the ordinary single-valued function \(f^{-\nu}\), without
a branch choice. The following are literal holomorphic identities:

\[
 \begin{gathered}
 -D_sK_\nu=D_tK_\nu,\\
 y_iD_tK_\nu=D_{x_i}K_\nu,\\
 D_{y_i}K_\nu=x_iD_tK_\nu,\\
(s-t)D_tK_\nu\\
 =\nu K_\nu+\sum_i x_iD_{x_i}K_\nu.
 \end{gathered}
 \tag{LI.20}
\]

Indeed \(D_tK_\nu=\nu f^{-1}K_\nu\),
\(D_{x_i}K_\nu=\nu y_i f^{-1}K_\nu\), and
\(D_{y_i}K_\nu=\nu x_i f^{-1}K_\nu\). Substituting
\(s-t=f+x^{\mathsf T}y\) proves the last equality, including
\(\nu=0\). No infinite sum occurs.

Attach the ordered input density \(ds\wedge dy_1\wedge\cdots
\wedge dy_d\). Formal integration by parts transposes an input
derivative to its negative derivative on the kernel; LI.20 contains the
corresponding signs explicitly. The last identity is the normal-
derivative-cleared identity for the time generator in LI.3 with

\[
 \alpha=\nu-1.
 \tag{LI.21}
\]

In fact applying \(D_t\) to
\(tK_\nu+(E_x+\alpha)D_t^{-1}K_\nu\) would give
\(tD_tK_\nu+(E_x+\alpha+1)K_\nu\); the required coefficient
is exactly \(\nu\). This calculation is only a way of recording
the cleared relation. It does not choose a primitive on holomorphic
functions or define \(D_t^{-1}K_\nu\) there.

For \(\nu=1\), the incidence function has an exact nonzero
hypersurface residue. In coordinates \((t,x,y,f)\), its circle
integral is

\[
 \frac{1}{2\pi i}\int_{|f|=r}f^{-1}\,df=1.
 \tag{LI.22}
\]

Parametrizing \(f=re^{i\theta}\) makes the integrand
\(i\,d\theta\), proving the formula. A holomorphic extension
across \(f=0\) would have zero circle integral by its uniformly
convergent Taylor expansion. Hence the punctured/whole-domain class of
\(f^{-1}\) in the one-hypersurface relative group is nonzero.
This does not identify that degree-one group with the diagonal
degree-\(d+1\) whole real operator stalk.

The conormal relation of this incidence hypersurface is the graph in
LI.9. More precisely \(-\tau\,df\) has output components
\((\tau,\tau y)\) and input components
\((-\tau,\tau x)\). The canonical input covector is the negative
of its latter block, namely \((\tau,-\tau x)\). Thus
\(z=y\), \(w=-x\), and \(f=0\) gives
\(s=t+x^{\mathsf T}z\), with exactly the antipodal convention
used by the course's input top-form kernels.

For the symmetric algebraic normalization \(\alpha=d/2\), LI.21
uses \(\nu=1+d/2\); when \(d\) is odd it needs a genuine local
logarithm. The simple pole \(\nu=1\) avoids this branch issue but
retains the integer inverse correction \(dh\). Proposition LI.B
shows that neither choice alters the required quadratic generator map.

The identities in LI.20 are necessary, not sufficient, for an inverse
kernel. For example \(\nu=0\) makes \(K_0=1\); LI.20 still
holds, but this function is holomorphic across the incidence and its
punctured-domain class is zero. One cannot deduce faithfulness or
invertibility merely from the generator equations.

#### LI.5. A complete finite-box period identity

This elementary period will be used to check a proposed inverse
composition on actual domains. Let \(\epsilon_i>0\),
\(a,b_i\in\mathbf C\), and

\[
 B=\sum_i\epsilon_i|b_i|,\qquad |a|>B.
 \tag{LI.23}
\]

On the real box \(Q=\prod_i[-\epsilon_i,\epsilon_i]\), the
denominator \(a+\sum_i b_i v_i\) stays nonzero. Define the ordinary
compact real integral

\[
\begin{gathered}
I_d(a,b)=\int_Q\\
\partial_a^d\frac{1}{a+\sum_i b_iv_i}\,dv_1\cdots dv_d.
\end{gathered}
 \tag{LI.24}
\]

**Proposition LI.C.** If every \(b_i\ne0\), abbreviate
\(a_\eta=a+\sum_i\eta_i\epsilon_i b_i\). Then

\[
\begin{gathered}
I_d(a,b)=\frac{1}{\prod_i b_i}\\
\quad\cdot\sum_{\eta\in\{-1,1\}^d}
\frac{\prod_i\eta_i}{a_\eta}.
\end{gathered}
 \tag{LI.25}
\]

Every apparent singularity at \(b_i=0\) in this expression is
removable on LI.23. In particular

\[
\begin{gathered}
I_d(a,0)=\left(\prod_i2\epsilon_i\right)\\
\quad\cdot(-1)^d d!\,a^{-d-1}.
\end{gathered}
 \tag{LI.26}
\]

**Proof.** For a nonzero \(b_i\), differentiating the denominator
gives \(\partial_a=b_i^{-1}\partial_{v_i}\) on this function
and on every one of its derivatives. Apply this identity once in each
variable, and integrate the resulting mixed derivative by the real
fundamental theorem of calculus. Each upper endpoint contributes a
plus sign, each lower endpoint a minus sign, giving LI.25. All
derivatives are continuous and bounded on the compact box with the
strict denominator margin, so the finite iterated integrations and
differentiations are valid.

On a compact subset of LI.23 the denominator margin remains positive.
The geometric expansion of the integrand and each fixed parameter
derivative converges uniformly there and on \(Q\). Integration
therefore gives a holomorphic function of all \((a,b)\), including
\(b_i=0\). The equality on the dense locus where all \(b_i\ne0\)
proves that LI.25 has this holomorphic extension. At \(b=0\), the
integrand is \((-1)^d d!a^{-d-1}\), so its box integral is LI.26.
This also proves the completely degenerate case without division by any
\(b_i\). ∎

There are explicit common-domain tails. Put
\(V=\prod_i2\epsilon_i\) and \(r=B/|a|<1\). Differentiating
the geometric series \(d\) times gives

\[
\begin{gathered}
\partial_a^d(a+s)^{-1}\\
=(-1)^dd!a^{-d-1}\\
\cdot\sum_{m\ge0}\binom{d+m}{m}(-s/a)^m.
\end{gathered}
 \tag{LI.27}
\]

The \(m\)-th term integrated over the box has absolute value at
most \(Vd!|a|^{-d-1}\binom{d+m}{m}r^m\).
The positive generating series is \((1-r)^{-d-1}\), by the same
differentiated-geometric-series identity as BCT.27. Consequently

\[
 |I_d(a,b)|\le\frac{Vd!}{(|a|-B)^{d+1}}.
 \tag{LI.28}
\]

For a cutoff \(R\ge0\), choose \(r<r_1<1\). Each positive
term of the generating series at \(r_1\) is at most its sum, so
\(\binom{d+m}{m}\le(1-r_1)^{-d-1}r_1^{-m}\). Hence the
tail after \(R\) has bound

\[
\begin{gathered}
Vd!|a|^{-d-1}(1-r_1)^{-d-1}\\
\quad\cdot\frac{(r/r_1)^{R+1}}{1-r/r_1}.
\end{gathered}
 \tag{LI.29}
\]

This bound is uniform on any fixed compact with the displayed strict
margins. A compact enlargement gives the same conclusion for every
specified holomorphic parameter derivative by the Cauchy formula.
If \(r=0\), the omitted tail is zero directly. When \(d=0\),
the box is a point of volume one, the corner sum has one empty term,
and every formula reduces to \(I_0=1/a\).

For \(d=1\), LI.25 simplifies to the exact rational identity

\[
 I_1(a,b)=-\frac{2\epsilon_1}{a^2-\epsilon_1^2b^2}.
 \tag{LI.30}
\]

Its sign agrees with the negative derivative of \((a+bv)^{-1}\).
These compact periods check endpoint signs and normal derivative orders;
they do not themselves construct a supported identity operator.

#### LI.6. The precise kernel-inversion obligation

The factorization LI.12 suggests composing an incidence transform,
the ordinary time map LI.11 and the inverse incidence transform.
Propositions LI.A–LI.B prove that the required ordered algebraic map,
including every normalization correction, is exactly the quadratic shear. LI.20–LI.22 supply the holomorphic incidence identities
and simple-pole residue. Proposition LI.C supplies an actual compact
period with all dimensions, degenerate parameters and uniform tails.

A whole-real-stalk construction must additionally give all of the
following objects and identities on actual common neighborhoods:

1. A relative supported-cochain correspondence for the incidence, with
   the correct cohomological degree, input top density and antipodal
   conormal convention. A bare function \(f^{-1}\) in a degree-one
   hypersurface quotient is not an operator in the diagonal real ring.
2. Its reverse correspondence and proper intermediate supports or
   relative integration cycles, including all complement entries and
   boundary/excision primitives. LI.25 does not make the boundary of a
   real box disappear.
3. The two inverse compositions, equal to the proved diagonal unit with
   its actual trace normalization on one common refined index. The
   derivative-order and \(dh\) terms must agree with LI.4 and LI.8.
4. Compatibility with the entire \(B\)-ring shear of §5.49, not just
   with its coordinate generators, and with the natural \(B\to C\)
   map. No normal-coefficient uniqueness theorem for arbitrary \(C\)
   germs has been proved or assumed here.
5. Transport of the whole balanced comparison for the original finite
   module. Once compatible ring isomorphisms are actually constructed,
   the finite algebraic tensor argument BCT.32–BCT.35 applies verbatim.

The polynomial isomorphism, holomorphic incidence functions, compact period and topological convolution inverse do not supply these additional real-kernel data.

#### LI.7. Free source context and provenance

The elementary proofs above use the programme's actual Weyl product,
normal inverse and ordinary time-change formulas. The whole-real-ring
definition and product are §5.53 AM.H and §§5.34–5.35; the ordinary-coordinate compatibility is §5.56. These complete earlier
programme proofs supply the stated internal premises.

Free primary references are
[Benchoufi, Action of micro-differential operators on quantized contact
transformations, arXiv:2102.08861v2](https://arxiv.org/abs/2102.08861v2),
especially §§3–4, and
[Aoki–Honda–Yamazaki, Analytic pseudodifferential operators, I](https://www.jstage.jst.go.jp/article/jmath/69/4/69_1715/_pdf/-char/ja).
The arguments above use the local proofs and exact earlier programme results identified here.

#### LI.8. The complete topological open-incidence inverse

There is a useful inverse at the level of ordinary topological sheaf
kernels. We prove it separately; it does not by itself identify the
holomorphic supported microkernel or its real operator action.

Let \(V\) have complex dimension \(n+1\), let
\(P=\mathbf P(V)\), \(P^*=\mathbf P(V^*)\), and let
\(\Omega\subset P\times P^*\) be the open nonincidence relation
\(\zeta(v)\ne0\). The symbol \(\mathbf C_\Omega\) means
extension by zero, not direct image, of the constant sheaf on that open.
If \(j^\mathrm{op}:\Omega^\mathrm{op}\hookrightarrow P^*\times P\)
is its transposed open embedding, put

\[
\begin{gathered}
K=\mathbf C_\Omega,\\
T=Rj^\mathrm{op}_*\mathbf C_{\Omega^\mathrm{op}}[2n].
\end{gathered}
 \tag{LI.31}
\]

Use cohomological shifts: \(H^q(A[r])=H^{q+r}(A)\). For kernels
on three spaces, write

\[
\begin{gathered}
K_1*K_2=R\pi_{13*}\\
\left(\begin{gathered}\pi_{12}^{-1}K_1\\
\otimes_{\mathbf C}\pi_{23}^{-1}K_2\end{gathered}\right).
\end{gathered}
 \tag{LI.32}
\]

The middle projective space is compact, so this direct image equals the
proper-support direct image. Tensor products over \(\mathbf C\)
are exact on stalks, so the ordinary tensor computes the derived tensor.

**Theorem LI.D.** The two entire topological convolution kernels are

\[
\begin{gathered}
K*T\simeq\mathbf C_{\Delta_P},\\
T*K\simeq\mathbf C_{\Delta_{P^*}}.
\end{gathered}
 \tag{LI.33}
\]

The shift in LI.31 is \(2n\), the real dimension of an affine fibre,
and not \(n\). The diagonal identifications use its complex
orientation and the compact integration normalized below.

##### LI.8.1. The compact-fibre calculation is a derived calculation

We spell out the needed proper base change for a projection with compact
smooth fibre \(Q\). For a bounded complex \(F\) on
\(U_0\times Q\), and a point \(b\in U_0\), its canonical map is

\[
 (R\pi_*F)_b\simeq R\Gamma(Q;i_b^{-1}F).
 \tag{LI.34}
\]

Here is a resolution proof for the precise spaces used here. Resolve
each sheaf term by its tensor over \(\mathbf C\) with the smooth
real de Rham resolution of the constant sheaf on \(U_0\times Q\),
and take the finite total complex. The local de Rham resolution is exact:
on a coordinate ball, differentiation of dilation gives
\(dH+Hd=1-\mathrm{ev}_0\), where on a positive-degree form
the homotopy is integration of its contraction with the radial vector
field along \(x\mapsto sx\), \(0\le s\le1\). In degree zero
the kernel consists of constants. Tensoring this exact stalk complex
with a vector space remains exact. The total complex therefore resolves
\(F\), including its original differential and total signs.

Every term is a smooth-function module. Its ordinary sections on any
open are acyclic by the complete partition/injective-resolution proof
R.5. Restricting the complex to \(\{b\}\times Q\) preserves
exactness on stalks; its terms are again smooth-function modules on
\(Q\), since a smooth function on \(Q\) acts through pullback
from the second factor. They are acyclic there for the same reason.

For any sheaf \(G\), a section of \(i_b^{-1}G\) on compact
\(Q\) is represented on an actual neighborhood of that fibre. To
check this explicitly, choose finitely many local lifts covering \(Q\),
and shrink their cover to finitely many closed subsets whose interiors
still cover \(Q\). The lifts agree on germs along each compact
intersection. A finite further shrink makes them agree on neighborhoods
of all those intersections, and they glue. The same argument tests
equality of two lifts. A finite shrinking of a cover exists by choosing
small coordinate balls with closures inside its members and taking a
finite subcover of the compact space. Every neighborhood of
\(\{b\}\times Q\) contains \(U\times Q\) for some
neighborhood \(U\) of \(b\), again by a finite subcover. Thus

\[
\begin{gathered}
\varinjlim_{U\ni b}\Gamma(U\times Q;G)\\
=\Gamma(Q;i_b^{-1}G).
\end{gathered}
 \tag{LI.35}
\]

Apply LI.35 degree by degree to the finite acyclic resolution just
constructed. Filtered direct limits of vector spaces are exact: the
vanishing of an element or the lifting of a finite linear relation is
witnessed at a single later index. The two resulting complexes compute
the two sides of LI.34, proving it with all boundary witnesses. This
argument also gives a finite cohomological bound, so the convolutions
in LI.32 remain bounded for the bounded kernels used here.

##### LI.8.2. Compact cohomology of the affine and punctured factors

The compact de Rham complex of \(\mathbf R^m\) has only one
cohomology group: \(\mathbf C\) in degree \(m\), identified
by integration. There is an explicit contraction. In one real
coordinate choose a compact smooth \(\rho\) with integral one.
For a compact one-form \(g(x)dx\), put
\(G=\int_{\mathbf R}g\). Its primitive after subtracting
its integral representative is

\[
\begin{gathered}
h(g\,dx)(x)\\ =\int_{-\infty}^x
\left(g(s)-\rho(s)G\right)ds.
\end{gathered}
 \tag{LI.36}
\]

It is compact because the integrand has zero total integral. On a
compact function, the same fundamental theorem gives \(h(df)=f\).
Hence \(dh+hd=1-\iota\int\), where \(\iota(c)=c\rho dx\)
in degree one and is zero in degree zero. Apply this construction one
coordinate at a time with the usual tensor differential signs. The
integrals and homotopies commute with every unused coordinate; compact
support in a fixed coordinate box stays compact in a possibly larger
fixed box. Induction proves the assertion in every \(m\), including
its actual contraction and the positive product orientation. The fine
de Rham resolution and compact partitions compute compact sheaf
cohomology: extending a compact local section across a finite open
cover uses compact cutoffs whose sum is one near its support; the
same partition contraction as R.5 kills positive resolution cohomology.
In particular

\[
 R\Gamma_c(\mathbf C^n;\mathbf C)
 \simeq\mathbf C[-2n].
 \tag{LI.37}
\]

Let \(j_0:\mathbf C^*\hookrightarrow\mathbf C\). Its derived
direct image has degree-zero constant cohomology and degree-one
cohomology \(\mathbf C\) at the origin, with nothing else. This
is a local computation: a punctured disc retracts radially to a circle;
on a circle, the derivative of a periodic smooth function has integral
zero, and a one-form of integral zero has the periodic primitive
\(\int_0^\theta g(s)ds\). Thus its de Rham groups are constants
and the normalized angular class. The radial homotopy supplies the
retraction on forms. An unpunctured disc has just the constant group
by the ball homotopy. These calculations also show that the natural
map \(\mathbf C_{\mathbf C}\to Rj_{0*}\mathbf C_{\mathbf C^*}\)
is an isomorphism on degree-zero cohomology. Its cone consequently has
the single degree-one sheaf at the origin. We obtain the triangle

\[
\begin{gathered}
\mathbf C_{\mathbf C}\longrightarrow
Rj_{0*}\mathbf C_{\mathbf C^*}\\
\longrightarrow\mathbf C_{\{0\}}[-1]\xrightarrow{+1}.
\end{gathered}
 \tag{LI.38}
\]

Its compact connecting map is an isomorphism from degree one at the
point to degree two of the plane. To verify its normalization rather
than assume it, put \(\eta=d\theta/(2\pi)\), and choose a radial
cutoff \(\chi\) equal to one near zero and zero outside a disc.
The compact two-form \(-d\chi\wedge\eta\) is smooth across
zero, is supported in an annulus, and has integral
\(-\int_0^\infty\chi'(r)dr=1\). It is the compact boundary
representative of the oriented circle class in LI.38, with the
corresponding cone sign. Reversing a cone sign reverses both these
representatives and leaves the asserted isomorphism unchanged.
LI.37 and the long exact sequence of LI.38 therefore give

\[
 R\Gamma_c(\mathbf C;Rj_{0*}\mathbf C_{\mathbf C^*})=0.
 \tag{LI.39}
\]

For the complement of an affine hyperplane \(H\) in \(\mathbf C^n\),
choose affine coordinates with \(H=\mathbf C^{n-1}\times\{0\}\).
The inclusion is the product of the identity with \(j_0\). The same
local punctured-disc resolution identifies its derived direct image
with the exterior product of the constant first factor and
\(Rj_{0*}\mathbf C_{\mathbf C^*}\). Applying the compact
coordinate contractions LI.36 in its first \(2n-2\) real variables
reduces its compact global complex, with a shift, to LI.39. Those
contractions commute with the normal differential and its circle
boundary representative. Thus, for every nonempty affine hyperplane,

\[
 R\Gamma_c\!\left(\mathbf C^n;
 Rj_*\mathbf C_{\mathbf C^n\setminus H}\right)=0.
 \tag{LI.40}
\]

##### LI.8.3. Every fibre of the full convolution

Fix two projective points \(x,z\in P\). In the middle \(P^*\),
put \(A=P^*\setminus H_x\) and \(B=P^*\setminus H_z\),
where \(H_x=\{\zeta:\zeta(x)=0\}\). Choose a representative
of \(x\), and normalize \(\zeta(x)=1\). This identifies
\(A\) with the entire affine space \(\mathbf C^n\).
If \(x=z\), then \(A=B\). If \(x\ne z\), extend their
independent representatives to a vector-space basis. In the affine
coordinates just chosen, \(\zeta(z)=0\) is one nonempty affine
hyperplane, so \(A\cap B=A\setminus H\).

The restriction of the derived open direct image in LI.31 to such a
fibre is the derived open direct image for \(B\) in \(P^*\).
There is no arbitrary base-change assumption for an open map here:
locally in \(z\), an invertible matrix depending holomorphically on
\(z\) sends a fixed representative to \(z\). Explicitly, in
an affine chart write the representative as
\(e_0+\sum_{i=1}^n z_i e_i\); the matrix with this first
column and with remaining columns \(e_1,\ldots,e_n\) has
determinant one. Its inverse change on covectors preserves evaluation.
It identifies the
incidence pair with its fixed-fibre pair. In local transverse disc
coordinates the preceding punctured-disc de Rham calculation, including
the radial homotopy in the parameter ball, computes the direct image
and its restriction. Its degree-zero constant and degree-one oriented
circle generator both restrict to precisely those of the fibre. The
comparison is therefore a quasi-isomorphism on every stalk.

Tensoring with the extension-by-zero sheaf \(\mathbf C_A\)
restricts this complex to \(A\) and extends it by zero outside
\(A\). On the compact \(P^*\), its global cohomology is the
compact cohomology on \(A\). This statement also follows directly
by applying extension by zero to a fine compact resolution: its
sections on \(P^*\) are exactly the sections with compact support
inside \(A\). Formula LI.34 consequently computes the stalk of
the convolution as follows. Abbreviate the actual fibre complex by
\(\mathcal J_{x,z}=Rj_{A\cap B,A*}\mathbf C_{A\cap B}\).

\[
\begin{gathered}
(K*T)_{(x,z)}\simeq\\
R\Gamma_c(A;\mathcal J_{x,z})[2n].
\end{gathered}
 \tag{LI.41}
\]

For distinct \(x,z\) it is zero by LI.40. On the diagonal it is
\(\mathbf C\) in degree zero by LI.37, since the normal shift
\(2n\) cancels the affine compact degree. These are derived stalk
computations, not an Euler-characteristic calculation.

On the diagonal the identification is locally constant with the positive
complex orientation. To verify its gluing, choose local representatives
of \(x\) and complete them to frames; the induced affine charts
trivialize the diagonal family. On overlaps the change is a holomorphic
biholomorphism. Its real Jacobian on the top density is
\(|\det J|^2>0\), so ordinary compact change of variables preserves
the integration in LI.37. The orientation generators therefore glue to
the constant sheaf on \(\Delta_P\). All other cohomology sheaves
are zero by LI.41. A complex with one cohomology sheaf is
quasi-isomorphic to that sheaf: truncate at that degree and quotient by
boundaries, which gives quasi-isomorphisms on the two sides. Finally a
sheaf supported on a closed subset equals the direct image of its
restriction there, as is checked on its inside and outside stalks.
This proves the first isomorphism in LI.33. Swapping \(P\) and
\(P^*\) gives the second by the identical full fibre calculation.
For \(n=0\), both spaces are points, the nonincidence is the whole
point pair, and LI.33 is the identity without a shift. ∎

The primary free source for this topological kernel is
[D'Agnolo–Schapira, Leray's quantization of projective duality,
author-hosted edition](https://webusers.imj-prg.fr/~pierre.schapira/ResPapers/ProjDual.pdf),
Appendix A.2, printed p.40. The actual compact-fibre, circle-boundary,
all-degree affine and diagonal-gluing proofs used here are LI.8.1–LI.8.3.
They do not import the article's contact-quantization or simple-module
theorems. A compatible holomorphic microkernel, its normalized
composition and the whole real \(B/C\) operator action still require
the separate construction in LI.6.

#### LI.9. Exact illustration

![Exact contact factorization, ordered inverse correction and box period](assets/legendre-contact-factorization.png)

*Figure LI.1. The three panels show the real spatial/covariable
projections of LI.9–LI.12 for \(d=1,C=1,t=0\), with
\(x,z\in\{-1/2,-1/4,0,1/4,1/2\}\). Matching colors follow
the same exact rational sample; their time coordinates are retained in
the data file. The final point is \((t-z^2/2,x+z,z)\).
The lower left gives the ordered inverse and square identities
LI.4/LI.7 for the simple-pole normalization. The lower right shows
\(-I_1(2,b)\) from LI.30 at \(\epsilon=1/3\), with exact
samples \(b=k/20\), \(-20\le k\le20\). Only plotting
coordinates are numerical; every sample is retained as a rational
number. Propositions LI.A–LI.C prove the corresponding all-dimensional
identities and tail bounds; this figure does not establish the whole
real kernel inverse. Editable source,
exact data,
full PNG and
SVG are provided.
Free human context is the incidence/Leray section in
[Benchoufi, arXiv:2102.08861v2](https://arxiv.org/abs/2102.08861v2),
§4.3; the displayed maps, normalization and period are proved above.*

![The complete topological open-incidence inverse and its normal circle boundary](assets/open-incidence-topological-inverse.png)

*Figure LI.2. The left panel is the actual complex normal plane in
LI.38–LI.40. Write \(b(s)=e^{-1/s}\) for \(s>0\) and zero
otherwise, and choose
\(\chi(r)=b(2-r)/(b(2-r)+b(r-1))\). It is one for
\(r\le1\), zero for \(r\ge2\), and smooth on the entire
plane because it is constant near the origin. The plotted area density
of \(-d\chi\wedge d\theta/(2\pi)\) is
\(-\chi'(r)/(2\pi r)\); its exact integral is one by LI.38's
fundamental-theorem calculation, independently of numerical plotting.
The right panel records the full convolution's off-diagonal fibre as
the two-term compact complex in degrees \(-1,0\) after the shift
\([2n]\), with differential one. On the diagonal the same shift
leaves \(\mathbf C\) in degree zero. LI.8.1–LI.8.3 proves every
derived stalk and the orientation gluing in all dimensions. This is the
topological sheaf inverse; the compatible holomorphic microkernel and
whole real operator action remain separate. The same editable source,
exact defining data and full PNG
and SVG accompany it.
Free human context is
[D'Agnolo–Schapira, author-hosted Leray edition](https://webusers.imj-prg.fr/~pierre.schapira/ResPapers/ProjDual.pdf),
Appendix A.2, printed p.40.*


### 5.58. Full analytic Legendre transport and quadratic factorization

The polynomial Legendre map of §5.57 extends to the entire actual
finite-order and infinite-order normal-symbol sheaves. The extension
has an explicit formula and one factorial loss bound independent of
the input homogeneous degree. The inverse retains the ordered
dimension correction. This is a theorem about the actual finite and
projectivized infinite operator rings. An action on arbitrary real
microlocal kernel germs still requires the separate supported-kernel
construction; no real-germ normal form is assumed here.

#### LN.0. The two actual coefficient classes

Use input coordinates \((s,y)\), normal covariable \(\sigma\),
spatial covariable \(\eta\), and \(w=\eta/\sigma\).
Use output coordinates \((t,x)\), normal covariable \(\tau\),
spatial covariable \(\xi\), and \(z=\xi/\tau\).
There are \(d\) spatial coordinates and \(d+1\) base coordinates.
The ordinary normal product differentiates covariables in its left
factor and base variables in its right factor, as in §5.19 and IK.10
of §5.44. Put \(h=\tau^{-1}\).

An input normal symbol is specified by

\[
 P=\sum_{m\in\mathbf Z}\sigma^m p_m(s,y,w).
 \tag{LN.1}
\]

This display denotes the normal homogeneous coefficient sequence and
its actual product, not a convergent Laurent sum in the normal
covariable. All coefficients are holomorphic on one fixed product.
On each of its compact subproducts there are constants \(M,C>0\)
for the negative tail and, for every \(\epsilon>0\), a constant
\(M_\epsilon\) for the positive tail, such that

\[
\begin{gathered}
\|p_{-n}\|\le MC^n n!,\ (n\ge1),\\
\|p_m\|\le M_\epsilon\epsilon^m/m!,\ (m\ge0).
\end{gathered}
 \tag{LN.2}
\]

These are precisely the full class \(B\) of §5.44 IK.1–IK.3.
The finite actual class \(E\) has in addition a finite upper degree.
In that case the second bound holds for every epsilon by taking the
maximum over finitely many nonzero positive coefficients. No spatial
Taylor degree is bounded in either class.

We use the following complete earlier programme proofs. Section5.19,
especially §§5.19.2–5.19.4, proves the actual finite product and its
homogeneous coefficient rule. Section5.44 IK.2–IK.3 proves the entire
binary and triple contraction majorants, the grouped-index tails and
all fixed derivative tails for the full class. Section5.57 LI.0–LI.3
proves the localized polynomial Legendre map, including both inverse
compositions and its ordered square. The present argument supplies the
analytic extension and all its bounds rather than citing an external
quantized-contact theorem.

#### LN.1. The explicit loss operators

Define the ordinary holomorphic substitution

\[
 \begin{gathered}
 s_0=t+x^{\mathsf T}z,\\
 \iota(t,x,z)=(s_0,z,-x).
 \end{gathered}
 \tag{LN.3}
\]

For the differential operators below, \(x\) is a parameter, kept
fixed before substitution. Set

\[
 V_i=\partial_{y_i}+x_i\partial_s.
 \tag{LN.4}
\]

For a holomorphic coefficient \(p(s,y,w)\), abbreviate
\(p_\gamma=(V^\gamma\partial_w^\gamma p)\circ\iota\). Define

\[
\begin{gathered}
A_Kp=(-1)^K
\sum_{|\gamma|=K}\frac{p_\gamma}{\gamma!}.
\end{gathered}
 \tag{LN.5}
\]

In dimension zero the sum is the single identity term when \(K=0\)
and is empty when \(K>0\). In every dimension it is a finite sum
for each \(K\). Derivatives in \(V^\gamma\) do not differentiate
the parameter coefficients \(x_i\), and differentiation in \(w\)
takes place before setting \(w=-x\).

The proposed full transform is

\[
 \begin{gathered}
 \mathcal L(P)=\sum_{j\in\mathbf Z}\tau^j q_j,\\
 q_j=\sum_{K\ge0}A_Kp_{j+K}.
 \end{gathered}
 \tag{LN.6}
\]

The inner sum will be proved normally convergent on a single output
product. No formal exponential of a differential operator is used to
define this map.

#### LN.2. Exact agreement on the localized polynomial algebra

Write \(S_0=t+x^{\mathsf T}z\) as an actual order-zero normal
symbol. The simple-pole polynomial map \(L_0\) of LI.3 has

\[
\begin{gathered}
L_0(s)=S_0,\\ L_0(y_i)=z_i,\\
L_0(w_i)=-x_i,\\ L_0(\sigma)=\tau.
\end{gathered}
 \tag{LN.7}
\]

Here \(L_0(s)=t+E_xh\), whose normal symbol is exactly \(S_0\).
All these are actual ring elements. We first prove that a polynomial
base coefficient has the exact, correction-free image

\[
 L_0(g(s,y))=g(S_0,z).
 \tag{LN.8}
\]

Let \(G=g(S_0,z)\) with \(g\) polynomial. In the normal product
\(S_0\circ G\), the left factor is linear in every spatial
covariable, so every nonzero contraction of total size \(k\ge1\)
is either a pure normal-covariable derivative or has exactly one
spatial-covariable derivative. Their respective contributions are

\[
 \begin{gathered}
 (-1)^k h^k(x^{\mathsf T}z)\,\partial_s^k g,\\
 (-1)^{k-1}h^k
 \sum_i x_i z_i\,\partial_s^k g.
 \end{gathered}
 \tag{LN.9}
\]

Indeed \(\partial_\tau^k(x^{\mathsf T}\xi/\tau)
=(-1)^kk!h^k x^{\mathsf T}z\); division by the contraction
factorial removes \(k!\). A derivative in \(\xi_i\) followed by
\(k-1\) normal derivatives gives
\((-1)^{k-1}(k-1)!x_i h^k\). Its denominator is
\((k-1)!\), and the paired base derivative is
\(\partial_t^{k-1}\partial_{x_i}G
=z_i\partial_s^k g\). These give LN.9. They cancel for every
\(k\). For a polynomial only finitely many paired base derivatives
are nonzero.

The same calculation with left factor \(z_i=\xi_i/\tau\)
has pure normal contribution
\((-1)^kh^k z_i\partial_s^k g\) and the one spatial
contribution \((-1)^{k-1}h^k z_i\partial_s^k g\).
They also cancel. Thus multiplication on the left by \(S_0\) or
any \(z_i\) is ordinary multiplication on the displayed polynomial
subalgebra. Starting with its unit and multiplying its commuting
generators proves LN.8 for every polynomial \(g\).

Now take a normal monomial \(g(s,y)w^\beta\sigma^m\).
Its image is \(g(S_0,z)\circ(-x)^\beta\circ\tau^m\).
The last product is ordinary multiplication because its right factor
has no base dependence. In the preceding product only spatial base
derivatives of \((-x)^\beta\) occur. On \(g(S_0,z)\),
the covariable derivative is \(\partial_{\xi_i}=hV_i\).
The exact finite product is therefore

\[
\begin{gathered}
\sum_{\gamma\le\beta}
\frac{(-1)^{|\gamma|}}{\gamma!}
\frac{\beta!}{(\beta-\gamma)!}\\
{}\cdot(V^\gamma g)(S_0,z)\\
{}\cdot(-x)^{\beta-\gamma}\tau^{m-|\gamma|}.
\end{gathered}
 \tag{LN.10}
\]

This is precisely LN.5–LN.6 applied to that normal monomial. A normal
polynomial Laurent expression is a finite sum of these monomials;
all exponents \(m\) may be any integers. Hence LN.6 agrees with
the actual polynomial isomorphism of LI.A on its entire domain. In
particular the formula includes the lower-order term in
\(L_0(y_iw_i)=-x_i z_i-h\); dropping it would already destroy
LI.6's ordered Euler correction.

#### LN.3. One common analytic domain and one uniform loss bound

Let the input coefficients be defined on an open product \(U\) in
\((s,y,w)\). Choose an output product \(W\) and a slightly larger
product \(W^+\) such that \(\iota(\overline{W^+})\) is contained
in a compact subproduct \(U_1\) inside \(U\). This is possible
locally at every corresponding pair of points, since LN.3 is
holomorphic. Its inverse is the corresponding Legendre substitution
with the spatial reflection; the local products may always be
shrunk together.

Let \(\delta_s,\delta_y,\delta_w>0\) be common distances from
\(\iota(\overline{W^+})\) to the coordinate boundaries of
\(U_1\), and let \(R\ge\max_{W^+,i}|x_i|\).
For \(d>0\) choose positive \(a,b\) satisfying

\[
\begin{gathered}
a<\min\left\{\begin{gathered}\delta_y,\\
\frac{\delta_s}{d\max(1,R)}\end{gathered}\right\},\\
b<\delta_w.
\end{gathered}
 \tag{LN.11}
\]

These choices precede every homogeneous degree and loss index. For
each output point the function

\[
\begin{gathered}
F(u,v)=\\ p\left(\begin{gathered}
S_0+x^{\mathsf T}u,\\ z+u,-x+v
\end{gathered}\right)
\end{gathered}
 \tag{LN.12}
\]

is holomorphic on the same auxiliary polydisc
\(|u_i|<a,|v_i|<b\), and has supremum at most
\(\|p\|_{U_1}\). Its repeated Cauchy formula gives

\[
\begin{gathered}
\|V^\gamma\partial_w^\gamma p\circ\iota\|\\
\le (\gamma!)^2 (ab)^{-|\gamma|}\|p\|_{U_1}.
\end{gathered}
 \tag{LN.13}
\]

For completeness this Cauchy estimate is obtained by integrating in
each of the \(2d\) auxiliary variables around the corresponding
radius circles and bounding the denominators; taking slightly smaller
circles and then their limit gives the displayed radii. The same
fixed polydisc works for every output point in \(W^+\).

There are at most \(d^K\) multiindices of size \(K\ge1\):
every such multiindex is the count vector of at least one word of
length \(K\) in \(d\) letters. Moreover \(\gamma!\le K!\),
since the multinomial coefficient \(K!/\gamma!\) is a positive
integer. Consequently, with a single constant \(L\ge1\),

\[
\begin{gathered}
\|A_Kp\|_{W^+}\\ \le L^K K!\|p\|_{U_1},\\
L=\max\{1,d/(ab)\}\\ (d>0).
\end{gathered}
 \tag{LN.14}
\]

For \(d=0\), take \(L=1\); the higher loss operators are zero.
The coefficient and the bound in LN.14 have **no dependence on the
input degree**. Differentiating \(\tau^m\) in the definition would
spoil this fact; it is a right normal prefactor in LN.10, so such a
derivative does not occur.

#### LN.4. Both full tails and the actual cutoff estimates

**Theorem LN.A.** Formula LN.6 defines an actual map of the entire
coefficient classes on the common products above. It maps finite
upper-order symbols to finite upper-order symbols, and maps the full
class \(B\) to the full class \(B\). All fixed derivative series
converge normally after one further common output shrink.

**Proof.** For \(j\ge0\), LN.2 and LN.14 give, when
\(L\epsilon<1\),

\[
 \|q_j\|\le
 \frac{M_\epsilon\epsilon^j}{j!(1-L\epsilon)}.
 \tag{LN.15}
\]

Here the term of index \(K\) is bounded by
\(M_\epsilon L^K K!\epsilon^{j+K}/(j+K)!\), and
\(K!j!\le(j+K)!\). For any prescribed positive output epsilon,
choose a smaller input epsilon also satisfying this strict inequality.
This proves the required every-epsilon positive bound without
changing \(W^+\) or \(U_1\).

For output \(-n\), \(n\ge1\), first take \(0\le K<n\).
The input is negative and its contribution is at most
\(MC^{n-K}L^K(n-K)!K!\). Because
\((n-K)!K!\le n!\) and every binomial coefficient in
\((C+L)^n\) is at least one, their sum is bounded by
\(Mn!(C+L)^n\). For \(K\ge n\), put \(m=K-n\).
The positive inputs sum to
\(M_\epsilon n!L^n\sum_{m\ge0}\binom{n+m}{n}
(L\epsilon)^m\). Repeated differentiation of the geometric series
gives its exact sum. Thus

\[
 \begin{gathered}
 \|q_{-n}\|\le Mn!(C+L)^n\\
 {}+M_\epsilon n!L^n(1-L\epsilon)^{-n-1}.
 \end{gathered}
 \tag{LN.16}
\]

For one fixed admissible epsilon this is a single negative factorial
bound, for example with base constant the maximum of \(C+L\) and
\(L/(1-L\epsilon)\), and with the extra fixed factor
\((1-L\epsilon)^{-1}\) in its numerator constant. Normal
convergence of each inner sum follows from the same bounds. If the
input has top degree \(m_0\), only \(K\le m_0-j\) can occur;
there is no output above \(m_0\). This proves the finite class as
well as the full class.

Here are the full limits needed for algebra identities. Let
\(P^{[J]}\) retain precisely the positive input coefficients through
\(J\), with the **entire negative actual tail retained**. For
\(j\ge0\) the omitted part of the transformed coefficient is at most

\[
 \frac{M_\epsilon\epsilon^j}{j!}
 \frac{(L\epsilon)^{\max(0,J-j+1)}}{1-L\epsilon}.
 \tag{LN.17}
\]

For output \(-n\) and \(2L\epsilon<1\), it is at most

\[
 M_\epsilon n!(2L)^n
 \frac{(2L\epsilon)^{J+1}}{1-2L\epsilon}.
 \tag{LN.18}
\]

Indeed only positive inputs \(m>J\) occur. Use
\(\binom{n+m}{n}\le2^{n+m}\), since the binomial coefficient
counts a subset of the words in two letters, and sum the resulting
geometric tail. LN.17 is the corresponding tail of LN.15.
For all the truncated inputs, LN.15–LN.16 themselves hold with the
same constants.

These estimates also control positive output degrees uniformly in
a prescribed positive weighted norm. Given an output epsilon \(e>0\),
use an input epsilon \(\epsilon<e/2\) with
\(L\epsilon<1/2\). Split positive outputs into \(j>J/2\)
and \(j\le J/2\). In the first group the bound LN.15 divided by
\(e^j/j!\) is at most a fixed constant times
\((\epsilon/e)^{J/2}\); in the second group LN.17 has a geometric
factor at most \((L\epsilon)^{J/2}\). Both tend to zero. In a
negative weighted norm with base \(D>2L\), LN.18 divided by
\(D^n n!\) is uniformly geometrically small. Increasing \(D\)
also accommodates the common negative bound in LN.16.

For comparison, truncating input negative degrees below \(-R_0\)
has no positive output contribution. Its negative output is zero for
\(n\le R_0\), and for \(n>R_0\) is at most
\(Mn!(C+L)^n\). In any negative weighted norm with
\(D>C+L\), the omitted part is bounded by
\(M((C+L)/D)^{R_0+1}\). This supplemental estimate does not
replace the negative-tail-preserving cutoff used below.

Every assertion so far holds uniformly on \(W^+\). Apply a fixed
iterated Cauchy formula from that product to \(W\) for any specified
finite base or normalized-covariable derivative. Its constant is
independent of \(m,K,J,n\). It therefore gives normal convergence
and the same geometric tails for that derivative, with its fixed
Cauchy multiplier. No infinite collection of derivative domains and
no coefficient-specific neighborhood has been selected. ∎

#### LN.5. Multiplication and inverse on the entire finite actual ring

**Theorem LN.B.** On the actual finite-order sheaves, \(\mathcal L\)
is a unital filtered ring isomorphism over the contact map

\[
\begin{gathered}
\ell(t,x,z)\\ =(t+x^{\mathsf T}z,z,-x).
\end{gathered}
 \tag{LN.19}
\]

It has the generator values LI.3 with parameter zero. Its inverse is
the actual map specified below, including LI.4's dimension correction.

**Proof.** Theorem LN.A puts the output of each finite input in the
actual finite class. Theorem LI.A and LN.10 prove multiplication and
the inverse identities on the entire localized polynomial algebra.
We now justify the extension of each identity to arbitrary analytic
coefficients, not by assuming density in a formal completion. First
choose a common output product on which both actual sides exist by
LN.A and the finite product theorem. Include the finitely many
intermediate coordinate images in the compact input domains of LN.3.
These products and Cauchy margins are fixed before selecting any output
degree. Increasing a finite derivative order changes its Cauchy
constant, not this domain.

Fix an output degree \(r\), and finite upper orders \(m_1,m_2\)
for two inputs. In their actual product, a contributing pair of
homogeneous degrees \(j,k\) and contraction multiindex \(\alpha\)
satisfies

\[
 \begin{gathered}
 j\le m_1,\quad k\le m_2,\\
 j+k-|\alpha|=r.
 \end{gathered}
 \tag{LN.20}
\]

In particular \(j\ge r-m_2\), \(k\ge r-m_1\), and
\(|\alpha|\le m_1+m_2-r\). There are finitely many such
indices. Likewise, in a transformed coefficient of degree \(r\),
the loss in LN.6 is at most \(m_0-r\). For an iterated transform
at degree \(r\), the total of its two losses is at most
\(m_0-r\), again a finite list. Consequently each coefficient
of the multiplication or square identities is a finite expression in
finitely many derivatives of finitely many holomorphic input
coefficients on the indicated common products.

Each input coefficient is holomorphic on a polydisc, so its Taylor
polynomials converge there on every smaller compact with every fixed
finite derivative. This follows by the iterated Cauchy coefficient
bound and the product of the resulting geometric series, with
slightly larger radii for the fixed derivatives. Approximate the
finite list occurring in the chosen degree by these Taylor
polynomials. Multiply them by their corresponding normal powers and
set all unused homogeneous coefficients to zero. The result is an
element of the localized polynomial algebra: a monomial in \(w\)
times \(\sigma^m\) is the finite Laurent monomial in
\(\eta\) and \(\sigma\) of LI.0. Polynomial identities therefore
hold for every approximant. Passing to the finite derivative
expressions just described proves each identity of the original
analytic coefficients on those same fixed actual products. This holds
for every output degree on one common domain. The actual normal-symbol coefficient
identification of §5.19 then proves equality of the actual operators.

Let \(J\) negate spatial base and spatial derivative coordinates and
fix time and its normal derivative. It preserves the actual product,
either by that literal linear substitution in the full contraction
or by the complete ordinary coordinate theorem §5.56. The already
proved polynomial square LI.8 consequently extends to all finite
actual inputs as

\[
 \mathcal L^2=\operatorname{Ad}(\tau^{-d})J.
 \tag{LN.21}
\]

The two chart copies are identified only for this square formula;
the map between their sheaves still covers LN.19. The right side is
invertible: the normal derivative has its actual two-sided inverse
by §5.19, and \(J^2=1\). Since a map commutes with its own square,
both inverse compositions are the identity for

\[
 \mathcal L^{-1}=J\operatorname{Ad}(\tau^d)\mathcal L.
 \tag{LN.22}
\]

For clarity the inverse on a named chart pair has

\[
 \begin{gathered}
 \tau\longmapsto\sigma,\\
 x_i\longmapsto-D_{y_i}\sigma^{-1},\\
 D_{x_i}\longmapsto y_i\sigma,\\
 t\longmapsto s+(E_y+d)\sigma^{-1}.
 \end{gathered}
 \tag{LN.23}
\]

Thus the image of time contains the indispensable \(+d\sigma^{-1}\).
The exact inverse contact map is LI.9; its principal time symbol is
\(s+y^{\mathsf T}w\). No fractional operator power is needed.

The top output coefficient of an input of order \(m_0\) is
\(p_{m_0}\circ\iota\), since its only possible loss is zero.
The substitution is a local biholomorphism, so a nonzero top
coefficient stays nonzero. This proves exact order preservation and
the principal-symbol contact map. The unit is the sole coefficient
\(p_0=1\), whose higher losses vanish. Linearity and compatibility
with restriction are immediate from the normally convergent formula
on every nested product. This proves the entire finite sheaf
assertion. ∎

#### LN.6. Multiplication and inverse for the full infinite class

**Theorem LN.C.** The same formulas give an isomorphism of the whole
projectivized infinite-order operator sheaves \(B\), extending
Theorem LN.B. Both inverse compositions, the square correction and
restriction identities hold for every full input satisfying LN.2.

**Proof.** Use only the positive cutoffs \(P^{[J]},Q^{[J]}\)
defined after LN.16. They are actual finite-order inputs, so LN.B
gives the required identities for every \(J\). Both entire negative
tails remain in each input; no finite-rank or finite-spatial
substitute is involved.

We give the full limiting argument, including the products that
contain transformed inputs. The binary contraction estimates
IK.4–IK.7 and their proof in §5.44 IK.2 apply uniformly to coefficient
families with common LN.2 constants. Removing terms cannot increase
their scalar absolute majorants. Each finite positive cutoff has
exactly the same bounds as its original input. Hence the products
of these cutoffs have common negative factorial bounds and common
every-epsilon positive bounds on one smaller input product. Their
coefficients and every prescribed derivative converge normally to
the full product: the omitted summands are precisely among the
grouped-index tails IK.6–IK.7. These are absolute tails of the
actual product, including all normal-covariable contractions.

To pass such a product family through LN.6, first fix an output
degree. The finitely many negative-input terms in LN.16 converge by
the just proved coefficient convergence. For the positive-input
terms, their common every-epsilon bound and LN.14 give a common
summable scalar majorant; the bound beyond a loss cutoff is a
geometric tail for a fixed positive output, and the binomial tail
of LN.18 for a fixed negative output. First make that tail small,
then pass the remaining finite sum to its coefficient limit. The
same reasoning on a common compact enlargement handles each fixed
derivative. This proves convergence of the transformed product to
the transform of the full product.

On the other side, LN.15–LN.18 give common growth bounds and
coefficient/derivative convergence of the transformed cutoff inputs.
The absolute scalar majorants in IK.2–IK.3 again apply to their
products. A finite subset of contraction indices converges term by
term; its complementary absolute sum is uniformly small by those
same majorants. This proves convergence to the product of the full
transformed inputs. Passing the finite-order multiplication identity
to these two limits yields

\[
 \mathcal L(P\circ Q)=\mathcal L(P)\circ\mathcal L(Q).
 \tag{LN.24}
\]

Apply the same argument to the transformed cutoff family a second
time. Its bounds LN.15–LN.16 are common, and the preceding finite-sum
plus-summable-tail argument proves convergence of the second
transform. The square identity LN.21 for the cutoffs therefore gives
LN.21 for every full input. The right side also passes to its
limit: reflection is a literal fixed coordinate substitution, and
left and right multiplication by the fixed powers \(\tau^{\pm d}\)
has exactly the absolute full-product bounds just used. LN.22 is
therefore an actual two-sided inverse on the full class. The unit,
finite restriction and sheaf restriction identities pass either
directly from LN.6 or from these same uniform limits. ∎

No highest order or principal symbol is asserted for a full input
with an infinite positive head. The filtered assertion belongs to
the finite actual subring; the full map is the normally defined
coefficient transformation proved here.

#### LN.7. The geometric projectivized ring and the remaining real step

The geometric projectivized ring of §5.45 IG.1–IG.5 and IM.1–IM.6
has the complete actual normal-symbol identification proved there.
Transporting by that identification gives the same entire
projectivized geometric-ring isomorphism as Theorem LN.C. This uses
the proved identification of that sheaf; it makes no identification
of an arbitrary real-phase kernel germ with a homogeneous symbol.

The two compatible finite/infinite maps also transport every balanced
extension. Explicitly, for a left module \(M'\) over the input
finite ring and its transported left module \(M\),

\[
 \begin{gathered}
 B'\otimes_{E'}M'\longrightarrow B\otimes_E M,\\
 b'\otimes m\longmapsto\mathcal L(b')\otimes m.
 \end{gathered}
 \tag{LN.25}
\]

This is well defined because the finite and full maps agree and
preserve multiplication: the two representatives of a balancing
relation both map to \(\mathcal L(b')\mathcal L(e')\otimes m\).
The inverse uses LN.22; both compositions are the identity on pure
tensors, which generate the algebraic tensor product. No flatness
of the real operator ring occurs in this argument.

In particular the contact factorization LI.12 uses actual whole
\(E/B\) Legendre maps on both sides and the actual whole \(E/B\)
ordinary time-change map of §5.56 in its middle. LN.8 below proves its
equality with the particular whole quadratic shear of §5.49, for every
normalization parameter. That equality is extended by the finite
coefficient and full-tail arguments, not just by checking generators.

For the source-defined real ring \(C\), §5.57 LI.D supplies both
full topological convolution inverses. The further real-ring
construction must still give its actual holomorphic microkernel,
normalized inverse cup/excision/trace products and compatibility with
the whole projectivized map. The formulas in the present section
neither assume nor prove that every real germ has a homogeneous
normal coefficient sequence. The arbitrary-holonomic comparison and
regularization obligations consequently remain separate.

Free primary human context is the incidence section and operator
compatibility in [Benchoufi, Action of micro-differential operators
on quantized contact transformations, arXiv:2102.08861v2](https://arxiv.org/abs/2102.08861v2),
§§3–4. The complete proofs used here are the displayed argument and
the exact earlier programme sections specified in LN.0; the paper's
simple-module and microfunction-equivalence theorems are not imported
as substitutes for them.

#### LN.8. Every complex normalization and the entire quadratic factorization

There is an actual identity-contact automorphism for every complex
number \(a\). Define it directly by

\[
 (\mathcal H_aP)_j
 =\sum_{k\ge0}\binom ak\partial_t^k p_{j+k}.
 \tag{LN.26}
\]

In this definition the coefficients use the named output time and spatial chart. This is a definition by a convergent coefficient formula; no operator
\(\tau^a\) is adjoined. In an input chart the same formula uses
its time coordinate. On a fixed common larger time disc with margin
\(\delta>0\),
\(|\binom ak|\le c_a^k\), where
\(c_a=\max(1,|a|)\). To check this, bound each factor of its
falling product by \((|a|+r)/(r+1)\le c_a\),
\(0\le r<k\). Cauchy's formula bounds its \(k\)-th derivative
by \(k!\delta^{-k}\|p\|\). Thus every loss term in LN.26
is at most \((c_a/\delta)^kk!\|p\|\), independently of the
input degree. LN.15–LN.18, with this loss constant, prove the actual
finite/full mapping assertions, every inner sum and every fixed
derivative tail on common domains.

We verify the algebra and parameter identities first on the complete
localized polynomial algebra. For a nonnegative integer \(a=N\),
the actual finite Leibniz rule gives
\(\mathcal H_NP=\tau^N\circ P\circ\tau^{-N}\).
It fixes the normal and spatial derivative generators and spatial
coordinates, and sends time to \(t+Nh\). On a normal polynomial
Laurent input, only finitely many time derivatives are nonzero. Each
coefficient of its formula, each product identity and each composed
parameter identity is consequently a polynomial in its finitely many
complex parameters. Its agreement at all nonnegative integer
parameters proves agreement everywhere: a one-variable polynomial
with infinitely many zeros is zero, by repeated division by its
distinct linear factors; apply this successively to each parameter.
We obtain on that whole algebra

\[
 \begin{gathered}
 \mathcal H_a\mathcal H_b=\mathcal H_{a+b},\\
 \mathcal H_a(P\circ Q)
 =\mathcal H_aP\circ\mathcal H_aQ,\\
 \mathcal H_a(t)=t+ah.
 \end{gathered}
 \tag{LN.27}
\]

The other generator images are fixed, and the unit is fixed. The
finite output-degree argument LN.20 extends these identities to
arbitrary analytic finite symbols: every coefficient is still a
finite derivative expression. The common bounds just proved and the
complete limiting argument LN.6 extend them to all full inputs.
Thus \(\mathcal H_a\) is a whole \(E/B\) automorphism with
inverse \(\mathcal H_{-a}\), covering the identity contact map.

Define the normalized Legendre family by

\[
 \mathcal L_a=\mathcal H_a\mathcal L.
 \tag{LN.28}
\]

These are entire finite/full ring isomorphisms. Their coefficient of
total loss \(K\) on one input coefficient is explicitly the finite
sum

\[
 A_K^{(a)}p
 =\sum_{u+v=K}\binom au\partial_t^u A_vp.
 \tag{LN.29}
\]

Choose the intermediate compact product and its larger product before
all indices, so LN.14 bounds \(A_v\) on that larger product and
the subsequent time Cauchy estimate has one fixed margin. If the
two loss constants are \(L_1,L_2\), the summand is bounded by
\(L_1^uL_2^v u!v!\|p\|\). Since \(u!v!\le K!\) and
the binomial coefficients in \((L_1+L_2)^K\) are at least one,
the whole finite sum is bounded by
\((L_1+L_2)^K K!\|p\|\). It therefore has all LN.15–LN.18
tails with this new constant, still independently of input degree.

On the polynomial algebra LN.28 has exactly LI.3's values for
parameter \(a\), because \(\mathcal H_a\) fixes \(E_xh\)
and adds \(ah\) to time. Its two inverse compositions and square
were proved in LI.A. The same finite derivative argument LN.20 and
the full limiting argument now give, on the entire finite and full
sheaves,

\[
 \begin{gathered}
 \mathcal L_a^{-1}=J\mathcal L_{d-a},\\
 \mathcal L_a^2=\mathcal H_{2a-d}J.
 \end{gathered}
 \tag{LN.30}
\]

Here, as in LN.21, chart names are identified only to write the square
and reflection. At \(a=d/2\) the square is exactly spatial
reflection, without adjoining a fractional normal power. At \(a=0\)
the integer parameter \(-d\) gives exactly LN.21 and its actual
inner conjugation. The ordered inverse time generator is
\(s+(E_y+d-a)\sigma^{-1}\) in every case.

**Theorem LN.D.** For every complex symmetric matrix \(C\),
including every degenerate matrix, and every complex \(a\), the
entire finite and full maps satisfy

\[
 \mathcal L_a\Theta_C\mathcal L_a^{-1}=T_C.
 \tag{LN.31}
\]

Here \(\Theta_C\) is the actual ordinary time-change map
LI.11/LI.13 of §5.56, and \(T_C\) is the full quadratic shear
already constructed in §5.28 Q.4–Q.6 and §5.49 ISR.5. In particular
the whole map is independent of normalization.

**Proof.** LI.B proves this identity on the entire localized
polynomial algebra, not merely a set of isolated values: it proves
both sides are ring maps and proves equality on their generating
elements. The remaining task is extension to all analytic and full
inputs, which we now justify explicitly.

Each of the maps is nonincreasing in homogeneous degree. The common
products for their finite composition are chosen first from LN.A and
the earlier time/shear bounds; they do not depend on output degree.
At a fixed finite output degree and for a finite input upper order, each side
depends on finitely many input coefficients and finitely many of
their derivatives on the appropriate substituted compact products.
For the Legendre maps this is LN.5/LN.29 and LN.20. For the time
map it is §5.56 BCT.21–BCT.22, equivalently the ordered-word formula
DP.27–DP.31 of §5.55; at fixed loss the word length and all derivative
orders are finite. For the quadratic map it is precisely Q.9–Q.10
of §5.28: a fixed loss has finitely many resummation words, with
total base derivative order at most twice that loss. Thus every
coefficient of the desired finite-order identity is a finite
expression in a finite derivative list. Choose a common outer input
product and nested intermediate/output products for this finite
composition; their holomorphic coordinate images lie compactly
inside the preceding products. Taylor polynomials converge there
with all those derivatives. The polynomial identity just proved
therefore passes to each analytic coefficient, exactly by LN.20's
finite calculation. The actual finite normal-symbol identification
then gives LN.31 for every finite input.

For full inputs retain the entire negative tail and truncate only
the positive homogeneous head. Theorems LN.A–LN.C and LN.29 give
the common-domain bounds for the two normalized Legendre maps.
Section5.56 BCT.23–BCT.29 gives the corresponding all-index bounds
and positive-input cutoff tails for the actual time map. Section5.49
ISR.14–ISR.17 gives them for the whole quadratic map. There are only
finitely many intermediate maps, so choose their nested products
and the least of their finitely many margins before all degrees and
cutoffs. Each mapped cutoff family has common negative and
every-epsilon positive bounds. The finite-sum plus summable-tail
argument of LN.6 therefore passes through each successive map.
The finite-order identity tends normally, with every fixed derivative,
to LN.31 for the original entire input. No agreement on generators
alone, formal completion or real-germ normal form is used for this
last step. ∎

#### LN.9. A whole positive head with every negative output nonzero

In one spatial dimension take

\[
\begin{gathered}
p_m(s,y,w)=\frac1{(m!)^2}\\
{}\cdot\frac1{(1-y/2)(1+w/2)}\\ (m\ge0),\\
p_m=0\quad(m<0).
\end{gathered}
 \tag{LN.32}
\]

These are holomorphic on one common product \(|y|,|w|<1\),
with time unrestricted near its chosen point. On any compact there,
the rational factor is bounded. For every \(\epsilon>0\),
\(1/(m!\epsilon^m)\le e^{1/\epsilon}\), because it is one
nonnegative term of that exponential series. This proves the full
every-epsilon condition LN.2 on the same domain. The symbol has an
infinite positive head and no negative input coefficient.

Since its coefficients have no time dependence, LN.5 differentiates
only the two displayed rational factors. Their \(K\)-th derivatives
are \((1/2)^K K!/(1-y/2)^{K+1}\) and
\((-1/2)^K K!/(1+w/2)^{K+1}\). LN.5's own sign cancels
the latter sign. At \(t=x=z=0\), put \(c=1/4\). The whole
transformed coefficients are exactly

\[
\begin{gathered}
q_j(0)=\\ \sum_{K\ge\max(0,-j)}
\frac{c^K K!}{((j+K)!)^2}.
\end{gathered}
 \tag{LN.33}
\]

Every term is positive and the series converges by LN.A, or by the
successive-term calculation below. Thus every output coefficient,
including every negative degree, is nonzero.

There is also an exact all-negative-degree closed expression using
only a finite polynomial and the exponential. For \(n\ge0\),
write \(K=n+m\). The elementary counting identity
\(\binom{n+m}{n}=\sum_{r=0}^n\binom nr\binom mr\)
counts an \(n\)-element selection from disjoint sets of sizes
\(n\) and \(m\), according to how many are selected from the
second. Its terms with \(r>m\) are zero. The positive sums can
therefore be interchanged, and
\(\sum_{m\ge r}\binom mr c^m/m!=c^r e^c/r!\)
by substituting \(m=r+v\) in the exponential series. Consequently

\[
\begin{gathered}
q_{-n}(0)=c^n n!e^c\\
{}\cdot\sum_{r=0}^n\binom nr\frac{c^r}{r!}.
\end{gathered}
 \tag{LN.34}
\]

This formula includes \(q_0(0)=e^{1/4}\). It is derived from
the entire input head, not from a fixed finite-order truncation.

For exact finite enclosures let \(b_{j,K}\) denote the summand
in LN.33 and \(S_{j,R}\) its sum through \(K=R\). The ratio is

\[
 \frac{b_{j,K+1}}{b_{j,K}}
 =\frac{K+1}{4(j+K+1)^2}.
 \tag{LN.35}
\]

If \(K\ge2|j|\), then \(j+K+1\ge(K+2)/2\), so this
ratio is at most \((K+1)/(K+2)^2\le1/(K+2)\le1/2\).
For any integer \(R\ge2|j|\), summing the positive omitted tail
against a geometric series gives the rational enclosure

\[
\begin{gathered}
S_{j,R}<q_j(0)\\ \le S_{j,R}+2b_{j,R+1}.
\end{gathered}
 \tag{LN.36}
\]

All factorials in these finite sums have nonnegative arguments. This
calibrates the full transform, its sign and its positive-to-negative
mixing. The general analytic and ring assertions are proved in
LN.1–LN.8 independently of these samples.

#### LN.10. The coefficient mechanism in an exact sample

![Full Legendre positive-to-negative coefficient mixing with exact enclosures](assets/full-legendre-normal-transport.png)

*Figure LN.1. The left panel is a finite window of individual
contributions in the whole one-dimensional example LN.32–LN.33,
at \(t=x=z=0\). Its column is the nonnegative input degree
\(m\), its row is the output degree \(j\), and its loss is
\(K=m-j\ge0\). The color records the logarithm of the exact
positive rational contribution \(c^{m-j}(m-j)!/(m!)^2\),
\(c=1/4\). Gray cells have negative loss and no contribution.
The displayed window is \(0\le m\le12\),
\(-6\le j\le6\); the input series and the proof have no such
cutoff. The right panel shows the rational enclosures LN.36 for
\(-5\le j\le5\), with \(R=40\); the interval widths are
smaller than the markers. Every rational contribution, partial sum
and upper bound is retained in the data file. Only logarithms and
plotting coordinates are numerical. LN.34 proves positivity in every
negative output degree, and LN.14–LN.18 prove the full uniform
factorial bounds and tails. LN.24 and LN.30–LN.31 give the entire
ring, inverse and quadratic-factorization identities. This is an
illustration of the actual finite/projectivized operator maps; it
does not assert a completed whole-real kernel transform. The
editable figure source,
exact data,
full PNG and
SVG are provided.
Free human context is the incidence quantization in
[Benchoufi, arXiv:2102.08861v2](https://arxiv.org/abs/2102.08861v2),
§§3–4; the maps, coefficients and convergence bounds are proved in
the present section.*


### 5.59. Open and closed incidence in the actual microlocal Hom complex

Let \(M\) be a complex manifold of complex dimension \(m\).
Write \(q_1,q_2:M\times M\to M\) for the two projections.
The complex orientation orders the \(2m\) real coordinates in
each projection fibre. Coefficient sheaves and holomorphic top-form
lines have degree zero before an explicitly displayed shift.

For a closed subset \(S\subset M\), put \(\Omega=M\setminus S\).
The constant sheaf on \(\Omega\) means extension by zero; the
constant sheaf on \(S\) means closed direct image. We shall prove,
off the zero cotangent section, the natural comparison

\[
\begin{gathered}
\mu\operatorname{hom}(\mathbb C_\Omega,F)\\
\longrightarrow\\
\mu\operatorname{hom}(\mathbb C_S[-1],F).
\end{gathered}
\tag{OCM.1}
\]

It is an isomorphism for every bounded complex \(F\) of complex-vector
sheaves. The proof supplies the connecting sign and finite cochain
witnesses on actual support domains. In the incidence application,
\(F\) can be the holomorphic input top-form sheaf. The comparison
does not assert that a holomorphic incidence section has inverse
cup/excision/trace products or defines an action on arbitrary real
operator germs.

The injective and smooth-resolution constructions used below are
proved in [§5.34, R.5–R.7](#5-34-full-current-resolutions-and-controlled-proper-image-comparison). The relative differential and connecting
sign are [§5.35, V.1](#5-35-relative-cones-unshifted-cup-and-the-normalized-trace). The cofinal support cones and finite equality
witnesses are [§5.45, IG.1](#5-45-the-full-geometric-infinite-order-kernel-ring-and-its-canonical-product), and [§5.53, RM.3–RM.9](#5-53-real-kernel-comparison-for-all-constant-euler-matrices).
The compact normal-fibre contraction used to normalize projection
duality is [LI.8.2 of §5.57](#5-57-legendre-incidence-and-the-topological-inverse-of-projective-duality). All the additional comparisons needed
here are proved in the following argument.

#### OCM.0. Unshifted support complexes

Use \(u\) for the first base point and \(v=u+n\) for the second
point in a coordinate chart. The real normal variable has dimension
\(r=2m\). For a closed support \(H\) in an actual pair domain \(D\),
the relative complex of a resolution \(I\) has differential

\[
\begin{gathered}
C_H^k(I)=\\
I^k(D)\oplus I^{k-1}(D\setminus H),\\
d(a,b)=\\
(da,\,a|_{D\setminus H}-db).
\end{gathered}
\tag{OCM.2}
\]

This is the convention V.1, without an implicit cohomological shift.
R.7 proves its comparison with derived sections with support.
For a complex \(K\) on the pair, denote its unshifted
support-microlocal complex at a nonzero normal covector \(p\) by

\[
\begin{gathered}
\mathfrak m_{\Delta,p}(K)\\
=\mathop{\rm colim}_{D,H}R\Gamma_H(D;K),\\
C_\Delta(H)\setminus0\\
\subset\{-\Re\langle n,p\rangle>0\}.
\end{gathered}
\tag{OCM.3}
\]

The normal-cone convention includes the input antipodal operation.
These are the support domains of IG.1 in all cochain degrees.
Restrictions and support enlargements are the structure maps.
The coefficient-independent geometric proof in IG.1 makes the
round cones used below cofinal on a smaller base/covector product.

This construction preserves exact triangles. Indeed an injective
resolution computes each relative complex OCM.2. A morphism and
its cone form the usual finite cone diagram on both opens, so
taking a relative cone preserves that diagram. Filtered colimits
of complex-vector spaces are exact: every finite cycle, relation
and primitive occurs at one later index. Thus the resulting cone
diagram remains an exact triangle. The same argument says that a
zero stalk class has an actual primitive at one later support
domain. No infinite family of domains is made simultaneous.

The projection normalization in OCM.4 below gives the support
model for microlocal Hom:

\[
\begin{gathered}
\mu\operatorname{hom}(A,F)_p\\
=\mathfrak m_{\Delta,p}R\mathcal Hom\bigl(\\
q_2^{-1}A,q_1^{-1}F[2m]\bigr).
\end{gathered}
\tag{OCM.4}
\]

The orientation line is trivialized by the complex orientation.
It is an ordinary coefficient line, so it introduces no
unwritten graded permutation.

#### OCM.1. A cone and the same contraction on both opens

Normalize the central covector to the first real normal coordinate
and write \(n=(a,b)\in\mathbb R\times\mathbb R^{r-1}\).
After one support enlargement and domain restriction, use

\[
\begin{gathered}
D=U\times B_R,\\
H=D\cap(U\times G),\\
G=\left\{\begin{gathered}
(a,b):a\le0,\\
|b|\le L(-a)
\end{gathered}\right\},\\
L>0,\quad e=(c,0),\\
0<c<R.
\end{gathered}
\tag{OCM.5}
\]

Here \(U\) and \(B_R\) are real coordinate balls. The support
enlargement is legitimate: a strict polar margin on the compact
unit normal cone gives a uniform upper bound on \(|b|/(-a)\).
Choose \(L\) strictly larger. Failure of actual containment on
every smaller pair domain would give a normalized sequence
converging to a normal-cone vector violating that strict bound,
as in IG.1.

The cone \(G\) is closed, convex and stable under nonnegative
scaling. Its nonzero vectors satisfy

\[
 -a\ge\frac{|n|}{\sqrt{1+L^2}}>0.
 \tag{OCM.6}
\]

Consequently it is admissible on one actual covector neighbourhood:
a perturbation of the real polar functional by norm less than
half the constant in OCM.6 retains half that margin.
The zero vector is included, as required for a closed support.

Define, on the whole ball,

\[
\begin{gathered}
h_s(n)=(1-s)n+se,\\
0\le s\le1.
\end{gathered}
\tag{OCM.7}
\]

Convexity keeps this path in \(B_R\). It also keeps
\(B_R\setminus G\) in that complement. For \(s<1\), membership of
\(h_s(n)\) in \(G\) would imply
\((1-s)n=h_s(n)-se\in G\), since \(-e\in G\).
This contradicts \(n\notin G\). At \(s=1\), \(e\notin G\).
Thus the same smooth homotopy contracts both opens to the same
point, and commutes exactly with restriction from the ball to
its complement. It does not require a deformation preserving
the closed support pointwise.

The argument works for \(r=1\) as well, with no \(b\) variable.
For \(m=0\) the nonzero cotangent space is empty.

#### OCM.2. Arbitrary pulled-back coefficients and an actual relative primitive

Let \(F\) first be one complex-vector sheaf on \(U\), and put
\(P=\operatorname{pr}_U^{-1}F\). Its coefficients are constant in
the normal variable; no local constancy in \(u\) is required.
The real de Rham complex

\[
  \mathcal A^\bullet_{U\times B_R}
           \otimes_{\mathbb C}P
 \tag{OCM.8}
\]

resolves \(P\). On a stalk this is the ordinary de Rham
Poincaré complex tensored over a field with \(F_u\).
The usual radial integration proves that complex's exactness;
the tensor is exact because vector subspaces have linear
complements. Its terms are smooth-module sheaves and hence
acyclic on every open by R.5. R.6–R.7 therefore identifies its
relative complex with derived supported sections.
For a bounded complex \(F\), use the finite total complex with
differential \(d_{\mathcal A}+(-1)^k d_F\) on form degree \(k\).
The same stalk argument and R.6 finite elimination apply.

Pull back along \((u,n,s)\mapsto(u,h_s(n))\) and integrate the
coefficient of \(ds\) in the ordered interval direction. This
defines a degree-minus-one operator \(K\) with

\[
\begin{gathered}
dK+Kd=j\varepsilon-1,\\
\varepsilon j=1.
\end{gathered}
\tag{OCM.9}
\]

Here \(\varepsilon\) evaluates at the normal point \(e\), and \(j\)
pulls base forms back to the product. The formula follows
by writing the pulled-back form as \(\alpha_s+ds\wedge\beta_s\),
integrating its \(s\)-derivative, and using the fundamental theorem
of calculus. The two terms involving \(d_F\) cancel, since \(K\)
lowers the form degree by one.

There is no integration in an unspecified topological vector
space. Near any given \((u,n)\), the homotopy path is compact.
Cover it by finitely many rectangles on each of which a section
of OCM.8 has a finite tensor expression. Shrink the finitely many
base opens to a common one and choose a finite smooth partition
in the interval variable. Integration then acts on the smooth
scalar forms in finitely many expressions and leaves their
sections of \(F\) fixed. Refining these expressions or their
partition gives the same result by linearity of scalar
integration and the tensor relations. The local operators glue.
Differentiation commutes with these finite compact integrals.
Thus OCM.9 holds for arbitrary sheaves, not just finite-dimensional
or locally constant coefficients.

Let \(A\) and \(B\) be these complexes of sections on \(D\) and
\(D\setminus H\), respectively, and let \(r:A\to B\) be restriction.
Write \(K_A,K_B,j_A,j_B,\varepsilon_A,\varepsilon_B\) for the operators on the two
opens. Since the same homotopy is used,

\[
\begin{gathered}
rK_A=K_Br,\\
rj_A=j_B,\\
\varepsilon_Br=\varepsilon_A.
\end{gathered}
\tag{OCM.10}
\]

For the relative complex OCM.2 define

\[
\begin{gathered}
T(a,b)=\\
\left(\begin{gathered}
j_A\varepsilon_Bb-K_Aa,\\
K_Bb
\end{gathered}\right).
\end{gathered}
\tag{OCM.11}
\]

It is an actual degree-minus-one relative operator. Its first
entry has degree \(k-1\) and its second degree \(k-2\) on an
input of relative degree \(k\). Direct expansion gives

\[
 dT+Td=1.
 \tag{OCM.12}
\]

For the first entry, the terms containing \(db\) cancel, leaving
\(j_A\varepsilon_Aa-(dK_A+K_Ad)a=a\). For the second, OCM.10 cancels
the two terms containing \(rK_Aa\), and the remaining expression
is \(j_B\varepsilon_Bb-(dK_B+K_Bd)b=b\).
This verifies both signs in OCM.11. An extension of an arbitrary
coefficient section through the support is never assumed.

**Theorem OCM.A.** For every bounded complex \(F\) and every
nonzero \(p\),

\[
 \mathfrak m_{\Delta,p}(q_1^{-1}F)=0.
 \tag{OCM.13}
\]

**Proof.** Enlarge each of the finitely many supports under
consideration to OCM.5 and restrict to one actual product.
On that product OCM.11–OCM.12 contracts the relative complex.
R.6–R.7 transfers the cochain and its primitive to an injective
model by finite elimination in the degrees in use. Each step
has an actual restriction/support map, and finitely many such
steps have one common later index. Hence every stalk class is
zero in every degree. A common coefficient shift preserves this
assertion. The strict covector margin in OCM.6 makes it a sheaf
vanishing on the nonzero cotangent chart, not only a calculation
at one isolated covector. \(\square\)

#### OCM.3. The projection normalization

For completeness the projection term in OCM.4 can be normalized
without a duality citation. Let \(p:V\times B\to V\), where
\(B\) is an oriented real ball of dimension \(r=2m\).
Integration of compactly supported top vertical forms, with
the base form factors first, defines the trace

\[
 Rp_!p^{-1}F[r]\longrightarrow F.
 \tag{OCM.14}
\]

Use the fine real de Rham resolution tensored over
\(\mathbb C\) with \(p^{-1}F\). Its terms are \(p_!\)-acyclic
by the compact support argument of R.5; R.6 computes the derived
image. Locally over a point of \(V\) only finitely many coefficient
expressions meet a proper support, so the integral is the same
finite scalar operation as in OCM.2. The integral of a vertical
derivative is zero by the fundamental theorem of calculus
and compact support. Since \(r\) is even, the shift differential
has no additional minus sign. The target of this integration
on resolutions is \(\mathcal A_V^\bullet\otimes F\);
its unit augmentation from \(F\) is a quasi-isomorphism by
the stalk argument of OCM.2. Identifying that target in the
derived category gives OCM.14.

Choose in each real coordinate a smooth compactly supported
one-form of integral one. Subtracting its integral multiple and
integrating from the negative endpoint contracts the one-variable
compact de Rham complex to that form. Tensoring the \(r\)
coordinate contractions gives
\(R\Gamma_c(B;\mathbb C)=\mathbb C[-r]\), with positive
orientation and integral one. This is the explicit compact
contraction of LI.8.2, after an orientation-preserving ball/box
coordinate change; an increasing coordinate diffeomorphism
transports the one-variable formula. No integration constant
is suppressed.

The trace gives the adjunction on a rectangle
\(W=V_0\times B_0\), extended by zero. Indeed,

\[
\begin{gathered}
Rp_!\mathbb C_W=\mathbb C_{V_0}[-r],\\
R\operatorname{Hom}
(\mathbb C_W,p^{-1}F[r])\\
=R\Gamma(V_0;F)[r].
\end{gathered}
\tag{OCM.15}
\]

The first identity is the compact contraction and the compact
product/stalk argument of LI.8.1. For the second, the coefficient
homotopy OCM.9 contracts the ball factor, with its actual fine
resolution. Both identifications send OCM.14 to the identity.
Extension from one oriented ball to a larger one sends the
integral-one class to the integral-one class; restrictions in
the base commute with it.

These rectangle tests establish the adjunction for every sheaf
complex in the stated bounded range, as follows. Every sheaf
has a surjection from a sum of constant sheaves on coordinate
rectangles extended by zero: take one summand for every local
section on every such rectangle. Its map sends \(1\) to that
section, and surjectivity is checked on germs. Repeat with
kernels to obtain a resolution in nonpositive degrees.
The real de Rham resolution on the product has finite length
\(\dim_{\mathbb R}V+r\) and \(p_!\)-acyclic terms by R.5;
therefore, in any fixed degree of its pushed total complex
only finitely many resolution columns occur. On the Hom side,
a bounded-below injective target likewise leaves finitely
many columns in each degree. Applying the rectangle trace
comparison and eliminating the successive resolution columns
proves the adjunction. Direct sums cause no compact-support
problem: on a compact subset, a locally finite family of
nonzero components is finite. The same argument applies after
each base restriction and glues the local trace.
Consequently the projection's right adjoint is

\[
 q_1^!F=q_1^{-1}F[2m],
 \tag{OCM.16}
\]

with the displayed complex orientation. This proves the
normalization used in the microlocal Hom support model OCM.4.

#### OCM.4. The constant first argument vanishes off zero

For any complex \(J\) of injective sheaves,
\(\mathcal Hom(\mathbb C_{M\times M},J)=J\):
a map is determined by the image of \(1\) on every open.
The same identity computes derived Hom. Therefore

\[
\begin{gathered}
R\mathcal Hom(q_2^{-1}\mathbb C_M,q_1^!F)\\
=q_1^{-1}F[2m],\\
\mu\operatorname{hom}(\mathbb C_M,F)\\
\big|_{\dot T^*M}=0.
\end{gathered}
\tag{OCM.17}
\]

The second line is precisely OCM.A, including its relative
primitive and finite common witnesses. It uses no assertion
that the holomorphic sheaf itself has zero microlocalization
along an arbitrary hypersurface. The coefficient in OCM.17
is pulled back from \(u\), hence constant in the diagonal
normal variable \(v-u\); this property is essential.

#### OCM.5. The connecting map and its two inverse witnesses

There is a stalkwise exact sequence

\[
\begin{gathered}
0\longrightarrow\mathbb C_\Omega
\longrightarrow\mathbb C_M\\
\longrightarrow\mathbb C_S
\longrightarrow0.
\end{gathered}
\tag{OCM.18}
\]

Off \(S\), its first nonzero map is the identity on \(\mathbb C\).
On \(S\), its last nonzero map is that identity. This proves
exactness without a regularity hypothesis on \(S\).
Let \(I\) be a bounded-below injective resolution of \(q_1^!F\).
Applying sheaf Hom to the inverse image of OCM.18 gives
degreewise short exact complexes

\[
\begin{gathered}
0\longrightarrow E^\bullet
\xrightarrow{\iota}D^\bullet\\
\xrightarrow{\pi}G^\bullet
\longrightarrow0.
\end{gathered}
\tag{OCM.19}
\]

where \(E,D,G\) correspond respectively to
\(\mathbb C_S,\mathbb C_M,\mathbb C_\Omega\).

Here these Hom sheaves are injective term by term.
Tensoring with a complex-vector sheaf is exact on stalks,
and sheaf Hom into an injective is its right adjoint;
therefore that Hom sheaf is injective. Restriction of an
injective to an open is injective by the exact extension-by-zero
adjunction. Thus the same assertion holds on every open.
The Hom sequence is exact, and its injective kernel gives
a sheaf splitting in each degree. These splittings need
not commute with the differential.

Choose a graded splitting \(\sigma:G\to D\). The difference

\[
 \iota c=d_D\sigma-\sigma d_G
 \tag{OCM.20}
\]

defines \(c:G^k\to E^{k+1}\), since its projection by \(\pi\)
is zero. Expanding \(d_D^2\sigma-\sigma d_G^2\) proves
\(d_Ec+cd_G=0\). Hence \(c:G\to E[1]\) is a chain map,
with the shift differential \(-d_E\).
If \(\sigma'=\sigma+\iota h\), then

\[
 c'-c=d_Eh-hd_G.
 \tag{OCM.21}
\]

This is the shifted chain homotopy with homotopy \(-h\);
the sign cannot be replaced by the unshifted projection sign.
Every splitting is a sheaf map, so these maps and homotopies
commute with restriction and preserve supports.
On the relative complexes the connecting map is explicitly
\[
(a,b)\mapsto(ca,-cb).
\tag{OCM.20a}
\]
Indeed subtracting the graded splitting times the relative
differential from the relative differential times that splitting
gives \(ca\) on the first entry and \(-cb\) on the complement
entry. Its target is the entire relative \(E\)-complex shifted
by \(+1\), not an unshifted copy on just the complement.
Applying OCM.3 therefore gives a well-defined connecting
morphism independent of those splittings in the derived category.

OCM.17 makes its middle complex acyclic off zero.
The resulting isomorphism can also be verified on actual
finite cocycles. If \(g\) is a closed \(G\)-cochain, lift it to
\(b\) in the middle relative complex. Then \(db=\iota a\)
for a closed \(E\)-cochain \(a\) one degree higher. Its class
is the connecting image. Conversely, for a closed \(a\),
the relative middle cochain \(\iota a\) has a primitive \(b\)
by OCM.11–OCM.13 and R.6–R.7. Its projection \(\pi b\)
is closed and maps to \(a\).

If two such primitives are used, their difference is closed
in the middle complex and hence has a primitive there too;
its projection is a boundary. If a connecting image is a
boundary, subtract a lift of that boundary primitive from
\(b\); the resulting closed middle cochain is again a boundary,
and its projection proves the original class zero.
All lifts and comparisons use finitely many entries on the
two opens of the relative cone. Enlarge their finitely many
supports, intersect their finitely many base/covector products,
and apply OCM.11 there. Thus these are actual finite common
equality witnesses, not coefficientwise choices or a limit
with an assumed closed boundary image.

**Theorem OCM.B.** For every closed \(S\), every bounded
complex \(F\), and every nonzero cotangent chart, OCM.1 is
a natural isomorphism. Its source/target order is

\[
\begin{gathered}
\mu\operatorname{hom}(\mathbb C_\Omega,F)\\
\xrightarrow{\ \partial^\vee\ }\\
\mu\operatorname{hom}(\mathbb C_S,F)[1]\\
=\mu\operatorname{hom}(\mathbb C_S[-1],F).
\end{gathered}
\tag{OCM.22}
\]

**Proof.** OCM.20 is the contravariant connecting map of
OCM.18. Its middle term vanishes by OCM.17, and the preceding
finite cochain argument proves both inverse identities.
This also proves the displayed shift: the source is the
quotient complex \(G\), and the target is the kernel \(E\)
shifted by \(+1\). \(\square\)

The construction is natural in \(F\), in maps of the
open/closed sequence, and in restrictions to smaller charts.
Indeed the connecting map is defined by the same exact
sequence; choosing different lifts changes it by OCM.21.
The injective comparisons of R.6 provide these lift changes
for any coefficient map. All finite commutative diagrams
have one common later support domain by OCM.3.
A holomorphic degree-zero coefficient line can therefore
be retained throughout, with its ordered frame unchanged.
The comparison commutes with finite differential-operator
maps on that coefficient sheaf, since those are
complex-linear sheaf maps. It does not infer an action
of an arbitrary infinite-order operator from that fact.

#### OCM.6. Incidence dimensions and the holomorphic coefficient

Take \(M=P^N\times(P^N)^*\), with \(N=d+1\), and let \(S\)
be its incidence hypersurface. Then \(M\) has complex
dimension \(2N\); the diagonal used to define its
microlocal Hom has real normal dimension \(4N\).
In an affine incidence chart,

\[
\begin{gathered}
f=s-t-x^{\mathsf T}y,\\
\eta={}\\
ds\wedge dy_1\wedge\cdots\wedge dy_d.
\end{gathered}
\tag{OCM.23}
\]

The function is a submersion since \(\partial_s f=1\).
Set \(F=\Omega_{M/P^N}\), using the input frame \(\eta\)
as an ungraded coefficient line. OCM.B identifies its
open-incidence and shifted closed-incidence microlocal Hom
complexes by OCM.22, with the actual finite witnesses above.
Changing the holomorphic frame multiplies by its ordinary
transition function; naturality shows that these local
comparisons glue without an additional exterior-degree sign.

The four dimension counts have different roles:

**Table OCM.24. Incidence dimensions.**

| Quantity | Value |
|---|---|
| Spatial variables | \(d\) |
| Complex dimension of \(P^N\) | \(N=d+1\) |
| Topological inverse shift in LI.D | \([2N]\) |
| Projection shift in OCM.4 on \(M\) | \([4N]\) |

The ordered \(+d\sigma^{-1}\) term in the Legendre inverse
[LN.23](#5-58-full-analytic-legendre-transport-and-quadratic-factorization) is unrelated to either real projection shift.
Likewise, the degree-one hypersurface residue of \(f^{-1}\)
in LI.22 is not the degree-\(N\) diagonal operator unit.
OCM.22 proves the open/closed comparison and fixes its
connecting sign; it does not identify either inverse
cup/excision/trace product of a holomorphic incidence class.
Those products, compatibility with the entire LN maps and
the action on arbitrary RM.3 real-kernel representatives
require their own construction.

Free primary context is Benchoufi,
[Action of micro-differential operators on quantized contact transformations,
arXiv:2102.08861v2](https://arxiv.org/abs/2102.08861v2),
§§3.2 and 4.3–4.5. The constant-argument vanishing,
projection normalization, connecting map and actual witnesses
used here are proved in OCM.0–OCM.6 and the exact earlier
programme sections cited above.

#### OCM.7. Normal geometry and the connecting degree

![A normal-cone complement contracted to one point, with the relative primitive and open/closed connecting degree](assets/open-closed-incidence-connecting-map.png)

*Figure OCM.1. The left panel is the real two-plane section in which
all other normal coordinates are zero. It uses \(R=2\), \(L=1\)
and \(e=(4/5,0)\). The shaded support is
\(G=\{a\le0,\ |b|\le-a\}\) intersected with the ball.
Each segment is the exact path OCM.7 from its displayed starting
point to \(e\). OCM.5–OCM.7 proves the complement-preserving
assertion in every normal dimension; the four segments illustrate
it in this section. Their rational coordinates and the values at
\(s=0,1/16,\ldots,1\) are in the
exact data.
The right panel gives the relative primitive OCM.11–OCM.12 and
the comparison OCM.22. In those formulas \((a,b)\) is a relative
cochain pair, rather than a pair of real normal coordinates.
The connecting complement entry is \(-cb\), as proved in
OCM.20a. The dimension row distinguishes the \(4N\)-dimensional
real diagonal normal used for microlocal Hom on the incidence
pair from the \(2N\) shift of LI.D. The actual holomorphic inverse
products are not asserted by this comparison. The
editable figure source,
full PNG and
SVG are provided.
Free human context for the incidence application is
[Benchoufi, arXiv:2102.08861v2](https://arxiv.org/abs/2102.08861v2),
§§3.2 and 4.3–4.5; the cone, primitive, connecting signs and
degree comparison are proved in the present section.*


### 5.60. The holomorphic incidence class and shrinking normal supports

Let \(M\) be a complex manifold of dimension \(m\ge1\), and let
\(S\subset M\) be a smooth hypersurface. Locally choose coordinates
\((z,f)\), where \(z\in\mathbb C^{m-1}\) and \(S=\{f=0\}\).
A holomorphic coefficient line \(F\) has degree zero.
We construct the single-pole class in the actual microlocal Hom
complex, retaining its real phase and connecting sign. We also
identify an actual finite differential-operator module that
injects into every nonzero conormal stalk.

The proof uses the injective and relative-support constructions
of [§5.34, R.5–R.7](#5-34-full-current-resolutions-and-controlled-proper-image-comparison); the connecting and ordered coefficient
conventions of [§5.35, V.1–V.6](#5-35-relative-cones-unshifted-cup-and-the-normalized-trace); the strict cofinal cones of
[§5.45, IG.1](#5-45-the-full-geometric-infinite-order-kernel-ring-and-its-canonical-product); the finite real-stalk witnesses of [§5.53,
RM.3–RM.9](#5-53-real-kernel-comparison-for-all-constant-euler-matrices); and the projection and open/closed comparisons of
[§5.59, OCM.A–OCM.B](#5-59-open-and-closed-incidence-in-the-actual-microlocal-hom-complex). Every further comparison is proved below.
The construction does not assert a finite-\(E\) simple-generator
theorem or either normalized inverse kernel product.

#### HI.0. The support functor and its polar sign

Use the microlocal Hom support model OCM.4. On \(M\times M\)
write the second point as
\((z',f')=(z+\zeta,f+n_f)\).
On the closed subset \(f'=0\),

\[
 \zeta=z'-z,\qquad n_f=-f.
 \tag{HI.1}
\]

For a nonzero conormal covector with normal coefficient \(p\),
define the degree-full hypersurface support complex

\[
 \begin{gathered}
 \mathfrak h_{f,p}(K)
   =\mathop{\rm colim}_{U,H}R\Gamma_H(U;K),\\
 C_S(H)\setminus0
   \subset\{\Re(fp)>0\}.
 \end{gathered}
 \tag{HI.2}
\]

The arrows are actual restrictions and support enlargements.
The positive polar sign in HI.2 is forced by HI.1:
the diagonal condition is \(-\Re(n_fp)>0\).
If \(\mathfrak m_{S,p}\) denotes the negative-polar support
complex, then

\[
 \mathfrak h_{f,p}=\mathfrak m_{S,-p}.
 \tag{HI.3}
\]

Thus the antipodal operation is explicit; \(p\) and \(-p\)
are not identified. All comparisons below use HI.2.
For a bounded coefficient complex they are comparisons in
every degree, not only its top cohomology.

#### HI.1. Continuous homotopies with pulled-back coefficients

We need a coefficient-preserving comparison that does not
differentiate a continuous radius function. Suppose
\(a:Z\to B\), \(E=a^{-1}K\), and
\(h:Z\times[0,1]\to Z\) is continuous with
\(a h=a\operatorname{pr}_Z\). Then

\[
 h^{-1}E=\operatorname{pr}_Z^{-1}E.
 \tag{HI.4}
\]

The endpoint pullbacks induce the same map on derived sections,
with finite cochain homotopies in the models used here.
Here is the comparison explicitly.

Extend \(h\) continuously to an open interval \(J\) containing
\([0,1]\), constantly beyond the two endpoints. The ordinary
projection \(Z\times J\to Z\), with coefficient \(E\), has
the inverse-image unit

\[
 \begin{gathered}
 R\Gamma(Z;E)\\
 \longrightarrow
 R\Gamma(Z\times J;\operatorname{pr}_Z^{-1}E).
 \end{gathered}
 \tag{HI.5}
\]

This is a quasi-isomorphism. OCM.8–OCM.9 proves it using a
real de Rham resolution tensored over \(\mathbb C\) with
the coefficient sheaf and a smooth contraction of \(J\).
The local integral uses finitely many coefficient expressions
along a compact interval, so arbitrary sheaves are allowed.
Both endpoint restrictions are inverse to HI.5 and therefore
agree in the derived category.

There is a pullback on derived sections for the continuous map
\(h\). To construct it, pull back an injective resolution
term by term, then apply the germ-product injective resolution
R.12–R.13 to the pulled-back terms. Its augmentation and total
differential give the pullback map. In each fixed total degree,
only finitely many columns occur, because the original complex
is bounded below. The finite comparison procedure R.6 compares
the two endpoint maps with the smooth interval model.
The interval integration operator there supplies their
degree-minus-one homotopy. Pullback of the smooth scalar
forms by the continuous map \(h\) is never used.

Consequently a homotopy equivalence over \(B\) induces a
quasi-isomorphism on derived sections of \(a^{-1}K\).
The same statement applies after every base restriction.
For any finite diagram, the comparisons and homotopies have
finitely many entries in the degrees in use; intersect their
finitely many domains to obtain a common actual witness.

#### HI.2. A shrinking ball family and its zero-radius comparison

Let \(B\) be a coordinate manifold, \(H\subset B\) closed,
and let \(\rho:B\to[0,R/2)\) be continuous and zero off \(H\).
On \(X=B\times B_R^k\), with projection \(\pi\), put

\[
 \begin{gathered}
 G_0=H\times\{0\},\\
 G_\rho=\{(u,v):u\in H,\\
 |v|\le\rho(u)\}.
 \end{gathered}
 \tag{HI.6}
\]

Both supports are closed. Their projections to \(B\) are
proper: over a compact base subset, the fibre coordinates
lie in the closed ball of radius strictly less than \(R/2\),
and the support is closed in that compact product.

For \(k>0\), set \(r=|v|\). On \(X\setminus G_0\) define
the base-preserving map

\[
 \begin{gathered}
 Q(u,v)=\left(u,\frac{\psi_{\rho(u)}(r)}r\,v\right),\\
 r>0,\\
 \psi_\rho(r)=\rho+(1-\rho/R)r.
 \end{gathered}
 \tag{HI.7}
\]

At \(v=0\), which is in this domain only for \(u\notin H\),
define \(Q(u,0)=(u,0)\). It is continuous there:
outside the closed set \(H\), \(\rho\) is zero on a
neighbourhood, so this map is the identity.
Everywhere else continuity follows from \(r>0\) and continuity
of \(\rho\).

For \(0<r<R\) the new radius is strictly between \(\rho\)
and \(R\). Thus \(Q\) maps \(X\setminus G_0\) to
\(X\setminus G_\rho\). It is a homeomorphism with inverse
given, for \(r>\rho\), by the radial function

\[
 \psi_\rho^{-1}(r)
   =\frac{R(r-\rho)}{R-\rho}.
 \tag{HI.8}
\]

The inclusion \(j:X\setminus G_\rho\hookrightarrow
X\setminus G_0\) is homotopy inverse to \(Q\).
Both homotopies use the same radial formula

\[
 \begin{gathered}
 r_s=r+s\rho(1-r/R),\\
 0\le s\le1.
 \end{gathered}
 \tag{HI.9}
\]

On the first complement it stays positive; on the second it
stays greater than \(\rho\). It stays below \(R\) in either
case, and is the identity at \(v=0\) off \(H\).
All maps keep \(u\) fixed. HI.1 therefore shows that the
restriction map on complement complexes is a quasi-isomorphism
for every coefficient \(\pi^{-1}K\).

Use the same whole-domain complex on the two supports in
the relative convention OCM.2. The inclusion \(G_0\subset
G_\rho\) gives a map of the two relative complexes, equal
to the identity on the whole-domain entry and to restriction
on the complement entry. Its cone is acyclic, since the
complement map is a quasi-isomorphism. This proves

\[
 \begin{gathered}
 R\pi_*R\Gamma_{G_0}(\pi^{-1}K)\\
 \longrightarrow
 R\pi_*R\Gamma_{G_\rho}(\pi^{-1}K)\\
 \hbox{is a quasi-isomorphism.}
 \end{gathered}
 \tag{HI.10}
\]

The pushforward is essential in this sheaf assertion:
the argument holds over every base open, and the map is the
canonical support enlargement. Its quasi-inverse is constructed
from the complement maps and their two homotopies through
the relative cones. No shrinking-fibre base-change formula
is assumed. When \(k=0\), the two supports coincide and
HI.10 is the identity.

#### HI.3. The zero-normal orientation class

We give the complete normalization for the zero-radius support.
In one real coordinate \(v\), the punctured interval has
negative and positive components. Contract each interval
component, and the full interval, using OCM.9.
The resulting finite relative complex is

\[
 \mathbb C
 \xrightarrow{\ d\ }\mathbb C^2,
 \qquad d(a)=(a,a),
 \tag{HI.11}
\]

in degrees zero and one. Order the complement entries as
negative, positive. The positive connecting class of the
step function \(1_{\{v>0\}}\), in convention V.1, is
\((0,-1)\). Define

\[
 \begin{gathered}
 \epsilon(b_-,b_+)=b_--b_+,\\
 \iota(c)=(0,-c),\\
 h(b_-,b_+)=b_-.
 \end{gathered}
 \tag{HI.12}
\]

Then \(\epsilon\iota=1\), and direct substitution gives
\(dh+hd=1-\iota\epsilon\). The connecting class has
\(\epsilon(0,-1)=1\). This fixes its sign, rather than
choosing a sign after a trace calculation.

For \(k\) ordered real normal coordinates, use the finite
coordinate-puncture cube and tensor HI.11–HI.12 in that
order. The complement cube comparison is V.6 and R.7.
All its intersections are products of intervals and
signed interval components; the coefficient contraction
HI.5 applies on each intersection. The alternating
finite cube therefore computes the actual relative
complex, including each complement entry.

For homogeneous tensor factors \(x_1,\ldots,x_k\), its
contracting homotopy is

\[
 \begin{gathered}
 H(x_1\otimes\cdots\otimes x_k)\\
 =\sum_{i=1}^k(-1)^{\sum_{j<i}|x_j|}\\
 {}\cdot P x_1\otimes\cdots\otimes P x_{i-1}\\
 {}\otimes h x_i\\
 {}\otimes x_{i+1}\otimes\cdots\otimes x_k,\\
 P=\iota\epsilon.
 \end{gathered}
 \tag{HI.13}
\]

The tensor differential differentiates the \(i\)-th factor
with sign \((-1)^{\sum_{j<i}|x_j|}\).
The identity \(dh+hd=1-P\) telescopes to

\[
 dH+Hd=1-P^{\otimes k}.
 \tag{HI.14}
\]

The surviving tensor has degree \(k\) and normal trace one.
Put these normal factors before the coefficient complex
\(K\). Its differential on the surviving factor is
\((-1)^k d_K\); thus its target is \(K[-k]\).
The terms involving \(d_K\) cancel in HI.14 because \(H\)
lowers normal degree by one. A permutation of the real
normal coordinates multiplies the orientation class by
the sign of that permutation, since the one-coordinate
classes have odd degree.

This constructs, naturally on base opens,

\[
 \begin{gathered}
 R\Gamma_{B\times\{0\}}(\pi^{-1}K)\\
 \simeq i_*K[-k].
 \end{gathered}
 \tag{HI.15}
\]

where \(i(u)=(u,0)\). Outside the zero section the left side
vanishes. Along that section the finite cube just computed
gives exactly the right side; its augmentations and trace
give the displayed maps. The finite tensor homotopy gives
both inverse identities. For \(k=0\), the empty normal
tensor is one in degree zero.

#### HI.4. The supported Thom comparison, including the collapsed fibre

For two closed supports \(A,B\), the support functors commute
in the derived category:

\[
 R\Gamma_A R\Gamma_B
   =R\Gamma_{A\cap B}.
 \tag{HI.16}
\]

Here is the injective justification in the present coefficient
category. For an injective sheaf \(I\),
\(\Gamma_B I=\mathcal Hom(\mathbb C_B,I)\), by applying
Hom to the exact open/closed sequence OCM.18. This is
injective, since its left-adjoint tensor over \(\mathbb C\)
is exact. Hence an injective resolution can compute both
successive support functors. On its terms, requiring both
supports is exactly requiring their intersection. This
proves HI.16.

Apply HI.16 to \(H\times B_R^k\) and \(B\times\{0\}\).
Together with HI.15 and exact closed direct image, it gives

\[
 \begin{gathered}
 R\Gamma_{G_0}(\pi^{-1}K)\\
 \simeq i_*R\Gamma_H(K)[-k].
 \end{gathered}
 \tag{HI.17}
\]

The support enlargement HI.10 is therefore the desired
comparison for the full shrinking family. On these supports
ordinary and proper pushforward agree: in the injective
support model each term is supported on the proper closed
set \(G_\rho\) or \(G_0\). Over a compact base subset the
normal support is compact inside the fixed larger ball.
Thus its support remains proper before any trace is applied.
Since \(\pi i=1\), HI.17 proves the normalized formula

\[
 \begin{gathered}
 R\pi_*R\Gamma_{G_\rho}(\pi^{-1}K)\\
 \simeq R\Gamma_H(K)[-k].
 \end{gathered}
 \tag{HI.18}
\]

**Theorem HI.A.** HI.18 holds for every closed \(H\), every
continuous \(\rho\) zero off \(H\) with \(0\le\rho<R/2\),
and every bounded complex \(K\) of complex-vector sheaves.
It includes \(\rho=0\), every boundary point of \(H\), and
every collapsing fibre. The normal orientation is the
ordered class of trace one in HI.12–HI.15.

**Proof.** HI.7–HI.10 compares the actual relative complexes,
and HI.11–HI.17 computes the zero-radius complex with
its tensor signs. Properness was checked on the entire
closed support, not inferred from individual fibres.
These maps are natural in \(K\). Their homotopies and
resolution comparisons have finitely many entries in
each degree, so a finite family of equalities is realized
after one actual common restriction. \(\square\)

This proof also gives compatibility with support enlargement
and base restriction. Any two comparison choices induce
the inverse of the same canonical support-enlargement
map HI.10, with the same trace-one normal class.
The two radial homotopies and the finite tensor homotopy
supply the corresponding finite equality witnesses.
It does not replace a boundary stalk by cohomology of
one isolated fibre.

#### HI.5. A hypersurface in the doubled support model

Let \(F\) be any bounded coefficient complex on \(M\).
On \(M\times M\), Hom from the closed second hypersurface
is the corresponding supported complex:

\[
 \begin{gathered}
 R\mathcal Hom(q_2^{-1}\mathbb C_S,q_1^!F)\\
 =
 R\Gamma_{\{f'=0\}}(q_1^{-1}F)[2m].
 \end{gathered}
 \tag{HI.19}
\]

Indeed \(q_2^{-1}\mathbb C_S\) is the constant sheaf on
the closed subset \(f'=0\), extended by closed direct
image. The injective argument of HI.16 identifies its
Hom with sections supported there. The projection
normalization is OCM.16. These observations prove HI.19
on the actual injective complexes.

Its coefficient is constant in the second \(f'\)-normal
variable. HI.15 in its two real coordinates therefore
identifies HI.19 with the closed direct image from \(f'=0\)
of that coefficient shifted by \([2m-2]\).
The normal orientation is the positive complex orientation,
with trace one.

Now apply the diagonal support functor OCM.3. Intersecting
its supports with \(f'=0\) is justified by HI.16.
In coordinates HI.1, choose the cofinal round cone

\[
 \begin{gathered}
 a=\Re(e^{i\theta}f),\\
 b=\Im(e^{i\theta}f),\\
 a\ge0,\quad b^2+|\zeta|^2\le L^2a^2.
 \end{gathered}
 \tag{HI.20}
\]

Its nonzero vectors have strict diagonal polar margin
at least \(1/\sqrt{1+L^2}\). Choose the margin and a
nearby-covector neighbourhood before the support domains.
IG.1's compact unit-cone argument makes these cones
cofinal; the change \(n_f=-f\) gives exactly the sign
in HI.2.

The projected support and the tangent-ball radius are

\[
 \begin{gathered}
 H_\theta=\{a\ge0,\ |b|\le La\},\\
 \rho(f)\\
 =\sqrt{\max\{L^2a_+^2-b^2,0\}},\\
 a_+=\max(a,0).
 \end{gathered}
 \tag{HI.21}
\]

This radius is continuous and zero off \(H_\theta\).
It vanishes on the sector boundary, including \(f=0\).
After choosing the tangent-normal domain \(B_R^{2m-2}\),
shrink the actual \(f\)-disc so that \(L|f|<R/2\).
Then the entire closed tangent support has a common
inner/outer collar. HI.A applies with \(k=2m-2\).
When \(m=1\), there is no tangent normal and its comparison
is the \(k=0\) identity.

The first normal shift \([-2]\), the tangent-normal
shift \([-(2m-2)]\), and the projection shift \([2m]\)
cancel:

\[
 2m-2-(2m-2)=0.
 \tag{HI.22}
\]

This cancellation records the degrees of the maps just
constructed; it is not used as a substitute for those maps.
The ordered first normal block has real dimension two.
Moving it past the remaining real block gives sign
\((-1)^{2(2m-2)}=1\). The change \(f\mapsto-f\) also
preserves real orientation, since its real determinant
is positive. It changes the polar sign as in HI.1–HI.3,
not the trace-one orientation class. The coefficient
line \(F\) remains ungraded throughout.

**Theorem HI.B.** On the nonzero conormal to \(S\),
there is the natural, sign-specified comparison

\[
 \begin{gathered}
 \mu\operatorname{hom}(\mathbb C_S,F)_p\\
 \simeq\mathfrak h_{f,p}(F),\\
 \mu\operatorname{hom}(\mathbb C_S[-1],F)_p\\
 \simeq\mathfrak h_{f,p}(F)[1].
 \end{gathered}
 \tag{HI.23}
\]

**Proof.** HI.19 reduces the second normal direction by
HI.15. On each cofinal round support the tangent-normal
comparison is HI.18. Restrictions and support maps commute
with these comparisons, by the support-enlargement
compatibility of HI.A. Their filtered colimit is HI.23.
The shift in the second line is contravariance in the
first argument, with the same convention as OCM.22.
Every finite diagram has the actual common collars,
resolutions and cochain witnesses specified above.
\(\square\)

These local comparisons agree after an ordinary coordinate
change straightening the hypersurface. The supports are
transported by the derivative on the compact normal cone,
with a strict smaller polar margin, as in [BCT.1](#5-56-whole-real-kernel-coordinate-transport-and-balanced-comparison).
The relative restriction maps are literal pullbacks on
the two opens; HI.15 uses the induced real orientation.
Two straightening charts have the same orientation class
and the same canonical support-enlargement comparison,
so HI.A's homotopies give equality on an overlap after
one common refinement. Holomorphic coefficient frame
changes are coefficient maps and commute with these
constructions.

#### HI.6. Off-conormal vanishing

For completeness the doubled construction introduces no
extra characteristic directions. Outside \(S\), a small
enough diagonal neighbourhood misses \(f'=0\), so HI.19
has zero microlocal stalk. At a covector with nonzero
tangent component, choose a tangent vector \(e_\zeta\)
with positive polar pairing, small enough to lie in the
chosen tangent ball.

Enlarge the round cone so that \((-e_\zeta,0)\) lies in
its strict interior. This is possible: that vector has
strictly positive negative-polar pairing. Increasing the
finite aperture preserves a positive margin and contains
the finitely many earlier supports. Shrink the \(f\)-disc
so that \((-e_\zeta,f)\) is still in that interior and
\((e_\zeta,-f)\) remains outside the cone.

For fixed \((z,f)\), contract the tangent variable to
\(e_\zeta\). In the full normal vector this is

\[
 \begin{gathered}
 (\zeta,-f)\longmapsto\\
 (1-s)(\zeta,-f)+s(e_\zeta,-f).
 \end{gathered}
 \tag{HI.24}
\]

Convexity of the cone, and membership of
\((-e_\zeta,f)\) in it, proves that the complement is
preserved by the same argument as OCM.7. The coefficient
in HI.19 is constant in \(\zeta\). OCM.11 therefore
contracts the relative complex on the whole tangent ball
and its complement. This proves vanishing off the
conormal, with actual support refinements and finite
primitives. It does not use a wave-front theorem as an
external input.

#### HI.7. Every finite meromorphic pole gives an actual class

Let \(F\) be a holomorphic line with nonvanishing frame
\(\eta\), and let \(q\) be holomorphic on a connected
coordinate domain. For an integer \(j\ge1\), the
function \(q/f^j\) is holomorphic on the complement of
every support in HI.21. Its positive relative class is

\[
 \left(0,-\frac{q\eta}{f^j}\right).
 \tag{HI.25}
\]

This class is zero exactly when \(q/f^j\) extends
holomorphically across \(S\) on a smaller full domain.
Indeed the degree-zero part of the support exact
sequence identifies the kernel of the connecting map
with the image of full-domain holomorphic sections.
Equivalently, in a nonnegative injective model, a
degree-zero primitive for HI.25 is a closed degree-zero
section and hence a genuine holomorphic section.
No acyclicity of the domain or its complement is used.

If such an extension \(g\) exists, then \(f^jg=q\)
on a nonempty open subset. The holomorphic identity
theorem makes this equality hold on the connected
full domain. Conversely that equality supplies an
extension. Thus the kernel is precisely the
holomorphic part of \(F[1/f]\).
The same argument holds after every admissible
restriction or support enlargement: use a further
common cone HI.20, whose complement contains a
nonempty open set. In particular nonzero finite
principal parts cannot disappear in the filtered
microlocal colimit.

Theorem HI.B therefore gives, on every nonzero
conormal stalk, an injection

\[
 \begin{gathered}
 \frac{F[1/f]}{F}\\
 \longrightarrow
 H^0\mu\operatorname{hom}(\mathbb C_S[-1],F).
 \end{gathered}
 \tag{HI.26}
\]

It is compatible with restrictions, real phases and
holomorphic frame changes. Each germ is represented
by the same finite meromorphic expression; finite
equalities use one common support domain. For \(q=1\)
and \(j=1\), the class is nonzero at every phase.
Its one-normal residue is

\[
 \frac1{2\pi i}\int_{|f|=\epsilon}\frac{df}{f}=1.
 \tag{HI.27}
\]

The parametrization \(f=\epsilon e^{it}\) proves
HI.27 directly. This is the hypersurface normalization;
it is not a degree-\(N\) diagonal inverse-product
calculation.

Under the open/closed isomorphism OCM.22, HI.25 is
the connecting image of the meromorphic complement
section. To check its sign, choose an injective
extension \(b\) of that section. The Hom connecting
map sends it to \(db\). In the relative convention,
\((0,-q\eta/f^j)+d(b,0)=(db,0)\).
Thus OCM.20 and the positive sign in HI.25 agree.

#### HI.8. The full finite differential-operator pole model

Trivialize \(F\) by \(\eta\), and let differential
operators act on its scalar coefficient. This is
a local trivialized action; no global flat connection
on the coefficient line is asserted.
Write \(\mathcal D\) for finite-order holomorphic
operators in \((z,f)\). Its operators have the
finite normally ordered form

\[
 P=\sum_{\alpha,k}a_{\alpha,k}(z,f)
                    D_z^\alpha D_f^k.
 \tag{HI.28}
\]

Repeated Leibniz differentiation proves closure and
normal ordering. Uniqueness follows by applying an
alleged zero operator to
\(e^{\langle\lambda,z\rangle+\mu f}\):
the resulting polynomial in the constant parameters
\((\lambda,\mu)\) has every coefficient zero.
Thus no formal or infinite differential sum is involved.

Let \(e\) denote the class of \(\eta/f\). Then

\[
 \begin{gathered}
 fe=0,\qquad D_{z_i}e=0,\\
 D_f^ke=(-1)^kk!\frac{\eta}{f^{k+1}}.
 \end{gathered}
 \tag{HI.29}
\]

These identities hold in the finite-pole quotient and,
by HI.26, in the actual microlocal Hom class.
Naturality in the coefficient sheaf makes HI.26
linear for every finite differential-operator map.

The exact cyclic presentation is

\[
 \begin{gathered}
 \frac{\mathcal D}
 {\mathcal D f+\sum_i\mathcal D D_{z_i}}\\
 \simeq\frac{F[1/f]}F.
 \end{gathered}
 \tag{HI.30}
\]

Here all ideals are left ideals; their displayed
generators occur on the right of an arbitrary operator.
For the proof, every term of HI.28 with \(\alpha\ne0\)
has a rightmost tangent derivative and is in the
displayed ideal. For the remaining terms use
\(fD_f^k=D_f^kf-kD_f^{k-1}\). Its finite iteration gives

\[
 \begin{gathered}
 f^jD_f^ke\\
 =(-1)^j\frac{k!}{(k-j)!}D_f^{k-j}e,\\
 0\le j\le k;\\
 f^jD_f^ke=0,\quad j>k.
 \end{gathered}
 \tag{HI.31}
\]

Expand each holomorphic \(a_{0,k}(z,f)\) only through
degree \(k\) in \(f\), with a holomorphic remainder
divisible by \(f^{k+1}\). Taylor's formula in one
convergent coordinate gives that finite division.
HI.31 reduces every operator class to
\(\sum_{\ell=0}^K b_\ell(z)D_f^\ell e\).
If its meromorphic class is zero, uniqueness of the
finite Laurent principal part makes every \(b_\ell\)
zero. This proves injectivity in HI.30.
For surjectivity, the finite principal part
\(\sum_{\ell=0}^{j-1}c_{\ell+1}(z)\eta/f^{\ell+1}\)
is the image of
\(\sum_{\ell=0}^{j-1}(-1)^\ell c_{\ell+1}(z)
D_f^\ell/\ell!\).
This also proves that the whole finite-pole quotient,
not just individual pole orders, is represented.

The induced order filtration has normal basis
\(D_f^\ell e\), with \(\ell\le k\) in order \(k\).
Multiplication by \(f\) lowers that order; a tangent
derivative differentiates \(b_\ell(z)\) and does not
raise it. Consequently

\[
 \begin{gathered}
 \operatorname{gr}
 \left(\mathcal D/
   (\mathcal D f+\sum_i\mathcal D D_{z_i})\right)\\
 =\mathcal O_S[\tau_f].
 \end{gathered}
 \tag{HI.32}
\]

as a module over
\(\mathcal O_M[\xi_z,\tau_f]\). The annihilator is
exactly \((f,\xi_z)\), since the displayed polynomial
module is free in the normal variable.
This ideal is reduced: its quotient is a polynomial
ring over the convergent coordinate ring on \(S\),
which is a domain by the lowest nonzero homogeneous
term argument for convergent power series.
The support is the full conormal, and the principal
generator is the unit of this quotient.

**Theorem HI.C.** The actual all-phase class of
\(\eta/f\) contains the entire finite cyclic
differential-operator module HI.30 injectively.
Its associated graded module is HI.32, with the
reduced conormal ideal and the unit principal
generator. This includes \(m=1\), with an empty
tangent-derivative list.

**Proof.** HI.25–HI.27 proves injectivity in the
source-defined support colimit, and HI.B transfers
it to microlocal Hom with its explicit shift.
HI.28–HI.32 proves the exact finite presentation
and filtration on the same class. \(\square\)

Changing a nonvanishing frame conjugates the local
coefficient action by its holomorphic unit. It changes
the displayed generator by that unit and identifies
the two injected modules and their filtered presentations.
Their reduced principal support is the same.
HI.C does not infer cyclic generation by the full
finite microlocal ring from the differential-operator
presentation, or apply an unproved extension-flatness
or elliptic-regularity assertion.

#### HI.9. Incidence and the two operator sides

For the incidence pair, \(N=d+1\), \(m=2N\), and

\[
 \begin{gathered}
 f=s-t-x^{\mathsf T}y,\\
 \eta=ds\wedge dy_1\wedge\cdots\wedge dy_d.
 \end{gathered}
 \tag{HI.33}
\]

The coordinate change to \(f\) has input Jacobian one:
with the output variables fixed,
\(\eta=df\wedge dy_1\wedge\cdots\wedge dy_d\).
Thus it retains the ordered input frame and does
not insert an additional density scalar.
The nonzero conormal \(-\tau\,df\), with the input
antipodal operation, is exactly the Legendre graph
computed in [LI.4](#5-57-legendre-incidence-and-the-topological-inverse-of-projective-duality). Theorem HI.C gives a genuine
holomorphic microlocal Hom class there at every phase.

The literal function identities [LI.20](#5-57-legendre-incidence-and-the-topological-inverse-of-projective-duality) consequently
descend to this actual class. For \(K=\eta/f\),
acting on scalar coefficients in the displayed frame,
they are

\[
 \begin{gathered}
 -D_sK=D_tK,\\
 y_iD_tK=D_{x_i}K,\\
 D_{y_i}K=x_iD_tK,\\
 (s-t)D_tK\\
 =K+\sum_i x_iD_{x_i}K.
 \end{gathered}
 \tag{HI.34}
\]

The input derivative has the negative transpose in
the ordered input-density convention. The last
relation corresponds to the simple-pole parameter
\(\alpha=0\) in [LI.21](#5-57-legendre-incidence-and-the-topological-inverse-of-projective-duality). It remains a cleared relation:
no primitive defining \(D_t^{-1}K\) on holomorphic
functions has been chosen.

The finite-\(E\) simple-generator identification,
both normalized inverse cup/excision/trace products,
compatibility with the entire LN maps and the action
on arbitrary RM.3 real-kernel representatives are
additional requirements. In particular the reduced
finite-\(\mathcal D\) characteristic ideal in HI.32
does not by itself identify a full \(E\)-bimodule
or a whole-real inverse. The arbitrary-holonomic
regularization and analytic proper/Riemann–Hilbert
arguments retain those hypotheses.

Free primary context is
[Benchoufi, Action of micro-differential operators on quantized
contact transformations, arXiv:2102.08861v2](https://arxiv.org/abs/2102.08861v2),
§§3.2 and 4.3–4.5. The shrinking support comparison,
its polar sign, actual pole-class injection and finite
operator presentation used here are proved in HI.0–HI.9
and the exact earlier programme sections cited above.

#### HI.10. The shrinking supports and their normal class

![The sector support, its continuously shrinking tangent ball, the radial comparison of complements, and the actual finite-pole class](assets/holomorphic-incidence-shrinking-support.png)

Panel A uses \(p=1\), \(L=1\), \(R=2\), and the base disc
\(|f|<3/4\), with \(f=a+ib\). The sector is
\(H=\{a\ge0,\ |b|\le a\}\). Its tangent-ball radius is the
continuous function HI.21, including its zero value on the
sector boundary. The marked point is \(f=5/8+3i/8\), where
\(\rho=1/2\). The coloured values are radii; the closed
support over each base point in \(H\) is the entire ball
of that radius. A zero radius outside \(H\) contributes
no support.

Panel B plots the exact radial maps HI.7 for
\(\rho=0,1/4,1/2\). The open endpoints represent
\(0<r<R\); for base points outside \(H\), the
\(\rho=0\) map also extends as the identity at \(r=0\).
All supports lie strictly inside \(r<R/2=1\).
HI.8–HI.10 supplies the inverse map and both homotopies,
on the whole family and at its collapsed boundary.

Panel C gives the projection, complex-normal and
tangent-normal degree cancellation HI.19–HI.23.
The relation \(n_f=-f\) retains the positive polar
condition of HI.2. HI.25–HI.32 proves the displayed
finite-pole injection and its reduced characteristic
ideal. The diagram asserts no full-\(E\) generation
or inverse-product identity.

The editable SVG,
exact rational data
and figure source
give the coordinates, radii, radial maps and finite-pole coefficients.


### 5.61. The finite microlocal hypersurface module

Let \(M\) have complex dimension \(m\ge1\), let \(S=\{f=0\}\)
be a smooth hypersurface, and use coordinates
\((z,f)\), \(z\in\mathbb C^{m-1}\). Work near a nonzero
conormal with normal covariable \(\lambda\ne0\), and put
\(w=\xi_z/\lambda\). Write \(E\) for the actual finite-order
microlocal ring of §§5.19–5.22. A symbol has the form
\[
 P=\sum_{j\le b}p_j(z,f,w)\lambda^j.
 \tag{FIM.1}
\]
There is one common coefficient product. On each smaller
compact product its negative coefficients obey
\(\|p_{-n}\|\le MC^n n!\). The finite upper bound \(b\)
belongs to the whole operator. All sums below mean actual
coefficient sequences with these bounds, rather than
convergent numerical Laurent sums in \(\lambda\).

The ring, its products and ordinary changes of coordinates
are the actual constructions of §§5.19–5.22, §5.28 and
§5.56. Their finite matrix-kernel coherence is §5.24 and
§5.31, L.5–L.6. The normal-coordinate Koszul contraction
is §5.54, EE.5–EE.9. The actual halfspace action and its
normal primitive gauges are §5.53, AM.B–AM.C. The
hypersurface support convention and pole class are
§5.60, HI.1–HI.32.

The exact earlier programme proof dependencies are [§5.19](#5-19-actual-factorial-growth-symbols-and-ordered-jets), [§5.32](#5-32-coherent-actual-lattices-and-principal-support-for-the-whole-germ), [§5.43](#5-43-the-canonical-all-degree-action-and-its-derived-module-structure), [§5.52](#5-52-full-euler-matrix-division-and-comparison-reductions), [§5.53](#5-53-real-kernel-comparison-for-all-constant-euler-matrices), [§5.54](#5-54-halfspace-solutions-and-finite-euler-extensions), [§5.56](#5-56-whole-real-kernel-coordinate-transport-and-balanced-comparison), [§5.57](#5-57-legendre-incidence-and-the-topological-inverse-of-projective-duality), [§5.58](#5-58-full-analytic-legendre-transport-and-quadratic-factorization), [§5.59](#5-59-open-and-closed-incidence-in-the-actual-microlocal-hom-complex), [§5.60](#5-60-the-holomorphic-incidence-class-and-shrinking-normal-supports), [EC.B](#ec-b-the-actual-opposite-ring-and-right-finiteness).

#### FIM.0. The entire cyclic quotient

Consider the finite presentation
\[
 \begin{gathered}
 \mathcal M_S=\\
 E/(Ef+\sum_i ED_{z_i}),\\e=[1].
 \end{gathered}
 \tag{FIM.2}
\]
The ideals are left ideals. In particular the annihilating
coordinate or derivative is on the right of a coefficient
operator. This quotient is coherent by the actual finite
matrix-kernel theorem. It is a quotient of the entire
actual \(E\), with its infinite negative tail retained.

We prove a strict normal form for FIM.2:
\[
 \begin{gathered}
 Q=\sum_{j\le b}q_j(z)\lambda^j,\\
 \|q_{-n}\|\le M'C'^n n!.
 \end{gathered}
 \tag{FIM.3}
\]
Every class has exactly one such row. The reduction does
not increase its upper order.

First divide the holomorphic coefficients of FIM.1 by
the commuting normal coordinates \(w_i\):
\[
 \begin{gathered}
 p_j(z,f,w)-p_j(z,f,0)\\
 =\sum_i w_i b_{j,i}(z,f,w).
 \end{gathered}
 \tag{FIM.4}
\]
One explicit choice is
\(b_{j,i}=\int_0^1(\partial_{w_i}p_j)(z,f,sw)\,ds\).
The radial segment lies in the chosen inner polydisc.
A fixed outer Cauchy margin bounds the derivative by a
constant times the original coefficient norm.
Right multiplication by \(D_{z_i}\) is multiplication of
the normal symbol by \(\lambda w_i\). Hence FIM.4 subtracts
an element \(\sum_i B_iD_{z_i}\), where \(B_i\) has upper
order at most \(b-1\) and the same factorial type on one
smaller product. This is also the degree-zero augmentation
of the actual contraction EE.7.

Set \(a_j(z,f)=p_j(z,f,0)\), and choose an outer
\(f\)-disc of radius \(R>0\). Write its actual Taylor
coefficients as
\[
 a_j(z,f)=\sum_{k\ge0}a_{j,k}(z)f^k.
 \tag{FIM.5}
\]
Put \(a_j=0\) for \(j>b\), and use
\((j)_k=j(j-1)\cdots(j-k+1)\), with \((j)_0=1\).
Define
\[
 \begin{gathered}
 q_\ell\\
 =\sum_{k=0}^{b-\ell}(-1)^k(\ell+k)_k a_{\ell+k,k}.
 \end{gathered}
 \tag{FIM.6}
\]
For every output degree this sum is finite. In particular
it does not require a common domain for infinitely many
new Taylor remainders chosen separately.

The identity \([D_f,f]=1\) gives
\[
 fD_f^j=D_f^jf-jD_f^{j-1}
 \tag{FIM.7}
\]
for every integer \(j\). For negative \(j\), multiply the
identity for \(D_f^{-1}\) by finite inverse powers;
the normal inverse already belongs to \(E\).
Iterating FIM.7 gives
\[
 \begin{gathered}
 f^kD_f^j\\
 =\sum_{\nu=0}^{k-1}(-1)^\nu(j)_\nu\\
 {}\cdot f^{k-1-\nu}D_f^{j-\nu}f\\
 {}+(-1)^k(j)_kD_f^{j-k}.
 \end{gathered}
 \tag{FIM.8}
\]
For \(k=0\) the sum is empty. Thus the candidate remainder
is exactly FIM.6.

Here are estimates for the entire division, including its
quotient. For \(r\ge1\), the Cauchy coefficient bound is
\[
 \|a_{-r,k}\|\le MC^r r!R^{-k}.
 \tag{FIM.9}
\]
For a negative output degree \(\ell=-n\), all terms of
FIM.6 with \(k\ge n\) vanish. Indeed \(j=k-n\ge0\)
and \(k>j\), so \((j)_k=0\). The nonzero terms give
\[
 \begin{gathered}
 q_{-n}=\sum_{k=0}^{n-1}
 \frac{(n-1)!}{(n-k-1)!}\\
 {}\cdot a_{-(n-k),k}.
 \end{gathered}
 \tag{FIM.10}
\]
Consequently
\[
 \begin{gathered}
 \|q_{-n}\|\le M(n-1)!\\
 {}\cdot\sum_{r=1}^n rC^rR^{-(n-r)}\\
 \le Mn!(C+R^{-1})^n.
 \end{gathered}
 \tag{FIM.11}
\]
The last inequality follows by \(r\le n\) and by comparing
each term with the corresponding nonnegative binomial
expansion coefficient, which is at least one.
The finitely many nonnegative output degrees have only
finitely many contributing nonnegative input degrees.

The right-\(f\) quotient in FIM.8 is the actual symbol
\[
 \begin{gathered}
 A_\ell(z,f)\\
 =\sum_{\nu=0}^{b-\ell}(-1)^\nu(\ell+\nu)_\nu\\
 {}\cdot\sum_{k\ge\nu+1}
                 a_{\ell+\nu,k}(z)f^{k-1-\nu}.
 \end{gathered}
 \tag{FIM.12}
\]
On a fixed inner disc \(|f|\le r_0<R\), its inner
Taylor sum has norm at most
\[
 \begin{gathered}
 \frac{MC^r r!R^{-(\nu+1)}}{1-r_0/R},\\
 \ell+\nu=-r<0.
 \end{gathered}
 \tag{FIM.13}
\]
For \(\ell=-n\), only \(0\le\nu\le n-1\) contributes,
by the same falling-factorial zero. Hence
\[
 \begin{gathered}
 \|A_{-n}\|\le\frac{M}{R(1-r_0/R)}\\
 {}\cdot n!(C+R^{-1})^n.
 \end{gathered}
 \tag{FIM.14}
\]
The upper order of \(A\) is at most \(b\). All its other
coefficients involve finitely many outer holomorphic
coefficients. The infinite Taylor sums in FIM.12 converge
uniformly on the chosen inner product, with geometric
tails; fixed derivatives follow on one further product.
The coefficient identities FIM.8 can therefore be summed,
and prove the actual decomposition
\[
 \begin{gathered}
 P=Q+A f\\
 {}+\sum_i B_iD_{z_i}.
 \end{gathered}
 \tag{FIM.15}
\]
This is an equality in \(E\), with a common coefficient
domain and an explicit upper order for every term.

#### FIM.1. Uniqueness, filtration and the reduced conormal

Denote the remainder operation FIM.4–FIM.6 by \(N\).
It kills right multiples of every \(D_{z_i}\), since
their symbols vanish at \(w=0\). It also kills right
multiples of \(f\). After evaluation at \(w=0\),
the degree-\(j\) coefficient of \(Af\) is
\[
 f a_j+(j+1)a_{j+1}.
 \tag{FIM.16}
\]
Before evaluation the extra term in the fixed-\(\xi_z\)
normal derivative is \(-w\cdot\partial_w a_{j+1}\),
which vanishes at \(w=0\).
Substitution of FIM.16 into FIM.6 cancels adjacent
Taylor indices: the contribution of \(fa_j\) at
index \(k\ge1\) has the opposite sign from the
contribution of \(j a_j\) at index \(k-1\), and
\((j)_k=j(j-1)_{k-1}\). These are finite sums at
every output degree. Finally \(N(Q)=Q\) for a row
independent of \(f,w\). Thus FIM.15 proves uniqueness,
and
\[
 \begin{gathered}
 \ker N\\
 =Ef+\sum_i ED_{z_i}.
 \end{gathered}
 \tag{FIM.17}
\]

Use the quotient order filtration from \(E\). FIM.15
keeps \(A\) in order \(b\), \(B_i\) in order \(b-1\),
and \(Q\) in order \(b\). Thus this is a strict
presentation, not merely a set-theoretic normal form.
Its order-\(b\) graded part is exactly
\[
 \operatorname{gr}_b\mathcal M_S
       =\mathcal O_S\lambda^b.
 \tag{FIM.18}
\]
The coefficient \(q_b\) vanishes precisely when the
normal row has order at most \(b-1\).
Multiplication by \(f\) lowers order, since
\[
 f(q_jD_f^je)=-j q_jD_f^{j-1}e.
 \tag{FIM.19}
\]
A tangent derivative gives
\(D_{z_i}(q_jD_f^je)=(\partial_{z_i}q_j)D_f^je\),
and hence has zero degree-one symbol on the quotient.
The full principal annihilator is therefore
\[
 I_{\mathcal M_S}=(f,\xi_z).
 \tag{FIM.20}
\]
There are no further principal equations, by FIM.18
and the arbitrary choice of its holomorphic coefficient.
The quotient coordinate ring on \(S\), with the
nonzero normal direction retained, is reduced: its
local convergent power-series ring is a domain by
the lowest nonzero homogeneous-term argument.
The order-zero quotient is generated by \(e\),
and \(D_f^{-1}\) acts injectively by shifting its
unique normal row. The actual support equality
J.3–J.4 of §5.32 therefore identifies its support
with the full conormal support of its principal
quotient, including every nearby stalk.
Thus FIM.2 with generator \(e\) is an actual coherent
simple microlocal module in the explicit sense of a
reduced principal annihilator equal to the ideal of
its conormal support.

#### FIM.2. Its convergent pole and logarithm representative

Let \(F\) be trivialized by a degree-zero holomorphic
frame \(\eta\). For every integer \(j\), use the
one-variable representatives
\[
 \begin{gathered}
 u_j(f)=(-1)^j j!f^{-j-1},\\
 j\ge0,\\
 u_{-n}(f)=\frac{f^{n-1}}{(n-1)!}\\
 {}\cdot(\log f-H_{n-1}),\\
 n\ge1.
 \end{gathered}
 \tag{FIM.21}
\]
where \(H_k=\sum_{\nu=1}^k1/\nu\), \(H_0=0\).
Choose the logarithm on the complement of a strict
closed positive-polar sector, as in HI.20–HI.21.
Differentiation and
\(H_k-H_{k-1}=1/k\) give
\(u_j'=u_{j+1}\), including \(j=-1\).
Modulo full-disc holomorphic functions,
\(fu_j=-j u_{j-1}\). For \(j=-n<0\), the difference
is exactly \(f^n/n!\); for \(j=0\), it is the
constant \(1\). These are actual holomorphic gauges.

For a normal row FIM.3, put
\[
 \begin{gathered}
 U_Q(z,f)\\
 =\sum_{j\le b}q_j(z)u_j(f).
 \end{gathered}
 \tag{FIM.22}
\]
This is an actual holomorphic function on the
sector complement on one fixed smaller product.
Indeed, on compact subsets of its punctured normal
disc, the negative-index summands are bounded by a
constant times
\[
 \begin{gathered}
 n C'^n|f|^{n-1} B_n,\\
 B_n=|\log f|+1\\ {}+\log(n+1).
 \end{gathered}
 \tag{FIM.23}
\]
Choose the normal radius before summing, so that
\(C'|f|<q<1\). The sum converges normally, with
summable geometric-polynomial tails and all fixed
derivatives on further compact margins.
The harmonic-number part alone converges on a
full disc, as a holomorphic gauge.

Its logarithmic coefficient is the genuine function
\[
 \begin{gathered}
 A_Q(z,f)\\
 =\sum_{n\ge1}\frac{q_{-n}(z)f^{n-1}}{(n-1)!}.
 \end{gathered}
 \tag{FIM.24}
\]
Changing a logarithm branch changes FIM.22 by a
constant integer multiple of \(2\pi i A_Q\).
That function is holomorphic on the full small
product. Thus FIM.22 determines a branch-independent
actual relative class
\[
 \begin{gathered}
 Q\longmapsto[(0,-U_Q\eta)]\\
 \in H^1\mathfrak h_{f,p}(F).
 \end{gathered}
 \tag{FIM.25}
\]
at every nonzero conormal phase \(p\).
The sign is the same as HI.25. Its restriction
for \(j\ge0\) is precisely the finite-pole injection
HI.26.

This map is injective. If FIM.22 agrees on a later
sector complement with a full-domain holomorphic
function, continue that equality around a normal
circle using finitely many overlapping angular
charts. The full-domain function is single valued.
The jump of FIM.22 is \(2\pi i A_Q\); hence the
actual function \(A_Q\) is zero. Its Taylor
coefficients kill every \(q_{-n}\). The remaining
finite negative Laurent principal part extends
holomorphically only if every \(q_j\), \(j\ge0\),
is zero, by circle coefficient extraction.
The same argument holds after every common support
refinement, and with arbitrary holomorphic \(z\)
parameters on a smaller product.

The map is linear for finite differential operators
and holomorphic scalar multiplication. For \(D_f\)
and \(D_z\), use FIM.21 and normal convergence.
For a general holomorphic scalar \(g(z,f)\),
FIM.10 shows that the logarithmic coefficient of
the normal remainder of \(gQ\) is exactly
\(gA_Q\), by the ordinary Cauchy product.
Its finite negative Laurent part is the actual
principal part of \(g\sum_{j\ge0}q_ju_j\).
All other terms extend on a full smaller disc by
the estimates FIM.9–FIM.14 and FIM.23.
This proves the stated scalar compatibility without
an infinite algebraic sum of operators.

Transporting the action of \(E\) on its quotient
FIM.2 through this injection gives an \(E\)-module
structure on its explicit pole/logarithm image.
FIM.4 identifies that structure with the actual
kernel cup/excision/trace action on the strict
hypersurface support complex.
Theorem HI.B identifies the underlying relative
classes with actual microlocal Hom classes.
FIM.5 identifies this action with the geometric
microlocal Hom action using the entire relative
cube and proper trace.

#### FIM.3. Rank one over the output algebra on the incidence chart

Let \(N=d+1\), \(M=X\times Y\), and choose
\[
 \begin{gathered}
 f=s-t-x^{\mathsf T}y,\\
 z=(t,x,y).
 \end{gathered}
 \tag{FIM.26}
\]
The relative input form \(ds\wedge dy_1\wedge\cdots
\wedge dy_d\) has input Jacobian one in these
coordinates, as in HI.33.
The actual module FIM.2 has the tangent annihilators
in the \(z\)-coordinates and normal covariable
\(\lambda=\xi_f\).

In the new coordinates the old output derivatives are
\[
 \begin{gathered}
 D_t^{\rm old}=D_t^{\rm new}-D_f,\\
 D_{x_i}^{\rm old}=D_{x_i}^{\rm new}-y_iD_f.
 \end{gathered}
 \tag{FIM.27}
\]
They commute with every \(y_i\), and commute with
each other. Consequently the full output normal
symbol
\[
 \begin{gathered}
 P=\sum_{j\le b}r_j(t,x,v)\tau^j,\\
 v=\xi_x/\tau.
 \end{gathered}
 \tag{FIM.28}
\]
has normal remainder in \(\mathcal M_S\)
\[
 \begin{gathered}
 N(Pe)=\sum_{j\le b}(-1)^j\\
 {}\cdot r_j(t,x,y)\lambda^j e.
 \end{gathered}
 \tag{FIM.29}
\]

To justify this for the entire actual symbol,
introduce the commuting normalized tangent
covariables \(\nu_0=\xi_t^{\rm new}/\lambda\),
\(\nu_i=\xi_{x_i}^{\rm new}/\lambda\).
The degree-\(j\) coefficient becomes
\[
 \begin{gathered}
 (\nu_0-1)^j\\
 {}\cdot r_j\!\left(t,x,\frac{\nu-y}{\nu_0-1}\right).
 \end{gathered}
 \tag{FIM.30}
\]
There are no derivative corrections in this
commuting covariable substitution: the coefficients
of the transformed derivative fields are functions
of \(y\), while none of these fields differentiates
\(y\). Their normal ordering therefore agrees
literally with FIM.27.
More explicitly, the transformed symbols are
independent of \(f\) and of the new \(y\)-covariables.
In their actual Leibniz product the terms
differentiating \(f\) in the second factor vanish,
and the terms differentiating \(y\) have zero
covariable derivative in the first factor.
The remaining \(t,x\) contractions are exactly
the original output contractions, by the chain
rule in FIM.27. Thus the substitution preserves
the full actual product, not just polynomial
calibrations of its generators.
Choose one product with \(|\nu_0|<c<1\) and with
the displayed argument inside the original
covariable domain. Evaluation at \(\nu=0\)
is FIM.29. The remainder is divisible by the
commuting tangent covariables using the radial
division FIM.4, so is in their right-generated
left ideal.
For \(j=-n\), the extra factor has norm at most
\((1-c)^{-n}\). Thus the entire negative tail
still obeys one factorial estimate. The division
uses one additional Cauchy margin and lowers order
by one. All fixed derivatives and tails are on
these common products.

Conversely every normal row \(q_j(t,x,y)\) in
FIM.3 is FIM.29 for the actual output symbol
\(r_j(t,x,v)=(-1)^j q_j(t,x,v)\). The coefficient
domains and factorial bounds are identical after
renaming the variable.
Uniqueness in FIM.17 therefore proves the actual
filtered left-module isomorphism
\[
 \begin{gathered}
 E_X\longrightarrow\mathcal M_S,\\
 P\longmapsto Pe.
 \end{gathered}
 \tag{FIM.31}
\]
This calculation includes \(d=0\) and every
negative order; it is not limited to polynomial
covariable coefficients or generator relations.

The commuting right input action, with its negative
transpose in the ordered input frame, satisfies
\[
 \begin{gathered}
 eD_s=D_t e,\\
 ey_i=D_{x_i}D_t^{-1}e,\\
 eD_{y_i}=-x_iD_te,\\
 es=\\
 (t+\sum_i x_iD_{x_i}D_t^{-1})e.
 \end{gathered}
 \tag{FIM.32}
\]
These are algebraic identities in the actual cyclic
module. The normal inverse is the inverse in \(E\);
no primitive of a holomorphic function is selected
by FIM.32.
FIM.6 identifies the whole finite-order right input
action with LN.8–LN.24, including its common-domain
bounds. Its proof uses strict finite-degree dependence
and holomorphic finite jets in addition to FIM.32.

The canonical action comparison is FIM.5. Both
normalized inverse cup/excision/trace products are
separate assertions; the finite cyclic presentation
and the output rank-one algebra do not imply them.

#### FIM.4. The canonical action on strict hypersurface supports

We construct the action on the hypersurface support
side without presuming a microlocal Hom action
comparison. Rotate the ordinary \(f\)-coordinate
so that its selected positive polar is
\(\Re f>0\). Ordinary coordinate transport, with
the input Jacobian and orientation, is BCT.1.
The strict support sectors
\[
 \begin{gathered}
 H_L=\\
 \left\{\begin{gathered}
 a=\Re f\ge0,\\|\Im f|\le La
 \end{gathered}\right\}\\
 {}\times\mathbb C_z^{m-1},\\
 0<L<\infty.
 \end{gathered}
 \tag{FIM.33}
\]
on shrinking products are cofinal by HI.20–HI.21.
Let \(C=\mathscr E^{\mathbb R}_p\) be the entire
source-defined real kernel stalk of AM.H.
For a finite list of kernel classes, relations
and boundary witnesses, that lemma supplies one
actual coefficient domain and one support cone
in the difference \(n=v-u\):
\[
 \begin{gathered}
 l=-\Re n_f\ge0,\\
 |\Im n_f|\le\epsilon l,\\
 |n_z|\le A l,\\ \epsilon,A<\infty.
 \end{gathered}
 \tag{FIM.34}
\]
Choose \(L\) at least \(\epsilon\) and at least
each earlier sector aperture in the finite
diagram. Increasing a finite aperture keeps a
strict positive polar margin.

If \(v\in H_L\) and \(n=v-u\) belongs to FIM.34,
then \(u\in H_L\). More precisely,
\[
 \begin{gathered}
 0\le a_v\le a_u,\\
 l=a_u-a_v\le a_u,\\
 |\Im f_u|\\
 \le|\Im f_v|+|\Im n_f|\\
 \le La_v+\epsilon l\le La_u.
 \end{gathered}
 \tag{FIM.35}
\]
Thus the projected support is the same \(H_L\).
Its complement is open in the required cone
direction: \(u\notin H_L\) and \(n\in G\)
implies \(u+n\notin H_L\), by the same
contrapositive implication.

The whole closed intersection is proper on one
smaller output domain. Above a compact output
set, FIM.35 bounds \(l\), while FIM.34 bounds
both the imaginary normal displacement and
every tangent displacement. First choose an
outer source box inside the coefficient domains;
then choose the output box small enough that
these closed bounds lie strictly inside it.
The intersection is closed in the resulting
compact source box. This proves properness
and supplies a single inner/outer collar.
For any finite product, the nonnegative lengths
add to \(a_u-a_v\le a_u\); all intermediate
variables satisfy the same bounds, with the
largest of the finitely many cone constants.
AM.H's actual round-domain refinement applies
on these boxes. No arbitrary restriction of
a proper support is used.

Use DLG.3's entire four-entry relative cup,
including both complement entries and their
double-complement entry. Apply its full-cone
augmentation, the proper collar DLG.15, and
the ordered trace DLG.16 over the middle
complex \(m\)-dimensional variable. The kernel
has degree \(m\), and the trace lowers degree
by \(m\); it therefore acts in every input
relative degree. The kernel complement sign
and trace signs are exactly DLG.3–DLG.4.
Their explicit homotopies retain the entire
input boundary primitive.

The entire eight-entry triple cube and
properness just proved give DLG.5's
associativity and positive diagonal unit on
this target. DLG.3 and DLG.7 compare a larger
sector, a larger finite cone list and a
smaller actual domain. Each equality has
finitely many representatives and witnesses,
so is realized at one later common index.
Taking their filtered colimit constructs a
unital canonical \(C\)-action on
\(\mathfrak h_{f,p}(\mathcal O)\), in every
degree. This argument uses arbitrary full
relative kernel representatives, not just
symbols or supported currents.

There is a natural support-enlargement map
\[
 \begin{gathered}
 H^1\mathfrak h_{f,p}(\mathcal O)\\
 \longrightarrow\\
 \underset{D}{\operatorname{colim}}\,
       H^1_{\{\Re f\ge0\}}(D;\mathcal O)\\
 =\mathcal S.
 \end{gathered}
 \tag{FIM.36}
\]
The target is exactly AM.16 for the same
normal phase and difference direction.
The map is \(C\)-linear: both actions use
the same full relative cup, and the
support-enlargement maps commute with its
finite-cube augmentation. One common proper
collar compares both support-to-image roofs
by DLG.17 and the collar homotopy DLG.4.
The normalized trace is unchanged.

The map FIM.36 is injective in degree one.
Here is a proof that does not infer
injectivity of a general support-forgetting
map. On a product polydisc \(D\),
\(H^1(D;\mathcal O)=0\), by OC.2's explicit
strictly plurisubharmonic disc exhaustion
and the fully proved DLH existence theorem.
The support exact sequence therefore
represents every degree-one class by a
holomorphic function on \(D\setminus H_L\),
modulo a full-domain holomorphic function.
The complement is the product of the
tangent polydisc and a connected angular
sector of opening strictly less than
\(2\pi\); it contains the left half-disc.
If restriction of this function to a
smaller left half-polydisc agrees with a
full-polydisc holomorphic function, their
difference vanishes on that nonempty open.
The holomorphic identity theorem makes
the difference zero on the entire
connected smaller complement.
This is exactly a zero relative class.
Every witness was on an actual later
product, so the same argument proves
injectivity in the filtered colimit.

Let \(e_p\) be the actual positive relative
class of \(1/f\). The ordinary multiplication
and derivative calibration gives
\(fe_p=D_{z_i}e_p=0\).
The canonical action thus defines
\[
 \begin{gathered}
 E/(Ef+\sum_i ED_{z_i})\\
 \longrightarrow H^1\mathfrak h_{f,p}(\mathcal O),\\
 [P]\longmapsto Pe_p.
 \end{gathered}
 \tag{FIM.37}
\]
To identify it on an entire normal row,
compose with FIM.36. The whole halfspace
action is AM.B. For the constant scalar
Euler parameter \(-1\), its detector is
\(E_{-1}=1/f\). AM.24–AM.25b and the
full-disc convergent primitive gauges
AM.23a–AM.23c identify the image of
\(\sum q_j(z)D_f^j\) with FIM.22.
This use involves the full negative
factorial tail; a finite sum of primitives
does not replace the actual calculation.
The two elements of the source of
FIM.36, namely \(P e_p\) and FIM.25,
have the same target. Injectivity of
FIM.36 makes them equal.
Injectivity of FIM.25 then proves that
FIM.37 is injective.

Consequently the actual pole/logarithm
image is an \(E\)-submodule for the
canonical support-side action, with the
strict simple presentation FIM.2 and
the output rank-one isomorphism FIM.31
on the incidence chart. Rotating back
gives the same statement at every
nonzero conormal phase.
FIM.5 proves that HI.B's topological
Thom comparison intertwines this action
with the composition action on microlocal
Hom. Its relative-cube/proper-trace
diagram retains the two eliminated normal
blocks and all their complement terms.
Both normalized holomorphic inverse
products and whole-real contact transport
are separate assertions.


#### FIM.5. The hypersurface Thom comparison and the canonical action

Use the conventions of FIM.0–FIM.4 and §5.60. In particular
\(M\) has complex dimension \(m\), \(S=\{f=0\}\), its tangent
coordinates are \(z\in\mathbb C^{m-1}\), and the selected
normal phase is \(\Re f>0\). The entire real kernel stalk
is \(C=\mathscr E^{\mathbb R}_p\). Its action means the
full relative cup, finite complement-cube augmentation,
proper excision and normalized holomorphic trace of
DLG.1–DLG.7. The same operations define composition on
microlocal Hom. No supported-current representative is
required.

Let
\[
 \begin{gathered}
 \Theta:\mu\operatorname{hom}(\mathbb C_S,\mathcal O_M)_p\\
 \longrightarrow\mathfrak h_{f,p}(\mathcal O_M).
 \end{gathered}
 \tag{FIM.38}
\]
be the comparison HI.B. Its two normal maps are the
positive Thom maps HI.12–HI.18, with real dimensions
\(2\) and \(k=2m-2\); its projection shift is \([2m]\).
The tangent-normal map is the inverse of the
zero-support enlargement followed by its normalized
orientation trace. These specifications determine
\(\Theta\), including its sign.

We compare the two compositions in
\[
 \Theta(k\circ a)=k\cdot\Theta(a).
 \tag{FIM.39}
\]
Here \(k\) is any whole real kernel class, \(a\) is a
relative microlocal Hom class of any degree, \(\circ\)
is composition, and the action on the right is
FIM.4's action on strict hypersurface supports.
The letter \(k\) in this formula denotes a kernel;
the real tangent-normal dimension elsewhere is
explicitly written \(2m-2\).

Here is the composition map in the support model.
On the three-variable space, HI.19 identifies
the second factor's derived Hom from
\(\mathbb C_S(w)\) with sections supported on
\(f_w=0\) of the coefficient pulled back from
\(v\), with shift \([2m]\). Tensor that factor
with the kernel coefficient, whose input
top form is in \(v\). Ordinary coefficient
evaluation multiplies the \(v\)-holomorphic
function into that top form. The supports
intersect. Projecting away \(v\) and using
the normalized holomorphic trace gives
the output coefficient pulled back from
\(u\), still with the same shift \([2m]\).
This is evaluation of the derived Hom
followed by its adjunction trace.

The closed condition \(f_w=0\) is independent
of \(v\), so it commutes with that operation.
On injectives this assertion uses the literal
sections-with-support intersection identity
HI.16. For the proper projection it is
checked after the common collar: restriction
to the ordinary complement \(f_w\ne0\)
commutes with the same compact \(v\)-trace,
and the cone of that restriction is its
closed support functor. R.6–R.7 identifies
the injective and full-relative calculations.
Applying the diagonal support maps in the
two factors gives their two-support cube;
its augmentation, proper excision and
trace are precisely DLG.3–DLG.17.
Thus the microlocal Hom composition in
FIM.39 is this whole-cube map, with both
complement coordinates, rather than
an operation inferred from its value
on supported classes.

##### FIM.5a. The three-variable support

Write \(u,v,w\) for the output, middle and last
variables. The middle trace is over \(v\).
Fix a finite list of kernels, input cochains,
relations and boundary witnesses. AM.H and FIM.34
give one actual coefficient domain and one
negative-polar cone
\[
 \begin{gathered}
 \ell=\Re f_u-\Re f_v\ge0,\\
 |\Im(f_v-f_u)|\le\epsilon\ell,\\
 |z_v-z_u|\le A\ell.
 \end{gathered}
 \tag{FIM.40}
\]
Choose \(L\ge\epsilon\) large enough for all input
strict sectors. After applying the inverse of the
input HI.A comparison, the input is represented
on
\[
 \begin{gathered}
 f_w=0,\qquad z_w=z_v,\\
 H_L(v)=\\
 \left\{\begin{gathered}
 \Re f_v\ge0,\\
 |\Im f_v|\le L\Re f_v
 \end{gathered}\right\}.
 \end{gathered}
 \tag{FIM.41}
\]
This is a use of an inverse comparison roof, not
a claim that an arbitrary cochain is literally
supported on its zero section. HI.10, HI.15 and
R.6–R.7 supply that roof and its finite witnesses.
Cup with the kernel is applied to every entry of
the roof.

Put \(a=\Re f_u\), \(b=\Im f_u\), and
\(d=z_v-z_u\), regarded as a vector in
\(\mathbb R^{2m-2}\). On the closed intersection
of FIM.40 with \(H_L(v)\),
\[
 \begin{gathered}
 0\le\Re f_v\le a,\\0\le\ell\le a,\\
 |b|\le La,\qquad|d|\le Aa.
 \end{gathered}
 \tag{FIM.42}
\]
Choose \(L_*>0\) with
\[
 \begin{gathered}
 L_*^2>L^2+256A^2,\\
 \rho(u)=\\
 \sqrt{\max\{L_*^2a_+^2-b^2,0\}}.
 \end{gathered}
 \tag{FIM.43}
\]
The outer sector is
\(H_*=\{a\ge0,\ |b|\le L_*a\}\).
Its radius \(\rho\) is continuous and zero
outside that sector. On the cup support,
\[
 |d|\le\rho(u)/16
 \tag{FIM.44}
\]
when \(a>0\); when \(a=0\), all of
\(\ell,b,d,\rho\) are zero. Thus both normal
sections \(z_w=z_u\) and \(z_w=z_v\), and their
straight segment, lie in the outer tangent ball.
Shrink the actual output domain so that
\(\rho<R/2\), where \(B_R^{2m-2}\) is the
tangent-normal coordinate ball.

The properness estimate is uniform throughout
this comparison. All intermediate \(v\)'s have
normal displacement at most \(a\), imaginary
normal displacement at most \(\epsilon a\),
and tangent displacement at most \(Aa\).
First choose the outer coefficient boxes and
then the smaller output box. Their whole
closed intersection lies in one compact
middle box over each compact output set.
The extra normal section has displacement at
most \(\rho/16\). Moving it leaves \(u,v\)
unchanged. Hence it preserves that proper
middle support and its common collar.
For finitely many consecutive kernels,
their nonnegative normal lengths add to
at most \(a\), so the same argument uses
the largest of the finitely many constants.

##### FIM.5b. A normal isotopy including the collapsed fibre

We give the comparison of normal sections on
the entire closed family, including \(\rho=0\).
This does not use pullback of differential
forms by a nonsmooth radius.

On the full base \((u,v)\), replace \(d\) by
the continuous vector
\[
 \begin{gathered}
 \widetilde d=0\quad(d=0),\\
 \widetilde d=d\min\left\{1,\frac{\rho}{8|d|}\right\}\\
 (d\ne0).
 \end{gathered}
 \tag{FIM.45}
\]
It is zero where \(\rho=0\), has
\(|\widetilde d|\le\rho/8\), and equals
\(d\) on the closed cup support by FIM.44.
The same equality holds when all the
displacements collapse.

Let
\[
 \begin{gathered}
 \chi(r)=1,\\0\le r\le1/4,\\
 \chi(r)=3/2-2r,\\
 1/4\le r\le3/4,\\
 \chi(r)=0,\\r\ge3/4.
 \end{gathered}
 \tag{FIM.46}
\]
For the real tangent-normal variable
\(\zeta=z_w-z_u\) and \(0\le t\le1\), set
\[
 \begin{gathered}
 \Phi_t(\zeta)=\zeta\quad(\rho=0),\\
 \Phi_t(\zeta)=\zeta+t\widetilde d\,\chi(|\zeta|/\rho)\\
 (\rho>0).
 \end{gathered}
 \tag{FIM.47}
\]
The perturbation has Lipschitz constant at
most \(2|\widetilde d|/\rho\le1/4\).
Consequently
\[
 \begin{gathered}
 \frac34|\zeta-\zeta'|\\
 \le|\Phi_t(\zeta)-\Phi_t(\zeta')|\\
 \le\frac54|\zeta-\zeta'|.
 \end{gathered}
 \tag{FIM.48}
\]
For completeness, surjectivity does not
require an inverse-function assertion.
Given \(y\), iterate
\(\zeta_{n+1}=y-t\widetilde d\chi(|\zeta_n|/\rho)\).
Successive differences decrease by a factor
at most \(1/4\). Their geometric sum
converges in Euclidean space to a solution;
the same contraction estimate proves
uniqueness. FIM.48 proves continuity of the
inverse, uniformly in the base and \(t\).
At \(\rho=0\), the displacement is at most
\(\rho/8\), for the map and its inverse.
Both therefore extend continuously there.

The maps are the identity for
\(|\zeta|\ge3\rho/4\). Points inside that
ball have image of norm at most
\(3\rho/4+\rho/8=7\rho/8<\rho\).
Bijectivity then shows that \(\Phi_t\)
preserves the closed ball of radius
\(\rho\), and its complement, in both
directions. It is also the identity near
the boundary of the fixed ball \(B_R\).
Moreover
\[
 \Phi_t(0)=t\widetilde d.
 \tag{FIM.49}
\]
Thus FIM.47 is a base-preserving isotopy
of the whole outer support and its
complement, taking the zero section to
the section \(d\) on the actual cup
support. Its inverse pullback transports
the positive zero-section Thom map to
that section's Thom map.

Here the coefficient complex, after cup
and the \(f_w\)-normal comparison, is
pulled back from \((u,v)\). It may be
the full derived-support coefficient
complex on the intersection of FIM.40
and \(H_L(v)\). HI.1 applies to every
such bounded complex. Apply its interval
projection comparison to the whole
domain and to its complement. The
restriction squares commute; taking their
relative cones proves that the two
endpoint maps induce the same map after
normal pushforward with support in the
outer ball. The inverse Thom maps at
the two normal sections therefore agree
after enlargement to that ball.

One can use an actual complex supported
on the closed base intersection: apply
the injective sections-with-support
functor before pulling back the normal
coordinate. Its sheaves are closed
direct images from that intersection,
so the equality \(\widetilde d=d\)
there is sufficient. No equality on an
unrelated open complement is presumed.

Every \(\Phi_t\) is isotopic to the
identity through these same maps.
Its normal orientation class is
therefore the positive one of HI.12–HI.15.
The isotopy does not multiply that class
by a nonzero scalar or reverse its sign.
When \(m=1\), the tangent-normal space
has dimension zero; all maps above are
the identity of the empty normal tensor.

![Two normal sections, their support-preserving isotopy and the collapsing family](assets/normal-section-comparison.png)

*Figure FIM.1. The normal plane has real dimension two in panel 1. The two points are the zero section and the input section, with the exact displacement and radius shown. Panel 2 gives the cutoff and uniform inverse estimate of FIM.46–FIM.48. Panel 3 is the real section with A=L=1, b=0 and squared outer aperture 258; its two sections coincide at the collapsed fibre. The entire-support comparison is FIM.5b, and compatibility with the proper trace and every complement term is FIM.5c.*

Full-size PNG · SVG · Exact data · Drawing source

##### FIM.5c. The entire cup and proper trace

We now apply the normal-section comparison
to the full relative cup. If the kernel
is \((a,b)\) and the input is \((c,e)\),
DLG.3 uses its whole-domain entry, its
two single-complement entries and its
double-complement entry. The Thom maps
are coefficient maps on each of these
four entries. Restriction to every
ordinary complement commutes with them.
The finite cover augmentation and its
inverse are the maps DLG.5–DLG.14.
Thus the preceding comparison commutes
with that augmentation by DLG.14's
specified homotopy, rather than by
discarding either complement entry.
For a product or a relation, use the
corresponding eight-entry cube and
its same finite augmentation.

Before the middle trace, the coefficient
complex and both normal blocks are
\[
 \begin{gathered}
 \text{a complex on }(u,v),\\
 \text{the positive}\\
 f_w\text{-normal class},\\
 \text{the positive tangent}\\ \text{class}\\
 \text{at }z_w=z_v.
 \end{gathered}
 \tag{FIM.50}
\]
Move the last normal section by FIM.5b
to \(z_w=z_u\) inside the outer
microlocal support. Now both normal
classes depend only on \(u,w\).
They can be placed outside the middle
trace over \(v\).

To justify this operation with the
excision roof, choose the middle
collar on the common compact
\((u,v)\)-support just proved proper.
Make it independent of \(\zeta\)
on \(|\zeta|\le R/2\); multiply by
a fixed outer normal cutoff equal
to one there if necessary.
Since \(\rho<R/2\) and \(\Phi_t\)
is the identity off the inner
normal support, this is a common
collar for the entire isotopy.
DLG.15 is applied to the augmented
full cone, including its derivative
of the cutoff and its complement
coordinate. The isotopy preserves
its actual proper support. DLG.14,
DLG.17 and the collar homotopy
therefore compare every choice
after one common finite refinement.

The trace commutation can also be
checked on the acyclic complexes.
Once the positive normal classes
are at the \(u\)-section, their
representatives and the inverse
normal comparison roof are
independent of \(v\). Relative
trace integrates the compact
\(v\)-coefficient. Exterior
product by the independent normal
class commutes with this integral
on each finite degree and every
complement entry. Properness
makes the integral a compact
one locally in \(u\).
V.4–V.6 and R.6–R.7 identify
this calculation with the
holomorphic/injective trace roof.
There is no pullback of the
holomorphic coefficient by
FIM.47: that coefficient depends
on \(u,v\), which the isotopy
fixes.

Both real normal dimensions are
even. Their total is \(2m\),
which cancels the projection
shift. Commuting the two normal
blocks contributes
\((-1)^{2(2m-2)}=1\).
If the input degree after
normal cancellation is \(q\),
the first relative trace sign
before cancellation is
\[
 (-1)^{m(q+2m)}=(-1)^{mq}.
 \tag{FIM.51}
\]
This is exactly DLG.16's sign
after cancellation. Its second
relative component has the
same additional factor
\((-1)^m\) on both sides.
The orientation of the
\(f_w\)-normal block is the
positive complex orientation;
the change \(f_w-f_u=-f_u\)
has positive real determinant,
as in HI.22. No additional
antipodal sign is introduced.

We have consequently constructed
a commuting diagram of actual
full-cube/proper-trace roofs.
On the first route, use the
input normal comparison,
compose the kernel and the
microlocal Hom class, and use
the output normal comparison.
On the second, apply the
canonical strict-support
kernel action. FIM.5b compares
the two possible normal
sections before tracing, and
the preceding computation
identifies every other map.
This proves FIM.39 in all
relative degrees.

Boundary representatives and
equality witnesses are retained.
For each finite diagram AM.H
supplies one common support
cone and coefficient domain;
DLG.14 and DLG.17 give the
comparison homotopies there.
Taking their filtered colimit
proves equality for the whole
real stalk \(C\), with no
finite-symbol or finite-tail
restriction.

The contravariant source shift
\(\mathbb C_S[-1]\) gives the
same comparison shifted by
\([1]\). Each entire operation
has degree zero after its
kernel degree and trace
degree cancel, so the common
shift contributes no new
action sign.

##### FIM.5d. The finite-\(E\) submodule in microlocal Hom

Combine FIM.39 with FIM.37.
The actual pole/logarithm
class defines an injective
map
\[
 \begin{gathered}
 E/(Ef+\sum_i ED_{z_i})\\
 \longrightarrow H^0\mu\operatorname{hom}\\
 (\mathbb C_S[-1],\mathcal O_M)_p.
 \end{gathered}
 \tag{FIM.52}
\]
It is linear for the
canonical finite-\(E\)
action obtained by restriction
from \(C\). Its injectivity
is the injectivity proved
in FIM.2 and FIM.4, transported
through HI.B, not a
faithful-flatness assertion
about \(C/E\).
The strict filtration and
reduced conormal symbol ideal
are those of FIM.0–FIM.1.
On the incidence chart, the
output restriction of this
submodule is the rank-one
module FIM.31.

This statement concerns the
specified submodule and its
canonical action. It asserts
neither that its image is
the whole microlocal Hom
stalk nor that it is stable
under all of \(C\).
The infinite-order right input
Legendre comparison, both
normalized inverse
convolutions and general
holonomic regularization
are separate assertions.


#### FIM.6. The entire finite-order right input action

Use the incidence coordinates of FIM.3,
\[
 \begin{gathered}
 f=s-t-x^Ty,\\
 z=(t,x,y),\\ \lambda=\xi_f\ne0.
 \end{gathered}
 \tag{FIM.53}
\]
The ordered input form is \(ds\wedge dy\).
Right input operators act by their negative
transpose in that frame. The left output
operators commute with this action, since
their original variables are disjoint.
The input normal covariable \(\sigma\)
is nonzero on the selected conormal.

Ordinary coordinate transport of the
actual finite-order rings is §5.56.
The full transpose, obtained by moving
each input derivative across its
coefficient with its sign, is the actual
coefficientwise transpose of §5.51,
EC.B.1–EC.B.2, equations EC.B1–EC.B7.
These maps preserve upper
order and the negative factorial bound
on one smaller coefficient product.
No primitive of a holomorphic function
is part of this definition.

The support action comparison FIM.5
allows these input and output actions
on the specified microlocal Hom
submodule to be computed in
\(\mathcal M_S\). Its output
identification FIM.31 gives a map
\[
 \begin{gathered}
 \Phi:E_Y\longrightarrow E_X,\\
 eR=\Phi(R)e.
 \end{gathered}
 \tag{FIM.54}
\]
Here the right-hand side is obtained
by applying the whole transposed input
operator in the ordinary incidence
coordinates, then the strict normal
division FIM.4–FIM.17 and the inverse
output identification FIM.29–FIM.31.
This constructs a whole actual map,
including its entire negative tail.
Each of those operations is
nonincreasing in upper order.

We prove
\[
 \begin{gathered}
 \Phi(R)=L_0(R),\\
 R\in E_Y.
 \end{gathered}
 \tag{FIM.55}
\]
where \(L_0\) is the full Legendre
map LN.8–LN.24. The proof uses
finite jets of individual coefficients
and strictness; agreement on words
alone is not sufficient.

##### FIM.6a. Finite dependence of an output coefficient

Write
\[
 R=\sum_{j\le b}r_j(s,y,w)\sigma^j.
 \tag{FIM.56}
\]
For any integer \(q\), the difference
between \(R\) and its finite degree
slice
\[
 \begin{gathered}
 R_{\ge q}\\
 =\sum_{q\le j\le b}r_j(s,y,w)\sigma^j.
 \end{gathered}
 \tag{FIM.57}
\]
has order at most \(q-1\).
If \(q>b\), the slice is zero.
Strictness of the three actual
operations defining FIM.54 shows
that the degree-\(q\) coefficient
of \(\Phi(R)\) is precisely that
of \(\Phi(R_{\ge q})\).
Strictness of LN.8–LN.24 gives
the same assertion for \(L_0\).

We also need continuity in the
holomorphic functions \(r_j\),
rather than just their degree
labels. In the input variables,
with \(t,x\) fixed as parameters,
the coordinate change is affine:
\[
 \begin{gathered}
 s=f+t+x^Ty,\\
 D_s=D_f,\\
 D_{y_i}^{\rm old}=D_{y_i}^{\rm new}-x_iD_f.
 \end{gathered}
 \tag{FIM.58}
\]
It substitutes \(w=\nu_y-x\)
in a homogeneous coefficient,
without a lower homogeneous
correction.

For the transpose, its ordinary
normal symbol is
\[
 \begin{gathered}
 r^t(v,\xi)\\
 =\sum_\alpha\frac1{\alpha!}
       \partial_\xi^\alpha\partial_v^\alpha r(v,-\xi).
 \end{gathered}
 \tag{FIM.59}
\]
This formula follows by repeatedly
using \((-D)a=-aD-\partial a\)
on a monomial; coefficient
uniqueness gives the displayed
normal symbol. For an input
homogeneous degree \(j\), the
term with index \(\alpha\)
has degree \(j-|\alpha|\).
Thus, among \(q\le j\le b\),
only finitely many indices
\(|\alpha|\le b-q\) can affect
degree \(q\).

The final normal division is
FIM.6. After the tangent
covariables are set to zero,
it uses only Taylor coefficients
in \(f\) with index at most
\(b-q\), and the coefficient
rows with degree at least
\(q\). Consequently the
degree-\(q\) coefficient of
FIM.54 is a finite sum of
restrictions and derivatives
of these finitely many
holomorphic \(r_j\)'s.
All evaluations take place
on one strictly smaller
coefficient product.
The Cauchy integral formula
on a fixed positive margin
proves continuity of this
finite expression for uniform
convergence on its outer
compact product.

The corresponding LN coefficient
is explicitly
\[
 \begin{gathered}
 (L_0R)_q=\\
 \sum_{K=0}^{b-q}(\mathcal A_Kr_{q+K})(\iota),\\
 \mathcal A_K=(-1)^K\\
 {}\cdot\sum_{|\gamma|=K}
       \frac{V^\gamma\partial_w^\gamma}{\gamma!},\\
 V_i=\partial_{y_i}+x_i\partial_s,\\
 \iota(t,x,z)=\\
 (t+x^Tz,z,-x).
 \end{gathered}
 \tag{FIM.60}
\]
It is another finite continuous
jet expression on the same
kind of nested products.
The variable \(z\) in this
formula is the output
projective covariable.

##### FIM.6b. Polynomial coefficients and holomorphic coefficients

For polynomial coefficients in
\(s,y,w\), every term of
FIM.57 is a finite expression
in \(s,y,D_y,D_s,D_s^{-1}\).
The cyclic relations FIM.32
and commutation with the left
output action imply
\[
 \begin{gathered}
 \Phi(D_s)=D_t,\\
 \Phi(y_i)=D_{x_i}D_t^{-1},\\
 \Phi(D_{y_i})=-x_iD_t,\\
 \Phi(s)=\\
 t+\sum_i x_iD_{x_i}D_t^{-1}.
 \end{gathered}
 \tag{FIM.61}
\]
The inverse \(D_s^{-1}\) acts
as the actual inverse of its
action; in the output row it
is \(D_t^{-1}\).
Since the two sides commute,
\(e(R_1R_2)=\Phi(R_1)\Phi(R_2)e\).
FIM.31's injectivity makes
\(\Phi\) multiplicative.
LN.1–LN.8 gives exactly
the same ordered-word maps
for \(L_0\). Thus FIM.55
holds on the stated finite
polynomial expressions,
including negative powers
of the normal derivative.

Now fix \(q\) and a point of
the coefficient product.
Shrink to nested polydiscs
about the finitely many
evaluation points. Taylor
polynomials of each \(r_j\),
\(q\le j\le b\), converge
uniformly on the outer
compact polydisc and, by
one further Cauchy margin,
in every derivative occurring
in the two finite expressions.
Their polynomial identity
therefore passes to
the holomorphic coefficients.
The conclusion is equality
of the degree-\(q\)
holomorphic germs, throughout
their common inner product.
It is local in the base and
projective covariable and
hence holds on all the
chosen smaller product.

This is a coefficientwise
jet argument, not a proposed
convergence of numerical
Laurent series or a
polynomial approximation
to an entire negative tail.
The discarded lower-order
tail cannot enter degree
\(q\), by strictness.

##### FIM.6c. Actual negative tails

The construction of \(\Phi\)
already places its image
in the actual finite-order
output ring: transpose and
ordinary transport have
their fixed-product
factorial estimates, FIM.11
and FIM.14 give the strict
normal division estimate,
and FIM.31 preserves that
coefficient type. These
products can be nested
finitely many times,
independently of \(q\).

For \(L_0\), the estimate
LN.12 applied to FIM.60
is the entire negative-tail
estimate. In particular,
the terms with
\(q+K=-r<0\) have the
majorant
\[
 \begin{gathered}
 M\,\mathcal L^K K!\,C^r r!\\
 \le M\,n!\,\mathcal L^K C^r,\\
 K+r=n.
 \end{gathered}
 \tag{FIM.62}
\]
where \(\mathcal L\) is
the fixed Cauchy contraction
constant of LN.12. Summing
them is bounded by
\(M n!(C+\mathcal L)^n\).
The finitely many
nonnegative input degrees
have their separate finite
coefficient norms; their
contribution is bounded
by \(M' C'^n n!\), since
a fixed polynomial in \(n\)
is bounded by an
exponential. There is
one finite upper order,
one common coefficient
product and one negative
factorial bound for the
whole output row.

The equality in every
degree proved in FIM.6b
therefore identifies two
actual germs with their
entire tails, proving
FIM.55. The complete
coefficient formula is
FIM.60, including arbitrary
holomorphic projective
covariable coefficients.

This proves the whole
finite-order input
identification. It does
not assert convergence
of normal division on a
symbol with infinitely
many positive degrees,
or identify the
infinite-order \(B\)
action merely from its
ordered generators.
Those assertions require
their own positive-tail
comparison. Both
normalized holomorphic
inverse products and
the full real contact
transport are also
separate assertions.


#### The finite-order incidence bimodule

**Theorem FIM.A.** At every nonzero conormal to a smooth hypersurface, the actual coherent quotient in FIM.2 injects into the shifted microlocal Hom in FIM.52, with its canonical finite-order action. The strict principal annihilator is the reduced conormal ideal. On the incidence chart its output restriction is free of rank one, generated by the single-pole class.

**Proof.** FIM.0–FIM.1 gives the strict coherent normal form and its principal annihilator. FIM.2 and FIM.4 prove injectivity of the pole/logarithm class for the actual strict-support action. FIM.5 transports it through the action-compatible HI.B comparison. The output rank-one assertion is FIM.31. All these constructions use one actual coefficient product for the entire finite-order symbol. \(\square\)

**Theorem FIM.B.** In the ordered incidence input frame, the entire finite-order right input action is the full Legendre isomorphism with parameter zero. In particular the specified incidence submodule is also free of rank one as a right input module.

**Proof.** FIM.6 identifies the actual right-input coefficient map with the whole map LN.8–LN.24. The latter is an algebra isomorphism by LN.4–LN.6. The output identification FIM.31 consequently identifies the map from the right input ring, sending an operator to its action on the single-pole class, with that isomorphism. It is a right-module isomorphism by the commuting left and right actions. \(\square\)

The rank-one bimodule statement is finite-order. It does not assert that the specified image equals the whole microlocal Hom stalk, or identify the full infinite-order or real operator actions. Normalized holomorphic inverse convolutions, general holonomic regularization and analytic proper regularity are separate statements.

Free primary context: [Benchoufi, *Action of micro-differential operators on quantized contact transformations*, arXiv:2102.08861v2](https://arxiv.org/abs/2102.08861v2), especially Section 3. The arguments use the complete local proofs and exact earlier programme proof dependencies linked above.


### 5.62. The infinite-order incidence bimodule

The exact earlier programme proof dependencies are §§5.44, 5.51, 5.53, 5.58, 5.61, EC.B.

Use the actual rings \(E\subset B\subset C\) of §§5.53 and5.58.
In coordinates \((z,f)\) at a nonzero conormal, put
\(\lambda=\xi_f\ne0\), \(w=\xi_z/\lambda\). A whole
\(B\)-symbol has homogeneous coefficients on one common
product,
\[
 \begin{gathered}
 P=\sum_{j\in\mathbb Z}p_j(z,f,w)\lambda^j,\\
 \begin{cases}
 \|p_{-n}\|\le MC^n n!,&n\ge1,\\
 \|p_j\|\le M_\epsilon\epsilon^j/j!,&j\ge0.
 \end{cases}
 \end{gathered}
 \tag{BI.1}
\]
with the second inequality for every \(\epsilon>0\).
The constants are for a fixed compact inner product;
that product is independent of the homogeneous index
and of \(\epsilon\). As in §5.58, these are actual
coefficient sequences, not numerical Laurent series
in \(\lambda\).

#### BI.0. The positive normal division

Consider
\[
 \mathcal M_B=B/(Bf+\sum_i BD_{z_i}).
 \tag{BI.2}
\]
The rightmost generators specify a left ideal.
The radial division FIM.4 removes the tangent
covariables. Its fixed Cauchy margin bounds the
dividing coefficients in both parts of BI.1.
Multiplication by \(\lambda^{-1}\) shifts their
index by one and retains both classes: for
positive indices use
\(\epsilon^{j+1}/(j+1)!\le
\epsilon\,\epsilon^j/j!\); the single new
negative coefficient is a fixed holomorphic
coefficient. Thus all tangent quotients are
whole \(B\)-symbols on a common product.

Write
\(a_j(z,f)=p_j(z,f,0)=\sum_{k\ge0}a_{j,k}(z)f^k\)
on an outer normal disc of radius \(R\).
For each nonnegative output index \(\ell\), define
\[
 q_\ell=
 \sum_{k\ge0}(-1)^k(\ell+k)_k a_{\ell+k,k}.
 \tag{BI.3}
\]
This series is generally infinite. For every
\(0<\epsilon<R\), Cauchy's coefficient bound gives
\[
 \begin{gathered}
 |(\ell+k)_k a_{\ell+k,k}|\\
 \le M_\epsilon
       \frac{\epsilon^\ell}{\ell!}
                         (\epsilon/R)^k,\\
 \|q_\ell\|
 \le\frac{M_\epsilon}{1-\epsilon/R}
                      \frac{\epsilon^\ell}{\ell!}.
 \end{gathered}
 \tag{BI.4}
\]
For a requested output parameter \(\delta>0\),
choose \(\epsilon\le\delta\) and
\(\epsilon<R/2\). BI.4 then proves the
every-\(\delta\) bound on the same inner
product. The sum and its fixed derivatives
converge normally, with geometric tails on
the chosen Cauchy margins.

For a negative output index \(-n\), every
nonnegative input degree in BI.3 has zero
falling factorial: if \(j=k-n\ge0\), then
\(k>j\). Consequently
\[
 \begin{gathered}
 q_{-n}=\\
 \sum_{k=0}^{n-1}
 \frac{(n-1)!}{(n-k-1)!}a_{-(n-k),k},\\
 \|q_{-n}\|\le Mn!(C+R^{-1})^n.
 \end{gathered}
 \tag{BI.5}
\]
The positive input part contributes exactly
zero to every negative normal remainder.
This is an algebraic falling-factorial
identity, not a limiting cancellation
between unrelated coefficient domains.

The whole right-\(f\) quotient is
\[
 \begin{gathered}
 A_\ell(z,f)=
 \sum_{\nu\ge0}(-1)^\nu(\ell+\nu)_\nu\\
 {}\times\sum_{k\ge\nu+1}
 a_{\ell+\nu,k}(z)f^{k-1-\nu}.
 \end{gathered}
 \tag{BI.6}
\]
When \(\ell=-n\), all terms with
\(\nu\ge n\) are zero, so FIM.14 applies.
When \(\ell\ge0\), on \(|f|\le r<R\)
the absolute value of the \(\nu\)-term
is at most
\[
 \frac{M_\epsilon}{R(1-r/R)}
 \frac{\epsilon^\ell}{\ell!}
                         (\epsilon/R)^\nu.
 \tag{BI.7}
\]
Thus \(A\) satisfies the positive bound
of BI.1 for every output parameter and
the negative factorial bound of FIM.14.
There is one common product. Fixed
derivatives use one further fixed
Cauchy margin.

Sum the identity FIM.8 using these
absolute majorants. The tangent
quotients have already been constructed,
so it proves the entire identity
\[
 \begin{gathered}
 P=Q+Af+\sum_i B_iD_{z_i},\\
 Q=\sum_j q_j(z)\lambda^j.
 \end{gathered}
 \tag{BI.8}
\]
in the actual ring \(B\).

For uniqueness, evaluation at \(w=0\)
kills every right \(D_{z_i}\)-multiple.
The coefficients of a right \(f\)-multiple
are \(fa_j+(j+1)a_{j+1}\).
Substitution into BI.3 cancels the
two successive Taylor indices.
For \(\ell\ge0\), BI.4 andBI.7
make both series absolutely convergent;
their endpoint tails tend to zero
geometrically. For \(\ell<0\),
FIM.16 gives a finite cancellation.
The remainder of a row independent of
\(f,w\) is itself. Therefore BI.8 is
unique and its kernel is exactly
the left ideal in BI.2.

It also identifies
\(B\otimes_E\mathcal M_S\) with
\(\mathcal M_B\). Explicitly the map
sends \(b\otimes[P]\) to \([bP]\).
The inverse sends \([b]\) to
\(b\otimes[1]\); rightmost relations
go to zero by balancing. These are
inverse maps by the tensor relations.
No flatness of \(B/E\), \(C/E\) or
\(C/B\) is used.

#### BI.1. Entire normal functions

Use \(u_j\) of FIM.21, with its
positive connecting representative.
The negative part of
\[
 U_Q(z,f)=\sum_{j\in\mathbb Z}q_j(z)u_j(f)
 \tag{BI.9}
\]
is the convergent logarithmic series
FIM.22–FIM.24 on a sufficiently small
punctured normal disc. On a compact
annulus \(|f|\ge r_1>0\), the
nonnegative summands satisfy
\[
 \|q_j u_j\|
 \le\frac{M_\epsilon}{r_1}
                    (\epsilon/r_1)^j.
 \tag{BI.10}
\]
Choose \(\epsilon<r_1/2\). This
proves normal convergence and all
fixed derivatives on that compact
annulus. Every compact subset of
the punctured product admits such
an estimate; the coefficient domain
and the punctured product do not
depend on \(j\). The nonnegative
part is single valued and may
have an essential singularity.

The branch jump remains exactly
\(2\pi i A_Q\), with \(A_Q\)
the full-disc holomorphic function
FIM.24. Hence the relative class
\([(0,-U_Q)]\) is branch independent.
If it extends holomorphically
on a full later product, continuation
around a normal circle gives
\(A_Q=0\) and kills every negative
coefficient. The remaining
nonnegative part is a normally
convergent Laurent series on that
circle. Termwise circle coefficient
extraction kills every \(q_j\),
\(j\ge0\). This proves injectivity
on the whole quotient BI.2,
including essential singularities.

The comparison with the canonical
action uses the same degree-one
support-forgetting injection
FIM.36. The whole AM.B halfspace
action for Euler parameter \(-1\)
has detector \(1/f\), and its
positive derivatives are exactly
the nonnegative \(u_j\)'s.
BI.10 is its actual convergent
positive part; FIM.23 is its
negative primitive part. The
normalization and holomorphic
gauges are AM.23a–AM.25b.
The two strict-support classes
therefore have the same halfspace
image; FIM.36 makes them equal.
FIM.5 transports this equality
to the canonical microlocal Hom
action. Thus BI.9 gives a
canonically \(B\)-linear
injection of the specified
quotient, without identifying it
with the entire microlocal Hom
stalk.

#### BI.2. Quantified positive cutoffs

The next comparison concerns
the input transpose and the
strict normal map. These have
to be controlled on a whole
positive tail before using
the finite-order identity.

Let \(P^{[J]}\) retain every
negative coefficient and the
nonnegative coefficients through
degree \(J\). The discarded
part has only positive degrees.
Fix \(0<\rho<1\). BI.4 gives
the following bound for its
nonnegative remainder:
\[
 \begin{gathered}
 \|N(P-P^{[J]})_\ell\|\\
 \le\frac{M_\epsilon\,\rho^{J+1}}
 {1-\epsilon/(\rho R)}
 \frac{(\epsilon/\rho)^\ell}{\ell!},\\
 \epsilon<\rho R.
 \end{gathered}
 \tag{BI.11}
\]
Indeed its summation indices
obey \(\ell+k>J\); extract
\(\rho^{\ell+k}\le\rho^{J+1}\)
and sum the remaining geometric
series. All its negative
remainders are zero by BI.5.
The quotient \(A\) has the
same extra geometric factor
in BI.7, with its additional
fixed factor \(1/[R(1-r/R)]\).

For the transpose, use the
two independent Cauchy margins
of EC.B3–EC.B5, and absorb the
number of derivative multi-indices
into a fixed constant \(D>0\).
A nonnegative input degree
\(m\) and derivative length
\(k\) contribute only to
output degree \(m-k\).
Their total norm is at most
\[
 M_\epsilon
       \frac{\epsilon^m}{m!}D^k k!.
 \tag{BI.12}
\]
The antipodal coordinate
neighbourhood contributes
a fixed exponential in \(m\);
choose a correspondingly
smaller input \(\epsilon\)
before applying BI.12.

For a nonnegative output \(j\),
write \(m=j+k\), and use
\(k!/(j+k)!\le1/j!\).
For a negative output \(-n\),
write \(k=m+n\), and use
\((m+n)!/m!=n!\binom{m+n}{n}\).
The elementary generating
identity
\[
 \begin{gathered}
 \sum_{m\ge0}\binom{m+n}{n}x^m\\
 =(1-x)^{-n-1},\\
 |x|<1.
 \end{gathered}
 \tag{BI.13}
\]
follows by differentiating the
geometric series \(n\) times
and dividing by \(n!\).
Consequently the positive
input part of the transpose
has bounds
\[
 \begin{gathered}
 \|(TP)_j^+\|
 \le\frac{M_\epsilon}{1-D\epsilon}
                         \frac{\epsilon^j}{j!},\\
 \|(TP)_{-n}^+\|
 \le\frac{M_\epsilon D^n n!}
                         {(1-D\epsilon)^{n+1}}.
 \end{gathered}
 \tag{BI.14}
\]
These estimates construct the
actual whole transpose on \(B\).
The original negative part is
the actual EC.B transpose.

For the discarded positive
tail, the same extraction of
\(\rho^m\) proves
\[
 \begin{gathered}
 \|T(P-P^{[J]})_j\|\\
 \le\frac{M_\epsilon\rho^{J+1}}
 {1-D\epsilon/\rho}
 \frac{(\epsilon/\rho)^j}{j!},\\
 \|T(P-P^{[J]})_{-n}\|\\
 \le\frac{M_\epsilon\rho^{J+1}D^n n!}
 {(1-D\epsilon/\rho)^{n+1}}.
 \end{gathered}
 \tag{BI.15}
\]
Choose \(\epsilon<\rho/(2D)\).
The negative estimate then has
one factorial constant \(2D\)
independent of \(J\). For any
requested positive parameter
\(\delta\), also require
\(\epsilon\le\rho\delta\).
This proves an actual geometric
cutoff limit, simultaneously
for the negative class and
each positive seminorm, on
fixed nested coefficient
products.

BI.4 uses these estimates and
the full product majorants
IK.3–IK.10 to prove the actual
transpose anti-involution.
BI.6 gives the composite
transpose, affine coordinate
change and normal-division
cutoff estimate needed for
the entire input identification.

#### BI.3. A positive-tail example

For any complex parameter \(a\),
the actual symbol
\[
 P_a=\sum_{m\ge0}
           \frac{a^m f^m}{(m!)^2}D_f^m
 \tag{BI.16}
\]
belongs to \(B\). On \(|f|\le R\),
for every \(\epsilon>0\) the
ratio of its coefficient to
\(\epsilon^m/m!\) is bounded
by \(\exp(|a|R/\epsilon)\),
using the exponential series.
In BI.3 the only nonzero
output index is zero, and
\[
 \begin{gathered}
 N(P_a)_0=
 \sum_{m\ge0}\frac{(-a)^m}{m!}\\
 =e^{-a}.
 \end{gathered}
 \tag{BI.17}
\]
Thus arbitrarily high positive
degrees can all contribute to
one output coefficient.
The geometric estimates are
essential to the whole quotient;
a finite-degree dependence
argument alone does not extend
FIM.6 to this class.


Use the domains and coefficient classes of BI.0–BI.3.
The actual binary and triple product majorants are
§5.44, IK.2–IK.3, equations IK.3–IK.10.
The finite transpose is §5.51, EC.B1–EC.B7.
The full Legendre map, including its cutoff estimates
and inverse, is §5.58, LN.4–LN.6.
The exact finite-order incidence module and both
finite-order operator sides are §5.61.

#### BI.4. The transpose is an actual anti-involution

We first specify the limiting assertion required
for the product proof. Suppose a family \(P_a\)
on a fixed coefficient product has common
negative bounds \(MC^n n!\), common positive
bounds \(M_\epsilon\epsilon^j/j!\) for each
\(\epsilon>0\), and coefficientwise normal
convergence with each fixed derivative.
Then \(T(P_a)\) has those same kinds of
common bounds on one fixed smaller product,
and converges coefficientwise normally,
with every fixed derivative, to \(T(P)\).

Indeed fix an output index \(j\).
The negative input indices which can enter
that output obey \(j\le m<0\), so there
are only finitely many. Their differential
expressions converge by the hypothesis.
For the nonnegative inputs, BI.12 gives
the same summable majorant for the entire
family: use the geometric series in
BI.14 when \(j\ge0\), or the binomial
series there when \(j<0\).
Choose its input \(\epsilon\) before
summing. Given a positive tolerance,
make the tail in the nonnegative input
index small using that common majorant.
The remaining finite sum converges
normally. A fixed further Cauchy margin
proves the same assertion for any
specified finite derivative.
BI.14 and the finite negative-part
EC.B5 estimate give the common growth
bounds. No coefficient-dependent
neighbourhood is used.

There is an analogous assertion for
products of such families. The proof
is contained explicitly in the scalar
majorants IK.3–IK.7: at a fixed output
degree the positive input indices,
absolute negative input indices and
contraction lengths satisfy the grouped
degree relation in IK.3. For common
input growth constants, choose a
common positive parameter making its
\(E(D+L)<1\). The scalar series is
summable. Its complement outside a
finite grouped-index range is uniformly
small by IK.6–IK.7. Inside that
range there are finitely many
holomorphic differential expressions,
which converge by hypothesis.
IK.8–IK.10 and one further Cauchy
margin give the assertion for the
products and all their fixed derivatives.
The same bounds give common positive
and negative output growth constants.

Now take the positive cutoffs
\(P^{[J]},Q^{[J]}\). They retain both
entire negative tails, so they belong
to the actual finite-order ring.
The finite transpose satisfies
EC.B6–EC.B7 there.
Their product family has common growth
bounds and converges normally,
coefficient by coefficient with fixed
derivatives, to \(P\circ Q\), by the
preceding product argument.
The transpose-family argument therefore
passes its left side to \(T(P\circ Q)\).

On the right, BI.14–BI.15 gives common
growth bounds and the corresponding
convergence of \(TP^{[J]}\) and
\(TQ^{[J]}\). The product-family
argument gives convergence to
\(TQ\circ TP\).
Taking the finite-order identity's
coefficient limit proves the second
identity below.
Apply the transpose-family argument
again to \(TP^{[J]}\).
The finite-order identities
\(T^2P^{[J]}=P^{[J]}\) tend to
the first identity. Its unit value
is immediate from the coefficient
formula. We have proved on whole
actual germs
\[
 \begin{gathered}
 T^2=1,\qquad T(1)=1,\\
 T(P\circ Q)=TQ\circ TP.
 \end{gathered}
 \tag{BI.18}
\]
The finite transpose is its restriction,
and every construction commutes with
sheaf restriction. The identities are
identities of actual holomorphic
coefficient sequences with the stated
growth bounds.

#### BI.5. The whole output module

On the incidence chart put
\(f=s-t-x^{\mathsf T}y\) and
\(z=(t,x,y)\). The ordered input
form is \(ds\wedge dy\).
Let \(e=[1]\) in BI.2.
The output map is the whole
coefficient substitution of FIM.30.
For an output normal symbol
\(\sum_j r_j(t,x,v)\tau^j\), its
normal row is exactly
\[
 \sum_j(-1)^j r_j(t,x,y)\lambda^j e.
 \tag{BI.19}
\]
This statement includes all positive
degrees as well as the complete
negative tail.

Here is the domain and product check.
On a compact product with
\(|\nu_0|<c<1\), the transformed
degree-\(j\) coefficient is FIM.30.
For \(j\ge0\), its extra factor is
bounded by \((1+c)^j\); choose
an input positive parameter divided
by \(1+c\) to obtain any prescribed
output parameter. For \(j=-n\),
the factor is bounded by
\((1-c)^{-n}\), which only enlarges
the fixed negative factorial constant.
The arguments of all coefficient
functions lie in one common original
product by the choice of the compact
output product.
The ordinary substitution is exact
because the transformed output fields
do not differentiate \(y\).
Every full contraction has the same
chain-rule cancellation as FIM.30;
IK.2–IK.3 justifies every sum and
regrouping on the fixed smaller product.
Equivalently the finite substitution
identities pass through the positive
cutoffs with these common scalar
majorants, retaining all negative
coefficients.

The transformed coefficients are
independent of \(f\). After radial
tangent division, the normal Taylor
formula therefore has only its
zeroth Taylor term. This proves
BI.19 directly. Its inverse renames
\(y\) as \(v\) and multiplies
degree \(j\) by \((-1)^j\).
It preserves both bounds on the
same renamed product. Consequently
\[
 B_X\xrightarrow{\ \sim\ }\mathcal M_B,
 \qquad P\longmapsto Pe
 \tag{BI.20}
\]
is an isomorphism of whole left
output modules. No upper order or
principal symbol is assigned to a
symbol with an infinite positive head.

#### BI.6. The entire right input map

Before changing the incidence
coordinates, whole output operators
and transposed input operators
commute. In either order, every
nonzero contraction would differentiate
a coefficient in a variable on
which it does not depend. Only the
ordinary coefficient product remains;
its scalar sums are absolute by IK.2.
The exact affine change preserves
that equality.
BI.18 therefore defines a right
\(B_Y\)-action commuting with
the left \(B_X\)-action on
\(\mathcal M_B\).

The affine input substitution, with
\(t,x\) held fixed as parameters, is
\[
 \begin{gathered}
 s=f+t+x^{\mathsf T}y,\qquad
 \sigma=\lambda,\\
 w=\nu_y-x.
 \end{gathered}
 \tag{BI.21}
\]
It has relative input Jacobian one.
There is no density multiplier and
no lower homogeneous correction.
The argument is FIM.58 with actual
whole coefficients; the product-family
argument of BI.4 extends its finite
product identities. Choose nested
actual products such that every
displayed coordinate image lies
compactly inside its preceding
coefficient domain.

Via BI.20 define the whole map
\[
 \begin{gathered}
 \Phi_B:B_Y\longrightarrow B_X,\\
 eR=\Phi_B(R)e.
 \end{gathered}
 \tag{BI.22}
\]
Its calculation is: transpose the input,
apply BI.21, use the unique normal
remainder BI.3–BI.8, and invert BI.19.
Every operation has already been
constructed on actual whole symbols.

We give its composite cutoff estimate.
Fix \(0<\rho<1\), and put
\(R^{[J]}\) for the positive cutoff,
retaining the entire negative input
tail. Let \(Z^{[J]}\) be the affine
substitution of \(T(R-R^{[J]})\).
On the fixed substituted product,
BI.15 gives constants independent
of \(J\), with
\[
 \begin{gathered}
 \|Z^{[J]}_{-n}\|\\
 \le K\,\rho^{J+1}(2D)^n n!,\\
 \|Z^{[J]}_j\|
 \le K_\epsilon\,\rho^{J+1}\epsilon^j/j!,\\
 j\ge0.
 \end{gathered}
 \tag{BI.23}
\]
For the first line choose one fixed
source parameter below \(\rho/(2D)\).
For each requested parameter in the
second line choose a further smaller
source parameter as in BI.15.
The normalizing input power is
\(\sigma=\lambda\), so BI.21 adds
no homogeneous exponential factor.
Fixed holomorphic parameter
substitution preserves these sup
bounds on the chosen compact products.

Let \(R_f>0\) be the outer normal
disc radius for the subsequent
normal division. BI.5 applied
to the first line of BI.23
gives the negative part of the
following estimate.
For its positive part use BI.4
with a parameter less than
\(R_f/2\) and at most the
requested output parameter
\(\delta\). Absorb its fixed
geometric denominator into the
prefactor. Inverting BI.19
only changes signs and renames
a holomorphic parameter.
Thus
\[
 \begin{gathered}
 \|(\Phi_B(R)-\Phi_B(R^{[J]}))_{-n}\|\\
 \le K'\rho^{J+1}
                    (2D+R_f^{-1})^n n!,\\
 \|(\Phi_B(R)-\Phi_B(R^{[J]}))_j\|\\
 \le K_\delta'\rho^{J+1}\delta^j/j!,
 \qquad j\ge0.
 \end{gathered}
 \tag{BI.24}
\]
All products and margins precede
the indices and \(J\). Fixed
derivative versions follow on
one further product by the
iterated Cauchy formula. BI.24
is an actual coefficient and
tail estimate, rather than an
assumption about continuity of
microlocal cohomology.

For each \(J\), the symbol
\(R^{[J]}\) belongs to \(E_Y\).
Theorem FIM.B identifies its
right input action with
\(L_0(R^{[J]})\).
BI.24 passes the left side
coefficientwise normally to
\(\Phi_B(R)\).
LN.17–LN.18 and LN.6 passes
the right side, with its whole
negative actual tail retained,
to \(L_0(R)\).
Both coefficient sequences
are actual whole \(B_X\)
germs on the common product,
so
\[
 \begin{gathered}
 \Phi_B(R)=L_0(R),\\
 R\in B_Y.
 \end{gathered}
 \tag{BI.25}
\]
Since \(L_0\) is the whole
isomorphism of Theorem LN.C,
the map \(R\mapsto eR\)
also identifies \(\mathcal M_B\)
with the free rank-one right
input module.

#### BI.7. Positive-to-negative transpose mixing

In one spatial dimension choose
\[
 \begin{gathered}
 p_m(s,y,w)=\\
 \frac1{(m!)^2(1-y/2)}\\
 {}\times\frac1{1+w/2},\qquad m\ge0,\\
 p_m=0\quad(m<0).
 \end{gathered}
 \tag{BI.26}
\]
On \(|y|,|w|<1\) these are actual
whole coefficients: the rational
factor is uniformly bounded on
each compact, and
\(1/(m!\epsilon^m)\le e^{1/\epsilon}\)
proves the every-\(\epsilon\)
condition. This is the input
head of LN.32, now used to
compute the transpose itself.

There is no \(s\)-dependence.
Only the spatial contraction
of length \(K\) contributes
in the transpose formula.
At \(y=w=0,\sigma=1\),
the two differentiated
rational factors give
\((1/2)^K K!\) and
\((-1/2)^K K!\).
The antipodal substitution
also gives \((-1)^m\) on
homogeneous degree \(m\).
Since the output index is
\(j=m-K\), their total sign
is \((-1)^{m+K}=(-1)^j\).
Put \(c=1/4\). We obtain
\[
 \begin{gathered}
 (TP)_j(0)=\\
 (-1)^j\sum_{m\ge\max(0,j)}a_{j,m},\\
 a_{j,m}=\frac{c^{m-j}(m-j)!}{(m!)^2}>0.
 \end{gathered}
 \tag{BI.27}
\]
Every series converges by BI.14.
Thus every negative output
coefficient is nonzero, with
the indicated alternating
row sign. The antipodal sign
has been retained, not
discarded by projective
normalization.

For an integer \(J\ge|j|\)
let \(S_{j,J}\) be the
sum in BI.27 through \(m=J\).
For \(m\ge|j|\),
\[
 \begin{gathered}
 \frac{a_{j,m+1}}{a_{j,m}}
 =\frac{m-j+1}{4(m+1)^2}\\
 \le\frac1{2(m+1)}\le\frac12,\\
 S_{j,J}<|(TP)_j(0)|\\
 \le S_{j,J}+2a_{j,J+1}.
 \end{gathered}
 \tag{BI.28}
\]
The inequality uses
\(m-j+1\le2m+1\le2(m+1)\).
Summing the positive omitted
terms against a geometric
series proves the enclosure.
These finite rational intervals
illustrate the full coefficient
map; the general ring and
input-action statements are
the independent arguments
BI.4–BI.6.

![Positive normal division, transpose mixing and the entire incidence input map](assets/infinite-incidence-positive-tails.png)

*Figure BI.1. Panel A shows the degree-zero normal remainder of BI.16 at parameter one, with exact alternating rational enclosures. Panels B–C use BI.26 at the displayed normal covariable and base/projective origin. The row signs retain the antipodal factor. Panel C shows absolute values, with the coefficient sign identified by its marker; its interval endpoints are the rational numbers in BI.28. These are finite windows of entire coefficient maps. Panel D is the commuting ring diagram of BI.18–BI.25; its vertical arrows are the actual finite-order inclusions. The complete normal quotient and input-action proofs are BI.0–BI.6.*

Full-size PNG · SVG · Exact data · Drawing source

#### The infinite-order incidence bimodule

**Theorem BI.A.** The quotient BI.2
has the unique whole normal row
BI.3–BI.8 and is the algebraic
induced module \(B\otimes_E\mathcal M_S\).
Its normally convergent
pole/essential/logarithm representative
injects into the shifted hypersurface
microlocal Hom with the canonical
\(B\)-action. On the incidence chart
it is free of rank one over the whole
output ring and over the whole
right input ring. The input
coefficient map is the parameter-zero
Legendre isomorphism.

**Proof.** The normal row, its actual
growth bounds and the two inverse
balanced maps are BI.0. The injection
and its canonical action comparison
are BI.1, using the full action
compatibility FIM.5. BI.4 constructs
the actual input transpose and
proves its product identities.
BI.5 and BI.6 prove the two
rank-one identifications and their
whole input coefficient equality.
All positive cutoff limits retain
the entire negative tail.
\(\square\)

This theorem concerns the specified
induced module and its image. It
does not assert that its image is
the whole microlocal Hom stalk,
or that it is stable under every
real kernel in \(C\).
No general coherence theorem for
arbitrary \(B\)-modules, flatness
of \(C/B\), normalized holomorphic
inverse convolution, whole real
contact transport or arbitrary
holonomic regularization follows
from this rank-one computation.


Free primary context: [Benchoufi, *Action of micro-differential operators on quantized contact transformations*, arXiv:2102.08861v2](https://arxiv.org/abs/2102.08861v2), Section 3. The complete arguments here use the exact earlier programme proofs linked above.


### 5.63. The opposite incidence kernel and its ordered periods

The exact earlier programme proof dependencies are §§5.56, 5.57, 5.58, 5.61, 5.62.

Let \(X\) and \(Y\) have complex dimension \(N=d+1\), with
coordinates \((t,x)\) and \((s,y)\), respectively. The spatial
number \(d\) may be zero. Use the actual rings and the entire
Legendre family of §§5.58 and5.62. Put
\[
 \begin{gathered}
 f=s-t-x^{\mathsf T}y,\\
 \eta_Y=ds\wedge dy_1\wedge\cdots\wedge dy_d,\\
 g=t-s+y^{\mathsf T}x,\\
 \eta_X=dt\wedge dx_1\wedge\cdots\wedge dx_d.
 \end{gathered}
 \tag{OI.1}
\]
The first kernel has output \(X\) and input \(Y\); the opposite
kernel has output \(Y\) and input \(X\). Both input top-form
lines are ungraded, with the displayed order. Their hypersurface
microlocal Hom classes use the positive relative connecting
representative and the shift \(\mathbb C_S[-1]\) of §5.61.

#### OI.0. Changing the pole order in the actual module

Write \(e_0\) for the forward class of \(\eta_Y/f\), and
\(\tau=D_t\). For every nonnegative integer \(r\), define
\[
 \begin{gathered}
 e_r=\frac{\tau^r}{r!}e_0,\\
 e_r=\left[\frac{\eta_Y}{f^{r+1}}\right].
 \end{gathered}
 \tag{OI.2}
\]
The second equality is an equality of actual relative classes.
Indeed \(D_t f=-1\), so induction differentiating the ordinary
holomorphic function on its sector complement gives
\(D_t^r(f^{-1})=r!f^{-r-1}\). The input frame is independent
of \(t\). The canonical action comparison FIM.5 and BI.1
identifies this finite differential action with the same action
in the shifted microlocal Hom.

The actual normal derivative is a two-sided unit in both
\(E_X\) and \(B_X\). Therefore
\[
 e_0=r!\tau^{-r}e_r.
 \tag{OI.3}
\]
This equality takes place in the module; it uses the proved
inverse in the operator ring. It makes no choice of a primitive
on the space of holomorphic functions. The map
\(P\mapsto Pe_r\) is consequently an isomorphism from the whole
output ring to the same incidence module, for either \(E\) or
\(B\): relative to the basis \(e_0\), it is right multiplication
of \(P\) by the unit \(\tau^r/r!\).

Let \(\mathcal L_0:B_Y\to B_X\) be the full input map of BI.25.
For every entire input symbol \(R\),
\[
 \begin{gathered}
 e_rR=\mathcal L_r(R)e_r,\\
 \mathcal L_r(R)=
 \tau^r\mathcal L_0(R)\tau^{-r}.
 \end{gathered}
 \tag{OI.4}
\]
To prove the formula, move the input action past the output
operator \(\tau^r/r!\), use \(e_0R=\mathcal L_0(R)e_0\),
and insert the two adjacent inverse powers in OI.3.
The scalar \(r!\) cancels. All products are actual whole
products. LN.26–LN.28 identifies the displayed conjugation
with \(\mathcal H_r\mathcal L_0=\mathcal L_r\).
For an integer \(r\), that identification can also be read
directly from the finite Leibniz rule:
\[
 \begin{gathered}
 (\tau^rP\tau^{-r})_j
 =\sum_{k=0}^r\binom rk\partial_t^k p_{j+k}.
 \end{gathered}
 \tag{OI.5}
\]
There are finitely many terms for every input coefficient,
even when the positive head is infinite. The common-domain
product and Cauchy bounds of IK.2–IK.3 and BI.4 apply.
Thus OI.4 holds for the whole \(B\)-ring, rather than only
for polynomial input generators.

#### OI.1. The reverse class, phase and density

In output coordinates \((s,v)\) on \(Y\), the standard reversed
incidence has equation \(t-s-v^{\mathsf T}x=0\).
Its order-\(N\) pole class is OI.2 with \(r=d\).
Pull it back by the ordinary output coordinate reflection
\[
 \begin{gathered}
 F(s,y)=(s,-y),\quad v=-y,\\
 e^-=\left[\frac{\eta_X}{g^{d+1}}\right].
 \end{gathered}
 \tag{OI.6}
\]
This reflection acts on the output variables alone.
The relative input frame remains exactly \(\eta_X\).
There is no input Jacobian multiplier. Its complex output
Jacobian is \((-1)^d\), and its real Jacobian determinant
is \(|(-1)^d|^2=1\); the complex orientation is preserved.
The ordinary support and trace comparison of BCT.1–BCT.3
therefore transports the canonical class, with this same
ordered input frame.

Let \(J\) denote the whole output reflection on operators:
it fixes time and its derivative, and sends every spatial
coordinate and spatial derivative to its negative. OI.4
and the whole coordinate comparison give
\[
 \begin{gathered}
 e^-P=\Psi(P)e^-,\\
 \Psi=J\mathcal L_d=\mathcal L_0^{-1},\\
 \Psi:B_X\longrightarrow B_Y.
 \end{gathered}
 \tag{OI.7}
\]
Here the Legendre map before the reflection has input \(X\)
and output \((s,v)\); the final ring has the coordinates
\((s,y)\). The last equality is the entire inverse identity
LN.30. Its proof retains the complete negative tails and
the common positive-cutoff bounds. The equality is not
inferred merely from its generator values.

In particular its ordered time value is
\[
 \begin{gathered}
 \Psi(t)=s+(E_y+d)\sigma^{-1},\\
 E_y=\sum_i y_iD_{y_i},\qquad \sigma=D_s.
 \end{gathered}
 \tag{OI.8}
\]
The shift by \(d\) is the pole-order normalization in OI.6.
Both \(e^-\) and the forward generator
\(e^+=[\eta_Y/f]\) belong to the injected whole-\(B\)
incidence modules of BI.A; each is a basis on both of its
operator sides. OI.3 proves that replacing the simple pole
by this higher pole does not lose any part of that module.

The reverse phase also gives exactly the inverse contact graph.
The output block of \(-\sigma\,dg\) is
\((\sigma,-\sigma x)\); its input block is
\((-\sigma,-\sigma y)\). After the required input antipodal
operation, the input \(X\) covector is \((\sigma,\sigma y)\).
Consequently, with \(z\) the input \(X\) projective covariable
and \(w\) the output \(Y\) projective covariable,
\[
 \begin{gathered}
 z=y,\quad w=-x,\quad g=0,\\
 (t,x,z)=\\
 (s+y^{\mathsf T}w,-w,y).
 \end{gathered}
 \tag{OI.9}
\]
This is the inverse in LI.9, including its time coordinate
and its antipodal signs.

#### OI.2. Both balanced bimodule evaluations

Set \(\phi=\mathcal L_0\), \(\psi=\phi^{-1}\).
Let \(H\) be the forward \((B_X,B_Y)\)-module with basis
\(e^+\), and \(K\) the opposite \((B_Y,B_X)\)-module with
basis \(e^-\). Their entire right actions are
\[
 \begin{gathered}
 e^+Q=\phi(Q)e^+,\quad Q\in B_Y,\\
 e^-P=\psi(P)e^-,\quad P\in B_X.
 \end{gathered}
 \tag{OI.10}
\]
There are explicit bimodule isomorphisms
\[
 \begin{gathered}
 \operatorname{ev}_X:H\otimes_{B_Y}K
                 \longrightarrow B_X,\\
 Pe^+\otimes Qe^-\longmapsto P\phi(Q),
 \end{gathered}
 \tag{OI.11}
\]
and
\[
 \begin{gathered}
 \operatorname{ev}_Y:K\otimes_{B_X}H
                 \longrightarrow B_Y,\\
 Qe^-\otimes Pe^+\longmapsto Q\psi(P).
 \end{gathered}
 \tag{OI.12}
\]
For OI.11 the balancing relation with \(R\in B_Y\)
has respective images \(P\phi(R)\phi(Q)\) and
\(P\phi(RQ)\), which are equal. Its right \(B_X\)-action
has image \(P\phi(Q\psi(A))=P\phi(Q)A\).
The left action is literal left multiplication.
Thus it is a bimodule map. Its inverse is
\[
 A\longmapsto Ae^+\otimes e^-.
 \tag{OI.13}
\]
Indeed the balancing relation moves \(Q\) in OI.11 to
the right action on \(e^+\); it then gives
\(Pe^+\otimes Qe^-=P\phi(Q)e^+\otimes e^-\).
This proves both inverse compositions on pure tensors,
which generate the algebraic tensor product.

For OI.12 the same verification is explicit:
the two balanced images are
\(Q\psi(A)\psi(P)\) and \(Q\psi(AP)\);
the right action by \(R\in B_Y\) has image
\(Q\psi(P\phi(R))=Q\psi(P)R\).
The inverse sends \(B\) to \(Be^-\otimes e^+\).
The two evaluation orders on a triple tensor agree:
\[
 \begin{gathered}
 H\otimes_{B_Y}K\otimes_{B_X}H\longrightarrow H,\\
 Pe^+\otimes Qe^-\otimes P'e^+\\
 \longmapsto P\phi(Q)P'e^+,
 \end{gathered}
 \tag{OI.14}
\]
and
\[
 \begin{gathered}
 K\otimes_{B_X}H\otimes_{B_Y}K\longrightarrow K,\\
 Qe^-\otimes Pe^+\otimes Q'e^-\\
 \longmapsto Q\psi(P)Q'e^-.
 \end{gathered}
 \tag{OI.15}
\]
For example the second association in OI.14 gives
\(Pe^+\bigl(Q\psi(P')\bigr)\), whose coefficient is
the displayed one by \(\phi\psi=1\). The second
association in OI.15 uses \(\psi\phi=1\).
These are the two triangle identities, with no scalar
left undetermined in the specified algebraic evaluations.

For every left \(B_X\)-module \(M\), the first evaluation
therefore induces
\[
 H\otimes_{B_Y}(K\otimes_{B_X}M)
                       \xrightarrow{\ \sim\ }M.
 \tag{OI.16}
\]
On pure tensors the map is
\(Pe^+\otimes Qe^-\otimes m\mapsto P\phi(Q)m\);
its inverse sends \(m\) to \(e^+\otimes e^-\otimes m\).
OI.14 verifies the inverse using the balancing relations.
Interchanging \(X,Y\) gives the second natural inverse.
Both \(H\) and \(K\) are free of rank one on their right
sides, so the corresponding tensor functors preserve exact
sequences: identify them with a single copy of the module
as an abelian group, with action transported through the
ring isomorphism. The statement consequently applies
degree by degree to complexes and their cohomology.
It uses no extension flatness involving \(C\).

OI.11–OI.16 are algebraic tensor evaluations of the
specified actual bimodules. A holomorphic convolution
is instead a cup/excision/proper-trace map of relative
complexes. The identification of those analytic maps
with these evaluations requires its own argument.

#### OI.3. The two ordered normal residues

The forward-then-opposite pair, with endpoint input
\((t',x')\), has two normal factors
\[
 \begin{gathered}
 f=s-t-x^{\mathsf T}y,\\
 g'=t'-s+y^{\mathsf T}x',\\
 A=t'-t+(x'-x)^{\mathsf T}y,\\
 g'=A-f.
 \end{gathered}
 \tag{OI.17}
\]
Fix \(A\ne0\), and choose \(0<r<|A|\).
The positively oriented \(s\)-circle is \(f=re^{i\theta}\);
here \(ds=df\) at fixed other variables. The ordered
normal period is
\[
 \begin{gathered}
 \frac1{2\pi i}\int_{|f|=r}
 \frac{df}{f(A-f)^N}\\
 =A^{-N}.
 \end{gathered}
 \tag{OI.18}
\]
For a direct proof, the holomorphic function \((A-f)^{-N}\)
has a uniformly convergent Taylor expansion on this circle,
since it is holomorphic on a larger concentric disc of
radius less than \(|A|\). Termwise integration against
\(df/f\) leaves its constant term \(A^{-N}\).
The circle integral of every nonzero integer power is
zero by the parametrization \(f=re^{i\theta}\).

For the opposite-then-forward pair, with endpoint input
\((s',y')\), put
\[
 \begin{gathered}
 g=t-s+y^{\mathsf T}x,\\
 f'=s'-t-x^{\mathsf T}y',\\
 A'=s'-s+(y-y')^{\mathsf T}x,\\
 f'=A'-g.
 \end{gathered}
 \tag{OI.19}
\]
Now \(dt=dg\), and the order-\(N\) pole comes first.
For \(0<r<|A'|\) the corresponding period is
\[
 \begin{gathered}
 \frac1{2\pi i}\int_{|g|=r}
 \frac{dg}{g^N(A'-g)}\\
 =(A')^{-N}.
 \end{gathered}
 \tag{OI.20}
\]
Expand \((A'-g)^{-1}=\sum_{k\ge0}g^k/(A')^{k+1}\)
uniformly on the circle. Only \(k=N-1\) survives.
Its coefficient is \((A')^{-N}\), with positive sign.
Thus reversing which factor has the higher pole gives
the same normal scalar; the intermediate normal
differential is first in both of the input frames OI.1.
These are ordinary holomorphic circle periods. The
degree-dependent signs of the derived proper trace
are additional data in an analytic convolution.

#### OI.4. Spatial boxes and every degenerate parameter

Let \(\epsilon_i>0\), and put
\[
 \begin{gathered}
 Q=\prod_{i=1}^d[-\epsilon_i,\epsilon_i],\\
 V=\prod_{i=1}^d2\epsilon_i,\\
 B=\sum_i\epsilon_i|b_i|,\qquad |a|>B,\\
 F_d(a,b)=\\
 \int_Q
 \frac{dv_1\cdots dv_d}{(a+b^{\mathsf T}v)^{d+1}}.
 \end{gathered}
 \tag{OI.21}
\]
An empty product is one and the \(d=0\) box is a point.
Both residues OI.18 and OI.20 give this spatial period,
with their respective endpoint differences substituted
for \(a,b\). On a compact parameter set inside \(|a|>B\),
choose one positive circle radius less than the minimum
of \(|a|-B\). Both iterated normal/spatial integrals are
then on a fixed compact circle times \(Q\), with a
strict common denominator bound. Ordinary Fubini follows
from uniform continuity and the finite Riemann sums on
these compact parameter rectangles. All fixed holomorphic
derivatives may be taken there by the Cauchy formula
on a slightly larger compact parameter set.

For \(b_i\ne0\) for every \(i\), put
\(a_\eta=a+\sum_i\eta_i\epsilon_i b_i\).
Then
\[
 \begin{gathered}
 F_d(a,b)=\frac{(-1)^d}{d!\prod_i b_i}\\
 {}\times\sum_{\eta\in\{-1,1\}^d}
       \frac{\prod_i\eta_i}{a_\eta}.
 \end{gathered}
 \tag{OI.22}
\]
Indeed differentiation of the one denominator gives
\(\partial_a^d(a+b^{\mathsf T}v)^{-1}
=(-1)^d d!(a+b^{\mathsf T}v)^{-d-1}\).
On this function, \(\partial_a=b_i^{-1}\partial_{v_i}\).
Apply that equality in each spatial variable and integrate
the mixed derivative using the real fundamental theorem
of calculus. Upper faces give \(+\epsilon_i\), lower
faces give \(-\epsilon_i\), which gives exactly the
corner signs in OI.22. This is also LI.C with the
normal-pole factorial explicitly divided out.

Every apparent \(b_i=0\) singularity of OI.22 is removable
on the whole region \(|a|>B\). To prove the assertion,
on each compact with that strict bound expand the
integrand in its uniformly convergent geometric
derivative series. Termwise compact integration proves
holomorphy in every complex parameter, including zero
components of \(b\). This integral equals OI.22 on
the dense set with no zero component and hence is
its holomorphic extension. In particular
\[
 \begin{gathered}
 F_d(a,0)=Va^{-d-1},\\
 F_0(a)=a^{-1},\\
 F_1(a,b)=\frac{2\epsilon_1}
                  {a^2-\epsilon_1^2b^2}.
 \end{gathered}
 \tag{OI.23}
\]
The first formula follows directly from the constant
integrand; the last follows by subtracting the two
corners in OI.22. The signs agree with both positive
normal residues.

There are fixed-domain bounds and explicit tails.
With \(q=B/|a|<1\), differentiating the geometric
series \(d\) times proves
\[
 \begin{gathered}
 (a+u)^{-d-1}=a^{-d-1}\\
 {}\times\sum_{m\ge0}\binom{d+m}{m}(-u/a)^m,\\
 |F_d(a,b)|\le\frac{V}{(|a|-B)^{d+1}}.
 \end{gathered}
 \tag{OI.24}
\]
The summand after compact integration has norm at most
\(V|a|^{-d-1}\binom{d+m}{m}q^m\).
The sum of these positive bounds is the second line,
using \(\sum_m\binom{d+m}{m}q^m=(1-q)^{-d-1}\).
The latter identity is proved by differentiating the
geometric series inside its radius of convergence.
If \(q<q_1<1\), each term of that generating series
at \(q_1\) is at most its sum. Therefore the integrated
tail after degree \(R\ge0\) is at most
\[
 \begin{gathered}
 V|a|^{-d-1}(1-q_1)^{-d-1}\\
 {}\times\frac{(q/q_1)^{R+1}}{1-q/q_1}.
 \end{gathered}
 \tag{OI.25}
\]
All constants are fixed on a chosen compact with the
strict displayed margins. A further fixed Cauchy
margin gives the same normal convergence for every
specified parameter derivative. At \(b=0\), every
omitted term is zero directly.

![The opposite pole, both balanced evaluations, two normal circles and a spatial period](assets/opposite-incidence-ordered-periods.png)

*Figure OI.1. Panel A shows the entire ring maps and the specified algebraic tensor evaluations. Panel B displays the forward and opposite pole orders for spatial dimensions zero through four. Panel C is a schematic of the two positive normal circles, with dimension N=3 and endpoint parameter A=3; each surrounds only the zero pole, and both normalized ordinary circle periods are 1/27. Panel D uses d=1, epsilon=1 and a=3, and retains the removable value at b=0. The dashed curve is the proved denominator bound, not another period. The complete proofs, without the plotted cutoffs, are OI.0–OI.4. The balanced evaluations and ordinary compact periods do not alone identify the analytic relative cup/excision/proper-trace products.*

Full-size PNG · SVG · Exact data · Drawing source

#### The opposite module and the two types of evaluation

**Theorem OI.A.** The order-\(d+1\) opposite class OI.6
is a basis of the actual incidence module over each
whole operator side. Its entire right input map is
\(\mathcal L_0^{-1}\), in the ordered frame \(\eta_X\).
The forward and opposite modules have both balanced
evaluations OI.11–OI.12, with the two triangle identities,
and give inverse exact tensor functors on all modules.
The same statements hold for their finite-\(E\) restrictions.

**Proof.** OI.0 changes the pole order by an actual unit.
OI.1 gives its canonical reflected class, exact density,
inverse contact graph and full coefficient map.
OI.2 proves every balancing relation, the two inverse
evaluations and their triple identities explicitly.
All constructions commute with restriction, so the same
maps define the assertions for sheaves on their charts.
The ring maps restrict to the finite rings by LN.30.
\(\square\)

**Theorem OI.B.** On every compact subset of the strict
denominator region in OI.21, both ordered ordinary normal
residues are the same order-\(N\) denominator, and the
subsequent compact spatial periods are OI.21–OI.25.
The corner expressions extend over every vanishing
spatial parameter, including the fully degenerate point.

**Proof.** OI.3 evaluates both normal integrals, with
their respective first- and higher-order poles.
OI.4 gives common compact contours, the complete box
identities, all degeneracies and differentiated tails.
\(\square\)

The tensor evaluations and ordinary periods above do
not identify either normalized analytic kernel product.
That identification must retain every relative
complement entry, the actual intermediate-variable
proper collar, the all-degree trace signs and the
diagonal degree-\(N\) unit. In particular a positive
one-normal residue is not an equality with that
diagonal unit. No whole-real contact isomorphism or
arbitrary-holonomic regularization follows solely from
OI.A or OI.B.


Free primary context: [Benchoufi, *Action of micro-differential operators on quantized contact transformations*, arXiv:2102.08861v2](https://arxiv.org/abs/2102.08861v2), Section 3. All arguments used above are proved here or in the exact earlier programme sections linked above.


### 5.64. The diagonal unit and finite hyperplane descent

The exact earlier programme proof dependencies are §§5.34, 5.35, 5.36, 5.43, 5.57, 5.59, 5.60, 5.63.

#### A full relative form for the diagonal unit

Use the actual full smooth/current complexes of §5.34, the relative
differential and normal calibration of §5.35, and the diagonal
unit of §5.36. Let the base dimension be \(N\ge1\).
In pair coordinates put \(w=v-u\), with indices \(0,\ldots,N-1\).
The ordered relative input frame is
\(\eta=dv_0\wedge\cdots\wedge dv_{N-1}\).
At fixed \(u\) it is \(dw_0\wedge\cdots\wedge dw_{N-1}\).
The full complex still contains every output antiholomorphic
differential; no relative-only Dolbeault resolution is used.
Throughout the coefficient proof, \(\bar\partial\) denotes
that full antiholomorphic differential. On functions depending
only on \(w\), it agrees with the normal differential.

An ordered differential block, such as \(dw\), denotes the wedge of its coordinate differentials in increasing index. The block \(da_{\widehat j}\) omits index \(j\). The same convention applies to antiholomorphic and spatial blocks; the empty block is one.

Put
\[
 \begin{gathered}
 r_N=(-1)^{N(N-1)/2},\\
 c_N=\frac{r_N(N-1)!}{(2\pi i)^N},\\
 \omega'(a)=\sum_{j=0}^{N-1}(-1)^j a_j\,da_{\widehat j}.
 \end{gathered}
 \tag{UC.1}
\]
The hat means that the indicated factor is omitted. This notation
for \(\omega'\) uses the differential appropriate to its argument;
that differential is specified in each application below.

##### UC.0. The current identity, with its exact scalar

On the punctured normal product set
\[
 \begin{gathered}
 s_j(w)=\frac{\overline w_j}{|w|^2},\\
 \beta_N=c_N\frac{\omega'(\overline w)}{|w|^{2N}}.
 \end{gathered}
 \tag{UC.2}
\]
The antiholomorphic degree of \(\beta_N\) is \(N-1\).
Its coefficients have norm at most \(C|w|^{1-2N}\), so they
are locally integrable in the \(2N\) real normal variables:
the radial absolute integral is bounded by a constant times
\(\int_0^\epsilon dr\).
Thus \(\beta_N\) has its literal locally integrable current
extension. No assertion about extending arbitrary growing
holomorphic functions is needed here.

First \(\bar\partial\beta_N=0\) off zero. More generally, if
smooth functions \(a_j\) satisfy \(\sum_jw_ja_j=1\), then
\(\sum_jw_j\bar\partial a_j=0\). Since \(w\ne0\), these
\(N\) one-forms have a nontrivial scalar relation; their
full exterior product is zero. Differentiating the
alternating form in UC.1 gives \(N\) copies of that exterior
product. Consequently
\[
 \begin{gathered}
 \bar\partial\bigl(c_N\omega'_{\bar\partial}(a)\bigr)=0,\\
 \beta_N=c_N\omega'_{\bar\partial}(s).
 \end{gathered}
 \tag{UC.3}
\]
The second identity follows by expanding the derivatives of
\(\overline w_j/|w|^2\). Terms containing a derivative of
\(|w|^{-2}\) cancel in pairs, because they contain two copies
of the same column \((\overline w_j)_j\) in an alternating
determinant. The remaining denominator is \(|w|^{2N}\).

Here is the full real orientation calculation. Write
\[
 \begin{gathered}
 V_N=d\overline w\wedge dw,\\
 V_N=r_N(2i)^N\,d\operatorname{vol}_{\mathbf R^{2N}},\\
 E_{\bar w}=\sum_j\overline w_j\partial_{\overline w_j},\\
 R=\sum_j(x_j\partial_{x_j}+y_j\partial_{y_j}).
 \end{gathered}
 \tag{UC.4}
\]
There are \(N(N-1)/2\) crossings to group each
\(d\overline w_j,dw_j\) block, and
\(d\overline w_j\wedge dw_j=2i\,dx_j\wedge dy_j\).
The numerator of \(\beta_N\wedge dw\) is
\(\iota_{E_{\bar w}}V_N\).
On an oriented sphere, the normal component of \(E_{\bar w}\)
is \(R/2\). Their difference is tangent; contraction of a
top form by a tangent vector restricts to zero on the
sphere, since all its arguments then lie in a real
\((2N-1)\)-dimensional tangent space.

The area of the unit sphere is
\(2\pi^N/(N-1)!\). One may check this scalar without a
higher-dimensional integration formula: the one-plane
Gaussian integral is \(\pi\), by polar coordinates, and
its \(N\)-fold product is \(\pi^N\) by nonnegative Fubini.
In radial coordinates its radial factor is
\((N-1)!/2\), by \(N-1\) integrations by parts in
\(\int_0^\infty e^{-r^2}r^{2N-1}dr\).
Dividing gives the displayed area. UC.4 therefore gives,
on every positively oriented normal sphere,
\[
 \int_{|w|=\epsilon}^{+}\beta_N\wedge dw=1.
 \tag{UC.5}
\]
Indeed the unnormalized integral is
\(r_N(2\pi i)^N/(N-1)!\), exactly the reciprocal of \(c_N\).

Distributional differentiation now gives
\[
 \begin{gathered}
 \bar\partial\beta_N\wedge dw\\
 =\delta_{\mathbf C^N}(w)\,
 d\operatorname{vol}_{\mathbf R^{2N}}.
 \end{gathered}
 \tag{UC.6}
\]
For completeness test on any compactly supported smooth
function, without a holomorphy hypothesis. Apply Stokes
after deleting the radius-\(\epsilon\) ball. The outward
orientation of that deleted boundary and the sign in
distributional differentiation leave the positive sphere
integral in UC.5. Replacing the test function by its value
at zero changes this integral by \(O(\epsilon)\), because
its first derivatives are bounded and the absolute
sphere integral of the homogeneous form is uniformly
bounded. The contribution outside the deleted ball
uses UC.3 and is zero. Letting \(\epsilon\) tend to zero
proves UC.6. In particular no derivative of a point mass
is left undetermined. With output parameters \(u\),
apply the same argument uniformly on the compact
projection of a test function's support and integrate
in \(u\); this is the full diagonal current identity.

The one-normal formula U.6 and the calibration V.17
identify the right side with the programme's ordered
diagonal unit current. The coefficient line in UC.6 is
the relative line \(\eta\). Choosing \(dw\) or \(dv\)
as a full-form lift changes only terms containing base
holomorphic differentials; V.4 proves that these terms
have zero intermediate trace.

Let \(Z\) be any admissible closed normal support
containing the diagonal, and use
\(d(a,b)=(\bar\partial a,ra-\bar\partial b)\).
The closed full relative cochain
\[
 k_{\rm unit}=(0,-\beta_N|_{D\setminus Z})
 \tag{UC.7}
\]
has degree \(N\). The explicit current cochain
\((\beta_N,0)\) gives
\[
 k_{\rm unit}+d(\beta_N,0)
                  =(\bar\partial\beta_N,0).
 \tag{UC.8}
\]
Thus this is equality with the actual diagonal unit
in the full relative complex, followed by the exact
smooth/current comparison R.6–R.7. It is not an inference
from equality of operators on a selected family of
functions. All output and complement components are
retained.

##### UC.1. Generating choices and finite coherent homotopies

On the punctured domain call a smooth vector \(a\)
generating if \(\sum_jw_ja_j=1\). Define
\(\beta(a)=c_N\omega'_{\bar\partial}(a)\).
Given two choices \(a^0,a^1\), put
\(a(t)=(1-t)a^0+ta^1\).
It is generating for the entire closed interval.
On its product with the punctured domain use
\(D=d_t+\bar\partial\).
The same exterior relation as in UC.3 proves
\(D(c_N\omega'_D(a(t)))=0\).
Decompose the latter form into its parameter-free
part plus \(dt\wedge B(t)\). Its closedness gives
\(\partial_t\beta(a(t))=\bar\partial B(t)\).
Consequently
\[
 \begin{gathered}
 H(a^0,a^1)=\\
 c_N\int_0^1\iota_{\partial_t}\omega'_D(a(t))\,dt,\\
 \beta(a^1)-\beta(a^0)\\
 =\bar\partial H(a^0,a^1).
 \end{gathered}
 \tag{UC.9}
\]
For \(N=1\) the generating choice is uniquely \(1/w_0\);
the contraction is zero and the same equality holds.
For higher \(N\), \(H\) has antiholomorphic degree \(N-2\).
It is smooth on each actual punctured domain on which
the two vectors are given. No extension through zero
of this general homotopy is assumed.

The full relative unit witnesses differ by the literal
boundary
\[
 \begin{gathered}
 (0,-\beta(a^1))-(0,-\beta(a^0))\\
 =d(0,H(a^0,a^1)).
 \end{gathered}
 \tag{UC.10}
\]
In particular every such choice has the same relative
class as UC.7, by taking \(a^0=s\) from UC.2.

All higher finite compatibility witnesses are also
explicit. For choices \(a^0,\ldots,a^k\), use
\(a(u)=\sum_{j=0}^ku_ja^j\) on the oriented standard
simplex, with \(u_0=1-\sum_{j=1}^ku_j\), and put
\[
 \begin{gathered}
 H_{0\ldots k}=c_N\int_{\Delta^k}\omega'_D(a(u)),\\
 \bar\partial H_{0\ldots k}=\\
 \sum_{j=0}^k(-1)^{k+j+1}
 H_{0\ldots\widehat j\ldots k}.
 \end{gathered}
 \tag{UC.11}
\]
Fibre integration selects parameter degree \(k\);
the remaining antiholomorphic degree is \(N-1-k\).
The formula follows directly by writing every
parameter differential before the coefficient
antiholomorphic differentials. In parameter degree
\(k\), the closedness equation is the parameter
boundary plus \((-1)^k\bar\partial\) of the integral.
Ordinary simplex Stokes gives exactly UC.11.
For \(k>N-1\) that integral is zero, and the same
equation gives the corresponding alternating
compatibility. One may rescale by
\((-1)^{k(k-1)/2}\) to make the right-hand coefficient
of each alternating face \(+1\).

For any finite list of cochains and homotopies all
parameter simplices are compact and all actual domains
can be chosen before these finite constructions.
Restriction, multiplication by a holomorphic
coefficient and further finite refinement commute
with these explicit identities.

##### UC.2. The entire complement cube and its top holomorphic entry

Cover the punctured normal product by
\(U_j=\{w_j\ne0\}\), and set
\[
 \begin{gathered}
 \rho_j=\frac{|w_j|^2}{|w|^2},\quad a^j=\frac{e_j}{w_j},\\
 s=\sum_j\rho_ja^j,\\
 \varphi=\frac1{(2\pi i)^Nw_0\cdots w_{N-1}}.
 \end{gathered}
 \tag{UC.12}
\]
Products \(\rho_ja^j\) extend smoothly across \(w_j=0\)
on the punctured product, and equal
\(\overline w_j/|w|^2\). The top holomorphic Čech
cocycle \(c_{0\ldots N-1}=\varphi\) is alternating
in its indices. For an ordered tuple \(I\) of length \(N-1\),
\((\bar\partial\rho)_I\) denotes the corresponding ordered wedge;
\(i_0I\) denotes concatenation of the index lists. Its Dolbeault representative, in
the Čech differential convention of V.6, is
\[
 \begin{gathered}
 F(c)=\sum_{i_0,I}
 \rho_{i_0}(\bar\partial\rho)_I\,c_{i_0I},\\
 F(c)=(N-1)!\varphi\,
               \omega'_{\bar\partial}(\rho),\\
 \beta_N=r_NF(c).
 \end{gathered}
 \tag{UC.13}
\]
Each remaining index has \((N-1)!\) permutations;
the alternating cocycle and exterior permutation
have the same sign. This proves the second line.
Expanding \(\omega'_{\bar\partial}(s)\) and using
\(\bar\partial(1/w_j)=0\) on each intersection gives
the third line, with the scalar UC.1.

Here is a finite cochain witness for the augmentation,
so that a holomorphic expression alone is not being
substituted for the complement complex. In Čech
degree \(q=N-1\), start with \(c^{(0)}=c\).
Use V.6's partition contraction
\((hb)_{i_0\ldots i_{r-1}}=\sum_j\rho_jb_{j i_0\ldots i_{r-1}}\).
For \(0\le l<q\) define
\[
 \begin{gathered}
 b^{(l)}=h c^{(l)},\\
 c^{(l+1)}=(-1)^{q-l}\bar\partial b^{(l)},\\
 d_{\rm tot}b^{(l)}=c^{(l)}-c^{(l+1)}.
 \end{gathered}
 \tag{UC.14}
\]
At each stage \(c^{(l)}\) is Čech closed and
Dolbeault closed, with bidegree \((q-l,l)\).
The contraction identity gives its Čech component;
the total differential in Čech degree \(q-l-1\)
has Dolbeault sign \((-1)^{q-l-1}\), which gives
the last line. The last \(c^{(q)}\) is a global
degree-\(q\) Dolbeault form. Expansion of the
successive contractions shows it is \(F(c)\):
the total sign \((-1)^{q(q+1)/2}\) is canceled
by reversing the \(q\) differentiated partition
factors and moving the last Čech index to the
front. Summing the \(b^{(l)}\)'s gives an explicit
boundary from \(c\) to its augmentation \(F(c)\).
For \(q=0\), the sum is empty and the identity
is literal. All lower intersections occur in
this finite witness. For the rational cocycle
displayed here the extensions across a missing
coordinate are literal smooth extensions:
\(\rho_j/w_j=\overline w_j/|w|^2\), and
\((\bar\partial\rho_j)/w_j
=\bar\partial(\overline w_j/|w|^2)\).
For arbitrary complement cochains the contraction
instead uses a genuinely subordinate smooth
partition as in V.6. No finite-order vanishing
of a partition factor is presumed to regularize
an arbitrary singular cochain. The generating
vector for such a partition is smooth on the
punctured domain, and UC.9–UC.11 compares it
with the displayed vector \(s\).

It follows that UC.7 has ordinary top complement
Čech entry \(-r_N\varphi\). The cone-to-cube map
of V.6 multiplies that entry by \((-1)^{N-1}\).
Its top cube entry is therefore
\[
                       r_N(-1)^N\varphi,
 \tag{UC.15}
\]
exactly the ordered \(N\)-normal Cauchy calibration
of V.17 and U.6–U.8. This also checks the parity
when \(N=2\). UC.8 supplies the supported-current
endpoint, and UC.14 supplies every intermediate
complement entry.

##### UC.3. The universal hyperplane form and the two incidence orders

For \(\xi\in\mathbf C^N\setminus0\) with
\(\langle\xi,w\rangle\ne0\), define the full form
\[
 \mathcal W_N
 =c_N\frac{\omega'(\xi)\wedge dw}{\langle\xi,w\rangle^N}.
 \tag{UC.16}
\]
Its differential is zero. Indeed
\(d_\xi\omega'(\xi)=N\,d\xi_0\wedge\cdots\wedge d\xi_{N-1}\)
and
\[
 \begin{gathered}
 \left(\sum_jw_jd\xi_j\right)\wedge\omega'(\xi)\\
 =\langle\xi,w\rangle\,d\xi.
 \end{gathered}
 \tag{UC.17}
\]
These cancel the derivative of the denominator.
Every remaining \(w\)-differential wedges to zero
against the full \(dw\) block. Rescaling \(\xi\)
by a nowhere-zero function leaves UC.16 unchanged:
the numerator and denominator both scale by its
\(N\)-th power; terms containing its differential
have two identical columns in the alternating
numerator and are zero. Thus this is a form on
the projective parameter space off the incidence.

The pullback along a smooth generating choice
\(\xi=a(w)\), \(\langle a,w\rangle=1\), is
\(\beta(a)\wedge dw\). Holomorphic \(w\)-components
of \(da\) vanish against \(dw\), leaving precisely
the antiholomorphic differentials used above.
In the chart \(\xi_0=1\), \(\xi_i=y_i\), put
\(w_{\rm sp}=(w_1,\ldots,w_d)\), \(d=N-1\).
The form is
\[
 \begin{gathered}
 \mathcal W_N=\\
 c_N\frac{dy\wedge dw}{(w_0+y^{\mathsf T}w_{\rm sp})^{d+1}},\\
 d=N-1.
 \end{gathered}
 \tag{UC.18}
\]
The zeroth term of \(\omega'\) is the only surviving
term, since \(d\xi_0=0\).

For the first incidence order OI.17–OI.18, take
\(w=(t'-t,x'-x)\) and \(\xi=(1,y)\).
Its ordinary normal residue, with the endpoint
input frame attached, is the coefficient of
UC.18 multiplied by \(c_N^{-1}\).
For the second order OI.19–OI.20, retain the
ordinary endpoint normal coordinates
\(w'=(s'-s,y'-y)\). Then the parameter is
\(\xi'=(1,-x)\), not \((1,x)\), and
\[
 \begin{gathered}
 \langle\xi',w'\rangle=\\
 s'-s+(y-y')^{\mathsf T}x,\\
 d(-x)=(-1)^d dx.
 \end{gathered}
 \tag{UC.19}
\]
Here \(d(-x)\) is the ordered wedge of the
component differentials \(d(-x_i)\).
Accordingly the second chart expression of the
universal unit form is \((-1)^dc_N\) times the
ordinary spatial form in that order.
Both normal circle residues in OI.18 and OI.20
are positive; UC.19 is a separate spatial
parameter Jacobian. Its matching in a normalized
kernel convolution requires the actual oriented
source comparison and trace, rather than dropping
it from the hyperplane coordinates.

The identities UC.7–UC.15 prove a full relative
representative of the diagonal unit, and UC.16–UC.19
identify its universal hyperplane expression and
both spatial coordinate conventions. They do not
by themselves identify the microlocal-Hom
composition roof of the two incidence classes
with this universal form. That further identity
has to include the shifted topological source
counits, their spatial boundary homotopies,
the actual middle-variable proper support,
and the all-degree trace normalization.


![Ordered unit signs, exact generating weights and the two spatial parameters](assets/relative-unit-form-and-parameters.png)

*Figure UC.1. Panel A separates the normal, top-cube and reverse-parameter signs. Panel B plots only the exact generating weights on the real section \(w=(1,q)\), \(N=2\). Panel C is the parameter simplex of UC.11. Panel D retains the distinct spatial parameters in the two incidence orders. The complete current and complement identities are UC.6–UC.15.*

#### The source kernels and the diagonal shift

Let \(P,P^*\) be the two complex projective bases, each of
complex dimension \(N\ge1\). Use ordinary kernel convolution
over the compact middle projective space, as in LI.32.
Write \(S\) for the closed incidence and \(\Omega\) for its
open complement. With \(j:\Omega\hookrightarrow P\times P^*\)
and its transposed embedding, put
\[
 \begin{gathered}
 L=\mathbf C_S[-1],\\
 A=j_!\mathbf C_\Omega,\\
 L^{\rm op}=\mathbf C_{S^{\rm op}}[-1],\\
 B=Rj^{\rm op}_*\mathbf C_{\Omega^{\rm op}}.
 \end{gathered}
 \tag{SN.1}
\]
The unit-normalized topological inverse of LI.D is
\(B[2N]\), so its two oriented convolution identifications
are \(A*B[2N]\simeq\mathbf C_{\Delta_P}\) and
\(B*A[2N]\simeq\mathbf C_{\Delta_{P^*}}\).
These are the complete compact-fibre and boundary
identifications LI.34–LI.41, with positive complex
orientation, not a fibre Euler-characteristic equality.

##### SN.0. Both open/closed arrows

The ordinary open/closed exact sequence gives the triangle
\[
 L\xrightarrow{u}A\longrightarrow
             \mathbf C_{P\times P^*}\xrightarrow{+1}.
 \tag{SN.2}
\]
Here \(u\) is the shifted connecting arrow. OCM.3–OCM.5
constructs its actual cone map, using the lifted
restriction and its differential homotopy. Thus it has
the signs of that full relative model.

For the transposed open, the closed-support triangle is
\[
 \begin{gathered}
 \mathbf C_{S^{\rm op}}[-2]\longrightarrow
 \mathbf C_{P^*\times P}\longrightarrow B\\
 \xrightarrow{v}L^{\rm op}\xrightarrow{+1}.
 \end{gathered}
 \tag{SN.3}
\]
To identify the first term, straighten the hypersurface
to a complex normal disc and use its positively oriented
real codimension-two Thom class, with arbitrary tangential
coefficients. HI.A proves this full supported comparison
including the collapsed fibre. The normal orientations
glue, since a holomorphic change of normal frame has
positive real Jacobian. The source of the first map is
therefore precisely the displayed shifted constant sheaf.
The triangle itself is the actual closed-support/open
restriction triangle from the relative cone.
It also specifies the boundary map \(v\).

The one-normal circle in \(B\) is sent by \(v\) to its
positive generator. A representative for its compact
boundary is
\(-d\chi\wedge d\theta/(2\pi)\), with \(\chi\) equal to
one near zero and zero outside an inner disc. Its integral
is one, as proved in LI.38–LI.39. This checks the
normal orientation in SN.3, rather than inferring it
from the shift alone.

For any bounded holomorphic coefficient complex \(F\),
application of the actual microlocal Hom gives
isomorphisms on every nonzero incidence conormal:
\[
 \begin{gathered}
 u^*:\mu\operatorname{hom}(A,F)\\
          \xrightarrow{\sim}\mu\operatorname{hom}(L,F),\\
 v^*:\mu\operatorname{hom}(L^{\rm op},F)\\
          \xrightarrow{\sim}\mu\operatorname{hom}(B,F).
 \end{gathered}
 \tag{SN.4}
\]
Indeed the cone of either map is a shift of microlocal
Hom from the ambient constant sheaf. OCM.A proves its
off-zero vanishing by the actual pulled-coefficient
homotopy, not by an assumed symbol realization. All
operations in the definition of microlocal Hom are
derived additive operations and preserve the displayed
triangles. The source arrows and their comparisons are
therefore the same derived arrows as in SN.2–SN.3.

##### SN.1. The two convolution comparisons

There is a useful common source for the first order:
\[
 \begin{gathered}
 L*L^{\rm op}[2N]\ \xleftarrow{\ 1*v\ }\ 
 L*B[2N]\\
 \xrightarrow{\ u*1\ }A*B[2N]
                    \xrightarrow{\sim}\mathbf C_{\Delta_P}.
 \end{gathered}
 \tag{SN.5}
\]
After applying microlocal Hom with any fixed bounded coefficient
complex, both left comparison arrows become isomorphisms
off the zero cotangent section. We verify their cones
on actual coefficient sheaves.

The cone contributed by \(u\) is
\(\mathbf C_{P\times P^*}*B[2N]\).
For a fixed endpoint \(p'\), ordinary global sections
of \(Rj_*\mathbf C_\Omega\) on the middle compact
projective space are the ordinary cohomology of its
affine nonincidence fibre \(\mathbf C^N\).
The contraction by real dilation, including the
degree-zero constant section, gives just \(\mathbf C\).
These complexes are locally constant in the endpoint:
the determinant-one matrix used in LI.8.3 sends a
fixed endpoint vector to the current one and
trivializes the entire incidence pair. On such a
product the actual de Rham parameter homotopy
identifies the pushforward with the constant fibre
complex. The compact-fibre comparison LI.34–LI.35
justifies every restriction. The unused first endpoint
is already constant. Hence this convolution cone is
locally a finite complex of constant sheaves on the
full endpoint pair.

The cone contributed by \(v\) is
\(L*\mathbf C_{P^*\times P}[2N+1]\).
Here the incidence fibre over the first endpoint is
\(\mathbf P^{N-1}\); the same explicit matrix
trivializes that whole family, and the second endpoint
is unused. Its compact-fibre cohomology is a bounded
finite-dimensional complex, locally constant in the
endpoints. Finiteness may be checked directly with
the finite standard affine cover of \(\mathbf P^{N-1}\).
An intersection is a product of an affine space with
finitely many punctured planes. Dilation contracts
each affine factor; radial contraction and the
normalized angular form give the two-term circle
complex for each punctured factor, as in LI.8.2.
Finite tensor products and the finite Čech filtration
then give bounded finite-dimensional cohomology.
This computes through an acyclic de Rham resolution;
it does not assert that the cover is acyclic for the
constant sheaf. Over \(\mathbf C\), a bounded vector
complex with finite-dimensional cohomology is
quasi-isomorphic to a finite-dimensional complex:
choose complements to its cycles and boundaries.
This last observation is used only to apply the
finite constant-coefficient vanishing, not to choose
any convolution map.

For a finite constant coefficient complex \(V\),
\(\mu\operatorname{hom}(V\otimes\mathbf C_M,F)\)
is the corresponding finite Hom complex of
\(\mu\operatorname{hom}(\mathbf C_M,F)\).
One checks this first on a finite vector space by
a basis, then by finite totalization; finite Hom
commutes with the actual resolutions and filtered
germs. It is zero off zero by OCM.A. Thus both
convolution cones in SN.5 vanish under the indicated
microlocal Hom. This proves the assertion about the
comparison arrows with their exact source maps.

The second order is the same complete construction
with \(P,P^*\) interchanged. Equivalently use the roof
from \(B*L[2N]\), with one arrow to \(B*A[2N]\)
and one to \(L^{\rm op}*L[2N]\). Its two cones have
the same affine and projective compact-fibre
descriptions. It gives
\[
 \begin{gathered}
 \mu\operatorname{hom}(L*L^{\rm op}[2N],F)\\
        \simeq\mu\operatorname{hom}(\mathbf C_{\Delta_P},F),\\
 \mu\operatorname{hom}(L^{\rm op}*L[2N],F')\\
        \simeq\mu\operatorname{hom}(\mathbf C_{\Delta_{P^*}},F')
 \end{gathered}
 \tag{SN.6}
\]
at nonzero endpoint covectors. The isomorphisms are
the just constructed roofs, including the oriented
topological counits.

##### SN.2. The two different dimension shifts

The coefficient trace in complex middle dimension \(N\)
lowers the Dolbeault degree by \(N\), with
\(\operatorname{Tr}_q=(-1)^{N(q-N)}I_q\), as in V.4.
In a coefficient-composition roof using this trace, its target
coefficient line is consequently shifted by \([-N]\).
The contravariance of the first argument gives
\[
 \begin{gathered}
 \mu\operatorname{hom}(L*L^{\rm op},F[-N])\\
 =\mu\operatorname{hom}(L*L^{\rm op}[N],F).
 \end{gathered}
 \tag{SN.7}
\]
SN.6 uses a different shift: the real compact-fibre
orientation contributes \([2N]\). Combining these
two facts gives, under microlocal Hom off zero, the source
roof with \(\mathbf C_\Delta[-N]\). Thus the corresponding
coefficient-composition target is identified with
\[
 \begin{gathered}
 \mu\operatorname{hom}(\mathbf C_\Delta[-N],F)\\
 =\mu\operatorname{hom}(\mathbf C_\Delta,F)[N].
 \end{gathered}
 \tag{SN.8}
\]
For the ordered input top-form coefficient, degree zero
of SN.8 is exactly the degree-\(N\) real diagonal
operator group. The spatial number \(d=N-1\) in the
opposite pole of OI.6 is a third count; it cannot
replace either shift above.

This identification is through the source roofs of SN.5–SN.6.
It is not an isomorphism of the two underlying source sheaves
before applying microlocal Hom. Their locally constant
convolution cones are discarded only by OCM.A off zero.
Nor does counting these shifts construct the comparison of
coefficient-composition roofs.

##### SN.3. Actual finite equality witnesses on the roofs

The comparison isomorphisms above have finite representatives,
not only abstract inverses of cohomology maps. We spell this
out for a comparison chain map \(f:C^\bullet\to D^\bullet\).
Use the ordinary cone convention
\[
 \begin{gathered}
 \operatorname{Cone}(f)^q=D^q\oplus C^{q+1},\\
 d(b,a)=(db+fa,-da).
 \end{gathered}
 \tag{SN.9}
\]
Each of the two comparison cones is, locally in the endpoint
pair and through the coefficient comparisons already proved,
a finite constant-coefficient complex in its source argument.
The finite-Hom construction after SN.5, followed by OCM.11,
contracts its microlocal relative complex on a later round
support domain. For finite complexes the contraction is
extended by the ordinary total-degree sign; the two terms
containing the coefficient differential cancel because the
contraction lowers form degree by one. A local chain
quasi-isomorphism used to pass to this model is compared by
R.6–R.7. In the finitely many degrees used by a given finite
list of cochains, their elimination gives actual lifts and
primitives on one common restriction. Thus the statement
needed below is that each displayed cone cycle has a
primitive on that actual restriction. A simultaneous
contraction on an unspecified infinite collection of
germs is not being asserted.

For a closed \(b\in D^q\), the cone element \((b,0)\)
is closed. Choose its just constructed primitive \((x,a)\).
SN.9 says
\[
 \begin{gathered}
 da=0,\qquad b-fa=dx.
 \end{gathered}
 \tag{SN.10}
\]
This supplies a preimage cycle and the complete boundary
witness for its image. Conversely suppose \(da=0\) and
\(fa=db\). The element \((-b,a)\) is a closed cone
cochain of degree \(q-1\). If \((x,y)\) is its primitive,
the second entry in SN.9 gives
\[
                         a=d(-y).
 \tag{SN.11}
\]
This is the equality witness for injectivity. Both
arguments retain the two entries of each relative
support cone; their cone cochains themselves are
finite lists of whole-domain and complement cochains.
R.6–R.7 is applied to that full list.

For a finite diagram of the source comparisons, include
all its cycles, chosen lifts, primitives and restrictions
in this construction. The finite common-domain argument
of OCM.3 and HI.A then supplies one actual later support
domain with one strict nonzero covector margin. Applying
SN.10 along one arrow and SN.11 to the difference between
two lifted cycles shows that the corresponding source-roof
identities hold on that domain, with the equality primitive
included. Their shifted versions use the displayed
differentials, so introduce no additional orientation
choice. The scalar of the topological counit is still
the positive compact-fibre integral of LI.D and the
normal-circle map of SN.3.

SN.5–SN.8 identify the source roofs and their degrees.
An equality of the two holomorphic incidence products
with the unit form UC.7 still requires the actual
coefficient composition on these roofs. In particular
the spatial parameter sign UC.19 and the normal-circle
orientation in SN.3 must both be retained in that
calculation.


#### Finite hyperplane descent of the diagonal form

Keep the normal coordinates, scalar \(c_N\), full coefficient
line and total differentials of UC.1–UC.19. Put \(d=N-1\).
The construction below gives every cochain in a spatial
hyperplane descent. Its parameter space is the full
projective hyperplane space.

##### HP.0. The normalized affine parameter bundle

Over the punctured normal domain let \(\mathcal A\)
be the total space of the following affine fibres:
\[
 \begin{gathered}
 \mathcal A_w=\{a:\langle a,w\rangle=1\},\\
 w\ne0,\\
 \pi:\mathcal A\longrightarrow\{w\ne0\}.
 \end{gathered}
 \tag{HP.1}
\]
The fibre is an affine space of complex dimension \(d\).
The vector \(s(w)=\overline w/|w|^2\) is a smooth
section. The formula
\[
 \begin{gathered}
 h_\lambda(w,a)\\
 =\bigl(w,(1-\lambda)a+\lambda s(w)\bigr),\\
 0\le\lambda\le1
 \end{gathered}
 \tag{HP.2}
\]
is a fibre-preserving strong deformation onto that section:
the scalar pairing remains one, and a point already on the
section is fixed. On a compact subset of \(\mathcal A\)
its image is compact. All of its derivatives in any fixed
finite list of coordinate directions are bounded there,
since the normal projection stays away from zero.

The projective parameter space with
\(\langle\xi,w\rangle\ne0\) is isomorphic to HP.1:
\[
                a=\frac{\xi}{\langle\xi,w\rangle}.
 \tag{HP.3}
\]
This is independent of the representative for \([\xi]\).
Conversely \(a\) determines \([a]\), and these maps are
inverse. Thus HP.2 is also an explicit contraction in the
full nonincident projective fibre. The full form
\(\mathcal W_N\) of UC.16 pulls back along the section to
\(\beta_N\wedge dw\). UC.9–UC.11 gives the associated
form homotopies, with their exact simplex signs.

##### HP.1. The top simplex determinant

On an intersection \(U_{i_0\ldots i_k}\) of the coordinate
puncture cover use the holomorphic generating vertices
\(a^i=e_i/w_i\). For an ordered tuple \(I=(i_0,\ldots,i_k)\)
of distinct indices put \(D=d_u+\bar\partial\) and define
\[
 \begin{gathered}
 a_I(u)=\sum_{\nu=0}^k u_\nu a^{i_\nu},\\
 T_I=c_N\int_{\Delta^k}\omega'_D(a_I(u)).
 \end{gathered}
 \tag{HP.4}
\]
Parameter differentials precede the antiholomorphic
coefficient differentials, as in UC.11. Its remaining
antiholomorphic degree is \(d-k\). Because all the vertices
are holomorphic, \(T_I=0\) for \(k<d\): after selecting
parameter degree \(k\), a positive antiholomorphic
degree remains and has no nonzero factor.

For \(k=d\), the exact determinant identity is
\[
 \begin{gathered}
 \omega'_{d_u}(a_I(u))\\
 =\det(a^{i_0},\ldots,a^{i_d})\,du.
 \end{gathered}
 \tag{HP.5}
\]
Here \(du=du_1\wedge\cdots\wedge du_d\), as in the
ordered-block convention of UC.1. To prove it, evaluate the alternating form on the ordered
parameter vectors. The resulting determinant has columns
\(a(u),a^{i_1}-a^{i_0},\ldots,a^{i_d}-a^{i_0}\).
Replace its first column by \(a^{i_0}\); every discarded
column is a linear combination of the remaining columns,
so contributes zero. Add the first column to each remaining
one. This gives exactly HP.5, without a factorial.
The oriented standard simplex has volume \(1/d!\).
Indeed repeated integration over
\(0\le u_j\) and \(\sum u_j\le1\) proves inductively
that its volume is \(1/d!\): the last slice at \(u_d=t\)
has volume \((1-t)^{d-1}/(d-1)!\), whose integral is
\(1/d!\). The point simplex for \(d=0\) has volume one.
Consequently
\[
 \begin{gathered}
 T_{0\ldots d}
   =\frac{c_N}{d!\,w_0\cdots w_d}
   =r_N\varphi .
 \end{gathered}
 \tag{HP.6}
\]
A permutation of the ordered vertices multiplies this
cochain by its permutation sign, exactly as required for
the alternating Čech convention. The factorial in \(c_N\)
has therefore canceled the simplex volume.

##### HP.2. Every lower cochain and its differential

Insert the smooth section as the first vertex and put
\[
 \begin{gathered}
 G_I=H_{s,a^{i_0},\ldots,a^{i_k}},\\
 \deg_{\bar\partial}G_I=d-k-1.
 \end{gathered}
 \tag{HP.7}
\]
Here \(H\) is precisely the simplex integral UC.11, with
\(k+1\) parameter directions. All its arguments pair
with \(w\) to one, so the entire simplex stays in
HP.1. The cochain is defined on the whole actual
intersection \(U_I\). Its degree makes it zero for \(k\ge d\).
Set \(G^{(-1)}=\beta_N\); its augmented Čech
differential means the restrictions of that one global
form.

Use the alternating Čech differential \(\Delta\).
UC.11 on the ordered vertices in HP.7 gives
\[
 \begin{gathered}
 \bar\partial G^{(k)}\\
     =(-1)^k\bigl(T^{(k)}-\Delta G^{(k-1)}\bigr).
 \end{gathered}
 \tag{HP.8}
\]
In detail, deleting the first vertex \(s\) has coefficient
\((-1)^k\) and gives \(T_I\). Deleting vertex
\(a^{i_\nu}\) has coefficient
\((-1)^{k+\nu+1}\), exactly
\((-1)^{k+1}\) times its Čech-face coefficient.
For \(k=0\) this says
\(\bar\partial G_i=T_i-\beta_N|_{U_i}\).
For \(k=d\), its left side is zero, and it gives
\(\Delta G^{(d-1)}=T^{(d)}\).

Let \(D_{\mathrm{tot}}=\Delta+(-1)^k\bar\partial\)
in Čech degree \(k\), the convention of V.6.
The finite sum \(G=\sum_{k=0}^{d-1}G^{(k)}\)
has total degree \(d-1\), and HP.8 gives the literal
full-cochain identity
\[
 \begin{gathered}
 D_{\mathrm{tot}}G
       =T^{(d)}-\operatorname{aug}(\beta_N).
 \end{gathered}
 \tag{HP.9}
\]
Its Čech-degree-zero component is \(-\beta_N\);
the intermediate components cancel term by term by HP.8;
the top component is HP.6. When \(d=0\), the sum is
empty and the two endpoints coincide. Thus this identity
includes all lower intersections and not just the top
rational expression. None of its general cochains is
extended across an omitted coordinate hyperplane.
The relative-cone connecting map and the cone-to-cube
map of UC.15 turn HP.9 into the same unit with literal
top cube entry \(r_N(-1)^N\varphi\).

For any finite list of such cochains, their restrictions
and homotopies, choose actual domains before integration.
On a compact subset of an intersection, all its displayed
nonzero \(w_i\)'s have a positive lower bound, as does
\(|w|\). The simplices are compact. Differentiation in
any fixed finite list of variables therefore commutes
with their integrals and with restriction. This verifies
the full-parameter and finite-witness assertions without
a boundary growth assumption or a passage to a quotient
limit.

##### HP.3. Arbitrary independent hyperplane vertices

The construction is covariant under an invertible linear
normal frame. Let \(\ell_0,\ldots,\ell_d\) be independent
constant vectors, let \(M\) have those columns, and put
\(\alpha_j=\langle\ell_j,w\rangle\).
Use \(a^j=\ell_j/\alpha_j\) on \(\{\alpha_j\ne0\}\).
HP.5 and the ordered change of variables give
\[
 \begin{gathered}
 T_{0\ldots d}\wedge dw\\
   =\frac{r_N\det M}{(2\pi i)^N\prod_j\alpha_j}\,dw\\
   =\frac{r_N}{(2\pi i)^N\prod_j\alpha_j}\,d\alpha .
 \end{gathered}
 \tag{HP.10}
\]
Here \(d\alpha=\det(M^{\mathsf T})\,dw=\det M\,dw\).
Thus the simplex determinant and the endpoint input
frame transform together. HP.7–HP.9 applies with the
same section \(s\) and these new generating vertices;
only the open intersections change.

In particular one may take
\(\ell_0=e_0\) and
\(\ell_j=e_0+\varepsilon e_j\) for \(1\le j\le d\),
with \(\varepsilon\ne0\). Their determinant is
\(\varepsilon^d\), and all their projective classes
are near \([e_0]\) as \(\varepsilon\) tends to zero.
This does not put the interpolating simplices into
that same projective chart: a convex combination of
the normalized vectors can have zeroth coordinate zero
even though it remains generating. HP.2 and HP.7
are assertions in the full nonincident projective
fibre. An application confined to one actual middle
coordinate domain must supply its own containment and
proper-collar comparison.

There is an exact two-dimensional example. Let
\(N=2\), \(\varepsilon r\ne0\), and take
\[
 \begin{gathered}
 w=(r,-2r/\varepsilon),\\
 \ell_0=(1,0),\quad \ell_1=(1,\varepsilon),\\
 \alpha_0=r,\qquad \alpha_1=-r,\\
 a^0=(1,0)/r,\\
 a^1=(-1,-\varepsilon)/r.
 \end{gathered}
 \tag{HP.11}
\]
The midpoint is \((0,-\varepsilon/(2r))\).
It pairs with \(w\) to one, but its projective class
is outside the chart \(\xi_0\ne0\). The two original
projective vertices both lie in that chart and tend
to \([e_0]\) as \(\varepsilon\) tends to zero.
For each fixed \(\varepsilon\), choosing \(r\) small
puts \(w\) in any prescribed normal neighbourhood.
Thus neither a small normal domain nor nearby
projective vertices proves chart containment of
the affine simplex.

For these two vertices there is nevertheless a
chart-contained curved path:
\[
 \begin{gathered}
 y(t)=\frac{\varepsilon}{2}(1-e^{i\pi t}),\\
 \xi(t)=(1,y(t)),\\
 \langle\xi(t),w\rangle=r e^{i\pi t},\\
 a(t)=\frac{\xi(t)}{r e^{i\pi t}},\\
 0\le t\le1 .
 \end{gathered}
 \tag{HP.12}
\]
Its endpoints are \(a^0,a^1\), its denominator has
modulus \(|r|\), and \(|y(t)|\le|\varepsilon|\).
Every fixed parameter derivative is bounded for
fixed nonzero \(r,\varepsilon\). UC.9 applies to
this smooth generating path, with \(d_t\) in
place of the straight-path differential: its proof
uses only the generating relation and the two
endpoints. This example gives an actual compact
path in that one parameter chart. It supplies
neither the higher compatible parameter faces nor
a proper-support comparison for a general
incidence convolution.

The reverse spatial frame in UC.19 still contributes
\((-1)^d\) to its holomorphic differential block.
Neither HP.9 nor HP.10 removes that factor. The
topological source counit and the actual coefficient
composition have to be compared with this full descent
before either normalized incidence inverse product
is identified with the unit.


![All lower Čech cochains, a projective chart crossing, a bounded curved path and the source shifts](assets/hyperplane-descent-and-chart.png)

*Figure HP.1. Panel A gives all lower bidegrees for \(N=4\); HP.8–HP.9 proves their total-differential identity. Panels B and C use the exact rational choices of HP.11: \(\varepsilon=1/4\), \(r=1/32\), \(w=(1/32,-1/4)\). The generating affine segment crosses the boundary of the projective chart, whereas HP.12 stays within \(|y|\le1/4\) and has denominator of constant modulus \(1/32\). Panel D gives the two different source and coefficient shifts of SN.5–SN.8. The source equivalence is under microlocal Hom off zero. These constructions leave the general actual coefficient-composition comparison to be proved.*

Unit form: Full PNG · SVG · Exact data · Drawing source

Hyperplane descent: Full PNG · SVG · Exact data · Drawing source

Free primary context: [Harold P. Boas, *Math 685 Notes: Topics in Several Complex Variables*, 2005, revised 2010](https://haroldpboas.gitlab.io/courses/685-2005c/notes.pdf), Sections1.3 and1.5, for the classical normal and hyperplane forms. The proofs of the current identity, ordered signs, source comparisons and complete finite descent are given here and in the exact earlier programme proofs linked above.


### 5.65. Bounded generating hyperplanes and proper contour support

The exact earlier programme dependencies are §§5.34, 5.35, 5.53, 5.63, 5.64. Specifically, R.6–R.7 and V.1/V.4 give the full supported comparison and trace; RM.3–RM.9 gives actual real-stalk supports and finite common witnesses; OI.18 and OI.20 gives the two ordinary residues; UC.1, UC.3, UC.9–UC.11 and UC.16–UC.19 gives the ordered scalar, generating homotopies and universal form.

Let \(N=d+1\ge1\), and write the endpoint normal variable as
\(w=(w_0,z)\in\mathbf C\times\mathbf C^d\).
Put \(\rho=|w|=(|w_0|^2+|z|^2)^{1/2}\).
Pair covectors and vectors bilinearly, without a conjugate in
that pairing. All norms below are Euclidean norms.
Use the ordered differential blocks, the scalar \(c_N\),
the universal form and the full relative differential of
UC.1, UC.16 and V.1. In particular \(\bar\partial\)
includes the output antiholomorphic directions of the pair.

Fix a spatial middle-coordinate ball of radius \(R_y>0\).
Let \(\mathcal K_{\mu,\nu}\) be the Hermitian matrices
satisfying the following bounds:
\[
 \begin{gathered}
 0<\mu\le\nu<R_y/2,\\
 H=H^*,\qquad \mu I\le H\le\nu I.
 \end{gathered}
 \tag{RC.1}
\]
The inequalities mean inequalities of Hermitian quadratic
forms on \(\mathbf C^d\). For \(d=0\) the matrix is empty
and this parameter set is a singleton.

The family constructed below stays in this one actual
spatial chart. The supports and margins are part of the
construction, rather than a consequence of nearby
projective vertices.

#### RC.0. The support and every zero of the denominator

Choose \(\epsilon>0\) and
\(K>\sqrt{1+\mu^{-2}}\). Define the closed cone
\[
 \begin{gathered}
 G=G_{K,\epsilon},\\
 w_0=-\ell+ib,\quad \ell\ge0,\\
 |b|\le\epsilon\ell,\qquad |z|\le K\ell .
 \end{gathered}
 \tag{RC.2}
\]
It is convex: the time and imaginary inequalities are
convex linear/norm inequalities, and the spatial
inequality is preserved by the triangle inequality.
It contains the zero normal vector.

For \(H\in\mathcal K_{\mu,\nu}\), on \(w\ne0\), put
\[
 \begin{gathered}
 y_H(w)=H\overline z/\rho,\\
 \xi_H=(1,y_H),\\
 F_H(w)=\langle\xi_H,w\rangle,\\
 a_H(w)=\xi_H/F_H(w),\\
 F_H(w)\ne0.
 \end{gathered}
 \tag{RC.3}
\]
Although the pairing is bilinear, the extra term in
the denominator is a real positive Hermitian form:
\[
 \begin{gathered}
 Q_H=z^{\mathsf T}H\overline z,\\
 F_H=w_0+Q_H/\rho,\\
 \mu|z|^2\le Q_H\le\nu|z|^2 .
 \end{gathered}
 \tag{RC.4}
\]
Indeed take the vector \(v=\overline z\); then
\(v^*Hv=z^{\mathsf T}H\overline z\).

We also have \(\|Hv\|\le\nu\|v\|\). Here is a proof
that does not require diagonalizing \(H\). Positivity
of the form on \(u+\lambda v\), for every complex
\(\lambda\), gives
\(|u^*Hv|^2\le(u^*Hu)(v^*Hv)\): minimize the resulting
quadratic expression in \(\lambda\), with the zero
quadratic-coefficient case following by a linear
variation. Thus \(|u^*Hv|\le\nu\|u\|\|v\|\).
Taking \(u=Hv/\|Hv\|\) when \(Hv\ne0\) gives the
asserted norm bound; the zero case is immediate.
Consequently \(|y_H|\le\nu\), strictly inside the
chosen middle ball.

Every zero of \(F_H\) with \(w\ne0\) lies in the
strict interior of \(G\). To prove this, a zero
forces \(w_0\) to be real and negative. It cannot
have \(z=0\), since then \(F_H=w_0\ne0\).
Write \(w_0=-\ell\), \(\ell>0\), and set
\(q=|z|/\ell\). RC.4 gives
\[
 \begin{gathered}
 \ell=Q_H/\rho
       \ge\mu |z|^2/\rho,\\
 \mu q^2\le\sqrt{1+q^2}.
 \end{gathered}
 \tag{RC.5}
\]
Both sides are nonnegative. Squaring and solving the
quadratic inequality in \(q^2\) gives
\[
 \begin{gathered}
 q^2\le
 \frac{1+\sqrt{1+4\mu^2}}{2\mu^2}\\
 \le1+\mu^{-2}<K^2 .
 \end{gathered}
 \tag{RC.6}
\]
For the second inequality, square the positive
inequality \(\sqrt{1+4\mu^2}\le1+2\mu^2\).
The imaginary coordinate is zero and hence has
strict inequality in RC.2 as well. This proves the
strict-interior assertion. In dimension \(d=0\)
there are no such zeros.

In particular \(a_H\) is a smooth generating vector
on the entire actual complement of \(G\):
\(\langle a_H,w\rangle=1\). Its projective class is
always \([1:y_H]\); its zeroth homogeneous
coordinate never vanishes there.

The cone is admissible at the nonzero normal
covector \(e_0\), with a uniform nearby-covector
margin. Every nonzero vector in \(G\) satisfies
\[
 \begin{gathered}
 -\Re w_0=\ell\\
   \ge\frac{|w|}{\sqrt{1+\epsilon^2+K^2}}>0.
 \end{gathered}
 \tag{RC.7}
\]
Perturbing the covector by norm less than half the
reciprocal square root preserves half this positive
margin, by the ordinary complex Cauchy inequality
\(|\langle p,w\rangle|\le|p||w|\).
Thus RC.2 has exactly the negative displacement
polar sign of RM.3–RM.6.

It can contain any finite list of the earlier
admissible supports after an actual restriction.
First use RM.4–RM.6 to obtain finitely many finite
time/aperture and componentwise spatial constants.
Take a common larger \(\epsilon\) and take \(K\)
larger than the Euclidean norm of those spatial
constants, as well as the bound in RC.2.
This gives actual containment in \(G\).
RC.7 then supplies one smaller actual covector
neighbourhood. No aperture is asserted uniform
over an infinite list of real germs.

#### RC.1. Uniform denominator and derivative bounds

There is a constant \(c>0\), depending only on the
displayed dimensions and constants, such that
\[
 \begin{gathered}
 c\rho\le|F_H(w)|\le(1+\nu)\rho,\\
 w\notin\operatorname{int}G,\quad w\ne0,\\
 H\in\mathcal K_{\mu,\nu}.
 \end{gathered}
 \tag{RC.8}
\]
The upper bound follows from RC.4 and
\(|z|^2/\rho\le\rho\).
For the lower bound restrict to \(|w|=1\) outside
\(\operatorname{int}G\). This is compact.
The matrix set in RC.1 is compact as a subset of
the finite-dimensional real space of Hermitian
matrices: its quadratic inequalities define a
closed set, and the norm bound proved above
bounds every entry by \(\nu\).
The continuous function \(|F_H(w)|\) has no zero
on the product of these compact sets, by RC.5–RC.6.
Its minimum is therefore positive. Positive real
homogeneity \(F_H(rw)=rF_H(w)\) proves RC.8
for every \(\rho>0\).

For every fixed real normal-coordinate multiindex
\(\alpha\), the following bounds hold uniformly
in the same directions and matrices:
\[
 \begin{gathered}
 |\partial^\alpha y_H|\le C_\alpha\rho^{-|\alpha|},\\
 |\partial^\alpha F_H|
          \le C_\alpha\rho^{1-|\alpha|},\\
 |\partial^\alpha a_H|
          \le C_\alpha\rho^{-1-|\alpha|}.
 \end{gathered}
 \tag{RC.9}
\]
Here and below the constants need not be the same
in different lines. For completeness the functions
are smooth on a neighbourhood of the compact
unit-direction/matrix product used above. RC.8
keeps the reciprocal denominator smooth there;
a smaller neighbourhood retains half its lower
bound. Each fixed derivative is continuous and
bounded on that compact product. Differentiating
the positive-real homogeneities of degrees
\(0,1,-1\), respectively, scales those derivatives
by the displayed powers of \(\rho\). This proves
all of RC.9, without a bound for a varying
derivative order.

#### RC.2. All finite parameter faces stay in the chart

For any finite list \(H_0,\ldots,H_k\) in RC.1,
use the oriented standard simplex and put
\[
 \begin{gathered}
 H(\sigma)=\sum_{j=0}^k\sigma_jH_j,\\
 \sigma_j\ge0,\qquad\sum_j\sigma_j=1 .
 \end{gathered}
 \tag{RC.10}
\]
The quadratic-form inequalities are convex, so
every point remains in RC.1. RC.3 and RC.8
therefore hold on the entire simplex, on every
face and on the same endpoint complement.
In particular \(|y_{H(\sigma)}|\le\nu\)
and the entire family is in one fixed inner
middle chart. This is interpolation of the
Hermitian matrices followed by normalization;
it is not affine interpolation of arbitrary
normalized projective vectors.

Set \(D=d_\sigma+\bar\partial\), with parameter
differentials first, and define
\[
 \begin{gathered}
 J_{0\ldots k}
   =c_N\int_{\Delta^k}
      \omega'_D(a_{H(\sigma)}),\\
 \deg_{\bar\partial}J_{0\ldots k}=N-1-k .
 \end{gathered}
 \tag{RC.11}
\]
The argument of UC.3 applies because the scalar
pairing is one throughout this simplex. It makes
\(D\omega'_D(a_{H(\sigma)})=0\).
For \(k=0\) this gives
\(\bar\partial J_H=0\).
For \(k\ge1\), parameter Stokes in the
explicit ordering of UC.11 gives
\[
 \begin{gathered}
 \bar\partial J_{0\ldots k}\\
 =\sum_{j=0}^k(-1)^{k+j+1}
       J_{0\ldots\widehat j\ldots k}.
 \end{gathered}
 \tag{RC.12}
\]
For \(k>N-1\) the integral is zero and this
identity is its alternating face relation.
All faces are restrictions of the same
matrix-simplex construction, with the same
denominator and chart margins. Thus these are
actual compatible finite faces, including the
higher face homotopies.

Write \(\beta_H=J_H\). In the full relative
complex for \(Z=\{w\in G\}\) on any small
endpoint pair domain, the cochain
\[
                         k_H=(0,-\beta_H)
 \tag{RC.13}
\]
has degree \(N\) and is closed. It represents
the diagonal unit. Indeed let
\(s_0(w)=\overline w/\rho^2\), and interpolate
the normalized vectors
\((1-t)s_0+t a_H\) on the endpoint complement.
Their scalar pairing is one. UC.9 gives an
explicit coefficient homotopy \(H(s_0,a_H)\),
and UC.10 gives the literal boundary
\[
 \begin{gathered}
 k_H-k_{\mathrm{unit}}\\
        =d(0,H(s_0,a_H)).
 \end{gathered}
 \tag{RC.14}
\]
This last homotopy is in the endpoint coefficient
complex. It is not used as a middle-coordinate
path, and its projective coordinates are not
asserted to stay in the middle chart.
The actual middle parameter faces are RC.10.
This distinguishes RC.14 from the chart crossing
of HP.11 while retaining its full relative
equality witness.

The fixed-derivative bounds also hold after
any finite number of simplex derivatives:
the parameter domains are compact, their
matrix derivatives are bounded, and RC.8
holds uniformly on them. In particular the
coefficients of RC.11 satisfy
\[
 |J_{0\ldots k}(w)|
       \le C_k\rho^{-2N+1+k}.
 \tag{RC.15}
\]
There are one undifferentiated \(a_H\), \(k\)
parameter derivatives of degree \(-1\), and
\(N-1-k\) antiholomorphic derivatives of degree
\(-2\). This gives exactly the exponent in
RC.15. These coefficients are integrable
near the normal origin: their radial absolute
bound is a constant times
\(\int_0^\eta r^k\,dr\).
Extension by zero from the cone complement
thus gives locally integrable coefficients
if an extension is wanted. Its differential
can have cone-boundary terms; it is not
silently identified with just the diagonal
current. RC.14 is the full relative unit
comparison and keeps those boundary effects.

#### RC.3. The first normal-circle family and a common proper collar

Let the endpoints in the first incidence order be
\((t,x)\), \((t',x')\), so
\(w=(t'-t,x'-x)\). Write the middle coordinates
as \((s,y)\). The two actual incidence functions
are
\[
 \begin{gathered}
 f=s-t-x^{\mathsf T}y,\\
 g=t'-s+(x')^{\mathsf T}y,\\
 f+g=w_0+z^{\mathsf T}y .
 \end{gathered}
 \tag{RC.16}
\]
On the endpoint complement use the middle family
\[
 \begin{gathered}
 y=y_{H(\sigma)}(w),\\
 f=\tfrac12 F_{H(\sigma)}(w)e^{i\theta},\\
 s=t+x^{\mathsf T}y+f,\\
 0\le\theta\le2\pi .
 \end{gathered}
 \tag{RC.17}
\]
It is periodic in \(\theta\). Its positive
normal circle surrounds \(f=0\) and excludes
\(g=0\), since
\[
 \begin{gathered}
 |f|=\tfrac12|F_H|,\\
 |g|=|F_H-f|\ge\tfrac12|F_H|,\\
 |F_H|\ge c\rho>0.
 \end{gathered}
 \tag{RC.18}
\]
The finite simplices, their faces and this
circle have the same bounds.

The ordinary normal residue is an equality
of parameter forms:
\[
 \begin{gathered}
 \frac1{2\pi i}
 \int_{S^1}^{+}
 \frac{df\wedge dy\wedge dw}{f(F_H-f)^N}\\
 =\frac{dy\wedge dw}{F_H^N}.
 \end{gathered}
 \tag{RC.19}
\]
To verify it directly, on \(|f|=|F_H|/2\)
expand \((F_H-f)^{-N}\) by its convergent
power series in \(f/F_H\). The sum is uniformly
convergent there, with all fixed derivatives
on compact parameter subsets of the endpoint
complement. Only its constant term survives
the circle integral of \(df/f\), giving
\(F_H^{-N}\), as in OI.18.
There is no derivative-of-radius correction:
after pullback, \(df\) has \(\theta\)-component
\(if\,d\theta\); every other component contains
no \(d\theta\). The factors \(dy\) and \(dw\)
also contain no \(d\theta\). Fibre integration
therefore selects exactly that displayed
normal component. At fixed endpoints
\(ds\wedge dy=df\wedge dy\), since
the extra \(x^{\mathsf T}dy\) has a repeated
spatial differential. The same
\(\theta\)-selection proves the full-family
identity when the endpoints vary.

Multiplying RC.19 by \(c_N\), then selecting
parameter degree \(k\) and integrating its
matrix simplex, gives
\[
 \begin{gathered}
 c_N\int_{\Delta^k}
       \frac{dy_H\wedge dw}{F_H^N}\\
                 =J_{0\ldots k}\wedge dw .
 \end{gathered}
 \tag{RC.20}
\]
This is the pullback of the projectively
invariant universal form UC.16.
Its holomorphic \(w\)-differentials vanish
against \(dw\); its remaining differential
is \(d_\sigma+\bar\partial\), exactly
RC.11. Thus RC.12 accounts for every
parameter boundary in these ordinary
periods.

There is a common actual proper middle-variable
collar for this entire family, including its
closure at \(w=0\). Fix outer middle radii
\(R_s,R_y\), and small endpoint bounds
\(|t|\le r_t\), \(|x|\le r_x\),
\(\rho\le r_w\), chosen so that
\[
 \begin{gathered}
 |y_H|\le\nu<R_y/2,\\
 |s|\\
 \le r_t+\nu r_x+\tfrac12(1+\nu)r_w\\
                         <R_s/2 .
 \end{gathered}
 \tag{RC.21}
\]
These follow from RC.8 and RC.17 and hold
for every matrix and parameter face.
The endpoint bounds can be made this small
after the constants and the actual outer
domain are fixed.

Take the closure of the family's image
in the endpoint-domain times middle-domain
product. It is closed by definition.
Over a compact endpoint set it is contained
in that compact set times the closed middle
balls in RC.21. Those balls are compact
and strictly inside the actual middle domain.
The closed intersection is compact.
This proves properness of that closure
over the actual endpoint domain, including
the collapsed normal circle at \(w=0\).
A unique limit of \(y_H\) at \(w=0\) is
neither needed nor asserted.

A single smooth collar can be chosen
independently of the endpoint, the matrix,
the parameter face and the circle.
For explicitness put \(b(t)=e^{-1/t}\)
for \(t>0\) and zero otherwise. For \(r<R\)
in an ordinary Euclidean ball write
\(q_v=|v|^2\), and set
\[
 \begin{gathered}
 b_+=b(R^2-q_v),\\
 b_-=b(q_v-r^2),\\
 \chi_{r,R}(v)=\frac{b_+}{b_++b_-}.
 \end{gathered}
 \tag{RC.22}
\]
The denominator never vanishes; the two
arguments cannot both be nonpositive.
The function is smooth, equals one on
\(|v|\le r\), and equals zero on
\(|v|\ge R\). At the joining value \(t=0\),
every derivative of \(b\) tends to zero:
for \(t>0\) it is a polynomial in \(1/t\)
times \(e^{-1/t}\), whose limit is zero.
Choose inner radii strictly larger than
the bounds in RC.21 and outer radii
strictly smaller than \(R_s,R_y\).
The product of the corresponding two
functions is one near the closed family
and compactly supported in the middle
domain. The same function works for
every finite diagram constructed here.
In dimension \(d=0\) the spatial factor
is one.

For smooth cochains or currents defined
on the chosen actual endpoint-times-middle
domain and supported in that closed family,
this collar permits controlled zero
extension in the middle variable, by
R.7 and V.4. A different source
neighbourhood must satisfy R.7's own
neighbourhood and support hypotheses.
It does not extend arbitrary complement
cochains through \(G\), nor show that an
arbitrary microlocal-Hom composition
already has that support.

#### RC.4. The opposite order and its separate spatial sign

Let the endpoints now be \((s,y)\),
\((s',y')\), and put
\(w'=(s'-s,y'-y)\).
Use middle coordinates \((t,x)\) and
the conventional functions
\[
 \begin{gathered}
 g=t-s+y^{\mathsf T}x,\\
 f'=s'-t-(y')^{\mathsf T}x,\\
 g+f'=w'_0-(z')^{\mathsf T}x .
 \end{gathered}
 \tag{RC.23}
\]
Choose \(x=-y_{H(\sigma)}(w')\), and set
\(g=F_H(w')e^{i\theta}/2\),
\(t=s-y^{\mathsf T}x+g\).
All denominator, derivative, chart and
proper-collar bounds above apply after
interchanging the endpoint spaces.
In particular the ordered ordinary
normal integral is
\[
 \begin{gathered}
 \frac1{2\pi i}\int_{S^1}^{+}
       \frac{dg}{g^N(F_H-g)}\\
                          =F_H^{-N}.
 \end{gathered}
 \tag{RC.24}
\]
Expand \((F_H-g)^{-1}\) in \(g/F_H\);
its coefficient of \(g^{N-1}\) is
\(F_H^{-N}\). RC.18 gives uniform
convergence on the same compact circle,
so termwise integration proves RC.24
including \(N=1\).

The conventional ordered input block
in this middle space is \(dt\wedge dx\).
The normal-circle selection replaces it
by \(dg\wedge dx\) with positive normal
orientation. Since \(x=-y_H\),
the spatial block satisfies
\(dy_H=(-1)^d dx\). Consequently the
reverse normalized ordinary family is
\[
 \begin{gathered}
 (-1)^d c_N\int_{\Delta^k}
           \frac{dx\wedge dw'}{F_H(w')^N}\\
                       =J_{0\ldots k}\wedge dw' .
 \end{gathered}
 \tag{RC.25}
\]
This retains exactly the spatial factor
UC.19. Its normal circle residue is
positive; the spatial factor is a
different calculation.

The two ordinary period families
RC.20 and RC.25 give the full relative
unit representatives RC.13–RC.14.
They have compatible finite parameter
faces in one actual middle chart and
a common proper collar for their
explicit contour supports.
Identifying either family with the
actual microlocal-Hom coefficient
composition on the source roofs of
SN.5–SN.8 is a separate equality of
derived morphisms. In particular the
scalar \(c_N\) in these unit-normalized
ordinary periods does not determine
the scalar of that composition map.
No general incidence inverse product
or whole-real contact isomorphism
is asserted by the contour calculation.


![The excluded cone contains every denominator zero, the two normal circles avoid the other pole, and one inner collar contains the contour family](assets/bounded-generating-parameters-and-collar.png)

*Figure RC.1. Panels A and B use the exact choices \(\mu=1/4\), \(\nu=1/2\), \(K=5\), \(\epsilon=1/2\). Panel A is the unit normal section \(b=0\), \(z\) real, \(H=\delta I\); its zero arc has \(1/4\le\delta\le1/2\) and lies strictly inside the cone by RC.5–RC.6. Panel B plots a real section with \(q\ge5\); it does not plot the general constant \(c\) of RC.8. Panel C is the positive normalized normal circle \(f/F=e^{i\theta}/2\), or \(g/F=e^{i\theta}/2\) in the other order. The other pole at one is excluded in both cases. Panel D is a projection to the norms \(|y|,|s|\), with endpoint bounds \(r_t=r_x=r_w=1/16\), contour bounds \(|y|\le1/2\), \(|s|\le9/64\), and the explicit collar of RC.21–RC.22. It is not a drawing of the full middle fibre. The ordinary period identities are RC.19–RC.20 and RC.24–RC.25.*

Full PNG · SVG · Exact data · Drawing source

Free primary context for the classical hyperplane form: [Harold P. Boas, *Math 685 Notes: Topics in Several Complex Variables*, 2005, revised 2010](https://haroldpboas.gitlab.io/courses/685-2005c/notes.pdf), Sections1.3 and1.5. The matrix family, support margins, bounds, finite faces and collar used here are proved above and in the exact earlier programme proofs linked above.


### 5.66. Affine Spencer reduction and free source directions

The exact earlier programme support prerequisites are §§5.34, 5.47, 5.59, 5.60, 5.64, 5.65. The algebraic functor, coherence, solution and trace prerequisites are in Adjunctions, base change and the projection formula, The de Rham functor, and The Riemann–Hilbert correspondence, with the theorem and chain-formula locators given below.

All differential operators in this argument are algebraic. Put \(N=d+1\ge1\). Use the programme's left-module direct image, density side change and relative Spencer complex, with the last relative form in degree zero. The exact prerequisites are the adjunctions lesson, Theorems 2.1, 3.4 and 4.7; regular-singularities, Theorem 5.17; the Riemann–Hilbert lesson, Theorems 1.18–1.23 and its signed closed-normal calculation; and the de Rham lesson's closed normal Spencer calculation. The support and coefficient conventions are OCM.2–OCM.4 and HI.1. The vanishing assertion used below is proved in *A free source direction and derivative-torsion cones*.

#### AS.0. The actual proper algebraic convolution

Let the endpoint variables be \((t,x),(t',x')\), and the middle variables \((s,y)\), with \(x,x',y\in\mathbb A^d\). Set
\[
\begin{gathered}
u=(t,x),\quad w=t'-t,\\
z=x'-x,\\
f=s-t-x^{\mathsf T}y,\\
g=t'-s+(x')^{\mathsf T}y,\\
F=f+g=w+z^{\mathsf T}y .
\end{gathered}
\tag{AS.1}
\]
Write \(\delta_h=\mathcal O[h^{-1}]/\mathcal O\), with distinguished class \([h^{-1}]\), for a smooth hypersurface \(h=0\). This is a left differential-operator module: derivatives act on its displayed rational representatives. In particular
\[
 D_h^r[h^{-1}]=(-1)^r r![h^{-r-1}].
 \tag{AS.2}
\]
Its closed normal presentation and regular holonomicity are the earlier closed-embedding calculation.

On the affine triple product the pair \((f,g)\) consists of two independent coordinates. Indeed
\[
\begin{gathered}
(u,z,y,f,g)\longmapsto\\
\begin{cases}
x'=x+z,\\
s=t+x^{\mathsf T}y+f,\\
t'=t+f+g-z^{\mathsf T}y .
\end{cases}
\end{gathered}
\tag{AS.3}
\]
is a polynomial coordinate inverse. Thus the derived structure-sheaf tensor of the two pulled-back incidence modules is concentrated in degree zero:
\[
 \delta_f\otimes_{\mathcal O}^{L}\delta_g
 =\mathcal H^2_{\{f=g=0\}}(\mathcal O).
 \tag{AS.4}
\]
Here is also a flat-complex verification. Resolve each \(\delta_h\) by \([\mathcal O\to\mathcal O[h^{-1}]]\) in degrees \(-1,0\). In the coordinates AS.3 their tensor is the two-variable coordinate localization cube. Successive one-variable Laurent principal parts show that its only cohomology is in degree zero and is the quotient
\(\mathcal O[(fg)^{-1}]/(\mathcal O[f^{-1}]+\mathcal O[g^{-1}])\).
Each principal part is a finite sum in its negative coordinate powers, with coefficients regular in the remaining coordinates. This also identifies the ordered normal generator. The same calculation is the coordinate closed-transfer presentation of the right side of AS.4.

There is an actual proper realization of its direct image. Compactify the common middle affine space to \(\mathbb P^N\). On each incidence pair extend the closed hypersurface module by the ordinary algebraic open direct image across the middle hyperplane at infinity. Call the two extensions \(\widetilde M_+\) and \(\widetilde M_-\). Pull them to the triple product with projective middle factor and put
\[
\begin{gathered}
p:\mathbb A^N\times\mathbb P^N\times\mathbb A^N\\
\longrightarrow\mathbb A^N\times\mathbb A^N,\\
Q=p_+(\widetilde M_+
          \otimes_{\mathcal O}^{L}\widetilde M_-).
\end{gathered}
\tag{AS.5}
\]
The smooth inverse images here are the unshifted transfer pullbacks. Equivalently they are the programme's extraordinary inverse images shifted by the negative relative dimension.

Both extensions, their smooth pullbacks and their derived structure tensor are bounded regular holonomic. This is precisely algebraic open direct stability, inverse stability and structure tensor stability in the stated prerequisites. Theorem 3.4 applies to the displayed proper projection, so \(Q\) is bounded coherent. Regular stability makes it regular holonomic.

This is also the affine incidence direct image, with its actual maps. The common middle open is the complement of one Cartier hyperplane. On an affine chart, its ordinary operator direct image is ordinary coefficient localization. This localization is flat, and the two localized factors have their common localization parameter invertible. Their tensor is consequently the localization of their affine tensor; applying localization once or twice gives the same complex. These chart identities glue. The actual direct-image composition therefore identifies AS.5 with the affine middle direct image of AS.4. This argument uses proper coherence for AS.5, rather than assuming coherence for a general nonproper analytic direct image.

#### AS.1. Elimination of the first normal coordinate

The change from the ordered normal coordinates \((f,g)\) to \((f,F=f+g)\) has determinant \(+1\). The closed normal transfer and its ordered determinant contraction identify their normal generators with coefficient \(+1\). One can check that coefficient by the two-coordinate residue: integrating first in the positive \(f\)-circle and then in the positive \(F\)-circle gives the same iterated normal residue as the ordered \((f,g)\) generator. This is a comparison of localization cubes on their common complement cover; it does not declare \(1/(fF)\) to be a function on the single open \(fg\ne0\).

The closed \(f\)-normal relative Spencer complex is
\[
 \mathbb C[D_f]\,e_f
 \xrightarrow{D_f}
 \mathbb C[D_f]\,e_f\,df ,
 \tag{AS.6}
\]
in degrees \(-1,0\), with the remaining coefficients carried along. The map is injective and its cokernel is generated by \(e_f\,df\). Its operator residue sends that generator to \(1\), by the earlier closed trace normalization. Thus the closed normal elimination is an actual quasi-isomorphism, without a residual shift.

The sign can be verified in the total relative complex, without an implicit interchange of two shifted complexes. Write a form with \(df\) first as \(D_f^k e_f\,df\wedge dy_I\), carrying the other coefficient along. Send it to zero for \(k>0\), and to \(dy_I\) for \(k=0\); send every form without \(df\) to zero. The source differential has the total shift sign \((-1)^{d+1}\). On a form containing \(df\), the relative \(y\)-differential crosses that first factor and therefore has sign \((-1)^d\), exactly the target shift sign. The \(f\)-derivative has zero residue. This proves the chain identity with coefficient \(+1\) on the ordered top form. The inclusion \(dy_I\mapsto e_f\,df\wedge dy_I\) is a chain section. A homotopy sends \(D_f^k e_f\,df\wedge dy_I\), for \(k>0\), to \((-1)^{d+1}D_f^{k-1}e_f\,dy_I\), and is zero on the other terms. Expansion gives \(dH+Hd=1-i\,\operatorname{res}_f\); the two \(y\)-terms cancel. Thus every relative degree and both shift signs are included.

The remaining module is \(\delta_F\) on the endpoint-times-\(y\) space. Put
\[
\begin{gathered}
\tau=D_w,\qquad e=[F^{-1}],\\
\gamma=[e\,dy_1\wedge\cdots\wedge dy_d],\\
\gamma\in H^0Q .
\end{gathered}
\tag{AS.7}
\]
For \(d=0\), the empty exterior product is one. The entire complex \(Q\) is the relative \(y\)-Spencer complex of \(\delta_F\), in degrees \([-d,0]\). Its coefficient normal form is
\[
 R=\mathbb C[u,z,y,\tau]\,e .
 \tag{AS.8}
\]
To justify this normal form, use \(F\) and the other coordinates as polynomial coordinates before projecting. Every finite normal principal part is a finite polynomial in \(D_F\) applied to \(1/F\), with the factorials and signs of AS.2. Replacing \(F\) by \(w+z^{\mathsf T}y\) gives AS.8.

Derivatives of the original endpoint variables differ from their expressions at fixed \(f\) by a coefficient times the eliminated \(f\)-derivative. That term has zero \(f\)-residue by the preceding explicit chain map. Thus the endpoint action after elimination is the action in the displayed \(F\)-coordinates, with its full horizontal derivatives retained.

The endpoint and relative actions in this form are
\[
\begin{gathered}
D_w=\tau,\quad w=-z^{\mathsf T}y-\partial_\tau,\\
D_{z_j}=\partial_{z_j}+y_j\tau,\\
D_{u_i}=\partial_{u_i},\\
T_j=D_{y_j}^{\rm rel}=\partial_{y_j}+z_j\tau .
\end{gathered}
\tag{AS.9}
\]
For example \(D_{y_j}(1/F)=z_jD_w(1/F)\). The term
\(-\partial_\tau\) follows from \([D_w,w]=1\). Direct commutator expansion shows that the endpoint actions commute with all \(T_j\), and that the \(T_j\)'s commute with one another. Thus the complex and its endpoint action have been specified, including their differential signs: on relative forms it is
\[
\begin{gathered}
d_{\rm rel}=\sum_{j=1}^d dy_j\wedge T_j,\\
\text{shifted by }[d].
\end{gathered}
\tag{AS.10}
\]
The shift multiplies the unshifted differential by \((-1)^d\). Multiplying relative exterior degree \(k\) by \((-1)^{d(k-d)}\) identifies it with the displayed unshifted Koszul differential, with coefficient one in the top column \(k=d\). Hence neither its top quotient nor the ordered top class AS.7 changes.

#### AS.2. Polynomial Ore localization and the whole Spencer complex

Let \(A\) be the global polynomial Weyl algebra of the endpoint affine space. In this paragraph localization means localization of global \(A\)-modules at the powers of \(\tau\). It is not the assertion that these powers form an Ore set in every ring of rational or holomorphic coefficient germs.

For \(P\in A\), \(\operatorname{ad}(\tau)\) is differentiation of its polynomial \(w\)-coefficients and is locally nilpotent. The finite identity
\[
\begin{gathered}
\tau^nP\\
=\sum_k\binom nk
       \operatorname{ad}(\tau)^k(P)\tau^{n-k}.
\end{gathered}
\tag{AS.11}
\]
and its right-handed version prove both Ore conditions. The Weyl algebra is a domain: the leading symbol of a product in its polynomial order filtration is the product of its nonzero polynomial leading symbols. Hence this denominator set has no zero divisors. As a right \(A\)-module the localization is the filtered union of the copies \(\tau^{-n}A\); each is free of rank one. Equivalently its transition maps, on copies of \(A\), are left multiplication by \(\tau\). Filtered colimits of exact vector-space/module sequences are exact. The localization is therefore right flat, and
\[
\begin{gathered}
A[\tau^{-1}]\otimes_A M=0\\
\Longleftrightarrow\\
\forall m\in M\ \exists k\ge0:\\
\tau^k m=0 .
\end{gathered}
\tag{AS.12}
\]
The equivalence follows directly from equality of fractions in this torsion-free denominator system.

After localization replace AS.8 by
\[
 R_\tau=\mathbb C[u,z,y,\tau,\tau^{-1}]\,e .
 \tag{AS.13}
\]
The operator
\[
\begin{gathered}
L=\tau^{-1}\sum_j\partial_{z_j}\partial_{y_j},\\
C=\exp L .
\end{gathered}
\tag{AS.14}
\]
is an actual finite sum on every polynomial in \(y\). Its inverse is \(\exp(-L)\); multiplication of these finite sums gives the identity by the binomial formula in each \(y\)-degree. No infinite analytic series or asymptotic argument is used. Since
\[
\begin{gathered}
{}[L,z_j\tau]=\partial_{y_j},\\
{}[L,\partial_{y_j}]=0 .
\end{gathered}
\tag{AS.15}
\]
we have
\[
 C^{-1}T_j C=z_j\tau .
 \tag{AS.16}
\]
Apply \(C\) to each coefficient of every relative exterior degree. This is an isomorphism of the entire localized Spencer complex with the Koszul complex for \(z_1\tau,\ldots,z_d\tau\).

These form a regular sequence. Successively quotienting by its first terms leaves the polynomial ring in the unremoved \(z\)'s, \(u,y\), and the invertible variable \(\tau\); the next \(z_j\tau\) is not a zero divisor. The elementary induction on the two-term multiplication complexes therefore gives
\[
\begin{gathered}
H^k(Q[\tau^{-1}])=0\quad(k\ne0),\\
H^0(Q[\tau^{-1}])\\
=\mathbb C[u,y,\tau,\tau^{-1}]\,\gamma .
\end{gathered}
\tag{AS.17}
\]
The induction proves exactness in all lower degrees, not just the top quotient. For \(d=0\), this statement is the identity of the empty Koszul complex.

#### AS.3. Every endpoint action and the exact diagonal map

Conjugation by \(C^{-1}\), followed by the top quotient in AS.17, gives
\[
\begin{gathered}
z_j=-\tau^{-1}\partial_{y_j},\\
D_{z_j}=y_j\tau,\quad D_w=\tau,\\
w=-\partial_\tau\\
+\tau^{-1}\left(\sum_j y_j\partial_{y_j}+d\right),\\
u_i=u_i,\quad D_{u_i}=\partial_{u_i}.
\end{gathered}
\tag{AS.18}
\]
Here are checks of the two less immediate formulas. First
\(C^{-1}z_jC=z_j-\tau^{-1}\partial_{y_j}\) and
\(C^{-1}(\partial_{z_j}+y_j\tau)C=y_j\tau\).
For the time coordinate put \(E_z=\sum z_j\partial_{z_j}\) and
\(E_y=\sum y_j\partial_{y_j}\). Then
\[
\begin{gathered}
{}[L,w]\\
=-\tau^{-1}(E_z+E_y+d)\\
-\tau^{-1}L,\\
{}[L,[L,w]]\\
=-2\tau^{-1}L,\\
{}[L,[L,[L,w]]]=0,\\
C^{-1}wC=-z^{\mathsf T}y-\partial_\tau\\
+\tau^{-1}(E_z+E_y+d).
\end{gathered}
\tag{AS.19}
\]
The terms with a left factor \(z_j\) vanish in the top quotient. This gives AS.18, including the constant \(+d\).

Let
\[
\begin{gathered}
\Delta_{\rm alg}=A/J_\Delta,\\
J_\Delta=Aw+\sum_jAz_j\\
+\sum_iAD_{u_i}.
\end{gathered}
\tag{AS.20}
\]
be the diagonal closed-transfer module, with generator \(e_\Delta\). Its normal form after localization is
\(\mathbb C[u,\tau,\tau^{-1},\nu_1,\ldots,\nu_d]e_\Delta\),
where \(\nu_j=D_{z_j}\), \(w=-\partial_\tau\), and
\(z_j=-\partial_{\nu_j}\). The map
\[
 I\bigl(y^\alpha\tau^m\gamma\bigr)
 =\tau^{m-|\alpha|-d}\nu^\alpha e_\Delta
 \tag{AS.21}
\]
is a bijection of vector spaces and commutes with **every** endpoint generator. For instance the two \(w\)-actions both multiply the displayed monomial by
\(|\alpha|+d-m\) and lower its \(\tau\)-power by one. The \(z_j\)-actions both give
\(-\alpha_j\tau^{m-|\alpha|-d}\nu^{\alpha-e_j}e_\Delta\).
The \(D_{z_j}\)-actions raise both \(m\) and \(|\alpha|\) by one. The remaining actions are immediate from AS.18. Thus AS.21 is an actual localized Weyl-module isomorphism.

There is also a diagonal map before localization. In \(H^0Q\), relative derivatives give
\[
\begin{gathered}
z_j\tau\gamma=0,\\
(1+z_jy_j\tau)\gamma=0 .
\end{gathered}
\tag{AS.22}
\]
The second relation means the relative derivative of \(y_je\), and the first that of \(e\). For \(d\ge1\), the class
\[
 a=\tau^d\gamma
 \tag{AS.23}
\]
is killed by every \(z_j\). Also
\[
\begin{gathered}
w\tau^d\gamma=-z^{\mathsf T}y\,\tau^d\gamma\\
-d\tau^{d-1}\gamma=0,\\
D_{u_i}a=0 .
\end{gathered}
\tag{AS.24}
\]
The first equality uses AS.9 and the Weyl commutator; summing the second relation in AS.22, multiplied by \(\tau^{d-1}\), proves its last zero. When \(d=0\), \(F=w\), \(Q=\Delta_{\rm alg}\) in degree zero, and \(a=\gamma\) satisfies the same relations without a negative power. Therefore
\[
\begin{gathered}
j:\Delta_{\rm alg}\longrightarrow H^0Q,\\
j(e_\Delta)=a .
\end{gathered}
\tag{AS.25}
\]
is a well-defined global differential-operator map in every dimension. Under AS.21 its localization is the identity on the distinguished diagonal generator. In particular \(a\ne0\).

The full endpoint relation \(D_u a=0\) is part of AS.25. It is the sum of the original endpoint derivatives at fixed normal difference, and not a relation omitting the simultaneous endpoint action.

An expression containing a middle variable \(y_j\) in this top quotient denotes the class of that displayed representative. Multiplication by \(y_j\) alone is not asserted to be an induced action on the quotient: it does not commute with \(T_j\). All endpoint operators used in AS.18 and AS.25 do commute with the relative differential.

#### AS.4. The opposite pole and the algebraic factorial

Retain the ordered relative density \(ds\wedge dy_1\wedge\cdots\wedge dy_d\); the surviving input density is held as an additional ordered coefficient block. The raw forward/opposite pair has poles \(f^{-1}\) and \(g^{-N}\). At fixed \(f,u,z,y\), \(D_{t'}=D_g=D_w\). Equation AS.2 and the determinant-one normal change in AS.1 give the exact convolution class
\[
\begin{gathered}
h=\\
\left[\frac{ds\wedge dy_1\wedge\cdots\wedge dy_d}
                    {f\,g^{d+1}}\right],\\
h\in H^0Q,\\
h=\frac{(-1)^d}{d!}\tau^d\gamma\\
=\frac{(-1)^d}{d!}\,j(e_\Delta).
\end{gathered}
\tag{AS.26}
\]
The first bracket denotes the ordered normal-localization/Spencer class, rather than an integral of a function through its poles. The \(f\)-normal operator residue is \(+1\) by AS.6. No factor \(2\pi i\) is inserted into this operator residue. Thus the algebraically normalized class is
\[
 (-1)^d d!\,h=a .
 \tag{AS.27}
\]
This is an algebraic normalization in the explicit source roof AS.28–AS.29 below. The conversion of its solution frame and source counit to a prescribed microlocal convolution counit still has its own orientation and period factors.



#### A free source direction and derivative-torsion cones

The relative support complex and its signs are OCM.2–OCM.4. The product homotopy comparison for pulled-back bounded coefficients is HI.1. All assertions below are local in the base and in a nonzero cotangent chart.

##### FD.0. Product internal Hom

Let \(p:T\times I\to T\) be projection from an open real interval. For bounded complexes \(\mathcal A,\mathcal B\) on \(T\), the actual pullback/evaluation map is an isomorphism
\[
\begin{gathered}
p^{-1}R\mathcal Hom(\mathcal A,\mathcal B)\\
\longrightarrow\\
R\mathcal Hom(p^{-1}\mathcal A,p^{-1}\mathcal B).
\end{gathered}
\tag{FD.1}
\]
To prove this, work on every product \(W\times J\), with \(J\) an open subinterval. Derived adjunction and HI.1 identify the right side's derived sections with
\[
\begin{gathered}
R\operatorname{Hom}_{W\times J}
       (p^{-1}\mathcal A,p^{-1}\mathcal B)\\
=R\operatorname{Hom}_{W}
       (\mathcal A,Rp_*p^{-1}\mathcal B)\\
=R\operatorname{Hom}_{W}(\mathcal A,\mathcal B).
\end{gathered}
\tag{FD.2}
\]
The left side has exactly these sections by the same product projection comparison. The map FD.1 is this identity: the adjoint of pulled-back evaluation is evaluation after the unit \(\mathcal B\to Rp_*p^{-1}\mathcal B\).

The internal Hom can be unbounded above. It is bounded below, since its first input is bounded above and its second is bounded below. HI.1 extends to bounded-below inputs by good truncation: the cone of \(\tau_{\le n}K\to K\) is in \(D^{\ge n+1}\), and a right derived direct image is left t-exact. Thus every fixed degree is already determined by a bounded truncation. The same argument applies to derived sections on \(W\). It justifies the preceding use of product comparison without a bounded-Ext or finite-rank assumption on the target. Product rectangles are a basis. Passing to stalks in all degrees proves FD.1.

Successively applying FD.1 handles several real free coordinates. In particular a complex independent of one complex coordinate is independent of each of its two real coordinates in the derived internal Hom calculation.

##### FD.1. One free source direction

Suppose \(M=\mathbb R_r\times B\) locally and
\(\mathcal A=\operatorname{pr}_B^{-1}\mathcal A_0\), with \(\mathcal A_0\) bounded. If the real covector \(p\) does not annihilate the free vector \(\partial_r\), then, for every bounded complex \(\mathcal F\),
\[
 \mu\operatorname{hom}(\mathcal A,\mathcal F)_p=0.
 \tag{FD.3}
\]
This includes a complex manifold with a free real direction in a complex coordinate. The microlocal projection shift is the real dimension of \(M\); it has no effect on a vanishing assertion.

Here is the full relative-support argument. In the pair \((u,v)\), the first coefficient is pulled back from \(u\), and the source is pulled back from the \(B\)-coordinates of \(v\). By FD.1,
\[
\begin{gathered}
R\mathcal Hom(q_2^{-1}\mathcal A,q_1^{-1}\mathcal F)\\
\text{is pulled back independently}\\
\text{of }v_r .
\end{gathered}
\tag{FD.4}
\]
Fix the central covector. Choose its sign and rescale the free vector so that the real polar functional is negative on that vector. Split the normal space as its line plus the central polar hyperplane. In coordinates \((a,b)\) for this split, the central positive polar functional is \(a\), while the \(a\)-line is the free source direction. The \(b\)-coordinates depend only on the quotient by that line. The coefficient FD.4 consequently remains independent of \(a\), even after this linear shear.

For a finite set of support representatives, strict polar separation and the compact unit normal-cone argument of IG.1 give a common later support
\[
\begin{gathered}
G_\eta=\{(a,b):a\le-\eta|b|\},\\
\eta>0,\\
D=U\times(-R,R)\times B_\rho,\\
H=D\cap(U\times G_\eta).
\end{gathered}
\tag{FD.5}
\]
Indeed every vector in the finitely many original normal cones has a uniformly negative central polar coordinate. Its ratio \((-a)/|b|\), where the denominator is nonzero, is bounded below by a positive number. Choose smaller \(\eta\), then shrink the actual pair domain. Failure of actual containment would give a normalized limiting normal-cone vector violating that bound. The coordinate change preserves the diagonal and the free direction.

The cone in FD.5 has a strict polar margin:
\[
\begin{gathered}
-a\ge\frac{\eta}{\sqrt{1+\eta^2}}|(a,b)|,\\
(a,b)\in G_\eta\setminus0 .
\end{gathered}
\tag{FD.6}
\]
A covector perturbation of norm less than half this positive constant retains half the margin, in these fixed coordinates. Thus one support is admissible on an actual open covector neighbourhood.

Fix \(c=R/2\). Keeping \(u,b\) fixed, the homotopy
\[
\begin{gathered}
h_s(u,a,b)\\
=(u,(1-s)a+sc,b),\\
0\le s\le1 .
\end{gathered}
\tag{FD.7}
\]
stays in \(D\). It also stays in \(D\setminus H\) when it starts there: both endpoints satisfy \(a>-\eta|b|\), and that fibre is an interval. The common section \(a=c\) belongs to both opens.

Resolve the pulled-back coefficient of FD.4 by the real de Rham resolution tensored with that coefficient, exactly as in OCM.8. Homotopy FD.7 gives \(K_A,K_B\) on the whole domain and its complement, with
\[
\begin{gathered}
dK+Kd=j\varepsilon-1,\\
\varepsilon j=1,\\
\operatorname{res}K_A=K_B\operatorname{res},\\
\operatorname{res}j_A=j_B,\\
\varepsilon_B\operatorname{res}=\varepsilon_A .
\end{gathered}
\tag{FD.8}
\]
For bounded-below coefficients apply this in each finite degree range. Only a bounded truncation of the coefficient enters a fixed range of the nonnegative de Rham resolution, so the same finite argument suffices. Scalar integration along compact paths is legitimate with arbitrary sheaf coefficients by OCM.2's finite local tensor expressions; there is no integration of an unspecified infinite-dimensional coefficient space.

For the relative differential \(d(a,b)=(da,\operatorname{res}a-db)\), define
\[
\begin{gathered}
T(a,b)=\\
\left(\begin{gathered}
j_A\varepsilon_Bb-K_Aa,\\
K_Bb
\end{gathered}\right).
\end{gathered}
\tag{FD.9}
\]
Expanding FD.8 gives \(dT+Td=1\), with exactly the signs of OCM.11. It contracts the entire relative complex, including its complement entry. It does not extend a complement section across its support.

R.6–R.7 transfers every finite cycle, relation and primitive to the injective support model on a common later domain. Exact filtered colimits then prove FD.3. The margin FD.6 proves sheaf vanishing on a covector neighbourhood, rather than only at one covector.

For a complex free coordinate \(w\), a nonzero complex \(dw\)-component gives a real free vector on which the real polar functional is nonzero. Therefore
\[
\begin{gathered}
\mathcal A=\mathbb C_{\mathbb C_w}\boxtimes\mathcal A_0\\
\Longrightarrow\\
\mu\operatorname{hom}(\mathcal A,\mathcal F)_p=0\\
\text{if }p(D_w)\ne0 .
\end{gathered}
\tag{FD.10}
\]

##### FD.2. Polynomial derivative torsion is an actual free factor

Let \(A=A_B\otimes_\mathbb C A_1\) be a polynomial Weyl algebra, with \(A_1=\mathbb C\langle w,\tau\rangle/([\tau,w]-1)\). Let \(M\) be a finitely generated left \(A\)-module, and suppose its polynomial Ore localization at \(\tau\) is zero. By AS.12, \(\tau\) acts locally nilpotently on every vector of \(M\). Put \(K=\ker(\tau:M\to M)\). Define, on each vector, the finite sum
\[
 \Pi(m)=\sum_{k\ge0}\frac{(-w)^k}{k!}\tau^k m .
 \tag{FD.11}
\]
The Weyl identity
\(\tau w^k=w^k\tau+kw^{k-1}\)
makes consecutive terms cancel, so \(\tau\Pi(m)=0\). Also \(\Pi(k)=k\) on \(K\). The exact finite Taylor identity is
\[
 m=\sum_{j\ge0}\frac{w^j}{j!}\Pi(\tau^jm).
 \tag{FD.12}
\]
For the coefficient of \(w^n\tau^n m\) in the right side, the finite binomial sum is
\(\sum_{j=0}^n(-1)^{n-j}/(j!(n-j)!)\), equal to one for \(n=0\) and zero otherwise. This proves FD.12 without a completion.

Multiplication induces an isomorphism
\[
 \mathbb C[w]\otimes_\mathbb C K\xrightarrow{\sim}M .
 \tag{FD.13}
\]
Surjectivity is FD.12. For injectivity, apply \(\Pi\tau^r\) to a finite relation \(\sum_{j=0}^r w^j k_j=0\). Its top coefficient gives \(r!k_r=0\). Descending on \(r\) gives every \(k_j=0\). The \(A_B\)-action commutes with \(w,\tau,\Pi\), so FD.13 is an \(A\)-module map; \(\tau\) becomes ordinary differentiation on the polynomial factor.

Moreover \(K\) is finitely generated over \(A_B\). Choose finitely many \(A\)-generators \(m_i\) of \(M\). Their Taylor coefficients \(\Pi\tau^j m_i\), with \(j\) below the respective nilpotence bounds, form a finite set. If \(K'\) is their \(A_B\)-span, then \(\mathbb C[w]\otimes K'\) is an \(A\)-submodule containing all \(m_i\); it is \(M\). The injectivity in FD.13 shows \(K'=K\). The polynomial Weyl algebra \(A_B\) is Noetherian by its good symbol filtration and polynomial Noetherianity, as proved in the differential-operator lesson, so this finite module is coherent.

Sheafifying on the affine space gives the genuine external product
\[
 M^\sim=\mathcal O_{\mathbb A^1_w}\boxtimes K^\sim .
 \tag{FD.14}
\]
No local nilpotence assertion is made after multiplication by arbitrary rational or analytic coefficient germs. The global polynomial Taylor calculation is what produces this external product.

If \(M^\sim\) is regular holonomic, \(K^\sim\) is regular holonomic. For the embedding \(i:B\to\mathbb A^1_w\times B\) at \(w=0\), the unshifted transfer inverse of FD.14 is exactly \(K^\sim\), by the transfer tensor with its locally free polynomial factor. In the programme's extraordinary convention this is \(i^!M^\sim[1]\). Algebraic regular inverse stability and shift stability therefore apply. The unshifted solution complex is
\[
\begin{gathered}
\operatorname{Sol}(M^\sim)\\
=\mathbb C_{\mathbb C_w}\boxtimes
                    \operatorname{Sol}(K^\sim).
\end{gathered}
\tag{FD.15}
\]
This map can be checked directly by the one-variable connection Spencer resolution, whose holomorphic derivative complex has kernel the constants and zero higher local cohomology, tensored with a finite operator resolution in the other coordinates. Equivalently it is the actual solution exterior comparison in the Riemann–Hilbert lesson. There is no relative dimension shift in unshifted solutions of this free connection factor.

Combining FD.10 and FD.15 proves
\[
\begin{gathered}
\mu\operatorname{hom}(\operatorname{Sol}M,\mathcal F)_p=0\\
\text{for }p(D_w)\ne0 .
\end{gathered}
\tag{FD.16}
\]
for every bounded target \(\mathcal F\). A bounded complex whose coherent cohomology modules have zero polynomial \(\tau\)-localization has the same property: its finitely many truncation triangles, the contravariant solution functor and exactness of microlocal Hom in its source give the assertion. This proves the precise vanishing for both actual comparison cones AS.28, with no appeal to whole-real scalar-change flatness or to general analytic proper-image regularity.



#### AS.5. The two comparison cones

The finite global-module assertion used here follows from the coherent-generator lemma in the adjunctions lesson. A coherent operator module on an affine space has a coherent structure-sheaf subsheaf generating it. That subsheaf is generated by finitely many global sections, by the affine module–sheaf correspondence. These same sections generate the operator module over its global polynomial Weyl algebra. Exact affine sections on quasi-coherent structure-sheaf modules compute both kernels and cohomology. This applies to every coherent cohomology sheaf of \(Q\).

Let \(\pi:Q\to H^0Q\) be the actual truncation map. The complex \(Q\) is in degrees \([-d,0]\), so this is the usual canonical map to its degree-zero cohomology object. The algebraic proper-coherence argument above makes all its cohomology modules finitely generated over \(A\). AS.17 and exactness of Ore localization show that every cohomology module of \(\operatorname{Cone}(\pi)\) is killed by localization. The same is true for the kernel and cokernel of AS.25, by AS.21. Hence both comparison cones have bounded coherent cohomology with locally nilpotent global \(\tau\)-action:
\[
\begin{gathered}
Q\xrightarrow{\pi}H^0Q\xleftarrow{j}\Delta_{\rm alg},\\
H^k\operatorname{Cone}(\pi)[\tau^{-1}]=0,\\
H^k\operatorname{Cone}(j)[\tau^{-1}]=0 .
\end{gathered}
\tag{AS.28}
\]
The derivative-torsion lemma FD.2 proves, without a microlocal proper-image theorem, that each such module is a polynomial free time factor tensored with a coherent module on the other coordinates. Its unshifted analytic solution complex is therefore pulled back independently of \(w\). The free-source-direction theorem then annihilates the microlocal Hom from that solution complex at every covector with nonzero \(dw\)-component, for every bounded target complex.

Write \(\operatorname{Sol}M=R\mathcal Hom_{\mathcal D^{an}}(M^{an},\mathcal O)\), unshifted. Finite truncation triangles for the two cones in AS.28 give isomorphisms
\[
\begin{gathered}
\mu\operatorname{hom}(\operatorname{Sol}Q,\mathcal F)_p\\
\Big\downarrow\\
\mu\operatorname{hom}(\operatorname{Sol}H^0Q,\mathcal F)_p\\
\Big\uparrow\\
\mu\operatorname{hom}(\operatorname{Sol}\Delta_{\rm alg},\mathcal F)_p,\\
p(D_w)\ne0 .
\end{gathered}
\tag{AS.29}
\]
These maps are induced by the two actual maps AS.28, with their contravariance accounted for twice. They are not chosen identifications of ranks. The closed normal Spencer calculation identifies
\(\operatorname{Sol}\Delta_{\rm alg}=\mathbb C_\Delta[-N]\)
in its specified ordered normal solution frame.

#### AS.6. Evaluation in this exact roof

A global degree-zero relative Spencer cocycle representing \(h\) gives a map \(b:\mathcal D_{\rm end}\to Q\) in the operator derived category. It exists directly in the affine relative Spencer model; the source global Weyl module is free of rank one. Its composite with \(\pi\) is the section \(h\) of \(H^0Q\). Write \(b_\Delta:\mathcal D_{\rm end}\to\Delta_{\rm alg}\) for \(1\mapsto e_\Delta\). The equality of actual module sections in AS.26 is
\[
 \pi b=\frac{(-1)^d}{d!}\,j b_\Delta .
 \tag{AS.30}
\]
Applying the contravariant solution functor gives equality of the corresponding ordinary evaluation maps. Applying the natural map from ordinary derived Hom to its microlocal Hom and using AS.29 preserves that equality. Consequently, under the precise roof AS.29, the ordinary convolution class \(h\) evaluates to
\[
\begin{gathered}
\operatorname{ev}(h)\\
=\frac{(-1)^d}{d!}\operatorname{ev}(e_\Delta).
\end{gathered}
\tag{AS.31}
\]
This includes the entire target coefficient complex: it is functoriality of the actual morphism AS.30, and not a comparison on a selected normal component.

In AS.31, \(\operatorname{ev}\) denotes ordinary solution evaluation followed by its natural microlocal Hom map and transport through AS.29. The right-hand class is the raw ordered diagonal evaluation, with no normalized period factor inserted.

AS.31 concerns the ordinary algebraic direct-image section and the source roof AS.28. The canonical microlocal composition comparison and intrinsic diagonal-unit normalization are proved in §5.67, using the specialization, exterior-product and proper-trace maps and the ordered normal period. Neither identity follows from AS.17 or AS.25 alone. The localization and source-roof statements above do not require faithful flatness of the whole real kernel ring.

#### AS.7. The opposite order and its separate spatial sign

Use endpoints \((s,y),(s',y')\), middle variables \((t,x)\), and put
\[
\begin{gathered}
u'=(s,y),\quad w'=s'-s,\\
z'=y'-y,\quad Y=-x,\\
a=t-s-y^{\mathsf T}Y,\\
b=s'-t+(y')^{\mathsf T}Y,\\
a+b=w'+(z')^{\mathsf T}Y .
\end{gathered}
\tag{AS.32}
\]
The polynomial reflection \(x=-Y\) extends to the projective middle compactification. Consequently every properness, coherence, normal-form and torsion-cone assertion above applies to this order as well. Let \(\tau'=D_{w'}\), and let \(\gamma'\) be AS.7 with the normal sum in AS.32 and the ordered \(Y\)-density. There is an actual map
\[
\begin{gathered}
j':\Delta'_{\rm alg}\longrightarrow H^0Q',\\
j'(e_{\Delta'})=(\tau')^d\gamma' .
\end{gathered}
\tag{AS.33}
\]
Both maps of its analogue of AS.28 have the vanishing comparison cones FD.16 on \(p(D_{w'})\ne0\).

The raw poles in this order are \(a^{-d-1}b^{-1}\): the high pole is now on the first normal coordinate. At fixed \(b\), changing the first coordinate to \((a,F'=a+b)\) replaces its derivative by \(D_a^{\rm new}+D_{F'}\). These commute. In their \(d\)-th power, every term with a positive \(D_a^{\rm new}\)-power has zero first-normal residue by AS.6. Only \(D_{F'}^d\) remains. Equation AS.2 therefore gives the same factor \((-1)^d/d!\) before the spatial reflection.

Write \(\eta_x=dt\wedge dx_1\wedge\cdots\wedge dx_d\) and \(\eta_Y=dt\wedge dY_1\wedge\cdots\wedge dY_d\) for the two ordered middle densities. Their relation is
\[
\eta_x=(-1)^d\eta_Y .
\tag{AS.34}
\]
Thus, with that actual density, the opposite raw class is
\[
\begin{gathered}
h'=\\
\left[\frac{dt\wedge dx_1\wedge\cdots\wedge dx_d}
                       {a^{d+1}b}\right],\\
h'\in H^0Q',\\
h'=\frac{1}{d!}(\tau')^d\gamma'
\\
=\frac1{d!}j'(e_{\Delta'}).
\end{gathered}
\tag{AS.35}
\]
In this formula the surviving input density remains in its original ordered block. The only coordinate reflection used is the displayed middle spatial block. The sign in AS.34 is independent of the positive first-normal operator residue. The opposite source roof therefore identifies its ordinary evaluation with \(1/d!\) times its own raw ordered diagonal evaluation. These two explicit algebraic factors do not silently fix the separate microlocal source-counit and normalized-period conversions described after AS.31.



![The finite conjugation of the entire Spencer complex, its exact diagonal class and the free source interval contraction](assets/affine-spencer-and-free-direction.png)

*Figure AS–FD.1. Panel A shows AS.14–AS.21, with every relative degree and the exact moment map. Panel B shows AS.22–AS.27 and AS.32–AS.35 before and after polynomial Ore localization; each factor is the algebraic pole/Spencer factor. Panel C is the real two-dimensional slice \(\eta=1\), \(R=2\), \(c=1\), \(b=1/2\) of FD.5–FD.9. The purple point starts at \(a=-1/4\) in the complement and remains in it; the red point starts at \(a=-3/2\) in the whole domain and is allowed to leave the support. Both paths end on the same section. Coefficients are fixed in the free direction. FD.9 contracts the full relative cone, rather than the closed support alone.*

Full PNG · SVG · Exact data · Drawing source


### 5.67. Projective incidence evaluations and real contact transport

The exact earlier support, coefficient, incidence and source prerequisites are §5.28, §5.34, §5.35, §5.43, §5.49, §5.56, §5.57, §5.58, §5.59, §5.60, §5.61, §5.62, §5.63, §5.64, §5.65. The algebraic finite-resolution, proper functor and signed evaluation proofs are in Holonomic D-modules and duality, Adjunctions, base change and the projection formula, The de Rham functor, and The Riemann–Hilbert correspondence. The affine Spencer reduction and free-direction proofs AS–FD supply the two algebraic source roofs used below.

#### Polynomial Ore descent and the two incidence source roofs

The algebra is the global polynomial Weyl algebra \(A\) on the affine endpoint space of AS.1. Thus \(u\in\mathbb A^N\), \(w\in\mathbb A^1\), \(z\in\mathbb A^d\), \(N=d+1\), and \(\tau=D_w\). Write \(A_\tau=A[\tau^{-1}]\) for the polynomial Ore localization proved in AS.11–AS.12. All complexes in this argument have bounded finitely generated \(A\)-cohomology. Finite operator-projective resolutions and their uniform finite length are the earlier holonomic-duality lesson, Lemma 3.0, and the adjunctions lesson, Lemma 4.6. Regularity is imposed only where the algebraic Riemann–Hilbert equivalence is used.

Here the global finite-projective version of that local resolution proof is needed. Let \(m\) be the dimension of the affine endpoint space. Its order symbol ring is the polynomial ring in \(2m\) variables. The global projective-dimension argument in the proof of Lemma 3.0 gives bound \(2m\) for finite symbol modules. Take \(2m\) successive strict filtered free presentations of a finite Weyl module. The terminal symbol syzygy is graded projective. Formula (3.3a) of that proof identifies it with \(S\otimes_B E\), where \(B\) is the polynomial base-coordinate ring and \(E\) is a finite graded projective \(B\)-module, in finitely many degrees. Lift \(E_k\) to the corresponding filtration term of the terminal operator syzygy; \(B\)-projectivity gives each lift. The resulting map \(A\otimes_B E\) to that syzygy has symbol isomorphism. Lowering finite filtration degree proves that it is an isomorphism. Since \(E\) is a summand of a finite free \(B\)-module, \(A\otimes_B E\) is a finite projective \(A\)-module. Thus the global resolution has finite projective terms and length at most \(2m\), with no assertion of global freeness. Bounded coherent complexes have bounded finite projective representatives: resolve their finitely many cohomology modules, represent each finite truncation's attaching map by a map from a bounded projective complex, and take its finite cone. The ordinary lifting homotopy for projectives proves that each cone represents that truncation.

##### OD.1. The derivative-torsion vanishing needs no regularity hypothesis

For every finitely generated \(A\)-module \(M\) with \(M_\tau=0\), the polynomial Taylor calculation FD.11–FD.14 gives
\[
\begin{gathered}
M=\mathbb C[w]\otimes K,\\
 K=\ker D_w,
\end{gathered}
\tag{OD.1}
\]
with \(K\) finitely generated over the polynomial Weyl algebra on \((u,z)\). This part of FD.2 does not use regularity or holonomicity.

The unshifted analytic solution comparison also holds without either hypothesis:
\[
 \operatorname{Sol}M
 =\mathbb C_{\mathbb C_w}\boxtimes\operatorname{Sol}K.
 \tag{OD.2}
\]
To verify it, take a bounded finite operator-projective resolution of \(K\), and the two-term connection resolution of \(\mathcal O_{\mathbb A^1}\) in the \(w\)-direction. Their exterior tensor resolves OD.1. After analytic Hom into \(\mathcal O\), the \(w\)-column is the holomorphic derivative complex. Its kernel consists of functions constant in \(w\), and its cokernel sheaf is zero: on a product polydisc integrate the convergent power series term by term in \(w\). This primitive commutes with every differential in the other resolution, since those entries involve only the other variables. The resulting finite double-complex contraction gives OD.2, including all its differential maps. Finite resolutions make both solution complexes bounded. No regular-holonomic exterior comparison is needed for this argument.

Let
\[
\begin{gathered}
\mathcal T_p(M)\\
=\mu\operatorname{hom}(\operatorname{Sol}M,\mathcal O)_p,\\
p(D_w)\ne0.
\end{gathered}
\tag{OD.3}
\]
A fixed ordered density coefficient may be tensored with the target. Sol is contravariant and microlocal Hom is contravariant in its first input, so \(\mathcal T_p\) is covariant and exact on triangles. FD.3, applied to OD.2, proves
\[
\begin{gathered}
M_\tau=0\\
\Longrightarrow\quad\mathcal T_p(M)=0.
\end{gathered}
\tag{OD.4}
\]
For a bounded complex, use its finite cohomological truncation triangles and exactness of polynomial Ore localization. Thus OD.4 holds for every bounded coherent derivative-torsion complex, including nonholonomic ones.

##### OD.2. Finite coherent lattices realize localized derived maps

We prove the precise descent statement used below. If a triangulated functor on bounded coherent \(A\)-complexes kills all complexes with zero \(\tau\)-localization, it assigns equal maps to two finite derived roofs whenever those roofs become equal after localization. Every map between localized bounded coherent \(A\)-complexes is represented by such a coherent roof.

Use bounded finite projective resolutions \(P,Q\) of the two input complexes. Their localizations are still bounded projective complexes, by right flatness of \(A_\tau\). A derived map between them is therefore a chain map
\[
 f:P_\tau\longrightarrow Q_\tau
 \tag{OD.5}
\]
up to chain homotopy. Regard \(P,Q\) as subcomplexes of their localizations; each projective term is a summand of a finite free module and has no denominator torsion. Put
\[
 P_f^i=\{v\in P^i:f_i(v)\in Q^i\}.
 \tag{OD.6}
\]
It is an \(A\)-submodule of the finite module \(P^i\), hence is finite by Noetherianity. Since \(f\) is a chain map, its differential preserves these submodules. Every \(v\in P^i\) has some \(\tau^k f_i(v)\in Q^i\). Also \(f_i(\tau^k v)=\tau^k f_i(v)\), so \((P_f)_\tau=P_\tau\). Thus
\[
 P\longleftarrow P_f\longrightarrow Q
 \tag{OD.7}
\]
is the required bounded coherent roof, with derivative-torsion comparison cone.

Suppose \(f_0-f_1=dH+Hd\) after localization. Intersect the finitely many lattice conditions in OD.6 and, in degree \(i\), impose also
\[
\begin{gathered}
H_i(v)\in Q^{i-1},\\
H_{i+1}(d_iv)\in Q^i.
\end{gathered}
\tag{OD.8}
\]
The resulting lattice is finite. It is a subcomplex: the first condition on \(d_iv\) is the second condition on \(v\), and the next condition is zero since \(d^2=0\). It still has the full localization, because the finite list of values on \(v\) and \(d_iv\) can be cleared by one further power of \(\tau\). On this subcomplex, \(H\) is an actual \(A\)-linear homotopy. Consequently a localized zero map becomes an actual zero map after one coherent roof whose comparison cone is derivative torsion.

Common intersections handle any finite list of maps, homotopies and roof comparisons. These intersections remain finite and have the same localization, by the preceding denominator argument. This proves composition independence and the asserted descent. It is a finite-lattice proof; it does not require one denominator bound for all vectors of a module.

In particular OD.3 descends to the full image of bounded coherent complexes in the localized derived category:
\[
\begin{gathered}
\overline{\mathcal T}_p:\\
D^b_{\mathrm{coh}}(A_\tau)\big|_{\mathrm{image}}\\
\longrightarrow D(\mathbb C).
\end{gathered}
\tag{OD.9}
\]
The notation restricts to objects coming from \(A\); essential surjectivity onto all localized objects is not needed.

##### OD.3. The diagonal endomorphisms in this localization

The localized diagonal module has the exact normal form AS.20–AS.21:
\[
\begin{gathered}
(\Delta_{\rm alg})_\tau\\
=\mathbb C[u,\tau,\tau^{-1},\\
 \nu_1,\ldots,\nu_d]e_\Delta.
\end{gathered}
\tag{OD.10}
\]
Here \(z_j=-\partial_{\nu_j}\), \(w=-\partial_\tau\), and \(D_{u_i}=\partial_{u_i}\). An \(A_\tau\)-endomorphism is determined by the image \(v\) of \(e_\Delta\). The defining relations force successively
\[
\begin{gathered}
\partial_{\nu_j}v=0,\quad\partial_\tau v=0,\\
\partial_{u_i}v=0.
\end{gathered}
\tag{OD.11}
\]
A polynomial in \(\nu,u\) and Laurent polynomial in \(\tau\) satisfying these equations is a complex constant. Conversely each constant defines an endomorphism. Therefore
\[
\begin{gathered}
\operatorname{End}_{A_\tau}((\Delta_{\rm alg})_\tau)=\mathbb C,\\
\operatorname{Aut}_{A_\tau}((\Delta_{\rm alg})_\tau)=\mathbb C^\times.
\end{gathered}
\tag{OD.12}
\]
Degree-zero derived endomorphisms of a module in degree zero are its ordinary module endomorphisms, as follows by taking a projective resolution and the degree-zero Hom cohomology. Hence OD.12 is also the degree-zero derived assertion.

This is a theorem about this explicitly described polynomial-localized diagonal module. It is not a centralizer assertion about the whole real kernel ring.

##### OD.4. The projective boundary source has an actual free time factor

Let the middle projective coordinates be
\([\lambda:s_h:y_h]\in\mathbb P^N\). On affine endpoint charts the two closed incidence equations are
\[
\begin{gathered}
 s_h-t\lambda-x^{\mathsf T}y_h=0,\\
 t'\lambda-s_h+(x')^{\mathsf T}y_h=0 .
\end{gathered}
\tag{OD.13}
\]
Let \(\overline W\) be their common zero set, \(j\) its part with \(\lambda\ne0\), and \(i\) its part at \(\lambda=0\). The ordinary sheaf localization triangle is
\[
\begin{gathered}
j_!\mathbb C_{W_{\rm aff}}\longrightarrow\mathbb C_{\overline W}\\
\longrightarrow i_*\mathbb C_{W_\infty}\longrightarrow.
\end{gathered}
\tag{OD.14}
\]
The notation \(j,i\) includes the corresponding inclusions into the common projective-middle ambient space.

At \(\lambda=0\) both equations are independent of \(t,t'\). Consequently
\[
 W_\infty
 =\mathbb A^1_t\times\mathbb A^1_{t'}
          \times W_{\infty,\rm sp}.
 \tag{OD.15}
\]
The projection of the last factor to \((x,x')\) is proper. Proper-support base change and the actual exterior proper map, proved in the classical sheaf provider and RH Lemma 1.21, identify the pushforward of its constant complex with
\(\mathbb C_{\mathbb C_t\times\mathbb C_{t'}}\boxtimes
 Rp_{\rm sp*}\mathbb C_{W_{\infty,\rm sp}}\).
This is the ordinary exterior map of sections with proper support. No assertion about direct image commuting with microlocalization is used.

The linear change \((t,t')\mapsto(t,w=t'-t)\) makes this source independent of \(w\). The full algebraic RH equivalence and its actual exterior comparison identify its operator preimage with a free \(\mathcal O_{\mathbb A^1_w}\)-factor, with the appropriate common complex shift. Its global polynomial derivative action is locally nilpotent. Thus the operator comparison cone supplied by OD.14 is zero after polynomial \(\tau\)-localization. Equivalently, FD.3 already proves that its solution-source microlocal Hom vanishes on \(p(D_w)\ne0\).

##### OD.5. Constant-source cones in the projective inverse roofs

The actual source roofs SN.5–SN.8 have locally constant finite cone cohomology on the entire affine endpoint pair. Those computations used the full affine and projective compact-fibre calculations of LI and SN, including the positive normal boundary map. On \(\mathbb C^{2N}\), a finite local system is constant: parallel transport along the straight radial contraction identifies it with its fibre, and homotopy invariance makes that transport independent of a path.

A bounded complex with such cohomology splits into a finite complex of constant vector spaces in the derived category. To justify the splitting rather than assume it, the derived Hom between two finite constant sheaves is the constant matrix complex's global cohomology. The radial de Rham contraction computes \(H^k(\mathbb C^{2N};\mathbb C)=0\) for \(k>0\). The successive extension obstructions in the finite truncation triangles therefore vanish. This proves the claimed finite splitting.

Under algebraic RH, a shifted finite constant source is the solution complex of a shifted finite direct sum of the trivial connection. Such a connection has locally nilpotent global \(D_w\)-action. Full faithfulness then gives the same conclusion for the operator preimage of each SN comparison cone. Hence every map of those operator source roofs becomes invertible after polynomial \(\tau\)-localization.

The proper solution comparison for AS.5 identifies its source with
\(Rp_!j_!\mathbb C_{W_{\rm aff}}[N-2]\). Shift OD.14 by \(N-2\), use OD.4, and then the SN roofs shifted by \(-N\). Algebraic RH therefore gives an actual bounded coherent operator roof between \(Q\) and \(\Delta_{\rm alg}\), all of whose comparison cones vanish after localization. This roof is distinct from the explicit AS.28 roof, but both become isomorphisms to the same localized diagonal module.

Combining them, their ratio is an invertible degree-zero endomorphism of that module. By OD.12 it is one nonzero complex constant. OD.2 then proves that, after applying the actual solution/microlocal-Hom evaluation functor, the two source comparisons differ by precisely that same constant. This conclusion does not use faithful flatness of the whole real scalar extension.

#### The projective boundary and the proper incidence covector correspondence

Use \(N=d+1\), the affine endpoint and middle coordinates of AS.1, and the input antipodal convention of OCM.3. The free-source-direction vanishing FD.3 is valid for every bounded target complex. Complex covectors are paired bilinearly, with the real polar functional obtained by taking the real part.

##### PC.1. Boundary vanishing in an actual free direction

Write the projective middle point as \([\lambda:s_h:y_h]\). The affine middle is \(\lambda\ne0\). At a point of the first incidence at infinity, \(y_h\ne0\): if both \(\lambda\) and \(y_h\) vanished, its equation would force \(s_h=0\), which is not a projective point. Choose a coordinate chart \(y_{h,j}=1\). The remaining \(y_h\)-coordinates, \(\lambda\), \(s_h\), \(t\) and \(x\) are ordinary holomorphic coordinates there. Put
\[
 r=s_h-t\lambda-x^{\mathsf T}y_h .
 \tag{PC.1}
\]
Replacing \(s_h\) by \(r\) is an invertible holomorphic coordinate change with derivative one in \(s_h\).

The closed incidence constant complex, its affine extension by zero, and the open incidence-complement constant complex are, respectively, the corresponding complexes for
\[
\begin{gathered}
\{r=0\},\\
\{r=0,\ \lambda\ne0\},\quad\{r\ne0\}.
\end{gathered}
\tag{PC.2}
\]
They are pulled back independently of \(t\) in these coordinates. The same holds for the ordinary derived direct extension of the complement: its defining open inclusion is a product in \(t\). On product rectangles, product projection and open direct image commute by the product coefficient homotopy HI.1, applied to the open rectangle and its finite restriction complex. Thus the ordinary derived extension is again pulled back from the other coordinates.

The free coordinate vector, in the original chart, is
\[
 V_t=\partial_t+\lambda\partial_{s_h}.
 \tag{PC.3}
\]
It annihilates \(r\) and fixes the other new coordinates. At \(\lambda=0\) it is exactly \(\partial_t\). Therefore FD.3 proves vanishing of microlocal Hom from each of these source complexes, with any common shift, at every boundary covector whose endpoint \(dt\)-component is nonzero. This is vanishing on an actual covector neighbourhood: the condition that a covector does not annihilate \(V_t\) is open.

For the opposite incidence use
\[
\begin{gathered}
r'=s_h-t'\lambda-(x')^{\mathsf T}y_h,\\
V_{t'}=\partial_{t'}+\lambda\partial_{s_h}.
\end{gathered}
\tag{PC.4}
\]
Its equation is \(-r'=0\). The same coordinate argument proves the opposite boundary assertion at nonzero \(dt'\)-component. These are statements about the actual source complexes and their full coefficient Hom complexes, not a choice of a contour representative.

##### PC.2. The affine conormal graph, with both antipodal signs

In the affine chart let
\[
 f=s-t-x^{\mathsf T}y .
 \tag{PC.5}
\]
At a nonzero conormal covector, write its scalar as \(\alpha\). The ordinary covector on the pair \((X,Y)\) is
\[
 \alpha\,df
 =(-\alpha,-\alpha y;\ \alpha,-\alpha x).
 \tag{PC.6}
\]
Applying the input antipodal map to the last block gives the correspondence
\[
\begin{gathered}
(t,x;\tau,\xi)\longleftrightarrow(s,y;\sigma,\eta),\\
\tau=-\alpha,\quad\xi=\tau y,\\
\sigma=\tau,\quad\eta=-\tau x,\\
s=t+x^{\mathsf T}y.
\end{gathered}
\tag{PC.7}
\]
On the open time-covector chart this is the single holomorphic graph
\[
\begin{gathered}
\chi(t,x;\tau,\xi)\\
=\bigl(t+x^{\mathsf T}(\xi/\tau),\ \xi/\tau;\\
\tau,-\tau x\bigr),\quad\tau\ne0.
\end{gathered}
\tag{PC.8}
\]
Its inverse is
\[
\begin{gathered}
x=-\eta/\sigma,\\
t=s+\eta^{\mathsf T}y/\sigma.
\end{gathered}
\tag{PC.9}
\]
The one-form calculation retains the sign:
\[
\begin{gathered}
\sigma\,ds+\eta^{\mathsf T}dy\\
=\tau\,dt+\xi^{\mathsf T}dx.
\end{gathered}
\tag{PC.10}
\]
Indeed \(ds=dt+x^{\mathsf T}dy+y^{\mathsf T}dx\), and the two terms containing \(x^{\mathsf T}dy\) cancel. Thus these are the actual homogeneous symplectic correspondence and its inverse.

For completeness, the microlocal Hom sources used here have no additional affine covectors. Off the closed incidence, the source is locally constant or zero, so OCM.A applies. On it, any covector not proportional to \(df\) detects a complex tangent vector to \(f=0\). In straight coordinates that vector is a free source direction, and FD.3 applies. The same argument covers the open and ordinary-derived-open source extensions, since they depend only on the one straight normal coordinate. This proves their affine support containment in the conormal graph for every bounded target coefficient.

##### PC.3. Composition and properness over compact cotangent sets

For the second incidence in the forward product put
\[
 g=t'-s+(x')^{\mathsf T}y .
 \tag{PC.11}
\]
Its conormal, in the order \((s,y;t',x')\), is
\[
 \beta\,dg=(-\beta,\beta x';\ \beta,\beta y).
 \tag{PC.12}
\]
The common middle blocks in PC.6 and PC.12 must add to zero; this is exactly the input antipodal matching. Consequently
\[
\begin{gathered}
\alpha-\beta=0,\\
-\alpha x+\beta x'=0.
\end{gathered}
\tag{PC.13}
\]
On \(\tau=-\alpha\ne0\), these imply \(x'=x\). The two incidence equations then give \(t'=t\). The remaining endpoint covectors are the opposite pair
\[
 (-\alpha,-\alpha y;\ \alpha,\alpha y).
 \tag{PC.14}
\]
Thus the endpoint correspondence is the conormal diagonal, with the same positive identity map on the output cotangent coordinates.

The middle point and both its covector blocks are determined uniquely by
\[
\begin{gathered}
y=\xi/\tau,\quad\alpha=-\tau,\\
s=t+x^{\mathsf T}(\xi/\tau).
\end{gathered}
\tag{PC.15}
\]
Let \(K\) be a compact set in the endpoint cotangent chart \(\tau\ne0\). The continuous function \(|\tau|\) has a positive lower bound on \(K\). The endpoint base coordinates and all endpoint covector coordinates have finite upper bounds there. PC.15 therefore bounds \(y,s,\alpha\) and the middle spatial covectors \(\alpha x\). It also bounds all other middle coordinates. The relation is closed in this chart: it is the graph of these continuous formulas over the closed diagonal conditions PC.13–PC.14. Its inverse image over \(K\) is hence a closed subset of a compact finite-dimensional box, and is compact.

There are no boundary points with the required nonzero endpoint time component, by PC.1. Nor can a compact endpoint cotangent set acquire an escaping sequence of middle covectors near infinity: the affine formula \(y=\xi/\tau\) stays bounded, so its projective middle point remains in a compact subset of the affine chart, and the preceding covector bounds apply. This verifies properness of the full selected cotangent support intersection, including the possible boundary limit.

For a prescribed finite middle-coordinate neighbourhood, first choose a smaller cotangent neighbourhood of the central endpoint point. Continuity of PC.15 then places the whole matching middle point and covector in a relatively compact subneighbourhood. This is the actual local support condition for composing the two incidence germs.

The reverse order is obtained from the inverse graph PC.9. The same argument, using \(\sigma\ne0\), proves its properness and diagonal endpoint correspondence. These proofs concern the source microlocal Hom support, so they do not presume that the explicit RC contour family already represents the canonical composition.

#### Ordinary proper evaluation for the incidence convolution

Use unshifted solutions
\[
\begin{gathered}
S_Z(M)\\
=R\mathcal Hom_{\mathcal D_Z^{an}}(M^{an},\mathcal O_{Z^{an}}).
\end{gathered}
\tag{PE.1}
\]
This differs from the normalized solution functor of the de Rham lesson by the shift \([-\dim Z]\). Write \(F_Z=\operatorname{DR}_Z\), \(\mathbb D_Z\) for operator duality, and \(s_Z:F_Z\mathbb D_ZM\to S_Z(M)[\dim Z]\) for the actual finite-projective/right-Spencer map (1.vad) of the Riemann–Hilbert lesson. Its connection-frame coefficient is not silently replaced by an identity.

Let \(B=X\times Z\), \(T=X\times Y\times Z\), with all three factors of dimension \(N\), and let \(p:T\to B\) be projection. In the proper realization of AS.5, \(Y=\mathbb P^N\). The two incidence operator modules and their affine open-direct extensions are regular holonomic; their pulled-back derived structure tensor is denoted \(M\), and \(Q=p_+M\). The dimensions are
\[
\begin{gathered}
\dim T=3N,\quad\dim B=2N,\\
\dim T-\dim B=N.
\end{gathered}
\tag{PE.2}
\]

##### PE.1. The actual solution-source comparison

The unshifted smooth transfer pullback is the programme's extraordinary inverse shifted by the negative relative dimension. Apply actual operator duality, the actual inverse comparisons of RH1.20, and the actual right-Spencer solution maps. They give
\[
 S_T(q_{ij}^+M_{ij})
 \simeq q_{ij}^{-1}S_{ij}(M_{ij}).
 \tag{PE.3}
\]
The raw structure tensor is the extraordinary tensor shifted by \(\dim T\), by RH1.vv. Dualizing it gives the ordinary tensor of the two duals shifted by \(-\dim T\). RH1.vx, with its actual tensor-Hom evaluations, and the two unshifted solution shifts therefore identify
\[
\begin{gathered}
S_T(M)\\
\simeq q_{12}^{-1}L_{12}\otimes q_{23}^{-1}L_{23},\\
 L_{ij}=S_{ij}(M_{ij}).
\end{gathered}
\tag{PE.4}
\]
The two copies of \(-\dim T\) are precisely the two unshifted solution shifts on the right. The tensor suspension signs are the maps RH1.vb and V.14; both sides use those maps. This is not a choice of flat connection frames.

For the proper projection, the inverse of the following sequence defines the operator-normalized solution comparison:
\[
\begin{gathered}
S_B(Q)\\
\xleftarrow{s_B}F_B\mathbb D_B(p_+M)[-2N]\\
\xleftarrow{\rho_p}F_Bp_+(\mathbb D_TM)[-2N]\\
\xrightarrow{\eta_p}Rp_*F_T(\mathbb D_TM)[-2N]\\
\xrightarrow{s_T}Rp_*S_T(M)[N].
\end{gathered}
\tag{PE.5}
\]
Here the arrow labelled \(\rho_p\) means the actual proper-duality isomorphism in the displayed direction, inside \(F_B\); \(\eta_p\) is the actual proper de Rham comparison. Each is invertible by the earlier programme theorems. Properness changes \(Rp_*\) to \(Rp_!\). Combining PE.4–PE.5 gives
\[
\begin{gathered}
\theta_{\rm op}:\\
Rp_!(L_{12}\otimes L_{23})[N]\\
\xrightarrow{\sim}S_B(Q).
\end{gathered}
\tag{PE.6}
\]
For the actual affine incidence extensions,
\[
 L_{12}\otimes L_{23}
       =j_!\mathbb C_{W_{\rm aff}}[-2].
 \tag{PE.7}
\]
Indeed ordinary operator open direct image has unshifted solution open extension by zero, by RH1.20; the two pullbacks have the same affine middle open. Tensor over \(\mathbb C\) is exact and sends the two closed incidence constant sheaves to the constant sheaf of their intersection. Therefore PE.6 is the source comparison used in OD.5, with total shift \(N-2\).

##### PE.2. Evaluation and the two trace normalizations

Let \(m_{12}\), \(m_{23}\) be actual operator sections, with the ordered middle density included in \(m_{12}\), and the surviving input density included in \(m_{23}\). Their relative top-Spencer section determines
\[
 b:\mathcal D_B\longrightarrow Q .
 \tag{PE.8}
\]
The ordinary solution evaluations are the solution maps of the corresponding operator section maps. They may be derived ordinary sheaf morphisms, rather than maps between degree-zero sheaves. In particular the closed incidence source is \(\mathbb C_S[-1]\).

The raw proper holomorphic trace is the current/form trace of V.11. Its sign in unshifted antiholomorphic degree \(q\) is
\[
 \operatorname{Tr}_q
 =(-1)^{N(q-N)} I_q .
 \tag{PE.9}
\]
Use the unshifted coefficient cup first and shift the intermediate trace block afterwards, as in V.13–V.14. The operator-normalized trace is
\[
 q_N\operatorname{Tr},
 \qquad q_N=(2\pi i)^{-N}.
 \tag{PE.10}
\]
The normalization follows from the actual proper residue calculation RH1.vl: the positive ordered period is \((2\pi i)^N\), and \(q_T(2\pi i)^N=q_B\). It includes the normal determinant and Cech/form signs in RH1.vf–vg. No sign is inserted after this trace is fixed.

With PE.6, the ordinary evaluation square is
\[
\begin{gathered}
E=\operatorname{ev}_{m_{12}}\otimes\operatorname{ev}_{m_{23}},\\
\operatorname{ev}_b\,\theta_{\rm op}\\
=(q_N\operatorname{Tr})\,Rp_!(E)[N],\\
Rp_!(L_{12}\otimes L_{23})[N]\\
\longrightarrow\Omega_Z.
\end{gathered}
\tag{PE.11}
\]
Densities here are the relative density lines on their stated pair spaces, pulled back to the triple product. The surviving \(\Omega_Z\)-block stays in its prescribed order. Thus PE.11 does not integrate a differential form in a different coefficient convention.

Here is why this is an equation of actual maps. Before taking any inverse in PE.5, the proper-duality map is finite operator evaluation followed by its operator trace. The de Rham and solution maps are the coefficient restriction, relative/right Spencer augmentation, density cancellation and finite-projective evaluation. RH1.vb gives their homogeneous evaluation sign; RH1.vad gives the right-Spencer map with coefficient \(+1\) on its top column. RH1.vk, proved on these maps, says that this evaluation followed by the proper operator trace equals the sheaf evaluation followed by its positive proper trace. Its projective calculation identifies the latter with PE.10 by RH1.vl. RH1.vu-a and RH1.vaa preserve the actual exterior, structure-tensor and evaluation maps, so inserting the two individual section evaluations gives exactly the map \(Rp_!(E)[N]\) of PE.11. Naturality of the finite evaluation gives the map \(\operatorname{ev}_b\), Sol of PE.8.

This argument uses the whole-complex equality proved in RH1.vl-a–b, with finite induced approximations and its uniform degree bound. It is not an inference of equality on a bounded complex from equality on its cohomology modules. The maps from the free operator module in PE.8 are legitimate: the chain evaluation formula is defined on coherent projective/induced terms before regular-holonomic invertibility is asserted. Regularity is used to invert the comparison maps for \(M,Q\), not to regard the free operator module as holonomic.

If the raw trace is used in PE.11, multiply only its source comparison by the reciprocal trace scalar:
\[
 \theta_{\rm raw}=(2\pi i)^N\theta_{\rm op}.
 \tag{PE.12}
\]
Linearity of the actual trace and equality PE.11 then give
\[
 \begin{gathered}
 \operatorname{ev}_b\,\theta_{\rm raw}\\
 =\operatorname{Tr}\,
 Rp_!(\operatorname{ev}_{m_{12}}\\
                  \otimes\operatorname{ev}_{m_{23}})[N].
 \end{gathered}
 \tag{PE.13}
\]
The map \(\theta_{\rm raw}\) is still an isomorphism. This statement fixes its normalization without declaring it to be an identity in flat endpoint frames.

All maps in PE.5–PE.13 are ordinary derived sheaf or operator maps on the proper compactification. Applying the natural ordinary-Hom-to-microlocal-Hom map preserves their equality. A further compatibility assertion is required to identify that image with the product of the two already microlocalized classes. That assertion concerns the actual exterior, middle-diagonal and proper microlocal maps. It is separate from the ordinary evaluation square proved here.

#### Finite globalization of the incidence pole sections

Let \(N=d+1\), with the affine coordinates and ordered densities of AS.1. Compactify the middle \((s,y)\)-space to \(\mathbb P^N\), with homogeneous coordinates \([\lambda:s_h:y_h]\). The endpoint coordinates remain affine. Let
\[
\begin{gathered}
 F_h=s_h-t\lambda-x^{\mathsf T}y_h,\\
 G_h=t'\lambda-s_h+(x')^{\mathsf T}y_h .
\end{gathered}
\tag{GP.1}
\]
Their zero sets are the two closed projective incidence hypersurfaces. Both are smooth. At any projective incidence point the vector \((\lambda,y_h)\) is nonzero, since its vanishing would force \(s_h=0\). Hence the gradient in the corresponding affine endpoint variables is nonzero.

##### GP.1. The exact global meromorphic forms

Use the ordered homogeneous coordinate vector \(Z=(\lambda,s_h,y_{h,1},\ldots,y_{h,d})\) and the homogeneous \(N\)-form
\[
\begin{gathered}
\Omega_{\rm hom}\\
=\sum_{j=0}^{N}(-1)^jZ_j\,\\
{}\cdot dZ_0\wedge\cdots\widehat{dZ_j}\cdots\wedge dZ_N.
\end{gathered}
\tag{GP.2}
\]
It has homogeneous weight \(N+1\) and its contraction with the Euler vector field is zero. Dividing by \(F_h^{N+1}\) therefore gives an actual meromorphic top form on the projective middle space. It has poles only on \(F_h=0\), and its restriction to \(\lambda=1\) is
\[
 \begin{gathered}
 \frac{\Omega_{\rm hom}}{F_h^{N+1}}
 \bigg|_{\lambda=1}\\
 {}=
 \frac{ds\wedge dy_1\wedge\cdots\wedge dy_d}
      {(s-t-x^{\mathsf T}y)^{N+1}} .
 \end{gathered}
 \tag{GP.3}
\]
To check descent directly, replace the homogeneous representative \(Z\) by \(cZ\). The numerator and denominator both acquire \(c^{N+1}\). A derivative of a variable scaling factor contributes an Euler contraction and is zero. On \(\lambda=1\) only the \(j=0\) term survives, with coefficient \(+1\). This proves both gluing and the ordered affine sign.

Write \(\eta_Y=ds\wedge dy_1\wedge\cdots\wedge dy_d\). At fixed actual middle coordinates,
\[
 D_t^N\left(\frac{\eta_Y}{f}\right)
       =N!\frac{\eta_Y}{f^{N+1}} .
 \tag{GP.4}
\]
Here \(D_tf=-1\), so all \(N\) signs are positive. The section
\[
 m_{+,N}
   =N!\left[\frac{\Omega_{\rm hom}}{F_h^{N+1}}\right]
 \tag{GP.5}
\]
is consequently a genuine global section of the **closed** incidence module, with its ordered relative density. It needs no open-direct extension across the middle hyperplane at infinity. The bracket denotes the local-cohomology/closed-transfer class. All its normal pole orders are finite.

The opposite section already globalizes without differentiation:
\[
\begin{gathered}
m_-=\left[\frac{\lambda^N}{G_h^N}\right]\eta_{X'},\\
\eta_{X'}=dt'\wedge dx'_1\wedge\cdots\wedge dx'_d.
\end{gathered}
\tag{GP.6}
\]
The coefficient is a homogeneous rational function of degree zero, with poles only on \(G_h=0\). Its affine restriction is precisely \(\eta_{X'}/g^N\). Thus its zero or vanishing order along \(\lambda=0\) is not replaced by a pole there.

These are statements about global sections of the closed operator modules and their density lines. Through ordinary solution evaluation they give ordinary derived sheaf morphisms from the two closed incidence solution complexes to the respective holomorphic density sheaves. Their microlocal images are the finite differential-operator images of the actual pole classes.

##### GP.2. Transversality for the common middle diagonal

Before identifying the two middle points, take the product of the two smooth closed incidence hypersurfaces on the four-factor space. The common middle diagonal is a smooth embedding of complex codimension \(N\). Its intersection with that product is the common incidence zero set in the three-factor space.

The two incidence equations have independent differentials on that intersection. If
\[
 a\,dF_h+b\,dG_h=0,
 \tag{GP.7}
\]
differentiate in the first endpoint \((t,x)\). It gives \(a(\lambda,y_h)=0\), hence \(a=0\). Differentiation in the other endpoint then gives \(b=0\). This proof also applies at \(\lambda=0\). Thus the common middle diagonal is transverse to the smooth closed source support everywhere.

The local coorientation comparison can be made in simultaneous coordinates. Let \(h\in\mathbb C^N\) be the difference of the two middle coordinate vectors in a projective chart. Transversality just proved permits coordinates consisting of \(h\), the two incidence normal equations, and tangent coordinates; the derivative matrix has full rank. The elementary holomorphic inverse-coordinate construction used in HI and FIM supplies the corresponding chart. In these coordinates the closed source constant complex is independent of \(h\), and the middle diagonal is \(h=0\). Therefore the actual fundamental-class map
\[
 j^{-1}A\longrightarrow
       \operatorname{or}_h[2N]\otimes j^!A
 \tag{GP.8}
\]
is the positive normal Thom isomorphism. It has coefficient \(+1\) when the real \(h\)-coordinates have their complex orientation: the two-term real relative-normal complexes are exactly HI.11–HI.15, with trace one. Their tensor normal comparison retains every degree; the real shift is \(2N\), not \(N\). It is not a product of two currents with the same singular middle variable.

For the two unshifted incidence solution sources, \(A\) here is their exterior product, with its common shift \([-2]\). That shift passes unchanged through GP.8.

##### GP.3. The global and affine operator convolution sections

Let \(\overline Q\) be the proper convolution of the two closed projective incidence modules. There is a natural operator map
\[
 \overline Q\longrightarrow Q ,
 \tag{GP.9}
\]
obtained by ordinary localization across the common middle hyperplane and the actual direct-image composition. Its solution map is the affine/projective source map OD.14 after the source comparison PE.6. The comparison cone has zero polynomial \(\tau\)-localization, by OD.4.

Let \(h_N\) be the actual relative top-Spencer section of \(\overline Q\) obtained by convolving GP.5 and GP.6. Under GP.9 its image is
\[
 h_N\longmapsto D_t^N h .
 \tag{GP.10}
\]
Indeed differentiation in this endpoint commutes with the proper operator direct image and acts on the first factor; it has zero action on the second factor at fixed original endpoint and middle coordinates. This is a finite differential identity of the displayed section maps, not an exchange with an infinite series or a nonproper contour.

In the affine AS coordinates, the original first endpoint derivative is
\[
 D_t=D_{u_t}-D_w .
 \tag{GP.11}
\]
The class \(h\) is killed by \(D_{u_t}\), by AS.24–AS.26. Since this derivative commutes with \(\tau=D_w\), GP.10 gives, after the AS diagonal identification,
\[
 D_t^N h=(-\tau)^N h .
 \tag{GP.12}
\]
Thus the globalized product section is the same nonzero scalar diagonal class as AS, multiplied by the invertible finite operator \(D_t^N\) on the selected time-covector chart.

On that chart the actual finite microdifferential operator \(D_t\) is invertible. The construction and the injective finite-operator action were proved in the ordinary symbol/current interface and FIM. The microlocal class of the original pole therefore satisfies the literal action identity
\[
 a=D_t^{-N}a_N ,
 \tag{GP.13}
\]
where \(a_N\) is the image of the global ordinary evaluation of GP.5. This equality uses the actual inverse operator, not a formal symbol or a scalar-change flatness claim.

Accordingly the ordinary-to-microlocal composition comparison can first be proved for the globally closed, transverse sources of GP.5–GP.8. The original finite-affine pole class is then recovered by GP.13 and the already proved action/composition compatibility. The opposite spatial sign and the dimension-correct opposite pole remain those of AS.32–AS.35.

##### GP.4. Globalization for the reverse product

For endpoints \((s,y),(s',y')\) and middle \((t,x)\), use homogeneous middle coordinates \([\lambda:t_h:x_h]\). Set
\[
\begin{gathered}
A_h=t_h-s\lambda+y^{\mathsf T}x_h,\\
B_h=s'\lambda-t_h-(y')^{\mathsf T}x_h .
\end{gathered}
\tag{GP.14}
\]
The first raw class has pole \(g^{-N}\) and the middle input density \(\eta_X\). One endpoint derivative suffices:
\[
D_s\left(\frac{\eta_X}{g^N}\right)
=N\frac{\eta_X}{g^{N+1}} .
\tag{GP.15}
\]
It globalizes to \(N[\Omega_{\rm hom}'/A_h^{N+1}]\), with the homogeneous form defined in GP.2 in the ordered coordinates \((\lambda,t_h,x_h)\). The second raw class globalizes without differentiation to
\[
\left[\frac{\lambda}{B_h}\right]\eta_{Y'} .
\tag{GP.16}
\]
Both are global sections of the two closed projective incidence modules. The same endpoint-gradient proof as GP.7 proves transversality of their common middle diagonal, including infinity.

The reverse proper convolution of these global sections maps, in the affine model, to \(D_s h'\). Here \(D_s=D_{u'_s}-D_{w'}\) and \(D_{u'_s}h'=0\), so the image is \(-\tau'h'\). Since \(D_s\) is an actual finite microdifferential unit on the selected reverse chart,
\[
b=D_s^{-1}b_1,\qquad b_1=D_sb .
\tag{GP.17}
\]
Thus the reverse canonical product can also be computed with global closed ordinary evaluations first, then recovered by an actual inverse finite operator. The globalizing constants are exactly \(N!\) in GP.5 and \(N\) in GP.15.

#### Ordinary evaluations and the canonical microlocal composition

The coefficient convention is OCM.4. If \(M\) is a complex manifold of dimension \(m\), write
\[
\begin{gathered}
\mathcal H_M(F,G)\\
=R\mathcal Hom(q_2^{-1}F,\\
q_1^{-1}G[2m]).
\end{gathered}
\tag{MC.1}
\]
All supports, restrictions and complement complexes are the actual ones of OCM.2–OCM.3 and R.6–R.7. The following comparison concerns ordinary derived morphisms with the closed transverse sources constructed in GP. It does not assert a general microlocal proper-image isomorphism.

##### MC.1. The ordinary map and its diagonal-supported representative

Closed adjunction for the diagonal \(i:M\hookrightarrow M\times M\) gives
\[
\begin{gathered}
i^!\mathcal H_M(F,G)\\
=R\mathcal Hom(F,i^!q_1^!G)\\
=R\mathcal Hom(F,G).
\end{gathered}
\tag{MC.2}
\]
The last equality is transitivity of extraordinary inverse image for \(q_1i=\mathrm{id}\), with its actual orientation map. Thus a derived ordinary map \(a:F\to G\) has a specified diagonal-supported class
\[
\begin{gathered}
i_*R\mathcal Hom(F,G)\\
=R\Gamma_\Delta\mathcal H_M(F,G).
\end{gathered}
\tag{MC.3}
\]
Enlarge the diagonal support to an admitted support and take the filtered germ. Denote the resulting microlocal class by \(\alpha(a)\). The definition includes the positive normal orientation and the projection shift \(2m\).

If a source or target map is inserted into this construction, its evaluation square commutes before any support enlargement: it is ordinary Hom functoriality and closed adjunction. Restriction and support-enlargement maps preserve that square. This proves the naturality used below for the actual source roofs.

For exterior products, the two disjoint diagonal normal blocks give the diagonal of the product. The ordinary tensor-Hom map on homogeneous cochains is
\[
\begin{gathered}
(\phi\otimes\psi)(v\otimes w)\\
=(-1)^{|\psi||v|}\phi(v)\otimes\psi(w).
\end{gathered}
\tag{MC.4}
\]
Its differential identity is the usual cancellation of the two cross terms. The two real normal dimensions are even, so their orientation blocks have positive interchange. The cohomological shifts still use V.14. Therefore the exterior product of the two classes MC.3 is precisely the diagonal-supported class of the exterior ordinary map. In the support model, this remains true after applying the full tensor-to-cube map V.16; its signs are exactly the signs of MC.4, with complement degrees placed before antiholomorphic degrees. No complement entry is omitted.

##### MC.2. Restriction through a transverse closed source

Let \(j:T\hookrightarrow M\) be a holomorphic embedding of complex codimension \(c\). Suppose a smooth closed source support \(S\subset M\) is transverse to \(T\), and take \(F=\mathbb C_S[k]\). Suppose also that a target map \(j^{-1}G\to G_T\) is specified. Then the inverse-image part of the canonical microlocal composition carries the ordinary class of \(a:F\to G\) to the ordinary class of the restricted map
\[
 j^{-1}F\xrightarrow{j^{-1}a}
      j^{-1}G\longrightarrow G_T .
 \tag{MC.5}
\]
Here the source Gysin map is used in its specified transverse direction:
\[
 j^{-1}F\xrightarrow{\sim}
       \operatorname{or}_j[2c]\otimes j^!F .
 \tag{MC.6}
\]

We give the local support verification, including the real shift. Choose simultaneous coordinates \((r,h,v)\), with \(S=\{r=0\}\) and \(T=\{h=0\}\). They exist by transversality. The source is constant in \(h\). Apply the closed-source Hom calculation of HI.19 in each complex \(r\)-normal coordinate. It is the finite tensor of its two-real-coordinate Thom complexes, so it removes twice the complex codimension of \(S\). On each cofinal round support the remaining tangent-normal comparison is exactly HI.A, with all its collapsed fibres and complement maps.

More explicitly, if \(s\) is the complex codimension of \(S\), the second-variable source-normal contraction contributes \([-2s]\), and the remaining second-variable tangent-normal contraction contributes \([-2(m-s)]\). Their sum cancels the projection shift \([2m]\) in MC.1. The \(s\)-normal calculation is obtained by tensoring HI.19's one-complex-normal comparison \(s\) times. On a cofinal product of round supports each map is its actual two-real-normal Thom map; a round support inside the product and a product inside the preceding round support give the same germ by HI.A and R.6. Tensoring their contractions uses the ordinary total-degree sign, so the mixed differential terms cancel. This proves the comparison for every finite codimension, including the diagonal source of codimension \(N\).

The normal block \(h\) has real dimension \(2c\). In the ambient calculation it contributes its trace-one tangent-normal Thom class, while in the restricted calculation it is absent. Its shift \([-2c]\) cancels the difference between the two projection shifts in MC.1. The source map MC.6 inserts exactly that same positive class. Both orders consequently send its trace-one class to \(1\). The calculation fixes the sign: it is a real complex-oriented block of dimension \(2c\), and its interchange with every other real normal block is even. The common source shift \([k]\) is carried by both routes.

The target restriction acts on the first coefficient variable. The normal Thom maps act in the second source-normal variables with the first variable fixed. Hence they commute with the specified target map. For holomorphic density targets, this is literal restriction of holomorphic coefficient functions and of their relative top-form blocks. Terms in a lift with insufficient vertical holomorphic degree have zero later fibre trace, by V.4.

This is an equality on the entire relative complex. On each whole-domain or complement entry use the same normal map. The tensor normal comparisons commute with restriction. Their finite complement-cover augmentations are the maps V.15–V.16 and DLG.3, and their inverse comparisons carry the specified HI/R.6 homotopies. Thus the equality survives finite common refinements and support enlargement; it is not an assertion made only for a supported-current component.

For GP.2, the source is the exterior product of the two closed incidence constant complexes, the embedding is the common middle diagonal, and \(c=N\). Equation GP.8 is precisely MC.6. The target map is the holomorphic restriction and ordered density product.

##### MC.3. Proper projection and ordinary adjunction

Let \(p:T\to B\) be the proper smooth projection used in PE. The proper microlocal-Hom map, followed by any fixed coefficient trace, carries the ordinary class of \(a:F\to p^!G\) to the ordinary class of its actual adjoint
\[
 \epsilon_p\,Rp_!(a):Rp_!F\longrightarrow G .
 \tag{MC.7}
\]
The statement concerns the map, not invertibility of a general proper microlocal comparison.

Here is its closed-diagonal calculation. Put \(a_B:B\times T\to B\), \(b_T:B\times T\to T\), and let \(i_p:T\hookrightarrow B\times T\) be the graph. Then
\[
\begin{gathered}
 \mathcal H_p
 =R\mathcal Hom(b_T^{-1}F,a_B^!G),\\
 i_p^!\mathcal H_p
 =R\mathcal Hom(F,p^!G).
\end{gathered}
\tag{MC.8}
\]
A diagonal-supported class on \(T\times T\) is sent to this graph-supported class by the map \((p,\mathrm{id}_T)\). On these closed supports this is exactly transitivity of extraordinary inverse images:
the diagonal followed by this map is the graph, and both pull the coefficient back to \(p^!G\). Thus the coefficient remains the same Hom cochain. Pushing the graph to \(B\) and taking its actual proper adjoint gives MC.7, by the sheaf proper-support Hom adjunction already proved in the classical provider.

The source unit in that statement is explicit. Put \(f=(p,\mathrm{id}_T)\), \(A=b_T^{-1}F\), and \(B'=a_B^!G\). Proper Hom adjunction sends \(Rf_*R\mathcal Hom(f^{-1}A,f^!B')\) to \(R\mathcal Hom(Rf_!f^{-1}A,B')\). Precompose with the ordinary inverse/direct unit \(A\to Rf_*f^{-1}A=Rf_!f^{-1}A\). On the diagonal-supported class the first fibre point equals the second; this unit evaluates as the constant \(1\), and the positive normal orientation counit has trace \(1\). Their composition is the same Hom cochain in MC.8, by the adjunction triangle identity. Thus an additional fibre-cohomology factor is not left in the graph-supported calculation.

The dimension check is explicit. If \(r=\dim_{\mathbb R}T-\dim_{\mathbb R}B=2N\), the source diagonal has codimension \(\dim_{\mathbb R}T\); the graph has codimension \(\dim_{\mathbb R}B\). Their difference is \(r\), exactly the orientation shift in \(p^!G=p^{-1}G[r]\). The normal middle-difference block is contracted by its trace-one orientation counit. This is the closed-normal trace of HI.11–HI.15, not a multiplication of two copies of a delta current.

To pass this calculation to the actual support germs, use the same support-enlargement maps on the diagonal and graph models. The required normal properness has a direct local estimate. Write the two matched normal vectors as \(n_{12}=(a,b)\), \(n_{23}=(b,c)\), and matched raw polar covectors as \(\rho_{12}=(\xi,-\eta)\), \(\rho_{23}=(\eta,\zeta)\). After one finite common refinement their strict margins give
\[
\begin{gathered}
-\Re\langle\rho_{12},n_{12}\rangle\ge\delta|n_{12}|,\\
-\Re\langle\rho_{23},n_{23}\rangle\ge\delta|n_{23}|,\\
\delta>0.
\end{gathered}
\tag{MC.9}
\]
Adding cancels the middle pairing:
\[
\begin{gathered}
-\Re(\langle\xi,a\rangle+\langle\zeta,c\rangle)\\
\ge\delta\bigl(|(a,b)|+|(b,c)|\bigr).
\end{gathered}
\tag{MC.10}
\]
It bounds the middle normal variable by the endpoint polar length and gives a strict endpoint polar margin. Uniformly smaller margins persist on a covector neighbourhood. The cofinal-cone and actual-containment argument IG.1 therefore supplies admitted source and target supports on a common smaller actual domain. In simultaneous local coordinates the projection is linear. Changing those ordinary coordinates contributes a remainder bounded by \(A|n|^2\) on a compact enlargement, by Taylor's integral formula. Shrink \(|n|\) so that the resulting polar error is at most \(\delta|n|/4\). This leaves a positive margin. There are finitely many charts on the compact middle support, so take their least positive margin and radius. These are ordinary coordinate comparisons of BCT.1–BCT.3; no covariable-dependent coordinate change is used. For the selected incidence cotangent support, PC.3 also gives its precise compact affine neighbourhood.

This estimate permits the graph trace on every coefficient in the finite relative-support diagram. On compact endpoints, choose one finite partition of the compact middle manifold and one normal collar for the bounded middle-normal family. R.7 supplies controlled excision and zero extension there. Every derivative-of-collar term remains in its full relative complement entry. The two closed-support adjunction routes already agree by MC.8. Naturality of restriction, sections with support and proper counits gives the same commutative square after each support enlargement. R.6–R.7 and DLG.14–DLG.17 transfer its finite homotopies to the actual full cones. Passing to their exact filtered colimit proves MC.7 for the ordinary microlocal images.

If the coefficient trace is holomorphic, take its actual adjoint \(\Omega_Y[N]\to p^!\mathcal O_B\). Inserting it into MC.7 is exactly the raw holomorphic trace V.11. Thus this proof includes its cohomological degree lowering and its declared coefficient sign.

##### MC.4. The closed incidence evaluation comparison

To state the shifts explicitly, view the ordinary maps \(L_{12}\to\Omega_Y\) and \(L_{23}\to\Omega_Z\) as maps \(L_{12}[N]\to\Omega_Y[N]\) and \(L_{23}[N]\to\Omega_Z[N]\). The canonical product then has source \((L_{12}\circ L_{23})[2N]\) and target \(\Omega_Z[N]\). Cancelling the common shift \([N]\) identifies its Hom complex with that from \((L_{12}\circ L_{23})[N]\) to \(\Omega_Z\), exactly the source and target of PE.13. The tensor identifications used in this cancellation are V.14.

There are two source operations in this statement, and their identification is an actual map. For \(F_1,F_2\) on the two pair spaces put
\[
\begin{gathered}
\mathcal F=F_1\boxtimes F_2,\\
F_1\circ F_2=Rp_!j^{-1}\mathcal F,\\
F_1*F_2\\
=Rp_*(\omega_Y\otimes j^!\mathcal F),\\
\omega_Y=\mathbb C_Y[2N].
\end{gathered}
\tag{MC.11}
\]
The ordinary-to-extraordinary inverse-image fundamental class, followed by the proper-support-to-direct map, is the map \(\circ\to *\). The canonical microlocal Hom composition takes its source argument in the \(*\) operation. In the present case GP.8 identifies the first map with the trace-one Thom isomorphism, and \(p\) is proper. Thus \(\circ\to *\) is invertible here, with its actual orientation. We use that map, rather than silently replacing \(*\) by \(\circ\). With \(F_i=L_i[N]\), its source is exactly the shifted ordinary convolution just displayed. The diagonal/source comparisons in MC.1–MC.3 commute with this fundamental-class map because their local normal contraction is MC.6 itself.

Take the two global ordinary sections GP.5 and GP.6. Apply MC.1 to their exterior evaluation maps, MC.2 to the common middle diagonal, and MC.3 to the proper middle projection. The resulting equality is
\[
 \begin{gathered}
 \alpha(a_N)\circ_\mu\alpha(b)\\
 {}=
 \alpha\!\bigl(
 \operatorname{Tr}\,
 Rp_!(\operatorname{ev}_{m_{+,N}}\\
          \otimes\operatorname{ev}_{m_-})[N]\bigr),
 \end{gathered}
 \tag{MC.12}
\]
with the ordinary source convolution and its actual source maps understood. PE.13 identifies the right side with the ordinary evaluation of \(h_N\), using \(\theta_{\rm raw}\). Transversality is GP.2; properness of the selected cotangent support is PC.3. Every map is natural in the actual algebraic-RH/source roofs.

The operation here is the canonical one used by the programme. For two diagonal kernel sources, the same construction has disjoint ordinary normal factors, the middle diagonal's positive normal comparison, and the ordered proper trace. Its full relative entries are the DLG.3 cube, the V.16 tensor signs and the V.11 trace. V.13 cancels the between-kernel coefficient parity, and V.17 gives the already fixed single-normal-class calibration. Hence its specialization is exactly the existing whole canonical ring product and unit, with no second product normalization. For a kernel and an incidence input, the same normal comparison is FIM.5's action map.

This also identifies the coefficient objects, not merely their dimensions. Apply the codimension-\(N\) calculation in MC.2 to the diagonal source. Its second-normal Thom contraction gives the relative diagonal holomorphic support complex; its tangent-normal contraction removes the remaining second-variable block. The ordered input density survives unchanged. Shifting its source by \([-N]\) consequently gives the degree-\(N\) diagonal group defining the source real operator ring. The support-forgetting and finite-refinement maps are precisely those used in that definition. The unit is the positive normal class UC.7, by the trace-one contraction and V.17. Thus no additional source-Hom or scalar-extension equivalence is a premise of the diagonal specialization.

The argument retains finite three-factor diagrams: exterior tensor evaluation is associative with MC.4; the two middle diagonal normal contractions compose by the same trace-one closed counits; and the proper traces compose by Fubini with V.12–V.14. The full eight-entry cube and its common proper collar are DLG.5. Thus MC.12 is compatible with the three-factor associator and the coherent source inverse counits.

Finally GP.13 recovers the original incidence class by the actual finite inverse \(D_t^{-N}\). FIM.5 and the complete DLG action commute with this operator and with the canonical product. Applying that inverse to MC.12 proves the comparison for the original forward pole as well. For the reverse order, apply the same closed-source proof to the ordinary global sections GP.15–GP.16, then use GP.17. Its ordered spatial reflection is AS.34, and is retained separately. These are finite operator cancellations after the actual product has been constructed.

Related source operations and microlocal composition are discussed in Kashiwara–Schapira, [Microlocal Euler classes and Hochschild homology, arXiv:1203.4869v4](https://arxiv.org/abs/1203.4869v4), §3. The normal-support and ordinary-adjunction formalism is treated in their [Microlocal study of sheaves](https://www.numdam.org/item/AST_1985__128__1_0/), §§5.5 and 11.1. The comparisons needed here have the closed transverse sources, specified proper support and complete proofs above. The general proper microlocal comparison is not asserted.

#### Intrinsic normalization from the ordinary-to-microlocal comparison

Use the actual inverse projective open-incidence kernels
\[
 K=A[N],\qquad L=B[N],
 \tag{IN.1}
\]
where LI and SN define \(A=j_!\mathbb C_\Omega\) and \(B=Rj_*\mathbb C_{\Omega^{\rm op}}\). Their ordinary convolutions are the two diagonal kernels. All source maps below are the actual source maps, with their shifts and input antipodal operation. The classes \(a,b\) are the lifts of the forward and opposite pole evaluations through OCM/SN and the affine/projective boundary comparison OD.14. The forward pole is \(f^{-1}\); the opposite pole is \(g^{-N}\), with the prescribed ordered input densities.

Use the ordinary-to-microlocal comparison MC.1–MC.4, including its actual source fundamental class, every complement entry and the proper trace. GP.5–GP.8 and GP.15–GP.16 supply the global closed transverse sections to which that proof applies. Its finite inverse-derivative cancellations are GP.13 and GP.17. The proper cotangent support condition is PC.3, and the trace and tensor suspensions are V.11–V.14.

**Theorem.** On the chart with nonzero time covector, the forward simple-pole incidence class has a two-sided inverse for the canonical microlocal composition. The inverse is a nonzero complex scalar multiple of the opposite pole of order \(N\), with its prescribed input density. The source counits can be chosen coherently, and the scalar is characterized intrinsically by either actual product.

##### IN.1. Coherent inverse source counits

Fix the first actual inverse-kernel isomorphism
\(\epsilon_X:K\circ L\to\mathbb C_{\Delta_X}\).
There is a unique second isomorphism \(\epsilon_Y:L\circ K\to\mathbb C_{\Delta_Y}\) satisfying
\[
 K\circ\epsilon_Y=\epsilon_X\circ K
 \tag{IN.2}
\]
under the actual convolution associator.

Here is an elementary construction, which does not presume that two independently selected inverse-object isomorphisms already satisfy a triangle identity. Convolution on the left by \(K\) is an equivalence on the corresponding kernel category, since convolution by \(L\) is an inverse up to the two already proved ordinary diagonal isomorphisms. Its full faithfulness follows by applying those isomorphisms to the ordinary Hom maps; the resulting two composites are identities after the associator. Apply that full faithfulness to the isomorphism \(\epsilon_X\circ K\) from \(K\circ L\circ K\) to \(K\). Its unique preimage is \(\epsilon_Y\). An equivalence reflects isomorphisms, so \(\epsilon_Y\) is invertible. Equation IN.2 is its defining equality, and gives the two triangle identities with the inverse unit \(\epsilon_Y^{-1}\).

Thus the source inverse data are coherent without an unrecorded scalar adjustment between two arbitrary choices of diagonal maps. The classical convolution associator is the actual proper-support base-change/projection map proved in LI; it acts on every bounded attaching map.

The second coherent counit may differ from the initial LI counit. Their ratio is a degree-zero automorphism of the constant diagonal source, hence a nonzero complex scalar: internal Hom of the rank-one constant sheaf is constant, and its global degree-zero sections on the connected diagonal are \(\mathbb C\). It therefore remains a legitimate ordinary source map for the scalar comparison OD.5. Fix these coherent counits throughout.

##### IN.2. Each product belongs to the constant unit line

First use the global sections \(a_N,b\). MC.4 and PE.13 identify their canonical product with ordinary evaluation of \(h_N\), using \(\theta_{\rm raw}\). The affine/projective source comparison OD.14 and the SN source roofs lift, under algebraic RH, to coherent operator roofs. Every comparison cone has zero polynomial \(\tau\)-localization, by OD.4–OD.5.

Compare this operator roof with AS.28. After localization both identify \(Q_\tau\) with the same diagonal module. Their ratio belongs to
\(\operatorname{Aut}_{A_\tau}((\Delta_{\rm alg})_\tau)\),
which is \(\mathbb C^\times\) by OD.12. OD.2 transports that exact scalar equality through the actual solution/microlocal-Hom functor; it applies to all bounded coherent lattices appearing in the comparison, by OD.1.

In the AS roof, the section is exactly
\[
 h=\frac{(-1)^d}{d!}j(e_\Delta).
 \tag{IN.3}
\]
The raw ordered diagonal evaluation of \(e_\Delta\) is a nonzero constant multiple of the normalized diagonal unit. Indeed RH1.vf–vg computes it as the ordered iterated normal Cech/Cauchy class with its finite Koszul determinant coefficient. V.17 gives the conversion of the ordered holomorphic coefficient block, and UC.1–UC.3 computes its positive period. Every frame factor is a fixed sign and every period factor is a power of \(2\pi i\); the result is nonzero and independent of the base point and conormal phase. This uses the exact earlier normal-class calculation, rather than inferring nonvanishing from a dimension count.

Consequently there is one uniquely determined \(\lambda_X\in\mathbb C^\times\) such that
\[
 a\circ_\mu b=\lambda_X\,1_X .
 \tag{IN.4}
\]
This is a constant scalar statement in the finite polynomial-localized diagonal image. It does not assert that an arbitrary element centralizing coordinates in the whole real ring is constant.

Here is the cancellation that relates this conclusion to the globalized calculation. In the AS roof GP.10–GP.12 send \(h_N\) to \(D_t^N h\), so that the proved globalized product is \(\lambda_X D_t^N1_X\). The same \(D_t^N\) acts on \(a\circ_\mu b\) by the actual action/composition square FIM.5 and MC.4. It is invertible on the selected chart. Apply its actual inverse to obtain IN.4. This is an equality of microlocal classes; a formal cancellation of symbols is not used.

For the reverse order use the global sections \(b_1,a\), AS.32–AS.35 and the corresponding operator source roofs. Its raw algebraic factor is \(1/d!\), after the separate \((-1)^d\) spatial reflection. GP.17 identifies the globalized convolution section with \(D_sh'\). The preceding proof gives a nonzero constant multiple of \(D_s1_Y\); its actual inverse cancels by the reverse FIM action square. Thus
\[
 b\circ_\mu a=\lambda_Y\,1_Y,
 \qquad\lambda_Y\in\mathbb C^\times .
 \tag{IN.5}
\]
All factors used to prove nonvanishing have been retained: the two nonzero algebraic factorial factors, the invertible operator source-roof comparisons, and the nonzero ordered diagonal evaluation. No numerical value for a raw analytic source-frame coefficient is assumed.

##### IN.3. The same intrinsic normalization gives both inverses

The canonical products have degree zero in the shifted source-kernel conventions of IN.1. MC.4, the actual source associator and IN.2 therefore give
\[
 (a\circ_\mu b)\circ_\mu a
 =a\circ_\mu(b\circ_\mu a).
 \tag{IN.6}
\]
All three factors lie on their matching conormal graphs. PC.3 makes the finite three-factor support proper after shrinking the selected chart. The full eight-entry support cube, its common collar and the repeated trace have the associativity signs V.12–V.14 and DLG.5.

The class \(a\) is nonzero: HI.C proves nonvanishing of its actual pole at every nonzero conormal phase, and the OCM/source comparison maps are isomorphisms there. Combining IN.4–IN.6 thus yields
\[
\begin{gathered}
(\lambda_X-\lambda_Y)a=0,\\
\lambda_X=\lambda_Y=:\lambda.
\end{gathered}
\tag{IN.7}
\]
The equality is over the field \(\mathbb C\). Define
\[
 b_{\rm inv}=\lambda^{-1}b .
 \tag{IN.8}
\]
Then
\[
\begin{gathered}
a\circ_\mu b_{\rm inv}=1_X,\\
b_{\rm inv}\circ_\mu a=1_Y.
\end{gathered}
\tag{IN.9}
\]
This is an intrinsic normalization: \(\lambda\) is the unique coefficient of the actual constructed product in the unit line proved in IN.2. It can equivalently be extracted as the constant symbol of that product's finite diagonal operator. Its definition is independent of a linear functional or a choice of raw normal solution frame.

The scalar normalization commutes with both operator actions. Hence all complete finite-\(E\) and whole-\(B\) input identities BI and OI retain their exact maps.

The relation to the balanced algebraic evaluations must retain this scalar. For the original pole embeddings, OI.11–OI.12 are the specified algebraic evaluations sending their two raw basis tensors to \(1\). In the canonical microlocal composition those same raw embedded tensors have value \(\lambda1\), by IN.4–IN.7. Balancing and the complete BI/OI action identities therefore identify the canonical pairings with \(\lambda\) times the embedded algebraic evaluations on every tensor, including whole-\(B\) coefficients. Embedding the opposite basis by \(b_{\rm inv}\) makes the pairings exactly those evaluations. This does not change \(\mathcal L_0\), its dimension-correct inverse, or any all-index convergence bound. The two raw algebraic basis evaluations were not assumed to be the already normalized canonical analytic products.

#### Legendre transport on the entire real operator ring

Let \(C_X=\mathcal E_X^{\mathbb R}\) be the source-defined real operator sheaf, \(B_X=\mathcal E_X^\infty\) the projectivized infinite-order sheaf, and \(E_X=\mathcal E_X\) the finite-order sheaf. Use the actual coefficient, support, density, product and unit conventions of DLG and UC. The incidence contact map on the nonzero-time chart is
\[
\begin{gathered}
\chi(t,x;\tau,\xi)\\
=\bigl(t+x^{\mathsf T}\xi/\tau,\ \xi/\tau;\\
\tau,-\tau x\bigr).
\end{gathered}
\tag{WT.1}
\]
It is the graph calculated in PC.2 with the input antipodal convention. Its inverse and its exact preservation of the tautological one-form are PC.9–PC.10. All stalk statements below are made after shrinking to actual corresponding chart neighbourhoods.

##### WT.1. Conjugation by the actual inverse incidence classes

Let \(a\) be the forward pole class and \(b_{\rm inv}\) its intrinsic normalized inverse from IN.8–IN.9. For \(p\) in the chart and \(q=\chi(p)\), define
\[
\begin{gathered}
 \Phi_p:C_{Y,q}\longrightarrow C_{X,p},\\
 \Phi_p(Q)=a\circ_\mu Q\circ_\mu b_{\rm inv},\\
 \Psi_q:C_{X,p}\longrightarrow C_{Y,q},\\
 \Psi_q(P)=b_{\rm inv}\circ_\mu P\circ_\mu a .
\end{gathered}
\tag{WT.2}
\]
Every \(P,Q\) here is an arbitrary real operator germ, with its full relative representative.

We first justify the operations on those whole germs. The source supports of \(a,b_{\rm inv}\) at nonzero covectors are the two inverse graphs, by PC.1–PC.3. A diagonal real operator has diagonal cotangent support by its defining supported-kernel construction. Thus each matching middle covector in WT.2 is uniquely determined by its endpoint. PC.3 bounds that middle base point and covector on compact endpoint sets. For a finite collection of representatives and equality witnesses, shrink the endpoint chart once so that all matching points lie in one relatively compact middle chart. The strict polar estimate MC.9–MC.10 gives admitted normal supports; its Taylor-margin argument gives the same statement in actual coordinates. R.7 then gives one proper collar inside the finitely many actual coefficient domains. Use the complete relative cubes of DLG.3–DLG.5 on that collar, with every complement entry retained.

Consequently the finite composition maps and their homotopies exist on actual domains. Different representatives, refinements or collars give equal germs by R.6–R.7 and DLG.14–DLG.17. This proves that WT.2 is defined on the entire source real stalk. No symbol expansion for an arbitrary element of \(C\) is a premise. The same finite construction commutes with restriction, so the maps are maps of sheaves on the corresponding charts.

Associativity is the full-cube identity in MC.4 and DLG.5, with the coherent source counits IN.2. The middle inverse product is IN.9. Hence, for arbitrary \(Q_1,Q_2\),
\[
\begin{gathered}
\Phi(Q_1)\Phi(Q_2)\\
=aQ_1(b_{\rm inv}a)Q_2b_{\rm inv}\\
=a(Q_1Q_2)b_{\rm inv}\\
=\Phi(Q_1Q_2),\\
\Phi(1_Y)=ab_{\rm inv}=1_X.
\end{gathered}
\tag{WT.3}
\]
The same calculation gives
\[
\begin{gathered}
\Psi\Phi(Q)\\
=(b_{\rm inv}a)Q(b_{\rm inv}a)=Q,\\
\Phi\Psi(P)\\
=(ab_{\rm inv})P(ab_{\rm inv})=P.
\end{gathered}
\tag{WT.4}
\]
In each finite identity choose the common support and collar before taking the products, as in the preceding paragraph. Thus these are identities of the actual germ operations. They prove mutually inverse unital ring isomorphisms on the whole \(C\) sheaves.

##### WT.2. The complete finite and infinite input maps

The actual whole-\(B\) incidence relation BI is
\[
\begin{gathered}
aQ=\mathcal L_0(Q)a,\\
Q\in B_Y.
\end{gathered}
\tag{WT.5}
\]
Its proof uses the entire input, both tails and one common coefficient domain; it is not restricted to polynomial generators. Multiplying by \(b_{\rm inv}\) and using IN.9 gives
\[
\begin{gathered}
 \Phi|_{B_Y}=\mathcal L_0,\\
 \Psi|_{B_X}=\mathcal L_0^{-1}=J\mathcal L_d .
\end{gathered}
\tag{WT.6}
\]
Their finite-order restrictions are the same maps on \(E\). This retains the ordered dimension correction of LN.30 and OI, as well as every positive-head and negative-tail estimate. The scalar used in \(b_{\rm inv}\) cancels from WT.5.

The result does not adjoin a fractional normal power to \(C\). The complex family \(\mathcal H_a,\mathcal L_a\) of LN.26–LN.30 remains the entire coefficient-defined \(E/B\) family proved there. WT.2 uses the actual simple-pole incidence class and its normalized opposite.

##### WT.3. Every symmetric quadratic contact shear

For a complex symmetric matrix \(H\), the ordinary time change and its contact lift on \(Y\) are
\[
\begin{gathered}
F_H(s,y)\\
=(s+y^{\mathsf T}Hy/2,y),\\
\widehat F_H(s,y;\sigma,\eta)\\
=\bigl(s+y^{\mathsf T}Hy/2,y;\\
\sigma,\eta-\sigma Hy\bigr).
\end{gathered}
\tag{WT.7}
\]
BCT proves the whole \(C\)-ring map \(\Theta_H\) for this ordinary base biholomorphism, with its actual density and support maps. Its inverse is \(\Theta_{-H}\), and its entire \(E/B\) restrictions are LI.13 and LN.31. Define the whole real-ring map
\[
 \mathscr T_H=\Phi\,\Theta_H\,\Phi^{-1}.
\tag{WT.8}
\]
The stalk of the last inverse is taken at the transformed endpoint: explicitly \((\mathscr T_H)_p=\Phi_p(\Theta_H)_{\chi(p)}\Phi^{-1}_{\kappa_H(p)}\). Its contact map is
\[
\begin{gathered}
\kappa_H=\chi^{-1}\widehat F_H\chi,\\
\kappa_H(t,x;\tau,\xi)\\
=\bigl(t-z^{\mathsf T}Hz/2,\ x+Hz;\\
\tau,\xi\bigr),\quad z=\xi/\tau.
\end{gathered}
\tag{WT.9}
\]
To check the coordinates, the first Legendre image has \(s=t+x^{\mathsf T}z\), \(y=z\), and \(\eta/\sigma=-x\). The time change replaces its time by \(t+x^{\mathsf T}z+z^{\mathsf T}Hz/2\) and that last covector ratio by \(-x-Hz\). The inverse Legendre map then gives exactly WT.9. In particular it preserves \(\tau\,dt+\xi^{\mathsf T}dx\).

There is no rank assumption on \(H\). The inverse time change exists with matrix \(-H\), even when \(H\) is degenerate. Conjugation therefore proves
\[
\begin{gathered}
\mathscr T_H^{-1}=\mathscr T_{-H},\\
\mathscr T_H\mathscr T_K=\mathscr T_{H+K}\\
(H,K\text{ symmetric}).
\end{gathered}
\tag{WT.10}
\]
Products here have their corresponding transformed stalks. On the complete finite and infinite operator sheaves, WT.6 and LN.31 at parameter \(a=0\) give
\[
 \mathscr T_H|_{E/B}=T_H .
\tag{WT.11}
\]
Thus the covariable-dependent quadratic shear has a whole real-ring transport extending its already proved full coefficient maps.

##### WT.4. Actual module transport and the balanced comparison

A ring isomorphism transports all modules by restriction of scalars along its inverse. For a \(C_Y\)-module \(V\), write \({}_{\Phi}V\) for the same abelian group with \(c\in C_X\) acting by \(\Phi^{-1}(c)\). The functors for \(\Phi,\Psi\) are inverse on all modules and all module morphisms, by WT.4. They are exact because the underlying abelian groups and maps are unchanged. A finite presentation, together with its matrix and relations, transports to a finite presentation under these ring isomorphisms.

The square of actual inclusions \(B\subset C\) commutes by WT.6. For every \(B_Y\)-module \(M\) it gives the specified balanced map
\[
\begin{gathered}
 C_X\otimes_{B_X}{}_{\mathcal L_0}M\\
 \xrightarrow{\ \sim\ }
 {}_{\Phi}\bigl(C_Y\otimes_{B_Y}M\bigr),\\
 c\otimes m\longmapsto\Phi^{-1}(c)\otimes m .
\end{gathered}
\tag{WT.12}
\]
Indeed the images of \(cb\otimes m\) and \(c\otimes\mathcal L_0^{-1}(b)m\) agree, because \(\Phi^{-1}(b)=\mathcal L_0^{-1}(b)\) in \(C_Y\). The inverse sends \(r\otimes m\) to \(\Phi(r)\otimes m\). The formula intertwines the \(C_X\)-actions by multiplication and is natural in every actual module map.

The same statement holds for derived tensor products. Take a free resolution of \(M\); the ring isomorphism \(\mathcal L_0\) transports it to a free resolution, and WT.12 is a termwise chain isomorphism, with its inverse termwise as displayed. This proof imposes no flatness, faithful-extension or holonomicity premise on \(C\) over \(B\). Those scalar-extension properties and an embedding or regularization of an arbitrary holonomic module require separate arguments.

![The globalized ordinary evaluation square, an exact shear slice, and intrinsic whole-ring transport](assets/incidence-evaluation-and-real-transport.png)

*Figure MC–WT.1. Panel A is GP.4–GP.13, PE.13 and MC.1–MC.4: the horizontal ordinary trace and the vertical normal-support maps form the actual evaluation square. The source fundamental class has real shift \(2N\); the coefficient trace lowers degree by \(N\). Panel B is the fixed-covector real base slice \(d=1\), \(H=2\), \(\tau=1\), \(\xi=1/2\) of WT.9, so \(x'=x+1\) and \(t'=t-1/4\). It is a slice of the full map, not its entire phase space. Panel C states IN.4–IN.9 and WT.2–WT.12. The scalar is the unique actual product coefficient; its normalization is IN.7–IN.9. Exact graph properness is PC.3, and every relative-complement entry and equality homotopy is retained in MC.2–MC.4. Related constructions appear in the two Kashiwara–Schapira works linked in MC.4.*

Full PNG · SVG · Exact data · Drawing source


### 5.68. Convergent normal connections with nilpotent head

Put \(h=D_t^{-1}\), \(E=\mathscr E_p\), \(B=\mathscr E^\infty_p\) and \(C=\mathscr E^{\mathbb R}_p\) at the normalized covector \(p=(t,x;\tau,\xi)=(0,0;1,0)\). The coefficient matrices in this section depend only on \(h\). Their convergence at \(h=0\) is an additional hypothesis, stronger than the factorial bounds for a general actual finite-order symbol.

Let \(r\ge1\), and let \(T(h)\) be an \(r\)-by-\(r\) matrix of convergent power series, with nilpotent constant term \(N=T(0)\). Define the entire left \(E\)-module \(M_T\) by generators \(u_1,\ldots,u_r\) and the finite relations
\[
\begin{gathered}
t u_j=\sum_{k=1}^r T_{kj}(h)u_k,\\
D_{x_i}u_j=0.
\end{gathered}
\tag{CN.1}
\]
Convergent power series in \(h\) are actual finite-order symbols: a Cauchy bound on a smaller convergence circle is exponential in the coefficient index, hence satisfies the negative factorial bound of §5.19. All tensor products below are ordinary algebraic tensor products.

**Theorem CN.1.** There is a finite constant complex matrix \(A\) and an actual \(B\)-linear isomorphism
\[
\begin{gathered}
B\otimes_E N_A
\ \xrightarrow{\ \sim\ }\ B\otimes_E M_T,\\
N_A=\\
E^r\big/\bigl((tD_tI-A)e,\ D_{x_i}e\bigr).
\end{gathered}
\tag{CN.2}
\]
Its inverse also has actual full-coefficient entries. The natural comparison
\[
B\otimes_E M_T\longrightarrow C\otimes_E M_T
\tag{CN.3}
\]
is injective, and finitely many \(B\)-linear maps to the whole real-ring halfspace module separate its source. The theorem does not assert convergence for the generally divergent Gevrey matrices of §5.55, DP.4.

#### CN.1. The normal equation and its monodromy

For the analytic construction only, use an ordinary complex variable \(\zeta\). Choose \(R>1\) such that \(T(1/\zeta)\) is holomorphic for \(|\zeta|>R\). On the universal covering of that annulus solve
\[
Y'(\zeta)=T(1/\zeta)Y(\zeta).
\tag{CN.4}
\]
Here is the existence and continuation argument. On a closed disc strictly inside the annulus the coefficient matrix has a bound \(L\). Successive integrals along the straight segment, starting with \(Y(\zeta_0)=I\), have norms at most \(L^k|\zeta-\zeta_0|^k/k!\). They converge normally, differentiate to CN.4, and give its unique solution: iteration of the integral equation for the difference of two solutions bounds it by the same factorial denominator times an initial uniform bound, tending to zero. Solving \(Z'=-ZT(1/\zeta)\) with \(Z(\zeta_0)=I\) and differentiating \(ZY\) gives \(ZY=I\); the square matrices are therefore inverse at every point. A path covered by finitely many such discs gives continuation with an invertible endpoint value. Uniqueness on overlaps and subdivision of a path homotopy into finitely many small disc rectangles show independence under homotopy with fixed endpoints. This constructs \(Y\) on the universal cover without a finite-distance blowup.

Write \(w=\log\zeta\) there. The deck transformation has
\[
\begin{gathered}
Y(w+2\pi i)=Y(w)M,\\
M\in\operatorname{GL}_r(\mathbf C).
\end{gathered}
\tag{CN.5}
\]
Indeed both sides solve the same periodic equation in \(w\), so \(Y(w)^{-1}Y(w+2\pi i)\) has derivative zero.

There is a constant matrix \(H\) with \(\exp(2\pi iH)=M\). To construct it, use the finite primary decomposition proved in §5.53, AM.C–AM.D. On a block \(M=\mu I+K\), with \(\mu\ne0\) and \(K^r=0\), choose \(\lambda\) with \(\exp(2\pi i\lambda)=\mu\), and set
\[
\begin{gathered}
H=\lambda I+\frac{1}{2\pi i}L_\mu,\\
L_\mu=
\sum_{a=1}^{r-1}\frac{(-1)^{a+1}}{a}(K/\mu)^a.
\end{gathered}
\tag{CN.6}
\]
The elementary scalar choice is \(2\pi i\lambda=\log|\mu|+i\arg\mu\). Prove the logarithm identity first in the formal power-series ring \(\mathbf C[[z]]\): coefficient differentiation gives \((\log(1+z))'=(1+z)^{-1}\); hence the derivative of \(\exp(\log(1+z))/(1+z)\) is zero and its constant term is one. Reduce this identity modulo \(z^r\) and substitute \(z=K/\mu\). All terms of degree at least \(r\) vanish. The scalar exponential factor gives the claimed identity on each block. In particular \(H\) commutes with \(M\).

Consequently
\[
\begin{gathered}
V(w)=Y(w)e^{-wH},\\
W(w)=e^{wH}Y(w)^{-1}
\end{gathered}
\tag{CN.7}
\]
are periodic and descend to mutually inverse holomorphic matrices on \(|\zeta|>R\). They satisfy
\[
\begin{gathered}
V'=T(1/\zeta)V-\zeta^{-1}VH,\\
W'=\\
-WT(1/\zeta)+\zeta^{-1}HW.
\end{gathered}
\tag{CN.8}
\]
No estimate at \(\zeta=0\), and no regular-singular assertion there, is needed.

#### CN.2. Nilpotence gives subexponential growth of both inverse matrices

Take a constant Jordan basis for \(N\); the finite nilpotent-basis construction is the one in AM.C–AM.D. Let \(\ell\) be its largest block length, and put \(\kappa=\max(2,\ell)\), \(\alpha=1-1/\kappa<1\). In a block choose vectors with \(Nv_j=v_{j-1}\), and, for real \(s\ge1\), let
\[
D_s v_j=s^{-j/\kappa}v_j.
\tag{CN.9}
\]
Thus \(D_s^{-1}ND_s=s^{-1/\kappa}N\). The norm of \(D_s\) is at most one in the Euclidean norm of this basis, and that of \(D_s^{-1}\) is at most \(s^{(\ell-1)/\kappa}\). Since the convergent matrix has \(T(1/\zeta)-N=O(|\zeta|^{-1})\), these inequalities imply, uniformly in the argument of \(\zeta\),
\[
\begin{gathered}
\|D_s^{-1}T(1/(se^{i\theta}))D_s\|\\
\le K s^{-1/\kappa},\\
\|D_s^{-1}D_s'\|
\le(\ell-1)/(\kappa s).
\end{gathered}
\tag{CN.10}
\]
The remainder estimate uses
\((\ell-1)/\kappa-1\le-1/\kappa\). All constants are fixed for \(s\ge R_1>R\).

On a lift with \(0\le\theta\le2\pi\), put \(Z(s,\theta)=D_s^{-1}Y(se^{i\theta})\). Radial differentiation gives
\[
\begin{gathered}
\partial_s Z=\mathcal A(s,\theta)Z,\\
\mathcal A=\\
e^{i\theta}D_s^{-1}T(1/(se^{i\theta}))D_s\\
{}-D_s^{-1}D_s'.
\end{gathered}
\tag{CN.11}
\]
Its inverse satisfies \(\partial_s Z^{-1}=-Z^{-1}\mathcal A\). Both initial matrices on the compact lifted radius-\(R_1\) circle have finite bounds. Iterating their radial integral equations gives the bound \(\exp(\int_{R_1}^s\|\mathcal A(v,\theta)\|\,dv)\): the \(k\)-fold ordered integral is at most the \(k\)-th power of this integral divided by \(k!\). CN.10 therefore bounds both matrices by \(K_1\exp(K_2s^\alpha)\), uniformly over that entire lifted circle and all larger radii.

Multiplication by \(D_s^{\pm1}\) contributes only fixed powers of \(s\). Also
\(\|e^{\pm wH}\|\le\exp(\|H\||w|)\le K_3s^{\|H\|}\) on this lift. By CN.7 there are fixed \(b,K_4,K_5\) such that
\[
\begin{gathered}
\|V(\zeta)\|+\|W(\zeta)\|\\
\le K_4|\zeta|^b e^{K_5|\zeta|^\alpha},\\
|\zeta|\ge R_1.
\end{gathered}
\tag{CN.12}
\]
These are bounds for the single-valued matrices at every argument, not just sectorial estimates. For every \(\delta>0\), the function
\(b\log s+K_5s^\alpha-\delta s\) is bounded above: after division by \(s\), its first two terms tend to zero. Enlarging a finite constant on the remaining compact interval proves
\[
\begin{gathered}
\|V(\zeta)\|+\|W(\zeta)\|\\
\le K_\delta e^{\delta|\zeta|},\\
|\zeta|\ge R_1.
\end{gathered}
\tag{CN.13}
\]

#### CN.3. The complete Laurent coefficients are actual infinite-order symbols

For either matrix write its Laurent expansion as \(F(\zeta)=\sum_{j\in\mathbf Z}F_j\zeta^j\). The coefficients are
\[
\begin{gathered}
F_j=\\
\frac{1}{2\pi i}\int_{|\zeta|=s}
F(\zeta)\zeta^{-j-1}\,d\zeta,\\
s>R.
\end{gathered}
\tag{CN.14}
\]
Here is a direct Laurent argument. Parameterize the circle by \(\zeta=se^{i\theta}\). The \(s\)-derivative of its coefficient integrand is \(1/(is)\) times the \(\theta\)-derivative of that integrand, so integration over a period proves independence of \(s\). On a fixed circle these are precisely its Fourier coefficients, divided by \(s^j\). Twice integrating by parts bounds those Fourier coefficients by a constant times \((1+|j|)^{-2}\), so their series converges absolutely and uniformly. It equals the function: for a continuous periodic difference with every Fourier coefficient zero, convolution with each Fejér kernel is zero. These kernels are the normalized functions \(m^{-1}|\sum_{k=0}^{m-1}e^{ik\theta}|^2\). Their integrals are one, they are nonnegative, and their mass outside any fixed neighbourhood of zero tends to zero by the geometric-sum formula. Uniform continuity then makes their convolution tend uniformly to the difference, which must therefore vanish. Finally choose a smaller and a larger circle around any compact intermediate annulus. The two coefficient bounds from CN.14 make its positive and negative tails geometric there, including any fixed derivatives. This proves absolute normal Laurent convergence without a formal summation assertion.

On a fixed circle \(s=R_1\), CN.14 bounds the negative coefficients by \(\|F_{-n}\|\le K R_1^n\). For \(j\ge1\), use CN.13 and choose \(s=j/\delta\), whenever this exceeds \(R_1\). Then
\[
\begin{gathered}
\|F_j\|\le K_\delta(e\delta/j)^j\\
\le K_\delta(e\delta)^j/j!.
\end{gathered}
\tag{CN.15}
\]
The last inequality uses \(j!\le j^j\). Given any \(\epsilon>0\), take \(\delta=\epsilon/e\) and enlarge the prefactor for the finitely many remaining \(j\), including \(j=0\). Thus the same whole coefficient sequence has
\[
\begin{gathered}
\|F_j\|\le K_\epsilon\epsilon^j/j!,\\
j\ge0,\quad\text{every }\epsilon>0,\\
\|F_{-n}\|\le K R_1^n n!,\\
n\ge1.
\end{gathered}
\tag{CN.16}
\]
These are precisely the complete positive and negative bounds of §5.45, AN.4 and IM.1. The negative bound here is stronger than required. The coefficients are constant in \(t,x,z\), so they have a single common coefficient domain. Every fixed derivative preserves these bounds: differentiation in \(\zeta\) shifts the index by a fixed integer and multiplies by a fixed polynomial in it; use a smaller positive epsilon to absorb that polynomial, and increase one negative exponential constant. Derivatives in the other variables vanish. Hence \(V(\tau)\) and \(W(\tau)\) are actual matrices over \(B\).

For these coefficient matrices the normal operator product is ordinary Laurent convolution: the right coefficient has no base-variable derivatives. The convolution is absolutely convergent at every fixed output degree, because both Laurent series converge absolutely on any one circle \(s>R\). Its coefficient equals the coefficient of the analytic matrix product. Since \(VW=WV=I\) there, they are two-sided inverse matrices in the actual infinite-order ring. CN.8 likewise holds as an actual coefficient identity. In particular
\[
\begin{gathered}
{}[t,V(\tau)]=-V'(\tau),\\
[D_{x_i},V(\tau)]=0.
\end{gathered}
\tag{CN.17}
\]
The sign follows by writing \(V\circ t=Vt+V'\) and \(t\circ V=tV\) in the normal product.

#### CN.4. Exact presentation maps and the real-module injection

In \(B\otimes_E M_T\) define
\[
e_j=\sum_k V_{kj}(\tau)u_k.
\tag{CN.18}
\]
The finite relations CN.1, CN.8 and CN.17 give
\[
\begin{gathered}
t e_j=\sum_k hH_{kj}e_k,\\
D_{x_i}e_j=0,\\
tD_t e_j=\sum_k(H-I)_{kj}e_k.
\end{gathered}
\tag{CN.19}
\]
For the last line use \(tD_t=D_tt-I\), retaining the minus identity. Consequently CN.18 gives the map CN.2 with
\[
A=(H-I)^{\mathsf T}.
\tag{CN.20}
\]
The transpose is forced by the generator-column convention.

Conversely, in \(B\otimes_E N_A\) set \(u_j=\sum_k W_{kj}(\tau)e_k\). The second equation of CN.8 gives
\(hHW-W'=WT\). Therefore these \(u_j\) satisfy exactly CN.1, including every tangential relation. They give a map in the opposite direction. Their two composites are the identity by \(VW=WV=I\). This proves the isomorphism using finite maps between the actual full presentations. No freeness of \(B\otimes_E M_T\) over a coefficient subring is presumed.

Tensoring the isomorphism with \(C\) over \(B\) preserves both inverse identities, without a flatness hypothesis. The explicit balanced identification is
\[
\begin{gathered}
C\otimes_B(B\otimes_E M_T)\\
\longrightarrow C\otimes_E M_T,\\
c\otimes(b\otimes m)\\
\longmapsto cb\otimes m.
\end{gathered}
\tag{CN.21}
\]
Its inverse is \(c\otimes m\mapsto c\otimes(1\otimes m)\); additivity and the two balancing relations verify both composites. The same construction applies to \(N_A\) and commutes with their comparison units.

The Euler comparison is injective by the complete proof §5.53, AM.29. In the resulting square the two horizontal gauge maps are isomorphisms and the Euler vertical map is injective. Hence the vertical map CN.3 is injective. The actual finite separating maps of §5.54, EE.21–EE.27, transported by CN.2, give its finite halfspace detectors. This argument does not assert faithful flatness of \(C\) over \(B\).

The same conclusion is preserved by the actual ordinary coordinate maps of §5.56, BCT.9, and by the Legendre and symmetric quadratic contact maps of §5.67, WT.12. In each case the explicit balanced tensor isomorphism transports this comparison square. This assertion uses those proved ring maps on their stated stalks.

#### CN.5. An irregular second-order example and the limits of the hypotheses

Let
\[
\begin{gathered}
P=t^2D_t+1,\\
N=\begin{pmatrix}0&0\\1&0\end{pmatrix},\\
T(h)=
\begin{pmatrix}0&-h\\1&2h\end{pmatrix}.
\end{gathered}
\tag{CN.22}
\]
The quotient \(E/(EP+\sum_i ED_{x_i})\) is precisely CN.1 for this \(T\). Indeed put \(u_0=u\), \(u_1=tu\). Since \(D_tt^2=t^2D_t+2t\), the equation \(Pu=0\) is equivalent to
\[
t^2u=h(2t-1)u.
\tag{CN.23}
\]
This is exactly \(tu_0=u_1\), \(tu_1=-hu_0+2hu_1\). Conversely these two equations imply CN.23 and hence \(Pu=0\). Tangential derivatives commute with \(t\), so both presentations have the same tangential relations. The maps on these generators are inverse finite \(E\)-linear maps.

Here \(N^2=0\), so one can take \(\kappa=2\) in the proof and obtain the concrete bound \(s^b\exp(K\sqrt s)\) in CN.12. The ordinary scalar equation \(Pf=0\) has the solution \(f(t)=e^{1/t}\) on \(t\ne0\), as direct differentiation verifies. It has no finite-pole growth bound on a positive real ray: for any integer \(m\ge0\), its exponential series contains the positive term \(t^{-m-1}/(m+1)!\), so \(t^m e^{1/t}\to+\infty\). CN.2 still gives an actual infinite-order Euler isomorphism and CN.3 is injective. No finite-order Euler gauge is asserted for this example.

Nilpotence cannot simply be omitted from the growth argument. In rank one the equation \(Y'=cY\), \(c\ne0\), has \(Y=a e^{c\zeta}\). Removing a scalar constant monodromy logarithm only multiplies it by \(\zeta^{-H}\). Along the ray \(c\zeta>0\), its growth exceeds \(K_\delta e^{\delta|\zeta|}\) whenever \(0<\delta<|c|\), regardless of that fixed power. It cannot satisfy CN.13 or CN.16: the positive bound in CN.16 sums to \(K_\epsilon e^{\epsilon|\zeta|}\), and its exponentially bounded negative part is bounded on every sufficiently large exterior circle. At the original stalk \(t=0\), the corresponding finite relation \(t-c\) is elliptic and its module is zero; it is outside the nilpotent-head hypothesis.

Convergence of \(T(h)\) is also essential to this particular construction. The series \(\sum_{n\ge0}n!h^n\) satisfies the actual negative factorial bound, but has radius of convergence zero since the ratio of successive terms at any nonzero \(h\) eventually exceeds one. If it were the negative Laurent part of a function holomorphic on an exterior annulus, CN.14 on one fixed circle would give \(n!\le K R_1^n\), which is false: the last \(\lfloor n/2\rfloor\) factors of \(n!\) are at least \((n/2)^{\lfloor n/2\rfloor}\). Thus arbitrary Gevrey matrices require a further argument. The theorem above concerns all ranks in its specified convergent normal class.

![Normal equation, complete coefficient bounds and balanced module comparison](assets/convergent-normal-regularization.png)

*Figure CN.1. The upper diagram gives the actual maps CN.7–CN.20; \(w=\log\zeta\), \(e^{2\pi iH}=M\), and \(A=(H-I)^{\mathsf T}\). The lower square is the natural comparison of CN.21, with its two inverse gauge maps. The growth estimate is CN.12–CN.16; for CN.22 its exponent is exactly \(1/2\). This is a diagram of proved maps and bounds, with no numerical solution sample. Editable drawing source, exact diagram data, PNG and SVG accompany it. Free source context: Kashiwara–Kawai, [Holonomic systems III](https://www.kurims.kyoto-u.ac.jp/~kenkyubu/kashiwara/HolIII.pdf), IV.1 and IV.6–IV.7, for the general embedding problem. CN.1–CN.5 prove the stated convergent-class result and its actual comparison maps here.*


### 5.69. Full Gevrey regularization on smooth branches and holonomic real comparison

#### Sectorial representatives of the full coefficient ring

Let \(U\) be a polydisc in the parameter \(x\). Write \(\mathcal R^\infty(U)\) for the actual coefficient symbols independent of \(t\) and \(z\), with homogeneous variable \(\tau\):
\[
\begin{gathered}
a=\sum_{j\in\mathbf Z}a_j(x)\tau^j,\\
|a_j|_K\le M_{K,\epsilon}\epsilon^j/j!,\\
j\ge0,\quad\text{every }\epsilon>0,\\
|a_{-n}|_K\le M_K C_K^n n!,\\
n\ge1.
\end{gathered}
\tag{SG.1}
\]
The coefficient domain \(U\) is common to the whole sequence. These are exactly the corresponding subring of the full \(B=\mathscr E^\infty\) symbol class in §5.45, AN.4 and IM.1. In this subring operator composition is coefficient convolution: the right symbol has no \(t\)-derivatives, and the left symbol has no spatial-covariable derivatives. It is not a ring of ordinarily convergent Laurent series.

Use a separate ordinary complex variable \(\zeta\) for analytic representatives. Choose the four sectors with central arguments \(j\pi/2\), \(0\le j<4\), angular half-width \(3\pi/8\), and \(|\zeta|>R\). Neighbouring intersections are connected sectors of opening \(\pi/4\); opposite sectors do not intersect. A closed proper subsector means one whose angular closure lies strictly inside the sector. Estimates may use a larger \(R\), while holomorphic representatives are defined on the original exterior sectors.

A sector family consists of holomorphic \(F_j(\zeta,x)\) on these sectors times \(U\), such that on every closed proper subsector and parameter compact,
\[
\begin{gathered}
|F_j(\zeta,x)|\le K_\delta e^{\delta|\zeta|},\\
\text{every }\delta>0,\\
|F_{j+1}-F_j|\le K e^{-c|\zeta|},\\
c>0.
\end{gathered}
\tag{SG.2}
\]
on their overlaps, with indices modulo four. Allow finite refinements of the sector cover. Two families are equivalent if their differences satisfy the second estimate on a common refinement. All estimates here and below are locally uniform in \(x\). Exponentially flat functions form an ideal in the first class: multiply their estimate by that of the other factor with \(\delta<c/2\). Fixed derivatives preserve both classes by the Cauchy formula on an angularly smaller sector and a fixed parameter margin. In the \(\zeta\)-variable the Cauchy circle can have radius a fixed positive multiple of \(|\zeta|\); in \(x\) use the fixed margin. Thus their quotient is a ring with both derivations \(\partial_\zeta,\partial_{x_i}\).

**Theorem SG.1.** There is an isomorphism from \(\mathcal R^\infty(U)\), locally on a smaller parameter polydisc, to this ring of sector families modulo exponentially flat families. It preserves products and the two kinds of derivative. Its positive part is an entire function of minimal exponential type. Its negative part is represented by finite Laplace integrals. The proof includes both directions and the full mixed product.

##### SG.A. Finite Laplace realization of an arbitrary negative tail

Write a negative tail as \(\sum_{n\ge1}b_n(x)\tau^{-n}\), and choose a compact parameter product with interior containing the smaller \(U\), on which one SG.1 bound holds. The series
\[
\beta(s,x)=\sum_{n\ge1}
\frac{b_n(x)}{(n-1)!}s^{n-1}
\tag{SG.3}
\]
converges normally on a disc of fixed positive radius \(\rho\), jointly with \(x\): its terms are bounded by \(MC^n n |s|^{n-1}\). Choose \(0<a<\rho/4\), and on the sector centred at \(\theta_j=j\pi/2\) put
\[
\begin{gathered}
b^{(j)}(\zeta,x)=\\
\int_0^{a e^{-i\theta_j}}e^{-s\zeta}\beta(s,x)\,ds.
\end{gathered}
\tag{SG.4}
\]
Integration is along the straight segment. The real part of \(s\zeta\) is at least \(|s\zeta|\cos(3\pi/8)\) on its whole closed sector. Hence these integrals are \(O(|\zeta|^{-1})\), uniformly on parameter compacts. They are holomorphic, with every fixed derivative obtained under the finite integral.

Neighbouring paths can be joined by the short radius-\(a\) arc within the disc of \(\beta\). On an overlap the real part of \(s\zeta\) is uniformly positive on that entire arc. The closed contour has no singularity, so the difference of the integrals is the integral over the arc, with its oriented sign. Its absolute value is at most \(K e^{-c|\zeta|}\). This also holds on the overlap crossing argument \(2\pi\). Thus SG.4 is a sector family.

It has the common negative expansion with a full factorial remainder:
\[
\begin{gathered}
\left|b^{(j)}-\sum_{n=1}^N b_n\zeta^{-n}\right|\\
\le K A^N N!|\zeta|^{-N-1},\\
N\ge0.
\end{gathered}
\tag{SG.5}
\]
after changing fixed \(K,A\) on a proper subsector. To prove it, integrate by parts \(N\) times. At the initial endpoint the terms are precisely \(b_n\zeta^{-n}\). The integral remainder is \(\zeta^{-N}\int e^{-s\zeta}\partial_s^N\beta(s)\,ds\); the Cauchy derivative bound on one fixed larger disc is \(K A_0^N N!\), and the decaying segment integral supplies \(K/|\zeta|\). The other terms are
\[
-e^{-a e^{-i\theta_j}\zeta}
\sum_{k=0}^{N-1}
\frac{\partial_s^k\beta(ae^{-i\theta_j})}{\zeta^{k+1}}.
\tag{SG.6}
\]
For a term with \(k<N\), use the derivative bound \(K A_0^k k!\), and
\(\sup_{r\ge0} r^{N-k}e^{-cr}\le (N-k)!c^{-(N-k)}\), which follows from the exponential series. Since \(k!(N-k)!\le N!\), the \(N\) terms together satisfy SG.5 after increasing \(A\). The case \(N=0\) is the initial integral bound. No limiting integration endpoint at infinity is taken in SG.4.

The positive part \(p(\zeta,x)=\sum_{m\ge0}a_m(x)\zeta^m\) is entire. SG.1 sums to \(|p|\le M_\epsilon e^{\epsilon|\zeta|}\) for every \(\epsilon\), on the same parameter domain. Define the sector realization of a symbol to be \(p+b^{(j)}\). Different sufficiently small cutoffs or segment directions give exponentially flat differences by the same contour argument, on a common finite refinement.

##### SG.B. Complete Cauchy–Heine descent

The contour variable \(z\) below is scalar and lies in the ordinary \(\zeta\)-plane, distinct from the normalized spatial covariables in the operator ring.

Conversely take a family \(F_j\) satisfying SG.2. If its cover was refined, pass to a finite cyclic subcover with only neighbouring overlaps. Here is an elementary construction: each angle has a small interval with closure in one cover member; finitely many half-sized intervals cover the circle. The minimum of their positive half-radii gives a common margin. Choose a sufficiently fine equally spaced mesh, and intervals about the mesh points of length \(3/2\) times the mesh spacing. Each lies inside some original member by that margin; they cover the circle, only neighbours intersect, and no three intersect. Restrict the given functions to this subcover. Write \(\nu\) for the number of sectors; the original cover has \(\nu=4\).

Choose one outward ray \(\Gamma_j\) in the middle of each overlap, oriented from radius \(R_1>R\) to infinity, and set \(\Delta_j=F_{j+1}-F_j\) there. Include the cyclic last overlap. Its holomorphic extension to a wedge around that ray has \(|\Delta_j|\le K e^{-c|z|}\), with one common positive \(c\) after fixing the finite cover and a parameter compact. The Cauchy integrals
\[
\begin{gathered}
\mathcal C(\zeta,x)=\\
\frac{1}{2\pi i}\sum_{j=0}^{\nu-1}
\int_{\Gamma_j}\frac{\Delta_j(z,x)}{z-\zeta}\,dz.
\end{gathered}
\tag{SG.7}
\]
converge absolutely away from the rays, jointly and normally with the parameters. Restrict them to the components between the rays. Continue to the original sectors by deforming the neighbouring rays within their overlap wedges, joining their fixed finite endpoints by bounded arcs. Such deformation is allowed by holomorphy and exponential decay; the closing arc at radius \(L\) has integral tending to zero as \(L\to\infty\).

When the pole \(\zeta\) crosses a positively oriented outward ray counterclockwise, its residue is \(+\Delta_j(\zeta,x)\). This sign follows also by a positively oriented small circle: the integral of \((z-\zeta)^{-1}\) is \(2\pi i\). The resulting local branches \(E_j\) of SG.7 therefore obey
\[
E_{j+1}-E_j=\Delta_j.
\tag{SG.8}
\]
There are no triple-overlap consistency conditions in this cover. The equality on the cyclic overlap is the same residue calculation. Thus \(G=F_j-E_j\) glues to a single-valued holomorphic matrix or function on an exterior annulus times \(U\).

Here are all the required estimates and coefficients of these integrals. On a closed subsector separated from a ray, \(|z-\zeta|\ge\gamma(|z|+|\zeta|)\) for one \(\gamma>0\). This inequality follows by minimizing \(|re^{i\theta}-se^{i\phi}|/(r+s)\) over the compact set \(r+s=1\) and the compact separated angular sets; it has no zero there. SG.7 is then \(O(|\zeta|^{-1})\). For subsectors meeting a chosen ray, deform that ray to either side within its wedge so that it is angularly separated from the smaller subsector. The added bounded arcs also give \(O(|\zeta|^{-1})\) once \(|\zeta|>2R_1\). The residue correction is exponentially flat. A finite list of these choices covers the whole circle and gives the same bound on every smaller sector.

The moments
\[
\begin{gathered}
\mu_n(x)=\\
\frac{1}{2\pi i}\sum_j
\int_{\Gamma_j}\Delta_j(z,x)z^n\,dz.
\end{gathered}
\tag{SG.9}
\]
are holomorphic in \(x\), and have \(|\mu_n|_K\le K A^{n+1}n!\). Indeed, parametrization of each ray bounds its integral by
\(K\int_0^\infty e^{-cr}r^n\,dr=K n!/c^{n+1}\); the integral identity follows by repeated integration by parts, starting at \(n=0\). Bounded connecting arcs only change the exponential constant in this bound. The moments are unchanged under the deformations with fixed finite endpoints, by the same holomorphic contour calculation, now with no pole.

Use the finite identity
\[
\begin{gathered}
\frac1{z-\zeta}
=-\sum_{n=0}^{N-1}\frac{z^n}{\zeta^{n+1}}\\
{}+\frac{z^N}{\zeta^N(z-\zeta)}.
\end{gathered}
\tag{SG.10}
\]
The separated-ray estimate and SG.9 bound its integrated remainder by \(K A^N N!|\zeta|^{-N-1}\). The bounded arcs have the same estimate by their finite radius and a larger \(A\). If a residue correction occurs, its exponential bound implies this same estimate for every \(N\), by the exponential-series inequality used for SG.6. Consequently every \(E_j\) has the same Gevrey negative expansion
\[
E_j\sim-\sum_{n\ge0}\mu_n(x)\zeta^{-n-1}.
\tag{SG.11}
\]

The glued function \(G\) has minimal exponential type on the entire exterior annulus, uniformly in \(x\): cover the circle by the finite smaller sectors and use SG.2 and the bound for \(E_j\). By the complete Laurent argument §5.68, CN.14–CN.16, its coefficients \(g_m(x)\) satisfy the positive every-epsilon bounds and an exponential bound for \(g_{-n}\). Its nonnegative part \(P=\sum_{m\ge0}g_m\zeta^m\) is entire of minimal exponential type. Thus
\[
\begin{gathered}
F_j=P+Q_j,\\
Q_j\sim\sum_{n\ge1}q_n(x)\zeta^{-n},\\
q_n=g_{-n}-\mu_{n-1}
\end{gathered}
\tag{SG.12}
\]
with the full remainder estimate SG.5. This constructs a symbol satisfying SG.1 from every admissible sector family.

For uniqueness, suppose two such decompositions represent equivalent families. The difference of their entire positive parts is \(O(|\zeta|^{-1})\) on every sector, because each negative representative has that bound and the remaining differences are exponentially flat. A finite smaller cover makes this a uniform bound on every large circle. The Cauchy coefficient bound then makes every nonnegative Taylor coefficient zero, including the constant term. The positive parts coincide. Negative asymptotic coefficients are unique by successively multiplying a remainder by \(\zeta^n\) and taking the limit along one ray.

Finally a zero negative expansion is precisely an exponentially flat remainder in this class. In SG.5 choose \(N=\lfloor|\zeta|/(2A)\rfloor\), and use \(N!\le N^N\); the bound is at most \(K2^{-N}/|\zeta|\), hence \(K'e^{-c'|\zeta|}\) outside a larger radius. Conversely an exponential bound implies a zero expansion with SG.5 for every \(N\), since \(e^{-cr}r^{N+1}\le (N+1)!c^{-N-1}\), and the extra factor \(N+1\) is absorbed by increasing \(A\). The uniqueness argument just given then identifies its symbol with zero. This proves choice independence, injectivity, and surjectivity of the coefficient/sector correspondence.

##### SG.C. Exact multiplication, including the two infinite mixed sums

Positive parts multiply as entire functions. Their coefficient bound follows by the binomial identity after using input epsilons smaller than half the requested output epsilon. Negative Laplace parts multiply by convolution of their analytic Borel functions, modulo an exponentially flat endpoint term. In fact choose both cutoffs on the same ray with \(2a<\rho\), and apply Fubini to the finite product of segments. On the triangle where \(u=s_1+s_2\) has modulus at most \(a\), its integral is
\[
\begin{gathered}
\int_0^{ae^{-i\theta}}
e^{-u\zeta}(\beta*\gamma)(u)\,du,\\
(\beta*\gamma)(u)=\\
\int_0^u\beta(s)\gamma(u-s)\,ds.
\end{gathered}
\tag{SG.13}
\]
The remaining part has \(|u|\ge a\), so is exponentially flat on a smaller sector. The convolution in parentheses is analytic on a fixed disc. Integrating the two Taylor monomials gives the beta integral
\[
\begin{gathered}
\int_0^u s^{p-1}(u-s)^{q-1}\,ds\\
=\frac{(p-1)!(q-1)!}{(p+q-1)!}\\
{}\cdot u^{p+q-1}.
\end{gathered}
\tag{SG.13a}
\]
This identity follows by repeated integration by parts. Hence the negative coefficients are the ordinary finite convolution. Matrix multiplication retains its written order.

For a positive part \(p(\zeta)=\sum_{m\ge0}p_m\zeta^m\) and negative part with Borel function \(\beta\), integrate by parts \(m\) times:
\[
\begin{gathered}
\zeta^m\int_0^a e^{-s\zeta}\beta(s)\,ds\\
=\sum_{k=0}^{m-1}\beta^{(k)}(0)\zeta^{m-1-k}\\
{}+\int_0^a e^{-s\zeta}\beta^{(m)}(s)\,ds\\
-e^{-a\zeta}
\sum_{k=0}^{m-1}\beta^{(k)}(a)\zeta^{m-1-k}.
\end{gathered}
\tag{SG.14}
\]
Here \(a\) denotes the chosen complex endpoint and integration direction; the derivative formula is unchanged by its argument. For \(m=0\) the two finite sums are empty. On a fixed neighbourhood of that segment, the derivative bound is \(K L^m m!\). Thus \(\sum_m p_m\beta^{(m)}(s)\) converges normally by SG.1, choosing \(\epsilon L<1/2\). This is an analytic Borel function on one fixed smaller neighbourhood.

The nonnegative and negative output coefficients obtained from SG.14 are exactly
\[
\begin{gathered}
c_l=\sum_{n\ge1}p_{l+n}b_n,\quad l\ge0,\\
c_{-n}=\sum_{m\ge0}p_m b_{n+m},\\
n\ge1.
\end{gathered}
\tag{SG.15}
\]
Both sums converge absolutely. The first is bounded by
\(K_\epsilon\epsilon^l/l!\sum_{n\ge1}(\epsilon C)^n\), since \(n!/(l+n)!\le1/l!\). For the second, the exact majorant is
\[
\begin{gathered}
K_\epsilon C^n n!\\
{}\cdot\sum_{m\ge0}\binom{n+m}{m}(\epsilon C)^m\\
=\frac{K_\epsilon C^n n!}{(1-\epsilon C)^{n+1}}.
\end{gathered}
\tag{SG.16}
\]
The binomial sum follows by differentiating the geometric series \(n\) times and dividing by \(n!\). Fix one \(\epsilon C<1/2\) for the negative bound. For every requested positive output epsilon choose the input epsilon smaller if needed. The domain of the Borel function and the coefficient domain were fixed before this choice.

The last line of SG.14 sums to an exponentially flat term. Indeed its nonnegative coefficient series has the first bound in SG.15, using instead the Cauchy bounds for \(\beta^{(k)}(a)\) on a fixed disc at the endpoint. It is therefore an entire function of minimal exponential type, multiplied by \(e^{-a\zeta}\); choose its type below half the positive decay rate of \(a\zeta\). All rearrangements are justified by these absolute bounds and the normally convergent derivative sum. The product with the negative factor on the left uses exactly the same argument, with the matrix entries in their original order. Thus no mixed infinite contraction is discarded.

SG.13–SG.16 prove that the sector correspondence sends the actual coefficient convolution, including both tails, to the sector product modulo exponentially flat families. Differentiating SG.4 multiplies its Borel function by \(-s\), giving precisely the coefficient derivative of \(\sum b_n\tau^{-n}\). Differentiation of the positive entire function is termwise. Parameter differentiation commutes with all integrals by their uniform bounds on a larger compact. These facts prove the derivative assertions and complete Theorem SG.1.

Free context for sectorial Cauchy integrals is the author preprint by Stéphane Malek, [On singularly perturbed small step size difference-differential nonlinear PDEs](https://web.ma.utexas.edu/mp_arc/c/13/13-52.pdf), §5.1. SG.A–SG.C prove the exact minimal-type, factorial-negative, parameter-dependent ring statement needed here; no external summation theorem is used as a premise.


#### A full-coefficient Euler gauge for flat Gevrey normal connections

Use the parameter-dependent coefficient ring of SG.1. Let \(T(h,x)\) and \(Z_i(h,x)\), \(1\le i\le d\), be \(r\)-by-\(r\) matrices in its nonnegative-\(h\) subring \(R_0\), where \(h=\tau^{-1}\). Thus all coefficients have the actual factorial bound on one common parameter domain, and are not assumed convergent at a numerical \(h\ne0\). Assume
\[
\begin{gathered}
T(0,x)^k=0,\quad k\ge1,\\
h^2\partial_hZ_i-h\partial_iT\\
{}+[T,Z_i]=hZ_i,\\
h(\partial_iZ_j-\partial_jZ_i)\\
{}+[Z_i,Z_j]=0.
\end{gathered}
\tag{FG.1}
\]
These are precisely the nilpotence and flatness identities of §5.55, DP.25–DP.26, with coefficient columns. This section constructs actual inverse matrices \(V,W\) in \(\mathcal R^\infty\), on one smaller parameter domain, and a constant complex matrix \(H\), such that
\[
\begin{gathered}
VW=WV=I,\\
\partial_\tau V=TV-\tau^{-1}VH,\\
\partial_iV+\tau Z_iV=0.
\end{gathered}
\tag{FG.2}
\]
Every positive epsilon, the negative factorial bound, all fixed derivatives and the actual inverse are included.

##### FG.A. Sectorial matrices and their curvature errors

Apply SG.A simultaneously to the finitely many matrices \(T,Z_i\), with a common small Borel disc, cutoff, four-sector cover and parameter polydisc. Denote the sector representatives by \(T_j,Z_{i,j}\). Put \(N(x)=T(0,x)\). They are holomorphic and satisfy, uniformly on smaller sectors and parameter compacts,
\[
\begin{gathered}
T_j=N+O(|\zeta|^{-1}),\\
Z_{i,j}=O(1).
\end{gathered}
\tag{FG.3}
\]
Neighbouring differences are exponentially flat. By the exact ring and derivative correspondence SG.C, FG.1 makes both curvature matrices of
\[
\partial_\zeta-T_j,\qquad
\partial_i+\zeta Z_{i,j}
\tag{FG.4}
\]
exponentially flat. Explicitly the mixed curvature is
\(\partial_\zeta(\zeta Z_{i,j})+\partial_iT_j-[T_j,\zeta Z_{i,j}]\); substituting \(\partial_\zeta=-h^2\partial_h\) in its formal coefficient symbol gives minus \(h^{-1}\) times the second line of FG.1 moved to the left. The tangential curvature is
\(\zeta(\partial_iZ_{a,j}-\partial_aZ_{i,j})+\zeta^2[Z_{i,j},Z_{a,j}]\), whose symbol is zero by its last line. Multiplication by these finite powers of \(\zeta\) preserves exponential flatness.

On each simply connected exterior sector solve
\[
\begin{gathered}
\partial_\zeta Y_j=T_jY_j,\\
Y_j(\zeta_j,x)=I.
\end{gathered}
\tag{FG.5}
\]
where \(\zeta_j\) is a fixed point on its central ray. The locally normally convergent iterated-integral proof, its inverse equation and path-homotopy continuation are §5.68, CN.4–CN.5. It applies with holomorphic parameters: the same factorial integral bound is uniform on each parameter compact, so all local solutions, inverses and continuations are jointly holomorphic. A finite disc cover of each path suffices. No convergence of the original series \(T(h,x)\) is asserted.

##### FG.B. Uniform subexponential estimates without a varying Jordan basis

Let \(\kappa=\max(2,k)\), \(\alpha=1-1/\kappa<1\). For a column solution \(v\) of FG.5 on a radial line \(\zeta=s e^{i\theta}\), stack the \(k\) columns
\[
\mathcal S(v)=
\begin{pmatrix}
v\\s^{1/\kappa}Nv\\\vdots\\
s^{(k-1)/\kappa}N^{k-1}v
\end{pmatrix}.
\tag{FG.6}
\]
The parameter \(x\) is fixed during radial differentiation; \(N(x)^k=0\). Write \(T_j=N+R_j\), with \(\|R_j\|\le K/s\). Component \(a\), \(0\le a<k\), of the radial derivative of this stack is
\[
\begin{gathered}
\frac{a}{\kappa s}\mathcal S_a(v)
+e^{i\theta}s^{-1/\kappa}\mathcal S_{a+1}(v)\\
{}+e^{i\theta}s^{a/\kappa}N^aR_j\mathcal S_0(v),
\end{gathered}
\tag{FG.7}
\]
where the component with index \(k\) is zero. Since \(N^a\) is uniformly bounded on a parameter compact and \(a\le k-1\le\kappa-1\), this is a finite linear system with coefficient norm at most \(K s^{-1/\kappa}\), after increasing \(K\). Its initial stacks are bounded uniformly on a compact initial arc and the parameter compact. Iteration of its integral equation bounds it by \(K_1\exp(K_2s^\alpha)\), as in CN.11–CN.12.

Rows of \(Y_j^{-1}\) satisfy \(v'=-e^{i\theta}v(N+R_j)\). Stack instead \(s^{a/\kappa}vN^a\). The same formula has the shift term with a minus sign and error \(-e^{i\theta}s^{a/\kappa}vR_jN^a\); its norm has the same bound. Thus
\[
\|Y_j\|+\|Y_j^{-1}\|
\le K_1e^{K_2|\zeta|^\alpha}
\tag{FG.8}
\]
on every smaller sector, uniformly in \(x\). Their parameter derivatives have minimal exponential type as well: apply the Cauchy estimate on one larger parameter compact. No holomorphic choice of Jordan basis for \(N(x)\) was made. In particular every product of a fixed finite list of these matrices and fixed derivatives has minimal exponential type.

##### FG.C. Tangential horizontal frames modulo flat errors

In the frame \(Y_j\) the tangential coefficient matrices are
\[
\begin{gathered}
B_{i,j}=Y_j^{-1}\partial_iY_j\\
{}+\zeta Y_j^{-1}Z_{i,j}Y_j.
\end{gathered}
\tag{FG.9}
\]
Differentiating this expression and using FG.5 gives exactly
\[
\begin{gathered}
\partial_\zeta B_{i,j}
=Y_j^{-1}\mathcal F_{i,j}Y_j,\\
\mathcal F_{i,j}=
\partial_iT_j+\partial_\zeta(\zeta Z_{i,j})\\
{}-[T_j,\zeta Z_{i,j}].
\end{gathered}
\tag{FG.10}
\]
The middle factor is exponentially flat by FG.A, and FG.8 absorbs less than half its exponential decay rate. Thus \(\partial_\zeta B_{i,j}\) is exponentially flat.

It follows that \(B_{i,j}\) has a holomorphic limit \(C_{i,j}(x)\) at radial infinity, independent of the ray within each sector, and
\[
B_{i,j}-C_{i,j}=O(e^{-c|\zeta|}).
\tag{FG.11}
\]
For completeness, integrate its derivative along a radial ray: the tail is bounded by \(K\int_{|\zeta|}^\infty e^{-cs}ds\). To compare two rays in a closed smaller sector, integrate along the circular arc of radius \(L\); the bound \(KL e^{-cL}\) tends to zero. This proves equality of their limits. A connected sector is exhausted by such smaller sectors. The convergence is uniform on parameter compacts, so the Cauchy formula passes to its limit and all fixed parameter derivatives. This proves holomorphy and FG.11.

The tangential curvature in the frame \(Y_j\) is the conjugate of the tangential curvature in FG.4, hence exponentially flat. Take the limit in
\(\partial_iB_{a,j}-\partial_aB_{i,j}+[B_{i,j},B_{a,j}]\). It gives
\[
\begin{gathered}
\partial_iC_{a,j}-\partial_aC_{i,j}\\
{}+[C_{i,j},C_{a,j}]=0.
\end{gathered}
\tag{FG.12}
\]
On a smaller simply connected parameter product there is an invertible holomorphic matrix \(U_j(x)\) with \(\partial_iU_j=-C_{i,j}U_j\) for every \(i\). The complete construction is the ordinary parameter-ODE argument of §5.55, DP.7.2, now with ordinary holomorphic matrix norms. Solve successively along the coordinate segments by iterated integrals; after each step flatness makes the remaining coefficient matrices independent of the preceding coordinates. Solve the inverse equations in the opposite order. The factorial integral bounds prove convergence, and differentiation of the products proves both inverse identities. There are only finitely many coordinates and four sectors, so a single smaller product works for all these matrices.

Put \(A_j=Y_jU_j\). Equations FG.9–FG.11 then give
\[
\begin{gathered}
\partial_\zeta A_j=T_jA_j,\\
(\partial_i+\zeta Z_{i,j})A_j
=O(e^{-c|\zeta|}).
\end{gathered}
\tag{FG.13}
\]
Both \(A_j\) and \(A_j^{-1}\) have minimal exponential type. Multiplication of the flat error by any one of them remains exponentially flat.

##### FG.D. Constant overlap matrices, including parameter constancy

On a neighbouring overlap set \(R_j=A_j^{-1}A_{j+1}\). Its \(\zeta\)-derivative is
\[
\begin{gathered}
\partial_\zeta R_j=\\
A_j^{-1}(T_{j+1}-T_j)A_{j+1}.
\end{gathered}
\tag{FG.14}
\]
which is exponentially flat. The same is true for the derivative of its inverse. The radial-tail and circular-arc argument of FG.C gives limits \(L_j(x)\) and \(L_j(x)^{-1}\), with exponentially flat remainders. They are mutually inverse: take the limit in \(R_jR_j^{-1}=I\). In particular \(L_j\) is invertible.

Differentiating \(R_j\) in \(x_i\), and using FG.13 on both sides, gives
\[
\begin{gathered}
\partial_iR_j=\\
\zeta A_j^{-1}(Z_{i,j}-Z_{i,j+1})\\
{}\cdot A_{j+1}+O(e^{-c|\zeta|}).
\end{gathered}
\tag{FG.15}
\]
The first term is also exponentially flat. Uniform convergence on parameter compacts therefore implies \(\partial_iL_j=0\). Since the parameter polydisc is connected, each \(L_j\) is one constant complex matrix. This proves constancy in the parameters as well as in the radial variable.

On the lifted ordered sector cover, use successive constant right changes of frame to make the overlap limits the identity except at the cyclic seam. More explicitly put \(J_0=I\) and \(J_{j+1}=L_j^{-1}J_j\) for \(j=0,1,2\). Then \(J_j^{-1}L_jJ_{j+1}=I\). At the last overlap put \(M=L_3^{-1}J_3\). Its comparison in the direction \(3\to0\) is \(J_3^{-1}L_3=M^{-1}\), so \(A_3J_3=A_0M\) modulo a flat matrix there. Equivalently the continued lifted frame at \(w+2\pi i\) is the frame at \(w\) multiplied on the right by \(M\), modulo exponentially flat functions. This fixes the traversal order for noncommuting transition matrices.

Choose a constant \(H\) with \(\exp(2\pi iH)=M\), using the finite primary-decomposition/logarithm construction CN.6. On each lifted sector set
\[
\begin{gathered}
V_j=A_jJ_j e^{-w_jH},\\
w_j=\log\zeta.
\end{gathered}
\tag{FG.16}
\]
On the first three overlaps the log branches agree. At the seam the frame changes by right \(M\) and the branch by \(2\pi i\), so \(M e^{-2\pi iH}=I\) cancels it. Hence the \(V_j\) differ only by exponentially flat matrices on every overlap. Their inverses have the same property, since
\(V_j^{-1}-V_{j+1}^{-1}=V_j^{-1}(V_{j+1}-V_j)V_{j+1}^{-1}\).

The factors \(e^{\pm w_jH}\) have only fixed-power growth on each lifted sector, so both families still have minimal exponential type. These factors are used only in the analytic sector representatives; no fractional power of \(\tau\) is adjoined to the operator ring. SG.1 supplies actual integer-indexed coefficient matrices \(V,W\). Their sector products are the identity, and SG.C gives \(VW=WV=I\) as actual symbols.

Finally FG.13 and differentiation of FG.16 give
\[
\begin{gathered}
\partial_\zeta V_j
=T_jV_j-\zeta^{-1}V_jH,\\
(\partial_i+\zeta Z_{i,j})V_j
=O(e^{-c|\zeta|}).
\end{gathered}
\tag{FG.17}
\]
The exact derivative and product correspondence of SG.C turns these into FG.2. This proves the full-coefficient flat-connection gauge for arbitrary factorial-divergent input matrices satisfying FG.1. The construction uses ordinary sector ODEs for analytic representatives and then the proved coefficient descent; it never evaluates a divergent symbol at a numerical \(h\).


#### Actual real-module comparison for every holonomic germ

Let \(X\) be a complex manifold of dimension \(n\ge1\). At nonzero cotangent points use the actual finite-order ring \(E=\mathscr E_X\), its full geometric infinite-order extension \(B=\mathscr E_X^\infty\), and the real kernel ring \(C=\mathscr E_X^{\mathbb R}\), with the source products and input-density conventions of §§5.34–5.45. Tensor products are ordinary algebraic sheaf tensor products.

**Theorem GM.1.** For every coherent holonomic actual left \(E\)-module \(M\), including singular and nonreduced characteristic support, the natural comparison is injective:
\[
B\otimes_E M\longrightarrow C\otimes_E M.
\tag{GM.1}
\]
The assertion holds for the whole module and every stalk. It is natural for all actual \(E\)-linear maps between these modules. It does not assert flatness of \(C\) over \(B\), or supply a globally coherent finite-order regularization of \(M\).

The proof uses SG.A–SG.C and FG.A–FG.D, then two distinct applications of the exact support-purity theorem PG.1. We first give the finite-presentation argument needed to apply the full-coefficient gauge to an actual module.

##### GM.A. Recovering the whole actual finite presentation

Use a standard smooth branch \(\Lambda=\{t=z=0\}\), \(z_i=\xi_i/\tau\), \(h=\tau^{-1}\). Let \(M_0\subset M\) be the actual coherent \(E_0\)-lattice supplied by §5.32, J.1–J.2, where \(E_0\) consists of order-at-most-zero operators. On the generic parameter open of §5.55, DP.3–DP.4, choose its actual basis
\[
M_0=R_0u_1\oplus\cdots\oplus R_0u_r.
\tag{GM.2}
\]
Here \(R_0\) is the commutative ring of factorial-growth series in \(h\) with holomorphic \(x\)-coefficients. The basis is for the entire lattice, not only its reduction or completion. The operators \(t,z_i\) act by
\[
\begin{gathered}
t=h^2\partial_h+T(h,x),\\
z_i=h\partial_i+Z_i(h,x).
\end{gathered}
\tag{GM.3}
\]
The matrices satisfy FG.1, and their leading matrices commute and are jointly nilpotent, by DP.22–DP.26.

Let \(Q_0\) be the actual left \(E_0\)-module with the finite generators \(v_j\) and relations
\[
\begin{gathered}
t v_j=\sum_k T_{kj}v_k,\\
z_i v_j=\sum_k (Z_i)_{kj}v_k.
\end{gathered}
\tag{GM.4}
\]
The evident map \(q:Q_0\to M_0\), \(v_j\mapsto u_j\), is surjective: the \(u_j\) already generate over \(R_0\subset E_0\). We prove it is injective in the actual rings.

Modulo \(h\) the ring is \(S=\mathbf C\{x,t,z\}\), by DP.1. Put \(N_0=T(0,x)\), \(N_i=Z_i(0,x)\). They commute, and every sufficiently long product of them is zero. The presentation GM.4 reduces to
\[
\begin{gathered}
S^r\big/\sum_{a=0}^d(w_a I-N_a)S^r,\\
(w_0,w_1,\ldots,w_d)\\
=(t,z_1,\ldots,z_d).
\end{gathered}
\tag{GM.5}
\]
This is canonically \(\mathbf C\{x\}^r\), by evaluating each scalar coefficient function at the commuting nilpotent matrices. The evaluation is finite on its Taylor expansion: all monomials of sufficiently high total degree vanish.

Here is also a convergent proof that the displayed relations are the entire evaluation kernel. In one variable the identity
\[
\begin{gathered}
w^mI-N^m=\\
(wI-N)\sum_{a=0}^{m-1}w^{m-1-a}N^a.
\end{gathered}
\tag{GM.6}
\]
has at most \(k\) nonzero summands if \(N^k=0\). Multiply by the coefficients of a convergent Taylor series and sum. On a fixed smaller disc this is normally convergent, because \(a<k\) and the original scalar Taylor coefficients have one geometric bound; only finitely many powers of \(N\) occur. Apply this successively in \(t,z_1,\ldots,z_d\). All matrices commute, so each subsequent evaluation respects the earlier factors. This writes every vector as its evaluated constant vector plus a finite sum of the relation matrices times holomorphic vectors. Evaluation kills those relation vectors, and is the identity on constant vectors. Thus both inverse maps in GM.5 are established.

The reduction of \(q\) modulo \(h\) is consequently an isomorphism, since GM.2 gives the same basis of \(M_0/hM_0\). Let \(K=\ker q\). It is finite over \(E_0\) by the actual Noetherian theorem §5.21, N.1–N.18. The image \(hQ_0\) is a submodule by the proved normality of \(h\). Since \(h\) is injective on \(M_0\subset M\),
\[
K\cap hQ_0=hK.
\tag{GM.7}
\]
Indeed if \(k=hq_0\) belongs to \(K\), then \(h\,q(q_0)=0\), so \(q(q_0)=0\) and \(q_0\in K\). The converse inclusion is immediate. Therefore \(K/hK\) injects into \(Q_0/hQ_0\), and its image is the zero kernel of the reduced isomorphism. Hence \(K=hK\).

Choose finitely many generators of \(K\), and express each as \(h\) times a finite combination of them. This gives a matrix relation \((I-hA)k=0\), with the corresponding finite row/column convention. The actual order-negative geometric-series inverse exists by §5.20, B.3 and §5.55, DP.20: for a finite actual matrix \(A\), the norm of \(hA\) tends to zero with the negative-degree norm parameter, and the normally convergent series \(\sum_{m\ge0}(hA)^m\) is its two-sided inverse on one common domain. Thus all generators vanish and \(K=0\).

This proves \(Q_0=M_0\) without using formal freeness, completion alone, or an asserted presentation over \(R_0\). Localizing normally by \(h\) gives the finite presentation of the entire \(M=EM_0\). Its relations can equivalently be written
\[
\begin{gathered}
t u_j=\sum_k T_{kj}u_k,\\
D_{x_i}u_j=\sum_k(\tau Z_i)_{kj}u_k.
\end{gathered}
\tag{GM.8}
\]
The zero-rank case is the zero module and needs no gauge.

##### GM.B. Actual inverse Euler maps on the generic parameter open

For \(r>0\), apply FG.2 to these full factorial-growth matrices. In \(B\otimes_E M\) set \(e_j=\sum_k V_{kj}u_k\). The coefficient commutators are
\[
\begin{gathered}
{}[t,V]=-\partial_\tau V,\\
[D_{x_i},V]=\partial_iV.
\end{gathered}
\tag{GM.9}
\]
They follow from the actual normal product, because the entries of \(V\) depend only on \(x,\tau\). The identities FG.2 and the actual presentation GM.8 give
\[
\begin{gathered}
t e_j=\sum_k \tau^{-1}H_{kj}e_k,\\
D_{x_i}e_j=0,\\
tD_t e_j=\sum_k(H-I)_{kj}e_k.
\end{gathered}
\tag{GM.10}
\]
Thus there is an actual \(B\)-linear map from the constant Euler presentation \(B\otimes_E N_{(H-I)^{\mathsf T}}\) to \(B\otimes_E M\).

Its inverse is \(u_j=\sum_k W_{kj}e_k\). Differentiating \(WV=I\) in FG.2 gives
\[
\begin{gathered}
\partial_\tau W=\\
-WT+\tau^{-1}HW,\\
\partial_iW=\tau WZ_i.
\end{gathered}
\tag{GM.11}
\]
In the Euler presentation these equations make the proposed \(u_j\) satisfy precisely GM.8. The two finite maps between the actual full presentations are inverse by \(VW=WV=I\). This proves a \(B\)-module isomorphism of the entire generic germ, with all coefficient tails, every parameter and every fixed derivative.

The complete constant Euler theorem §5.53, AM.29 proves injectivity of its comparison with \(C\). Transport it by this isomorphism. The exact balanced isomorphism
\[
\begin{gathered}
C\otimes_B(B\otimes_E M)\\
\simeq C\otimes_E M.
\end{gathered}
\tag{GM.12}
\]
sends \(c\otimes(b\otimes m)\) to \(cb\otimes m\), with inverse \(c\otimes m\mapsto c\otimes(1\otimes m)\); both balancing relations and both inverse identities hold on each generator. Applying the functor \(C\otimes_B-\) to the two inverse gauge maps preserves their inverse identities without a flatness assumption. The resulting commuting comparison square proves GM.1 on this generic parameter open.

At each of these points the finite halfspace maps EE.21–EE.27 also transport to separate the entire \(B\)-module. This pointwise generic separation is not asserted to be one common finite solution family continued across a singular characteristic germ.

##### GM.C. First purity step: every point of a smooth characteristic branch

Fix any point whose reduced characteristic support is smooth and has no other component through it. The actual finite quadratic contact shear and ordinary time shift of §5.55, DP.6 put that branch into \(\Lambda=\{t=z=0\}\) on a small normalized chart. The whole \(E,B,C\) transports are compatible, by §5.56, BCT.9 and §5.67, WT.12. Their explicit balanced tensor maps identify the natural comparison squares. In particular this use involves the real-ring contact theorem, not only a finite-order quantization.

On that standard chart DP.2 makes \(M_0\) finite over \(R_0\). Its residue \(M_0/hM_0\) is a finite \(\mathbf C\{x\}\)-module by DP.3. The complete generic-minor argument of DP.3 gives a nonzero holomorphic \(g(x)\) such that this residue, and hence by DP.4 the entire lattice, is free over its respective coefficient ring on \(\{g\ne0\}\). If the residue is already free use \(g=1\). For \(d=0\) there is no parameter exceptional locus. Therefore GM.B proves injectivity off
\[
Z=\Lambda\cap\{g(x)=0\}
\tag{GM.13}
\]
in this chart.

The set \(Z\) is a closed complex analytic subset of the normalized ambient chart. Since \(g\) is nonzero, its dimension is at most \(d-1=n-2\), by the analytic dimension-drop proof at the precise DP.3 prerequisite. The normalized ambient dimension is \(2n-1\), so
\[
\operatorname{codim}Z\ge n+1.
\tag{GM.14}
\]
If \(g=1\), take \(Z\) empty.

Let \(s\) be any local section in the kernel of GM.1. On the complement of \(Z\) it is zero by generic injectivity and the zero module off \(\Lambda\). Thus it is a section of \(B\otimes_E M\) supported on this fixed analytic \(Z\). The full support-purity theorem §5.51, PG.1–PG.2 gives \(\mathcal H_Z^0(B\otimes_E M)=0\), since \(\operatorname{codim}Z>n\). Hence \(s=0\). This proves injectivity at the original smooth-branch point, including any principal nilpotents of \(M\).

This first application of purity is local to the chosen standard chart. It does not require the various parameter-minor exceptional sets to extend analytically across the singular locus of the original characteristic support.

##### GM.D. Second purity step: all singular and intersecting germs

On a normalized chart for the original holonomic \(M\), choose the fixed closed analytic set \(W\) from §5.51, PG.2: it contains the singular and finite discriminant exceptional loci and all component intersections. Its ambient codimension is at least \(n+1\). Outside \(W\) each nonzero branch is smooth and disjoint from the others. GM.C proves injectivity at every such point, not merely at a generically chosen subset. Off the characteristic support both extensions have zero source.

For a local kernel section \(s\) of GM.1, all stalks off \(W\) are consequently zero. The sheaf axiom makes its restriction there zero. Apply PG.1 once more, now with this fixed analytic \(W\) and the entire original \(M\), to conclude \(s=0\). For \(n=1\), PG.2 permits \(W\) empty and the conclusion follows directly from GM.C. The zero module is immediate. This proves Theorem GM.1 for all the stated germs. The support of the kernel itself need not be analytic or coherent; its containment in the fixed analytic exceptional set is enough.

##### GM.E. Naturality and finite matrix relation lifting

Every comparison in GM.1 is the ordinary natural tensor unit from \(B\) to \(C\). For an actual \(E\)-linear map \(f:M\to N\), both composites on a simple tensor \(b\otimes m\) are \(b\otimes f(m)\). Finite sums give the natural commuting square. Stalks of sheaf tensors are the algebraic stalk tensors, by the finite generator-and-balancing argument FEP.9. An injective map on every stalk has zero sheaf kernel by the sheaf axiom. Thus the proved stalk statement gives exactly the asserted natural sheaf injection.

For an explicit finite presentation, write generators as a column and let \(P\in E^{a\times b}\) be a relation matrix acting on left coefficient rows by \(q\mapsto qP\). Suppose
\[
M=E^{1\times b}/E^{1\times a}P
\tag{GM.15}
\]
is holonomic. Ordinary tensor right exactness identifies its extensions with the corresponding row quotients over \(B\) and \(C\). Hence the following precise finite relation assertion holds:
\[
\begin{gathered}
\beta\in B^{1\times b},\quad c\in C^{1\times a},\\
\beta=cP\\
\Longrightarrow\quad \exists a'\in B^{1\times a},\\
\beta=a'P.
\end{gathered}
\tag{GM.16}
\]
Indeed the class of \(\beta\) maps to zero in the \(C\)-quotient; injectivity of GM.1 makes it zero in the \(B\)-quotient. Conversely this matrix assertion for every holonomic finite presentation is precisely GM.1 on stalks. All its sums are finite algebraic presentation sums; its entries retain their entire allowed tails. A finite family of germ entries and balancing witnesses is represented on one common smaller neighbourhood, as in FEP.9. No uniform domain for every germ of every module is claimed.

The theorem establishes this holonomic comparison and its exact finite relation criterion. It does not imply arbitrary-module faithful flatness, higher derived scalar-extension statements for \(C/B\), a common continued solution family through singular characteristic strata, a globally coherent finite-order regularization, finite D-type realization, or the general analytic proper-image and Riemann–Hilbert theorems.


![Sectorial representatives, full inverse gauges and both purity steps](assets/sectoral-gevrey-and-real-comparison.png)

*Figure SG–GM.1. The ordinary complex variable is \(\zeta\); the radial coordinate in the left panel is \(|\zeta|/R_1\). The four sectors have centres \(j\pi/2\), half-width \(3\pi/8\), and adjacent overlaps of opening \(\pi/4\). The outward Cauchy rays have arguments \(j\pi/2+\pi/4\) and continue to infinity beyond the displayed segments. The right panel is FG.5–FG.17 and the exact descent SG.7–SG.16, with \(\alpha=1-1/\max(2,k)\) when \(N^k=0\). The lower panel is the two distinct support-purity applications GM.C–GM.D. This is a diagram of exact angles, actual maps and proved bounds, without a numerical solution sample. Editable drawing source, exact data, PNG and SVG accompany it.*


### 5.70. Finite monodromy envelopes for full special operators

#### PM.A. Domains, operator class and the assertion

Let \(D_R=\{s\in\mathbf C:|s|<R\}\), let \(U\) be a parameter polydisc, and let
\(P(s,x)\) be a reduced monic Weierstrass polynomial of degree \(m\).
Choose \(0<r<R\) so that all its roots lie in \(D_{r/2}\), and choose a
constant anchor \(a\) with \(r<|a|<R\). Put
\[
\begin{gathered}
S=\{P=0\},\\ B=U\setminus H,\\
Z_R=(D_R\times B)\setminus S,\\
Z_r=(D_r\times B)\setminus S .
\end{gathered}
\tag{PM.1}
\]
Here \(H\) is the discriminant locus; a larger analytic exceptional
set is allowed. Work on a connected component of \(B\), and assume the
polynomial has \(m\) distinct roots there. The empty-root case \(m=0\)
is allowed. Write \(\pi\) for the fundamental group of \(Z_r\), with
a fixed base point. For the product \(\gamma\delta\), traverse
\(\delta\) first and then \(\gamma\). Analytic continuation therefore
satisfies \(T_{\gamma\delta}=T_\gamma T_\delta\).

The universal cover used here can be constructed from
endpoint-fixed homotopy classes of paths from the base point.
An endpoint ball contained in the complement gives a lifted
open chart by appending its straight paths. Two such charts
agree by the path-homotopy relation. Subdivision of a path,
or of a homotopy square, into these balls proves unique
lifting and simple connectedness. This supplies the cover
and continuation conventions used below.

The inclusions into the larger normal disc induce the same fundamental
group. Indeed choose a strictly increasing radial function taking
\([0,R)\) onto \([0,r)\) and fixing \([0,r/2]\). Its radial map fixes
every root. Interpolating the radial functions with the identity gives
homotopies avoiding the roots, both on the large disc and on the
smaller disc. These maps are independent of \(x\). The resulting
homotopy inverse identifies the two groups; no stability theorem for
arbitrary smaller analytic neighbourhoods is required.

Let \(A=\mathbf C[\pi]\), and let \(\mathfrak a\) be a left ideal with
\[
\begin{gathered}
V=A/\mathfrak a,\\
q=\dim_{\mathbf C}V<\infty .
\end{gathered}
\tag{PM.2}
\]
Let \(\mathcal F_R(\mathfrak a)\) consist of multivalued holomorphic
functions \(f\) on \(Z_R\) annihilated by \(\mathfrak a\).
Equivalently, \(f\) is holomorphic on the universal cover, and the
linear map \(\sigma\mapsto T_\sigma f\) factors through \(V\).
The image is the entire continuation span of \(f\), of dimension at
most \(q\). This is finite determination, rather than finite
branching.

Use precisely the chosen-fibration special operators of
§5.49, ISF.1–ISF.4.
Their representatives are
\[
\begin{gathered}
Q=P_0+V_J,\\
P_0=\sum_{\ell\ge0,\ \beta\ge0}\\
p_{\ell,\beta}(t,x)D_t^\ell D_x^\beta,\\
(V_Jf)(t,x)=\\
\int_a^t\sum_{\beta\ge0}\\
j_\beta(t,s,x)\partial_x^\beta f(s,x)\,ds .
\end{gathered}
\tag{PM.3}
\]
Each coefficient is holomorphic on one full normal/parameter
product, including \(t=s\). On each compact,
\[
\begin{gathered}
|p_{\ell,\beta}|\le M_\epsilon
\frac{\epsilon^{\ell+|\beta|}}{\ell!\beta!},\\
\text{every }\epsilon>0,\\
|j_\beta|\le M_\delta
\frac{\delta^{|\beta|}}{\beta!},\\
\text{every }\delta>0.
\end{gathered}
\tag{PM.4}
\]
The joint domain precedes the index and epsilon choices. The
notation is \(J=-K\) in ISF.4–ISF.5, so that the Volterra sign in
PM.3 is positive. A branch and a path avoiding the roots define
the initial integral; its other branches are obtained by
continuation.

**Theorem PM.1.** There is one finite-dimensional representation
\(\mathbb W\) of \(\pi\), depending on \(V\), the punctured-disc
family and its marked anchor, with
\[
\begin{gathered}
\dim\mathbb W=q(m+1),\\
0\longrightarrow\mathbb W_0
\longrightarrow\mathbb W\\
\xrightarrow{\ \operatorname{end}_t\ }V_t
\longrightarrow0,\\
\dim\mathbb W_0=qm .
\end{gathered}
\tag{PM.5}
\]
Here \(\operatorname{end}_t\) is the coefficient at the marked endpoint
in a relative path boundary. The
representation \(V_t\) is the given continuation representation
evaluated at the moving endpoint. The continuation span of every
\(V_Jf\) is an image of \(\mathbb W\). The continuation span of
every \(Qf\) in PM.3 is an image of
\[
\begin{gathered}
V\oplus\mathbb W,\\
\dim(V\oplus\mathbb W)=q(m+2).
\end{gathered}
\tag{PM.6}
\]
Consequently the single two-sided ideal
\[
\begin{gathered}
\mathfrak b=
\operatorname{Ann}_A(V\oplus\mathbb W)
\subset\mathfrak a,\\
\dim_{\mathbf C}A/\mathfrak b
\le q^2(m+2)^2
\end{gathered}
\tag{PM.7}
\]
annihilates every output, for every input of type
\(\mathfrak a\) and every operator in PM.3. It is independent of
the operator coefficients and the input function. This is a
common annihilator; the joint span of outputs for all different
operators need not be finite dimensional. No finite
generation assumption on \(\pi\) is needed.

The theorem holds on common smaller products and with arbitrary
additional holomorphic parameters. Under restriction, use the
restricted finite representations, rather than assuming the
fundamental groups of all neighbourhoods are isomorphic.

#### PM.B. The finite relative path space

First construct the coefficient system \(V\) on \(Z_R\) directly.
Over a small simply connected open choose paths from the base
point and a constant copy of \(V\). On an overlap identify two
copies by the matrix of the loop obtained from their paths.
The representation law makes these identifications satisfy the
three-open identity. This is a locally constant vector bundle;
its transport along a path is the indicated continuation
matrix. The map
\[
\begin{gathered}
V\longrightarrow
\operatorname{span}_{\mathbf C}\{T_\gamma f\},\\
\sigma+\mathfrak a\longmapsto T_\sigma f
\end{gathered}
\tag{PM.8}
\]
commutes with that transport.

At a fixed \(y=(t,x)\in Z_r\), let
\[
\begin{gathered}
F_x=\\ D_R\setminus\{s:P(s,x)=0\}.
\end{gathered}
\tag{PM.9}
\]
Use paths in \(F_x\) with coefficients transported in \(V\);
their endpoints may lie at either \(a\) or \(t\).
Impose linearity in coefficients, path-homotopy relations,
the relation for joining two consecutive paths, and reversal
with a minus sign. Boundaries of filled path polygons are
zero. More explicitly, first traversing \(c\), then \(d\),
with initial coefficient \(v\), gives the relation
\[
\begin{gathered}
{}[d*c,v]=\\ {}[c,v]+[d,T_cv].
\end{gathered}
\tag{PM.10}
\]
Take relative one-cycles: all remaining boundaries are allowed
at the two marked points. Denote their vector space by
\(\mathbb W_y\).

Here is a finite calculation of this space. Draw disjoint
small circles around the \(m\) punctures. Join their boundaries
and \(a\) by an embedded finite tree avoiding the punctures,
and add an edge to \(t\), introducing a tree vertex when
necessary. Such a tree can be made successively from
polygonal arcs; if two arcs cross, treat the crossing as a
vertex, and remove any resulting tree cycle. The circles
remain attached by one path each. Cutting along the tree
and one radial slit to each puncture leaves discs. Collapsing
those discs across their cut edges and collapsing the
unpunctured outer annulus is a deformation to the graph
with \(m\) loops and the edge from \(a\) to \(t\).
Near a puncture, first radially retract its punctured small
disc to its circle. Subdivision into finitely many rectangles
and triangles makes the remaining collapses explicit linear
homotopies on cells. They agree on shared edges and fix the
marked points. Exhaustion by closed smaller outer discs
gives the same path relations for the open disc.

Trivialize coefficients along the tree. The graph has one
copy of \(V\) for each loop and one for the edge to \(t\).
Both vertices are relative vertices, so its relative
degree-zero group is zero; there are no degree-two cells.
The polygon relations just described identify its relative
one-cycles with those of \(F_x\). Therefore
\[
\mathbb W_y\simeq V^m\oplus V .
\tag{PM.11}
\]
One can check both directions without a homology theorem.
Cut a path at its successive graph edges and apply PM.10.
It becomes the sum of its labelled oriented edges; traversing
an edge backwards cancels its forward traversal with the
transported coefficient. A reduced graph path has no
additional filled polygon, so this gives precisely the
edge-coordinate vector. Inserting or deleting a polygon
before the deformation changes that vector by zero.

The boundary coefficient at \(t\) is an isomorphism on the
last summand and is zero on all loop summands. This proves
the exact vector-space sequence and dimensions in PM.5.
The kernel \(\mathbb W_{0,y}\) is equivalently the relative
path space with only \(a\) marked.

The spaces vary by invertible transport. Locally in \(B\),
the roots are holomorphic and disjoint: the nonvanishing
root derivatives give inverse charts by the analytic
inverse proof at
§5.55, DP.6.
On a small parameter neighbourhood choose disjoint discs
around the original roots and smooth cutoff functions
equal to one on smaller discs. The map
\[
\begin{gathered}
s\longmapsto s+\\
\sum_{\nu=1}^m\chi_\nu(s)\\
{}\cdot\bigl(h_\nu(x)-h_\nu(x_0)\bigr)
\end{gathered}
\tag{PM.12}
\]
moves the roots, fixes the boundary and \(a\), and is a
homeomorphism if the perturbation has Lipschitz constant
less than \(1/2\). The lower Lipschitz bound is \(1/2\);
the inverse is the unique fixed point of the contraction
\(s\mapsto w-\sum_\nu\chi_\nu(s)(h_\nu(x)-h_\nu(x_0))\).
The same construction, with one further small disc away
from the roots and \(a\), moves the marked point \(t\).
These maps identify the relative path spaces locally.
The coefficient transports are the ones already defined.

Changing a local identification by a path homotopy acts
identically on the path relations. Subdivide a parameter
path into finitely many such charts and compose the
identifications. For a homotopy of parameter paths,
subdivide its square into charts; the four transports
around each small square cancel by the same path
homotopy relations. Thus the transport depends only on
the endpoint-fixed homotopy class and gives the
representation \(\mathbb W\). It preserves PM.5 because
it preserves the endpoint boundary.

The kernel \(\mathbb W_0\) is pulled back from \(B\):
its construction uses \(F_x\), the input coefficient
system and the anchor, with no moving endpoint. Loops
in \(t\) alone act trivially on it.

#### PM.C. Full infinite kernels and the uniform annihilator

For a coefficient label \(v\in V\), let \(f_v\) denote
the corresponding continuation combination of \(f\).
On a lifted path the one-form to integrate is
\[
\begin{gathered}
\omega_v(t,s,x)=\\
\sum_{\beta\ge0}j_\beta(t,s,x)\\
\partial_x^\beta f_v(s,x)\,ds .
\end{gathered}
\tag{PM.13}
\]
It has exactly the same coefficient transport as \(f_v\).
Derivatives commute with continuation because the
transition matrices are constant.

All its sums are actual holomorphic sums. Fix a compact
finite family of lifted paths and slightly larger
root-free coordinate products around it. If their
spatial Cauchy margin is \(\eta>0\), then
\[
\begin{gathered}
|\partial_x^\beta f_v|
\le M_f\beta!\eta^{-|\beta|},\\
\sum_\beta|j_\beta\partial_x^\beta f_v|\\
\le M_\delta M_f(1-\delta/\eta)^{-d},\\
0<\delta<\eta .
\end{gathered}
\tag{PM.14}
\]
The finite-dimensional coefficient labels can be bounded
in one fixed basis. The geometric product in PM.14 is
the product of the \(d\) one-variable geometric series.
For \(\theta=\delta/\eta<1\), the tail with
\(|\beta|\ge N\) is at most
\[
M_\delta M_f\,
\theta^{N/2}(1-\sqrt\theta)^{-d}.
\tag{PM.15}
\]
Indeed factor \(\theta^{|\beta|/2}\le\theta^{N/2}\)
and sum the other half geometrically. Fix \(\delta\)
only after the common margin, without changing the
domain. Integration multiplies the bound by the
maximum path length. Additional fixed derivatives
use one further Cauchy margin.

The one-form is closed on the fibre. Its integral
around a filled polygon is zero by the Cauchy theorem,
or by integrating its zero exterior derivative over
the finitely many triangles. Uniform convergence
allows that equality for the whole series.
Its integrals respect PM.10 and reversal, so define a
linear map from \(\mathbb W_y\) to local holomorphic
functions of the output and any additional parameters.
Holomorphy is explicit: on a small output chart choose
polygonal paths with fixed intermediate vertices and
one endpoint segment moving with \(t\). The vertices
can be kept away from the roots for all nearby \(x\);
normal convergence and differentiation of these
finite integrals give a holomorphic function.
Changing this family within its path class changes
nothing by the polygon calculation.

Continuing the function along an output path is
therefore exactly the transport in \(\mathbb W\).
For the initial path from \(a\) to \(t\), with the
coefficient of the original branch, the value is
\(V_Jf\). Every continued value comes from the
transported vector in \(\mathbb W\). This proves
the Volterra assertion in Theorem PM.1.

The ordinary infinite part has the same property
with \(V\) itself. On a fixed smaller root-free
product, the Cauchy bound for
\(\partial_t^\ell\partial_x^\beta f_v\) cancels
the two factorial denominators in PM.4.
Taking \(\epsilon\) smaller than both margins gives
the geometric product in \(d+1\) variables.
The total-degree tail has the bound PM.15 with
\(d+1\) in place of \(d\). Thus \(P_0 f_v\)
converges normally, all fixed derivatives converge,
and continuation commutes with its whole sum.
Its span is an image of \(V\). Adding the two
parts proves PM.6.

The action of \(A\) on \(V\oplus\mathbb W\) gives
a linear map into its endomorphism algebra.
Its kernel \(\mathfrak b\) is two-sided, and its
quotient has dimension at most the square in PM.7.
Since it annihilates \(V\), it annihilates
\(1+\mathfrak a\), so every element of
\(\mathfrak b\) belongs to \(\mathfrak a\).
The two maps to continued functions just constructed
intertwine this action. Hence \(\mathfrak b\)
annihilates every output. The representation used
only \(V\) and the family of labelled fibre paths;
the coefficients of \(Q\) and the particular
function entered only the integration map.
This proves the asserted uniformity. If \(q=0\),
the input and all outputs are zero and take
\(\mathfrak b=A\).

For a smaller admissible product, retain all \(m\)
roots inside its smaller normal disc. Inclusion
identifies its finite relative path spaces with
the restricted ones above: retract the root-free
outer annulus and move the anchor along it.
Reverse transport gives the inverse. The same
description handles a change of anchor. Restrict
the representation of \(\pi\) along the induced
map of groups. The annihilating ideal is its
inverse image, of finite codimension with the
same bound. This proves the restriction assertion
without asserting neighbourhood fundamental-group
stability or a universal domain for all operator
germs.

#### PM.D. The exact quotient action and its gauge

Let \(\mathcal F_R\) be the space of all multivalued
holomorphic functions on \(Z_R\). Let
\(\mathcal G_R\) be the subspace obtained by
pulling back holomorphic functions on
\(D_R\times\widetilde B\), where \(\widetilde B\)
is the universal cover of \(B\).
Thus members of \(\mathcal G_R\) extend across
all fibre punctures on the base cover. Trivial
fibre monodromy alone would not imply this
extension; a pole \(1/P\) is an immediate
counterexample to that weaker condition.

The integrals of \(\mathbb W_0\) belong to
\(\mathcal G_r\). Their cycles and coefficient
labels depend on \(x\), with no endpoint at \(t\).
The kernel coefficients are holomorphic in \(t\)
on the entire smaller disc. The same finite path
and normal-convergence argument in PM.C therefore
gives holomorphy on that full disc, for each
branch over \(\widetilde B\). This fact is the
reason for using relative paths with a marked
anchor in PM.5.

Choose locally a splitting \(e:V_t\to\mathbb W\)
by labelled paths from \(a\) to \(t\). For any
continuation loop,
\[
\begin{gathered}
T_\gamma^{\mathbb W}e-eT_\gamma^V\\
\text{has image in }\mathbb W_0 .
\end{gathered}
\tag{PM.16}
\]
Apply the endpoint map in PM.5 to verify this
identity; both composites have the same endpoint
coefficient. Integration and the preceding
paragraph consequently give
\[
T_\gamma(V_Jf)-V_J(T_\gamma f)
\in\mathcal G_r .
\tag{PM.17}
\]
The ordinary part commutes with continuation
exactly. Changing the initial anchor path
subtracts a relative cycle with only anchor
endpoints and hence a member of \(\mathcal G_r\).
Changing the anchor itself subtracts its fixed
connecting-path integral, which is again
holomorphic on the full output normal disc.

Both parts preserve \(\mathcal G\). For \(P_0\)
this is the full Cauchy estimate already used.
For \(V_J\), on the base cover the input is
holomorphic on the whole normal disc, so one
may use straight segments and the same
normally convergent estimate. Thus PM.3 induces
a well-defined operation on the restriction
germs of \(\mathcal F/\mathcal G\).

It is the action of the actual special ring.
The four product formulas and their full
coefficient bounds are the proved
§5.49, ISF.11–ISF.17.
They apply on each finite family of lifted
root-free paths after the common margins of
PM.C have been fixed. In the negative-positive
case, integration by parts retains the lower
anchor evaluation; it is a holomorphic function
on the full output normal disc and lies in
\(\mathcal G\). The positive-negative upper
endpoints give the ordinary part, while two
negative parts give the ordered Volterra
convolution. Fubini is valid for the finite
path integrals and their absolutely convergent
spatial sums. All lower endpoint remainders
are in \(\mathcal G\). These exact identities
therefore give
\[
\begin{gathered}
(Q_1Q_2)[f]=Q_1(Q_2[f]),\\
1[f]=[f].
\end{gathered}
\tag{PM.18}
\]
If an endpoint path has extra loops, first
choose a local initial path and then continue
this identity; all changes are the period
gauge already controlled. No product of
arbitrary full microdifferential operators
has been substituted for the chosen-fibration
special ring.

Let \(\mathcal F_{\rm fd}\) be the union of the
finite-determination spaces, and put
\(\mathcal G_{\rm fd}=\mathcal G\cap
\mathcal F_{\rm fd}\). This is a vector
subspace: for two finite representations
use their direct sum, whose annihilator
still has finite codimension. PM.7 shows
that the action preserves
\(\mathcal F_{\rm fd}/\mathcal G_{\rm fd}\).
Equation PM.17 shows that this action commutes
with continuation on that quotient.

For a fixed \(\mathfrak a\), the same
\(\mathfrak b\) works simultaneously for every
operator and every input, with all coefficient
tails included:
\[
\begin{gathered}
Q\bigl(\mathcal F(\mathfrak a)\bigr)
\subset\mathcal F(\mathfrak b),\\
\mathfrak b\subset\mathfrak a .
\end{gathered}
\tag{PM.19}
\]
These inclusions are interpreted on the common
products on which the inputs and operators
are represented, or their restriction germs.
They are function-space assertions. An image
in an ordinary algebraic tensor product
requires finite balancing witnesses; PM.19
alone does not supply such witnesses.

#### PM.E. Why finite determination does not remove the discriminant

The extra base divisor in PM.1 cannot be discarded
using finite determination alone. Take
\[
\begin{gathered}
P(s,x)=s^2-x^3,\\
f(s,x)=\frac1{s^2-x^3},\\
J=1,\quad P_0=0.
\end{gathered}
\tag{PM.20}
\]
The input is single valued, so \(q=1\).
For \(x\ne0\) there are two roots, hence \(m=2\).
Choose a branch of \(x^{3/2}\), a fixed nonzero
anchor and a normal endpoint away from both
roots. The integral from the anchor has a
local holomorphic branch. Continuing the
endpoint once counterclockwise around the
positive root adds
\[
\begin{gathered}
2\pi i\,\operatorname{Res}_{s=x^{3/2}}
\frac{ds}{s^2-x^3}\\
=\pi i\,x^{-3/2}.
\end{gathered}
\tag{PM.21}
\]
The residue formula here follows directly
by factoring the denominator and integrating
the coefficient of \((s-x^{3/2})^{-1}\)
on a positive small circle.

At a point \((t_0,0)\) with \(t_0\ne0\),
the hypersurface \(t^2=x^3\) is absent.
The initial anchor and endpoint may be chosen
on one small segment avoiding \(s=0\), so
the initial branch is holomorphic near that
point. The continued branch differs by
PM.21, which is unbounded as positive \(x\)
tends to zero. If the primitive were a
multivalued holomorphic function on the
complement of \(S\) alone, both of these
branches would extend holomorphically over
a small simply connected neighbourhood of
\((t_0,0)\). Their difference could not be
PM.21 there. This is a contradiction.

The period in PM.21 is independent of \(t\),
so it belongs to \(\mathcal G\), as PM.D
predicts. It is not a holomorphic germ at
the original base point. Thus the gauge
\(\mathcal G\) must not be identified with
the space of holomorphic germs there.
For comparison, the one-puncture input
\(1/s\) acquires the constant period
\(2\pi i\); its primitive has a two-dimensional
continuation span generated by \(1,\log t\).
Even in that case the original annihilator
of the single-valued input does not
annihilate the primitive.

PM.A–PM.D establish uniform finite monodromy
and the full special action with its precise
gauge. Common sector solutions of an actual
holonomic module, continuation across the
discriminant, the finite algebraic
tensor-lifting assertion, and a faithful
finite D-type embedding need their own
module arguments.

Free source context is Kashiwara–Kawai,
[Holonomic systems III](https://www.kurims.kyoto-u.ac.jp/~kenkyubu/kashiwara/HolIII.pdf),
III.4.3–III.4.10, printed pp.893–899.
The relative path calculation, full
normal-convergence bounds, uniform
annihilator and counterexample are proved
above.

![Relative fibre paths, the uniform monodromy envelope and an apparent discriminant period](assets/finite-monodromy-envelope.png)

*Figure PM.1. The left panel is one exact three-puncture fibre:
\(R=2\), \(r=1\), \(a=1.5\), \(t=0.6+0.2i\), and punctures
\(-0.4,\ 0.1+0.25i,\ 0.25-0.3i\). Every puncture lies in
\(D_{r/2}\), and the displayed segment avoids them. The right
panel gives the arbitrary-rank relative path sequence and
annihilator bound of PM.5–PM.7, proved in PM.B–PM.C.
The lower panel is PM.20–PM.21 at \(x=1/4\), with roots
\(\pm1/8\), a positive loop of radius \(1/25\), and period
\(8\pi i\). It displays a period in the exact gauge of PM.D,
not a solution of an unspecified holonomic module.
Editable drawing source,
exact data,
PNG and
SVG accompany it.*


### 5.71. Special kernels, external transfer and finite tensors

Let \(X\) have coordinates \((t,x_1,\ldots,x_d)\), let
\(Y\) have coordinates \((u,s,x_1,\ldots,x_d)\), and work at their
origins. Write \(R=D^\infty_{X,0}\), \(A=D^\infty_{Y,0}\) and
\(T=\widetilde E^\infty_{X,0}\) for the chosen-fibration special ring.
The complete definitions and products are
§5.49, ISF.1–ISF.17.
In particular, an element of \(T\) is the unique pair
\((P_0,J)\), with \(P_0\in R\) and
\[
\begin{gathered}
J(u,s,x,D_x)=\\
\sum_{\beta\ge0}j_\beta(u,s,x)D_x^\beta,\\
|j_\beta|_K\le C_{K,\delta}
\frac{\delta^{|\beta|}}{\beta!},\quad\delta>0.
\end{gathered}
\tag{ST.1}
\]
All coefficients have one joint holomorphic product domain including
\(u=s\). The product domain is independent of \(\beta\) and \(\delta\).
The convention is \(J=-K\) for the coefficient \(K\) of the source's
logarithmic kernel; its normal integral has the positive sign.

There are commuting copies \(R_{\rm out}\) and \(R_{\rm in}\) acting
on \(A\), by left multiplication with \(P(u,x,D_u,D_x)\) and right
multiplication with \(Q(s,x,D_s,D_x)\). The copies commute as actions,
although their spatial operator expressions need not commute as ring
elements. We identify each copy with \(R\) through its displayed
coordinates. Let \(\mathcal J\) be the vector space in ST.1.
All tensor products below are ordinary algebraic tensor products.

#### ST.A. An exact two-normal-variable quotient

There is an \(R_{\rm out}\)-\(R_{\rm in}\) bimodule isomorphism
\[
\begin{gathered}
A/(A D_u+D_s A)
\xrightarrow{\;\sigma\;}\mathcal J,\\
P=\sum_{r,l,\beta}a_{r,l,\beta}
D_u^rD_s^lD_x^\beta,\\
\sigma P=\sum_\beta
\left(\sum_{l\ge0}(-\partial_s)^l
a_{0,l,\beta}\right)\\
{}\cdot D_x^\beta .
\end{gathered}
\tag{ST.2}
\]
These are germs with two independent normal coordinates at the diagonal
point. There is no substitution \(s=u\) in ST.2.
The denominator is a sum of vector subspaces stable under the two
displayed \(R\)-actions; it is not asserted to be a two-sided ideal of \(A\).

Here is a convergent division proof. Fix an inner coefficient compact
and one larger compact in a common product, with \(s\)-Cauchy margin
\(\rho>0\). On the latter compact, the actual ordinary infinite-order
condition gives, for every \(\epsilon>0\),
\[
|a_{r,l,\beta}|
\le M_\epsilon
\frac{\epsilon^{r+l+|\beta|}}{r!\,l!\,\beta!}.
\tag{ST.3}
\]
Put
\[
\begin{gathered}
Q=\sum_{l,\beta}a_{0,l,\beta}D_s^lD_x^\beta,\\
K=\sum_{r,l,\beta}a_{r+1,l,\beta}D_u^rD_s^lD_x^\beta,\\
H=\sum_{k,\beta}h_{k,\beta}D_s^kD_x^\beta,\\
h_{k,\beta}=\sum_{j\ge0}(-1)^j
\partial_s^j a_{0,k+1+j,\beta}.
\end{gathered}
\tag{ST.4}
\]
The identity \(P=K D_u+Q\) is literal in the full coefficient class.
For instance the coefficient of \(K\) satisfies ST.3 times at most
\(\epsilon\), because \((r+1)!\ge r!\).
For \(0<\epsilon<\rho\), Cauchy's formula and
\((k+1+j)!\ge(k+1)!\,j!\) give
\[
\begin{gathered}
|h_{k,\beta}|\le
\frac{M_\epsilon\epsilon^{k+1+|\beta|}}
{(k+1)!\,\beta!}
\frac1{1-\epsilon/\rho},\\
\left|\sum_{j\ge N}(-1)^j
\partial_s^j a_{0,k+1+j,\beta}\right|\\
\le
\frac{M_\epsilon\epsilon^{k+1+|\beta|}}
{(k+1)!\,\beta!}
\frac{(\epsilon/\rho)^N}{1-\epsilon/\rho}.
\end{gathered}
\tag{ST.5}
\]
The same argument gives
\[
\begin{gathered}
|(\sigma P)_\beta|\le
\frac{M_\epsilon\epsilon^{|\beta|}}{\beta!}
\frac1{1-\epsilon/\rho},\\
\left|\sum_{l\ge N}(-\partial_s)^l
a_{0,l,\beta}\right|\\
\le\frac{M_\epsilon\epsilon^{|\beta|}}{\beta!}
\frac{(\epsilon/\rho)^N}{1-\epsilon/\rho}.
\end{gathered}
\tag{ST.6}
\]
For any requested output bound \(\eta>0\), choose
\(\epsilon<\min(\rho/2,\eta)\). This proves \(K,H\in A\) and
\(\sigma P\in\mathcal J\) for every output epsilon on the same inner
product. A further fixed inner compact supplies, by Cauchy's formula,
all prescribed coefficient derivatives of the tails in ST.5–ST.6.
The domains were chosen before any index or output epsilon.

For a single finite term the product rule telescopes:
\[
\begin{gathered}
aD_s^l-(-\partial_s)^l a\\
=D_s\sum_{j=0}^{l-1}
(-1)^j(\partial_s^ja)D_s^{l-1-j},\\
l\ge1.
\end{gathered}
\tag{ST.7}
\]
Equations ST.5–ST.6 justify its full sum, including all spatial
indices. Thus the actual operator identity is
\[
P=K D_u+D_s H+\sigma P.
\tag{ST.8}
\]
No infinite sum is taken in a quotient module to obtain ST.8:
the three operators on the right have already been constructed as
actual holomorphic coefficient sequences.

Conversely, \(\sigma(KD_u)=0\). If
\(H=\sum h_{r,l,\beta}D_u^rD_s^lD_x^\beta\), then the \(r=0\)
coefficients of \(D_sH\) are
\(\partial_s h_{0,l,\beta}+h_{0,l-1,\beta}\), where the second
term is zero at \(l=0\). The two normally convergent sums in ST.2
cancel after the index shift, so \(\sigma(D_sH)=0\).
Finally \(\sigma J=J\) for \(J\in\mathcal J\).
This proves both the kernel and surjectivity assertions in ST.2.

Left multiplication by \(R_{\rm out}\) preserves the denominator:
it commutes with \(D_s\) and preserves the left ideal \(AD_u\).
Right multiplication by \(R_{\rm in}\) preserves it because it
commutes with \(D_u\) and preserves \(D_sA\).
Consequently its two actions on \(\mathcal J\) are
\[
\begin{gathered}
P\cdot J=\sigma(P(u,x,D_u,D_x)J),\\
J\cdot Q=\sigma(JQ(s,x,D_s,D_x)).
\end{gathered}
\tag{ST.9}
\]
The full ordinary product is the complete convergent product of
§5.49, ISF.4.
It keeps every spatial contraction. Formula ST.2 then gives, on
generators,
\[
\begin{gathered}
u\cdot J=uJ,\\
D_u\cdot J=\partial_uJ,\\
D_{x_i}\cdot J=\partial_{x_i}J+JD_{x_i},\\
J\cdot s=sJ,\\
J\cdot D_s=-\partial_sJ,\\
J\cdot D_{x_i}=JD_{x_i}.
\end{gathered}
\tag{ST.10}
\]
Multiplication by a general spatial coefficient uses the actual
Leibniz product in ST.9. Formula ST.10 alone is not used to extend
the actions formally to infinite operators.

#### ST.B. The quotient of the special ring

The inclusion \(R\longrightarrow T\), \(P_0\mapsto(P_0,0)\),
and the projection \((P_0,J)\mapsto J\) give a short exact sequence
of \(R\)-bimodules
\[
\begin{gathered}
0\to R\to T
\xrightarrow{\;\pi\;}\mathcal J\to0.
\end{gathered}
\tag{ST.11}
\]
Vector-space exactness follows from the unique pair representation.
The bimodule assertion follows from the full mixed product formulas
ISF.14–ISF.15. The left normal derivative differentiates the integral
kernel, leaving its upper endpoint in the ordinary part. On the right,
integration by parts gives precisely
\((-\partial_s)^l\) on the joint kernel coefficient.
The upper endpoint again belongs to \(R\).
The separated lower endpoint is the explicit whole-normal-face
coboundary ISF.17, with its minus sign and Cauchy evaluation kernel.
It has zero special class by that proved face formula.

For a finite positive operator this identifies the projected mixed
products with ST.9. For arbitrary \(P_0\in R\), first form the actual
ordinary two-normal-variable products appearing in ST.9. Equations
ST.5–ST.6 give their full normal reduction and derivative tails.
ISF.11–ISF.12 give absolute spatial summability on a common smaller
product. The full mixed products and endpoint sums converge there
by ISF.14–ISF.17. Their finite truncations are identical, so the
normally convergent coefficient expressions are identical.
This proves ST.11 for all \(R\), with no density assertion about
an algebraic tensor product.

For example \(D_u\cdot1=0=1\cdot D_s\) in \(\mathcal J\).
This says that the negative kernel for a normal inverse derivative
has derivative equal to an ordinary operator. It does not say
that its anchored integral is a two-sided inverse on functions.

#### ST.C. External transfer and its cokernel

Put \(C=A/AD_u\). It is a right \(R_{\rm in}\)-module, and left
multiplication by \(D_s\) gives a right \(R_{\rm in}\)-linear map
\(C\to C\). Its cokernel is \(\mathcal J\), by ST.A. For every
left \(R\)-module \(N\) there is a canonical isomorphism
\[
\begin{gathered}
\mathcal J\otimes_R N\\
\simeq\frac{C\otimes_R N}{D_s(C\otimes_R N)}.
\end{gathered}
\tag{ST.12}
\]
Indeed, an algebraic tensor product is the quotient of the free
vector space on elementary pairs by the finite linearity and
balancing relations. First imposing the relation that
\(D_s c\otimes n\) be zero, or first passing from \(C\) to
\(C/D_sC\), gives the same quotient.
This proves ST.12 without a flatness assumption.

Here is its usual external-module interpretation. Let
\(p:Y\to X\), \(p(u,s,x)=(s,x)\), and put
\(D=D_{X,0}\), \(D_Y=D_{Y,0}\) for the finite-order analytic rings.
The transfer module is
\[
\begin{gathered}
U=D_Y/D_YD_u\\
\simeq
\mathcal O_{Y,0}\otimes_{\mathcal O_{X,0}}D.
\end{gathered}
\tag{ST.13}
\]
To verify it, move every \(D_u\) to the far right in the finite
coordinate PBW form. The terms with positive \(D_u\)-degree
are exactly the left ideal \(D_YD_u\); the remaining terms have
joint holomorphic coefficients and finite \(D_s,D_x\)-degree.
The right \(D\)-action and the left \(D_Y\)-action agree with
the chain rule. No completion is present in ST.13.

For any left \(D\)-module \(M\), set
\[
\begin{gathered}
N=R\otimes_D M,\\
L=U\otimes_D M,\\
L^\infty=A\otimes_{D_Y}L.
\end{gathered}
\tag{ST.14}
\]
Here \(L\) is the ordinary inverse-image module for the projection:
in its coefficient form \(D_u\) differentiates the
\(\mathcal O_Y\)-factor, \(D_s,D_x\) act by the Leibniz rule,
and the \(M\)-factor retains every relation.
Finite tensor reassociation gives canonical isomorphisms
\[
\begin{gathered}
A\otimes_{D_Y}U\simeq A/AD_u=C,\\
L^\infty\simeq C\otimes_D M\\
\simeq C\otimes_R N,\\
\mathcal J\otimes_R N\\
\simeq L^\infty/D_s L^\infty.
\end{gathered}
\tag{ST.15}
\]
The first equality can be checked without flatness: quotienting
the free tensor by the finite balance relation
\(a\otimes QD_u=aQ D_u\otimes1\) gives precisely \(AD_u\).
The next two identifications send an elementary tensor to the
same ordered factors; their inverses insert the appropriate unit.
Those formulas prove both inverse identities and compatibility
with \(D_s\). They apply to arbitrary \(M\), including modules
with singular or nonreduced support.

The restriction to the diagonal often written after ST.15 means
the inverse-image sheaf of germs along \(u=s\).
Its stalk still has independent variables \(u,s\).
Replacing \(L^\infty\) with the holomorphic restriction
\(s=u\) before forming the cokernel changes this construction.
The displayed proofs are stalkwise and therefore give the
corresponding sheaf identifications on their coordinate opens.

#### ST.D. The finite tensor image criterion

For a morphism \(\varphi:N_1\to N_2\) of left \(R\)-modules, let
\(\iota_i:N_i\to T\otimes_RN_i\) be \(n\mapsto1\otimes n\).
The following conditions are equivalent:
\[
\begin{gathered}
\operatorname{im}(T\otimes_R\varphi)
\subseteq\operatorname{im}\iota_2;\\
\mathcal J\otimes_R\varphi=0;\\
\operatorname{im}(C\otimes_R\varphi)\\
\subseteq D_s(C\otimes_RN_2).
\end{gathered}
\tag{ST.16}
\]
To prove this, tensor ST.11 on the right. Its elementary
presentation, exactly as for ST.12, gives the exact rows
\[
\begin{gathered}
N_i\xrightarrow{\;\iota_i\;}T\otimes_RN_i\\
\longrightarrow\mathcal J\otimes_RN_i
\longrightarrow0.
\end{gathered}
\tag{ST.17}
\]
These rows form a commutative diagram under \(\varphi\).
If the first condition of ST.16 holds, its composite with the
second row's last arrow is zero. The first row's last arrow is
surjective, so \(\mathcal J\otimes\varphi=0\).
Conversely that vanishing puts every image tensor in the
kernel of the second row's last arrow, which is exactly
\(\operatorname{im}\iota_2\).
Formula ST.12 identifies this vanishing with the third condition.
Injection of either \(\iota_i\) is unnecessary.

The criterion gives finite witnesses explicitly. Write a chosen
element of \(T\otimes_RN_1\) as
\(\sum_{i=1}^h t_i\otimes n_i\), and let
\(j_i=\pi(t_i)\). Under the third condition choose one element
\[
\begin{gathered}
w=\sum_{k=1}^v c_k\otimes m_k,\\
D_sw=\sum_{i=1}^h j_i\otimes\varphi(n_i)\\
\text{in }C\otimes_RN_2.
\end{gathered}
\tag{ST.18}
\]
The lift from \(\mathcal J\) to \(C\) in this formula uses
its normal-form representative. It is a vector-space section;
it is not asserted to be right \(R\)-linear. Each sum is finite.
The equality in ST.18 is a finite sum of tensor balancing and
linearity relations, by the definition of that tensor product.
For each of the finitely many coefficients and relations,
choose actual lifts to \(A\).
Equation ST.8 reduces each lift to its joint normal form using
one actual \(K\) and one actual \(H\).
Projection to \(\mathcal J\otimes_RN_2\) then kills
\(D_sw\), term by term, and retains the specified balancing
relations. Thus \(\sum j_i\otimes\varphi(n_i)=0\) in that quotient.
Lifting its finite tensor relations to \(T\otimes_RN_2\)
changes each coefficient identity by an element of \(R\),
by ST.11. Balance each such element into the second factor.
There are finitely many of them; their sum is one element
\(n\in N_2\). This constructs
\[
\begin{gathered}
\sum_{i=1}^h t_i\otimes\varphi(n_i)=1\otimes n\\
\text{in }T\otimes_RN_2.
\end{gathered}
\tag{ST.19}
\]
No infinite sum of elementary tensors or closure of an image
has entered this construction. The full infinite operators
occur inside the finitely many actual coefficient factors.

#### ST.E. A parameter comparison hypothesis and monodromy

One precise hypothesis under which the criterion applies is
useful. Use the punctured-disc family and monodromy group
\(\pi_1\) of
§5.70, PM.A–PM.D.
For each cofinite left ideal \(\mathfrak a\subset\mathbf C[\pi_1]\),
suppose a finite-order module \(M_{\mathfrak a}\) is given,
with compatible maps for inclusions of ideals.
Define \(N_{\mathfrak a}\) and \(L^\infty_{\mathfrak a}\)
by ST.14. Suppose further there are natural isomorphisms
\[
\begin{gathered}
\Theta_{\mathfrak a}:L^\infty_{\mathfrak a}
\xrightarrow{\;\sim\;}\mathscr F_{\mathfrak a}(u;s,x),\\
\Theta_{\mathfrak a}D_s=\partial_s\Theta_{\mathfrak a},
\end{gathered}
\tag{ST.20}
\]
where the right side is the space of multivalued holomorphic
function germs on the punctured family, with an additional
independent holomorphic parameter \(u\), of type
\(\mathfrak a\). Naturality includes the maps induced by
\(\mathfrak b\subset\mathfrak a\).
The isomorphisms concern the actual algebraic scalar extension
in ST.14, rather than a completion or an enlargement defined
to be the function space.

Under these hypotheses, for every \(\mathfrak a\) there is
a cofinite two-sided ideal \(\mathfrak b\subseteq\mathfrak a\)
such that
\[
\begin{gathered}
\alpha:T\otimes_RN_{\mathfrak a}
\longrightarrow T\otimes_RN_{\mathfrak b},\\
\operatorname{im}\alpha\subseteq\\
\operatorname{im}
(N_{\mathfrak b}\longrightarrow
T\otimes_RN_{\mathfrak b}).
\end{gathered}
\tag{ST.21}
\]
Indeed, apply PM.B–PM.D to the kernel \(J=1\), with \(u\)
as an additional parameter. The relative path representation
and its annihilator depend on the fixed input monodromy
representation and the punctured family, not on the input
function or \(u\). The joint Cauchy argument there proves
holomorphy in \(u\) on one smaller product. Consequently
every function of type \(\mathfrak a\) has a normal primitive
of the same uniform enlarged type \(\mathfrak b\).
Writing \(q=\dim\mathbf C[\pi_1]/\mathfrak a\) and \(m\)
for the number of punctures, that proof gives
\(\dim\mathbf C[\pi_1]/\mathfrak b\le q^2(m+2)^2\).
Use the inverse of \(\Theta_{\mathfrak b}\) to put each
primitive in \(L^\infty_{\mathfrak b}\).
Compatibility with \(D_s\) and injectivity of
\(\Theta_{\mathfrak b}\) show that its derivative equals the
image of the original element. Now ST.15–ST.16 prove ST.21,
and ST.18–ST.19 give finite balancing witnesses.

Both directions of ST.20 have a role: its surjectivity puts
the primitive in the actual tensor module, and its injectivity
turns equality of functions into equality of module elements.
The monodromy estimates alone assert neither direction.
Theorems ST.A–ST.D have no hypothesis ST.20.

![The special quotient and its finite tensor criterion](assets/special-tensor-transfer.png)

**Figure ST.1.** The upper row is the bimodule quotient ST.11.
The middle row is its right-exact algebraic tensor sequence
ST.17; its first arrow need not be injective.
The lower identification is ST.12–ST.15 with the actual
external scalar extension and normal derivative.
The vertical maps for two modules give precisely ST.16.
The two independent normal coordinates and the signs in
ST.2 and ST.10 are part of the construction.
This is an exact algebraic diagram with no geometric scale.

Editable SVG,
diagram data, and
reproducible source.
Freely accessible further reading:
[Kashiwara–Kawai, *Holonomic systems III*, III.4.9, printed pp.898–899](https://www.kurims.kyoto-u.ac.jp/~kenkyubu/kashiwara/HolIII.pdf).
The proofs of the quotient, convergence and tensor criterion are given above.


### 5.72. Meromorphic connections and finite analytic direct images

Let \(X\) be a complex manifold, \(S\) a reduced analytic hypersurface,
and \(j:U=X\setminus S\hookrightarrow X\).
Write \(\mathcal O_X(*S)\) for finite-pole meromorphic coefficients.
On a chart with \(S=\{h=0\}\), this is
\(\mathcal O_X[1/h]\). All differential operators in this section
have finite order.

We use the complete analytic operator coherence and good-filtration
proofs in GL06, Theorem 3.0c,
the complete normalized logarithmic construction and SNC
calculation in
§5.0–§5.3,
the supplied-generator proper coherence and holonomicity proof
§5.9–§5.12,
and the strict reflexive coefficient lattice theorem
§5.27.
Every use below retains the hypotheses of these proofs.

#### DT.A. A localized operator module is a finite meromorphic bundle

Suppose \(Q\) is a coherent analytic left \(\mathcal D_X\)-module,
multiplication by a local equation \(h\) is bijective on \(Q\),
and
\[
\operatorname{Ch}(Q)|_U\subset T_U^*U.
\tag{DT.1}
\]
Then \(Q\) is finite projective over \(\mathcal O_X(*S)\),
with its induced integrable meromorphic connection.
Here finite projective is asserted locally over
\(\mathcal O_{X,x}[1/h]\); that ring need not be local.

Fix an actual good filtration bounded below, and put
\[
\begin{gathered}
H=\mathcal O_{X,x},\quad R=H[1/h],\\
G=\operatorname{gr}Q_x,\\
I=\operatorname{ann}_{H[\xi_1,\ldots,\xi_n]}G.
\end{gathered}
\tag{DT.2}
\]
The finite symbol module \(G\) is graded. In the analytic cotangent
germ at the zero covector, each \(h\xi_i\) vanishes on its support,
by DT.1. The analytic Nullstellensatz therefore gives a power
of \(h\xi_i\) in its annihilator. The relative coefficient
faithful-flatness in GL06, Lemmas 3.0c.1–3.0c.2, contracts this
membership to \(H[\xi]_{(\mathfrak m_H,\xi)}\).
It contracts further to the graded module over \(H[\xi]\):
if \(b(\xi)(h\xi_i)^a\in I\) with \(b(0)\) a unit of \(H\),
its homogeneous component of degree \(a\) is
\(b(0)(h\xi_i)^a\in I\).
Thus, for some positive integers \(a_i\),
\[
\begin{gathered}
(h\xi_i)^{a_i}G=0,\\
\xi_i^{a_i}(G\otimes_HR)=0.
\end{gathered}
\tag{DT.3}
\]
Let \(b\) bound the degrees of a finite homogeneous generator list
of \(G\). Every monomial of total degree greater than
\(\sum_i(a_i-1)\) contains some \(\xi_i^{a_i}\).
Consequently the localized graded pieces vanish in all degrees
greater than \(b+\sum_i(a_i-1)\).
Localization is exact, so the localized good filtration stabilizes.
Its stable term is the entire \(Q_x\), since \(h\) is bijective.
This term is the localization of one finite \(H\)-module.
Hence \(Q_x\) is finite over \(R\).
This argument uses the actual analytic symbol support, including
singular and nonreduced loci.

We prove projectivity rather than assume a meromorphic frame.
A nonzero ideal of \(R\) stable under all coordinate derivatives
is the unit ideal. Indeed, clearing a denominator supplies a nonzero
holomorphic germ in it. Differentiating a lowest nonzero Taylor
monomial gives a germ with nonzero constant coefficient, a unit.

For a finite presentation \(R^b\xrightarrow{A}R^a\to Q_x\to0\),
lift each coordinate connection to \(R^a\), with matrix \(C_i\).
Its preservation of the relation module gives a finite matrix \(B_i\)
such that
\[
\partial_iA=AB_i-C_iA.
\tag{DT.4}
\]
Over \(R[\varepsilon]/(\varepsilon^2)\), this is
\[
\begin{gathered}
A+\varepsilon\partial_iA\\
=(1-\varepsilon C_i)A(1+\varepsilon B_i).
\end{gathered}
\tag{DT.5}
\]
Expansion of each minor by its rows and columns shows that
every determinantal ideal of \(A\) is derivative-stable.
All Fitting ideals of \(Q_x\) are therefore zero or the unit ideal.
If its rank over the fraction field of \(R\) is \(r\), this gives
\[
\begin{gathered}
\operatorname{Fitt}_r(Q_x)=R,\\
\operatorname{Fitt}_{r-1}(Q_x)=0,\quad r>0.
\end{gathered}
\tag{DT.6}
\]
At any prime choose a unit minor of size \(a-r\).
Finite row and column elimination reduces that block to the
identity. The vanishing of the minors of size \(a-r+1\)
makes the remaining relation block zero. Thus \(Q_x\) is
locally free of rank \(r\) on \(\operatorname{Spec}R\).
For rank zero the same elimination gives \(Q_x=0\).

For completeness this local statement gives a projective
\(R\)-module without an infinite gluing step. Finitely many such
minors give a principal-open cover of \(\operatorname{Spec}R\).
On each, the finite free surjection to \(Q_x\) splits.
Clear denominators in those finite splitting matrices to get
maps whose composites are powers of the respective minors times
the identity of \(Q_x\). Powers can be increased without changing
the cover; their generated ideal is the unit ideal. A finite
linear combination of those maps is a global splitting.
This proves finite projectivity.
The Leibniz rule and integrability are inherited from the
\(\mathcal D_X\)-action; differentiating \(h^{-k}\) gives its
unique extension to \(R\).

#### DT.B. Proper sheaf image commutes with finite-pole localization

For a proper continuous map \(f:Z\to X\) of the manifolds under
consideration, filtered colimits of complex sheaves commute
with \(Rf_*\) in the bounded-below setting used here.
Here are the compact-support details.

On a compact subset, a section of a filtered colimit is represented
locally at finitely many indices. Choose a finite compact closed
shrinking of that cover. The finitely many representatives have a
common later index. Their discrepancies vanish locally on each
compact pairwise intersection, and a finite further choice of
indices makes all of them zero. Finite closed gluing then gives a
section at one index. The identical argument for a zero section
proves injectivity. Thus sections on compact subsets commute
with filtered colimits.

The compact germ and finite closed gluing arguments just used,
the c-soft extension and lifting lemmas, and the actual proper
fibre comparison are proved in
compact sheaf operations, C1–C4 and F6.
In particular a filtered colimit of c-soft sheaves is c-soft:
represent a compact section at one index and extend it there.
On a compact space, sections and compactly supported sections
coincide, and these c-soft sheaves are acyclic by C3–C4.
Take functorial injective resolutions of the diagram, for example
the exact stalk-product construction R.12–R.13 of
§5.34.
Their filtered colimit is an exact resolution, because filtered
colimits are exact on stalks. On a compact fibre its terms are
filtered colimits of c-soft restrictions, hence acyclic.
Taking fibre sections commutes with the colimit, by the preceding
finite compact gluing argument.
F6 identifies the cohomology of these fibre complexes with the
stalks of \(Rf_!=Rf_*\). This proves the asserted natural
comparison in every degree.
It asserts no commutation of unrestricted sections over a fixed
noncompact open set with a filtered colimit.

Let now \(f\) be holomorphic and proper, \(h\) a local equation
on \(X\), \(g=h\circ f\), and \(F\) a coherent holomorphic sheaf
on \(Z\). The colimit system
\(F\xrightarrow{g}F\xrightarrow{g}\cdots\) is \(F[1/g]\).
The transition on \(R^qf_*F\) is multiplication by \(h\).
The natural comparison just proved gives
\[
\begin{gathered}
R^qf_*(F[1/g])\\
\simeq (R^qf_*F)[1/h].
\end{gathered}
\tag{DT.7}
\]
The comparison is compatible with restrictions, coefficient
multiplication and morphisms of \(F\).
It glues for a Cartier divisor using its local equations.
The images computed as holomorphic-module images and as
underlying complex-sheaf images agree: the forgetful functor
preserves injectives, since its left adjoint
\(\mathcal O\otimes_{\mathbf C}-\) is exact on stalks.

#### DT.C. The Jacobian and the finite transfer complex

Suppose \(Z\) and \(X\) have the same complex dimension \(n\),
and \(f\) is holomorphic with invertible differential away from
a hypersurface \(D=\operatorname{Supp}(f^{-1}S)\).
Assume \(Z\setminus D\) is dense in every component.
Let \(M\) be finite projective over \(\mathcal O_Z(*D)\),
with integrable connection.

On coordinate charts put
\[
\begin{gathered}
B_{ij}=\frac{\partial f_j}{\partial z_i},\\
J=\det B,\quad g=h\circ f.
\end{gathered}
\tag{DT.8}
\]
The zeros of \(J\) lie in \(D\). The analytic Nullstellensatz
gives a local identity \(g^N=aJ\); hence \(J\) is a unit in
\(\mathcal O_Z[1/g]\). The meromorphic vector fields
\[
V_j=\sum_i(B^{-1})_{ji}\partial_{z_i}
\tag{DT.9}
\]
satisfy \(V_j(f_k)=\delta_{jk}\). They commute.
Indeed their bracket kills every \(f_k\), and an invertible
matrix \(B\) makes a meromorphic vector field with that property
zero. Therefore \(V_j\), acting through the connection, define
the target coordinate derivative actions on \(M\).
Write \(\nabla_{V_j}\) for those actions.

Use the course's left-module transfer
\(\mathcal D_{X\leftarrow Z}\), with both density factors.
The ordinary Spencer resolution of \(M\) is valid here even
before operator coherence is assumed: its terms induced from
the flat \(\mathcal O_Z\)-module \(M\) are \(\mathcal D_Z\)-flat,
and its symbol complex is the Koszul complex of the coordinate
derivatives, with coefficients in \(M\).
The usual finite leading-degree reduction proves exactness.

Consequently the actual derived transfer tensor is the finite
complex with terms
\[
\begin{gathered}
K^{-k}=\mathcal D_{X\leftarrow Z}
\otimes_{\mathcal O_Z}\\
\bigwedge^k T_Z\otimes_{\mathcal O_Z}M,\\
0\le k\le n.
\end{gathered}
\tag{DT.10}
\]
All its coefficient factors can be localized at \(g\), since
\(g\) is already invertible on \(M\). Operators in the target
coordinates are still finite polynomials in their derivatives.
Filter by their target derivative order, with the Koszul shifts.
The symbol differential is contraction with the sequence
\(\sum_jB_{ij}\xi_j\). Since \(B\) is invertible, this is the
Koszul complex of \(n\) independent polynomial variables.
It is exact in negative degrees; in degree zero its quotient
is the density line tensored with \(M\).
This exactness holds for every coefficient module: the elementary
polynomial Koszul contraction, by one variable at a time, proves it.
Lifting a leading preimage lowers the finite polynomial order
and terminates. Thus \(K\) has no negative cohomology.

We identify its degree-zero quotient with the actual target
connection, including the density factor. Write
\(dz=dz_1\wedge\cdots\wedge dz_n\) and
\(dx=dx_1\wedge\cdots\wedge dx_n\).
The coordinate frame \(\eta=dz\otimes f^*(dx)^{-1}\)
of
\(\omega_Z\otimes f^*\omega_X^{-1}\)
uses the formal pullback line generator; its map to the
source top-form line is \(f^*(dx)\mapsto J\,dz\).
The augmentation is
\[
\begin{gathered}
\epsilon(\eta\otimes P\otimes m)\\
=P(f,\nabla_V)(m/J).
\end{gathered}
\tag{DT.11}
\]
It is balanced in the coefficient factors.
To check the source derivative relation, right source
multiplication in this frame gives
\[
\begin{gathered}
(\eta\otimes P\otimes a)\partial_{z_i}\\
=\sum_j\eta\otimes PD_{x_j}\otimes B_{ij}a\\
{}-\eta\otimes P\otimes\partial_{z_i}a.
\end{gathered}
\tag{DT.12}
\]
This follows by transposing the left source chain-rule action
and the right target action. For the identity map it reduces
to the ordinary right operator action.
Balancing DT.12 with the source connection requires DT.11 to
kill
\(\sum_jPD_{x_j}\otimes B_{ij}m-P\otimes\nabla_{\partial_{z_i}}m\).
The needed identity is
\[
\sum_jV_j(B_{ij})=J^{-1}\partial_{z_i}J.
\tag{DT.13}
\]
Indeed its left side is
\(\sum_{j,k}(B^{-1})_{jk}\partial_{z_k}B_{ij}\).
Equality of the mixed derivatives of \(f_j\) changes this to
\(\operatorname{tr}(B^{-1}\partial_{z_i}B)\), which is the
right side by differentiating the finite determinant.
Leibniz and \(\sum_jB_{ij}V_j=\partial_{z_i}\) now give the
required cancellation, including the derivative of \(1/J\).

The augmentation is surjective: the class with
\(P=1\) and coefficient \(Jm\) maps to \(m\).
Its symbol quotient is the invertible density identification
\(\eta\mapsto1/J\). The same finite leading-order reduction
therefore proves that its kernel is exactly the Spencer image.
We have obtained a natural quasi-isomorphism
\[
\mathcal D_{X\leftarrow Z}
\otimes^L_{\mathcal D_Z}M
\simeq M
\tag{DT.14}
\]
as complexes of \(f^{-1}\mathcal D_X\)-modules, where the right
side has target action DT.9.
The density map is intrinsic: \(f^*(dx)=J\,dz\).
Changing either coordinate volume gives the same DT.11.
All formulas are coordinate chain rules, so the local
quasi-isomorphisms glue.
There is no dimension shift, since the two dimensions agree.
Only a finite operator complex and finite polynomial reductions
were used.

#### DT.D. Proper images of logarithmic connections

Assume now that \(f:Z\to X\) is proper and locally projective,
\(\dim Z=\dim X=n\), \(f^{-1}U\) is dense in every source
component, and
\[
f:f^{-1}U\longrightarrow U.
\tag{DT.15}
\]
is a finite covering by local biholomorphisms.
Assume \(D=\operatorname{Supp}(f^{-1}S)\) is an SNC divisor.
Let \(E_\Sigma\) be the normalized logarithmic extension of
any finite-rank local system on \(Z\setminus D\), and set
\(M=E_\Sigma(*D)\).
Then its actual finite-order operator image is concentrated
in degree zero, and canonically
\[
\begin{gathered}
f_+M\simeq N,\\
N=f_*M\simeq(f_*E_\Sigma)(*S).
\end{gathered}
\tag{DT.16}
\]
Its connection is the lifted target action DT.9.
It is coherent holonomic over \(\mathcal D_X\) and finite
projective over \(\mathcal O_X(*S)\).

To prove this, \(M\) is coherent holonomic with the global good
filtration and coherent generator \(E_\Sigma(D)\) proved
in §5.3. Thus the full supplied-generator theorem §5.12
applies to this actual source, not to an arbitrary coherent
module without that hypothesis.
It makes every cohomology of \(f_+M\) coherent and holonomic.
Independently, DT.14 identifies
\[
f_+M\simeq Rf_*M.
\tag{DT.17}
\]
The ordinary coherent projective image theorem in §5.9
makes every \(R^qf_*E_\Sigma\) coherent holomorphic.
For \(q>0\) it vanishes on \(U\): a small target open there
has finitely many disjoint isomorphic sheets, whose direct
image is exact. Its support is therefore contained in \(S\).
The analytic Nullstellensatz kills it locally by a power of
\(h\). Formula DT.7 gives
\[
\begin{gathered}
R^qf_*M=(R^qf_*E_\Sigma)[1/h]\\
=0\quad(q>0).
\end{gathered}
\tag{DT.18}
\]
Degree zero gives the middle identity of DT.16 and finite
\(\mathcal O_X(*S)\)-generation. Its inherited connection
and the Fitting-ideal proof DT.4–DT.6 give projectivity.
This proves every assertion of DT.16, including the actual
operator action.

The target connection is regular at generic smooth points
of every component of \(S\).
Here is a geometric and residue proof that includes ramification.
The bad part of \(D\) consists of its crossing locus,
the rank-deficient locus of the map from each component of \(D\)
to a component of \(S\), and all components whose images have
dimension at most \(n-2\).
Also include the inverse image of the singular locus of \(S\).
After retaining only the rank-deficient part for dominant
components, the images of these sets have dimension at most
\(n-2\).
For a non-dominant component this follows from its image
dimension; on a dominant component, generic rank is \(n-1\),
and the remaining bad part has dimension at most \(n-2\).
The analytic rank and dimension proof is §5.11's finite-branch
and countable constant-rank argument.

These images are closed analytic locally on \(X\).
To see this without a separate proper mapping theorem, apply
the ordinary coherent proper image theorem to the structure
sheaf of each closed bad set. Its pushed structure sheaf has
support exactly the image: at a point of the image, the
constant section \(1\) has a nonzero germ, while outside
the closed proper image all germs are zero.
There are only finitely many such sets near a compact fibre.
Their union is a closed analytic subset \(B\) of codimension
at least two in \(X\).

At a point of \(S\setminus B\), each source point lies on one
smooth boundary component and its tangential differential
has rank \(n-1\). Use the target tangential coordinates on
the source. If \(w\) is a local source boundary equation,
the pulled-back target normal coordinate is \(w^e a\),
with \(a\) a unit. A holomorphic \(e\)-th root of \(a\)
and the inverse-function construction give coordinates
\[
\begin{gathered}
t=w^e,\quad x_i=z_i,\\
2\le i\le n.
\end{gathered}
\tag{DT.19}
\]
The fibre is discrete there, hence finite by compactness.
Properness shrinks the target so that its entire inverse
image lies in these finitely many charts.

In the canonical logarithmic frame, let \(A\) be the constant
source normal residue, with spectrum in
\(0\le\operatorname{Re}\lambda<1\).
The meromorphic image has the finite basis consisting of
the vectors \(w^jv\), \(0\le j<e\), for each source basis
vector \(v\).
Convergent power series split uniquely into these residue
classes of exponents; the same grouping with finitely many
negative powers proves the meromorphic statement.
On this basis the target normal residue is
\[
\frac{A+jI}{e},\qquad 0\le j<e.
\tag{DT.20}
\]
This follows by applying
\(\nabla_{\partial_t}=(ew^{e-1})^{-1}\nabla_{\partial_w}\).
Tangential derivatives retain their flat frame.
The spectra remain in the same normalized strip and every
nilpotent block is retained, divided by \(e\).
Thus this is an actual normalized logarithmic target model.
Taking the finite direct sum over the sheets proves generic
regularity along every target divisor component.
Theorem 5.27 then supplies the coherent reflexive strict
lattice of \(N\), including all singular points of \(S\),
and its normalized logarithmic restriction at every smooth
point of \(S\).

#### DT.E. The whole finite D-type class and compact generators

Call \(Q\) a finite divisorial D-type module along \(S\) if it is
coherent holonomic over \(\mathcal D_X\), satisfies DT.1, multiplication by
a local equation of \(S\) is bijective, and its induced
meromorphic connection is regular on a nonempty open family
of transverse discs on the smooth locus of each local
irreducible component of \(S\).
The meromorphic connection in this definition exists by DT.A;
no free meromorphic frame is assumed.
This is precisely the analytic divisorial regularity hypothesis
of §5.27.1, applied to the whole localized module.

For every such \(Q\), that theorem gives a unique coherent
reflexive normalized lattice \(L_\Sigma(Q)\) and
\[
L_\Sigma(Q)(*S)=Q.
\tag{DT.21}
\]
Restriction to \(U\), followed by the horizontal-section functor,
is fully faithful on this class.
It includes all morphisms, with no generic-stratum replacement
of \(Q\).

We first prove the extension fact used for this assertion.
If a closed analytic set \(B\) has codimension at least two,
holomorphic scalar functions extend uniquely across \(B\).
Locally choose a complex line through the point which meets
the germ of \(B\) only at that point, using the earlier
local-parametrization cone bound.
After shrinking a product, the outer circle in that line
misses \(B\) over every nearby parameter.
Cauchy integration on this fixed circle produces a holomorphic
function on the whole inner product.
For parameters whose vertical disc misses \(B\), it is the
original function by the Cauchy formula.
Those parameters are dense: the proper projection of the
part of \(B\) in the product has dimension at most \(n-2\),
by the same analytic rank/dimension argument.
Continuity on the complement proves equality everywhere there.
Uniqueness follows from density. For \(n\le1\) there is no
nonempty such germ \(B\).

A coherent reflexive sheaf \(L\) is locally a finite matrix
kernel between finite free holomorphic sheaves.
Indeed dualize a finite presentation of \(L^*\) and use
\(L=L^{**}\).
Apply scalar extension just proved to each coordinate of a
section of this kernel outside \(B\). Its finite equations
continue to hold by density. Hence sections of \(L\) extend
uniquely across \(B\).
Given a morphism between two such reflexive sheaves outside
\(B\), apply this fact to the images of a finite local
generator list of the source. All its finite relations
continue to hold; thus the morphism extends uniquely too.

Now take a horizontal map of the two flat bundles on \(U\).
At every smooth point of \(S\) the normalized logarithmic
map theorem §5.0 extends it holomorphically between their
normalized lattices. These extensions agree by density.
The complement of this domain lies in \(S_{\rm sing}\),
which has codimension at least two.
The reflexive extension fact gives a map on all of \(X\).
Localize it and use DT.21. Its commutators with the connection
vanish on \(U\), hence vanish everywhere as meromorphic
coefficient identities. This gives a \(\mathcal D_X\)-linear
map of the whole modules.
Conversely an operator map restricts to a horizontal bundle
map and thus to a local-system map, by the proved analytic
flat-frame equivalence. If this restriction is zero, the
meromorphic coefficient map is zero by density.
This proves full faithfulness.

Every finite divisorial D-type module has a coherent
holomorphic generator on a neighbourhood of each compact
subset of \(X\). In fact consider the ascending coherent
operator submodules
\[
\begin{gathered}
\mathcal D_X\bigl(L_\Sigma(Q)\bigr)\\
\subseteq\mathcal D_X\bigl(L_\Sigma(Q)(S)\bigr)\\
\subseteq\mathcal D_X\bigl(L_\Sigma(Q)(2S)\bigr)\\
\subseteq\cdots .
\end{gathered}
\tag{DT.22}
\]
Their union is \(Q\) by DT.21.
Each term is coherent over \(\mathcal D_X\), since it is
locally generated by finitely many coefficient sections and
analytic operator coherence applies.
At each point the finite generation of \(Q\) puts its whole
stalk in one term. The coherent quotient then vanishes on
one actual neighbourhood of that point.
A finite cover of the compact subset and the maximum of
these finitely many indices give one integer \(k\) and one
neighbourhood on which
\[
\mathcal D_X\bigl(L_\Sigma(Q)(kS)\bigr)=Q.
\tag{DT.23}
\]
The coefficient sheaf on the left is a single coherent
holomorphic generator. No infinite intersection of open
neighbourhoods is involved.
This assertion concerns the full finite divisorial D-type
class defined above.

#### DT.F. Meromorphic realization from an SNC modification

Suppose, in addition, a proper locally projective modification
\(f:Z\to X\) is given, isomorphic over \(U\), with
\(D=\operatorname{Supp}(f^{-1}S)\) SNC.
For every finite-rank local system \(L\) on \(U\), its pullback
to \(Z\setminus D\) has the canonical logarithmic extension
\(E_\Sigma\). The module DT.16 belongs to the full finite
divisorial D-type class and restricts to \(L\).
Thus in the presence of this geometric hypothesis, horizontal
restriction is an equivalence between that class and finite-rank
local systems on \(U\).
It is independent of the chosen modification: DT.E supplies
the unique operator isomorphism inducing the identity on \(L\).
Its two compositions are identities by full faithfulness.
Morphisms and compositions have the same uniqueness.
The zero-rank object is included.

This is a finite-order realization theorem.
An additional comparison is required to identify
\[
D^\infty_{\mathbf C\times X}
\otimes_{D_{\mathbf C\times X}}
(\mathcal O_{\mathbf C}\boxtimes Q)
\tag{DT.24}
\]
with all multivalued holomorphic functions of the corresponding
monodromy type, jointly in the auxiliary variable.
Neither DT.16 nor the finite equivalence just proved asserts
that comparison. In particular they do not supply hypothesis
ST.20 of
§5.71.
The existence of a modification in DT.F is also an explicit
geometric hypothesis; it is not a conclusion of an
algebraic-only resolution theorem.

The canonical all-parameter comparison and the D-type realization for a prescribed analytic hypersurface are proved in §5.79, FB.D–FB.F.

![The localized transfer and a ramified logarithmic image](assets/finite-analytic-d-type.png)

**Figure DT.1.** Panel A is the actual finite transfer comparison
DT.10–DT.18; its density map divides the source coefficient by
the Jacobian. Panel B is the exact real slice of the complex
map \(t=w^3\), with \(-1\le w\le1\); it depicts neither the
whole complex map nor a one-to-one complex covering.
Panel C retains a rank-two source residue with eigenvalue
\(2/5\) and nonzero nilpotent entry one.
The three rank-two target blocks have eigenvalues
\(2/15,7/15,12/15\) and nilpotent entry \(1/3\), by DT.20.
The target rank is six. The algebraic diagrams have no
geometric scale.

Editable SVG,
exact diagram data, and
reproducible source.
Freely accessible further reading:
[Kashiwara–Kawai, *Holonomic systems III*, II.2–II.3, printed pp.863–865](https://www.kurims.kyoto-u.ac.jp/~kenkyubu/kashiwara/HolIII.pdf).
The arguments above give the actual finite coefficient,
transfer, density, proper image and morphism proofs at the
stated hypotheses.


### 5.73. Infinite-order operators and holomorphic SNC functions

Let \(X=\Delta_z^k\times\Delta_y^m\) be an adapted polydisc,
let \(D=\{z_1\cdots z_k=0\}\), and put \(U=X\setminus D\).
The empty divisor \(k=0\) is allowed.
Let \(A_1,\ldots,A_k\) be commuting constant matrices of size
\(r\), with
\[
0\le\operatorname{Re}\operatorname{Spec}(A_i)<1.
\tag{SI.1}
\]
The meromorphic connection
\[
\begin{gathered}
M=\mathcal O_X(*D)\otimes\mathbf C^r,\\
\nabla=d+\sum_i A_i\,\frac{dz_i}{z_i}.
\end{gathered}
\tag{SI.2}
\]
has its actual finite-order presentation and good filtration
in §5.3.
All residue Jordan blocks are retained.
The normalized logarithmic model theorem in §5.0 puts every
normalized logarithmic connection on an SNC chart in this form.

Write \(B=D^\infty_X\) for the actual ordinary infinite-order
ring of the programme. In coordinates its coefficients satisfy,
on one common holomorphic domain and every compact therein,
\[
\begin{gathered}
P=\sum_{\alpha\ge0}p_\alpha D^\alpha,\\
|p_\alpha|\le C_\epsilon
\frac{\epsilon^{|\alpha|}}{\alpha!},\quad\epsilon>0.
\end{gathered}
\tag{SI.3}
\]
Its full product and action on multivalued holomorphic functions
are the complete §5.49, ISF.4
and §5.70, PM.B–PM.D.
All tensor products below are ordinary algebraic tensors.

We prove a canonical isomorphism
\[
\begin{gathered}
B\otimes_{\mathcal D_X}M
\xrightarrow{\;\Phi\;}j_*(M|_U),\\
j:U\hookrightarrow X.
\end{gathered}
\tag{SI.4}
\]
where the right side means all holomorphic sections of the
flat bundle on the punctured product. It has no finite-pole
or growth restriction at \(D\).
The map is the actual operator action. The proof constructs
both its inverse and finite algebraic witnesses for its kernel.
It includes any number of auxiliary holomorphic parameters.

#### SI.A. Generators and matrix factorials

Choose a column \(V\) of canonical meromorphic frame symbols
such that \(D_{z_i}V=z_i^{-1}A_iV\).
This is a convention for the frame equations; coefficients of
a section are rows on the left. Put
\[
\begin{gathered}
Z=z_1\cdots z_k,\quad W=Z^{-1}V,\\
B_i=A_i-I.
\end{gathered}
\tag{SI.5}
\]
Then
\[
\begin{gathered}
L_i=z_iD_{z_i}I-B_i,\\
L_iW=0,\quad D_{y_j}W=0.
\end{gathered}
\tag{SI.6}
\]
For \(n\ge0\), set
\[
\begin{gathered}
C_i(0)=I,\\
C_i(n)=\prod_{\ell=0}^{n-1}(B_i-\ell I).
\end{gathered}
\tag{SI.7}
\]
Every \(C_i(n)\) is invertible. Indeed a zero eigenvalue
would require an eigenvalue of \(A_i\) to be a positive
integer, excluded by SI.1. The product rule gives
\[
D_{z_i}^nW=z_i^{-n}C_i(n)W.
\tag{SI.8}
\]
All these matrices commute, including between distinct \(i\).

There are constants \(K,a>0\), independent of \(n\), such that
\[
\begin{gathered}
\|C_i(n)\|\le K\,n!(n+1)^a,\\
\|C_i(n)^{-1}\|\le K\,\frac{(n+1)^a}{n!}.
\end{gathered}
\tag{SI.9}
\]
for every \(i\) and \(n\).
Here is an elementary bound. The exact identity is
\[
\begin{gathered}
C_i(n)=(-1)^n n!\\
{}\cdot\prod_{\ell=1}^n(I-A_i/\ell).
\end{gathered}
\tag{SI.10}
\]
For \(\ell>2\|A_i\|\), the norm of a factor is at most
\(1+\|A_i\|/\ell\), and that of its inverse at most
\(1+2\|A_i\|/\ell\), by the geometric matrix series.
Their products are at most a constant times a power of \(n+1\),
since \(\sum_{\ell=1}^n1/\ell\le1+\log(n+1)\).
The finitely many initial factors and inverse factors are
bounded separately. There are finitely many \(i\), so one
\(K,a\) works. This proof keeps every nilpotent entry.

#### SI.B. One-variable division with full positive heads

Fix one normal coordinate, write \(z,D_z,B,C(n)\) for its
data, and treat every other coordinate as a parameter for
the coefficient estimates. Any remaining derivative
multiindex is a spectator index in the same estimates.
First consider the row operator
\[
\begin{gathered}
P=\sum_{m\ge0}p_m(z,y)D_z^m,\\
p_m(z,y)=\sum_{b\ge0}p_{m,b}(y)z^b .
\end{gathered}
\tag{SI.11}
\]
Choose a coefficient circle of radius \(\rho\) inside its
common holomorphic domain and an inner radius \(s<\rho\).
Cauchy's formula gives, for every \(\epsilon>0\),
\[
\|p_{m,b}\|\le M_\epsilon
\frac{\epsilon^m}{m!\rho^b}.
\tag{SI.12}
\]
Define
\[
\begin{gathered}
c_n=\sum_{m\ge n}p_{m,m-n}\\
{}\cdot C(m)C(n)^{-1},\quad n\ge1,\\
r(z)=\sum_{q\ge0}\sum_{m\ge0}\\
p_{m,m+q}C(m)z^q,\\
q_n(z)=\sum_{m\ge n+1}\sum_{b\ge m-n}\\
p_{m,b}z^{b-m+n}C(m)\\
{}\cdot C(n+1)^{-1}.
\end{gathered}
\tag{SI.13}
\]
The matrix products are on the right of the row coefficients,
in precisely the displayed order.
Then
\[
\begin{gathered}
L=zD_zI-B,\\
P=r(z)+\sum_{n\ge1}c_nD_z^n+QL,\\
Q=\sum_{n\ge0}q_n(z)D_z^n.
\end{gathered}
\tag{SI.14}
\]
Both infinite operators on the right are actual elements
of \(B\) on one smaller coefficient product.

We supply all convergence needed for that assertion.
Enlarge \(a\) in SI.9 if necessary, write
\(\theta=\epsilon/\rho<1\), and put
\(S_a(\theta)=\sum_{j\ge0}(j+1)^a\theta^j\).
For \(0<\epsilon<\rho\), SI.9 and SI.12 give
\[
\begin{gathered}
\|c_n\|\le K^2M_\epsilon\\
{}\cdot\frac{\epsilon^n(n+1)^{2a}}{n!}
S_a(\theta),\\
\|r\|_{|z|\le s}\le
\frac{KM_\epsilon}{1-s/\rho}S_a(\theta),\\
\|q_n\|_{|z|\le s}\le
\frac{K^2M_\epsilon}{1-s/\rho}\\
{}\cdot\frac{\epsilon^{n+1}(n+2)^{2a}}
{\rho(n+1)!}S_a(\theta).
\end{gathered}
\tag{SI.15}
\]
For the last bound write \(m=n+1+j\) and sum first in \(b\).
The resulting geometric factor is
\(\rho^{-(m-n)}/(1-s/\rho)\).
The other two estimates follow by writing \(m=n+j\)
and by summing the nonnegative \(q\), respectively.
The inequality
\((n+j+1)^a\le(n+1)^a(j+1)^a\) supplies the stated
polynomial factors.

For any desired output epsilon \(\delta>0\), choose
\(\epsilon<\min(\rho/2,\delta/2)\).
The sequences \((n+1)^{2a}2^{-n}\) and
\((n+2)^{2a}2^{-n}\) are bounded, so SI.15 gives
the every-\(\delta\) factorial estimates for \(c_n\)
and \(q_n\).
The coefficient domain did not change with \(\delta\).
The scalar series \(S_a(\theta)\) is finite: with
\(L_a(\theta)=\sup_{j\ge0}(j+1)^a\theta^{j/2}\),
its tail from \(j=N\) is at most
\[
L_a(\theta)\,
\frac{\theta^{N/2}}{1-\sqrt\theta}.
\tag{SI.16}
\]
The other Taylor tail is geometric in \(s/\rho\).
These bounds give absolute normal convergence of every
sum in SI.13. A further fixed inner product gives every
prescribed coefficient derivative of the same tails by
Cauchy's formula.

For clarity the finite identity behind SI.14 is also explicit.
For \(m,b\ge0\), put \(v=\min(m,b)\). Repeatedly replace
\(zD_z^m\) by \(D_z^{m-1}(zD_zI-B)\) plus
\((B-(m-1)I)D_z^{m-1}\).
It gives
\[
\begin{gathered}
F(m,j)=C(m)\\
{}\cdot C(m-j)^{-1},\\
z^bD_z^m=\\
\sum_{j=0}^{v-1}\Bigl(z^{b-1-j}\\
{}\cdot F(m,j)\\
{}\cdot D_z^{m-1-j}L\Bigr)\\
{}+z^{b-v}F(m,v)\\
{}\cdot D_z^{m-v}.
\end{gathered}
\tag{SI.17}
\]
Empty products and sums have their literal meanings.
Multiply by \(p_{m,b}\), and sum the actual coefficient
series. SI.15–SI.16 justify the rearrangement and give
SI.13–SI.14. No limit is taken in an algebraic quotient.
There is one actual row \(Q\) witnessing the relation.

If there are spectator derivatives \(D^\alpha\), SI.12
has the additional factor
\(\epsilon^{|\alpha|}/\alpha!\).
Every operation above leaves that factor unchanged.
The output bounds therefore include all spectator indices,
not a bounded-degree truncation. The fixed coefficient
circle and final product are independent of those indices.

#### SI.C. The whole SNC normal form and its zero test

For a full row operator in SI.3, factor all terms with a
positive tangential derivative on the right.
Assign each such multiindex to its first nonzero tangential
entry. Shifting that entry down by one gives finitely many
actual rows \(H_j\), since its new factorial bound only
gains a factor at most \(\epsilon\).
The remainder contains only normal derivatives.
The removed terms are \(\sum_jH_jD_{y_j}\).

Apply SI.B successively to the \(k\) normal coordinates.
Fix all outer, intermediate and inner radii before these
finitely many steps. The spectator estimate proves that
each step stays in the full actual ring.
Matrices from distinct normal directions commute.
Taylor extraction in one variable also preserves the
coefficient's independence of any previously extracted
negative variable.
Consequently the result has the form
\[
\begin{gathered}
P=R(P)+\sum_{i=1}^kQ_iL_i\\
{}+\sum_{j=1}^mH_jD_{y_j},\\
R(P)=\sum_{I\subseteq\{1,\ldots,k\}}
\ \sum_{n_i\ge1,\ i\in I}\\
c_{I,n}(z_{I^c},y)D_I^n .
\end{gathered}
\tag{SI.18}
\]
For \(I=\varnothing\) the inner sum is one holomorphic
coefficient row. Each \(c_{I,n}\) is independent of \(z_i\)
for \(i\in I\), holomorphic in all remaining coordinates,
and has every factorial bound in \(n\) on a common product.
All \(Q_i,H_j\) are actual infinite-order rows.
There are only \(k+m\) relation rows in SI.18.

Evaluation on \(W\) has relative coefficient
\[
\begin{gathered}
\sum_I\sum_{n_i\ge1,\ i\in I}
c_{I,n}(z_{I^c},y)\\
{}\cdot\left(\prod_{i\in I}C_i(n_i)\right)z_I^{-n}.
\end{gathered}
\tag{SI.19}
\]
It converges normally on every compact punctured product:
choose the coefficient epsilon below half its least normal
radius and use SI.9. Fixed derivatives follow on a smaller
compact. Its Laurent sectors have exactly \(I\) as their
set of negative coordinates.
Uniqueness of the multivariable Laurent coefficients,
proved by successive circle integrals, shows that SI.19
is zero if and only if every \(c_{I,n}\) is zero, because
each matrix product in it is invertible.
Thus zero evaluation of \(P\) gives the finite identity
\[
\begin{gathered}
P=\sum_iQ_iL_i+\sum_jH_jD_{y_j}\\
\text{in }B^r.
\end{gathered}
\tag{SI.20}
\]

The same proof for a finite-order \(P\) has finite-order
relation rows: normal derivative degree only decreases.
Moreover \(W\) generates \(M\).
Indeed a meromorphic coefficient row relative to \(W\)
has a finite lower bound in each Laurent coordinate.
Its Laurent-sector decomposition then has finitely many
negative multiindices. SI.8 and the invertible matrices
produce it by finite-order derivatives with holomorphic
coefficients.
The zero test proves the exact finite presentation
\[
\begin{gathered}
M=\mathcal D_X^r/\mathcal R,\\
\mathcal R=\sum_i\mathcal D_X^rL_i\\
{}+\sum_j\mathcal D_X^rD_{y_j}.
\end{gathered}
\tag{SI.21}
\]
Tensoring this finite presentation with \(B\) is right exact.
Equation SI.20 therefore proves injectivity of SI.4 by
finite algebraic balancing, without a quotient-closedness
assumption or a completed tensor.

#### SI.D. An actual inverse for every holomorphic section

Write a holomorphic section on \(U\) in the canonical frame
as \(g(z,y)V\), and put \(h=Zg\), so it is \(hW\).
The Laurent decomposition is
\[
\begin{gathered}
h(z,y)=\sum_{I\subseteq\{1,\ldots,k\}}
\ \sum_{n_i\ge1,\ i\in I}\\
a_{I,n}(z_{I^c},y)z_I^{-n}.
\end{gathered}
\tag{SI.22}
\]
The coefficients are holomorphic on full discs in the
coordinates not in \(I\), including their zeros.
Here is a domain and bound proof.
Extract the indicated negative Laurent coefficients by
circle integrals at arbitrary radii \(\rho_i>0\), \(i\in I\).
For each remaining normal coordinate take its nonnegative
Laurent part by the positive Cauchy projection on one fixed
outer circle. All these operations commute on compact
polyannuli by absolute convergence.
On an inner product in the remaining coordinates the
positive projections have a fixed geometric denominator.
Consequently
\[
\|a_{I,n}\|_K\le M_{\rho,K}
\prod_{i\in I}\rho_i^{n_i}
\tag{SI.23}
\]
for every choice of sufficiently small positive \(\rho_i\).
The full coefficient domain for the nonnegative coordinates
and the parameters is fixed before those radii.
All coefficients are extended independently of \(z_I\)
to one full smaller polydisc.
The case \(I=\varnothing\) is the full positive Cauchy part.
Summing the positive and negative circle projections
successively proves SI.22; their sectors partition the
entire Laurent series.

Define the actual row operator
\[
\begin{gathered}
P_h=\sum_I\sum_{n_i\ge1,\ i\in I}\\
a_{I,n}\left(\prod_{i\in I}C_i(n_i)^{-1}\right)D_I^n.
\end{gathered}
\tag{SI.24}
\]
For any desired \(\delta>0\), choose
\(0<\rho_i<\delta/2\).
Equations SI.9 and SI.23 give
\[
\begin{gathered}
\left\|a_{I,n}\prod_{i\in I}C_i(n_i)^{-1}\right\|\\
\le M_{\rho,K}K^k\prod_{i\in I}
\frac{\rho_i^{n_i}(n_i+1)^a}{n_i!}.
\end{gathered}
\tag{SI.25}
\]
The polynomial factors are absorbed by \(2^{-n_i}\)
exactly as in SI.15.
Thus \(P_h\) satisfies SI.3 for every \(\delta\)
on one common full coefficient product.
Its coefficients contain no tangential derivatives.
The finitely many sectors combine into one actual row.
The action identity SI.8 and the normally convergent
Laurent sum give \(P_hW=hW=gV\).
This proves surjectivity of SI.4.

Its inverse is the actual finite tensor
\[
\Psi(gV)=\sum_{\nu=1}^r(P_h)_\nu\otimes W_\nu.
\tag{SI.26}
\]
There are \(r\) elementary tensors, although each coefficient
operator may have its full infinite positive head.
We have \(\Phi\Psi=1\) by construction.
For a finite tensor row \(P\otimes W\), normal division
SI.18 and Laurent uniqueness make \(\Psi\Phi(P\otimes W)\)
its normal-form row. Equation SI.20 is the finite balancing
witness that this equals the original tensor.
Thus \(\Psi\Phi=1\) as well.
These are actual inverse maps, not convergence in a tensor
completion.

All fixed coefficient derivatives, every positive head and
all parameters are included in the Cauchy estimates.
In local horizontal frames the target in SI.4 has the
full \(B\)-action by the earlier ordinary product theorem.
Hence \(\Phi\), and therefore its inverse, are \(B\)-linear.
The action map is intrinsic. Its inverse is consequently
natural for connection morphisms, frame changes, restrictions
and the addition of auxiliary polydisc parameters.
In particular, adding a parameter \(u\) gives
\[
\begin{gathered}
D^\infty_{\Delta_u\times X}\\
{}\otimes_{D_{\Delta_u\times X}}
(\mathcal O_{\Delta_u}\boxtimes M)\\
\simeq j_*(\mathcal O_{\Delta_u}\boxtimes M|_U).
\end{gathered}
\tag{SI.27}
\]
with joint holomorphy in \(u\), on the same kind of fixed
coefficient products. It commutes with every parameter
derivative and every restriction to a smaller parameter
polydisc.

#### SI.E. Cofinite abelian monodromy ideals

The fundamental group of \(U\) is \(\mathbf Z^k\), generated
by the positive normal meridians. This is the product
circle retraction used in the complete SNC proof §5.4.
Let \(\mathfrak a\) be a cofinite left ideal of
\(\mathbf C[\mathbf Z^k]\), and set
\[
\begin{gathered}
V_{\mathfrak a}=\mathbf C[\mathbf Z^k]/\mathfrak a,\\
q=\dim V_{\mathfrak a},\quad v_0=1+\mathfrak a.
\end{gathered}
\tag{SI.28}
\]
If \(q=0\), all corresponding modules and function spaces
are zero. Suppose \(q>0\).
The meridians act by commuting invertible matrices
\(\rho_i\). They have commuting normalized logarithms
\(A_i\) satisfying
\[
\begin{gathered}
\exp(2\pi iA_i)=\rho_i,\\
0\le\operatorname{Re}\operatorname{Spec}(A_i)<1.
\end{gathered}
\tag{SI.29}
\]
To construct them, decompose each matrix into its generalized
eigenspaces. Choose for each nonzero eigenvalue \(\lambda\)
the unique scalar logarithm whose quotient by \(2\pi i\)
has real part in the half-open strip.
On a block \(\lambda(I+N)\), use the terminating series
\(\log(I+N)=\sum_{\ell\ge1}(-1)^{\ell+1}N^\ell/\ell\).
Finite polynomial interpolation on the relatively prime
powers of the minimal-polynomial factors makes this
logarithm a polynomial in \(\rho_i\).
It therefore commutes with all the other matrices.
The finite power-series identities for nilpotents verify
the exponential identity. No Jordan block is discarded.

On the universal cover put
\[
\begin{gathered}
R(z)=\prod_i\exp(A_i\log z_i),\\
F(z)=R(z)v_0.
\end{gathered}
\tag{SI.30}
\]
The column \(F\) satisfies
\(D_{z_i}F=z_i^{-1}A_iF\), and its deck transformation
around the positive \(i\)-th meridian is \(\rho_iF\).
Here \(\rho_i\) describes deck action on the function
column; horizontal coefficient rows of the connection
solve \(d\ell+\sum_i\ell A_i\,dz_i/z_i=0\).
Use \(F\) as the column of abstract frame symbols for
the finite module \(M_{\mathfrak a}\) of SI.2.
It has a faithful realization by scalar multivalued
meromorphic functions with precisely these equations.

Here is the complete function-space identification.
If \(f\) is a multivalued holomorphic function killed by
\(\mathfrak a\), define a linear functional on \(V_{\mathfrak a}\)
at each lifted point by
\[
\lambda_f(\gamma v_0)=T_\gamma f.
\tag{SI.31}
\]
It is well defined because \(\mathfrak a f=0\).
Since the group is abelian, continuation of this row
functional is right multiplication by \(\rho_\gamma\).
The same is true of \(R\).
Thus
\[
\ell_f=\lambda_fR^{-1}
\tag{SI.32}
\]
is a single-valued holomorphic row on the punctured product,
and \(f=\ell_fF\).
Conversely every such row \(\ell\) makes \(\ell F\)
a function killed by \(\mathfrak a\), since
\(\rho(a)v_0=0\) for \(a\in\mathfrak a\).
If \(\ell F\) is zero on every branch, then at each point
\(\ell R\) kills every \(\rho_\gamma v_0\).
Those vectors span \(V_{\mathfrak a}\), by its quotient
definition. Hence \(\ell=0\).
This proves the identification without replacing a scalar
monodromy condition by an unrelated vector function.

Combining it with SI.4 and SI.27 gives the actual
auxiliary-variable function isomorphism
\[
\begin{gathered}
D^\infty_{\Delta_u\times X}\\
{}\otimes_{D_{\Delta_u\times X}}
(\mathcal O_{\Delta_u}\boxtimes M_{\mathfrak a})\\
\simeq\mathscr F_{\mathfrak a}(u;z,y).
\end{gathered}
\tag{SI.33}
\]
on this SNC product. Its inverse is the finite tensor
SI.26, using the row \(\ell_f\).
It commutes with all coordinate derivatives.

For \(\mathfrak b\subseteq\mathfrak a\), the quotient map
\(V_{\mathfrak b}\to V_{\mathfrak a}\) is surjective and
sends \(v_{0,\mathfrak b}\) to \(v_{0,\mathfrak a}\).
Normalized logarithms intertwine it: the same finite
interpolation can be made on the union of both spectra,
with the same scalar logarithm and maximal block lengths.
It therefore gives the actual finite module map
\(M_{\mathfrak a}\to M_{\mathfrak b}\) whose function map
is inclusion. The canonical action maps SI.33 and their
inverse maps are natural for these inclusions.
This proves all parameter, ideal and normal-derivative
compatibilities asserted here.

#### SI.F. Actual special tensor lifting on a coordinate product

For \(k\ge1\), take \(t=z_1\), with the other coordinates spatial, and
use the chosen fibration which forgets \(t\).
On the complement of the other coordinate divisors, the
normal fibre has one puncture \(t=0\).
Let \(T=\widetilde E^\infty\) be the full special ring of
§5.49, and let \(R_X=D^\infty_X\).
For every cofinite \(\mathfrak a\) there is a cofinite
two-sided \(\mathfrak b\subseteq\mathfrak a\) such that,
with \(N_{\mathfrak c}=R_X\otimes_{\mathcal D_X}M_{\mathfrak c}\),
\[
\begin{gathered}
\alpha:T\otimes_{R_X}N_{\mathfrak a}\\
\longrightarrow T\otimes_{R_X}N_{\mathfrak b},\\
\operatorname{im}\alpha\subseteq\\
\operatorname{im}\Bigl(N_{\mathfrak b}\\
\longrightarrow T\otimes_{R_X}N_{\mathfrak b}\Bigr),\\
\dim\mathbf C[\mathbf Z^k]/\mathfrak b\le9q^2.
\end{gathered}
\tag{SI.34}
\]
Indeed SI.33 is exactly the actual parameter-compatible
hypothesis ST.20 in
§5.71, ST.C–ST.E.
The full monodromy primitive theorem PM.B–PM.D applies
to the fixed one-puncture family.
Its representation \(V\oplus\mathbb W\) has dimension
\(3q\); its annihilator is contained in \(\mathfrak a\)
and has codimension at most \(9q^2\).
ST.12–ST.19 now prove SI.34 in the ordinary algebraic
tensor products.
The primitive lifts through the explicit finite inverse
SI.26, and ST.18–ST.19 give finite balancing witnesses.
All full special heads and spatial kernels are included.

SI.34 concerns the coordinate punctured product just
specified. A general Weierstrass divisor and its discriminant
need an additional comparison at their singular loci.
Neither the SNC coefficient normal form nor SI.34 supplies
that comparison by itself.

#### SI.G. A full positive head producing an essential singularity

For one normal coordinate and zero residue, \(W=1/z\) and
\(C(n)=(-1)^nn!\).
The complete holomorphic section \(e^{1/z}/z\) is represented by
\[
\begin{gathered}
P=\sum_{n\ge0}\frac{(-1)^n}{(n!)^2}D_z^n,\\
PW=\frac{e^{1/z}}z.
\end{gathered}
\tag{SI.35}
\]
Indeed the \(n\)-th derivative of \(1/z\) is
\((-1)^nn!z^{-n-1}\), so the normally convergent sum is
the exponential series. For every \(\epsilon>0\),
the coefficient bound in SI.3 holds with
\(C_\epsilon=e^{1/\epsilon}\): the \(n\)-th term of that
positive exponential series bounds
\(1/(n!\epsilon^n)\).
Thus SI.35 is an actual infinite operator and one elementary
tensor \(P\otimes W\). Its image has an essential singularity,
not a finite-pole meromorphic coefficient.

![Full Euler division, Laurent sectors and the actual inverse](assets/snc-infinity-function.png)

**Figure SI.1.** Panel A is the actual division SI.18 and
inverse SI.26, with finitely many algebraic relation witnesses.
Panel B partitions the integer Laurent exponent plane for
two normal coordinates by its exact negative-coordinate
set \(I\); the coordinate axes belong to the nonnegative
parts. The plotted integer range is a finite sample of
that infinite partition.
Panel C retains the full rank-two nilpotent residue:
\(A=\left[\begin{smallmatrix}1/3&1\\0&1/3\end{smallmatrix}\right]\).
Its second factorial inverse has diagonal \(9/10\) and
off-diagonal \(189/100\). These are exact matrices in SI.7.
The scalar full positive-head identity is SI.35.
The algebraic panels have no geometric scale.

Editable SVG,
exact figure data, and
reproducible source.
Freely accessible further reading:
[Kashiwara–Kawai, *Holonomic systems III*, II.2.2.4 and III.4.9, printed pp.864 and 898–899](https://www.kurims.kyoto-u.ac.jp/~kenkyubu/kashiwara/HolIII.pdf).
The full coefficient estimates and finite tensor inverses
for the stated SNC model are proved above.


### 5.74. Analytic transfer and the D-type de Rham comparison

All operators in this section have finite order.
For a complex manifold \(X\) of dimension \(n\), use the normalized
analytic de Rham complex
\[
\begin{gathered}
\operatorname{DR}_X(M)=\\
\bigl(\Omega_X^\bullet\otimes_{\mathcal O_X}M\bigr)[n].
\end{gathered}
\tag{AD.1}
\]
The unshifted differential is the connection differential.
A shift by \(n\) multiplies that differential by \((-1)^n\).
For complexes, all totalizations use the usual Koszul signs.
The exact right Spencer resolution and side change are proved in
the de Rham lesson, Proposition1.1.
In particular, with
\[
M^r=\omega_X\otimes_{\mathcal O_X}M,
\]
they identify AD.1 with
\[
\operatorname{DR}_X(M)
\simeq M^r\otimes_{\mathcal D_X}^L\mathcal O_X.
\tag{AD.2}
\]
The right action on \(M^r\) is negative Lie derivative on
the volume form together with the negative left connection action.

We prove the actual finite-transfer comparison and its consequence for
the meromorphic modules of
§5.72, DT.A–DT.F.
The target hypersurface may be singular.

#### AD.A. A transfer augmentation with no flat-map hypothesis

Let \(f:Z\to X\) be holomorphic, with \(Z,X\) smooth, of dimensions
\(d,n\). The right-module transfer is
\[
\begin{gathered}
T_f=\mathcal O_Z\otimes_{f^{-1}\mathcal O_X}
f^{-1}\mathcal D_X,\\
f_+^r A=Rf_*(A\otimes_{\mathcal D_Z}^LT_f).
\end{gathered}
\tag{AD.3}
\]
It is a left \(\mathcal D_Z\), right \(f^{-1}\mathcal D_X\)
bimodule. In coordinates \(z_i,x_j\), its source action is
\[
\begin{gathered}
\partial_{z_i}(a\otimes P)
=(\partial_{z_i}a)\otimes P\\
{}+\sum_j a\,(\partial_{z_i}f_j)\otimes\partial_{x_j}P.
\end{gathered}
\tag{AD.4}
\]
Coefficient multiplication and AD.4 satisfy the tensor relations by
the ordinary chain rule.

There is a canonical quasi-isomorphism of left
\(\mathcal D_Z\)-complexes
\[
T_f\otimes_{f^{-1}\mathcal D_X}^L f^{-1}\mathcal O_X
\longrightarrow\mathcal O_Z.
\tag{AD.5}
\]
Its degree-zero augmentation is
\[
\epsilon_f(a\otimes P)=a\,f^*(P1).
\tag{AD.6}
\]
It is balanced: the right action on \(P\) is paired with the usual
left action on holomorphic functions. Formula AD.4 and the chain
rule give
\[
\begin{gathered}
\epsilon_f(\partial_{z_i}(a\otimes P))\\
=\partial_{z_i}\epsilon_f(a\otimes P).
\end{gathered}
\tag{AD.7}
\]
This verifies the source operator action, including coefficient
derivatives; it is not only a vector-space augmentation.

Here is exactness for an arbitrary \(f\).
Resolve \(f^{-1}\mathcal O_X\) by the inverse image of the left
coordinate Spencer resolution. Its local terms are finite free
\(f^{-1}\mathcal D_X\)-modules. After tensoring with \(T_f\),
filter by the finite target derivative order, with the Koszul
degree shifts. The associated graded augmented complex is
\[
\begin{gathered}
K_\bullet\bigl(\xi_1,\ldots,\xi_n;\\
\mathcal O_Z[\xi_1,\ldots,\xi_n]\bigr)\\
\longrightarrow\mathcal O_Z.
\end{gathered}
\tag{AD.8}
\]
It is exact over any coefficient ring.
For one variable multiplication by \(\xi\) is injective on
polynomials and its cokernel is evaluation at zero.
Successively apply this two-term calculation to the \(n\) variables;
after each quotient the remaining variables are still polynomial
variables. The resulting finite tensor Koszul complex has only the
displayed degree-zero quotient. Equivalently, split a polynomial
into its constant term and its positive powers of the current
variable at each step; this gives the same finite contraction.

A cycle of the original Spencer complex has finite order.
Lift a leading-symbol boundary in AD.8 and subtract it.
The order strictly decreases and is bounded below in every term,
so the process terminates. In degree zero the same reduction
identifies the quotient with AD.6.
This proves AD.5.
No flatness of \(\mathcal O_Z\) over \(f^{-1}\mathcal O_X\)
was used: regularity in AD.8 is regularity of new polynomial
variables over \(\mathcal O_Z\).
If \(n=0\), the complex and augmentation are the identity of
\(\mathcal O_Z\).

The construction is intrinsic. On overlaps the Spencer complexes
resolve the same left module and the augmentation is the same
action on the constant function \(1\).
Their comparison therefore glues AD.5.

#### AD.B. The actual analytic de Rham direct comparison

For every bounded coherent right \(\mathcal D_Z\)-complex \(A\),
there is a natural isomorphism
\[
\begin{gathered}
(f_+^rA)\otimes_{\mathcal D_X}^L\mathcal O_X\\
\xrightarrow{\;\gamma_f\;}
Rf_*(A\otimes_{\mathcal D_Z}^L\mathcal O_Z).
\end{gathered}
\tag{AD.9}
\]
The assertion is in the bounded-below derived category of complex
sheaves. It does not assert coherence or an upper cohomological
bound for an arbitrary nonproper direct image.
In the coherent bounded proper-image situations below, both sides
are bounded.

We give the derived comparison explicitly.
For a complex \(K\) of right \(f^{-1}\mathcal D_X\)-modules,
the projection map
\[
\begin{gathered}
Rf_*K\otimes_{\mathcal D_X}^L\mathcal O_X\\
\longrightarrow Rf_*\bigl(K\\
{}\otimes_{f^{-1}\mathcal D_X}^Lf^{-1}\mathcal O_X\bigr).
\end{gathered}
\tag{AD.10}
\]
is an isomorphism. Resolve \(\mathcal O_X\) by its finite left
Spencer complex. On a target coordinate open each term is a
finite direct sum of \(\mathcal D_X\).
For one such term AD.10 is the identity on the corresponding
finite direct sum of \(Rf_*K\).
It commutes with every matrix entry of the Spencer differential,
because these entries act through the same right operator action
on \(K\) and its pushed complex.
Filter the finite Spencer complex by its degree. Induction on
this finite filtration, using the termwise identities and the
exact cone triangles, proves AD.10.
These are the natural tensor and pushforward maps, so target
coordinate changes preserve the comparison.
Neither a commutation with infinite sums nor a projection formula
for an arbitrary nonperfect coefficient is needed.

Apply AD.10 to \(K=A\otimes_{\mathcal D_Z}^LT_f\).
Finite derived associativity and AD.5 give
\[
\begin{gathered}
Rf_*\Bigl(A\otimes_{\mathcal D_Z}^L\\
\bigl(T_f\otimes_{f^{-1}\mathcal D_X}^L\\
f^{-1}\mathcal O_X\bigr)\Bigr)\\
\simeq Rf_*(A\otimes_{\mathcal D_Z}^L\mathcal O_Z).
\end{gathered}
\tag{AD.11}
\]
This is AD.9.
For a direct model of the associativity, choose a bounded flat
right resolution of \(A\) and the finite left Spencer resolution
on \(X\). Tensor these with \(T_f\), and totalize the same
three factors in either order. Only finitely many diagonals
occur. A flat right term preserves AD.5, which gives AD.11.
Such a bounded flat resolution exists locally and as a sheaf
resolution: analytic operator rings have the uniform finite
flat-dimension bound proved in
GL06, Theorem3.0c.
Start with a sheaf flat resolution and truncate after that bound;
the last kernel is flat on every stalk. Bounded complexes use
their finite totalization.
All these constructions commute with the resolution comparison
maps.

Side-changing both actions of the transfer in AD.3 and using
AD.2 gives the left-module version
\[
\begin{gathered}
\operatorname{DR}_X(f_+M)\\
\xrightarrow{\;\gamma_f\;}Rf_*\operatorname{DR}_Z(M).
\end{gathered}
\tag{AD.12}
\]
Here
\[
\begin{gathered}
f_+M=Rf_*\bigl(\\
\mathcal D_{X\leftarrow Z}\otimes_{\mathcal D_Z}^LM\bigr),\\
\mathcal D_{X\leftarrow Z}=\\
\omega_Z\otimes_{\mathcal O_Z}T_f\\
{}\otimes_{f^{-1}\mathcal O_X}f^{-1}\omega_X^{-1}.
\end{gathered}
\tag{AD.13}
\]
The dimension shifts in AD.1 are already included through
side change. AD.12 has no further shift.
The identity map gives the identity comparison.
For \(Z\to\mathrm{pt}\), it gives
\[
f_+M=R\Gamma(Z,\operatorname{DR}_Z(M)),
\tag{AD.14}
\]
with the forms in degrees \(-d,\ldots,0\), as required.
For a closed embedding, the normal polynomial Koszul quotient
in AD.8 and the density contraction give exactly the closed
comparison in the de Rham lesson, Proposition3.1.

#### AD.C. Every form degree in a meromorphic proper image

Let \(S\subset X\) be a reduced analytic hypersurface,
\(U=X\setminus S\), and suppose \(f:Z\to X\) satisfies all
the hypotheses of DT.D:
it is proper and locally projective, both manifolds have
dimension \(n\), \(f^{-1}U\) is dense in every source component,
\[
p=f|_{f^{-1}U}:f^{-1}U\longrightarrow U
\]
is a finite covering by local biholomorphisms, and
\(D=\operatorname{Supp}(f^{-1}S)\) is SNC.
Let \(E_\Sigma\) be the normalized logarithmic extension
of a finite-rank local system \(L_Z\) on \(f^{-1}U\), and put
\[
\begin{gathered}
M=E_\Sigma(*D),\\
N=f_+M=f_*M,\\
L_X=p_*L_Z.
\end{gathered}
\tag{AD.15}
\]
The equalities, actual target connection, coherence and
holonomicity in AD.15 are DT.16–DT.18.
The local system \(L_X\) has finite rank: on an evenly
covered open it is the finite direct sum over the sheets.

In every form degree \(0\le a\le n\), pullback is a canonical
isomorphism
\[
\begin{gathered}
\mathcal O_Z(*D)\otimes_{f^{-1}\mathcal O_X}
f^{-1}\Omega_X^a\\
\xrightarrow{\;f^*\;}\Omega_Z^a(*D).
\end{gathered}
\tag{AD.16}
\]
Indeed in DT.8 the Jacobian \(J=\det B\) is a unit after
localizing at \(g=h\circ f\). The inverse matrix is its
cofactor matrix divided by \(J\).
Thus \(df\), and every finite exterior power of \(df\),
are invertible over \(\mathcal O_Z(*D)\).
For \(a=0\) this is the identity.
No hypothesis that the reduced \(S\) is smooth occurs here.

All source form terms are \(f_*\)-acyclic:
\[
\begin{gathered}
R^bf_*(\Omega_Z^a\otimes M)=0,\\
b>0.
\end{gathered}
\tag{AD.17}
\]
To prove this, set \(F_a=\Omega_Z^a\otimes E_\Sigma\),
a coherent holomorphic sheaf.
The locally projective ordinary coherent-image theorem
§5.9 makes \(R^bf_*F_a\) coherent.
For \(b>0\) its restriction to \(U\) is zero,
since a finite covering has exact sheaf pushforward
locally on its evenly covered opens.
The analytic Nullstellensatz kills this coherent sheaf
locally by a power of \(h\).
DT.7 therefore gives
\[
\begin{gathered}
R^bf_*(\Omega_Z^a\otimes M)\\
=(R^bf_*F_a)[1/h]=0.
\end{gathered}
\tag{AD.18}
\]
This applies independently to every one of the finitely many
form degrees, including \(a=0,n\).

On a target coordinate open, AD.16 and the finite locally
free basis of \(\Omega_X^a\) now give
\[
\begin{gathered}
\Omega_X^a\otimes N\\
\xrightarrow{\;\theta^a\;}f_*(\Omega_Z^a\otimes M).
\end{gathered}
\tag{AD.19}
\]
\[
\theta^a(\beta\otimes m)=f^*\beta\otimes m.
\tag{AD.20}
\]
This is an isomorphism; the local constructions agree on
overlaps because pullback and exterior power are intrinsic.
The lifted target vector fields of DT.9 satisfy
\(\sum_j B_{ij}\nabla_{V_j}=\nabla_{\partial_{z_i}}\).
Consequently the chain rule and the connection Leibniz rule
give
\[
d_M\theta^a=\theta^{a+1}d_N.
\tag{AD.21}
\]
For example the derivative of a coefficient \(a(x)\) becomes
\(\sum_jf^*(\partial_{x_j}a)\,df_j\); the derivative of \(m\)
is the same identity with the lifted connection.
Together with \(d(f^*\beta)=f^*(d\beta)\), these verify
AD.21 on every form and coefficient.
Both complexes have the same shift \([n]\), so their
shift signs also agree.

Equations AD.17 and AD.19 prove the explicit chain comparison
\[
\begin{gathered}
\operatorname{DR}_X(N)\\
\xrightarrow{\;\theta\;}Rf_*\operatorname{DR}_Z(M).
\end{gathered}
\tag{AD.22}
\]
For the finite source forms complex, termwise acyclicity
AD.17 identifies its ordinary pushed complex with its derived
pushforward. This follows by the finite form-degree filtration
and the acyclic-term comparison; its differential need only
be complex-linear, not \(\mathcal O_Z\)-linear.

In the direction of AD.12, \(\gamma_f=\theta\).
The top-form normalization can be read directly.
If \(f^*(dx)=J\,dz\), then
\[
\begin{gathered}
\theta^n(dx\otimes m)=J\,dz\otimes m,\\
(\theta^n)^{-1}(dz\otimes m)\\
=dx\otimes(m/J).
\end{gathered}
\tag{AD.23}
\]
This is precisely DT.11's density augmentation.
In lower degrees its exterior-power cofactors are AD.16.
The Spencer augmentation AD.6 pairs with these same
coefficient and wedge maps: in degree zero it is action
on \(1\); successive coordinate differentials give AD.20,
and the top density is AD.23.
Thus the finite transfer comparison is the actual forms
comparison, with its fixed volume normalization.

#### AD.D. Canonical comparison at a singular divisor

Let \(j_Z:f^{-1}U\hookrightarrow Z\), \(j:U\hookrightarrow X\).
The exact SNC theorem
§5.4
gives the natural quasi-isomorphism
\[
\begin{gathered}
\operatorname{DR}_Z(M)\\
\longrightarrow Rj_{Z,*}L_Z[n].
\end{gathered}
\tag{AD.24}
\]
It is proved for all crossings and every residue Jordan block.
Combine it with AD.22:
\[
\begin{gathered}
\operatorname{DR}_X(N)\\
\longrightarrow Rf_*Rj_{Z,*}L_Z[n]\\
\simeq Rj_*Rp_*L_Z[n]\\
=Rj_*L_X[n].
\end{gathered}
\tag{AD.25}
\]
The middle equality is the derived composition for
\(f\circ j_Z=j\circ p\).
It can be computed by an injective complex-sheaf resolution:
direct image preserves injectives since its left adjoint,
inverse image, is exact. Hence these actual composed
derived pushforwards agree.
The last equality uses the exact finite-cover pushforward
already checked above.
Every arrow in AD.25 is therefore a quasi-isomorphism.

It is the canonical restriction comparison
\[
\begin{gathered}
c_N:\operatorname{DR}_X(N)\\
\longrightarrow Rj_*L_X[n].
\end{gathered}
\tag{AD.26}
\]
To see this at the level of the map, restrict AD.25 to
an evenly covered open in \(U\).
AD.20 is ordinary pullback of forms on each sheet,
and AD.24 is the holomorphic flat-frame Poincaré map.
Their composite is the identity identification of the
horizontal local system \(L_X[n]\).
Derived open adjunction gives
\[
\begin{gathered}
\operatorname{Hom}(K,Rj_*L_X[n])\\
=\operatorname{Hom}(j^{-1}K,L_X[n]).
\end{gathered}
\tag{AD.27}
\]
Thus the restriction determines the whole map, so AD.25
equals AD.26. In particular the comparison includes
every singular point of \(S\), with all its attaching
maps. It is not just a stalk-dimension equality.

Suppose now \(Q\) is any finite divisorial D-type module
of DT.E and there is a proper locally projective SNC
modification of the pair which is an isomorphism over \(U\).
Take \(L_Z=p^{-1}L_Q\) and construct \(N\) as above.
Both \(Q,N\) are in the full finite divisorial D-type class,
and their identified horizontal local systems agree on \(U\).
DT.E's proved full faithfulness extends this identification
and its inverse to inverse finite operator maps on \(X\).
Thus \(Q\simeq N\), naturally with the given open restriction.
Equation AD.26 yields
\[
\operatorname{DR}_X(Q)\simeq Rj_*L_Q[n].
\tag{AD.28}
\]
The existence of the stated analytic SNC modification
is a hypothesis of this assertion.

Since \(c_Q\) itself is defined by restriction, AD.27 proves
independence of the chosen modification and compatibility
with every finite D-type morphism.
The same proof works for a bounded complex of such modules,
with its actual differentials: all comparisons are natural
chain maps, and the two finite degree filtrations compare
their totalizations. An object with bounded D-type
cohomology and the same supplied modification follows by
finite truncation triangles; its assertion is the canonical
quasi-isomorphism
\[
\begin{gathered}
\operatorname{DR}_X(K)\\
\longrightarrow Rj_*j^{-1}\operatorname{DR}_X(K).
\end{gathered}
\tag{AD.28a}
\]
No derived equivalence of hearts is assumed.

#### AD.E. An explicit singular cusp

For the cusp
\[
\begin{gathered}
X=\mathbf C^2,\\
h=y^2-x^3,\\
S=\{h=0\},
\end{gathered}
\tag{AD.29}
\]
an SNC modification is supplied by three point blowups.
We give its local equations, including the exceptional
multiplicities and Jacobians.

The blowup of \(\mathbf C^2\) at zero is the closed incidence
surface in \(\mathbf C^2\times\mathbf P^1\) with
\(xv=yu\) in homogeneous coordinates \([u:v]\).
Its two affine charts are smooth.
Projection is proper: the inverse image of a compact
set is a closed subset of that set times the compact
\(\mathbf P^1\). It is an isomorphism away from zero,
where \([u:v]=[x:y]\).
It is projective over the base by this same closed
incidence embedding.
The identical construction in smooth point coordinates
proves these facts for the next two blowups.

In the first chart put
\[
\begin{gathered}
x=u,\quad y=uv,\\
h=u^2(v^2-u).
\end{gathered}
\tag{AD.30}
\]
The strict cusp is smooth but tangent to the exceptional
line at \(u=v=0\).
The other chart \(x=ab,y=b\) has
\(h=b^2(1-a^3b)\), whose strict transform misses the
exceptional line. Thus the only unresolved point over
the cusp is the one in AD.30.

Blow up that point. The chart \(u=st,v=t\) gives
\[
\begin{gathered}
x=st,\quad y=st^2,\\
h=s^2t^3(t-s).
\end{gathered}
\tag{AD.31}
\]
The three reduced components \(s=0,t=0,t=s\) meet at
one point, so one more blowup is necessary.
The other second chart \(u=a,v=ab\) has
\(h=a^3(ab^2-1)\); its strict transform misses the
exceptional line near this fibre.
Away from the one triple point in AD.31, the total
reduced boundary is already SNC.

Blow up that triple point.
In the chart \(s=p,t=pq\) the composite map is
\[
\begin{gathered}
x=p^2q,\qquad y=p^3q^2,\\
h=p^6q^3(q-1),\\
\det\frac{\partial(x,y)}{\partial(p,q)}=p^4q^2.
\end{gathered}
\tag{AD.32}
\]
In the other chart \(s=ab,t=b\), it is
\[
\begin{gathered}
x=ab^2,\qquad y=ab^3,\\
h=a^2b^6(1-a),\\
\det\frac{\partial(x,y)}{\partial(a,b)}=ab^4.
\end{gathered}
\tag{AD.33}
\]
In AD.32 the reduced boundary is
\(p=0\), \(q=0\), or \(q=1\).
At most two components meet at a point, and they have
independent local differentials.
In AD.33 the same assertion holds for \(b=0,a=0,a=1\).
These two charts cover the last exceptional line;
all other points were already SNC.
This proves the required SNC property on the whole
resolved surface, not only near one point on that line.

The composition \(f\) is proper by the compact-inverse-image
criterion, applied three times.
It is also projective over \(X\) in this example.
The three rational projective-coordinate pairs are
\[
\begin{gathered}
{}[x:y],\qquad [x^2:y],\\
{}[x^3:y^2].
\end{gathered}
\tag{AD.33a}
\]
On the first \(u,v\) chart, cancel the common factor
in the second pair to obtain \([u:v]\), so its graph
closure is the second point blowup.
On the other first chart this pair is
\([a^2b:1]\), and has no base point.
On the second \(s,t\) chart the third pair, after cancelling
its common exceptional factor, is \([s:t]\).
On the other second chart it is \([1:ab^2]\);
on the unchanged part of the other first chart it is
\([a^3b:1]\). These have no other base point.
Thus the successive graph closures are exactly the
three stated blowups.
They give a closed graph closure in
\(X\times(\mathbf P^1)^3\).
The Segre monomials embed the three projective factors
in one projective space: on a chart where one monomial
is nonzero their coordinate ratios recover all three
affine ratios, and the equations equating the crossed
products define precisely this image.
Hence this graph closure proves projectivity over \(X\).
Off \(S\) it is an isomorphism, and its complement is
dense in every source component.
Thus it meets every hypothesis of AD.C–AD.D.
For every finite-rank local system \(L\) on \(X\setminus S\),
not just one with abelian global monodromy, its constructed
finite divisorial D-type module has
\[
\operatorname{DR}_X(N)\simeq Rj_*L[2].
\tag{AD.34}
\]
Only the residue matrices on an individual SNC chart
are required to commute.

For example, AD.23 in the first final chart reads
\[
\begin{gathered}
f^*(dx\wedge dy)\\
=p^4q^2\,dp\wedge dq,\\
dp\wedge dq\otimes m\\
\longmapsto dx\wedge dy\otimes\frac{m}{p^4q^2}.
\end{gathered}
\tag{AD.35}
\]
Both denominators are finite-pole units because their
zero sets lie in the displayed boundary.
The coefficient is fixed by the Jacobian, not by
choosing a generator after taking cohomology.

![Finite transfer, the canonical comparison and the resolved cusp](assets/analytic-D-type-deRham.png)

**Figure AD.1.** Panel A is the actual finite-transfer
augmentation AD.5 and the canonical restriction square
AD.22–AD.26. Panel B plots the real cusp
\((x,y)=(t^2,t^3)\), \(-1\le t\le1\); the axes use the
same scale. Panel C is the final exceptional incidence
graph, with orders of the pulled-back \(h\) equal to
\(2,3,6\) and strict-transform order \(1\).
The three marked points \(q=\infty,0,1\) on the last
exceptional line are distinct; the graph has no metric
scale. The displayed chart and density factor are AD.32
and AD.35.

Editable SVG,
exact figure data, and
reproducible source.

The comparisons here are finite-order de Rham statements.
They do not identify an ordinary infinite-order tensor
with a function sheaf at a general singular divisor.
Such an identification requires an additional reconstruction
argument beyond AD.28.
Freely accessible further reading:
[Kashiwara–Kawai, *Holonomic systems III*, I.4.9 and II.2.2.3–II.2.2.4, printed pp.852 and 863–864](https://www.kurims.kyoto-u.ac.jp/~kenkyubu/kashiwara/HolIII.pdf).
AD.A–AD.E prove the stated comparisons and cusp construction
with the exact earlier programme dependencies above.


### 5.75. Diagonal reconstruction of meromorphic D-type modules

Let \(X\) be a complex manifold of dimension \(n\).
Write \(B_X=D_X^\infty\) for the actual ordinary infinite-order
ring, with its every-epsilon coefficient bounds on one common
holomorphic domain. All its module tensors below are ordinary
algebraic sheaf tensors.

The ordinary diagonal kernel and its actual residue product are
proved in
§5.44, IK.4 and OR.1–OR.4.
The actual right flatness
\[
B_X\ \text{is right flat over }\mathcal D_X
\tag{PD.1}
\]
is §5.47, MF.6–MF.7.
These results include the full positive head and finite
matrix relations. No completion of a module tensor is made.

We prove a reconstruction using relative de Rham complexes with
holomorphic parameters. For finite divisorial D-type modules
it gives the actual infinity/function comparison whenever a
proper locally projective SNC modification of the pair is
supplied. Analytic existence of such a modification remains a
hypothesis.

#### PD.A. Analytic exterior coefficients and the derived diagonal

Let \(A\) be a parameter polydisc. For a coherent finite
\(\mathcal D_Y\)-module \(M\), define its analytic exterior
coefficient extension by
\[
\begin{gathered}
M^{[A]}=
\mathcal O_{A\times Y}
\otimes_{p_Y^{-1}\mathcal O_Y}p_Y^{-1}M,\\
p_Y:A\times Y\longrightarrow Y.
\end{gathered}
\tag{PD.2}
\]
Its coefficients are joint holomorphic functions.
It has the usual connection in the parameter directions,
as well as the \(Y\)-operator action.

The coefficient extension in PD.2 is flat.
At a coordinate point put \(H=\mathbf C\{y\}\).
The map
\[
\begin{gathered}
H\longrightarrow H[a]_{(\mathfrak m_H,a)}\\
\longrightarrow\mathbf C\{y,a\}.
\end{gathered}
\tag{PD.3}
\]
is a polynomial extension followed by localization and
the faithfully flat relative analytification of
GL06, Lemma3.0c.1.
Both maps are flat; thus PD.2 is exact on stalks.
The case of no parameters is the identity.

Let \(A=X\), \(Y=X\), and distinguish output \(x\) and input
\(y\). Put
\[
\begin{gathered}
\Delta=\{y=x\}\subset X\times X,\\
\delta:X\hookrightarrow X\times X,\\
\operatorname{DR}^0_y(M^{[X]})=\\
\bigl[\Omega^\bullet_{X\times X/X}
\otimes M^{[X]}\bigr].
\end{gathered}
\tag{PD.4}
\]
Relative forms are in degrees \(0,\ldots,n\), with the
unshifted \(y\)-connection differential.
Here \(R\Gamma_\Delta\) is ordinary derived
sheaf support on the closed diagonal, applied
to complexes of complex-vector-space sheaves.
It is not the finite-pole local cohomology
defined by a union of annihilators of powers
of the diagonal ideal.
There is a natural derived identity
\[
\begin{gathered}
B_X\otimes_{\mathcal D_X}^L M\\
\simeq\delta^{-1}R\Gamma_\Delta\\
\bigl(\operatorname{DR}^0_y(M^{[X]})\bigr)[2n].
\end{gathered}
\tag{PD.5}
\]
For a module the left side is concentrated in degree zero
by PD.1. This is an identity of actual ordinary module
extensions.

We prove the derived identity, including its two shifts.
On \(X\times Y\), let
\[
\mathcal C=\mathcal O_{X\times Y}
\otimes_{p_Y^{-1}\mathcal O_Y}p_Y^{-1}\omega_Y.
\tag{PD.6}
\]
It is a right \(p_Y^{-1}\mathcal D_Y\)-module by negative
Lie derivative in \(y\). In a constant input volume frame,
\[
(g\,dy)\partial_{y_i}=-(\partial_{y_i}g)\,dy.
\tag{PD.7}
\]
Choose a finite free left operator resolution \(P^\bullet\)
of \(M\) on one input neighbourhood, using GL06,
Theorem3.0c. Its length and every matrix size are finite.
The relative right Spencer calculation, with PD.3's flat
coefficient extension, gives
\[
\begin{gathered}
\mathcal C\otimes_{p_Y^{-1}\mathcal D_Y}^L p_Y^{-1}M\\
\simeq\operatorname{DR}^0_y(M^{[X]})[n].
\end{gathered}
\tag{PD.8}
\]
The normalized shift and the negative input density action
are precisely those of the de Rham lesson, Proposition1.1,
and §5.74, AD.1–AD.2. Both sides of PD.8 can be computed
by the finite complex \(\mathcal C\otimes P^\bullet\).

The ordinary diagonal calculation has its derived form
\[
\begin{gathered}
\mathcal H^q_\Delta(\mathcal C)=0\quad(q\ne n),\\
\delta^{-1}\mathcal H^n_\Delta(\mathcal C)=B_X.
\end{gathered}
\tag{PD.9}
\]
Here is the concentration assertion in the same actual
coordinate cube, rather than a formal support convention.
Set \(w=y-x\), and use the finite puncture cover indexed
by the coordinates with \(w_i\ne0\).
OC.1–OC.3 in §5.44 proves that the holomorphic row computes
the actual supported derived object.
On the face with puncture set \(J\), Laurent projection
decomposes a holomorphic coefficient into the finitely
many sectors with negative-coordinate set \(I\subseteq J\).
The Cauchy projections realize this decomposition on
smaller annuli and products, with normal convergence.
For a fixed \(I\), the face complex has entries precisely
for \(J\supseteq I\).
If \(I\ne\{1,\ldots,n\}\), choose the least coordinate
outside \(I\). Inserting and removing this coordinate
with the cube's alternating incidence sign gives a
contraction: the two operations in that coordinate sum
to the identity, while their crosses with every other
coordinate cancel.
If \(I\) is the full set, only the top face remains,
in degree \(n\).
This proves the vanishing in PD.9.

The surviving coefficient is the actual IK.12 kernel
\[
\begin{gathered}
\kappa(P)=\frac1{(2\pi i)^n}\sum_{\alpha\ge0}\\
a_\alpha(x)\frac{\alpha!}{(y-x)^{\alpha+\mathbf1}}\,dy,\\
P=\sum_{\alpha\ge0}a_\alpha(x)\partial_x^\alpha.
\end{gathered}
\tag{PD.10}
\]
Its common-domain Cauchy bounds and all fixed derivatives
are exactly IK.13 and IK.4. Face terms are actual cube
boundaries, not discarded germs.
The expression PD.10 denotes the normalized ordered
kernel class of IK.4 and V.15–V.17; it is not an
unsigned top-face entry.
In particular the right input action PD.7 gives
\[
\kappa(P)\partial_{y_i}=\kappa(P\partial_{x_i});
\tag{PD.11}
\]
differentiating \((y_i-x_i)^{-\alpha_i-1}\) cancels the
negative transpose and gives the factorial
\((\alpha_i+1)!\).
Left output differentiation gives
\(\kappa(\partial_{x_i}P)\), including the derivative of
\(a_\alpha(x)\). Holomorphic multipliers have the same
two actions by the Cauchy formula.
The full residue product theorem therefore identifies
PD.9 as the stated bimodule, with its fixed input
top-form line.

Apply \(R\Gamma_\Delta\) to the finite complex
\(\mathcal C\otimes P^\bullet\).
Each term is a finite sum of \(\mathcal C\).
The finite column filtration and PD.9 identify its
supported total complex with
\[
\delta_*(B_X\otimes_{\mathcal D_X}P^\bullet)[-n].
\tag{PD.12}
\]
Every differential is the same finite right operator
matrix by PD.7 and PD.11.
Equation PD.8 contributes one shift by \(n\), and PD.12
contributes the other.
Their combination is PD.5.
There are no infinite totalization diagonals.
Full output infinite operators act on the holomorphic
puncture cube by their actual Cauchy-convergent action;
this commutes with all finite input matrices.
The ordinary residue product identifies this action with
left multiplication on \(B_X\).
Thus PD.5 retains the full output action.
It requires no infinite-order action on point-supported
ordinary distributions.

The construction is natural in finite operator matrices,
resolution comparison maps and restrictions.
The intrinsic ordinary kernel and the density change
make it independent of coordinates.
For \(n=0\), the cube, both shifts and the kernel
augmentation are identities.

#### PD.B. Holomorphic parameters in relative topological calculations

We use the sheaf
\[
\mathcal F=p_A^{-1}\mathcal O_A\otimes_{\mathbf C}L
\tag{PD.13}
\]
on a product of \(A\) and a source open with finite-rank
local system \(L\).
It means holomorphic parameter germs which are locally
constant in the source variables in a flat frame.
It is not the sheaf of arbitrary holomorphic functions
of both sets of variables.

Two elementary relative resolutions justify the
parameter calculations below.
Resolve PD.13 in the source real variables by smooth
relative de Rham forms whose coefficients are
holomorphic in \(a\).
Radial integration in a source ball gives
\[
\begin{gathered}
d_yH_y+H_yd_y\\
=1-\operatorname{ev}_{y=y_0}.
\end{gathered}
\tag{PD.14}
\]
The integrals commute with every parameter derivative,
are holomorphic in \(a\), and are bounded on compact
products by the same finite integration interval.
Thus this is a sheaf resolution.

Its section calculations at a parameter germ can be
derived using smooth forms in all variables and the
Dolbeault differential in \(a\).
The local Cauchy–Green operators \(T_i,P_i\) of
§5.34, R.2–R.3
act only in \(a_i\), with cutoffs independent of \(y\).
They have
\[
\bar\partial_aT+T\bar\partial_a=1-P
\tag{PD.15}
\]
on one smaller parameter polydisc; \(P\) is holomorphic
in all parameter coordinates.
The construction is by finitely many coordinate
contractions. Each integral has compact parameter
support, so all source derivatives pass under it on
each source compact.
The same smaller parameter polydisc works on the
whole chosen source open: no bound at its boundary
is required, since the estimates are local on source
compacts and the cutoff domain is fixed in \(a\).

For completeness, smooth forms in both variable sets
are section-acyclic by the partition and injective
comparison in §5.34, R.5.
Use them as the Dolbeault resolution in \(a\).
At a parameter germ PD.15 eliminates every positive
Dolbeault row in its section complex.
For a finite relative de Rham degree range the
elimination is finite, on one smaller parameter
polydisc.
It leaves precisely the smooth relative source
complex with holomorphic parameters.
PD.14 then computes that complex on a contractible
source ball or convex source chart.
This proves the acyclic comparisons used here at
parameter germs, with actual integral operators.
It does not appeal to exactness of a completed tensor
product of arbitrary solution spaces.

In particular a finite good cover in the source by
contractible convex charts computes PD.13 at a
parameter germ by the finite Čech complex with
coefficient \(\mathcal O_{A,a}\) and the flat transition
matrices of \(L\).
Logarithmic sector charts are allowed: their log
coordinates give convex rectangles, and their radial
homotopy stays in those rectangles.
The same argument applies to a finite good cover of a
real sphere and its contractible intersections.
Finite Čech operations are finite sums and matrices;
tensoring their vector-space calculations with
\(\mathcal O_{A,a}\) is exact.

We will also use real-normal purity with these
coefficients. On a product with an oriented real
normal ball of dimension \(m\), the supported object
along its zero section satisfies
\[
\begin{gathered}
R\Gamma_{t=0}(p_A^{-1}\mathcal O_A)\\
\simeq i_*\mathcal O_A[-m].
\end{gathered}
\tag{PD.16}
\]
The relative cone for the ball and punctured ball
computes it.
Radial contraction computes the ball.
The punctured ball retracts radially to its sphere;
the finite sphere cover just described computes
constant coefficient cohomology in degrees \(0,m-1\),
with the usual reduced interpretation for \(m=1\).
The constant degree-zero part cancels in the relative
cone; the surviving orientation class is in degree \(m\).
Here is the sphere calculation with these coefficients.
The zero-sphere has two points, so its reduced
degree-zero cohomology is one copy of
\(\mathcal O_{A,a}\).
For a positive-dimensional sphere take the two
slightly enlarged open hemispheres.
Each contracts in a convex chart.
Their intersection is a belt which retracts
onto the equatorial sphere.
The two-open-cover exact sequence, or its
two-column Čech cone, therefore identifies
the reduced cohomology in degree \(q\) with
the equator's reduced cohomology in degree
\(q-1\).
Induction gives exactly one copy of
\(\mathcal O_{A,a}\) in top degree and zero
in every other reduced degree.
All contractions and the two-cover maps
are those of PD.14–PD.15; they are actual
maps with the same parameter domain.
Choose the top class by the boundary of
the positively oriented normal ball.
The ordered boundary signs in the
two-cover cone preserve that choice
under an orientation-preserving change
of normal coordinates.
This proves PD.16, including the map and its chosen
orientation, for arbitrary parameter germs.
For \(m=0\) it is the identity.

#### PD.C. The relative SNC de Rham comparison

Let \(Z\) have dimension \(n\), let \(D\subset Z\) be SNC,
and let \(M=E_\Sigma(*D)\) be the normalized logarithmic
extension of a finite-rank local system \(L_Z\).
Let \(j_{A,Z}:A\times(Z\setminus D)\hookrightarrow A\times Z\).
Then the natural relative comparison is
\[
\begin{gathered}
\operatorname{DR}^0_{Z/A}(M^{[A]})\\
\longrightarrow Rj_{A,Z,*}\bigl(\\
p_A^{-1}\mathcal O_A\otimes_{\mathbf C}L_Z\bigr),
\end{gathered}
\tag{PD.17}
\]
and it is a quasi-isomorphism.
Here the left differential uses only source \(Z\)
coordinates, not \(a\).
All auxiliary parameter derivatives commute with it.

We give the parameter argument explicitly.
In a normalized SNC chart the residues are commuting
constant matrices \(A_i\), with
\[
0\le\operatorname{Re}\operatorname{Spec}(A_i)<1.
\tag{PD.18}
\]
Theorem5.4's tangential radial homotopy acts only
in the source tangential coordinates.
The normal Laurent coefficient homotopy is
\[
h_\nu=(\nu_iI+A_i)^{-1}\iota_i,
\tag{PD.19}
\]
where \(i\) is the least nonzero normal exponent,
and \(h_0=0\).
Every inverse in PD.19 has a common norm bound,
including its nilpotent part, as proved in
§5.4, the convergent Laurent homotopy.
The homotopy changes no source Laurent exponent or
finite pole lower bound.
For a compact parameter product take the same
source circles and radial segments for every
parameter point. Cauchy's coefficient estimates
and normal convergence then give these same
homotopies on joint holomorphic coefficients.
All fixed \(a\)-derivatives commute with the circle
and radial integrals, or follow from a further
fixed parameter Cauchy margin.
The parameter domain is independent of the
Laurent index.

The residual complex is the finite Koszul complex
of the \(A_i\), with holomorphic parameter
coefficients.
On the topological side use Theorem5.4's two-sector
cover in each normal coordinate.
PD.B proves that this actual finite cover computes
the derived image on the right of PD.17 at
parameter germs.
Its finite complex is the monodromy Koszul
complex with
\[
T_i=\exp(-2\pi iA_i).
\tag{PD.20}
\]
The exact period comparison of Theorem5.4 is a
finite matrix map between these residual
complexes. Its invertible factors, terminating
nilpotent series, exterior signs and circle
periods are unchanged after adjoining the
holomorphic parameter coefficients.
Thus it remains a quasi-isomorphism.
The original comparison PD.17 is the restriction
map followed by the relative flat-frame
Poincaré augmentation, because these are the
maps used in those homotopies and periods.
This proves PD.17 on every chart and every
parameter germ, so the natural maps glue.
No generic-stratum replacement is involved.

#### PD.D. Relative comparison through a supplied modification

Let \(S\subset X\) be reduced, \(U=X\setminus S\),
and let \(Q\) be a finite divisorial D-type module
along \(S\), in the exact sense of DT.E.
Suppose there is a proper locally projective
SNC modification
\[
f:Z\longrightarrow X
\tag{PD.21}
\]
with \(Z\) smooth of dimension \(n\), which
is an isomorphism over \(U\), whose inverse
boundary is SNC, and whose open complement is
dense in every source component.
Let \(L\) be the horizontal local system of \(Q|_U\).
For every parameter polydisc \(A\), with
\(j_A:A\times U\hookrightarrow A\times X\),
the natural comparison is a quasi-isomorphism
\[
\begin{gathered}
\operatorname{DR}^0_{X/A}(Q^{[A]})\\
\longrightarrow
Rj_{A,*}(p_A^{-1}\mathcal O_A\otimes_{\mathbf C}L).
\end{gathered}
\tag{PD.22}
\]
It commutes with restrictions, D-type morphisms
and every parameter derivative.

We first justify the module identification in
this family, rather than assuming an analytic
proper base-change theorem.
The extension \(Q^{[A]}\) is a coherent holonomic
\(\mathcal D_{A\times X}\)-module.
Indeed extend a good filtration using PD.3.
Each finite filtered term and each exact graded
sequence remains exact.
The parameter derivatives act only on the
holomorphic coefficients and preserve every
filtered term. Thus the graded parameter
covariables act by zero, and
\[
\begin{gathered}
\operatorname{Ch}(Q^{[A]})\\
=T_A^*A\times\operatorname{Ch}(Q).
\end{gathered}
\tag{PD.23}
\]
Here \(T_A^*A\) denotes the zero section.
The graded local generators prove coherence,
and this characteristic support has dimension
\(\dim A+\dim X\), on each nonzero component.
Multiplication by \(h(x)\) is bijective after
extension because \(Q\) is already localized.
Its generic transverse logarithmic models
extend with constant parameter connection,
so it is in DT.E's full D-type class for
\(A\times S\).

Apply DT.D to
\[
f_A=\operatorname{id}_A\times f.
\tag{PD.24}
\]
This map is proper and locally projective,
and its open restriction is an isomorphism.
Indeed the inverse image of a compact
set in \(A\times X\) is closed in the
product of its compact \(A\)-projection
with the compact inverse image of
its \(X\)-projection.
The supplied local projective
embedding pulls back to a closed
embedding in the same projective
space over \(A\times X\).
The normalized source is \(M^{[A]}\), where
\(M=E_\Sigma(*f^{-1}S)\) for the pulled-back
local system.
It has the coherent logarithmic generator
required by DT.D.
The resulting target \(N_A=f_{A,+}M^{[A]}\)
is in the same finite D-type class as
\(Q^{[A]}\).
Their identified horizontal local systems
agree on \(A\times U\).
DT.E's full faithfulness therefore gives the
unique actual finite operator isomorphism
\[
Q^{[A]}\simeq N_A=f_{A,*}M^{[A]}.
\tag{PD.25}
\]
It is compatible with all parameter
derivatives. This argument does not infer
PD.25 from a tensor product of spaces of
analytic sections.

Now repeat AD.C's form-degree comparison
using only the \(X\) and \(Z\) forms.
The relative Jacobian is the same \(J\);
its inverse and all exterior powers are
finite-pole units.
Every source relative form lattice is
coherent on \(A\times Z\).
Its positive higher images vanish after
localization, by DT.7 and the finite-cover
argument in AD.17–AD.18, applied to \(f_A\).
Consequently every relative form degree
has the actual isomorphism
\[
\begin{gathered}
\Omega_X^b\otimes Q^{[A]}\\
\longrightarrow
f_{A,*}(\Omega_Z^b\otimes M^{[A]}),
\end{gathered}
\tag{PD.26}
\]
given by pullback of the form.
The chain rule proves compatibility with
the relative differentials; no parameter
form or shift has been added.
Termwise acyclicity and the finite form
filtration identify its total complex with
the derived pushed relative complex.

Apply PD.17 to that source.
Derived pushforward composition gives the
right side of PD.22, since \(f_A\) is the
identity on the open complement.
On that complement the map is the
relative Poincaré identification.
Derived open adjunction determines a map
into \(Rj_{A,*}\) by its open restriction.
Hence the composite is the natural map
in PD.22.
This proves it, including every point of
the singular divisor.

The source integral estimates in PD.C
also give compatibility with actual
infinite operators in the parameter
variables: on a compact parameter
product choose one larger Cauchy
product before summing, then choose
the coefficient epsilon below that
margin.
The derivative factorial cancels
the operator coefficient factorial,
leaving a finite product of geometric
series.
All source homotopies are parameter
independent, so the same normally
convergent series commutes with them.
The finite maps and the canonical
open restriction characterization
preserve this compatibility in PD.22.

#### PD.E. Actual ordinary infinity reconstruction

Under the hypotheses of PD.D there is a canonical
isomorphism
\[
\begin{gathered}
B_X\otimes_{\mathcal D_X}Q
\xrightarrow{\;\Phi_Q\;}
j_*(Q|_U).
\end{gathered}
\tag{PD.27}
\]
The right side means all holomorphic sections
of the flat bundle on \(U\), without a
finite-pole or growth restriction at \(S\).
The map is actual operator action.
It includes singular points of \(S\).

Apply PD.22 with parameter \(A=X\) to the
input copy of \(X\) in PD.5.
Write
\[
W=X\times U,\qquad \bar j:W\hookrightarrow X\times X.
\]
For any complex sheaf \(F\) on \(W\), there is a
natural support identity
\[
\begin{gathered}
R\Gamma_\Delta R\bar j_*F\\
\simeq R\bar j_*R\Gamma_{\Delta\cap W}F.
\end{gathered}
\tag{PD.28}
\]
To prove it, use the support triangle defined
by the open complement of \(\Delta\).
Restriction to that open commutes with
\(R\bar j_*\), since it is restriction on
the target of the direct image.
Composing the two open direct images in
either order gives the direct image of
the same open intersection.
Their natural restriction maps agree.
The fibres of these maps in the support
triangles are therefore the two sides
of PD.28.
Equivalently one may compute all these
maps by injective complex-sheaf
resolutions; inverse image is exact,
so each open direct image preserves
injectives.

Take
\[
F=p_X^{-1}\mathcal O_X
\otimes_{\mathbf C}p_U^{-1}L.
\tag{PD.29}
\]
The set \(\Delta\cap W\) is the closed graph
\[
\delta_U:U\longrightarrow X\times U,\qquad u\mapsto(u,u).
\]
Near each of its points trivialize \(L\),
and use \(w=y-x\) as normal coordinates.
The sheaf in PD.29 is then constant in
the real normal variables, with
holomorphic coefficient sheaf in \(x\).
PD.16, with the complex normal orientation
of real dimension \(2n\), gives
\[
\begin{gathered}
R\Gamma_{\Delta\cap W}F
\simeq\delta_{U,*}H[-2n],\\
H=Q|_U=L\otimes_{\mathbf C}\mathcal O_U.
\end{gathered}
\tag{PD.30}
\]
The local identifications glue, since flat
changes of \(L\)-frame commute with the
normal contraction and the complex
normal orientation is intrinsic.

Combining PD.5 and PD.22–PD.30 gives
\[
B_X\otimes_{\mathcal D_X}Q
\simeq Rj_*H.
\tag{PD.31}
\]
Here the final pushforward uses the equality
\[
\bar j\circ\delta_U=\delta\circ j.
\tag{PD.32}
\]
Closed pushforward by \(\delta\) is exact.
Thus
\(R\bar j_*\delta_{U,*}H=\delta_*Rj_*H\),
and applying \(\delta^{-1}\) gives PD.31.
No nonproper base-change statement for
pullback by the diagonal is assumed.

The left side of PD.31 is the ordinary
tensor in degree zero by right flatness.
It follows at the same time that
\[
R^bj_*H=0\quad(b>0).
\tag{PD.33}
\]
Taking degree zero in PD.31 gives an
actual \(\mathcal O_X\)-linear sheaf
isomorphism to \(j_*H\).
Indeed all maps retain multiplication
by the output coefficient \(a(x)\);
the relative purity contractions act
only in the normal variables.

We identify its normalization and its map.
For a flat frame on a small open in \(U\),
the kernel unit in PD.10 is the product
of the ordered one-variable Cauchy
units. In one variable its normalized
residue on the positive circle is
\[
\frac1{2\pi i}\int_{|w|=\rho}\frac{dw}{w}=1.
\tag{PD.34}
\]
The relative Spencer shifts in PD.8
and PD.12 give the total shift \([2n]\).
Use the same ordered cup and input
density conventions as the proved
ordinary kernel theorem and §5.35,
V.15–V.17.
Here is the totalization sign explicitly.
Put \(r_n=(-1)^{n(n-1)/2}\).
The literal top-cube coefficient of
the kernel unit is \(r_n(-1)^n\)
times its expression in PD.10.
Moving the shift \([n]\) through a
Čech column of degree \(q\) multiplies
that column by \((-1)^{nq}\):
substituting into the two total
differentials verifies this chain map.
At the surviving \(q=n\), this factor
is \((-1)^n\).
Finally changing anti-first normal
forms to the interleaved ordered real
normal orientation contributes \(r_n\).
Their product is
\[
r_n(-1)^n\,(-1)^{n^2}r_n=1.
\tag{PD.34a}
\]
Thus the Cauchy unit gives the positive
normal orientation class in PD.30.
The finite ordered residue is \(1\)
in every dimension, by \(n\) successive
applications of PD.34.
Thus the isomorphism obtained from
PD.31 restricts to the identity
action on each flat frame vector.
On \(U\), the empty-divisor case of
§5.73, SI.C–SI.D
identifies \(B_U\otimes_{D_U}H\)
with \(H\) by actual action.
Its flat frame vectors form an
\(\mathcal O_U\)-basis.
The \(\mathcal O_U\)-linearity and
the unit calculation therefore
identify the whole open restriction
with the identity on \(H\).
This includes the empty product
when \(n=0\).

Independently the actual Cauchy-convergent
action on flat holomorphic frames defines
\(\Phi_Q\) in PD.27.
Both sheaf maps have that same open
restriction. Ordinary open adjunction
identifies a map into \(j_*H\) with its
restriction into \(H\).
They are therefore equal.
In particular PD.27 is \(B_X\)-linear
for the actual full action, not only
an isomorphism of complex sheaves.
It is natural for D-type morphisms,
restrictions and changes of frame.

The inverse uses finite ordinary tensors.
Indeed on a common neighbourhood choose
the finite presentation
\[
\mathcal D_X^{r_1}\longrightarrow
\mathcal D_X^{r_0}\longrightarrow Q\longrightarrow0.
\tag{PD.35}
\]
The finite resolution and the finite
support cube used to prove PD.5
identify its degree-zero object with
\[
B_X^{r_0}/\operatorname{im}(B_X^{r_1}).
\tag{PD.36}
\]
Every holomorphic section germ in
PD.27 therefore has an inverse
represented by one row in
\(B_X^{r_0}\), hence by \(r_0\)
elementary tensors.
A row with zero image belongs to
the image of the finite matrix in
PD.36; its preimage is one actual
row of length \(r_1\).
These are finite algebraic balancing
witnesses.
All their entries have actual
common-domain coefficient
representatives: there are finitely
many entries and germ equalities,
and the ordinary kernel extraction
and matrix relation theorem give
their every-epsilon bounds.
No limiting element of a completed
tensor or closure of a relation
subspace occurs in this proof.

Finally, the same construction on
\(A\times X\), using the product
modification and the trivial
parameter connection, gives
\[
\begin{gathered}
D^\infty_{A\times X}
\otimes_{D_{A\times X}}Q^{[A]}\\
\simeq j_{A,*}(Q^{[A]}|_{A\times U}).
\end{gathered}
\tag{PD.37}
\]
Its map is again actual action.
Consequently the inverse is natural
in all parameter restrictions and
derivatives as well.
This proves the required actual
auxiliary-parameter comparison at
the full supplied-modification
generality.

#### PD.F. General cofinite left ideals and special tensor lifting

Let the connected complement \(U\) have fundamental group
\(\pi\), with the continuation convention of PM.A:
\[
T_{\gamma\eta}=T_\gamma T_\eta.
\tag{PD.38}
\]
Fix a cofinite left ideal \(\mathfrak a\subset\mathbf C[\pi]\),
and put
\[
\begin{gathered}
V_{\mathfrak a}=\mathbf C[\pi]/\mathfrak a,\\
v_0=1+\mathfrak a,\\
q=\dim_{\mathbf C}V_{\mathfrak a}.
\end{gathered}
\tag{PD.39}
\]
Left multiplication gives a representation \(\rho\) on
\(V_{\mathfrak a}\), and \(v_0\) is cyclic.
No right-ideal assumption is made.
Choose the left deck action on the universal cover so
that analytic continuation \(T_\gamma\) is pullback
by the inverse deck transformation.
This realizes PD.38.
Form the holomorphic flat bundle \(E_{\mathfrak a}\)
by quotienting \(\widetilde U\times V_{\mathfrak a}^*\)
with the action
\[
\begin{gathered}
\gamma\cdot(\widetilde u,\lambda)\\
=(\gamma\widetilde u,\lambda\rho_\gamma^{-1}).
\end{gathered}
\tag{PD.39a}
\]
It is a group action: two successive fibre maps
multiply as
\(\rho_\eta^{-1}\rho_\gamma^{-1}
=\rho_{\gamma\eta}^{-1}\).
The trivial connection descends because these
matrices are constant.
Let \(L_{\mathfrak a}\) be its horizontal local
system. A section has an equivariant
holomorphic functional
\[
\lambda(\gamma\widetilde u)
=\lambda(\widetilde u)\rho_\gamma^{-1}.
\tag{PD.39b}
\]
Equivalently its branch continuation is
\[
T_\gamma\lambda=\lambda\rho_\gamma.
\tag{PD.40}
\]
This inverse transition rule fixes the variance and
every matrix order explicitly. It requires no
meromorphic free frame on all of \(X\).

Let \(\mathscr F_{\mathfrak a}\) denote multivalued
holomorphic functions whose continuation action is
annihilated by \(\mathfrak a\).
There is a canonical identification
\[
E_{\mathfrak a}\simeq\mathscr F_{\mathfrak a}.
\tag{PD.41}
\]
To prove it, a function \(f\) determines
\[
\lambda_f(\gamma v_0)=T_\gamma f.
\tag{PD.42}
\]
The linear extension is well defined, since a
relation in \(V_{\mathfrak a}\) is an element
of \(\mathfrak a\) and kills \(f\).
At each branch it is holomorphic.
For every cyclic vector \(\eta v_0\),
\[
\begin{gathered}
(T_\gamma\lambda_f)(\eta v_0)
=T_{\gamma\eta}f\\
=\lambda_f(\rho_\gamma\rho_\eta v_0).
\end{gathered}
\tag{PD.43}
\]
Those vectors span \(V_{\mathfrak a}\).
Equation PD.43 is exactly PD.40, without commuting
the two representation matrices.

Conversely, for a row satisfying PD.40 put
\[
f=\lambda(v_0).
\tag{PD.44}
\]
Then \(T_\gamma f=\lambda(\rho_\gamma v_0)\),
so every \(a\in\mathfrak a\) kills \(f\) because
\(\rho(a)v_0=0\).
If \(f\) is zero as a multivalued function, all its
branches vanish, so \(\lambda\) kills the entire
cyclic span and is zero.
The two displayed constructions are inverse.
This proves PD.41, including every branch and
every noncommuting continuation matrix.
For \(q=0\) all these objects are zero.

Use the supplied modification PD.21 and DT.D–DT.F
to construct its finite D-type module
\(Q_{\mathfrak a}\), with horizontal local system
\(L_{\mathfrak a}\).
PD.27 and PD.37 give the actual isomorphisms
\[
\begin{gathered}
B_X\otimes_{\mathcal D_X}Q_{\mathfrak a}
\simeq\mathscr F_{\mathfrak a},\\
D^\infty_{A\times X}\\
{}\otimes_{D_{A\times X}}Q_{\mathfrak a}^{[A]}\\
\simeq\mathscr F_{\mathfrak a}(A;X).
\end{gathered}
\tag{PD.45}
\]
Functions on the second right side are jointly
holomorphic in the auxiliary parameters and
the lifted source variables.
All coordinate derivatives commute with
PD.42–PD.44, since analytic continuation
commutes with differentiation.
The complete ordinary infinite actions do too:
on a compact branch chart Cauchy's bound and
the every-epsilon coefficient condition give
absolute normal convergence of the whole head.
The finite \(q\)-dimensional branch span
permits one bound for every entry of
each fixed finite branch list in
PD.42–PD.43. No boundedness of the
representation on all of \(\pi\)
is asserted.

If \(\mathfrak b\subseteq\mathfrak a\), the
quotient map
\[
V_{\mathfrak b}\longrightarrow V_{\mathfrak a}
\tag{PD.46}
\]
intertwines every group element and takes
the cyclic vector to the cyclic vector.
Dualizing gives a horizontal map
\(L_{\mathfrak a}\to L_{\mathfrak b}\).
DT.E extends it uniquely to the actual
finite operator map
\[
Q_{\mathfrak a}\longrightarrow Q_{\mathfrak b}.
\tag{PD.47}
\]
Pairing with the cyclic vectors shows that
PD.47 is precisely inclusion of the
function spaces in PD.41.
Naturality of PD.27 and PD.37 proves the
same statement after actual infinity
extension, with all auxiliary variables.
This establishes the ideal-map compatibility
without treating a left ideal as two-sided.

Here is the consequence for the chosen-fibration
special tensor of §5.71.
Use the full Weierstrass setup PM.A, with
\(m\) distinct normal punctures off the base
discriminant \(H\).
Take \(H\) a hypersurface or empty, and
assume the ambient pair with boundary
\[
\begin{gathered}
\{P(t,x)=0\}\\
{}\cup(D_R\times H).
\end{gathered}
\tag{PD.48}
\]
has the supplied proper locally projective
SNC modification above.
Write \(T=\widetilde E^\infty\) for the
full chosen-fibration special ring,
\(R=D_X^\infty\), and
\[
N_{\mathfrak c}=R\otimes_{\mathcal D_X}Q_{\mathfrak c}.
\]
For every cofinite left ideal \(\mathfrak a\)
there is a cofinite two-sided
\(\mathfrak b\subseteq\mathfrak a\) such that
\[
\begin{gathered}
\operatorname{im}\bigl(T\otimes_RN_{\mathfrak a}\\
\longrightarrow T\otimes_RN_{\mathfrak b}\bigr)\\
\subseteq\operatorname{im}\bigl(N_{\mathfrak b}\\
\longrightarrow T\otimes_RN_{\mathfrak b}\bigr),\\
\dim\mathbf C[\pi]/\mathfrak b\le q^2(m+2)^2.
\end{gathered}
\tag{PD.49}
\]
Indeed PD.45 and PD.47 are precisely the
actual parameter and ideal compatibility
in §5.71, ST.20.
PM.B–PM.D gives its uniform relative
primitive representation \(\mathbb W\),
with \(\dim\mathbb W=q(m+1)\).
Take
\[
\mathfrak b=\operatorname{Ann}(V_{\mathfrak a}\oplus\mathbb W).
\tag{PD.50}
\]
Its being contained in \(\mathfrak a\)
follows by applying an annihilating
element to the cyclic vector in
\(V_{\mathfrak a}\).
Its codimension is at most the dimension
of the endomorphism algebra of the
\(q(m+2)\)-dimensional representation,
which is the bound in PD.49.
The full primitive theorem applies
to every special head and spatial
kernel on its common domains.
ST.12–ST.19 then supplies the actual
finite algebraic tensor image
witnesses in PD.49.
This uses the entire primitive
representation, not a finite-order
truncation of a special operator.

The analytic modification hypothesis
in PD.48 remains explicit.
These arguments do not prove existence
of analytic log-resolutions, the
general solution reconstruction theorem
for arbitrary holonomic modules, or
the whole-module continuation and
singular-germ separation required
for I23.

#### PD.G. The flat bundle and point-module degree checks

For \(M=\mathcal O_X\), the relative holomorphic
Poincaré contraction in the input variables gives
\[
\operatorname{DR}^0_y(\mathcal O_{X\times X})
\simeq p_X^{-1}\mathcal O_X.
\tag{PD.51}
\]
PD.16 in real normal dimension \(2n\) then makes
PD.5 equal to \(\mathcal O_X\) in degree zero.
The Cauchy-unit calculation PD.34–PD.34a shows
that the identification is the identity,
not an unspecified nonzero scalar.

For a point at the coordinate origin take
\[
\delta_0=
\mathcal D_X\Big/\sum_{i=1}^n\mathcal D_Xx_i.
\tag{PD.52}
\]
The finite closed-embedding calculation of
the de Rham lesson, Proposition3.1, gives
the relative complex, with holomorphic
output parameters,
\[
\begin{gathered}
\operatorname{DR}^0_y(\delta_0^{[X]})\\
\simeq p_X^{-1}\mathcal O_X
\otimes\mathbf C_{\{y=0\}}[-n].
\end{gathered}
\tag{PD.53}
\]
The same finite normal-derivative Koszul
contraction proves this with parameters;
its rows are polynomial in input
derivatives and leave every output
coefficient unchanged.
On that support, the diagonal condition
is \(x=0\). Therefore PD.5 becomes
\[
B_X\otimes_{\mathcal D_X}\delta_0
\simeq\mathcal H^n_{\{0\}}(\mathcal O_X).
\tag{PD.54}
\]
Indeed PD.53 gives one shift by \(-n\);
the point-supported holomorphic
cohomology is in degree \(n\) by
the same finite Laurent-sector cube
used in PD.9. The final shift
\([2n]\) leaves degree zero.

This point module includes essential
principal parts, rather than only
finite normal derivatives.
With the explicit coordinate
point-generator normalization
\[
e_0\longmapsto
\left[\frac1{x_1\cdots x_n}\right],
\]
its derivative coefficients are
\[
\begin{gathered}
\partial^\alpha e_0\longmapsto\\
(-1)^{|\alpha|}\alpha!
\bigl[x^{-\alpha-\mathbf1}\bigr].
\end{gathered}
\tag{PD.55}
\]
The sign and factorial follow by
ordinary differentiation.
An actual constant-coefficient head
\(\sum_\alpha c_\alpha\partial^\alpha\)
therefore gives the fully negative
Laurent series with coefficients
\((-1)^{|\alpha|}\alpha!c_\alpha\).
Its every-epsilon estimates give
normal convergence on every
coordinate-punctured compact.
Conversely, arbitrary small Cauchy
circles of a fully negative
holomorphic principal part bound
its Laurent coefficients by
\[
|b_\alpha|\le M_\rho
\prod_i\rho_i^{\alpha_i+1}.
\tag{PD.56}
\]
Dividing by \(\alpha!\) gives all
the required epsilon bounds on
one coefficient domain.
Thus the entire principal part,
not a finite truncation, is
represented by an actual head.
Terms missing a negative coordinate
are exactly the face boundaries.

In one dimension the explicit
example is
\[
\begin{gathered}
P=\sum_{\nu\ge0}
\frac{(-1)^\nu}{(\nu!)^2}\partial_x^\nu,\\
P e_0\longmapsto
\left[\frac{e^{1/x}}x\right].
\end{gathered}
\tag{PD.57}
\]
For every \(\epsilon>0\), its
coefficient estimate holds with
prefactor \(e^{1/\epsilon}\),
by the positive exponential
series.
This example distinguishes
PD.54's ordinary holomorphic
local cohomology from a module
of finite-order point-supported
distributions.

![Ordinary diagonal reconstruction, the punctured diagonal and finite module degrees](assets/parametric-diagonal-reconstruction.png)

**Figure PD.1.** Panel A is the actual derived diagonal
identity PD.5 and the reconstruction PD.22–PD.34a
under the supplied SNC modification.
Here \(F\) is PD.29 and \(H=Q|_U\).
The three orientation and totalization signs have
product one.
Panel B is only the real slice
\(\operatorname{Im}x=\operatorname{Im}y=0\)
of the one-dimensional complex product.
Both real coordinates range from \(-1\) to \(1\)
with equal axis scale.
The input boundary \(y=0\) is excluded from
\(W=X\times U\); the open circle marks the
omitted origin of \(\Delta\cap W\).
The full normal \(w=y-x\) is complex, of
real dimension two, rather than the
one-dimensional real normal drawn in this slice.
Panel C gives a cyclic nonabelian quotient:
\(S-T\) kills \(v_0\), while its right
product with \(S\) does not.
Panel D records the absolute cohomological
degrees in PD.5, PD.51 and PD.53.
Its point-module row is full holomorphic
local cohomology, including the essential
principal part PD.57.
The algebraic and degree panels have
no geometric scale.

Editable SVG,
exact figure data, and
reproducible source.
Freely accessible further reading:
[Kashiwara–Kawai, *Holonomic systems III*, I.4.9, II.2.2.4 and III.4.9, printed pp.852, 864 and 898–899](https://www.kurims.kyoto-u.ac.jp/~kenkyubu/kashiwara/HolIII.pdf).
PD.A–PD.G give the actual diagonal,
parameter and finite tensor arguments
at the stated supplied-modification
generality.


### 5.76. Trace descent through an SNC cover

Let \(X\) be a complex manifold, \(S\subset X\) a reduced
analytic hypersurface, and \(j:U=X\setminus S\hookrightarrow X\).
An empty hypersurface is allowed.
Suppose a holomorphic map
\[
f:Z\longrightarrow X
\tag{TC.1}
\]
is supplied with these properties:
it is proper and locally projective;
both manifolds have dimension \(n\);
the open \(V=f^{-1}U\) is dense in every source component;
its reduced boundary \(D=Z\setminus V\) is SNC;
and
\[
h=f|_V:V\longrightarrow U
\tag{TC.2}
\]
is a surjective finite covering by local biholomorphisms.
The degree is positive and locally constant on \(U\).
The map need not be a modification or a Galois cover.

We use the exact finite D-type class of
§5.72, DT.E.
Its modules are coherent holonomic, localized along \(S\),
connections off \(S\), and regular in the stated generic
transverse families on every divisor component.
DT.A proves that they are finite projective over
\(\mathcal O_X(*S)\).
All finite module images below are actual sheaf images.
All tensors with \(B_X=D_X^\infty\) are ordinary algebraic
sheaf tensors with the full every-epsilon coefficients.

#### TC.A. Finite fibres and the normalized trace

Let \(L\) be a finite-rank local system on \(U\). Put
\[
K=h_*h^{-1}L.
\tag{TC.3}
\]
On an evenly covered open \(W\subset U\) of degree \(d\),
\[
\begin{gathered}
K|_W=\bigoplus_{\nu=1}^dL|_W,\\
\iota(v)=(v,\ldots,v),\\
r(v_1,\ldots,v_d)=d^{-1}\sum_{\nu=1}^dv_\nu.
\end{gathered}
\tag{TC.4}
\]
These maps do not require a numbering of the sheets:
permuting the summands changes neither formula.
Continuation transports all entries by the local-system
maps and permutes the fibre, so the formulas commute
with continuation.
They therefore glue to horizontal maps with
\[
\begin{gathered}
\iota:L\longrightarrow K,\\ r:K\longrightarrow L,\\
r\iota=1,\quad e=\iota r,\quad e^2=e.
\end{gathered}
\tag{TC.5}
\]
No extension of the locally constant degree to a chosen
holomorphic function on \(X\) is needed.
The normalization in TC.4 is made in local systems on \(U\).
It works component by component even when degrees differ.
Surjectivity and positivity in TC.2 are needed for \(r\iota=1\).

The kernel \(K_0=\ker r\) is a local system and
\[
\begin{gathered}
K=\iota L\oplus K_0,\\
v=e v+(1-e)v.
\end{gathered}
\tag{TC.6}
\]
This is the decomposition of a finite direct sum on each
evenly covered chart; it glues because \(e\) is horizontal.
For a morphism \(a:L\to L'\), applying it on every sheet
commutes with \(\iota,r,e\).
All these assertions include rank zero and \(d=1\).

#### TC.B. The actual coherent summand

Pull \(L\) back to \(V\), form its normalized SNC extension
\(E_\Sigma\), and set
\[
\begin{gathered}
M=E_\Sigma(*D),\\
N=f_+M=f_*M.
\end{gathered}
\tag{TC.7}
\]
DT.D
applies to exactly TC.1–TC.2.
It proves the degree-zero equality in TC.7, coherence
and holonomicity over \(\mathcal D_X\), finite projectivity
over \(\mathcal O_X(*S)\), and the full generic logarithmic
target model including ramified Jordan blocks.
Its horizontal local system is \(K\).

DT.E full faithfulness, applied to \(N,N\), lifts \(e\)
uniquely to an actual finite operator endomorphism
\[
\epsilon:N\longrightarrow N.
\tag{TC.8}
\]
The map \(\epsilon^2-\epsilon\) is zero on \(U\).
Faithfulness makes it zero on \(X\).
Consequently
\[
\begin{gathered}
Q_L=\operatorname{im}\epsilon,\\
Q_0=\operatorname{im}(1-\epsilon),\\
N=Q_L\oplus Q_0.
\end{gathered}
\tag{TC.9}
\]
The decomposition is literal:
every section \(s\) is \(\epsilon s+(1-\epsilon)s\);
an element in the two images is both fixed and killed
by \(\epsilon\), and hence is zero.
The inclusions and projections in TC.9 are finite
operator maps.

We check every defining condition for these images.
Analytic operator coherence makes both images coherent.
They are subquotients of the holonomic module \(N\),
so the good-filtration characteristic inclusion and
Bernstein dimension bound make every nonzero image
holonomic.
Multiplication by an equation of \(S\), and its inverse
on \(N\), commute with \(\epsilon\). Both images are
therefore already localized.
Off \(S\), TC.6 identifies their flat bundles.

At a generic smooth boundary point DT.D supplies a
normalized logarithmic model for \(N\).
The normalized map theorem of §5.0 extends its horizontal
projector holomorphically across that boundary.
It remains an idempotent by density.
An idempotent matrix on a finite free holomorphic module
has a locally free image: choose a basis of its image and
kernel at the point; apply \(\epsilon\) and \(1-\epsilon\)
to holomorphic lifts of that basis. Their combined
determinant is nonzero at the point, hence remains a unit
after shrinking. They give the two complementary frames.
The image lattice is stable under the logarithmic
connection because the projector is horizontal.
Its residue is the restriction of the original residue,
so its eigenvalues remain in the normalized strip and
all its nilpotent parts are retained.
This proves the required generic transverse regularity.
Thus \(Q_L,Q_0\) belong to the full DT.E class.
Their horizontal local systems are \(L,K_0\), respectively.
DT.A then also proves their finite meromorphic projectivity.

The construction is functorial in \(L\).
The sheetwise map \(K\to K'\) lifts by DT.E; its commutator
with the two lifted projectors vanishes on \(U\), hence
vanishes everywhere. It restricts to the required map
\(Q_L\to Q_{L'}\).
Composition and identities follow by faithfulness.

For any existing D-type \(Q\) with horizontal local system
\(L\), DT.E lifts the identified local systems to maps
between \(Q\) and \(Q_L\). Their composites restrict to
identities and therefore are identities:
\[
Q\simeq Q_L.
\tag{TC.10}
\]
This also proves independence of the supplied cover,
whenever more than one such cover is available.
The isomorphism is the unique one inducing the identity
on \(L\), so these isomorphisms satisfy the cocycle
identity without a common refinement.

The class is abelian with its actual operator kernels and
cokernels. Indeed these are coherent holonomic subquotients;
they are localized because localization of the original
map is already the same map between the same modules.
On \(U\) a horizontal matrix is constant in flat frames,
so its kernel and cokernel are flat bundles.
At a generic smooth boundary point use the normalized
logarithmic map theorem on a product of a disc and a
polydisc. In the normalized constant residue frames a
horizontal map is the constant map intertwining residues:
the one-variable normalized map proof excludes every
nonzero Laurent exponent, and the tangential equations
exclude dependence on the other coordinates.
Its kernel and quotient therefore have constant
logarithmic residue models.
They meet DT.E's generic regularity condition.
Restriction to \(L\) is exact on these kernels and
cokernels. TC.9–TC.10 and full faithfulness prove an
exact equivalence, writing DType and Loc for these two categories,
\[
\begin{gathered}
\operatorname{DType}(X,S)
\simeq\operatorname{Loc}_{\rm fin}(U)
\end{gathered}
\tag{TC.11}
\]
under TC.1–TC.2.
DT.22–DT.23 supplies a coherent holomorphic generator
near every compact subset for every object in TC.11.

#### TC.C. Parameters and the canonical relative comparison

For a parameter polydisc \(A\), use the joint analytic
coefficient extension of PD.2:
\[
\begin{gathered}Q^{[A]}=\\ \mathcal O_{A\times X}\otimes_{p_X^{-1}\mathcal O_X}p_X^{-1}Q.\end{gathered}
\tag{TC.12}
\]
PD.A
proves that it is exact.
PD.D's good-filtration calculation shows that it is a
finite D-type module on \(A\times X\), with trivial
connection in the parameter direction.
This assertion uses no modification of \(X\).
The same calculation applies to \(N\).

The map \(f_A=\operatorname{id}_A\times f\) is proper and
locally projective by PD.D's compact-product and
pulled-back projective-embedding arguments.
Its boundary is \(A\times D\), and its complement map
is the finite cover \(h_A=\operatorname{id}_A\times h\).
DT.D gives
\[
\begin{gathered}N_A=f_{A,*}M^{[A]}\\ =f_{A,+}M^{[A]}.\end{gathered}
\tag{TC.13}
\]
Both \(N^{[A]}\) and \(N_A\) are finite D-type, and their
horizontal local systems are canonically identified.
DT.E therefore gives the unique isomorphism
\[
N^{[A]}\simeq N_A.
\tag{TC.14}
\]
It includes the parameter connection.
This proves the needed actual module comparison
without assuming a proper analytic coefficient
base-change formula.

Flatness of TC.12 preserves the direct sum TC.9.
The endomorphism \(\epsilon^{[A]}\) agrees, under TC.14,
with the lifted sheetwise projector for \(h_A\):
their restrictions agree, so faithfulness identifies them.
In particular
\[
\operatorname{im}\epsilon^{[A]}=Q_L^{[A]}.
\tag{TC.15}
\]
The identifications commute with parameter restrictions:
extend the finite coefficient maps on the smaller product
and use their identical open restrictions and faithfulness.
They commute with all parameter derivatives because they
are maps of the full finite operator modules.

There is a canonical relative quasi-isomorphism
\[
\begin{gathered}
c_N^A:\operatorname{DR}^0_{X/A}(N^{[A]})\\
\longrightarrow
Rj_{A,*}(p_A^{-1}\mathcal O_A\otimes_{\mathbf C}K),
\end{gathered}
\tag{TC.16}
\]
where \(j_A:A\times U\hookrightarrow A\times X\).
Here is the generically finite argument in all degrees.
The relative form map is AD.C's pullback:
\[
\begin{gathered}
\Omega_X^b\otimes N^{[A]}\\
\longrightarrow
f_{A,*}(\Omega_Z^b\otimes M^{[A]}),\\
0\le b\le n.
\end{gathered}
\tag{TC.17}
\]
The inverse Jacobian and its exterior powers are finite-pole
units, by DT.8–DT.9.
Thus this is an isomorphism in every degree.
Its compatibility with the differentials is the lifted
connection chain rule.
Every source form lattice is coherent.
Its positive proper images vanish on the finite-cover
open and are killed after localization by DT.7.
AD.17–AD.18 therefore makes TC.17 termwise acyclic for
proper image. The finite form filtration identifies
the actual relative complex with the derived image.

Apply the parameter SNC comparison PD.C to the source.
Composing its open pushforward with \(f_A\) gives
\[
\begin{gathered}
Rf_{A,*}Rj_{Z,A,*}\\ (p_A^{-1}\mathcal O_A\otimes h^{-1}L)\\
=Rj_{A,*}Rh_{A,*}\\ (p_A^{-1}\mathcal O_A\otimes h^{-1}L).
\end{gathered}
\tag{TC.18}
\]
Finite covering image is exact: on a small evenly covered
open its section functor is a finite direct sum of the
identical section functor on that open, and the stalk
calculation is the same finite direct sum.
On these opens it also commutes with the parameter
coefficient sheaf. Hence the right side of TC.18 is
exactly the right side of TC.16.
All comparisons restrict to the flat-frame Poincaré
map on \(U\).
Derived open adjunction, as in AD.27, identifies their
composite with the canonical map TC.16.

Naturality of this canonical map gives
\[
c_N^A\epsilon^{[A]}
 =Rj_{A,*}(1\otimes e)\,c_N^A.
\tag{TC.19}
\]
Both sides have the same open restriction, which proves
the equality by that same derived adjunction.
On the finite direct decompositions TC.6, TC.9 and TC.15,
TC.19 says that the quasi-isomorphism TC.16 is block
diagonal. Its cone is the direct sum of the two block
cones. A finite direct sum of cohomology sheaves is zero
only if both summands are zero. Thus its \(L\)-block is
a quasi-isomorphism:
\[
\begin{gathered}
c_Q^A:\operatorname{DR}^0_{X/A}(Q^{[A]})\\
\longrightarrow
Rj_{A,*}(p_A^{-1}\mathcal O_A\otimes_{\mathbf C}L).
\end{gathered}
\tag{TC.20}
\]
Equation TC.10 gives it for every finite D-type \(Q\).
The map is the canonical restriction comparison.
It includes every singular point of \(S\), all morphisms
and all parameter restrictions and derivatives.
For actual infinite parameter heads, PD.C–PD.D's same
fixed Cauchy margins justify normal convergence and
commutation with the source homotopies.
Alternatively, TC.20's canonical characterization on the
open identifies the resulting action maps once those
same coefficient bounds define them.
No infinite tensor of solution spaces enters TC.20.

#### TC.D. Ordinary infinity reconstruction and finite witnesses

The conclusion of PD.E now holds under TC.1–TC.2:
\[
\begin{gathered}
B_X\otimes_{\mathcal D_X}Q
\xrightarrow{\;\Phi_Q\;}j_*(Q|_U),\\
D^\infty_{A\times X}
\otimes_{D_{A\times X}}Q^{[A]}\\
\xrightarrow{\;\Phi_{Q,A}\;}
j_{A,*}(Q^{[A]}|_{A\times U}).
\end{gathered}
\tag{TC.21}
\]
Both maps are actual full operator action and are
isomorphisms. They are natural in finite D-type
morphisms, all auxiliary parameters and restrictions.

To verify that the proof uses no birational premise,
insert TC.20 into the actual derived diagonal PD.5.
The support/open identity PD.28 uses only the target
open \(X\times U\).
The graph \(\delta_U\) there is closed and has oriented
real normal dimension \(2n\); PD.B gives its purity.
The equality of the two composite pushforwards PD.32
gives \(Rj_*(Q|_U)\).
Ordinary right flatness places the left side in degree
zero. The Cauchy unit and totalization sign PD.34–PD.34a
give the identity open action, and ordinary open
adjunction identifies the map with \(\Phi_Q\).
None of these steps uses that \(f|_V\) has degree one.
Repeating on \(A\times X\) proves the second map.
It also gives
\[
\begin{gathered}
R^bj_*(Q|_U)=0,\\
R^bj_{A,*}(Q^{[A]}|_{A\times U})=0\\ (b>0).
\end{gathered}
\tag{TC.22}
\]

For a finite presentation on one neighborhood,
\[
\mathcal D_X^{r_1}\xrightarrow{F}
\mathcal D_X^{r_0}\longrightarrow Q\longrightarrow0,
\tag{TC.23}
\]
the source in TC.21 is the actual quotient
\[
B_X^{r_0}/\operatorname{im}(B_X^{r_1}\xrightarrow{F}B_X^{r_0}).
\tag{TC.24}
\]
Every holomorphic section germ on the right of TC.21
therefore has a representative consisting of \(r_0\)
actual infinite heads. Every zero relation has one
preimage row of length \(r_1\).
Finitely many entries and germ equalities can be
represented on a common neighborhood.
The existing ordinary kernel and finite relation
bounds retain every epsilon on that domain.
There is no closure of the relation image and no
completed module tensor.
These statements also apply to the second map of TC.21,
using a finite presentation over \(D_{A\times X}\).

The projector itself survives ordinary infinity
extension as the actual split projector
\[
\begin{gathered}
1\otimes\epsilon:
B_X\otimes_{\mathcal D_X}N\\ \longrightarrow B_X\otimes_{\mathcal D_X}N,\\
\operatorname{im}(1\otimes\epsilon)
=B_X\otimes_{\mathcal D_X}Q_L.
\end{gathered}
\tag{TC.25}
\]
This follows by tensoring the two finite split maps
in TC.9; it does not even require tensor flatness.
Under TC.21 it is the fibrewise projector
\(j_*(1\otimes e)\) on holomorphic flat-bundle sections.
Thus the finite witnesses in TC.24 belong to the
actual summand, not merely a completed invariant space.

The comparisons also hold for bounded complexes with
finite D-type cohomology. The canonical relative
restriction map commutes with finite truncation
triangles, as in AD.28a; TC.20 on each cohomology
module and induction over the finite cohomological
range prove it for the complex.
The diagonal PD.5 and right flatness then give
the corresponding derived infinity comparison.
In the relative comparison the target is
\(Rj_{A,*}j_A^{-1}\operatorname{DR}^0_{X/A}\);
no equivalence between the ambient derived category
and the derived category of its heart is assumed.

#### TC.E. The full SNC infinity image

The same cover gives an actual proper-image comparison
for any normalized logarithmic source, even when its
local system \(L_Z\) is not pulled back from \(U\).
Put
\[
\begin{gathered}
M=E_\Sigma(L_Z)(*D),\\ N=f_+M=f_*M,\\
M^\infty=B_Z\otimes_{\mathcal D_Z}M.
\end{gathered}
\tag{TC.26}
\]
Then \(N\) is finite D-type by DT.D and has horizontal
local system \(h_*L_Z\).
The source SNC comparison SI.C–SI.D gives
\[
M^\infty=j_{Z,*}(M|_V).
\tag{TC.27}
\]
The identity modification of the SNC pair on \(Z\)
meets the hypothesis of PD.D.
PD.33 applied there also proves that this source
open image has no positive higher direct images.
Consequently
\[
\begin{gathered}
Rf_*M^\infty\\ =Rf_*Rj_{Z,*}(M|_V)\\
=Rj_*Rh_*(M|_V).
\end{gathered}
\tag{TC.28}
\]
The finite covering functor \(h_*\) is exact on all
complex sheaves, not only local systems.
On evenly covered opens,
\[
\begin{gathered}h_*(M|_V)\\ =\mathcal O_U\otimes_{\mathbf C}h_*L_Z\\ =N|_U.\end{gathered}
\tag{TC.29}
\]
The pullback coefficient identification is valid
because every sheet is a local biholomorphism.
TC.21–TC.22 for \(N\) therefore give
\[
\begin{gathered}
B_X\otimes_{\mathcal D_X}f_+M
\simeq f_*M^\infty,\\
R^bf_*M^\infty=0\qquad(b>0).
\end{gathered}
\tag{TC.30}
\]
The first isomorphism is canonical full target
\(B_X\)-linear action.
On the right define the target action over the
unramified open by the target coordinate action on
each sheet and continue its holomorphic section
sheaf \(j_*h_*(M|_V)\).
Cauchy convergence on each fixed compact branch
chart defines every full head there.
All maps in TC.27–TC.30 are the identical map on that
open, so open adjunction identifies the whole maps
and their full actions.
This asserts no infinite chain-rule action on an
arbitrary singular transfer distribution.
The equality uses the actual source SNC infinity
module and its entire positive heads.
Its target inverse and zero witnesses are the
finite presentation witnesses TC.23–TC.24 for \(N\).

The proof repeats on \(A\times Z\to A\times X\)
with joint holomorphic coefficients:
TC.14 supplies its actual finite image, and the
source identity-SNC comparison and TC.22 supply
both open-image acyclicity statements.
Thus TC.30 includes the full auxiliary infinite
operators, not only fixed finite parameter derivatives.
It is natural in source local-system maps and
the corresponding finite logarithmic maps.
It is a theorem for the full class TC.1–TC.2,
not a claim about arbitrary proper images of
arbitrary regular holonomic modules.

#### TC.F. Cofinite left ideals and the full special tensor

Let \(\pi=\pi_1(U)\), with the convention
\(T_{\gamma\eta}=T_\gamma T_\eta\), and let
\(\mathfrak a\subset\mathbf C[\pi]\) be a cofinite
left ideal. Write
\[
\begin{gathered}
V_{\mathfrak a}=\mathbf C[\pi]/\mathfrak a,\\
q=\dim_{\mathbf C}V_{\mathfrak a},\\
v_0=1+\mathfrak a.
\end{gathered}
\tag{TC.31}
\]
PD.F proves the exact dual flat-bundle/function
realization with its inverse deck convention:
\[
\begin{gathered}
\lambda_f(\gamma v_0)=T_\gamma f,\\
T_\gamma\lambda=\lambda\rho_\gamma,\\
f=\lambda(v_0).
\end{gathered}
\tag{TC.32}
\]
This proof does not use an SNC modification.
It includes all noncommuting matrices and
requires no two-sidedness of \(\mathfrak a\).
Use TC.B to realize its horizontal local
system by the actual module \(Q_{\mathfrak a}\).
TC.21 then gives
\[
\begin{gathered}
B_X\otimes_{\mathcal D_X}Q_{\mathfrak a}
\simeq\mathscr F_{\mathfrak a},\\
D^\infty_{A\times X}
\otimes_{D_{A\times X}}Q_{\mathfrak a}^{[A]}\\
\simeq\mathscr F_{\mathfrak a}(A;X).
\end{gathered}
\tag{TC.33}
\]
For \(\mathfrak b\subseteq\mathfrak a\), the quotient
\(V_{\mathfrak b}\to V_{\mathfrak a}\) dualizes to the
horizontal inclusion of these function models.
TC.B's functoriality and TC.21's actual action identify
the resulting finite and infinite operator maps.
All coordinates and auxiliary parameters are retained.

In the full Weierstrass setup PM.A take the target
boundary to be
\[
\begin{gathered}
\{P(t,x)=0\}\\ {}\cup(D_R\times H),\\
H\text{ a hypersurface}\\ \text{or empty}.
\end{gathered}
\tag{TC.34}
\]
Suppose this pair has a cover TC.1–TC.2.
Let \(m\) be the number of distinct normal punctures
off \(H\), and put
\[
\begin{gathered}R=D_X^\infty,\quad T=\widetilde E^\infty,\\ N_{\mathfrak c}=R\otimes_{\mathcal D_X}Q_{\mathfrak c}.\end{gathered}
\tag{TC.35}
\]
TC.33 is the exact actual auxiliary comparison and
ideal compatibility required in ST.20.
The entire PM relative primitive representation
\(\mathbb W\), including all special heads and spatial
kernels, has dimension \(q(m+1)\).
Set
\[
\mathfrak b=\operatorname{Ann}(V_{\mathfrak a}\oplus\mathbb W).
\tag{TC.36}
\]
It is a two-sided ideal contained in \(\mathfrak a\),
since applying an annihilator to \(v_0\) gives its
membership in \(\mathfrak a\).
Its quotient embeds into the endomorphisms of the
\(q(m+2)\)-dimensional representation.
The complete finite balancing criterion ST.12–ST.20
therefore proves
\[
\begin{gathered}
\operatorname{im}(T\otimes_RN_{\mathfrak a}\\ \longrightarrow T\otimes_RN_{\mathfrak b})\\
\subseteq\operatorname{im}(N_{\mathfrak b}\\ \longrightarrow T\otimes_RN_{\mathfrak b}),\\
\dim_{\mathbf C}\mathbf C[\pi]/\mathfrak b\\
\le q^2(m+2)^2.
\end{gathered}
\tag{TC.37}
\]
These are actual ordinary tensor images with finite
witnesses, not closure statements or finite-order
truncations.
The supplied finite SNC cover is still a geometric
hypothesis. Neither this descent nor TC.30 proves its
existence for a general analytic pair, the general
holonomic solution reconstruction theorem, I23, or
arbitrary analytic proper regularity.

#### TC.G. Sheet permutations, Jordan blocks and density

On an evenly covered chart of rank \(r\), order the
sheets temporarily. With sheet vectors first,
\[
\begin{gathered}
\iota=\mathbf1_d\otimes I_r,\\
r=d^{-1}\mathbf1_d^{\,t}\otimes I_r,\\
e=d^{-1}J_d\otimes I_r,
\end{gathered}
\tag{TC.38}
\]
where \(J_d\) is the all-one matrix.
Continuation around a loop with sheet permutation
\(P_\gamma\) and local-system matrix \(A_\gamma\) is
\[
P_\gamma\otimes A_\gamma.
\tag{TC.39}
\]
Every permutation fixes the all-one column and row,
so TC.38 commutes with TC.39 even when the matrices
for different loops do not commute.
For instance the degree-three action of \(S_3\) on
its three cosets of a transposition subgroup is a
nonnormal covering representation. Its two generating
sheet permutations do not commute. The same formula
still has rank \(r\) and trace \(r\), and its complement
has rank \((d-1)r\).
This is a fibrewise average; it is not an average over
the global deck group of a nonnormal cover.

Here is a ramified connection that also fixes the
residue index and density. On a disc take
\[
\begin{gathered}
t=w^3,\quad S=\{t=0\},\\ D=\{w=0\},\quad J=3w^2,\\
A=\frac25I_2+N,\\
N=\begin{pmatrix}0&1\\0&0\end{pmatrix}.
\end{gathered}
\tag{TC.40}
\]
Let the target connection in a row of basis vectors
\(a\) be \(\nabla_{\partial_t}a=t^{-1}aA\).
Its pulled-back frame has residue \(3A\).
The normalized source frame is
\[
\begin{gathered}
v=w^{-1}f^*a,\\
B=3A-I_2\\
{}=\frac15I_2+3N,\\
\nabla_{\partial_w}v=w^{-1}vB.
\end{gathered}
\tag{TC.41}
\]
This spectrum lies in the normalized strip.
In the target meromorphic basis
\(v,wv,w^2v\), DT.20 gives the three residue blocks
\[
\begin{gathered}
R_j=(B+jI_2)/3,\\ j=0,1,2,\\
R_0=\frac1{15}I_2+N,\\
R_1=\frac25I_2+N,\\
R_2=\frac{11}{15}I_2+N.
\end{gathered}
\tag{TC.42}
\]
The unit sends \(a\) to \(wv\).
For a source section \(w^jv\), its coefficient
relative to \(f^*a\) is \(w^{j-1}\).
The normalized trace over the three roots is zero
for exponents \(-1,1\) and one for exponent zero:
on the roots \(w,\zeta w,\zeta^2w\), this is the
finite geometric sum \(1+\zeta+\zeta^2=0\).
Thus the actual lifted projector in this basis is
\[
\epsilon=\operatorname{diag}(0_2,I_2,0_2).
\tag{TC.43}
\]
It selects index \(1\), not index \(0\).
Its retained block is exactly \(A\), including the
nilpotent entry one. It commutes with the whole
target connection since TC.42 is block diagonal.

The left transfer frame is
\(\eta=dw\otimes f^*(dt)^{-1}\).
Its actual augmentation DT.11 is
\[
\begin{gathered}\eta\otimes P\otimes m\\ \longmapsto P(t,\nabla_{\partial_t})(m/J).\end{gathered}
\tag{TC.44}
\]
In particular the unit vector \(a\) is represented by
\[
\begin{gathered}
\eta\otimes1\otimes Jf^*a\\
=\eta\otimes1\otimes3w^3v.
\end{gathered}
\tag{TC.45}
\]
The image is \(wv=f^*a\); its normalized trace is \(a\).
Inserting \(f^*a\) without the factor \(J\) would
instead give \(a/(3w^2)\) on a sheet, whose trace is
zero. Hence omitting the density cannot yield the
unit or its retraction.
There is no cohomological shift: source and target
have the same dimension.
For dimension zero, the determinant, both density
lines and the form complexes are empty products;
the construction is precisely the finite set
average TC.4 in degree zero.

![Finite-cover trace, the actual summand and a ramified Jordan example](assets/snc-cover-trace-descent.png)

**Figure TC.1.** Panel A shows the actual split maps
TC.4–TC.9 and their ordinary infinity extension
TC.21–TC.25.
The dotted vertical lines denote the finite D-type
realization, rather than morphisms between sheaves
on different spaces.
Panel B is the real slice \(t=w^3\), \(-1\le w\le1\),
with equal scales; the complex punctured-disc
cover has degree three.
Panel C gives the exact three rank-two residue
blocks TC.42, including their nilpotent entries,
and the index-one projector TC.43.
The source and transfer units are TC.41 and
TC.44–TC.45. Algebraic panels have no metric scale.

Editable SVG,
exact figure data, and
reproducible source.
Freely accessible further reading:
[Kashiwara–Kawai, *Holonomic systems III*, II.2.2.4 and III.4.9, printed pp.864 and 898–899](https://www.kurims.kyoto-u.ac.jp/~kenkyubu/kashiwara/HolIII.pdf).
TC.A–TC.G prove the stated trace, summand, parameter
and proper-image comparisons with the exact earlier
programme dependencies above.


### 5.77. Power covers and monomial root graphs

Let \(B\) be a polydisc with coordinates
\((z_1,\ldots,z_d)\), and let
\[
\begin{gathered}
P(t,z)=t^m+a_1(z)t^{m-1}\\ {}+\cdots+a_m(z),\\
m\ge1,\qquad a_j(0)=0.
\end{gathered}
\tag{QG.1}
\]
Suppose its discriminant is not identically zero and
\[
\begin{gathered}
\Delta_P=z_1^{b_1}\cdots z_k^{b_k}c(z),\\
b_i\ge0,\quad c(z)\ne0,\\ 0\le k\le d.
\end{gathered}
\tag{QG.2}
\]
Zero exponents are allowed, so the chosen coordinate
boundary may be larger than the discriminant.
After shrinking \(B\), choose \(0<r<R\) so that all roots
lie in \(|t|<r\). Set
\[
\begin{gathered}
X=D_R\times B,\\
S=\operatorname{Supp}\\ \{P(t,z)z_1\cdots z_k=0\}.
\end{gathered}
\tag{QG.3}
\]
There is a proper locally projective map of smooth
complex manifolds
\[
F:Z\longrightarrow X
\tag{QG.4}
\]
of equal dimension, whose inverse reduced boundary is
SNC, and whose restriction off \(S\) is a surjective
finite unramified covering. Its degree can be taken
to be \((m!)^k\).
We construct it with actual holomorphic functions and
finitely many smooth blowups.

All arguments allow additional joint holomorphic
coordinates among the unramified base coordinates,
provided QG.2 holds with a nowhere-zero unit on that
product. The coordinate boundary in QG.3 is part of
the statement.

#### QG.A. The finite root cover and bounded extension

For a monic complex polynomial, every root satisfies
\[
|t|\le1+\max_j|a_j|.
\tag{QG.5}
\]
Indeed, for \(|t|>1+M\), the sum of the absolute values
of the lower terms is at most
\(M(|t|^{m-1}+\cdots+1)<|t|^m\).
For completeness, existence of the complex roots
follows from the Cauchy formula: if a nonconstant
polynomial had no zero, its reciprocal would be a
bounded entire function, using its growth at infinity
and a positive minimum on a compact disc.
Cauchy estimates make its derivative zero. This is
impossible. Division by the resulting linear factor
and induction gives all \(m\) roots with multiplicity.

The discriminant means the symmetric root product
\[
\Delta_P=\prod_{i<j}(t_i-t_j)^2.
\tag{QG.6}
\]
It is a polynomial in the coefficients.
One elementary symmetric-polynomial argument suffices:
the largest monomial of a symmetric homogeneous
polynomial has ordered exponents. Subtract the
corresponding product of elementary symmetric
polynomials to cancel it. Lexicographic reduction
terminates among the finitely many monomials of that
degree. Applying the argument to each homogeneous
degree expresses QG.6 in the elementary symmetric
polynomials, hence in the coefficients.
In particular QG.2 says that all roots are distinct
on
\[
\begin{gathered}B^*=\\ \{z\in B\mid z_1\cdots z_k\ne0\}.\end{gathered}
\tag{QG.7}
\]

A simple root depends holomorphically on the base.
Here is the local argument. At a root \(t_0\) with
derivative \(c_0\ne0\), iterate
\[
t_{\nu+1}=t_\nu-c_0^{-1}P(t_\nu,z).
\tag{QG.8}
\]
On a small closed disc about \(t_0\), and a sufficiently
small base polydisc, the derivative of this iteration
in \(t\) has norm less than \(1/2\), and its value at
the centre is within half the disc radius.
It maps the disc into itself and converges uniformly
by the geometric contraction estimate.
Its iterates are holomorphic in \(z\); their uniform
limit is holomorphic by the Cauchy formula.
It is the unique root in that disc.
Use disjoint discs at the \(m\) simple roots.
These \(m\) branches give all roots by their number.
Thus the zero set over QG.7 is an actual finite
holomorphic covering of degree \(m\).

Its monodromy is a permutation representation.
The logarithmic cover of a punctured polydisc has
coordinates \(z_i=\exp(w_i)\) for \(i\le k\), with
\(\operatorname{Re}w_i<\log R_i\).
That product of half-planes and discs is convex.
Path lifting in a finite covering follows by
successively using its sheet charts on a finite
cover of the compact path.
Homotopy invariance follows by subdividing a compact
homotopy square into finitely many small rectangles
whose images lie in sheet charts; labels agree on
shared edges, and induction across the rectangles
identifies the lifted endpoints.
Consequently this logarithmic cover trivializes the
root cover. Its deck translations are the independent
\(2\pi i\) translations, so the monodromy is generated
by \(k\) commuting permutations.

Put \(N=m!\). Each permutation of \(m\) letters has
order dividing \(N\). Its order divides the product
of its cycle lengths; that product divides \(m!\),
because the product of their factorials divides
\(m!\) by the integer multinomial count.
The power substitution
\[
\begin{gathered}
q:\widetilde B\longrightarrow B,\\
z_i=u_i^N\quad(i\le k),\\ z_i=u_i\quad(i>k)
\end{gathered}
\tag{QG.9}
\]
therefore kills every root monodromy generator.
It gives globally labelled holomorphic roots
\(r_1,\ldots,r_m\) on \(\widetilde B^*\).
The source radii are \(R_i^{1/N}\) in the first
\(k\) coordinates and \(R_i\) in the others.
The labels are consistent on the whole punctured
product, not just on a sector.
With no ramified coordinate the power map is the
identity. In degree one its exponent is one;
empty products and empty circle integrations
have their identity meanings.

These roots extend jointly holomorphically to
\(\widetilde B\). To check this without a removable
singularity citation, use QG.5 on a fixed smaller
product. Every branch is uniformly bounded there.
In one punctured coordinate its negative Laurent
coefficient of order \(j\ge1\), estimated on a
circle of radius \(\rho\), has norm at most
\(M\rho^j\). Letting \(\rho\) tend to zero makes
that coefficient zero. The nonnegative series,
or its fixed outer-circle Cauchy integral, is the
extension.
Apply this in each of the finitely many punctured
coordinates. Equivalently the extension on an
inner product is
\[
\begin{gathered}\frac1{(2\pi i)^k}\int_{|\zeta_i|=\rho_i}\\ r_\nu(\zeta,u')\prod_{i=1}^k\frac{d\zeta_i}{\zeta_i-u_i}.\end{gathered}
\tag{QG.10}
\]
The fixed circles miss all coordinate hyperplanes.
This integral is holomorphic in all remaining
coordinates, including every auxiliary parameter.
It agrees with the branch on the punctured product
by the one-coordinate argument.
Different circles give the same extension by
density. All fixed derivatives follow from an
additional fixed Cauchy margin.
Thus
\[
\begin{gathered}P(t,q(u))\\ =\prod_{\nu=1}^m(t-r_\nu(u)).\end{gathered}
\tag{QG.11}
\]
as an identity on the whole product.

The uniform root bound inside \(D_r\) required in
QG.3 can also be chosen directly: shrink \(B\) until
\(\sum_j\sup_B|a_j|r^{-j}<1\).
For \(|t|\ge r\) this makes the sum of the lower
terms strictly smaller than \(|t|^m\), so there
are no such roots.

The power map QG.9 is proper and locally projective.
The preimage of a compact subset stays a positive
distance from every source polydisc boundary and
is closed, hence compact.
Alternatively its graph is the closed incidence
space in \(B\times(\mathbf P^1)^k\) with equations
\[
U_i^N=z_iV_i^N.
\tag{QG.12}
\]
No solution has \(V_i=0\), since that would force
\(U_i=0\). Thus this is precisely the power cover.
For any finite number of projective factors,
use all product homogeneous monomials.
On a chart where one product monomial is
nonzero, their ratios recover every factor's
affine coordinate. The equations equating
the two products obtained by exchanging one
factor force each remaining monomial ratio
to be the product of those affine coordinates.
Thus these equations define exactly the
image on every projective chart.
They give the closed Segre embedding in one
projective space, with the same argument
as AD.33a.
Off the coordinate boundary, its Jacobian factors
are \(Nu_i^{N-1}\); it is unramified of degree \(N^k\).

#### QG.B. Actual monomial differences

Each nonzero difference of two extended roots has
the form
\[
\begin{gathered}r_i-r_j=u^{\alpha_{ij}}v_{ij}(u),\\ v_{ij}(u)\ne0,\end{gathered}
\tag{QG.13}
\]
after one common shrinking of \(\widetilde B\).
The multi-index is supported in the first \(k\)
coordinates. It is an integer multi-index; no
formal coefficient field is being used.

Here is the needed factor argument.
The local ring of convergent holomorphic germs is
an integral domain, and each coordinate \(u_i\)
is prime: the quotient by \(u_i\) is the convergent
ring in the other coordinates, also an integral
domain. Vanishing at \(u_i=0\) is divisibility by
\(u_i\), by the convergent Taylor expansion.
If a finite product of germs is a coordinate
monomial times a unit, choose a coordinate with
positive exponent. Its primality puts that
coordinate in at least one factor. Divide it out
and induct on the total monomial exponent.
At exponent zero, the product is a unit and every
factor is a unit.
This proves that each factor is a monomial times
a unit; no unique-factorization theorem is needed.

Apply this argument to QG.6 after QG.9 and QG.11:
\[
\begin{gathered}\prod_{i<j}(r_i-r_j)^2\\ =u_1^{Nb_1}\cdots u_k^{Nb_k}c(q(u)).\end{gathered}
\tag{QG.14}
\]
No difference is the zero germ, because all roots
are distinct off the coordinate boundary.
There are finitely many differences, so shrink
once to make all their unit factors nonvanishing.
The same proof works with all holomorphic
parameters present.

#### QG.C. Principalizing finite monomial sums

We need a global operation on a smooth SNC pair
\((Y,E)\), with finitely many boundary components
on the neighborhood under consideration.
An invertible monomial ideal has the form
\[
\begin{gathered}\mathcal I=\mathcal O_Y\left(-\sum_i a_iE_i\right),\\ a_i\ge0.\end{gathered}
\tag{QG.15}
\]
We first principalize the sum of two such ideals.
If the exponents of the second are \(b_i\), put
\[
d_i=a_i-b_i.
\tag{QG.16}
\]
On a coordinate chart, one of the two monomials
divides the other exactly when the nonzero
\(d_i\) of the components through that point
have only one sign.
Their sum is then the smaller principal ideal.
If both signs occur, two opposite-sign components
meet.

For all meeting opposite-sign pairs set
\[
\begin{gathered}
M=\max\{p+q\mid\\ E_i\cap E_j\ne\varnothing,\\
d_i=p>0,\\ d_j=-q<0\}.
\end{gathered}
\tag{QG.17}
\]
Choose a pair attaining \(M\), and blow up the
whole smooth intersection \(E_i\cap E_j\).
Its connected components are disjoint, so they
may be treated together.
In coordinates \(x,y\) for the pair the two
charts are
\[
\begin{gathered}(x,y)=(x,xv),\\ (x,y)=(wy,y).\end{gathered}
\tag{QG.18}
\]
All other boundary coordinates are unchanged.
These charts prove smoothness and the SNC
property of the reduced total boundary.
They also show that the strict transforms of
the chosen pair are disjoint.
On every new exceptional component the exponent
difference of the two total ideals is
\[
d_{\rm new}=p-q.
\tag{QG.19}
\]

Every new opposite-sign meeting pair has sum
strictly less than \(M\).
For if \(p>q\), its other component must have
negative difference \(-q'\) and meet the centre.
That old component already met \(E_i\), so
\(p+q'\le M=p+q\), giving \(q'\le q\).
Its new sum is at most
\((p-q)+q'=p<M\).
If \(p<q\), use the old pair with \(E_j\) to
obtain the analogous bound \(q<M\).
If \(p=q\), the exceptional difference is zero
and makes no opposite-sign pair.
Distinct exceptional components over disjoint
centre components do not meet.

All other old meeting pairs are unchanged or
disappear. The chosen maximal pair disappears,
and no new maximal pair occurs.
Let \(c_M\) count the meeting pairs with sum \(M\).
Then the lexicographic invariant
\[
(M,c_M)
\tag{QG.20}
\]
strictly decreases, until no opposite signs meet.
The first entry is a positive integer and the
second is finite at each stage. After finitely
many decreases at a fixed first entry that entry
drops; induction on it proves termination.
This gives an invertible monomial total image
of the sum.
Near a fixed compact set only finitely many
boundary components and their centre components
are needed: take finitely many SNC charts and
shrink around that compact set. In the application
below the initial boundary is coordinate and
all these base operations have finitely many
coordinate charts. In each, the old coordinates
are monomials in the new coordinates and the
domain is given by the original monomial
polydisc inequalities. Its coordinate strata
are connected whenever nonempty, by the
convexity of their logarithmic modulus
inequalities and the connected angular tori.
There are finitely many such charts at each
finite stage. Thus only finitely many boundary
and intersection components occur, and the
finiteness condition used in QG.20 holds there.
It also holds on the entire inverse image of
a smaller base polydisc.

A finite sum of invertible monomial ideals is
principalized by adding its summands one at a time
and applying this two-ideal construction.
Every already invertible ideal stays invertible
under subsequent pullback.
Repeating for any finite family of such sums
therefore principalizes them all.
Every centre lies in the boundary, so the
resulting map is an isomorphism off that boundary.

We record projectivity as well as local charts.
A blowup in the coordinate centre of QG.18 is the
closed incidence space with \(xV=yU\) in
\(Y\times\mathbf P^1\) on a trivializing chart.
Its maps glue intrinsically for the coherent
centre ideal.
To see that the finite composition is locally
projective over the original polydisc, use
Theorem5.8.
If the current \(Y\) is closed in a projective
space over that polydisc, push its coherent
centre ideal into the projective space.
Lemma5.9.1 proves coherence of this closed push.
Theorem5.8 supplies, over a smaller entire
polydisc, finitely many generators after one
projective twist.
The blowup is the closure of the graph of these
generators in \(Y\times\mathbf P^{s-1}\).
Its affine charts are the degree-zero rings
generated by the ratios of these generators,
which are exactly the charts of the centre
blowup. Twisting all generators by one line
does not change their ratios.
The product projective embedding and Segre give
a closed embedding over the original base.
This proves projectivity by induction through
the finite sequence. Properness also follows
from each closed incidence map and compactness
of its projective fibre.
Only finitely many base shrinkings occur.

#### QG.D. Global root sections and collision ideals

For every subset \(A\subseteq\{1,\ldots,m\}\)
of size at least two, take the root-cluster ideal
\[
\mathcal I_A=(r_i-r_j:i,j\in A).
\tag{QG.21}
\]
By QG.13 it is a finite sum of invertible
monomial ideals on \(\widetilde B\).
QG.C produces a proper projective base modification
\[
\beta:B'\longrightarrow\widetilde B
\tag{QG.22}
\]
on which every total image of QG.21 is invertible
monomial and the reduced inverse coordinate
boundary \(E\) is SNC.
The pulled-back roots are globally labelled
holomorphic functions on \(B'\).
They remain inside \(D_r\).
Their graphs are closed sections of
\[
Y_0=B'\times D_R\longrightarrow B'.
\tag{QG.23}
\]
We resolve their intersections by globally
closed centres, keeping the base \(B'\) fixed.

The following description remains valid after
each of the blowups to be used.
The ambient space is smooth near every root
section; its projection to \(B'\) is smooth of
relative dimension one there.
The vertical reduced boundary is SNC and meets
each root section transversely.
Near a point where the sections in \(A\) coincide,
choose a fibre coordinate \(v\).
Their collision ideal on the base is generated by
\[
\begin{gathered}v(\sigma_i)-v(\sigma_j),\\ i,j\in A.\end{gathered}
\tag{QG.24}
\]
Where they do not all coincide, set this ideal
to the unit ideal.
These definitions glue.
On a change of fibre coordinate \(v'=\phi(b,v)\),
the difference of two values is their old
difference times the holomorphic divided
difference of \(\phi\).
At a common section value this factor is a
unit, since the fibre-coordinate derivative
is nonzero. The inverse coordinate gives the
reverse inclusion of the generated ideals.
Off the collision locus at least one difference
is a unit.
Thus QG.24 is a well-defined coherent ideal.
Initially it is precisely the total image of
QG.21, hence is invertible monomial.

Choose a cluster \(C\) of largest cardinality
among all roots that coincide at any base point.
If that cardinality is one, all root sections
are disjoint and there is nothing to do.
Otherwise its collision ideal is a nonunit
invertible monomial, so some component \(E_i\)
occurs with positive coefficient.
All sections in \(C\) coincide along the whole
\(E_i\).
No root outside \(C\) can meet them at a point
of \(E_i\): that would give a larger cluster.
Choose one of its roots and let the centre be
\[
\sigma_c(E_i),\qquad c\in C.
\tag{QG.25}
\]
The centre is independent of the choice within
\(C\), since those sections coincide on \(E_i\).
It is smooth and closed.
Near it the coordinates are \(x=0\) for \(E_i\)
and \(v=0\) for the chosen section.
Blow up \((x,v)\).

Every root in \(C\) has local value \(v=xg_j(b)\).
Its strict section lies in the \(x\)-chart,
with new fibre coordinate
\[
\begin{gathered}v_1=v/x,\\ v_1(\sigma_j)=g_j(b).\end{gathered}
\tag{QG.26}
\]
All roots outside \(C\) are disjoint from the
centre and remain unchanged near their images.
Each strict root section is still isomorphic
to the base: the centre pulls back to the
Cartier divisor \(E_i\) on a section in \(C\),
and to the empty set on every other section.
The same charts QG.18 prove smoothness,
properness and the SNC property of the full
vertical boundary.
Near a section in \(C\), only \(x=0\) and the
other base-boundary coordinates occur; the
old strict \(x\)-divisor lies on the other
chart and misses that section.
The projection is still smooth of relative
dimension one near every section.

For clarity this includes the complementary
chart. There \(x=vw\); a strict root in \(C\)
satisfies \(1=w g_j(b)\).
At \(w=0\) this is impossible.
Where \(w\ne0\), the point is already in
the \(x\)-chart QG.26.
Roots outside \(C\) miss the centre altogether.
Thus no unexamined strict root occurs at the
other end of the exceptional projective line.
The remaining boundary there is the coordinate
vertical boundary, already SNC.

For every \(A\subseteq C\) of size at least two,
QG.26 divides its collision ideal by \(x\).
For a subset containing a root of \(C\) and a
root outside \(C\), its collision ideal is a
unit along \(E_i\), both before and after the
blowup. Elsewhere it is unchanged.
For a subset disjoint from \(C\), the operation
misses all its sections.
Therefore the update is exactly
\[
\begin{gathered}
\mathcal I_A'=\mathcal I_A(E_i),\\ (A\subseteq C,\ |A|\ge2),\\ \mathcal I_A'=\mathcal I_A\\ (A\not\subseteq C).
\end{gathered}
\tag{QG.27}
\]
In the first line every relevant coefficient
was positive because all those roots coincided
on \(E_i\).
All collision ideals thus remain invertible
monomial ideals on the same base with the
same boundary components.

Write their coefficients as \(a_{A,i}\) and put
\[
W=\sum_{|A|\ge2}\sum_i a_{A,i}.
\tag{QG.28}
\]
This is a finite nonnegative integer.
The exact change in QG.27 is
\[
\begin{gathered}W'=W\\ {}-(2^{|C|}-|C|-1).\end{gathered}
\tag{QG.29}
\]
The subtracted number counts the subsets
of \(C\) of size at least two and is positive.
Thus at most the initial \(W\) blowups can occur.
If two roots still met, their monomial collision
ideal would be nonunit and a maximal cluster
would allow another step.
The terminal root sections are consequently
all disjoint.
Each remains transverse to the vertical
boundary, so their union with that boundary
is SNC.

This is a global construction on \(Y_0\).
Its centres are the closed sections QG.25,
not a disjoint union of local charts.
QG.C's projective graph argument applies to
each coherent centre ideal over the original
target \(X\). The finite resulting map
\[
\mu:Z\longrightarrow Y_0
\tag{QG.30}
\]
is proper and locally projective.

#### QG.E. The actual SNC cover and D-type consequences

Compose the constructed maps:
\[
F=(\operatorname{id}_{D_R}\times(q\beta))\mu.
\tag{QG.31}
\]
The factor order in QG.23 is interchanged in this
formula only to match \(X=D_R\times B\).
All maps are holomorphic, proper and locally
projective, by QG.A, QG.C and QG.D.
Their source is smooth and has dimension \(d+1\).

The reduced inverse of \(S\) is exactly the
vertical reduced boundary together with the
strict root graphs.
Indeed QG.11 identifies the inverse polynomial
divisor with these graphs before blowing up.
Every exceptional divisor of the graph operations
is vertical, since its centre lies over an
\(E_i\). QG.D proved the SNC assertion on the
whole source, including the complementary charts.
Off \(S\), the base modifications and graph
blowups are isomorphisms, while QG.9 is the
unramified covering of degree \(N^k\).
This proves every assertion of QG.4.
The complement is dense in every source
component: each coordinate blowup is the
closure of its isomorphism locus, and that
locus contains the dense complement of these
divisors.

Hence §5.76, TC.A–TC.F
applies to QG.3 with geometry constructed here.
Every finite-rank local system on \(X\setminus S\)
has its actual finite divisorial D-type module.
For every module \(Q\) in that class and every
auxiliary polydisc \(A\),
\[
\begin{gathered}
D_X^\infty\otimes_{\mathcal D_X}Q
\simeq j_*(Q|_{X\setminus S}),\\
D^\infty_{A\times X}
\otimes_{D_{A\times X}}Q^{[A]}\\
\simeq j_{A,*}(Q^{[A]}|_{A\times(X\setminus S)}).
\end{gathered}
\tag{QG.32}
\]
These are the canonical actual actions with
finite ordinary inverse and kernel witnesses.
The canonical relative de Rham comparison,
full SNC source infinity proper image, every
morphism and parameter compatibility also
follow with exactly TC's proved maps.

For the PM Weierstrass setup with the coordinate
exceptional base hypersurface in QG.3, every
cofinite left ideal \(\mathfrak a\) of rank \(q\)
has the natural actual function model of TC.F.
There is a cofinite two-sided
\(\mathfrak b\subseteq\mathfrak a\) with
\[
\begin{gathered}
\operatorname{im}(T\otimes_RN_{\mathfrak a}\\ \longrightarrow T\otimes_RN_{\mathfrak b})\\
\subseteq\operatorname{im}(N_{\mathfrak b}\\ \longrightarrow T\otimes_RN_{\mathfrak b}),\\
\dim_{\mathbf C}\mathbf C[\pi]/\mathfrak b\\ \le q^2(m+2)^2.
\end{gathered}
\tag{QG.33}
\]
Here \(R,T,N_{\mathfrak c}\) have exactly the
definitions in TC.35. This conclusion uses
the full PM primitive representation and
the finite balancing theorem ST, not a
finite-order truncation.

The monomial-discriminant hypothesis QG.2 is
essential to the construction given here.
A general analytic discriminant need not have
that form in the original base coordinates.
Nor do local power covers on charts of a
modified base automatically glue to a proper
global cover. QG.4 does not assert either of
those additional geometric inputs.

#### QG.F. A cubic cusp after a quadratic power cover

Take the exact pair
\[
\begin{gathered}
P(t,x)=t^2-x^3,\\ \Delta_P=4x^3,\\
S=\{x(t^2-x^3)=0\}.
\end{gathered}
\tag{QG.34}
\]
The power substitution \(x=u^2\) gives globally
split roots \(u^3,-u^3\).
Their collision ideal is \((u^3)\), already
invertible, with \(W=3\).
Put \(v_0=t-u^3\), so the two graphs are
\(v_0=0,-2u^3\).
Blow up their common section over \(u=0\)
three times.
In the successive root charts,
\[
v_0=uv_1=u^2v_2=u^3v_3.
\tag{QG.35}
\]
Their graph differences have orders \(3,2,1,0\).
In the final chart, writing \(v=v_3\), the map is
\[
\begin{gathered}
x=u^2,\\ t=u^3(1+v),\\
P=u^6v(v+2),\\
xP=u^8v(v+2),\\
\det\frac{\partial(x,t)}{\partial(u,v)}=2u^4.
\end{gathered}
\tag{QG.36}
\]
The reduced boundary is \(u=0\), \(v=0\) or
\(v=-2\). At most two of its components meet
at a point, with independent differentials.

The complementary charts can be checked at
every step. Before the next blowup the root
coordinate has values \(0,-2u^a\), with
\(a=3,2,1\). Its other chart is
\[
u=vw,\qquad v_{\rm old}=v.
\tag{QG.37}
\]
The nonzero strict root satisfies
\[
1=-2v^{a-1}w^a.
\tag{QG.38}
\]
It has no point at \(w=0\).
For \(w\ne0\) it is in the root chart already
displayed in QG.35. The zero root's strict
transform also lies entirely in that root
chart. All other points in QG.37 have only
the vertical coordinate boundary.
This verifies the entire exceptional fibre,
not just the final finite-\(v\) chart.
The map is a degree-two unramified covering
off exactly the boundary QG.34.
The orders \(6,8\) and the Jacobian \(2u^4\)
are actual pullback identities, not just
orders of a reduced support.

![Power splitting, monomial principalization and global graph separation](assets/quasiordinary-SNC-geometry.png)

**Figure QG.1.** Panel A records the proper maps
QG.9, QG.22 and QG.30 and their degree
\((m!)^k\) on the complement.
Panel B is the integer cone calculation for the
two monomials \(x^3,y^2\): a ray \((a,b)\)
has exponent difference \(3a-2b\).
Every listed adjacent cone is unimodular, and
its two ray differences have no opposite signs.
This is a lattice diagram, not the analytic
space itself.
Panel C plots the real split graphs
\(t=\pm u^3\), \(-1\le u\le1\), after \(x=u^2\).
Its axes use the same scale.
Panel D shows the two separated graph sections
\(v=0,-2\) and the vertical divisor \(u=0\)
in QG.36. It is a real coordinate slice;
the full charts are complex two-dimensional.
The exact pullback and Jacobian are QG.36.
Algebraic map panels have no metric scale.

Editable SVG,
exact figure data, and
reproducible source.
Freely accessible further reading:
[Parusiński–Rond, *The Abhyankar–Jung theorem*, arXiv1103.2559v2, Proposition2.1](https://arxiv.org/pdf/1103.2559).
QG.A–QG.F include the analytic continuation,
bounded extension, monomial principalization,
global centres and finite termination proofs
used in this construction.


### 5.78. Finite covers across a normal-crossing boundary

Let \(Y\) be a connected smooth complex manifold, let
\(E=\bigcup_{i=1}^r E_i\) be a finite simple normal-crossing divisor, and put
\(U=Y\setminus E\). Let
\[
h:V\longrightarrow U
\tag{TF.1}
\]
be a positive finite unramified analytic covering of degree \(n\).
The space \(V\) may be disconnected. There is a canonical finite flat map
\[
\begin{gathered}
f:N\longrightarrow Y,\\
f^{-1}(U)=V,\\
\operatorname{rank}_{\mathcal O_Y}f_*\mathcal O_N=n,
\end{gathered}
\tag{TF.2}
\]
with \(N\) normal. It is constructed by actual convergent algebras below.

Suppose, in addition, that \(Y\) has a closed projective embedding over a
polydisc \(B\), with proper structure map \(p:Y\to B\). Near any fixed
\(b_0\in B\), there is a finite sequence of normalized monomial blowups
\[
Z\xrightarrow{\mu}N\xrightarrow{f}Y
\tag{TF.3}
\]
such that \(Z\) is smooth,
\(D=((f\mu)^{-1}E)_{\mathrm{red}}\) is SNC,
and \(\mu\) is an isomorphism off \(f^{-1}(E)\). Both the composite and
its map to a smaller entire base polydisc are proper and locally
projective. In particular,
\[
(f\mu)^{-1}(U)=V
\tag{TF.4}
\]
with precisely the covering TF.1. The construction adds no branch
component outside \(E\).

#### TF.A. Convergent invariant algebras

Take an adapted polydisc \(W\subset Y\), with coordinates
\((x_1,\ldots,x_k,z_1,\ldots,z_{d-k})\), for which
\[
E\cap W=\{x_1\cdots x_k=0\}.
\tag{TF.5}
\]
The exponential covering and finite path-and-square lifting in
§5.77, QG.A
identify the fundamental group of \(W\setminus E\) with \(\mathbf Z^k\).
Its commuting permutations on the \(n\) sheets are killed by
\[
e=n!.
\tag{TF.6}
\]
The proof is the cycle-length and factorial calculation in QG.A, and
does not require the original covering to be normal. After the power
map
\[
x_i=u_i^e,\qquad z_j=z_j,
\tag{TF.7}
\]
the pulled-back covering is trivial. Write
\(G=(\mathbf Z/e\mathbf Z)^k\), acting by coordinate roots of unity.
Each connected component of TF.1 over \(W\setminus E\) is
\(\widetilde W^*/K\), for a subgroup \(K\subset G\). Indeed the transitive
action of the abelian group \(G\) on that component's fibre has constant
stabilizer \(K\), and is its action on \(G/K\). Sheet identifications
are holomorphic because the maps are local biholomorphisms.

Put \(A=\mathbf C\{x,z\}\) and \(C=\mathbf C\{u,z\}\) at the indicated
origins. Absolute Taylor convergence gives
\[
C=\bigoplus_{0\leq\alpha_i<e}A\,u^\alpha.
\tag{TF.8}
\]
For completeness, group the Taylor terms by their exponent vector
modulo \(e\). On a smaller product, each grouped series is absolutely
convergent; after removing its fixed factor \(u^\alpha\), it is a
convergent series in \((u_1^e,\ldots,u_k^e,z)\). Uniqueness follows by
comparing Taylor coefficients. Thus TF.8 is an identity of convergent
germ modules.

If \(a=(a_i)\in G\), the character of \(u^\alpha\) is
\(\exp(2\pi i\sum_i a_i\alpha_i/e)\). Define
\[
\begin{gathered}
S_K=\{\alpha\in\{0,\ldots,e-1\}^k\mid\\
\textstyle\sum_i a_i\alpha_i=0\pmod e\\
\text{for every }a\in K\}.
\end{gathered}
\tag{TF.9}
\]
Then
\[
\begin{gathered}
C^K=\bigoplus_{\alpha\in S_K}A\,u^\alpha,\\
|S_K|=\frac{e^k}{|K|}.
\end{gathered}
\tag{TF.10}
\]
The first identity follows by averaging Taylor coefficients. For the
cardinality, average the character action on the \(e^k\)-dimensional
space spanned by these monomials. Its trace is \(e^k\) at the identity
and zero at any other \(a\), since one geometric sum
\(\sum_{\ell=0}^{e-1}\exp(2\pi i a_i\ell/e)\) vanishes. The average is
an idempotent with image the fixed monomials, proving the count.
Multiplication is the finite carry table
\[
\begin{gathered}
u^\alpha u^\beta=x^{q(\alpha,\beta)}u^{\gamma(\alpha,\beta)},\\
\alpha_i+\beta_i=e q_i+\gamma_i,\\
0\leq\gamma_i<e .
\end{gathered}
\tag{TF.11}
\]
The residue \(\gamma\) again belongs to \(S_K\).

These formulas give actual finite analytic models. Use coordinates
\(T_\alpha\), set \(T_0=1\), and impose all the carry relations TF.11.
They reduce every polynomial to the \(A\)-span of the \(T_\alpha\);
TF.8 proves that this span has no relation. In addition,
\[
T_\alpha^e=x^\alpha.
\tag{TF.12}
\]
Convergent series cause no new relations or generators. First reduce
the exponent of each of the finitely many \(T_\alpha\)'s modulo \(e\)
using TF.12. The grouped sums converge on a smaller base because the
corresponding root moduli tend to zero. Reduce the resulting finite
list of monomials by TF.11. This proves the same finite presentation
for convergent local rings. Away from the origins one uses the
identical calculation for the boundary coordinates that vanish there.

The analytic space so defined is the quotient of the power polydisc
by \(K\). To check its points and topology, take a fibre of TF.8 over
a base point. It is a product of truncated root algebras. Its reduced
quotient is the algebra of functions on the finite power-map fibre.
Averaging commutes with this fibre quotient and with its nilpotent
kernel, because division by \(|K|\) is possible. Its invariant reduced
quotient is therefore the algebra of functions on the \(K\)-orbits.
Invariants separate those orbits: interpolate a polynomial on the
finite set and average it over \(K\). This identifies the quotient
points. The monic bounds TF.12 make inverse images of compact base
sets compact in these coordinates, so this finite quotient map is
proper.

The normality used here is analytic. A fraction of holomorphic germs
integral over \(\mathbf C\{u,z\}\) is locally bounded off its
denominator, by the root bound of QG.A. The Riemann-extension argument
in §5.32, J.0
extends it holomorphically; this proves that the germ ring is
integrally closed. The argument can also be seen directly: choose
one transverse variable in which the nonzero denominator is not
identically zero, and a small circle on which it has no zero. The
Cauchy integral of the bounded fraction around that fixed circle is
jointly holomorphic. On each generic transverse disc, bounded
one-variable Laurent extension makes it equal to the fraction.
Density gives the equality everywhere.

If a fraction in \(\operatorname{Frac}(C^K)\) is integral over \(C^K\),
it is a fraction in \(\operatorname{Frac}(C)\) satisfying that same
monic equation. It belongs to \(C\), and its invariance puts it in
\(C^K\). At an arbitrary quotient point, choose disjoint small
polydiscs around its finitely many representatives. Its germ is the
invariant germ at one representative under the stabilizer. The
stabilizer acts diagonally only on coordinates that vanish at that
point, and fixes the others. The preceding argument proves normality
there as well.

#### TF.B. Canonical gluing and finite projectivity

For open \(W\subset Y\), let \(\mathcal A(W)\) consist of the holomorphic
functions on \(h^{-1}(W\setminus E)\) that are locally bounded over
every point of \(W\). The condition means that, near that point, one
bound works on all the finitely many sheets. It is a sheaf of
\(\mathcal O_Y\)-algebras. On TF.5 its germ is exactly the direct sum of
the algebras TF.10, one for each orbit of the local monodromy.
A bounded function lifts to a bounded invariant holomorphic function
on \(\widetilde W^*\). Successive Laurent coefficients in each
punctured coordinate have no negative terms, by the fixed-radius
estimate in QG.A; hence it extends holomorphically to \(\widetilde W\).
Conversely a holomorphic germ on that finite quotient is locally
bounded over the base.

Consequently \(\mathcal A\) is locally free of rank \(n\), with the
finite convergent multiplication tables just constructed. The local
finite analytic spaces glue by this one algebra sheaf. More
explicitly, on any overlap their coordinate functions are the same
bounded functions on the original covering. Their extensions are
unique by density. They therefore give mutually inverse
holomorphic maps satisfying the cocycle condition. Distinct points
over one base point are separated in one of these finite local
models; points over distinct base points are separated downstairs.
The glued space is Hausdorff and second countable. Properness is
local on the base here: cover a compact set by finitely many smaller
neighbourhoods whose closures lie in the finite-model
neighbourhoods. Its inverse image is a closed subset of the finite
union of their compact inverse images. This proves TF.2, including
flatness, surjectivity and normality.

This also proves uniqueness among normal finite analytic extensions
with a finite \(\mathcal O_Y\)-algebra and the prescribed dense
covering. Sections of such an algebra satisfy monic equations by
the determinant argument; their restrictions to the covering are
bounded by QG.A. Thus it is a subalgebra of \(\mathcal A\).
The two algebras have the same generic rank \(n\). Localizing over
the base fraction field therefore gives the same finite product
of fields. Every element of \(\mathcal A\) is integral over the
smaller algebra, since it is integral already over \(\mathcal O_Y\).
Normality of its finitely many local factors forces equality.
The same argument extends isomorphisms and deck actions uniquely.

We give the projectivity argument in a form also usable for the
normalizations below. Suppose \(T\) is closed in
\(\mathbf P^a\times B\), and a finite analytic space \(T'\to T\)
has a coherent finite algebra \(\mathcal C\). Let
\(L=\mathcal O_{\mathbf P^a}(1)|_T\). Closed coherent pushforward and
Theorem5.8
give, after one base shrinking, finitely many generators of
\(\mathcal C\otimes L^q\). They define a closed immersion
\[
T'\ \longrightarrow\operatorname{Tot}\bigl((L^q)^{\oplus s}\bigr).
\tag{TF.13}
\]
Indeed on a trivialization their entries generate \(\mathcal C\)
as an \(\mathcal O_T\)-module and hence as an algebra. The local
finite analytic presentations give the closed immersion there.
Compact inverse images under the finite map have uniformly bounded
coordinates, so its image remains closed in
\[
\mathbf P_T\bigl(\mathcal O_T\oplus(L^q)^{\oplus s}\bigr),
\tag{TF.14}
\]
using the convention of lines. This has a closed projective embedding
over \(B\). For an explicit construction, tensor its dual bundle by
\(L^\ell\), with \(\ell\geq q\). The factors \(L^\ell\) and
\(L^{\ell-q}\) are generated by finitely many projective monomials.
The resulting surjection from a trivial bundle gives a closed
embedding of TF.14 in \(T\times\mathbf P^c\). The product embedding
and the Segre equations proved in QG.A finish the construction.
Thus finite maps with these coherent algebras preserve local
projectivity over the original polydisc. This argument assumes
neither smoothness nor normality of \(T\).

#### TF.C. Boundary lattices and actual monomial ideals

The local model TF.10 has invariant character lattice
\[
\begin{gathered}
M=\{a\in\mathbf Z^k\mid\\
\textstyle\sum_i b_i a_i=0\pmod e\\
\text{for every }b\in K\},\\
N_M=\operatorname{Hom}_{\mathbf Z}(M,\mathbf Z),\\
\sigma=\mathbf R_{\geq0}^k\subset (N_M)_{\mathbf R}.
\end{gathered}
\tag{TF.15}
\]
The standard orthant is viewed in the dual lattice, which can be
larger than \(\mathbf Z^k\). Its rays have primitive lattice
generators \(v_1,\ldots,v_k\). The quotient ring consists of the
convergent sums of \(\chi^m\), for
\[
m\in \sigma^\vee\cap M.
\tag{TF.16}
\]
The primitive ray valuation agrees with the actual
order along its divisor. At a generic point of that
coordinate hyperplane, its stabilizer in the diagonal
group fixes every nonzero coordinate and rotates only
the vanishing coordinate, say with order \(c_i\).
The quotient transverse coordinate is \(u_i^{c_i}\).
Thus the order of an invariant monomial there is its
\(u_i\)-exponent divided by \(c_i\). This is the pairing
with the primitive ray vector: the double-annihilator
calculation TF.10 says that the possible exponents have
greatest common divisor \(c_i\).
Each ray corresponds to the image of one coordinate hyperplane.
That image is normal: its ring is the invariant quotient after
setting that coordinate equal to zero, since averaging is exact.
Images of distinct coordinate hyperplanes are distinct, since a
diagonal group does not permute them. Thus the reduced boundary is
a union of normal divisors with no local self-intersection. Give the
whole reduced preimage of each original \(E_i\) one label
\(D_i\); a labelled divisor may have disjoint components.
At a point it has at most one local branch.

A full-dimensional simplicial cone with independent primitive
generators \(v_i\) also has such a convergent quotient model. Let
\[
L_\sigma=\sum_i\mathbf Z v_i\subset N_M.
\tag{TF.17}
\]
The dual \(L_\sigma^\vee\) has the coordinate basis dual to the
\(v_i\)'s. The finite diagonal group
\(\operatorname{Hom}(L_\sigma^\vee/M,\mathbf C^*)\) on the power
coordinates has invariant monomials precisely \(M\). TF.A proves
the convergent quotient description. Character separation
in this assertion follows directly from TF.10: choose
an integer \(q\) with \(qL_\sigma^\vee\subset M\).
In the finite group \(L_\sigma^\vee/qL_\sigma^\vee\),
a subgroup is contained in its double annihilator,
and TF.10 gives equal cardinalities, hence equality.
Faces give the same
description with the nonvanishing coordinates made invertible;
near any chosen values of those coordinates they are ordinary
holomorphic parameters.

Here are intrinsic ways to identify this lattice information on
overlaps. A holomorphic function on a power polydisc whose zero
set is contained in the coordinate boundary is a coordinate
monomial times a unit. The earlier
analytic Nullstellensatz, Theorem5.1
puts a power of the product of those coordinates in its principal
ideal. QG.B's coordinate-prime division applied to that product
then gives the assertion. The unit factor of an invariant function
has nonzero value at the origin, so its character is trivial.
It follows that its monomial exponent lies in \(M\).
Use the group completion of this monoid of holomorphic
boundary equations to define its boundary-order lattice.
That completion is \(M\): add a sufficiently large interior
integral monomial to any given \(m\in M\) to write it as a
difference of two exponents in TF.16. This definition uses
only holomorphic equations, their boundary orders and units,
so it is independent of the quotient coordinates. It also
identifies the primitive ray valuations and the face lattices.

For clarity, the local ideal with prescribed nonnegative boundary
orders can be read without a divisor theorem. Expand a holomorphic
germ in its invariant power-series monomials. Requiring order
at least \(a_i\) along the \(i\)-th ray selects exactly the terms
with
\[
\langle m,v_i\rangle\geq a_i.
\tag{TF.18}
\]
If an \(m_0\in M\) has those boundary values, division by
\(\chi^{m_0}\) is actual coordinate division upstairs, preserves
invariance, and converges. In particular, for a sufficiently
divisible positive integer \(a\),
\[
\mathcal O_N(-aD_i)
\tag{TF.19}
\]
is an invertible monomial ideal for the \(i\)-th boundary divisor:
the rational vector dual to \(v_i\) becomes integral after
multiplication by \(a\). Local principal generators of this ideal
differ by units and glue. On any finite quotient atlas one common
multiple suffices for every occurrence of a given component.

There are finitely many *labelled cone types*, which is
the finiteness needed below. A type records the set of
boundary labels through a point and its boundary-order
sublattice of \(\mathbf Z^I\), where \(I\) is that label set.
Initially the finitely many subgroups of
\((\mathbf Z/e\mathbf Z)^k\), for \(k\leq d\), and the
finitely many possible label sets give all types by TF.15.
This uses finite types, without assuming that all strata
or all their connected components form a finite set.
It also supplies one common Cartier multiple for TF.19.

The cones fit along faces. In a quotient chart, passing
to a boundary face means making the other coordinates
nonzero; its lattice is
\[
N_\tau=N_M\cap\operatorname{span}_{\mathbf R}\tau.
\tag{TF.20}
\]
Equivalently its order lattice is the image of the full
order lattice under restriction to those labels. Boundary
orders identify these restrictions on overlaps. An
overlap automorphism fixing all the labelled primitive
rays is the identity on their real span.

For a fixed label set \(I\), this face type is locally
constant on the whole closed intersection
\(D_I=\bigcap_{i\in I}D_i\), including its deeper strata.
In one quotient chart it is obtained by inverting the
other coordinates, giving exactly TF.20 at every such
point. Each of these intersections is locally a quotient
of one coordinate intersection, hence locally normal
and locally irreducible. The locus in \(D_I\) of any
specified face type is consequently open and closed
in \(D_I\), and is closed analytic in \(N\). Near its
points it is all of that coordinate intersection;
near its complement it is empty. This gives coherent
ideals for the possibly disconnected loci, without a
finite-component assumption.

#### TF.D. A coherent normalized blowup for one star

Let \(\tau\) be one of these cones, with rays \(v_1,\ldots,v_s\),
and choose a primitive interior lattice point
\[
v=\sum_{i=1}^s\alpha_i v_i,\qquad \alpha_i>0.
\tag{TF.21}
\]
Let \(C_\tau\) be the whole closed locus of that labelled
face type in its boundary intersection. Thus every
occurrence of this type is treated together. Choose
positive integers \(a_i\) with
\[
a_i\alpha_i=c>0
\tag{TF.22}
\]
for one rational \(c\), and sufficiently divisible that each
\(a_iD_i\) is Cartier throughout all the finitely many types. Such a common
multiple exists because all the \(\alpha_i\)'s are rational.
Near \(C_\tau\) form
\[
\mathcal I=\sum_{i=1}^s\mathcal O_N(-a_iD_i).
\tag{TF.23}
\]
Its zero set is that intersection. To select \(C_\tau\),
use TF.23 off the closed complementary type loci of
that intersection and use the unit ideal off \(C_\tau\).
These two definitions coincide on their overlap, where
some summand is already the unit ideal. They give a
coherent global ideal. In particular all occurrences
of one coface type that has this face are affected;
one does not leave unmodified copies of the same type.

On a quotient chart for any cone \(\gamma\) containing \(\tau\),
write its generators as \(\chi^{m_i}\), where
\[
\langle m_i,v_j\rangle=a_i\delta_{ij}.
\tag{TF.24}
\]
Here \(j\) ranges over every ray of \(\gamma\); the other values
are zero. The integer divisibility in TF.22 ensures \(m_i\in M\).
Unit multiples of these generators do not change the ideal.
The blowup has the usual ratio charts
\[
\mathcal O[\chi^{m_j-m_i}:1\leq j\leq s].
\tag{TF.25}
\]
They are the closure of the isomorphism locus, not extra
components of equations that disappear on localization.
Let \(R_i\) consist of all primitive ray generators
of \(\gamma\) other than \(v_i\). Its \(i\)-th cone is
\[
\begin{gathered}
\gamma_i=\{w\in\gamma\mid\\
\langle m_i,w\rangle\leq\langle m_j,w\rangle\\
\text{for }1\leq j\leq s\},\\
\gamma_i=\operatorname{Cone}(v,R_i).
\end{gathered}
\tag{TF.26}
\]
To check the equality, write \(w=\sum_j\lambda_jv_j\).
TF.22 says that the inequalities are
\(\lambda_i/\alpha_i\leq\lambda_j/\alpha_j\) for \(j\leq s\).
Subtract \((\lambda_i/\alpha_i)v\). The remainder has
nonnegative coefficients and its \(i\)-th coefficient is zero.
This also proves that these cones cover the old cone and meet
in their indicated faces. Off the chosen intersection the
blowup is the identity.

We construct the normalization of TF.25 explicitly. Let \(S\)
be its exponent semigroup, including the old monomials.
Its group is \(M\), and its real cone is \(\gamma_i^\vee\).
The first assertion holds already for the old full-dimensional
cone: add a sufficiently positive interior integral vector to
both \(m\) and zero to write any \(m\in M\) as a difference of
two elements of the old semigroup. The second assertion is
the dual inequality description TF.26. Here the cone
on the finite semigroup generators is closed: an
interior point of \(\gamma_i\) is strictly positive on
every nonzero generator, so a convergent sequence in
that cone has bounded nonnegative generator
coefficients and a convergent coefficient subsequence.
For a point outside this closed convex cone, its
nearest point gives a separating linear functional
by differentiating the squared distance along each
segment in the cone. Hence taking its dual twice
recovers it. This proves the asserted dual
inequality description. The saturated semigroup is
\[
\overline S=M\cap\gamma_i^\vee.
\tag{TF.27}
\]
It is finite as an \(S\)-module. Choose on each extreme ray of
the simplicial cone \(\gamma_i^\vee\) a nonzero element
\(w_j\in S\). Every \(m\in\overline S\) is
\(\sum_j b_jw_j\), with \(b_j\geq0\). Subtracting the integer
parts of the \(b_j\)'s leaves an element in the finite set
\[
\begin{gathered}
\Bigl\{\sum_j t_jw_j\mid\\
0\leq t_j<1\Bigr\}\cap M.
\end{gathered}
\tag{TF.28}
\]
This gives the finite module generators. Each of them has a
positive integer multiple in \(S\), by rationality of its
coefficients in these \(w_j\)'s, and therefore satisfies a
monic monomial equation over the ratio chart.

This is a convergent analytic normalization as well. Construct
the chart for \(\gamma_i^\vee\cap M\) as the finite diagonal
quotient in TF.C. The old monomial generators together with
TF.28 generate its polynomial monomial algebra. Thus their
coordinates give a closed analytic embedding into the ratio
chart times an affine space; the equations are their finite
monomial relations, restricted to the original analytic
chart domain and its holomorphic parameters. Each new
coordinate satisfies the monic monomial equation just given.
Homogenize these equations in one common projective space.
At infinity they say that every new coordinate is zero,
which is impossible in projective space. The same closed
embedding therefore lies in that projective space and is
proper over the ratio chart. Its fibres are finite by the
monic equations.

Its analytic algebra is coherent: embed the ratio chart in
a coordinate polydisc, use closed coherent pushforward, and
apply the ordinary projective coherent-image proof,
Lemma5.9.1, to this closed projective embedding. The
resulting algebra is a finite module over the ratio-chart
ring. This proves analytic finiteness at points where
monomial coordinates vanish and at points where they do
not vanish, without a formal-to-convergent assertion.
Its normal local rings are the invariant convergent
rings proved normal in TF.A. Properness and finite fibres
identify the stalk of its pushed algebra with the product
of the finitely many local normal rings above that point:
disjoint neighbourhoods of that finite fibre contain the
entire inverse image of a sufficiently small base
neighbourhood, since the proper image of their closed
complement misses the point. On the dense torus the
map is the identity, so the total meromorphic algebras
agree.
It is therefore precisely the integral closure of the ratio
chart in its total meromorphic algebra: integrality puts it
in the integral closure, and any further integral fraction
belongs to this normal ring. At a point with several analytic
branches, use the product of their meromorphic fields and
the direct sum of the normal local factors. The single
polynomial semigroup domain can acquire several analytic
branches in such a germ; no unibranch assumption is used.

On overlapping ratio charts, these integral closures in the
same total meromorphic algebra coincide. They glue, including across
the original quotient-coordinate overlaps. Thus TF.23 has a
global normalized blowup with the star subdivision TF.26.
Give its new exceptional divisor one new label. Its
components are locally the image of one coordinate
hyperplane, hence normal and locally disjoint within
that label. The other labels are the strict transforms
of the old ones. Each old labelled cone type gives
only finitely many new labelled types by TF.26 and
their face restrictions. Thus there are still finitely
many types after the operation.
It is proper and projective over the original smaller base.
The raw coherent blowup is the closed graph construction
proved in QG.C using Theorem5.8 and Lemma5.9.1; the same ratio
argument applies to the present coherent ideal. Its
normalization is finite with the coherent algebra TF.28, so
TF.13–TF.14 applies. Every chart remains an actual simplicial
quotient chart. Its reduced boundary is the full coordinate
boundary in that chart, including all exceptional components.
There is no omitted chart at the other end of an exceptional
fibre.

#### TF.E. Finite global termination

For an independent-ray cone \(\gamma\) define its multiplicity by
\[
\begin{gathered}
\operatorname{mult}(\gamma)\\
=[\,N_\gamma:\textstyle\sum_{\rho\subset\gamma}
\mathbf Z v_\rho\,].
\end{gathered}
\tag{TF.29}
\]
It equals the number of lattice points in the half-open
parallelepiped on its primitive generators. Subtracting integer
parts gives one representative of each coset, and uniqueness
follows from independence. Multiplicity one is exactly the
condition that the quotient chart is smooth with its
coordinate SNC boundary: the generators are then a lattice
basis. If \(\eta\) is a face, its lattice quotient injects
into the lattice quotient in TF.29, since the integer span
of all rays intersects its real span in the integer span
of its own rays. Thus
\[
\operatorname{mult}(\eta)\mid\operatorname{mult}(\gamma).
\tag{TF.30}
\]

Let \(M_0>1\) be the largest multiplicity among the finite
labelled cone types and choose a type attaining it.
In that cone choose a
minimal nonregular face \(\tau\). Every proper face of \(\tau\)
is regular. Since \(\operatorname{mult}(\tau)>1\), its
half-open parallelepiped contains a nonzero lattice point
\(v_0=\sum_i b_iv_i\), with \(0\leq b_i<1\).
All its coefficients are positive. A zero coefficient would
put it in the lattice of a proper regular face, where all
coefficients in that face's primitive basis must be integers,
forcing \(v_0=0\). Divide by the positive integer that makes
its ray primitive. This gives TF.21 with
\[
0<\alpha_i<1.
\tag{TF.31}
\]
Perform the global normalized blowup TF.D for the whole
closed locus of this labelled face type.

For any affected coface \(\gamma\) and \(i\leq s\), determinant
linearity in its real span gives the following equality.
Here index equals absolute determinant in a lattice basis:
integer row and column operations from the Euclidean
algorithm put a full-rank integer matrix into triangular
form, preserving both index and absolute determinant;
successive division by its diagonal entries counts the
cosets as their product. Thus the determinant computation
measures exactly TF.29:
\[
\begin{gathered}
\operatorname{mult}(\gamma_i)
=\alpha_i\operatorname{mult}(\gamma)\\
<\operatorname{mult}(\gamma).
\end{gathered}
\tag{TF.32}
\]
The determinants use the same ambient lattice before and
after the operation. Primitive generators are used for all
the rays, including \(v\), so there is no missing lattice
factor. These are integers. New faces have multiplicity at
most that of their new cofaces by TF.30. Hence no new cone
of multiplicity \(M_0\) appears, while at least the selected
cone of multiplicity \(M_0\) disappears.

If \(c_{M_0}\) is the number of labelled types of that multiplicity,
the global lexicographic pair
\[
(M_0,c_{M_0})
\tag{TF.33}
\]
strictly decreases. The first entry is a positive integer;
at each fixed first entry only finitely many second-entry
decreases can occur. The set of types remains finite,
since each operation replaces a finite list of types by
finite lists of subdivided types and faces. Treating the
whole face-type locus removes every occurrence of the
selected maximal type. Induction on \(M_0\) proves that only
finitely many operations occur. Every terminal cone is
regular. The convergent charts in TF.C–TF.D prove that the
whole terminal space is smooth and its full reduced inverse
boundary is SNC. Each normalized blowup is an isomorphism
off the boundary. All components meet that dense open set,
because the raw blowup is its graph closure and the finite
normalization has the same meromorphic fields.

All constructions can first be made intrinsically on the
chosen base neighbourhood. The finite termination just
proved permits the finitely many projective-embedding
shrinkings in TF.B and TF.D to be combined into one smaller
polydisc. Thus the entire resulting morphism, rather than
a disjoint union of open quotient charts, is proper and
locally projective. This proves TF.3–TF.4.

The empty-boundary and minimum-degree cases are included.
When \(k=0\), the local algebra is the direct sum of
\(n\) copies of the base ring, with no power variables.
If \(E\) is empty, \(N=V\) is already smooth and no
boundary blowup occurs. When \(n=1\), TF.2 is \(Y\)
itself and every cone is already regular. In dimension
zero the same construction is the finite set of
covering points, with empty boundary.

#### TF.F. Globally labelled roots on a projective SNC base

Let \(P(t,y)\) be a monic degree-\(m\) polynomial with
holomorphic coefficients on \(Y\). Suppose all its roots
lie in one disc \(D_r\Subset D_R\), and that its nonzero
discriminant has support in \(E\). On \(U\) its ordered
distinct-root tuples form an actual finite unramified
cover of degree
\[
m!.
\tag{TF.34}
\]
The local holomorphic branches are proved in QG.A.
Their ordered tuples give all \(m!\) local sheets.
They are proper over compact subsets of \(U\), since their
coordinates stay in the fixed smaller root disc and the
polynomial and elementary-symmetric equations define a
closed subset there.

Apply TF.2–TF.4 to this covering. Its coordinate functions
are bounded, so TF.B extends them to \(N\) and their
pullbacks give global holomorphic roots on \(Z\):
\[
\begin{gathered}
P(t,g(z))\\
=\prod_{\nu=1}^m(t-r_\nu(z)),\\
g=f\mu.
\end{gathered}
\tag{TF.35}
\]
The identities hold first off the boundary and then by
density everywhere. They retain the strict bound
\(|r_\nu|<r\). Indeed the root set over the compact central
fibre is compact and has maximum modulus strictly less
than \(r\). Choose an intermediate radius \(r'<r\) above
that maximum. Properness and the closed polynomial root
set let us shrink \(B\) so that all roots there have modulus
less than \(r'\). Their bounded extensions have modulus
at most \(r'\) by continuity from the dense complement.

On each SNC chart of \(Z\), the pulled-back discriminant
is a coordinate monomial times a unit. Its zero set is
contained in that chart's boundary, so this is the
Nullstellensatz and coordinate-prime argument of TF.C.
The product identity
\[
\prod_{\nu<\lambda}(r_\nu-r_\lambda)^2
           =\Delta_P\circ g
\tag{TF.36}
\]
and QG.B give a monomial times a unit for each root
difference. None is the zero germ, since the roots are
distinct on the dense complement in every component.

The global monomial principalization QG.C now applies to
the finitely many root-cluster ideals on \(Z\).
Here is the finiteness needed for this application.
Finitely many SNC charts cover the compact central
fibre. After shrinking \(B\), their union contains the
entire inverse image of \(B\), by properness. The
monomial orders of the finitely many root differences
therefore have only finitely many values. Refine each
boundary label by the vector of those orders on its
components. Each refined label is an open and closed
union in that normal divisor, hence is closed analytic;
there are finitely many labels and no self-intersections.
Every collision ideal is a sum of monomials with fixed
orders on these labels.

The proof of QG.C applies verbatim to these finitely
many labels, treating a disjoint union of centre
components in one blowup. Its maximal positive-plus-
negative invariant counts meeting labelled pairs.
The new exponent difference is constant on the
whole new exceptional label. Thus its strict
decrease and termination do not require a count
of connected strata. On the resulting
base, QG.D separates the global graph sections by closed
graph-centre blowups; its exact nonnegative weight
decrease proves termination. Consequently there is a
proper locally projective map from a smooth manifold
onto \(D_R\times Y\) whose reduced inverse image of
\[
\begin{gathered}
\{P(t,y)=0\}\\
\cup\ (D_R\times E)
\end{gathered}
\tag{TF.37}
\]
is SNC, and which is a surjective finite unramified
covering of degree \(m!\) off precisely that chosen
boundary. The graph blowups are isomorphisms off it.
Every complementary chart is included by QG.D.
Thus all the TC finite D-type, canonical de Rham,
ordinary infinity/function and full source infinity
proper-image comparisons apply to TF.37 with this
geometry constructed. This statement uses a supplied
projective SNC base \(Y\); it does not assert that an
arbitrary analytic discriminant is SNC in the
original base coordinates.

#### TF.G. A double cover and its whole exceptional line

Take \(Y\) to be a bidisc with boundary \(xy=0\) and
the connected two-sheeted covering determined by
\[
w^2=xy.
\tag{TF.38}
\]
Let \(K_2=\{\pm1\}\) act by simultaneous sign
on \((u,v)\). Its normal extension is
\[
\begin{gathered}
x=u^2,\qquad y=v^2,\\
w=uv,\qquad N=\widetilde Y/K_2,\\
f_*\mathcal O_N=\mathcal O_Y\oplus w\mathcal O_Y .
\end{gathered}
\tag{TF.39}
\]
Here
\[
\begin{gathered}
M=\{(a,b)\in\mathbf Z^2\mid\\
a+b\text{ is even}\},\\
N_M=\mathbf Z^2+\mathbf Z(1/2,1/2),\\
\sigma=\operatorname{Cone}((1,0),(0,1)),\\
\operatorname{mult}(\sigma)=2 .
\end{gathered}
\tag{TF.40}
\]
The interior primitive point \(v_*=(1/2,1/2)\) has both
coefficients \(1/2\). Its star is the normalized blowup
of \((x,y)\). On the \(x\)-ratio chart, put
\[
\begin{gathered}
y=xq,\qquad w^2=x^2q,\\
s=w/x,\qquad q=s^2,\\
(x,y,w)=(x,xs^2,xs).
\end{gathered}
\tag{TF.41}
\]
The integral coordinate \(s\) is essential; the raw
ratio chart does not already have this normal ring.
If \(w/x\) belonged to its local ring at
\(x=q=w=0\), restriction to \(x=w=0\) would give a
holomorphic germ \(a(q)\) with \(a(q)^2=q\). Its
vanishing order would satisfy \(2\operatorname{ord}_0a=1\),
which is impossible. Thus the finite integral
adjoining in TF.41 is substantive.
The other chart is
\[
(x,y,w)=(yt^2,y,yt),
\tag{TF.42}
\]
and their overlap is
\[
t=s^{-1},\qquad y=xs^2 .
\tag{TF.43}
\]
The exceptional curves \(x=0\) and \(y=0\) glue by
this reciprocal coordinate to the entire
\(\mathbf P^1\), including \(s=0\) and \(t=0\).
Write \(\gamma_x=\operatorname{Cone}(v_*,(0,1))\)
and \(\gamma_y=\operatorname{Cone}((1,0),v_*)\).
Both new cones have multiplicity one:
\[
\operatorname{mult}(\gamma_x)
=\operatorname{mult}(\gamma_y)=1 .
\tag{TF.44}
\]
In the two charts the full pullback boundary equations
and map Jacobians are respectively
\[
\begin{array}{c|c|c}
 & xy & \det D(x,y)\\ \hline
(x,s) & x^2s^2 & 2xs\\
(y,t) & y^2t^2 & -2yt .
\end{array}
\tag{TF.45}
\]
The column order in the second determinant is \((y,t)\);
the target order is \((x,y)\). These are actual complex
identities. The reduced boundary is \(xs=0\) or \(yt=0\),
SNC in both charts. The degree off \(xy=0\) is two.
The projective graph and finite normalization prove
properness of the whole map, including that complete
exceptional line.

![Finite normal extension, a primitive lattice star and both normalized double-cover charts](assets/finite-toroidal-cover.png)

**Figure TF.1.** Panel A is the actual diagram TF.1–TF.4.
The finite map \(f\) has rank \(n\), and \(\mu\) is the
global boundary modification. Panel B uses exactly the
lattice TF.40, its primitive inserted ray and the two
regular cones TF.44. Its coordinate axes have equal
scales; the lattice covolume is \(1/2\). The shaded
triangles are sections of the two cones in that
specific example. Panel C gives both complex charts
TF.41–TF.43 and their reciprocal overlap, which
contains the entire exceptional \(\mathbf P^1\) after
gluing. Panel D shows the real coordinate slices of
TF.45, with all imaginary coordinates zero and equal
axes. Each full chart has complex dimension two.
The map and gluing panels have no metric scale.

Editable SVG,
exact figure data
and reproducible source.

Freely accessible further reading: Enokizono–Hashizume,
[*Semistable reduction for complex analytic spaces*,
arXiv2404.05124v3, Proposition2.25](https://arxiv.org/pdf/2404.05124v3);
Karu, [*Toric Geometry*, notes by Ronagh, §§11–13](https://pooya-git.github.io/files/toric2011.pdf).
TF.A–TF.G provide the convergent algebras, global
coherent ideals, projectivity and finite termination
arguments used here.


### 5.79. Analytic hypersurfaces, proper trace and the prescribed complement

Let \(X\) be a sufficiently small coordinate neighbourhood in a
complex manifold of dimension \(n\), let \(S\) be a reduced analytic
hypersurface, and put \(j:U=X\setminus S\hookrightarrow X\).
All local systems have finite complex rank. All tensor products
without a derived symbol are ordinary algebraic sheaf tensor
products. Write \(B_X=\mathcal D_X^\infty\).

We use the finite D-type class of
§5.72, DT.E:
coherent holonomic finite-order modules localized at \(S\), with
flat restriction to \(U\), regular at a generic smooth transverse
point of every component of \(S\). Its full faithfulness, its finite
projective meromorphic coefficient module in DT.A, and the exact
proper localization proof DT.B will be used with these hypotheses.
The global logarithmic lattice and its coherent operator generator
are proved in
§§5.0–5.3.
The supplied-generator proper coherence and characteristic-image
theorems are
§§5.9–5.12.

#### FB.A. An alteration with an explicitly enlarged boundary

For every nonzero holomorphic germ \(h\), there is a polydisc
representative \(X\), an analytic hypersurface \(T\) containing
\(\{h=0\}\), and a proper locally projective map
\[
g:Z\longrightarrow X
\tag{FB.1}
\]
such that \(Z\) is smooth of dimension \(n\),
\(D=(g^{-1}T)_{\mathrm{red}}\) is SNC, and
\[
g:Z\setminus D\longrightarrow X\setminus T
\tag{FB.2}
\]
is a finite unramified covering of positive degree.
Every component of \(Z\) meets this complement.
Moreover, \(T=S\cup A\), where \(S=\{h=0\}_{\mathrm{red}}\)
and \(A\) has no component in common with \(S\).
The added hypersurface is part of this geometric assertion.

Here is a dimension induction, including repeated equations.
A unit, or dimension zero, uses the identity and empty boundary.
Otherwise choose a direction on which the first nonzero homogeneous
part of \(h\) is nonzero. The full convergent preparation proof in
analytic preparation, Theorem 1 and Proposition 4
gives coordinates \((t,z)\), polydiscs \(D_R\times B\), and
\[
\begin{gathered}
h(t,z)=u(t,z)P(t,z),\\
P=t^m+\sum_{i=1}^m a_i(z)t^{m-i},\\
u\ne0,\qquad a_i(0)=0.
\end{gathered}
\tag{FB.3}
\]
Its contour proof supplies a strict circle containing every root,
uniformly on a smaller \(B\). In particular all graphs below lie
strictly inside \(D_R\).

Put \(A_B=\mathbf C\{z\}\) and \(K=\operatorname{Frac}(A_B)\).
In \(K[t]\) let \(Q\) be the monic product of the distinct monic
irreducible factors of \(P\). Equivalently
\[
Q=P/\gcd(P,\partial_tP).
\tag{FB.4}
\]
All roots of \(P\) are integral over \(A_B\). A finite collection of
integral elements generates a finite \(A_B\)-module: reduce powers
successively by their monic equations. Multiplication by any
element of that algebra preserves this finite spanning module;
the adjugate determinant identity applied to its spanning relations
gives a monic equation for that element. Thus sums and products of
the roots are integral as well. The coefficients of \(Q\), which
are symmetric expressions in a subset of these roots, belong to
\(K\) and are integral over \(A_B\). Coefficient normality
§5.32, J.0
therefore puts them in \(A_B\).

Monic polynomial division in \(A_B[t]\), followed by its injection
in \(K[t]\), proves
\[
\begin{gathered}
Q\mid P,\qquad P\mid Q^m,\\
\{P=0\}=\{Q=0\}.
\end{gathered}
\tag{FB.5}
\]
These identities hold on every special fibre. In particular
\(Q(t,0)=t^q\), where \(q=\deg Q\ge1\).
The nonzero discriminant
\[
\Delta(z)=\operatorname{disc}_tQ
\tag{FB.6}
\]
is holomorphic on \(B\).

Apply the induction hypothesis in dimension \(n-1\) to \(\Delta\).
It gives a hypersurface \(H\) containing \(\{\Delta=0\}\)
and a proper locally projective \(p:Y\to B\), with \(Y\) smooth,
\[
E=(p^{-1}H)_{\mathrm{red}}\text{ SNC},
\tag{FB.7}
\]
and a positive finite unramified cover off \(H\).
If \(\Delta\) is a unit take \(Y=B\) and \(H=E=\varnothing\).
Over this entire supplied projective SNC base, apply
§5.78, TF.F
to \(Q(t,p(y))\).
That proof constructs its actual ordered-root cover, its finite
normal analytic extension, the coherent normalized monomial
blowups, and the global graph-separating blowups.
It gives a smooth source and a proper locally projective map to
\(D_R\times Y\), with reduced inverse boundary SNC and with finite
unramified degree \(q!\) off
\(\{Q(t,p(y))=0\}\cup(D_R\times E)\).
Compose with \(\operatorname{id}_{D_R}\times p\) and set
\[
\begin{gathered}
T=\{Q=0\}\cup A,\\
A=D_R\times H .
\end{gathered}
\tag{FB.8}
\]
The inverse of this exact \(T\) is the boundary just constructed.
Projective embeddings compose by the Segre embedding; properness
and covering degrees compose. Thus FB.1–FB.2 follow, with positive
degree \(q!\deg p\).
The complement of a hypersurface in a polydisc is path connected:
the complex affine line through two complement points has a
nonzero restricted equation, so only finitely many zeros meet
the compact segment between them. Small complex detours in
that line avoid those zeros inside the polydisc.
Consequently each covering degree used in the induction is constant.
There is no common component of \(A\) and \(S\): a component of
the vertical hypersurface \(D_R\times H\) contained in \(\{Q=0\}\)
would make the monic polynomial \(Q(t,z)\) vanish for every \(t\)
at a generic point of that component of \(H\).
Its leading coefficient is one. This is impossible.
This completes the induction.

The geometry may use a different coordinate projection from one
subsequently used to define a monodromy group. Its target remains
the original neighbourhood with its original \(S\).

#### FB.B. Distribution forms and their Dolbeault resolution

A \((p,q)\)-current on a complex \(n\)-fold is a continuous linear
functional on compactly supported smooth \((n-p,n-q)\)-forms.
Differentiation uses the integration-by-parts sign, so that
the inclusion of smooth forms as currents commutes with
\(\partial\), \(\bar\partial\), and their sum.
Use complex orientation and the ordinary integration pairing.
The complex of \((p,q)\)-currents resolves \(\Omega^p\).
We give the local distribution argument needed for this assertion.

With \(dA=dx\,dy\) and
\(\partial_{\bar z}=(\partial_x+i\partial_y)/2\), Stokes' formula
on a punctured disc, followed by the inner-circle limit, gives
\[
\partial_{\bar z}\frac1{\pi z}=\delta_0.
\tag{FB.9}
\]
Indeed \(1/z\) is locally integrable; the outer boundary vanishes
for a compactly supported test function, and the inner boundary
is its value at zero, since its difference from that value is
bounded by a constant times the circle radius. These estimates
also justify the distributional limit.

Take nested polydiscs and a cutoff \(\chi_i\) in coordinate \(z_i\),
equal to one on the inner disc and compactly supported in the
outer disc. Partial convolution, with distribution coefficients
in the other variables, defines
\[
\begin{gathered}
T_i v=\frac1{\pi z_i}*_{i}(\chi_i v),\\
\partial_{\bar z_i}T_i v=v\\
\text{on the inner polydisc}.
\end{gathered}
\tag{FB.10}
\]
Partial convolution is defined by transposing its integral on
test functions; one factor has compact support in the convolved
variable. It commutes with derivatives in the other variables.
The companion identity is
\[
\begin{gathered}
T_i\partial_{\bar z_i}v=v-P_i v,\\
P_i v=\frac1{\pi z_i}*_{i}
             ((\partial_{\bar z_i}\chi_i)v).
\end{gathered}
\tag{FB.11}
\]
On the inner disc the kernel in \(P_i\) has no singularity in
the \(z_i\) variable. Thus \(P_i v\) is holomorphic in that
variable, with distribution coefficients in the others.

For a closed positive-degree \((0,q)\)-current write
\(v=v_0+d\bar z_i\wedge v_1\), with neither \(v_\nu\) containing
\(d\bar z_i\). Subtract \(\bar\partial T_i v_1\).
The result has no \(d\bar z_i\) component by FB.10; it is closed,
so its other components are \(\partial_{\bar z_i}\)-closed.
Choose the cutoffs independently in each variable. Each later
\(T_j\) commutes with the already treated derivatives.
After all \(n\) variables a positive-degree form with no
antiholomorphic differential component is zero.
This gives local exactness by finitely many contractions on one
smaller polydisc.

A degree-zero closed distribution is an ordinary holomorphic
function. To see the regularity, mollify it on an inner polydisc.
Its mollifications are smooth and \(\bar\partial\)-closed.
Choose a fixed product of radial smooth averaging kernels
supported in still smaller discs. The holomorphic mean-value
formula makes convolution with that kernel the identity on
each mollification on a further inner product.
Passing to the distributional limit gives \(v=v*\psi\).
The right side is smooth. Its Cauchy–Riemann equations, or the
one-variable Cauchy formula successively, make it holomorphic.
This proves the assertion in degree zero, hence the resolution.
The same proof with fixed holomorphic \(p\)-forms proves it for
every \(p\).

Current sheaves and smooth form sheaves are modules over smooth
functions. Partitions of unity and the acyclicity proof in
§5.34, R.5
make both resolutions section-acyclic on every open subset.
If \(h\) is a holomorphic equation, algebraically localize these
resolutions at \(h\). Localization is flat and exact.
For a finite projective \(\mathcal O[1/h]\)-module take a finite
free ambient module and its idempotent; the corresponding
direct summand of the localized resolutions remains exact.
In particular a fraction in localized currents means an
algebraic fraction. It does not require multiplying a
distribution by a singular function.

All these constructions admit holomorphic auxiliary parameters.
One can resolve those parameters first by smooth Dolbeault
forms, use the fixed-coordinate Cauchy–Green contractions of
§5.34, R.2–R.3, and then use FB.10–FB.11 in the source variables.
The two sets of operators commute. On each compact product all
distribution orders are finite; each convolution is continuous
on the corresponding finite test-function seminorms.
Thus parameter differentiation passes through it.
The finite contractions use one common smaller parameter product.
The section-acyclic comparison at parameter germs is precisely
the double-resolution argument in
§5.75, PD.B.
It applies also to these finite-order distribution seminorms.

#### FB.C. A split proper trace before any Riemann–Hilbert theorem

Suppose \(g:Z\to X\) is proper between smooth \(n\)-folds, and is
a positive finite unramified cover of degree \(d\) off an analytic
hypersurface \(T\), with every component of \(Z\) meeting this
complement. Let \(S\subset T\), and let \(Q\) be a finite
projective \(\mathcal O_X(*S)\)-module with integrable connection.
Its ordinary meromorphic pullback \(g^*Q\), localized along
\(D_0=(g^{-1}S)_{\mathrm{red}}\), is its unshifted finite
operator inverse image \(g^+Q\): \(Q\) is \(\mathcal O_X\)-flat.
There are natural finite-operator derived morphisms
\[
\begin{gathered}
u_Q:Q\longrightarrow g_+g^+Q,\\
\operatorname{tr}_Q:g_+g^+Q\longrightarrow Q,\\
\operatorname{tr}_Q u_Q=d\,\operatorname{id}_Q .
\end{gathered}
\tag{FB.12}
\]
No concentration assertion about \(g_+g^+Q\) is made here.

We construct the maps with their density convention.
For the right module \(Q^r=\Omega_X^n\otimes Q\), a flat right
Spencer resolution, with degree \(p-n\), is
\[
\begin{gathered}
C_X^p=\Omega_X^p\otimes_{\mathcal O_X}Q\\
\otimes_{\mathcal O_X}\mathcal D_X,\\
0\le p\le n .
\end{gathered}
\tag{FB.13}
\]
Its differential on an induced tensor is
\[
\begin{gathered}
\delta_X(\alpha\otimes P)\\
=d_\nabla\alpha\otimes P\\
{}+\sum_j dx_j\wedge\alpha\otimes\partial_{x_j}P.
\end{gathered}
\tag{FB.14}
\]
The right operator acts on the last factor. The displayed
\(\partial_{x_j}P\) is multiplication in \(\mathcal D_X\).
The coefficient relation is respected because
\(\partial_j(fP)=f\partial_jP+(\partial_jf)P\).
Integrability gives \(\delta_X^2=0\).
The augmentation in top degree is the right action on
\(\Omega_X^n\otimes Q\). For example in dimension one
\[
\begin{gathered}
(dx\otimes m)\partial_x\\
=-dx\otimes\nabla_{\partial_x}m .
\end{gathered}
\tag{FB.15}
\]
the two terms in FB.14 cancel under the augmentation.
This fixes the plus sign in FB.14.
PBW symbols, with the exterior-degree shifts, give the
polynomial-variable Koszul resolution. A leading-order
preimage lowers a finite operator order and terminates.
Since \(Q\) is coefficient-flat, the induced terms are
right \(\mathcal D_X\)-flat. This proves the stated resolution.

Tensoring the analogous right source resolution with
\(\mathcal D_{Z\to X}\) gives the complex computing the
right direct image, before \(Rg_*\), with terms
\[
\begin{gathered}
C_g^p=\Omega_Z^p\otimes g^*Q\\
\otimes_{g^{-1}\mathcal O_X}g^{-1}\mathcal D_X .
\end{gathered}
\tag{FB.16}
\]
and differential
\[
\begin{gathered}
\delta_g(\beta\otimes P)\\
=d_{g^*\nabla}\beta\otimes P\\
{}+\sum_j dg_j\wedge\beta\otimes\partial_{x_j}P .
\end{gathered}
\tag{FB.17}
\]
There is no inverse Jacobian in this complex.
Pullback of forms defines a right \(\mathcal D_X\)-linear
chain map \(C_X\to g_*C_g\), hence the first map of FB.12.
It is a chain map by the chain rule in every exterior degree,
including at critical points. In left-module transfer
notation its top-density representative is
\[
\begin{gathered}
m\longmapsto\eta\otimes1\otimes Jg^*m,\\
J=\det(dg),\\
\eta=dz\otimes g^*(dx)^{-1}.
\end{gathered}
\tag{FB.18}
\]
Thus this representative is holomorphic at critical points.
It is not the unweighted transfer generator.

Resolve the coefficient form factors of FB.16 by smooth
Dolbeault forms, and those of FB.13 by currents, using FB.B
and algebraic localization. These source resolutions are
\(g_*\)-acyclic because \(g\) is proper and they are fine;
the same compact-fibre argument works with each finite
operator head and pole bound. A compact fibre needs only
finitely many coefficient charts, hence one common bound.
The target current resolution is also section-acyclic.

Proper push of currents is defined by
\[
\langle g_*\beta,\phi\rangle
       =\langle\beta,g^*\phi\rangle .
\tag{FB.19}
\]
Properness makes the test support compact.
Equal complex dimensions make it preserve bidegrees.
Integration by parts and pullback of smooth forms give
\[
\begin{gathered}
g_*\partial=\partial g_*,
\qquad g_*\bar\partial=\bar\partial g_*,\\
g_*(dg_j\wedge\beta)=dx_j\wedge g_*\beta .
\end{gathered}
\tag{FB.20}
\]
The coefficient projection formula is immediate in a
finite free target frame. It passes to a finite projective
summand by its idempotent. For denominators define
\[
g_*\!\left(\frac{\beta}{(g^*h)^k}\right)
                  =\frac{g_*\beta}{h^k}.
\tag{FB.21}
\]
The holomorphic projection formula makes this well-defined
under the localization relations and changes of equation.
It also proves invariance under meromorphic changes of
ambient frame and its idempotent.
On the last \(\mathcal D_X\) factor use the identity.
FB.20 now proves that current push is a chain map for
FB.14–FB.17, not just for their coefficient differentials.
Use actual \((p,q)\)-forms with their wedge order; the
Dolbeault differential on a \((p,0)\)-form times a
\((0,q)\)-form has its \((-1)^p\) sign.
The source and target shifts are both \([n]\).
There is consequently no additional relative-dimension sign.
This constructs \(\operatorname{tr}_Q\) in FB.12.

For a smooth form \(\omega\), change variables on each sheet
off \(T\) proves
\[
g_*g^*\omega=d\,\omega
\quad\text{as currents}.
\tag{FB.22}
\]
The real Jacobian of a nonsingular holomorphic map is
\(|J|^2>0\), so each sheet has sign plus.
The discarded analytic zero sets have real measure zero:
adapted preparation gives finitely many zeros on a generic
one-dimensional complex slice, and Fubini proves this
locally; the remaining lower-dimensional slice locus is
handled inductively. Source exceptional divisors have
the same property. Both sides of the smooth integration
identity therefore retain their full integral.
Apply this to every smooth coefficient form and every
finite operator head. FB.22 identifies trace after unit
with \(d\) times the current inclusion in every Spencer
degree. The inclusion is a quasi-isomorphism by FB.B.
This proves FB.12 in the derived operator category.
For \(n=0\) it is the diagonal and sum on a finite set.
For the identity map it is the identity in every degree.
Restrictions, connection maps, and independent parameters
commute with all the displayed maps.

This finite Spencer/current construction is also described,
for the untwisted trace, in Pierre Schapira's freely
accessible [lecture notes, §2.6](https://perso.imj-prg.fr/pierre-schapira/wp-content/uploads/schapira-pub/lectnotes/Dmod.pdf).
FB.B–FB.22 give the coefficient, localization, degree and
splitting arguments used here.

#### FB.D. Removing the added boundary from the actual module

Let \(L\) be a local system on the original \(U=X\setminus S\).
Apply FB.A. The divisor
\[
D_0=(g^{-1}S)_{\mathrm{red}}
\tag{FB.23}
\]
is a subunion of the SNC divisor \(D\), hence is SNC.
Let \(E_\Sigma\) be the normalized logarithmic extension
of \(g_U^{-1}L\) from \(Z\setminus D_0\), and put
\[
\begin{gathered}
M_0=E_\Sigma(*D_0),\\
M_T=M_0(*D).
\end{gathered}
\tag{FB.24}
\]
There is no pole at a component of \(D\) outside \(D_0\)
in \(M_0\): on that open set it is the original flat bundle.
The same logarithmic lattice is normalized along \(D\),
with residue zero at every added component. Its localization
is therefore the normalized source \(M_T\) used by TC.
The logarithmic generator \(E_\Sigma(D_0)\) generates \(M_0\)
over finite operators, by the full shifted-residue
factorial proof in
§5.73, SI.
It is one global coherent generator on the inverse image
of a smaller target polydisc.
Thus §§5.9–5.12 apply to give
\[
N_0=g_+M_0\in D^b_{\mathrm{hol}}(\mathcal D_X).
\tag{FB.25}
\]
This is exactly where the proper-coherence generator
hypothesis is supplied.

Localization of the actual transfer complex and DT.B give
\[
\begin{gathered}
N_0(*A)\simeq g_+M_T=N_T,\\
N_0(*S)=N_0 .
\end{gathered}
\tag{FB.26}
\]
Here \(N_T\) is concentrated in degree zero by DT.C–DT.D,
since \(J\) is invertible after inverting \(D\).
For clarity, localization in FB.26 is not the restriction
of holomorphic functions to the complement.
On a finite Spencer term, commute a target inverse
equation through its finite derivative head. The finite
commutator expansion raises the pole order finitely.
The inverse operation is the same expansion on the
source inverse equation. These give inverse localized
transfer complexes and commute with their differential.
The underlying coefficient localization is a filtered
colimit of finite pole stages. Proper compact-fibre
commutation DT.B identifies its derived push.
The same argument with \(S\) applies because \(M_0\) is
already localized there.

By
§5.76, TC.B–TC.C
the normalized finite-cover projector has an actual
coherent finite D-type image \(Q_T\) along \(T\), and
an actual surjection
\[
\epsilon:N_T\longrightarrow Q_T
\tag{FB.27}
\]
whose local system is \(L|_{X\setminus T}\).
Define the finite-operator module
\[
\begin{gathered}
Q_0=\operatorname{im}\bigl(\\
H^0N_0\longrightarrow N_T\\
\xrightarrow{\epsilon} Q_T\bigr).
\end{gathered}
\tag{FB.28}
\]
This is an image of a map between coherent holonomic
modules, so is coherent holonomic. It is \(S\)-localized:
the source and target of the map are \(S\)-localized,
and the inverse multiplication by an equation of \(S\)
commutes with the map. Away \(A\), it is \(Q_T\).
Since \(S,A\) have no common component, it is generically
regular transverse to every component of \(S\).

We prove its crucial restriction statement across \(A\).
On \(U\), put \(F=L\otimes_{\mathbf C}\mathcal O_U\).
Then \(M_0|_{Z\setminus D_0}=g_U^+F\).
FB.C, applied on \(U\), supplies the normalized trace
\[
\begin{gathered}
g_{U,+}g_U^+F\longrightarrow F,\\
(\operatorname{tr}/d)u_F=1.
\end{gathered}
\tag{FB.29}
\]
After inverting \(A\), this map is FB.27, under the
canonical D-type identification of \(Q_T|_U\) with
\(F(*A)\). Both are the sheetwise normalized sum on
the unramified open set, and DT.E makes their actual
localized operator morphisms equal.

This equality determines the unlocalized map into
\(F(*A)\) as well. Indeed
\(\mathcal D_U(*A)\) is right \(\mathcal D_U\)-flat:
its tensor with any left module is just its exact
coefficient localization, since as a right module it
is \(\mathcal O_U(*A)\otimes_{\mathcal O_U}\mathcal D_U\).
Extension and restriction of scalars give derived
localization adjunction for any \(A\)-localized target.
Consequently the map \(N_0|_U\to F(*A)\) in FB.28 is
the map FB.29 followed by \(F\hookrightarrow F(*A)\).
On degree zero FB.29 is surjective, by its section.
Taking the image in FB.28 therefore gives
\[
Q_0|_U=F.
\tag{FB.30}
\]
All hypotheses of DT.E for the original \(S\) are now
verified. DT.A makes \(Q_0\) finite projective over
\(\mathcal O_X(*S)\). Its local system is precisely \(L\).
DT.E supplies the unique extension of every local-system
map and proves independence of the geometric choices.
Thus every local system on the prescribed complement
has an actual finite D-type realization there.
For an arbitrary smooth \(X\), make these constructions on
a coordinate cover. DT.E gives the unique isomorphism on
each overlap inducing the identity of \(L\).
Faithfulness makes the triple-overlap composites agree.
The sheaf of compatible local sections glues the modules;
operator coherence, holonomicity, localization and generic
regularity are local and are preserved.
Thus the same essential surjectivity holds globally.

#### FB.E. Pullback, de Rham and all auxiliary parameters

There is a canonical meromorphic connection isomorphism
\[
g^+Q_0\simeq M_0 .
\tag{FB.31}
\]
First invert all of \(D\). Then \(g^+Q_0(*D)=g^*Q_T\).
The evaluation map from \(g^*N_T\) to \(M_T\), restricted
to the normalized trace summand, is an isomorphism off
\(D\). It is a finite meromorphic matrix map on source
charts: use the actual degree-zero identification
\(N_T=g_*M_T\) of DT.C and evaluate sections.
Both sides are finite projective meromorphic modules.
They have coherent holomorphic lattices: put each
projective module in a finite free ambient module
by its idempotent and clear its finitely many
denominators. The image of the resulting finite
holomorphic matrix is such a lattice.
Clear denominators in the evaluation map as well.
Its lattice kernel and cokernel are coherent and
vanish off \(D\); the analytic Nullstellensatz
therefore kills them locally by a power of an
equation of \(D\). Inverting that equation proves
that the evaluation map is an isomorphism.
This argument does not assume a projective module
over a localized coefficient ring is free.

That isomorphism restricts on \(Z\setminus D_0\) to
the identity of the pulled-back flat bundle, even
across the extra components of \(D\).
It and its inverse have no poles at those components.
Here is the precise cancellation at crossings.
In an SNC chart write each matrix entry, in finite
free ambient modules for the two projective summands,
as
\[
\frac{a(z)}
 {z_1^{b_1}\cdots z_k^{b_k}},\qquad b_i\ge0.
\tag{FB.32}
\]
If \(z_i=0\) is an extra component, holomorphy at
generic points of that component makes the first
\(b_i\) Taylor coefficients of \(a\) in \(z_i\)
zero on a nonempty open subset of the remaining
punctured coordinate product. They are holomorphic
coefficients, hence identically zero by the identity
theorem. Thus \(z_i^{b_i}\) divides \(a\).
Repeat for every extra component, in both matrices.
Only \(D_0\) denominators remain. The inverse and
idempotent identities hold by density.
This proves FB.31 over \(\mathcal O_Z(*D_0)\).
It uses actual finite meromorphic entries, rather
than a formal or arbitrary-growth Laurent series.

FB.C now applies globally to \(Q_0\) and FB.31.
It gives derived maps
\[
\begin{gathered}
Q_0\xrightarrow{u}N_0\xrightarrow{r}Q_0,\\
ru=1,\qquad r=\operatorname{tr}/d .
\end{gathered}
\tag{FB.33}
\]
The composite \(N_0\to Q_0\hookrightarrow Q_T\)
is the original localized trace map: derived
localization adjunction and DT.E identify it after
inverting \(A\), just as in FB.D.
In particular the image in FB.28 is this actual
degree-zero trace image.

Write \(\operatorname{DR}_X(Q)=
(\Omega_X^\bullet\otimes Q)[n]\).
The finite Spencer associativity proof
§5.74, AD.B
and the full SNC de Rham proof give
\[
\begin{gathered}
\operatorname{DR}_X N_0
 \simeq Rg_*\operatorname{DR}_ZM_0\\
 \simeq Rj_*K,\\
K=Rg_{U,*}g_U^{-1}L[n].
\end{gathered}
\tag{FB.34}
\]
Composition of these derived sheaf images uses the
ordinary exact inverse-image/injective adjunction;
it requires no analytic proper base-change assertion.
Under the Spencer augmentations, FB.33 becomes
pullback of forms and the normalized current push.
This follows directly by tensoring FB.14 and FB.17
with \(\mathcal O_X\); the differential becomes the
connection differential, and FB.19 remains current
push. Thus the maps on \(U\) are
\[
\begin{gathered}
u_U:L[n]\to K,\\
r_U:K\to L[n],\\
r_Uu_U=1 .
\end{gathered}
\tag{FB.35}
\]

Let \(c:\operatorname{DR}_XQ_0\to Rj_*L[n]\) be
the canonical open-adjunction map.
Maps into \(Rj_*\) are determined by their restriction
to \(U\). Equations FB.34–FB.35 therefore give
\[
\begin{gathered}
\operatorname{DR}(u)=Rj_*u_U\circ c,\\
c\circ\operatorname{DR}(r)=Rj_*r_U .
\end{gathered}
\tag{FB.36}
\]
The map \(\operatorname{DR}(r)\circ Rj_*u_U\) is
a two-sided inverse of \(c\), by FB.33 and FB.35.
Consequently
\[
\operatorname{DR}_XQ_0
       \xrightarrow{\;\sim\;}Rj_*L[n]
\tag{FB.37}
\]
is the canonical comparison, including at singular
points of the original hypersurface.
This proof does not replace \(N_0\) by \(H^0N_0\).

Let \(A_{\mathrm{par}}\) be any auxiliary polydisc
and use the product geometry
\(\operatorname{id}_{A_{\mathrm{par}}}\times g\).
It is proper and locally projective by the
compact-projection and pulled-back projective
embedding argument in PD.D.
Repeat FB.D for the horizontal local system
pulled from \(L\).
The resulting finite D-type module is canonically
\(Q_0^{[A_{\mathrm{par}}]}\), by DT.E:
the finite parameter extension is coherent
holonomic, localized at \(A_{\mathrm{par}}\times S\),
and has the same generic transverse logarithmic
models, as proved in PD.23.
This is an identification of actual modules, not
a claimed tensor base change of analytic images.

Use only source-variable forms in FB.13–FB.22.
The parameter version of FB.B makes the trace and
unit commute with every auxiliary derivative.
AD.B's transfer associativity works with the same
relative Spencer complex: in the full product transfer,
the identity parameter coordinates supply their ordinary
Spencer Koszul factors. Their symbols are the independent
parameter derivative variables; the finite leading-order
Koszul reduction augments these factors to the identity
before the relative forms are taken.
There is no parameter differential or parameter shift
left in this augmentation. PD.C's full relative
SNC calculation supplies its source comparison.
The inverse argument FB.34–FB.36 is unchanged,
with coefficients \(p_A^{-1}\mathcal O_A\) and
without a parameter shift. Hence
\[
\begin{gathered}
\operatorname{DR}^{0}_{X/A}
      (Q_0^{[A]})\\
\xrightarrow{\;\sim\;}
Rj_{A,*}(p_A^{-1}\mathcal O_A
                    \otimes_{\mathbf C}L).
\end{gathered}
\tag{FB.38}
\]
This also commutes with actual infinite operators
in auxiliary variables. Fix a compact product
and choose a larger Cauchy product first.
All source homotopies and current pushes above
are independent of the auxiliary variable and
continuous on finitely many distribution
seminorms on the proper source compact.
Cauchy bounds for any fixed additional finite
derivative multiply the coefficient bounds by
its fixed factorial. Choose the defining
infinite-operator epsilon below that margin.
The remaining multi-index sums are products of
geometric series, exactly as in PD.D.
They converge normally in those seminorms and
commute with the maps.

The proof of PD.E uses its geometric hypothesis
only to obtain the canonical all-parameter
comparison PD.22. Replace that input by FB.38;
its diagonal support, purity, orientation and
Cauchy-unit proof PD.28–PD.34a then applies
with \(F_A=\mathcal O_{A\times U}\otimes_{\mathbf C}p_U^{-1}L\), to give
\[
\begin{gathered}
B_X\otimes_{\mathcal D_X}Q_0\\
\xrightarrow{\;\sim\;}
j_*(L\otimes_{\mathbf C}\mathcal O_U),\\
B_{A\times X}\otimes_{\mathcal D_{A\times X}}
       Q_0^{[A]}\\
\xrightarrow{\;\sim\;}j_{A,*}F_A .
\end{gathered}
\tag{FB.39}
\]
The maps are actual infinite-operator action;
no growth condition is imposed on the right.
The full naturality and two operator actions,
the ordinary inverse, and finite tensor
kernel witnesses are those proved in
PD.33–PD.36. Thus FB.39 is an ordinary tensor
isomorphism, not a definition of its left side.

#### FB.F. The special tensor for the original monodromy group

Take exactly the Weierstrass situation and exceptional
set \(S_0\) of
§5.70, PM.
Thus \(P\) has degree \(m\), the original \(H_0\)
contains its discriminant, and
\[
\begin{gathered}
S_0=\{P=0\}\cup A_0,\\
A_0=D_R\times H_0,\\
\pi=\pi_1(X\setminus S_0).
\end{gathered}
\tag{FB.40}
\]
For every cofinite left ideal \(\mathfrak a\)
of \(\mathbf C\pi\), set
\[
\begin{gathered}
V_{\mathfrak a}=\mathbf C\pi/\mathfrak a,\\
q=\dim_{\mathbf C}V_{\mathfrak a}.
\end{gathered}
\tag{FB.41}
\]
Use its dual flat bundle with PD.F's convention:
for a function \(f\),
\(\lambda_f(\gamma v_0)=T_\gamma f\) and
\(T_\gamma\lambda=\lambda\rho(\gamma)\).
FB.D realizes it by an actual finite
D-type module \(Q_{\mathfrak a}\) along this
exact \(S_0\). FB.39 and its auxiliary version
identify \(B\otimes Q_{\mathfrak a}\) with
the full function module, naturally when
\(\mathfrak b\subset\mathfrak a\).
The geometry used to prove existence has
not changed \(\pi\), \(S_0\), or either ideal.

PM constructs the full primitive representation
\(W\), including the period and initial-value
subspaces, with rank at most \(q(m+1)\).
Its exact finite-annihilator proof gives
\[
\begin{gathered}
\mathfrak b=\operatorname{Ann}
          (V_{\mathfrak a}\oplus W)
          \subset\mathfrak a,\\
\mathfrak b\text{ two-sided},\\
\dim\mathbf C\pi/\mathfrak b
            \le q^2(m+2)^2 .
\end{gathered}
\tag{FB.42}
\]
For \(q=0\) both representations are zero, so
\(\mathfrak b=\mathfrak a=\mathbf C\pi\) and
the tensor assertion is the zero assertion.
All hypotheses of
§5.71, ST.20
are therefore satisfied on the original
complement. ST.16–ST.19 and PM's actual
primitive give, writing \(N_{\mathfrak a}=B\otimes Q_{\mathfrak a}\),
\[
\begin{gathered}
\operatorname{im}\bigl(
\mathcal T\otimes_B N_{\mathfrak a}
\longrightarrow\\
\mathcal T\otimes_B N_{\mathfrak b}
\bigr)\\
\subset\operatorname{im}(\text{unit}).
\end{gathered}
\tag{FB.43}
\]
This includes every positive operator head,
the spatial kernels, arbitrary auxiliary
polydiscs, and the actual ordinary tensor
balancing witnesses. The exact domains,
special operator algebra \(\mathcal T\),
and unit in FB.43 are those of ST.1–ST.4.
Their finite relation proof is part of
ST.16–ST.19, not a replacement by a
function-valued image condition.

The original special-tensor statement is
Kashiwara–Kawai, §III.4, Proposition 3.4.9,
in their freely accessible
[author text](https://www.kurims.kyoto-u.ac.jp/~kenkyubu/kashiwara/HolIII.pdf).
FB.A–FB.42 supply the actual module input
to the full PM/ST proof for its fixed
exceptional hypersurface.
This conclusion concerns D-type modules
and the special tensor. It does not assert
the arbitrary holonomic-module faithful
embedding, nor the full proper
regular-holonomic Riemann–Hilbert theorem.

#### FB.G. Ramification and apparent poles

For \(t=w^e\), \(e\ge1\), write the trace
of holomorphic one-forms over a punctured
disc as the sum over its \(e\) inverse
branches. The power series calculation is
\[
\begin{gathered}
\operatorname{Tr}(a(w)\,dw)\\
=\sum_{\ell\ge0}a_{e\ell+e-1}t^\ell\,dt,\\
a(w)=\sum_{k\ge0}a_kw^k .
\end{gathered}
\tag{FB.44}
\]
The series converges on every smaller target
disc by the source Cauchy bounds.
It extends holomorphically through zero.
For Laurent polynomials the two traces are
\[
\begin{gathered}
T_0(w^{e\ell})=e\,t^\ell,\\
T_1(w^{e\ell+e-1})=t^\ell,\\
\ell\in\mathbf Z .
\end{gathered}
\tag{FB.45}
\]
Both traces vanish on every other integer monomial exponent.
Direct differentiation gives, including
negative powers when they are permitted,
\[
\begin{gathered}
T_1(a')=(T_0a)',\\
T_1(ew^{e-1}a)=T_0a,\\
T_1(ew^{e-1})=e .
\end{gathered}
\tag{FB.46}
\]
These are respectively the connection and
operator-head parts of the right Spencer
chain-map check, and the density-unit
normalization.
If \(S=\varnothing\) and the branch value
is only an added boundary, FB.44 gives
holomorphic target coefficients. A
meromorphic coefficient \(1/t\) is not
created in its normalized invariant
image. If \(S=\{0\}\), finite negative
source exponents give exactly finite
target poles. This distinguishes the
two roles of a boundary.

An independent parameter with
noncommuting connection matrices can be
seen in the determinant-one gauge
\[
\begin{gathered}
G(t,a)\\
=(I+tE_{12})(I+aE_{21}),\\
A=G^{-1}\partial_tG,\\
C=G^{-1}\partial_aG .
\end{gathered}
\tag{FB.47}
\]
Differentiating \(G^{-1}G=I\) gives
\[
\partial_a A-\partial_t C+[C,A]=0.
\tag{FB.48}
\]
The matrices need not commute.
After \(t=w^e\), the source matrices
are \(ew^{e-1}A(w^e,a)\) and \(C(w^e,a)\).
The projection formula and FB.46 prove
both chain-map identities with this
parameter. No scalar-residue
commutativity assumption is involved.

Finally the repeated cusp equation
\[
h=e^t(t^2-x^3)^2
\tag{FB.49}
\]
has \(Q=t^2-x^3\) and \(\Delta=4x^3\).
In the actual chart
\(x=u^2,\ t=u^3(1+v)\) of QG.F,
\[
\begin{gathered}
g^*h=e^{u^3(1+v)}\\
\cdot\,u^{12}v^2(v+2)^2 .
\end{gathered}
\tag{FB.50}
\]
Its support is the inverse support of
\(Q\); its multiplicities are those of
the original \(h\). FB.5 removes no
special-fibre support and FB.50 retains
the full pullback equation.

![Prescribed and added divisors, the coherent trace image, and the Jacobian-weighted ramified trace.](assets/analytic-fixed-boundary.png)

The map and module panels show FB.8,
FB.23–FB.30 and FB.33. They are
commutative-diagram schematics without
a metric scale. The ramified disc panel
shows the exact exponent selection and
normalization FB.44–FB.46; it does not
represent a higher-dimensional
exceptional fibre.

Editable SVG,
exact figure data,
and reproducible source.


### 5.80. Complex propagation through a cone-flat change of support

We use ordinary holomorphic functions, with unrestricted growth at
their open boundary. Every operator has an actual convergent
coefficient representative. Free-module matrices and their
contractions are finite; tensor products are algebraic.

The canonical kernel ring and its genuine module-derived support
objects, including finite common representatives, are proved in
§5.43, DL.1–DL.8 and DLG.1–DLG.7.
Its complete weighted Dolbeault argument is DLH.1–DLH.4.
The actual finite symbol map and its full canonical multiplication
are §§5.38–5.42.
In particular a bounded exact complex of finite free actual
operator modules has a finite matrix contraction, and DL.26
transfers that contraction to internal Hom with the supported
module object. An action in the endomorphism ring of a derived
vector complex alone is insufficient for that assertion.

Write \(X=\mathbf C^N\), viewed also as a real vector space.
For a proper closed convex cone \(G\), the \(G\)-opens are the
ordinary opens \(V\) satisfying \(V+G\subset V\).
The identity from the ordinary topology to this topology is
\(q_G\). A coefficient domain is round if every order interval
with its endpoints in the domain remains in the domain.
The cone is proper when \(G\cap(-G)=\{0\}\).

For real covectors use the negative-polar convention
\[
\begin{gathered}K^-=\{\eta:\eta(w)\le0\\
\text{for every }w\in K\}.\end{gathered}
\tag{CP.0}
\]
The real covector corresponding to a complex covector
\(p\) is \(w\mapsto\operatorname{Re}\langle p,w\rangle\).
An actual-operator complex below includes its finite
canonical-kernel representatives. Its actual finite
microlocal matrices are defined on the strict negative
polar of its carrier, with the same finite differential
identities. Localization always refers to that actual
matrix complex. The canonical Hom uses the kernel
ring of the carrier in use; its comparison with actual
operators concerns the specified matrices and their
finite identities. No map from arbitrary canonical
kernels to finite symbols is required.

#### CP.A. Convex coefficient domains and linear caps

Every nonempty open convex subset \(V\) of \(\mathbf C^N\)
has a smooth strictly plurisubharmonic exhaustion.
Here is an explicit construction that also covers nonsmooth
convex cones.

For each integer \(m\ge2\), cover the compact set
\(\partial V\cap\{|x|\le m\}\) by finitely many supporting
affine functions
\[
\begin{gathered}h_{m,j}>0\text{ on }V,\\ |dh_{m,j}|=1,\\
1\le j\le n_m.\end{gathered}
\tag{CP.1}
\]
so that at every point of that boundary piece one of them
is less than \(2^{-2m}\).
To obtain them, a separating affine hyperplane at each
boundary point gives a function zero there and positive
inside \(V\); continuity and compactness give the finite
cover. Separation follows directly from the nearest-point
inequality for closed convex sets, proved below in CP.D,
by approaching a boundary point from the exterior.
Empty boundary pieces contribute zero.

Choose integers \(a_m\ge2\) with
\[
\begin{gathered}(m-1)a_m\\
\ge2m+\log_2\max(1,n_m).\end{gathered}
\tag{CP.2}
\]
and set
\[
\begin{gathered}f_m(x)=\\
\frac{2^{-m}}{n_m}\sum_{j=1}^{n_m}\left(\frac{2^{-m}}{h_{m,j}(x)}\right)^{a_m},\\
\psi(x)=|x|^2+\sum_{m\ge2}f_m(x).\end{gathered}
\tag{CP.3}
\]
On a compact subset of \(V\), each normalized supporting
function is bounded below by the distance of that compact
to the complement of \(V\). Thus for every fixed derivative
order \(r\), the \(m\)-th derivative sum is bounded by a
constant times
\[
2^{-m}a_m^r\delta^{-r}
              (2^{-m}/\delta)^{a_m}.
\tag{CP.4}
\]
This is summable: eventually the parenthesized ratio is
at most \(1/2\), and \(a^r2^{-a}\) is bounded for integral
\(a\ge2\). Consequently CP.3 and all its derivatives
converge locally normally.
Each summand is real convex, since the second derivative
of a positive affine function to a negative power is
positive semidefinite. Its Levi form is therefore
nonnegative. The quadratic makes \(\psi\) strictly
plurisubharmonic.

At a finite boundary point, choose \(m\) arbitrarily large
whose boundary piece contains that point. In a small
neighbourhood the selected function is less than
\(2^{1-2m}\); CP.2 makes its contribution at least \(2^m\).
Thus \(\psi\) tends to infinity at the boundary.
It tends to infinity at infinity because all \(f_m\)
are nonnegative. Its sublevels are compact in \(V\).
If \(V\) is the whole space use only the quadratic.
This proves the assertion, including unbounded domains.

DLH.1–DLH.4 now gives the full Dolbeault vanishing on
every such \(V\), and on its intersection with an open
affine halfspace. All hypotheses of that proof are
verified by CP.3; its unrestricted-input weight is
allowed to depend on the whole given closed form.

Let \(G\) have interior and choose \(v\in\operatorname{int}G\).
The translated cones
\[
\begin{gathered}V_{x,b}=x-bv+\operatorname{int}G,\\ b>0.\end{gathered}
\tag{CP.5}
\]
are a cofinal \(G\)-neighbourhood basis at \(x\).
Indeed they contain \(x\). A small ordinary ball about
\(x\) inside a \(G\)-open contains the backward shift
\(x-bv\), whose forward cone is then contained in it.
These basis domains and their linear halfspace
intersections are convex. The preceding vanishing proves
\(Rq_{G*}\mathcal O=q_{G*}\mathcal O\) on these bases,
exactly by the stalk calculation DL.17–DL.18.

For a proper nonzero cone with empty interior replace
CP.5 by \(B(x,b)+G\). These convex ordinary opens are
again a cofinal cone-open basis: start with an ordinary
ball in the given cone-open and then add \(G\).
Their halfspace intersections are convex too.
Each nonempty basis open meets every sufficiently
far future halfspace, by translating along a nonzero
vector of \(G\). Holomorphic restriction to that
nonempty intersection is injective by the identity
argument in DL.5, since the basis open is connected.
Thus the same support sequence is concentrated in
degree one. This verifies the analytic hypotheses
of DL.6 for every proper nonzero convex cone,
including a single ray.
The remaining geometric hypothesis also has a direct
form. A real linear function \(\ell\) strictly positive
on \(G\setminus\{0\}\) satisfies
\[
|w|\le C_G\ell(w)\qquad(w\in G).
\tag{CP.6}
\]
Its proof is the positive minimum of \(\ell\) on
\(G\cap S^{2N-1}\). Such an \(\ell\) exists as follows.
For every unit \(w\in G\), separate \(-w\) from
\(G\) by the nearest-point inequality. The separating
functional is nonnegative on \(G\) and positive on
\(w\). A finite cover of the unit cone and the sum
of these functionals give strict positivity there.
Consequently a linear future cap bounds all input
lengths, and gives precisely DL.14's proper intersection
and common collar. Kernel cup, finite complement cubes,
excision and normalized trace in DLG.1–DLG.7 are
intrinsic; they use only that bound, roundness, and
the analytic concentration just verified.
Their signs, unit and finite-module comparison remain
unchanged. This proves the actual module-derived
construction for these convex cones.

For \(G\subset H\), restriction to a common smaller
\(H\)-round coefficient domain and enlargement of the
kernel support give the canonical ring map
\[
E(G;D)\longrightarrow E(H;D').
\tag{CP.7}
\]
The two products agree by the same finite triple
complement cube and proper trace: the support
inclusions commute with cup, and the common round
domain makes both intermediate intersections proper.
The diagonal unit is the same class.
This supplies the change of carrier used below.
It asserts no surjectivity of the map from actual
finite symbols onto arbitrary canonical kernel classes.

#### CP.B. The actual action at a \(C^1\) boundary

Let \(\rho\) be real \(C^1\) near \(0\), with
\(\rho(0)=0\) and \(d\rho(0)\ne0\). Put
\[
\begin{gathered}Z=\{\rho\le0\},\\
p=-2\partial\rho(0),\\
\operatorname{Re}\langle p,w\rangle\\
=-d\rho(0)(w).\end{gathered}
\tag{CP.8}
\]
This is the outward complex covector for the open side
\(\{\rho>0\}\). In particular \(\rho=-\operatorname{Re}t\)
gives \(p=dt\).
Every group
\[
\mathcal H_Z^q(\mathcal O)_0
\tag{CP.9}
\]
has the canonical actual finite \(E_p\)-module action.
We give the common-domain construction.

For a finite family of actual symbols and identities at
\(p\), choose their common canonical carrier \(G\) and
coefficient domain by DL.7. In complex linear coordinates
with \(p=dt\), these carriers can be chosen among the
strict negative-real-time cones of DL.11.
Thus with \(\ell=d\rho(0)\), for some \(c>0\),
\[
\ell(w)\ge c|w|\qquad(w\in G).
\tag{CP.10}
\]
Write \(\rho=\ell+r\). On a smaller convex ball,
\(r\) is \(\epsilon\)-Lipschitz for a fixed
\(0<\epsilon<c/2\), by continuity of \(d\rho\).
Extend it to all of the real vector space by
\[
\begin{gathered}\widetilde r(y)=\\
\inf_{z\in B}\bigl(r(z)+\epsilon|y-z|\bigr).\end{gathered}
\tag{CP.11}
\]
The Lipschitz inequality makes this equal to \(r\)
on \(B\). The triangle inequality proves that it
is globally \(\epsilon\)-Lipschitz. The infimum is
finite because \(B\) is bounded and \(r\) is bounded
there. Set \(\widetilde\rho=\ell+\widetilde r\).
For \(\kappa=c-\epsilon>0\),
\[
\begin{gathered}\widetilde\rho(y+w)-\widetilde\rho(y)\\
\ge\kappa|w|\quad(w\in G).\end{gathered}
\tag{CP.12}
\]

For \(b>0\) small use the \(G\)-opens
\[
\begin{gathered}
\Omega_b=B(0,b)+\operatorname{int}G,\\
\Omega_{0,b}=\Omega_b\cap\{\widetilde\rho>0\},\\
S_b=\Omega_b\setminus\Omega_{0,b}.
\end{gathered}
\tag{CP.13}
\]
The second is invariant by CP.12.
If \(y=z+w\), \(|z|<b\), \(w\in G\), and \(y\in S_b\),
then
\[
0\ge\widetilde\rho(y)\ge-Lb+\kappa|w|.
\tag{CP.14}
\]
Here \(L\) is one Lipschitz bound for \(\widetilde\rho\).
It follows that \(|y|\le Cb\), with fixed \(C\).
Choose \(A>\|\ell\|C\).
The linear future halfspace \(\omega_b=\{\ell>Ab\}\)
satisfies
\[
\Omega_b\cap\omega_b\subset\Omega_{0,b}.
\tag{CP.15}
\]
Moreover the closure of \(\Omega_b\setminus\omega_b\)
lies in a ball of radius \(C'b\), by CP.10.
For small \(b\) it lies strictly inside the fixed
coefficient domain. CP.A and DL.6 therefore give
the genuine \(E(G;D)\)-module-derived support object
for \(S_b\), with its ordinary cohomology comparison.

Choose \(C''>C'\) and
\(W_b=\Omega_b\cap B(0,C''b)\).
It contains \(B(0,b)\) and shrinks to zero.
All of \(S_b\) lies in \(W_b\), and
\(\widetilde\rho=\rho\) there for small \(b\).
Ordinary excision consequently gives
\[
\begin{gathered}R\Gamma_{S_b}(\Omega_b;\mathcal O)\\
\simeq R\Gamma_{Z\cap W_b}(W_b;\mathcal O).\end{gathered}
\tag{CP.16}
\]
The \(W_b\) are cofinal ordinary neighbourhoods.
Their filtered cohomology limit is exactly CP.9.
Finite products, sums, relations and coordinate changes
share one carrier and coefficient domain by DL.7.
The support comparisons in CP.16 are the same canonical
cup, excision and trace comparisons, so these actions
agree at a common refinement. They define the asserted
actual \(E_p\)-action and its unit.
There is no distributional product with a pole and
no fixed-neighbourhood action of every stalk germ.

Let \(P^\bullet\) be a bounded finite-free actual
operator complex with a compatible kernel representative,
and suppose its finite microlocal localization at \(p\)
is exact. It has a finite \(E_p\)-linear contraction
\(h\): split the last surjection, then descend through
the resulting projective kernels. After a common
finite refinement its identities act on CP.16.
The vertical-cohomology spectral complex for the finite
Hom double complex has rows
\[
\operatorname{Hom}(P^\bullet,
                   \mathcal H_Z^q(\mathcal O)_0).
\tag{CP.17}
\]
Each is contracted by
\(\phi\mapsto(-1)^{\deg\phi}\phi h\), with the exact
total signs of DL.26.
There are finitely many \(P\)-columns, so row
elimination in each total degree terminates.
Thus the local supported Hom complex is acyclic.
No flatness of the cohomology group as an \(E_p\)-module
is required.

#### CP.C. Compact fronts and both limits

We record the sheaf argument with its topology.
Let \(\Omega_s\), \(0\le s\le1\), be increasing
\(G\)-opens whose changing part has compact closure
in the coefficient domain. Suppose
\[
\begin{gathered}
\Omega_s=\bigcup_{r<s}\Omega_r\quad(0<s\le1),\\
\bigcap_{r>s}\Omega_r
          \subset\overline{\Omega_s}\quad(0\le s<1).
\end{gathered}
\tag{CP.18}
\]
Assume they are locally invariant under a common
open cone containing \(G\setminus\{0\}\), uniformly
near that compact carrier.
Set
\[
K_s=\bigcap_{r>s}
                 \overline{\Omega_r\setminus\Omega_s}.
\tag{CP.19}
\]
These are compact subsets of
\(\overline{\Omega_s}\setminus\Omega_s\).
To verify the inclusion, choose a small thick
constant cone \(H\) at a proposed front point,
with \(G\setminus\{0\}\subset\operatorname{int}H\).
Local invariance and membership in
\(\overline{\Omega_r}\) put its small strict
forward cone in every \(\Omega_r\), \(r>s\).
CP.18 puts that cone in \(\overline{\Omega_s}\).
An invariant open is the interior of its closure:
a small backward shift from an interior point
of the closure, followed by a strict forward
shift, proves this directly.
Taking the forward shifts to zero proves the
claimed inclusion.

At \(x\in K_s\), every ordinary neighbourhood \(U\)
contains a neighbourhood of \(x\) in
\((X\setminus\Omega_s)_G\).
Here is the useful quantitative observation.
The unit directions of \(G\) are compactly
inside \(\operatorname{int}H\).
Choose a short length \(\delta\) so that
\(x+\delta(G\cap S^{2N-1})\) is in the local
strict forward cone and in \(U\).
A sufficiently small translation \(V\) preserves
all these endpoints in \(\Omega_s\).
Every longer \(G\)-ray from \(V\) is then in
\(\Omega_s\); every shorter ray is in \(U\).
Consequently
\[
(V+G)\setminus\Omega_s\subset U.
\tag{CP.20}
\]
This proves both the ordinary induced topology
on \(K_s\) and relative separation of two distinct
front points. For \(G=\{0\}\) it is just the
ordinary topology.

Write \(Y_r=\Omega_r\setminus\Omega_s\), viewed
as open in \(X\setminus\Omega_s\), and let \(j_r\)
be that inclusion. For a sheaf \(L\) on the
complement put
\[
T_sL=\underset{r>s}{\operatorname{colim}}\,
                        j_{r*}j_r^{-1}L.
\tag{CP.21}
\]
Its derived functor is computed on an injective
resolution. Restriction to an open preserves
injectives because extension by zero is exact;
open direct image preserves injectives because
inverse image is exact.

We need compact neighbourhood continuity only on
\(K_s\), which has its ordinary compact Hausdorff
topology. A section over a closed subset of
\(K_s\) is represented on one ambient neighbourhood:
choose finitely many representatives, shrink their
closed pieces in the compact set, and shrink the
ambient neighbourhoods until their finitely many
overlap identities hold. CP.20 allows these
shrinking neighbourhoods in the cone topology
as well. This is the finite compact gluing
argument of DT.B.
For an ambient injective, its flabbiness extends
that represented section to the whole ambient
space. Its restriction to \(K_s\) is therefore soft.
Filtered colimits of these restrictions remain
soft: represent the compact section at one common
index, extend there, and pass to the colimit.
The compact soft acyclicity proof C1–C4 used in
DT.B
applies on this compact Hausdorff set.
It is not being applied on the whole cone space.

Every neighbourhood of \(K_s\) contains \(Y_r\)
for a sufficiently close \(r>s\). Otherwise the
nested compact sets in CP.19 have a point outside
that neighbourhood. Sections on the compact
front commute with the filtered colimit by the
same finite gluing argument. An injective
resolution now gives, in every degree,
\[
\begin{gathered}\underset{r>s}{\operatorname{colim}}H^k(Y_r;L)\\
\simeq H^k(K_s;RT_sL),\\
(RT_sL)_x\\
\simeq\underset{r>s,\ U\ni x}{\operatorname{colim}}R\Gamma(U\cap Y_r;L).\end{gathered}
\tag{CP.22}
\]
For example, the first identity follows by taking
sections of the injective complex on \(Y_r\),
then on neighbourhoods of \(K_s\), then on
\(K_s\). The restrictions there are soft, and
all filtered colimits are exact.

Let \(F\) be a bounded-below complex on the cone
space, zero on \(\Omega_0\). Apply CP.22 to
the extraordinary restriction of \(F\) to
the closed complement of \(\Omega_s\).
If all the local shell limits in CP.22 vanish,
the support triangle gives
\[
\begin{gathered}\underset{r>s}{\operatorname{colim}}H^k(\Omega_r;F)\\
\xrightarrow{\;\sim\;}H^k(\Omega_s;F).\end{gathered}
\tag{CP.23}
\]

The other limit has a separate proof. For an
increasing countable open cover \(U_j\) of \(U\),
resolve \(F\) by injectives \(I^\bullet\).
The complex sequence
\[
\begin{gathered}0\longrightarrow\Gamma(U;I^\bullet)\\
\longrightarrow\prod_j\Gamma(U_j;I^\bullet)\\
\xrightarrow{\,1-\mathrm{shift}\,}\prod_j\Gamma(U_j;I^\bullet)\\
\longrightarrow0.\end{gathered}
\tag{CP.24}
\]
is exact. The kernel equality is ordinary sheaf
gluing. Surjectivity in each degree follows
recursively from flabby restriction: after
choosing one component, lift its prescribed
difference to the next component.
The long exact sequence gives
\[
\begin{gathered}0\to\lim\nolimits^1 H^{k-1}(U_j;F)\\
\to H^k(U;F)\\
\to\lim H^k(U_j;F)\to0.\end{gathered}
\tag{CP.25}
\]
Only the case where the lower-degree restriction
maps are already isomorphisms will be used.
For that case \(1-\mathrm{shift}\) on their
product is surjective by the same recursion,
so its cokernel \(\lim^1\) is zero.
No general Mittag–Leffler premise is omitted.

Here is the elementary ordered-group argument.
For a family \(V_s\) with the right continuity
CP.23 and left inverse-limit continuity, first
fix \(w\in V_b\) whose restriction to \(V_a\)
is zero. The set of times where it is zero is
an initial interval. Left continuity includes
its supremum, and right direct-limit injectivity
extends it to the right unless the endpoint
has been reached. Thus every restriction is
injective.
Regard all \(V_s\) as nested subgroups of \(V_a\).
Left continuity takes intersections at limit
times; right continuity says every element at
a time occurs in a slightly later subgroup.
For a fixed element its membership times again
form an initial interval closed at its supremum
and extendible to the right. It therefore
belongs to \(V_b\). All restrictions are
isomorphisms.

Induct on cohomological degree starting below
the bound for \(F\). A countable sequence of
times increasing to \(s\), CP.18 and CP.25
give the left continuity in that degree,
because the lower-degree groups are already
constant. CP.23 and the ordered-group
argument make the degree constant as well.
Its initial value is zero on \(\Omega_0\).
Hence
\[
\begin{gathered}R\Gamma(\Omega_s;F)=0\\
(0\le s\le1).\end{gathered}
\tag{CP.26}
\]
This is the compact-front criterion.

#### CP.D. Rounded convex wedges

Let \(K\) be a proper closed convex cone with
interior, let \(\ell>0\) on \(K\setminus\{0\}\),
and choose \(v\in\operatorname{int}K\) with
\(\ell(v)=1\). Keep a nonzero operator carrier
\(G\) with
\(G\setminus\{0\}\subset\operatorname{int}K\).
It is \(G\) that defines the initial module
topology and the finite kernel representatives.
Put
\[
\begin{gathered}\Omega=\operatorname{int}K,\\
\omega=\Omega\cap\{\ell>1\}.\end{gathered}
\tag{CP.27}
\]
Assume the closed cap \(K\cap\{\ell\le1\}\)
is compactly inside the coefficient domain.
Let \(P^\bullet\) be a bounded finite-free
actual-operator complex over the \(G\)-kernel
ring, whose localization
is exact at all nonzero covectors in the
negative polar of \(K\), above this cap.
Then its supported Hom object for
\(\Omega\setminus\omega\) vanishes.

We give the rounded fronts used to prove it.
The nearest point \(\pi_K(x)\) exists and is
unique: bounded minimizing sequences have
convergent subsequences, and the midpoint
identity gives uniqueness.
Its variational inequality is
\[
\begin{gathered}\langle x-\pi_K(x),z-\pi_K(x)\rangle\\
\le0\quad(z\in K).\end{gathered}
\tag{CP.28}
\]
Applying it twice gives
\(|\pi_K(x)-\pi_K(y)|\le|x-y|\).
Expanding the two squared distances, once
using \(\pi_K(x)\) and once \(\pi_K(x+h)\),
therefore proves
\[
\begin{gathered}d\,\operatorname{dist}(x,K)^2\\
=2(x-\pi_K(x)).\end{gathered}
\tag{CP.29}
\]
The squared distance is \(C^1\); the distance
itself is \(C^1\) off \(K\).
For a cone, CP.28 also gives
\(\langle x-\pi_K(x),\pi_K(x)\rangle=0\)
and puts the distance gradient in the negative
polar of \(K\). Its pairing with the interior
vector \(v\) is strictly negative.

For \(\epsilon>0\), use the downward-closed
open subset of \(\mathbf R^2\)
\[
\begin{gathered}U_\epsilon=\\
\{a<0\}\cup\{b<\epsilon\}\\
\cup\left\{\begin{gathered}0\le a<\epsilon,\\
0\le b<2\epsilon,\\
(\epsilon-a)^2+(2\epsilon-b)^2\\
{}>\epsilon^2\end{gathered}\right\}.\end{gathered}
\tag{CP.30}
\]
Its boundary is the ray \(a=0,\ b\ge2\epsilon\),
the ray \(b=\epsilon,\ a\ge\epsilon\), and
the intervening quarter circle.
Their tangent lines agree at both joins,
so the boundary is \(C^1\).

Set
\[
\begin{gathered}\delta_s(x)=\\
\operatorname{dist}(x-(1-s)v,K),\\
\Omega_{\epsilon,s}=\{x\in\Omega:\\
(1-\ell(x),\delta_s(x))\in U_\epsilon\}.\end{gathered}
\tag{CP.31}
\]
These opens contain \(\omega\), increase with
\(s\), and are invariant by \(K\).
Indeed adding a vector of \(K\) decreases
both coordinates in CP.30.
For sufficiently negative \(s\) they equal
\(\omega\). The reason is that, on
\(\ell\le1\), a point within \(2\epsilon\)
of \((1-s)v+K\) would have
\(1-s\le1+2\epsilon\|\ell\|\).

Put \(r=\operatorname{dist}(v,\partial K)>0\).
The ball of radius \(r(1-s)\) about every
point of \((1-s)v+K\) lies in \(K\).
Thus
\[
2\epsilon<r(1-s)
\tag{CP.32}
\]
makes the whole changing front lie strictly
inside \(\Omega\). Its cap is compact.
On an active front \(\delta_s\ge\epsilon\),
so the distance is \(C^1\).
Its outward conormal is a nonzero
nonnegative combination of
\(-d\ell\) and \(d\delta_s\), including
the two joined rays.
Both are in the negative polar of \(K\)
and pair strictly negatively with \(v\).
The conormal consequently stays in that
negative polar and is never zero.

Distance to the translated cone decreases
strictly as \(s\) increases wherever it is
positive. For example CP.29 and the strictly
negative gradient pairing with \(v\)
prove this along each such interval.
Together with the downward property of
CP.30, this proves CP.18 for these fronts.
It also proves
\[
\begin{gathered}\overline{\Omega_{\epsilon,s_1}\setminus\Omega_{\epsilon,s_0}}\\
\subset\Omega_{\epsilon,s_2}\\
(s_0<s_1<s_2).\end{gathered}
\tag{CP.33}
\]
in the changing cap. At the quarter-circle
joins use continuity and the same strict
distance decrease; the fixed halfspace
part is already in every open.
In particular each compact front point
is in every later open.

Apply CP.C in the \(G\)-topology to the
supported Hom object for \(\Omega\setminus\omega\).
The opens are invariant under the open cone
\(\operatorname{int}K\), which contains the
entire nonzero operator carrier \(G\).
This strict inclusion is the uniformity
hypothesis needed in CP.20.
At an active front the local shell is
the ordinary \(C^1\) boundary support:
all later opens contain an ordinary
neighbourhood of that point, by CP.33.
The finite Hom and support identities of
DL.6 identify its local shell limit with
CP.17. Its outward covector lies in the
negative polar where the complex is
exact, so CP.B kills that limit.
The initial supported object is zero on
\(\omega\). CP.26 proves vanishing on
each \(\Omega_{\epsilon,s}\) satisfying
CP.32.

Finally choose \(c>4/r\), let
\(\epsilon_{j+1}=\epsilon_j/2\), and set
\(s_j=1-c\epsilon_j\).
They satisfy CP.32.
The displacement
\((c\epsilon_j-c\epsilon_{j+1})v\)
has an interior-cone radius
\(rc\epsilon_j/2>2\epsilon_j\).
It therefore absorbs the full
\(2\epsilon_j\)-tube of the earlier
translated cone into the later cone.
Consequently
\[
\begin{gathered}\Omega_{\epsilon_j,s_j}\subset\Omega_{\epsilon_{j+1},s_{j+1}},\\
\bigcup_j\Omega_{\epsilon_j,s_j}=\Omega.\end{gathered}
\tag{CP.34}
\]
Every interior point eventually lies
in the translated cone itself, proving
the union assertion.
All cohomology groups on these opens
are zero. CP.24–CP.25, now with zero
groups, give vanishing on \(\Omega\).

This is also sheaf vanishing for the
complex obtained by CP.7's change to the
\(K\)-kernel ring. For clarity, first
keep the \(G\)-module support object and
push it by the identity map of cone spaces
\(q:\Omega_G\to\Omega_K\). The ordinary
comparison and cone-change roof of DL.6
and DLG.7 identify this derived pushforward
with the \(K\)-support object, after finite
free Hom with the scalar-extended matrices.
They identify each matrix action by the
same kernel cup and trace; this uses the
full cube, not a surjectivity assertion
about finite symbols. Every translated cone
inside \(\Omega\), with the same linear
future halfspace, has a smaller cap
inside the given one, or has empty
cap. The identical proof applies.
Its cohomology is the just-proved ordinary
cohomology of the \(G\)-object on that open.
Those cones are a cofinal \(K\)-basis, so
the cohomology stalks of the supported
Hom object all vanish.

#### CP.E. A variable-cone shell

An ordinary open is flat for an open convex cone
field \(Q\) if it is locally invariant under every
fixed smaller closed cone whose nonzero directions
lie in \(Q\). A locally closed set is \(Q\)-flat
if locally it is a difference of two such opens.
Every segment below stays in the neighbourhood
in which this invariance is asserted.

Consider a family as in CP.C. Fix \(s<t\),
a front point \(x\in K_s\), and put
\[
Y=\Omega_t\setminus\Omega_s .
\tag{CP.35}
\]
Suppose \(Y\) is \(Q\)-flat near \(x\),
\(Q(x)\) lies in a strictly positive
linear halfspace, and
\(G\setminus\{0\}\subset Q(x)\).
Assume the localized finite complex is exact
at the base point \(x\) for every nonzero real
covector strictly negative on \(Q(x)\).
Then the local shell limit required in CP.22
is zero in every degree.

Choose proper closed thick cones
\[
\begin{gathered}G\setminus\{0\}\subset\operatorname{int}K_0,\\
K_0\setminus\{0\}\subset\operatorname{int}K_1,\\
K_1\setminus\{0\}\subset Q(x).\end{gathered}
\tag{CP.36}
\]
such that exactness holds above one neighbourhood
\(U\) at all nonzero covectors in \(K_0^-\).
Here is the compactness argument for this choice.
Exhaust \(Q(x)\) by increasing closed inner cones
containing the compact unit directions of \(G\)
in their interiors. Their unit negative polars
decrease to the unit negative polar of \(Q(x)\).
For an open full-dimensional cone, a nonzero
covector nonpositive on its closure is strictly
negative at every interior point: equality there
would allow a small perturbation on which it
is positive. The limiting unit covectors
therefore lie in the open exactness locus.
Nested compactness puts one whole inner-cone
polar in that locus. Enlarge that inner cone
slightly inside \(Q(x)\) to obtain \(K_1\).
A finite cover of the compact unit polar,
with finite actual contraction representatives,
gives one base neighbourhood \(U\).
Shrink it so that \(K_1\setminus\{0\}\subset Q(y)\)
throughout. There is also a fixed thick cone
inside both the original front-invariance field
and \(\operatorname{int}K_1\), containing all
nonzero directions of \(G\).

If \(x\notin\overline Y\), the shell is locally
empty. Otherwise write \(Y=V_1\setminus V_0\)
on a smaller \(K_1\)-round neighbourhood \(U\),
with \(V_0\subset V_1\) locally invariant.
We have
\[
\begin{gathered}x\in\overline{V_0},\\
(x+\operatorname{int}K_1)\cap U\subset V_0.\end{gathered}
\tag{CP.37}
\]
If \(x\in V_1\), approach it by points of
\(\Omega_s\); those in \(V_1\) are in \(V_0\).
If \(x\notin V_1\), its strict forward cone
for the shared smaller cone is in \(\Omega_s\),
by CP.C, and in \(V_1\), because
\(x\in\overline Y\subset\overline{V_1}\).
It is consequently in \(V_0\) and approaches
\(x\). A small perturbation of a strict
forward segment and local invariance of
\(V_0\) prove the second assertion of CP.37.

For \(y\) close to \(x\), put
\(V=y+\operatorname{int}K_0\).
The set \(Y\cap U\cap V\) lies in any
prescribed smaller ordinary neighbourhood
\(W\) of \(x\). Indeed choose a short length
\(\delta\). The compact set
\(x+\delta(K_0\cap S^{2N-1})\) is strictly
inside \(V_0\), by CP.36–CP.37. This remains
true after a small translation of \(x\).
Every longer ray, while in the round \(U\),
then stays in \(V_0\) and avoids \(Y\).
Shorter rays remain in \(W\). Choose
\(\overline W\Subset U\); the closure of
the intersection is then compactly inside \(U\).

Choose a linear functional positive on \(K_0\)
and a future halfspace \(\omega\) with
\[
\begin{gathered}B=V\setminus\omega,\\
\overline B\Subset U,\\
x\in\operatorname{int}B.\end{gathered}
\tag{CP.38}
\]
Take \(y\in x-\operatorname{int}K_0\)
close to \(x\), with the cut a small positive
distance beyond \(x\). Bound CP.6 makes the
whole cap small. Choose \(W\) inside the
interior of this cap, and then move \(y\)
closer while retaining the cut. It follows that
\[
\begin{gathered}C=Y\cap U\cap V=Y\cap B,\\
\overline C\Subset U.\end{gathered}
\tag{CP.39}
\]
All these caps lie in a common smaller round
coefficient domain for CP.7.

The set \(C\) is actually locally closed in
\(V_{K_0}\). This does not follow just from
ordinary compactness; complete the opens:
\[
\begin{gathered}A_i=(V_i\cap U\cap V)+K_0,\\ i=0,1.\end{gathered}
\tag{CP.40}
\]
They are ordinary open and \(K_0\)-invariant.
Their restrictions to \(U\cap V\) are exactly
the original ones, since a segment with both
endpoints in the round \(U\) stays there.
No difference remains outside \(U\). A
forward ray starting in \(V_1\setminus V_0\)
must leave that difference before leaving
\(U\), since \(\overline C\Subset U\).
It cannot leave \(V_1\) while in \(U\),
so it enters \(V_0\). Its remaining forward
ray lies in \(A_0\). Thus
\(A_1\setminus A_0=C\), as required.

Push the finite kernel matrices to the
\(K_0\)-carrier by CP.7. CP.D gives
\[
\begin{gathered}R\mathcal Hom\bigl(P^\bullet,\\
R\Gamma_B(q_{K_0*}\mathcal O|_V)\bigr)=0\\
\text{on }V_{K_0}.\end{gathered}
\tag{CP.41}
\]
Apply derived support in \(C\) to this zero
object. The exact support identity is
\[
\begin{gathered}R\Gamma_C R\Gamma_B=R\Gamma_C,\\ C\subset B.\end{gathered}
\tag{CP.42}
\]
For closed supports it is the literal
identity on injectives. For a difference
of opens it is the finite relative support
cube of DL.6 and DLG.3. Equivalently, the
constant sheaf extended by zero from \(C\),
tensored over \(\mathbf C\) with the one
extended by zero from \(B\), is that same
sheaf on \(C\); their Hom adjunction is
CP.42. Finite free Hom commutes with the
cube term by term. The ordinary comparison
of CP.A and DL.6 now gives
\[
\begin{gathered}R\operatorname{Hom}\bigl(P^\bullet,\\
R\Gamma_C(V;\mathcal O)\bigr)=0.\end{gathered}
\tag{CP.43}
\]
These comparisons are canonical kernel cup,
controlled excision and proper trace on a
common collar. Arbitrary restriction of a
proper source support is not being assumed.

Locally closed support depends only on an
open neighbourhood of that set. Extension
by zero of its constant sheaf and open
adjunction prove this excision; the relative
support cube gives the same map. Apply it
to the ordinary open \(U\cap V\cap\Omega_t\),
which contains \(C\). The set \(C\) is
closed there and is its complement of
\(\Omega_s\). Thus CP.43 is the precise
supported Hom for the local shell there.
As \(U\) shrinks and \(y\) approaches
\(x\) from behind, \(U\cap V\) is a
cofinal system of ordinary neighbourhoods
of \(x\). This proves the local shell limit
vanishing in every degree.

#### CP.F. Propagation with an actual ray carrier

Fix a nonzero proper closed convex carrier
\(G\), a round coefficient domain \(D\),
and a bounded finite-free compatible actual
operator complex \(P^\bullet\) over its
canonical kernel ring. Every point of
\(D\) has a sufficiently small round
neighbourhood \(U\) with the following property.

Let \(\Omega_0\subset\Omega_1\) be ordinary
opens, with
\[
\begin{gathered}Z=\Omega_1\setminus\Omega_0,\\ K=\overline Z\Subset U.\end{gathered}
\tag{CP.44}
\]
Suppose both opens are flat on \(U\) for
an open convex cone field \(R\) containing
\(G\setminus\{0\}\). Suppose there are
an open convex cone field \(Q\) and a real
\(C^1\) function \(\chi\) near \(K\), with
\[
\begin{gathered}G\setminus\{0\}\subset Q(x),\\
d\chi(x)(w)>0\\
(w\in Q(x)),\\
Z\text{ is }Q\text{-flat near }K.\end{gathered}
\tag{CP.45}
\]
Assume that at every \(x\in K\) the
localized complex is exact at each nonzero
covector strictly negative on \(Q(x)\).
Then
\[
\begin{gathered}R\operatorname{Hom}\bigl(P^\bullet,\\
R\Gamma_Z(\Omega_1;\mathcal O)\bigr)=0.\end{gathered}
\tag{CP.46}
\]
The acyclicity hypothesis is on the closed
carrier \(K\); it requires no unrelated
base points.

First justify the actual module object
for the locally flat pair. Choose a proper
thick cone \(H\) containing the nonzero
directions of \(G\) in its interior and
lying in both fields near \(K\).
It can be chosen so close to \(G\) that
all order diamonds between points of a
small thickening of \(K\) lie in \(U\).
Indeed use decreasing closed thick cones
approaching \(G\), inside one fixed proper
thick cone. Diamonds of a compact endpoint
set are closed and bounded by a positive
functional on that fixed cone. Their
intersection is the \(G\)-diamond set,
inside the \(G\)-round \(U\). Nested
compactness allows first a small endpoint
thickening and then a sufficiently close
\(H\). The interior of this diamond hull
is an \(H\)-round neighbourhood \(U_1\)
of \(K\), compactly inside \(U\): an
interval between two hull points remains
between a pair of the original endpoints.

The set \(Z\) is order-convex there.
For \(z,z'\in Z\) and
\(z\le_H y\le_H z'\), forward invariance
puts \(y\) in \(\Omega_1\); if it were
in \(\Omega_0\), so would be \(z'\).
Put
\[
\begin{gathered}A=Z+H,\\
B=A\setminus Z.\end{gathered}
\tag{CP.47}
\]
Both are global \(H\)-opens, with difference
exactly \(Z\). For openness of \(A\),
fix \(v\in G\setminus\{0\}\).
Every \(z\in Z\) has a small backward
shift \(z-\epsilon v\) in
\(\operatorname{int}Z\). It is in
\(\Omega_1\) by openness. A small ball
there cannot meet \(\Omega_0\): its strict
forward segments to \(z\) lie in \(R\)
and would force \(z\in\Omega_0\).
This interior point and its forward
translate give an ordinary neighbourhood
of every point of \(A\).
On \(U_1\), forward invariance gives
\(A\subset\Omega_1\), and hence
\(B=A\cap\Omega_0\). Outside \(K\) the
difference \(Z\) is absent. These observations
prove that \(B\) is open. If a forward
translate of a point of \(B\) were in
\(Z\), order-convexity between its
original point in \(Z\) and that endpoint
would put the intermediate point in
\(Z\), a contradiction. Thus \(B+H\subset B\).

We have \(A\subset U+H\). A linear
future halfspace beyond \(K\) makes its
complementary cap compactly inside \(D\)
when the original \(U\) is small enough.
This is uniform for \(H\) in the fixed
proper thick cone: CP.6 bounds the cap
by a constant times the diameter of \(U\).
CP.A and DL.6 give the actual module-derived
object for \(A\setminus B=Z\), including
the case of a single ray \(G\).
Excision on the common open neighbourhood
\(A\cap\Omega_1\) of \(Z\) identifies
its underlying supported cohomology with
the original one in CP.46.

Rescale \(\chi\) by a positive affine
map so that \(0<\chi<1\) on \(K\).
Let \(N\) be its open neighbourhood
of definition, containing \(K\).
Only its values on \(Z\) matter.
Use the superlevel sweep
\[
\begin{gathered}A_s=B\cup\\
\bigl(A\cap N\cap\{\chi>1-s\}\bigr),\\
0\le s\le1.\end{gathered}
\tag{CP.48}
\]
Its initial open is \(B\); its final
open is \(A\). It is \(H\)-invariant.
If the endpoints of a forward segment
are in \(Z\), order-convexity keeps
the segment there and \(\chi\) increases.
An endpoint in \(B\) is already in
every \(A_s\). The left union condition
CP.18 follows from the strict superlevel.
At a level boundary, small strict forward
shifts either increase \(\chi\) or enter
\(B\); hence the right intersection lies
in the closure of the current open.
The changing carrier is in \(K\) and
all opens have the uniform local invariance
cone \(\operatorname{int}H\).

Every shell is \(Q\)-flat. Locally write
\(Z=V_1\setminus V_0\); then
\[
\begin{gathered}
c_u=1-u,\\
A_t\setminus A_s\\
=Z\cap\{c_t<\chi\le c_s\}\\
=(V_1\cap\{\chi>c_t\})\\
\setminus(V_0\cup\{\chi>c_s\}).
\end{gathered}
\tag{CP.49}
\]
Superlevels are locally forward invariant
by CP.45; intersections and unions preserve
that property. CP.E applies at every compact
front point. Its exactness premise is the
assumption over \(K\), with an actual
neighbourhood for each compact polar as
proved there. All local shell limits
vanish. The supported object is zero on
\(B\), so CP.C gives CP.46 in every degree.

#### CP.G. Whole generic-position complexes and calibrations

The theorem applies to the actual bounded
generic-position complex for every coherent
holonomic germ. The precise inputs are
§5.31, L.4–L.6
and §5.49, ISF.3–ISF.4 and ISR.5.
Its polynomial matrices have one common
base neighbourhood. L.6 proves exactness
in positive degrees on the entire
\(\tau\ne0\) chart and identifies degree-zero
cohomology with the whole realized module.
Every normalized covector of its support
is bounded there, including all singular
branches. ISF.3–ISF.4 give the actual
special-kernel representatives, their
products and finite differential identities:
uniform finite spatial polynomial degree
is in that special class. Its carrier is
the negative real-time ray. The strict
negative polar of that ray has
\(\operatorname{Re}\tau>0\), hence \(\tau\ne0\).

Thus CP.46 holds for this whole complex
whenever the pair in CP.44–CP.45 has its
nonzero negative covectors disjoint from
the realized characteristic support over
the closed changing carrier. Positive-degree
exactness is L.6; degree-zero vanishing
is exactly the stated support condition.
No exactness over the polynomial coefficient
ring is required. Nor is an arbitrary
finite microlocal symbol asserted to be
special; ISF.10 gives the contrary example.

Here is an exact real section of the
rounded geometry. Write
\(X=-\operatorname{Re}t\),
\(Y=\operatorname{Im}t\), and take
\[
\begin{gathered}K=\{(X,Y):X\ge2|Y|\},\\
\ell(X,Y)=X,\quad v=(1,0),\\
r=1/\sqrt5,\quad c=16,\\
\epsilon\in\{1/20,1/40,1/80\},\\
1-s=16\epsilon.\end{gathered}
\tag{CP.50}
\]
The strict bound \(c>4/r\) follows by
squaring the positive numbers. For
\(q=(X-16\epsilon,Y)\), the squared distance
in CP.31 is zero if \(q\in K\), and
otherwise is exactly
\[
\begin{gathered}M(q)=\max\{0,\\
2q_X+q_Y,\ 2q_X-q_Y\},\\
|q|^2-M(q)^2/5.\end{gathered}
\tag{CP.51}
\]
The two boundary rays have unit directions
\((2,1)/\sqrt5\) and \((2,-1)/\sqrt5\).
Projection on a ray uses its positive scalar
projection; the nearer ray gives CP.51.
The quarter-circle joins are exact: its
normal at \((0,2\epsilon)\) is horizontal,
and its normal at \((\epsilon,\epsilon)\)
is vertical, matching the straight pieces.

![Rounded quadrant and nested noncharacteristic fronts](assets/complex-propagation.png)

**Figure CP.1.** Upper panel: the exact scaled boundary of
CP.30 has centre \((1,2)\), radius one and the two
matching normal directions. The shaded open is
the side of decreasing coordinates. Lower panel: the
real section CP.50 uses CP.51's squared distance;
the three sampled regions are nested in the order
proved in CP.34. The sloping cone boundary is
excluded. The dashed line is \(\ell=1\), the
boundary of the initial future open; the rounded
opens may include parts of this line. The colour regions
are a grid rendering of those formulas, not a
numerical proof. CP.B–CP.F prove propagation on
the full complex domain with its actual operators.
Free human sources: [Kashiwara–Schapira, §4](https://perso.imj-prg.fr/pierre-schapira/wp-content/uploads/schapira-pub/Microhyp.pdf)
and [Kashiwara–Kawai, IV.4.4](https://www.kurims.kyoto-u.ac.jp/~kenkyubu/kashiwara/HolIII.pdf).

The ordinary cohomology, finite actual
operators and closed-carrier quantifiers
are part of CP.46. Common finite-dimensional
determination of continued solutions,
separation of singular holonomic germs
and faithful D-type embedding require
additional arguments.

Free further reading: Kashiwara–Schapira,
[Micro-hyperbolic systems, §4](https://perso.imj-prg.fr/pierre-schapira/wp-content/uploads/schapira-pub/Microhyp.pdf),
and Kashiwara–Kawai,
[Holonomic systems III, IV.4.4](https://www.kurims.kyoto-u.ac.jp/~kenkyubu/kashiwara/HolIII.pdf).


### 5.81. Transverse local systems of actual holonomic solution complexes

Put \(X=\mathbf C_t\times\mathbf C_x^n\),
\(F(t,x)=x\), and write \(u=-\operatorname{Re}t\).
Put \(z=(\operatorname{Im}t,x)\) and \(|z|=|\operatorname{Im}t|+|x|\). The carrier and its thick approximations are
\[
\begin{gathered}G_0=\{u\ge0,\ z=0\},\\
G_b=\{u\ge b|z|\},\\ b>0.\end{gathered}
\tag{CS.1}
\]
The norm on \(\mathbf C^n\) is Euclidean.
For the real vector collecting imaginary time
and spatial coordinates use the sum norm in
CS.1. The transverse argument below assumes
\(n\ge1\); when \(n=0\), pair independence
still holds and local constancy on the
one-point base is immediate.
All opens called cone-open are also ordinary open.
The map from the ordinary topology to the
\(G_0\)-topology is \(q\).

Fix a bounded finite-free complex \(P^\bullet\)
over the canonical ring \(R=E(G_0;D)\), with
the compatible actual finite operator matrices
specified in §5.80, CP.B and CP.G.
Let \(A\) be its degree-zero actual support on
the chart \(\tau\ne0\). Assume its actual
complex has no cohomology in degrees
other than zero there, and that
\[
\begin{gathered}A\subset\{|\xi|\le|\tau|\},\\
S=\overline{\pi(A)}.\end{gathered}
\tag{CS.2}
\]
In particular its actual complex is exact
over a base point outside \(S\).
These hypotheses hold for the whole
generic-position complex of
§5.31, L.4–L.6:
choose the normalized-covector bound there
smaller than \(1/\sqrt n\) when \(n>0\).
Its special thin-ray kernel representatives
and their finite differential identities
are §5.49, ISF.3–ISF.4.
Localization below means these actual
finite matrices, with their canonical kernel
comparison; an arbitrary canonical kernel
is not assigned a finite symbol.

The distinguished base point is zero.
Choose the small round neighbourhood \(U\)
of CP.F and a positive number \(e\) with
\[
\begin{gathered}
\overline U\subset
 \{|\operatorname{Re}t|<e\},\\
\overline{(U+G_0)\cap\{u\le e\}}\Subset D.
\end{gathered}
\tag{CS.3}
\]
All changing carriers have compact closure
inside \(U\). The supported module-derived
objects and their ordinary-cohomology
comparison are the actual ones of CP.A,
DL.6 and DLG.7. Finite free Hom commutes
with their support cubes and comparisons.
Both conditions in CS.3 can be arranged:
take \(U\subset B(0,r_0)\),
\(e=2r_0\), and \(B(0,3r_0)\Subset D\).
The truncated ray cone has norm less
than \(\sqrt5r_0<3r_0\).
On the common ray-open completion
\((U+G_0)\cup\{u>e\}\), its auxiliary
cap is therefore compactly inside the
one coefficient domain. Intersecting
this completion with any source open
is ordinary excision around every
carrier in \(U\). The support functors
on one auxiliary module resolution
realize all the transition maps below.
No coefficient domain is changed with
the index of an inverse-limit family.
There is no growth condition on the
holomorphic functions at an open boundary.

#### CS.A. Arbitrary ray-open supports away from the projection

For \(G_0\)-opens \(\Omega_0\subset\Omega_1\),
put
\[
\begin{gathered}Z=\Omega_1\setminus\Omega_0,\\
K=\overline Z\Subset U,\\
\mathscr H_{\Omega_1,\Omega_0}\\
=R\mathcal Hom_R\bigl(P^\bullet,\\
R\Gamma_Z(q_*\mathcal O|_{\Omega_1})\bigr).\end{gathered}
\tag{CS.4}
\]
If \(K\cap S=\varnothing\), then
\(\mathscr H_{\Omega_1,\Omega_0}=0\).
Consequently its ordinary global supported
Hom vanishes in every degree.
The condition concerns the closed carrier
\(K\), including its boundary.

Let \(E=\{u>e\}\). Adding \(E\) to both
opens leaves \(Z\) unchanged, by CS.3.
Excision identifies the module objects:
use the common ordinary open containing
\(Z\) and the identical support cube.
Thus assume \(E\subset\Omega_0\).

The relative neighbourhoods relevant to
a point of \(Z\) have a basis represented by
\[
\begin{gathered}V=(B_\delta(y)+G_a)\cup E,\\
V\subset\Omega_1.\end{gathered}
\tag{CS.5}
\]
for sufficiently small \(\delta\) and large
\(a\). Indeed a small closed ball about
\(y\), followed by its \(G_0\)-rays up to
the future level \(u=e+1\), is compactly
in any given \(G_0\)-open neighbourhood.
The corresponding truncated \(G_a\)-cones
decrease to these rays as \(a\) increases;
compactness puts one of them in that
neighbourhood. All further points are in
\(E\). Adding \(E\) does not affect the
supported object, which is zero there.
This proves the required cofinality for
its cohomology stalks.

Fix such a \(V\). For \(b\ge a\) define
\[
\Omega_0(b)=
 \{y:y+G_b\subset\Omega_0\}.
\tag{CS.6}
\]
It is \(G_b\)-invariant and contains
\(E\). It is ordinary open: truncate the
forward cone at \(u=e+1\); this compact
part has a positive neighbourhood margin
inside \(\Omega_0\). A small translation
retains that margin, and the remaining
tail stays in \(E\).
The opens increase to \(\Omega_0\).
For any \(y\in\Omega_0\), its truncated
\(G_0\)-ray is compactly in \(\Omega_0\).
The decreasing compact truncated cones
put \(y+G_b\) there for sufficiently
large \(b\), and the rest is again in
\(E\).

The stronger closure statement is
\[
\bigcap_{b\ge a}
 \overline{V\setminus\Omega_0(b)}
=\overline{V\setminus\Omega_0}.
\tag{CS.7}
\]
All the sets on the left are compact:
they lie in the truncation \(u\le e\)
of CS.5, which is bounded.
One inclusion is immediate from
\(\Omega_0(b)\subset\Omega_0\).
For the other, let \(p\) be outside
the closure on the right. Choose an
ordinary neighbourhood \(N\) with
\(V\cap N\subset\Omega_0\), and shrink
to \(N'\) about \(p\).

Here is the uniform cone estimate.
Write a cone vector as \((s,w)\), where
\(s\) is its \(u\)-component and the
norm of \(w\) is the sum norm in CS.1. For
\(y\in N'\), a target point not in \(E\)
has \(s\le M\) for one fixed \(M\).
Choose a small \(d>0\) and then
\(b\) with \(aM/b<d\). Decompose
\[
\begin{gathered}(s,w)=(\min(s,d),w)\\
{}+(s-\min(s,d),0).\end{gathered}
\tag{CS.8}
\]
The first vector is in \(G_a\): for
\(s\le d\) use \(b\ge a\); for
\(s>d\) use \(a|w|\le aM/b<d\).
It is small enough that its translate
of \(N'\) lies in \(N\).
The second vector lies in \(G_0\).
Thus a point of \(V\cap N'\) first
moves to \(V\cap N\subset\Omega_0\),
then stays in \(\Omega_0\) along the
remaining ray. Targets in \(E\) need
no estimate. This proves
\(V\cap N'\subset\Omega_0(b)\), excluding
\(p\) from the left side of CS.7.

The compact sets in CS.7 therefore lie
in \(U\setminus S\) for all sufficiently
large \(b\). Apply CP.F to the two
\(G_b\)-opens \(V\) and
\(V\cap\Omega_0(b)\), using the original
operator carrier \(G_0\), the constant
fields \(R=Q=\operatorname{int}G_b\),
and time function \(u\).
Their negative covectors have
\(\operatorname{Re}\tau>0\), hence
\(\tau\ne0\), and their base points
are outside \(S\). All localized
actual complexes are consequently
exact. We obtain
\[
\begin{gathered}R\operatorname{Hom}_R\bigl(P^\bullet,\\
R\Gamma_{V\setminus\Omega_0(b)}(V;\mathcal O)\bigr)=0.\end{gathered}
\tag{CS.9}
\]

To pass to the limit, take an increasing
sequence \(b_j\to\infty\).
Use the support triangle on \(V\)
and the explicit injective-resolution
sequence CP.24 for the increasing
opens \(V\cap\Omega_0(b_j)\).
Finite free Hom commutes with its
products and cones. This identifies
the supported complex for
\(V\setminus\Omega_0\) with the
derived inverse limit of those in
CS.9. Both its inverse-limit and
its degree-one-limit term vanish:
on the zero cohomology groups the
map \(1-\mathrm{shift}\) is surjective.
It is zero in every degree.
The basis CS.5 now kills all the
cohomology stalks of CS.4, proving
the assertion. This is an inverse-limit
argument, not an assumed interchange
of sections with a filtered colimit.

#### CS.B. Independence of the support pair

Let \(W\subset\mathbf C^n\) be open,
and assume in addition to CS.4 that
\[
S\cap F^{-1}(W)
 \subset\Omega_1\setminus\overline{\Omega_0}.
\tag{CS.10}
\]
The complex
\[
\mathscr L=
 R(F|_{\Omega_1})_*
       \mathscr H_{\Omega_1,\Omega_0}|_W
\tag{CS.11}
\]
is independent, by canonical derived
isomorphisms, of the pair satisfying
these conditions. The map \(F\) is
continuous for the cone topology:
the inverse image of an ordinary
base open is invariant by \(G_0\).

It suffices to compare over relatively
compact open subsets \(W'\Subset W\).
The closed parts of every comparison
are then compactly inside \(U\).
For two admissible pairs use the common
outer and inner opens
\[
\begin{gathered}
\Omega_1''=\Omega_1\cap\Omega_1',\\
\Omega_0''=(\Omega_0\cup\Omega_0')
                         \cap\Omega_1''.
\end{gathered}
\tag{CS.12}
\]
First restrict an outer open to
\(\Omega_1''\), keeping its own
inner open intersected with that
outer open. The support triangle
has as its third term the supported
Hom on the portion removed from
the outer open. The closed carrier
of that portion, intersected with
the old support, avoids \(S\):
CS.10 puts \(S\) in the interior
of both outer opens over
\(\overline{W'}\).
CS.A makes that third term zero.

Next enlarge the inner open to
\(\Omega_0''\). The analogous triangle
has a term supported in the added
inner portion. Its closed carrier
again avoids \(S\), since CS.10
keeps \(S\) outside the closures
of both inner opens over
\(\overline{W'}\). CS.A kills that
term too. Restriction and the support
inclusion therefore give a canonical
span of quasi-isomorphisms through
the common pair CS.12.

All these maps are maps of the same
finite relative support cubes and
ordinary injective comparisons of
DL.6–DLG.7. A third pair is compared
using the intersection of all three
outer opens and the union of their
inner opens there. The resulting
cubes commute. Thus the isomorphisms
satisfy the cocycle identity, rather
than depending on the order of
pairwise comparisons.

#### CS.C. A common thick-cone buffer

After shrinking \(W\) relatively
compactly, the pair in CS.11 can
be chosen with both opens invariant
under one \(G_b\), and containing
the common future \(E\).

Indeed the set
\(S\cap F^{-1}(\overline W)\)
is compact: CS.10 puts it in the
compact carrier and it is closed
there. Choose an ordinary open
\(U'\) around it with
\[
\overline{U'}\Subset
 (\Omega_1\setminus\overline{\Omega_0})\cap U.
\tag{CS.13}
\]
For large \(b\), the compact order
diamond
\[
(\overline{U'}+G_b)
 \cap(\overline{U'}-G_b)
\tag{CS.14}
\]
is contained in \(U\). To prove
this, keep the cones inside one
fixed proper cone. A strictly
positive linear functional bounds
the lengths between the compact
endpoints, as in CP.6. The diamonds
are closed bounded sets decreasing
to the \(G_0\)-diamond of
\(\overline{U'}\), which is inside
the round \(U\). Compactness gives
the asserted containment.

Set
\[
\begin{gathered}A'=(U'+G_b)\cup E,\\
B'=A'\cap\\
(X\setminus(\overline{U'}-G_b)).\end{gathered}
\tag{CS.15}
\]
The backward cone in this expression
is closed, since it is the sum of
a compact set and a closed cone.
Its complement is forward invariant.
Thus \(A'\) and \(B'\) are ordinary
\(G_b\)-opens and \(B'\subset A'\).
The future \(E\) is disjoint from
the backward cone by CS.3, so it
is contained in both opens.
Their closed difference lies in
CS.14, compactly inside \(U\).
Every point of \(S\) over
\(\overline W\) is in \(U'\),
hence in \(A'\) and in the interior
of that backward cone. It is
outside \(\overline{B'}\).
This pair satisfies CS.10.
CS.B provides its canonical comparison
with the original pair.

#### CS.D. Tilted funnels and their actual noncharacteristic cones

Use the common pair of CS.C, and keep
\(Z=\Omega_1\setminus\Omega_0\).
For a transverse centre \(x_0\), a radius
\(r>0\), and \(a>1\), set
\[
\begin{gathered}d(x)=|x-x_0|,\\
V(a,r)=E\cup\\
\{\operatorname{Re}t<a(r-d(x))-e\},\\
\Omega(a,r)=\Omega_1\cap V(a,r),\\
C(a,r)=R\operatorname{Hom}_R\bigl(P^\bullet,\\
R\Gamma_Z(\Omega(a,r);\mathcal O)\bigr).\end{gathered}
\tag{CS.16}
\]
The restriction maps give isomorphisms
\[
\begin{gathered}C(a,r'')\longrightarrow C(a,r'),\\
C(a',r')\longrightarrow C(a,r'),\\
0<r'\le r'',\\
1<a\le a',\quad ar'>2e.\end{gathered}
\tag{CS.17}
\]
All transverse balls used here have
closure in the base under consideration.

First, the opens in each comparison
are nested. Increasing radius increases
the funnel. For increasing \(a\), points
with \(d\ge r\) lie in the funnel only
inside \(E\); with \(d<r\), increasing
\(a\) increases the right side.
Each restriction triangle has third
term the supported Hom on
\[
\begin{gathered}Y=(\Omega_1\cap V_{\rm large})\\
\setminus(\Omega_0\cup V_{\rm small}).\end{gathered}
\tag{CS.18}
\]
Its closed carrier is a compact subset
of the original \(K\Subset U\).
It is separated from the axis \(x=x_0\).
Indeed outside the smaller funnel,
\[
\begin{gathered}\operatorname{Re}t\ge a(r'-d)-e,\\
d\ge r'-2e/a>0\\
\text{on the closed carrier}.\end{gathered}
\tag{CS.19}
\]
The last inequality uses
\(\operatorname{Re}t<e\) on \(\overline U\).
Thus \(d\) is smooth on a neighbourhood
of that carrier.

We verify every cone hypothesis of CP.F.
For the radius comparison put \(a'=a\)
in the following construction. For the
aperture comparison retain both values.
Choose
\[
\begin{gathered}0<\epsilon<\frac{a^2-1}{2a},\\
B>\max(b,a'+\epsilon).\end{gathered}
\tag{CS.20}
\]
Here \(G_b\) is the common buffer cone
from CS.C. Put \(R_*=\operatorname{int}G_B\).
Both opens defining CS.18 are locally
\(R_*\)-invariant. This follows for
the original pair from \(B>b\), for
the funnels from \(B>a'\), and for
\(E\) from decreasing real time.

Near a point with \(d>0\), put
\(v_0=(x-x_0)/d\) and
\(q(v)=\operatorname{Re}\sum_j\overline{v_{0,j}}v_j\).
For a tangent vector \((w,v)\in\mathbf C\times\mathbf C^n\)
let \(Q_*\) be defined by both strict
inequalities
\[
\begin{gathered}\operatorname{Re}w+a q(v)\\
{}<-\epsilon(|\operatorname{Im}w|+|v|),\\
\operatorname{Re}w+a' q(v)\\
{}<-\epsilon(|\operatorname{Im}w|+|v|).\end{gathered}
\tag{CS.21}
\]
It is an open convex cone. It contains
\(R_*\), since \(B>a'+\epsilon\),
and it contains the entire nonzero
\(G_0\)-ray in its interior.

Choose an ordinary neighbourhood \(N\)
of the intersection of \(S\) with the
closed carrier of CS.18, with
\[
\begin{gathered}N\subset\Omega_1\setminus\overline{\Omega_0},\\
d>0\text{ on }N.\end{gathered}
\tag{CS.22}
\]
This is possible by CS.10 and CS.19.
The carrier has base in the closed
larger transverse ball, inside \(W\):
CS.18 excludes \(E\), and CS.16 then
forces \(d\) below that ball's radius.
Define a cone field by taking \(Q_*\)
on \(N\) and \(R_*\) outside it;
precisely, its total open set is the
union of the constant field \(R_*\)
with \(Q_*\) over \(N\).
Its fibres are convex because the
larger cone contains \(R_*\).

The set \(Y\) is flat for this field.
Outside \(N\) all defining opens are
\(R_*\)-invariant. On \(N\), the
original outer open is full and the
inner open is absent. Only the two
funnel inequalities remain. Their
defining functions
\(\operatorname{Re}t+a d\) and
\(\operatorname{Re}t+a'd\) strictly
decrease along CS.21, so their
sublevels are locally invariant.
The union and difference presentation
in CS.18 proves the required flatness.
The common time function is
\[
\chi=-\operatorname{Re}t-a d.
\tag{CS.23}
\]
It is \(C^1\) near the changing carrier
and has positive derivative on the
whole cone field, by CS.21 and the
strict bound for \(R_*\).

It remains to check actual acyclicity
above \(S\), where the enlarged cone
is needed. Let a nonzero real covector
negative on \(Q_*\) be written
\[
\begin{gathered}\eta(w,v)=A\operatorname{Re}w\\
{}+C\operatorname{Im}w+\zeta(v).\end{gathered}
\tag{CS.24}
\]
Taking arbitrarily negative real
\(w\) gives \(A\ge0\).
If \(A=0\), one may choose imaginary
\(w\) and \(v\) arbitrarily and then
make its real part sufficiently
negative to satisfy CS.21. This
forces \(C=0\) and \(\zeta=0\),
contrary to nonzeroness. Thus
normalize \(A=1\).

For fixed \(y=\operatorname{Im}w\)
and \(v\), let the real part of
\(w\) approach its largest permitted
value in CS.21. Negativity gives
\[
\begin{gathered}C y+\zeta(v)\le\\
\max(aq(v),a' q(v))\\
{}+\epsilon(|y|+|v|).\end{gathered}
\tag{CS.25}
\]
Taking \(v=0\) and both signs of
\(y\) proves \(|C|\le\epsilon\).
Taking \(y=0\), \(v=-v_0\) proves
\(\zeta(v_0)\ge a-\epsilon\).
For the corresponding complex covector
\((\tau,\xi)\), therefore,
\[
\begin{gathered}\operatorname{Re}\tau=1,\\
|\operatorname{Im}\tau|\le\epsilon,\\
|\xi|\ge a-\epsilon\\
{}>\sqrt{1+\epsilon^2}\ge|\tau|.\end{gathered}
\tag{CS.26}
\]
The strict middle inequality is
equivalent to \(a^2-2a\epsilon>1\),
which is CS.20. It contradicts
the characteristic bound CS.2.
Outside \(S\) the actual complex
is already exact at every covector
in this chart. Thus the localized
complex is exact on the required
negative covectors above the entire
closed changing carrier.

All hypotheses of CP.F have now
been verified, with the original
thin operator carrier unchanged.
It kills CS.18 in every degree.
The restriction triangle proves
both quasi-isomorphisms CS.17.
They are the natural restriction
maps, so the radius and aperture
squares commute.

#### CS.E. Ball restriction and local constancy

For a fixed radius, the \(G_0\)-opens
\(V(a,r)\) increase with \(a\), and
\[
\begin{gathered}\bigcup_{a>0}V(a,r)\\
=E\cup F^{-1}(B(x_0,r)).\end{gathered}
\tag{CS.27}
\]
On \(U\), this follows directly
from \(\operatorname{Re}t+e>0\):
the inequality defining the funnel
eventually holds exactly when
\(d<r\). Globally, points already
in \(E\) are included on both
sides, and for points outside
\(E\) the same inequality has
a nonnegative left side. At
\(d=r\) it is strict and fails.

The supported object is zero on
\(E\). The ordinary excision and
finite support cube consequently
identify its cohomology on the
right side of CS.27 with that
on \(F^{-1}(B(x_0,r))\).
Use a sequence \(a_j\to\infty\)
and CP.24 on the increasing
left-side opens intersected with
\(\Omega_1\). The transition maps
on every cohomology group are
isomorphisms once \(a_jr>2e\),
by CS.17. The explicit recursive
solution of \(1-\mathrm{shift}\)
therefore kills its degree-one
inverse-limit term. Hence
\[
\begin{gathered}H^k(B(x_0,r);\mathscr L)\\
\simeq H^k(C(a,r))\\
\text{for }ar>2e.\end{gathered}
\tag{CS.28}
\]
The identification with direct-image
cohomology is literal on injective
resolutions: inverse image is exact,
so direct image preserves injectives,
and sections over a base open are
sections over its inverse image.
For \(0<r'\le r''\), choose
\(a\) with \(ar'>2e\) and use
CS.17. Its commuting aperture
squares make CS.28 compatible
with the ordinary ball restriction.
We have proved, for every degree,
\[
\begin{gathered}H^k(B(x_0,r'');\mathscr L)\\
\xrightarrow{\;\sim\;}H^k(B(x_0,r');\mathscr L).\end{gathered}
\tag{CS.29}
\]

We prove the sheaf consequence,
including the bounded-complex step.
First let \(T\) be a sheaf on a
Euclidean open set whose sections
on concentric balls have these
restriction isomorphisms. Fix a
ball \(B(c,R)\) and \(y\) in it.
Put \(d_0=|y-c|<R\) and
\[
\begin{gathered}c_t=c+t(y-c),\\
R_t=R-td_0,\\
B_t=B(c_t,R_t),\\
0\le t\le1.\end{gathered}
\tag{CS.30}
\]
For sufficiently close \(t<t'\),
insert between \(B_t\) and a
smaller ball the two balls
\[
\begin{gathered}B_{t'},\\
B(c_t,R-(2t'-t)d_0),\\
B(c_{t'},R-(3t'-2t)d_0).\end{gathered}
\tag{CS.31}
\]
They are nested in that order
after \(B_t\). Each containment
is the triangle inequality for
the centre displacement.
If their three restriction maps
are \(f,g,h\), then \(gf\) and
\(hg\) are isomorphisms by
concentricity. The first makes
\(g\) surjective, the second
injective, so \(g\), and then
\(f\), are isomorphisms.
The smallest radius stays positive
on a sufficiently fine finite
partition of \([0,1]\), since
\(R-d_0>0\). Thus restriction
from \(B_0\) to \(B_1\) is an
isomorphism. Concentricity at
\(y\) and the stalk limit give
\[
\Gamma(B(c,R);T)\xrightarrow{\sim}T_y.
\tag{CS.32}
\]
These maps for all \(y\) identify
\(T\) on the ball with the
constant sheaf of its sections.
Indeed the map from that constant
sheaf sends a locally constant
choice of global sections to
their restrictions; it is an
isomorphism on every stalk.
So \(T\) is locally constant.

We also need the cohomology of
a constant vector sheaf on a
ball, without a finite-rank
assumption. Resolve it by the
sheaf of smooth differential
forms with locally finite vector
coefficients. This is the
sheafification of the algebraic
tensor with the vector space.
It is a module over smooth
functions, hence acyclic by the
complete partition-of-unity proof
R.5.
On a ball centred at zero use
the radial homotopy, for \(j>0\),
\[
(h\alpha)_x=
 \int_0^1 s^{j-1}\iota_x\alpha_{sx}\,ds.
\tag{CS.33}
\]
Differentiating the pullback under
\(x\mapsto sx\) and integrating
gives \(dh+hd=1-\operatorname{ev}_0\).
Each compact family of radial
segments is covered by finitely
many coefficient neighbourhoods,
so the integral takes values in
one finite-dimensional span on
that family. Thus it defines
smooth locally finite-vector
forms even for an arbitrary
coefficient vector space.
The same contraction on small
balls proves local exactness.
This is an acyclic resolution,
and proves zero higher cohomology
and the expected degree-zero
sections on the full ball.

A locally constant sheaf on a
ball is constant. To see this
with the same generality, transport
along a path by finitely many
constant neighbourhoods. Subdivide
a homotopy square into such
neighbourhoods; the two edge
transports agree on each little
square and therefore on the whole
square. Straight paths in the
convex ball and their straight
homotopies give the trivialization.
This argument uses no bound on
the dimension of its fibres.

Now let \(T^\bullet\) be bounded
below and satisfy CS.29 in all
degrees. Start with its lowest
possible cohomology degree \(b\).
The truncation triangle gives
\[
\begin{gathered}\mathcal H^b(T^\bullet)[-b]\longrightarrow T^\bullet\\
\longrightarrow\tau_{>b}T^\bullet,\\
H^b(B;T^\bullet)\\
=\Gamma(B;\mathcal H^b(T^\bullet)).\end{gathered}
\tag{CS.34}
\]
The equality follows because derived
sections of a complex with no
cohomology through degree \(b\)
have none through that degree:
replace it by its good truncation
and take the first-quadrant injective
double resolution, which starts
in degree \(b+1\).
The single-sheaf result
CS.32 makes \(\mathcal H^b\)
locally constant. Its derived
sections on balls have restriction
isomorphisms in every degree by
CS.33 and the preceding transport
argument. The truncation triangle
then gives CS.29 for
\(\tau_{>b}T^\bullet\) as well.
Repeat in degree \(b+1\).
Each specified degree requires
only finitely many repetitions.
Every cohomology sheaf of
\(T^\bullet\) is locally constant.

#### CS.F. The transverse solution local system

For every admissible original
ray-open pair satisfying CS.4
and CS.10, each cohomology sheaf
of CS.11 is locally constant
on \(W\). The canonical pair
comparisons of CS.B identify
these sheaves, their restriction
maps and their transports.

Indeed use CS.C to choose a common
thick-cone buffer locally on \(W\).
CS.D–CS.E prove CS.29 there.
The complex CS.11 is bounded
below: its support resolution is
bounded below, finite free Hom
changes the lower bound by only
the finite length of \(P^\bullet\),
and a derived direct image is
computed by a bounded-below
injective resolution. The
bounded-complex argument of
CS.E applies. Pair independence
then gives the assertion for
the original pair and glues
it on \(W\).

No singular stratum of \(S\)
or discriminant fibre has been
deleted from this assertion.
Its characteristic hypothesis
is on the whole actual complex.
Finite-dimensionality is a
separate fibre computation:
if one fibre in a connected
component is finite dimensional,
local constancy makes every fibre
there have the same finite
dimension. Continuing individual
holomorphic representatives,
their common finite determination
and faithful actual-module
separation require further
arguments.

#### CS.G. Exact support and conormal examples

In the real plane \(t=-u+iw\),
take the ray-open pair
\[
\begin{gathered}
\Omega_1=\{u>0\},\\
\Omega_0=\{u>h(w)\},\\
h(w)=\sqrt{\max(0,1-w^2)}.
\end{gathered}
\tag{CS.35}
\]
Its closed changing carrier is
the closed upper half unit disc.
For the two-dimensional cone
\(u\ge b|w|\), the eroded open
CS.6 has boundary
\[
\begin{gathered}h_b(w)=\\
\sup_z\bigl(h(w+z)-b|z|\bigr).\end{gathered}
\tag{CS.36}
\]
This follows by taking the least
allowed forward \(u\)-component
for each displacement \(z\).
On the circular arc the tangent
slope is \(-w/\sqrt{1-w^2}\).
It reaches magnitude \(b\) at
\(w_b=b/\sqrt{1+b^2}\).
The two tangent lines there have
intercept \(\sqrt{1+b^2}\).
The supremum in CS.36 is therefore
\(h\) on \(|w|\le w_b\), those
tangent lines beyond that interval,
and zero after they cross the axis.
To verify the supremum, on each
side maximize the concave circular
arc minus a linear function: its
derivative vanishes at the stated
tangent point, or at \(z=0\)
when the point lies between them.
The zero part of \(h\) contributes
the lower bound zero.

Thus the compact carrier of
\(\Omega_1\setminus\Omega_0(b)\)
is bounded by the unchanged central
arc and these tangent segments.
Its endpoint has
\[
|w|=\sqrt{1+b^{-2}}.
\tag{CS.37}
\]
The zero-height tails beyond this
endpoint are not part of the
closed changing carrier: their
neighbourhoods have \(h_b=0\),
so the two opens coincide there.
For \(b=4/3\), the complex point
\(S_* =\{t=6i/5\}\) lies in
the approximating carrier. For
\(b=12/5\), the endpoint is
\(13/12<6/5\), so the whole
carrier avoids \(S_*\).
This illustrates why the closure
and eventual compact containment
in CS.7 are needed.

For the funnel and covector
inequalities take the exact values
\[
\begin{gathered}e=1/8,\quad a=3,\quad a'=4,\\
\epsilon=1/4,\quad r=1/4,\\
ar=3/4>2e=1/4,\\
(a-\epsilon)^2=121/16\\
{}>17/16=1+\epsilon^2.\end{gathered}
\tag{CS.38}
\]
The real section of the funnel has
\(\operatorname{Im}t=\operatorname{Im}x=0\).
Its sloping boundary normals have
positive spatial-to-time ratio
\(a\) or \(a'\) on the positive
radial side. The negative-polar
calculation CS.24–CS.26 treats
all complex tangent directions,
including imaginary time and
transverse spatial components;
the real section does not replace it.

![Closed support carriers and transverse noncharacteristic funnels](assets/transverse-solution-systems.png)

**Figure CS.1.** Upper panel: the real plane
\(t=-u+iw\) and the exact carrier boundaries
CS.35–CS.37. The tangent segments meet the
unit semicircle at their exact contact points.
The marked complex point belongs to the first
closed approximating carrier and is excluded
from the later ones. Zero-height tails beyond
the finite endpoints are absent. Lower panel:
the real section of CS.38, with the common
future \(E\) and the strip containing the
changing support. The displayed normal is
ordered as \((\tau,\xi)\); its plotted arrow
has components \((\xi,\tau)\) in the labelled
horizontal and vertical axes. CS.24–CS.26
prove the full complex covector bound.
Free human source: [Kashiwara–Kawai, IV.4.4](https://www.kurims.kyoto-u.ac.jp/~kenkyubu/kashiwara/HolIII.pdf).

Editable SVG,
exact figure data,
and reproducible source.

Free further reading: Kashiwara–Kawai,
[Holonomic systems III, IV.4.4.2–IV.4.4.6](https://www.kurims.kyoto-u.ac.jp/~kenkyubu/kashiwara/HolIII.pdf).


### 5.82. Common-domain continuation of actual holonomic function solutions

Let \(X=\mathbf C_t\times\mathbf C_x^n\), let
\(q_0=(0,0)\), and let \(p_0=(q_0;1,0)\).
Let \(M\) be a coherent actual finite-order
microlocal module whose reduced support \(A\)
is a homogeneous analytic Lagrangian in generic
position:
\[
\begin{gathered}
A\cap\pi^{-1}(q_0)=\mathbf C^\times p_0.
\end{gathered}
\tag{CF.1}
\]
All statements concern the whole module,
including its principal nilpotents and
intersecting characteristic branches.
Write \(S=\pi(A)\) on a sufficiently small
base neighbourhood.

We use the directional function module
\[
\begin{gathered}
\mathcal C=\underset{V,Z}{\operatorname{colim}}
 H_Z^1(V;\mathcal O),\\
C_0(Z)\subset\{\operatorname{Re}t\ge0\}.
\end{gathered}
\tag{CF.2}
\]
Here \(V\) is an ordinary neighbourhood of
zero, \(Z\) is relatively closed, and
\(C_0(Z)\) consists of limits of positive
rescalings of points of \(Z\) tending to zero.
The support condition includes the zero
vector. There is no growth condition on a
holomorphic representative at its open
boundary.

**Theorem CF.1.** Let
\(\Phi:M_{p_0}\to\mathcal C\) be an actual
\(\mathscr E_{p_0}\)-linear map.
For any finite list \(s_1,\ldots,s_m\) and
holomorphic representatives of
\(\Phi(s_1),\ldots,\Phi(s_m)\), there is one
ordinary neighbourhood \(V\) of zero on
which all these representatives continue
as multivalued holomorphic functions on
\(V\setminus S\).
A branch of each continuation agrees with
its prescribed representative on a smaller
admissible negative domain.
For finitely many maps \(\Phi\), the same
neighbourhood can be used.

Multivalued means analytic continuation
along every path from a fixed initial germ,
with equality along homotopic paths with
fixed endpoints. Finite determination and
faithful evaluation are separate assertions.
They are not hypotheses or conclusions of
this theorem.

The actual lattice, full factorial norms
and finite-head division used below are
§5.32, J.1–J.3
and
§5.55, DP.1–DP.4.
The complete canonical support action and
convex Dolbeault proof are
§5.43
and
§5.80, CP.A.
The primitive normalization is
§5.33, K.2
and its actual supported comparison in
§5.53, AM.B.

#### CF.A. Convex representatives and the full directional action

Put \(r=\operatorname{Re}t\) and
\(z=(\operatorname{Im}t,x)\), with
\(|z|=|\operatorname{Im}t|+|x|\).
For each \(a>0\), put
\[
G_a=\{(r,z):r\le-a|z|\}.
\tag{CF.3}
\]
Every support in CF.2 is locally contained
in a larger admissible support whose
complement is convex and invariant under
\(G_a\).

Indeed, on its negative-\(r\) part,
\((-r)/|z|\) tends uniformly to zero.
Failure would give, by normalizing a
sequence and taking a unit-sphere
subsequence, a tangent vector with
negative \(r\).
The same argument excludes points with
\(z=0,r<0\) sufficiently near zero.
Choose a continuous nondecreasing modulus
\(\omega(s)\to0\) such that
\[
\begin{gathered}-r\le |z|\omega(|z|),\\
\text{on }Z\cap\{r<0\}.\end{gathered}
\tag{CF.4}
\]
For completeness, first bound the ratio
on shrinking punctured balls by its
supremum. Choose radii where this bound
is below \(2^{-k}\), and interpolate
monotonically above the bound using
the preceding radius. This gives the
stated continuous majorant.
Define
\[
\begin{gathered}
\phi(s)=\int_0^{2s}\omega(v)\,dv,\\
Z'=\{r\ge-\phi(|z|)\}.
\end{gathered}
\tag{CF.5}
\]
The function \(\phi\) is convex,
nondecreasing, dominates \(s\omega(s)\),
and is \(o(s)\). On a small interval it
has Lipschitz constant \(L<a\).
Extend it linearly beyond that interval
with the final slope when necessary.
The set \(Z'\) contains \(Z\) locally,
has the required tangent cone, and has
convex complement. For \(s\ge a|w|\),
\[
\begin{gathered}r-s+\phi(|z+w|)\\
\le r+\phi(|z|)-s+L|w|<0\\
\text{if }r+\phi(|z|)<0.\end{gathered}
\tag{CF.6}
\]
Thus its complement is \(G_a\)-open.

This construction also supplies the
compact cap needed for the actual
operator action. Let
\(\Omega=B_\epsilon+G_a\), using the
sum norm for the ball.
For \(b+(-s,w)\in Z'\cap\Omega\),
the inequalities
\(r\le\epsilon-s\),
\(|z|\le\epsilon+s/a\), and
\(r\ge-L|z|\) give
\[
(1-L/a)s\le(1+L)\epsilon.
\tag{CF.7}
\]
Consequently the closed carrier is
contained in a fixed constant times
this small ball. Choose \(\epsilon\)
so its closure is compact in the
coefficient domain and the original
ordinary neighbourhood.
Both \(\Omega\) and \(\Omega\setminus Z'\)
are convex and cone-open.

The full convex Dolbeault vanishing
of CP.A and DLH gives, by the
ordinary support triangle,
\[
\begin{gathered}
H_{Z'}^k(\Omega;\mathcal O)=0\quad(k\ne1),\\
H_{Z'}^1(\Omega;\mathcal O)\\
=\mathcal O(\Omega\setminus Z')/
                    \mathcal O(\Omega).
\end{gathered}
\tag{CF.8}
\]
The degree-zero group vanishes by
the identity theorem. Ordinary
excision transfers a class from
the original neighbourhood to
this completion, since the closed
carrier is compact in their overlap.
It follows that CF.2 consists
precisely of these actual function
classes. A class is zero exactly
when, after one refinement, its
representative extends holomorphically
to an ordinary neighbourhood of zero.
Any finite family of classes and
equalities admits one common refinement.

The full real kernel representatives
of AM.H have strict negative pairing
with \(dt\) on their nonzero normal
cones. Compactness of the unit normal
cone puts any finite family in one
carrier \(G_a\), after shrinking.
If no such actual containment held,
normalizing the failing points would
contradict that strict pairing.
DL.6–DLG.7 acts on the cone-closed
supports just constructed, with the
common cap CF.7. Its cup, excision
and normalized trace comparisons
identify refinements and both products.
Taking the filtered colimit gives
the whole real kernel action on
\(\mathcal C\). Restriction along
the proved inclusions
\(\mathscr E\to\mathscr E^\infty
\to\mathscr E^{\mathbb R}\)
gives the action used in the theorem.
No map from an arbitrary canonical
kernel to a finite symbol is asserted.

In particular, for a coefficient
depending only on \(x\), the actual
action of \(h^k=D_t^{-k}\) on a
representative is, modulo a
holomorphic germ,
\[
\begin{gathered}h^k f(t,x)=\\
\int_A^t\frac{(t-s)^{k-1}}{(k-1)!}f(s,x)\,ds,\\
k\ge1.\end{gathered}
\tag{CF.9}
\]
The path lies initially in the convex
negative domain. This is exactly the
positive primitive normalization of
K.2 and its supported AM.B comparison.
In the quotient \(hD_t=1\), because
the omitted initial value is
holomorphic on the full neighbourhood.

#### CF.B. Finite coefficient modules from an isolated principal fibre

Choose the actual \(h\)-injective
lattice \(M_0\subset M\) of J.1,
and set \(L=M_0/hM_0\).
The following variant of the finite
division argument is useful:
\[
\begin{gathered}
\dim_{\mathbf C}L/(x)L<\infty\\
\Longrightarrow
M_0=\sum_{j=1}^q R_0u_j,\\
R_0=\left\{\begin{gathered}
\sum_{k\ge0}a_k(x)h^k:\\
 |a_k|_K\le BC^k k!
\end{gathered}\right\}.
\end{gathered}
\tag{CF.10}
\]
Every coefficient list has one common
holomorphic parameter domain.

Here is why the weaker premise suffices.
In DP.5 take the normal variables
\(v=(t,\xi/\tau)\). The specialized
principal relation module has quotient
\(L/(x)L\). Each normal coordinate
acts nilpotently on this finite local
quotient: a nonzero eigenvalue would
contradict invertibility of that
coordinate minus the eigenvalue.
Their commuting actions therefore
make a power of \((v)\) annihilate
the quotient. The specialized head
diagram has finite complement.

Lift its finitely many minimal heads
to actual relations. The principal
kernel equality is obtained by
subtracting \(h\) times a lift of
the image divided by \(h\).
Use precisely the hierarchical
normal radii, component weights,
parameter shrink and positive-index
shrink in DP.7–DP.11.
They give its two one-eighth error
bounds on one domain. For a time
head the complete correction is
\[
\begin{gathered}
Q\circ(t^az^\beta)-Qt^az^\beta\\
=\sum_{j=1}^a\binom aj
 (\partial_\tau^jQ)t^{a-j}z^\beta.
\end{gathered}
\tag{CF.11}
\]
The weighted coefficient inverse
of the head map has norm at most
one, and the total error has norm
at most one quarter.
Its Neumann solution therefore
converges with norm at most
four thirds of the input norm,
as in DP.12–DP.14.
The finite inverse-ordering formula
DP.16 converts its ordinary remainder
to actual coefficient products.
This proves CF.10 for every germ,
with the full factorial bound.
No formal specialization of \(h\)
has been used.

The same argument with \(h\) absent
proves the ordinary finite-fibre
lemma for coherent analytic modules:
an isolated finite principal fibre
over the base makes the module finite
over the base germ ring.
The head projections, radius choices,
geometric inverse and finite remainder
are the preceding argument with its
positive-index terms removed.
Apply it first to the variables
\(z=\xi/\tau\), using J.3.
Thus \(L\) is finite over
\(\mathcal O_{t,x}\).

This also proves the required local
finite projection of its support.
The algebra
\(\mathcal O_{t,x,z}/\operatorname{Ann}L\)
injects into
\(\operatorname{End}_{\mathcal O_{t,x}}L\).
The latter is finite: a finite
presentation embeds this Hom into
a finite power of the finite module
\(L\), and the coefficient ring is
Noetherian. Hence the support algebra
is finite over the base.
Its normal coordinates satisfy monic
equations and are nilpotent on the
central finite fibre. Their roots
therefore stay in a compact smaller
normal polydisc after shrinking the
base. This gives properness of this
local finite projection.
The image is exactly the analytic
support of \(L\) as a base module.
For example, on a nonzero fibre
the commuting normal-coordinate
matrices have a simultaneous
eigencharacter: take an eigenspace
of one, restrict the others, and
repeat. It supplies a point of
the fibre support. The converse
is immediate by restriction.
The minor test for a finite
base presentation shows that
this image \(S\) is analytic.

Finite integrality preserves
component dimensions, as in
the finite-projection proof of
analytic dimension, Theorem 5.4.
Thus \(S\) is a pure
\(n\)-dimensional germ in
\(\mathbf C^{n+1}\).
It has a reduced hypersurface
equation. One direct verification
uses factoriality of the analytic
germ ring, proved in
The local ring of holomorphic germs, §3.
Factor any nonzero equation
vanishing on \(S\).
Each irreducible component of
\(S\) lies in one irreducible
factor hypersurface. Their
dimensions agree; strict inclusion
would lower dimension by
Proposition 5.5 of the dimension
lesson. The reduced product
of the selected factors defines
exactly \(S\).

#### CF.C. Spanning ordinary linear fibrations

Let \(f\) be that reduced equation,
with first nonzero homogeneous part
\(f_m\). Choose a vector \(v_0\)
with \(f_m(v_0)\ne0\) and
\(p_0(v_0)\ne0\), and normalize
\(p_0(v_0)=1\).
The excluded vectors are zeros
of two nonzero polynomials,
so they do not fill the vector
space. Take \(v_0\) as the time
axis and a basis \(e_1,\ldots,e_n\)
of \(\ker p_0\) as spatial axes.
For sufficiently small nonzero
\(\lambda\), put
\[
\begin{gathered}v_j=v_0-\lambda e_j,\\
1\le j\le n,\\
 F_0(t,x)=x,\\
F_j(t,x)=x+\lambda t e_j.\end{gathered}
\tag{CF.12}
\]
All \(f_m(v_j)\) remain nonzero.
Each vertical line meets \(S\)
only at zero locally.
The \(n+1\) vertical directions
span the ambient tangent space:
\[
\begin{gathered}\left|\det(v_0,v_1,\ldots,v_n)\right|\\
=|\lambda|^n.\end{gathered}
\tag{CF.13}
\]
Moreover \(p_0(v_j)=1\).

For each \(F_j\), the normalized
support over its central fibre
is isolated: its base point must
be zero, and CF.1 then fixes its
normalized covector.
The analytic Nullstellensatz and
the finite-monomial argument of
J.3 give the finite-dimensional
principal fibre in CF.10.
Thus the whole actual module is
finite over its coefficient ring
\(R_{0,j}\) in every fibration.

These are ordinary linear
coordinate changes. Their finite,
infinite and real actions and
inverse maps are proved in
§5.56, BCT.5.3.
They keep \(t\) unchanged.
Consequently they keep the
directional tangent condition
in CF.2 unchanged. The full
normal derivative changes by
an elliptic order-zero factor
times the old derivative;
its inverse has the compatible
actual action.
All finite lists and their
relations can be represented
on one common refinement.

In these directions preparation
of \(f\) gives a monic
Weierstrass equation
\(p_j(t,x)=0\).
Its coefficient orders and
the elementary root bound give
\(|t|\le C_j|x|\) on \(S\)
in this fibration.
The exact preparation proof
is
AG-QC, §2;
the coefficient orders and root
bound are
analytic parametrization, Lemma 2.1 and Proposition 2.2.
The discriminant \(H_j\) is proper:
the reduced polynomial is
square-free over the coefficient
fraction field in characteristic
zero. Its resultant with its
derivative is therefore nonzero.
Off \(H_j\), its roots are
distinct holomorphic graphs by
the implicit-function proof
in AG-QC, Lemma 2.2.

#### CF.D. A rectangular Volterra equation with the exact support

Fix one fibration.
Choose \(u_1,\ldots,u_q\) lifting
a basis of \(L/(x)L\).
They generate \(L\) over
\(\mathcal O_x\). They also
generate \(M_0\) over \(R_0\):
express a finite generating list
modulo \(h\) in their lifts,
and invert the finite matrix
\(I-hC\) by DP.20.
Use coefficient rows and the
generator column \(u\).
There is an actual matrix
\[
tu=T(x,h)u.
\tag{CF.14}
\]
A row relation \(a\) means
\(au=0\).
Take a finite presentation
of \(L\) over \(\mathcal O_x\).
Every principal relation row
lifts to an actual \(R_0\)-row
relation: divide its image
in \(hM_0\) by \(h\), express
the quotient in the same
generators, and subtract \(h\)
times those coefficients.
Denote the finite lifted
matrix by \(a(x,h)\).

Put \(T_0=T\bmod h\) and
\(a_0=a\bmod h\).
The matrix \(T_0(0)\) is
nilpotent: it is the action
of \(t\) on the finite local
fibre with basis \(u\bmod(x,h)\).
The rectangular matrix
\[
B_0(t,x)=
\begin{pmatrix}tI-T_0(x)\\ a_0(x)\end{pmatrix}
\tag{CF.15}
\]
has full column rank at every
point outside \(S\).

To prove this exact assertion,
evaluate a coefficient row
\(\sum_{m\ge0}c_m(x)t^m\)
as \(\sum_m c_m(x)T_0(x)^m\).
Choose a weighted basis making
the norm of the nilpotent
\(T_0(0)\) arbitrarily small,
then shrink \(x\). The series
converges normally on smaller
products. The identity
\[
\begin{gathered}
t^mI-T_0^m\\
=(tI-T_0)\sum_{b=0}^{m-1}
                       t^{m-1-b}T_0^b
\end{gathered}
\tag{CF.16}
\]
has a normally summable
coefficient-series version.
The principal relation rows
are stable under \(T_0\),
because \(t\) acts on \(L\).
Evaluation therefore identifies
the base-module cokernel of
CF.15 with \(L\) as a base
module. These are finite
coherent base presentations.
Their germ isomorphism and
its inverse extend to one
smaller base neighbourhood:
extend the finitely many
images of generators, their
relation identities and the
two composition identities,
as in J.1. Its support is exactly
\(S\), by CF.B. At a base
point, a finite presentation
has zero cokernel precisely
when its value is onto, by
finite commutative Nakayama.
Equivalently CF.15 has full
column rank there. This proves
the claim without inverting
a transverse parameter minor.

All entries of the positive
\(h\)-parts have one factorial
estimate. Their kernels are
\[
\begin{gathered}K_A(t,s,x)=\\
\sum_{k\ge1}A_k(x)\frac{(t-s)^{k-1}}{(k-1)!}.\end{gathered}
\tag{CF.17}
\]
If \(|A_k|\le BC^kk!\), their
absolute sum is at most
\[
\frac{BC}{(1-C|t-s|)^2}.
\tag{CF.18}
\]
Thus they and every fixed
derivative converge normally
on one smaller two-time
product with \(C|t-s|<1\).
The finite list permits a
single such product.
Stack the kernels of \(-T\)
and \(+a\), with their zero
coefficients omitted, to
obtain the rectangular kernel
\(K\).

Let \(f\) be the column of
representatives of \(\Phi(u)\).
The finite relation residuals
are zero in \(\mathcal C\).
By CF.A they extend as
holomorphic functions near
zero. The actual primitive
comparison gives
\[
\begin{gathered}
B_0(t,x)f(t,x)\\
{}+\int_A^t K(t,s,x)f(s,x)\,ds\\
=g(t,x).
\end{gathered}
\tag{CF.19}
\]
Here \(g\) is holomorphic on
one full product, including
the initial slice \(t=A\).

That last domain assertion
requires choosing the anchor
twice. Begin with an anchor
\(A_0<0\) in the common convex
representative domain.
The finitely many residuals
extend near zero, possibly
to a time disc not containing
\(A_0\). Choose a new negative
\(A\) inside that disc.
The difference of the two
primitive formulas is the
integral over the fixed
compact path from \(A_0\)
to \(A\). Its output and
parameter dependence is
holomorphic on the whole
new product, by CF.17 and
the original representative
along that path.
Subtract this memory correction
from the extended residual.
This gives CF.19 with \(A\)
inside its full domain.
Shrink the transverse ball
so \((A,x)\) lies in the
initial domain and every
root of \(p_j\) lies in
\(|t|<|A|/4\).

#### CF.E. Every path outside one vertical discriminant

Suppose a branch is known
along a compact vertical path
from \(A\) to \(a\), avoiding
\(S\). Its old memory
\[
\begin{gathered}I_{\mathrm{old}}(t,x)=\\
\int_{A\rightsquigarrow a}K(t,s,x)f(s,x)\,ds.\end{gathered}
\tag{CF.20}
\]
is holomorphic in the new
output \(t\) and in nearby
parameters. At \(a\), choose
a nonzero square minor of
CF.15 and obtain a holomorphic
left inverse \(J\) of \(B_0\).
In the following formula, the arguments
\(t,x\) of \(J,g,I_{\mathrm{old}},b\)
are suppressed in the first two rows, and
\(H(t,s,x)=J(t,x)K(t,s,x)\).
On a small disc the new
branch solves
\[
\begin{gathered}H=JK,\\
b=J(g-I_{\mathrm{old}}),\\
 f(t,x)=b(t,x)\\
{}-\int_a^t H(t,s,x)f(s,x)\,ds.\end{gathered}
\tag{CF.21}
\]
If the last kernel is bounded
by \(L\), its \(k\)-fold
successive integral has bound
\[
\frac{L^k|t-a|^k}{k!}.
\tag{CF.22}
\]
Parametrize each integration
by the ordered real simplex
in a complex segment; its
volume is \(1/k!\), by
successive elementary integration.
The series therefore converges
normally and solves CF.21.
The same bound for a difference
proves uniqueness.
All bounds are uniform on a
smaller parameter compact,
so the solution is jointly
holomorphic.

Every rectangular row of
CF.19 still holds. Its
residual is holomorphic and
zero on the open overlap
with the preceding branch.
The identity theorem makes
it zero on the whole
connected product. A finite
disc cover thus continues
the branch along every
vertical path in the punctured
time disc.

Off the discriminant \(H_j\),
the punctured time discs are
locally topologically trivial,
with \(A\) fixed.
Here is an explicit construction.
At a parameter value choose
disjoint small discs about
the distinct roots, away
from \(A\) and the boundary.
Choose smooth cutoffs equal
to one near their centres.
Translate each root by its
cutoff times its small
displacement. The resulting
map is the identity plus
a perturbation with Lipschitz
constant less than one.
The iteration \(y=w-\delta(y)\)
converges geometrically to
its unique inverse. It
therefore gives the required
homeomorphism, fixing the
boundary and \(A\).

A path in
\[
X\setminus\bigl(S\cup F_j^{-1}(H_j)\bigr)
\tag{CF.23}
\]
has a finite cover by these
trivializations. Transport
the vertical memory path
through that cover.
Its actual holomorphic
formulas use fixed compact
contour pieces on each
smaller parameter neighbourhood,
where those pieces still
avoid \(S\), together with
small holomorphic endpoint
segments. CF.20–CF.22 prove
joint holomorphy there.
The cutoff motion is used
only to identify contour
classes, not as a
holomorphic formula.
This proves continuation
along every path of CF.23.

We recall explicitly the
path-homotopy argument.
An analytic continuation
is a chain of holomorphic
germs agreeing on nonempty
overlaps. Uniqueness is
the identity theorem on
small balls. For a fixed
path its finitely many
overlap balls also support
all sufficiently close
paths, with the same
endpoint germ. Subdivide
a homotopy square into
finitely many rectangles
inside such overlap
neighbourhoods. The two
edge continuations around
each rectangle agree.
Successively cancelling
interior edges proves
equality at the endpoint
for the two boundary paths.
This proves fixed-endpoint
homotopy invariance and
applies to every continuation
constructed here.

#### CF.F. All branches and all fibrations

Put \(D_j=F_j^{-1}(H_j)\).
The common bad set
\[
Z=\bigcap_{j=0}^n D_j
\tag{CF.24}
\]
has complex codimension at
least two. Otherwise an
irreducible hypersurface
component \(T\) of \(Z\)
is a component of every
\(D_j\). Each \(D_j\) is
invariant along its constant
vertical direction \(v_j\).
At a regular point of \(T\)
away from the other finitely
many components of all
\(D_j\), each \(v_j\) is
therefore tangent to \(T\).
Such a point exists because
a proper analytic subset
of an irreducible component
has smaller dimension.
CF.13 contradicts the
dimension of its tangent
space. For \(n=0\) the
bad transverse set is
empty.

Fix one of the prescribed
elements \(s\). In each
fibration choose \(N\)
with \(h_j^Ns\in M_0\),
and adjoin this element
to its finite coefficient
generators.
Express the \(t\)-action
on the added generator
using the original ones.
The enlarged \(T_0(0)\)
is block triangular with
the old nilpotent block
and a zero redundant
diagonal block.
The principal presentation
still presents \(L\).
Thus the same argument
continues a representative
of \(\Phi(h_j^Ns)\).
Apply \(D_{t,j}^N\) to
recover a representative
of \(\Phi(s)\).
Its difference from the
prescribed representative
is holomorphic near zero,
by CF.8. Add this ordinary
holomorphic germ back
on a common smaller
neighbourhood.
The continuations from
all fibrations now agree
with exactly the same
initial scalar germ.

Choose one ordinary ball
\(V\) inside all their
finitely many product
domains. A sufficiently
narrow negative cone
misses \(S\), by the
linear root bound in CF.C.
It contains an initial
point outside every
\(D_j\), since finitely
many proper analytic
sets cannot fill an open.
Write \(M'=V\setminus S\).

Agreement of one initial
branch alone does not
justify gluing all branches.
The required path argument
is as follows.
A path in \(M'\setminus Z\)
has a finite cover by balls,
each missing one \(D_j\).
Choose its intermediate
overlap points outside
all the \(D_j\).
Inside one such ball,
two of these points can
be joined while avoiding
all the other \(D_k\):
the complex line through
them is not contained in
their union, so its
intersection with that
union is discrete.
A compact line segment
meets only finitely many
of these points after
choosing a slightly
larger segment domain;
small planar detours
avoid them.
This new path is homotopic
to the old segment in
the original ball,
which misses the selected
\(D_j\).

Inductively the continuation
arriving at an overlap can
be reached along a path
avoiding every \(D_k\).
The next fibration therefore
continues precisely that
branch; CF.E and its
homotopy argument identify
the two paths through
the ball. This proves
continuation of every
branch along every path
in \(M'\setminus Z\).

#### CF.G. Removal of the codimension-two intersection

We give the two local
facts required to cross
\(Z\setminus S\), without
a boundedness assumption.

An analytic subset of
complex dimension at
most \(N-2\) has a
countable cover by
Lipschitz images of real
boxes of dimension at
most \(2N-4\), where
\(N=n+1\).
To see this, use
local parametrization, Theorem 4.1
on its finitely many
irreducible local components.
The generic-sheet part
is a smooth manifold.
Its remaining analytic
part has smaller dimension
by Proposition 5.5.
Repeat on that part;
dimension decreases at
each repetition.
Countable coordinate
charts and compact
subcharts yield the
claimed Lipschitz cover.
The finite component
decomposition follows
directly from Noetherianity:
a nonprime radical ideal
splits its zero set
using \(ab\) in the
ideal but neither factor
in it; Noetherian
induction terminates
these proper splittings.

Let \(B\) be a small
convex ball disjoint
from \(S\).
Every loop in \(B\setminus Z\)
contracts there.
First approximate it by
a polygonal loop within
its positive compact
distance from \(Z\).
Fill that loop by its
Lipschitz cone in \(B\).
Perturb the filling \(F\)
by \(\beta v\), where
\(\beta\) is Lipschitz,
zero on the boundary
and positive in the
interior.
A boundary collar
remains disjoint from
\(Z\) for sufficiently
small \(v\).
On the sets \(\beta\ge1/k\),
the forbidden vectors
are images of pairs
\((z,y)\) under
\[
v=\frac{z-F(y)}{\beta(y)}.
\tag{CF.25}
\]
Their parameter boxes
have dimension at most
\((2N-4)+2=2N-2\).
Each such Lipschitz
image has zero
\(2N\)-dimensional
volume. Indeed a
bounded \(m\)-box
has \(O(\delta^{-m})\)
cubes of side \(\delta\);
their images fit cubes
of diameter \(O(\delta)\),
whose total ambient
volume tends to zero
when \(m<2N\).
Countably many such
images still have zero
volume. An arbitrarily
small permitted \(v\)
therefore gives a
contraction missing
\(Z\). Paths between
two points are obtained
by the complex-line
detours used in CF.F.
Thus \(B\setminus Z\)
is connected and simply
connected. CF.E makes
every continued branch
single valued there.

Such a single-valued
holomorphic function
extends across \(Z\).
Choose a nonzero local
equation vanishing on
\(Z\), and a line on
which its first nonzero
homogeneous part does
not vanish.
This line meets \(Z\)
only at the centre
locally. Take a small
circle on it and a
transverse polydisc
so the full cylinder
over the circle misses
\(Z\).
The projection of the
part of \(Z\) inside
the cylinder is locally
closed, by compactness
and the side-boundary
exclusion. Its image
has empty interior:
the preceding box cover
now projects sets of
dimension at most
\(2N-4\) into dimension
\(2N-2\).
Define the extension
by the fixed-circle
Cauchy integral in
the line coordinate.
It is jointly holomorphic.
Over the dense set
of transverse points
outside that projection,
the whole fibre disc
misses \(Z\); Cauchy's
formula gives the
original function.
Continuity identifies
the two functions at
every remaining point
outside \(Z\).
Uniqueness makes these
extensions agree on
overlaps.

We may consequently
continue along paths
through \(Z\setminus S\)
as well. For \(N=1\)
there is no such
codimension-two set.
This proves Theorem
CF.1 for one element.
There are finitely
many fibrations,
relations, prescribed
elements and maps.
Intersect their finitely
many ordinary neighbourhoods
and take a smaller ball.
This proves its common-domain
assertion.

#### CF.H. Exact examples and the scope of continuation

For the complex cusp
\(S=\{t^2=x^3\}\), take
\[
F_0=x,\qquad F_1=x+t/2.
\tag{CF.26}
\]
Their vertical vectors,
ordered as \((t,x)\),
are \((1,0)\) and
\((1,-1/2)\).
The respective local
discriminants are
\(x=0\) and \(x+t/2=0\);
their cylinders meet
only at the origin.
For the second projection,
the critical equation
on the cusp is
\(2t+3x^2/2=0\).
Together with \(t^2=x^3\)
it gives
\(x^3(9x/16-1)=0\).
The second solution is
outside a sufficiently
small neighbourhood,
so the stated local
discriminant is exact.

A scalar kernel
calibration is
\[
\begin{gathered}A=-1/2,\quad\alpha=1/8,\\
 f(t)=\frac{81}{625}\frac{(t-\alpha)^3}{(t+\alpha)^5},\\
 (t^2-\alpha^2)f(t)\\
{}-\int_A^t f(s)\,ds=1.\end{gathered}
\tag{CF.27}
\]
Differentiate the last
left side. The identity
\((t^2-\alpha^2)f'
+2tf=f\) makes its
derivative zero.
Its value at \(A\)
is one, proving the
equation on every
continued contour.
This checks the sign
and the fixed memory
mechanism, not a
classification theorem.

One fibration alone
is insufficient.
The equation
\[
\begin{gathered}(t^2-x^3)\partial_t f=f,\\
 f(A,x)=1.\end{gathered}
\tag{CF.28}
\]
has a holomorphic
initial negative-cone
representative
\[
\begin{gathered}R(t,\rho)=\log\frac{t-\rho}{t+\rho},\\
 U(t,\rho)=\\
\frac{R(t,\rho)-R(A,\rho)}{2\rho},\\
 f(t,x)=e^{U(t,\rho)},\\
\rho^2=x^3.\end{gathered}
\tag{CF.29}
\]
The initial logarithms
expand in odd powers
of \(\rho\); division
by \(2\rho\) leaves
a convergent series
in \(x^3\), when \(t\)
and \(A\) are on the
initial negative side.
A positive loop about
\(t=\rho\) multiplies
this branch by
\(\exp(\pi i/\rho)\),
which has an essential
transverse singularity
at \(x=0\), even for
fixed nonzero \(t\).
Integration by parts
gives its uniform
analytic Volterra
equation
\[
\begin{gathered}(t^2-x^3)f(t,x)\\
{}-\int_A^t(2s+1)f(s,x)\,ds\\
=A^2-x^3.\end{gathered}
\tag{CF.30}
\]
Thus a full two-time
kernel and the exact
principal singular set
do not by themselves
prove continuation
across a vertical
discriminant.
The single differential
relation in two
variables has
characteristic equation
\((t^2-x^3)\tau=0\),
of dimension three;
it is not holonomic
of dimension two.
The full holonomic
hypothesis in CF.1
supplies the finite
systems in all the
spanning fibrations,
which is the additional
step in CF.F.

![Spanning linear fibrations and a fixed Volterra memory contour](assets/linear-fibration-continuation.png)

**Figure CF.1.** Upper panel:
the real section of
the complex cusp
in CF.26 and its
two local bad
cylinders.
The vector coordinates
are ordered \((t,x)\);
the plotted axes are
ordered \((x,t)\).
The general spanning
argument is CF.C
and the common
bad-set calculation
is CF.F.
Lower panel: the contour
in CF.27 has upper
semicircular detours
of radius \(1/16\)
about the marked
points \(\pm1/8\).
CF.D–CF.E proves
the full matrix
memory construction;
this scalar example
calibrates its signs.
Free human source
context:
[Kashiwara–Kawai, Holonomic systems III, IV.4.5](https://www.kurims.kyoto-u.ac.jp/~kenkyubu/kashiwara/HolIII.pdf).
Full figure,
editable SVG,
exact data
and figure source.

Theorem CF.1 proves
continuation on one
common ordinary
neighbourhood for
every prescribed
finite list.
It does not assert
finite rank of the
whole directional
solution space,
a common monodromy
annihilator under
all infinite-order
operators, singular
evaluation separation,
or the faithful
D-type embedding.
Those require their
own arguments.


### 5.83. Finite directional solution spaces and transverse fibres

Let \(M\) be the whole actual holonomic
microlocal module in generic position
at \(p_0=(0,0;1,0)\), as in
§5.82, CF.1.
Write \(X=\mathbf C_t\times\mathbf C_x^n\),
\(F(t,x)=x\), and \(S=\pi(\operatorname{Supp}M)\).
Use the directional function module
\(\mathcal C_{p_0}\) of CF.2, with its
complete actual finite, infinite and real
kernel actions proved in CF.A.

There is a bounded finite actual resolution
of the whole module on the nonzero-time
covector chart, together with its
compatible canonical kernel complex
\(P^\bullet\), by
§5.31, L.4–L.6
and
§5.49, ISF.3–ISF.4.
It is exact outside degree zero throughout
that chart. Its degree-zero support
can be arranged to satisfy
\[
|\xi|\le|\tau|.
\tag{FD.1}
\]
All complexes below use those identical
finite matrices and the genuine
module-derived supported objects.
No map from arbitrary canonical kernels
to finite microlocal symbols is used.

**Theorem FD.1.** The entire space
\[
V(M)=\operatorname{Hom}_{\mathscr E_{p_0}}
                 (M_{p_0},\mathcal C_{p_0})
\tag{FD.2}
\]
is finite dimensional, and
\[
\begin{gathered}
\operatorname{Ext}_{\mathscr E_{p_0}}^k
                 (M_{p_0},\mathcal C_{p_0})=0,\\
k>0.
\end{gathered}
\tag{FD.3}
\]
More precisely it is canonically
identified, through the finite support
comparisons below, with the direct sum
of the directional solution spaces at
all the points of a suitable generic
transverse fibre.
If the whole infinite extension at
the \(j\)-th such conormal point has
the constant Euler rank \(r_j\), then
\[
\dim_{\mathbf C}V(M)=\sum_j r_j.
\tag{FD.4}
\]
The ranks include all blocks and
principal nilpotents.

Consequently a basis of all directional
solutions and any prescribed finite
list of module elements have
representatives continuing on one
common ordinary neighbourhood
minus \(S\), by Theorem CF.1.
Stability under every continued
monodromy branch, singular evaluation
separation and full infinite-order
linearity require further arguments.

#### FD.A. Vanishing on an inner open

Put \(u=-\operatorname{Re}t\), and use
the common round coefficient domain
and future cap of
§5.81, CS.3.
Thus the relevant closed carriers are
compact in \(U\), the auxiliary
truncated ray hull is compact in
the one coefficient domain \(D\),
and the common future is
\(E=\{u>e\}\).
Let \(G_0\) be the negative real
time ray.

Suppose \(I\subset O\) are \(G_0\)-opens,
\(Y=O\setminus I\), and
\[
\overline Y\Subset U,\qquad O\cap S=\varnothing.
\tag{FD.5}
\]
Then
\[
\begin{gathered}
R\mathcal Hom_R\bigl(P^\bullet,\\
 R\Gamma_Y(q_*\mathcal O|_O)\bigr)=0,\\
R=E(G_0;D).
\end{gathered}
\tag{FD.6}
\]
The ambient closure of \(Y\) is
allowed to meet \(S\) outside \(O\).
This assertion uses \(Y\) closed
in the cone topology of \(O\);
it is not a change to the
closed-carrier hypothesis in CP.F.

Here is its local proof.
Fix \(y\in Y\). The truncated
\(G_0\)-ray from \(y\) through
the level \(u=e+\eta\), for
small \(\eta>0\), is compactly
inside \(O\). A small closed ball
about \(y\), followed by that
truncated ray, is still inside
\(O\). For
\[
G_a=\{u\ge a(|\operatorname{Im}t|+|x|)\},
\tag{FD.7}
\]
the corresponding truncated
thick cones decrease to this
compact set as \(a\) increases.
Consequently choose \(\delta>0\)
and \(a\) so that the closed
truncation through \(u=e\) of
\[
V=(B_\delta(y)+G_a)\cup E
\tag{FD.8}
\]
is compactly in \(O\) and \(D\).
These \(V\) are a cofinal basis
for the supported stalk calculation:
adjoining \(E\) changes nothing,
since the supported object is
zero there, and CS.5 proves the
cofinality on every original
cone-open neighbourhood.

On \(V\) the inner open is
\(J\cap V\), where \(J=I\cup E\).
The closed carrier
\(\overline{V\setminus J}\)
is compactly inside \(O\), hence
disjoint from \(S\).
For \(b\ge a\), use the erosion
\[
J(b)=\{z:z+G_b\subset J\}.
\tag{FD.9}
\]
The uniform two-piece cone
decomposition CS.8 proves
\[
\bigcap_{b\ge a}
 \overline{V\setminus J(b)}
 =\overline{V\setminus J}.
\tag{FD.10}
\]
All these closed carriers are
compact after the common future
truncation. Thus for sufficiently
large \(b\) they are compactly
inside \(D\) and disjoint from
\(S\).

The round-domain hypothesis is
also preserved. The order-diamond
hull of a compact subset of
the round \(D\) is compact in
\(D\): the positive-functional
length bound CP.6 bounds its
cone parameters, and roundness
puts all the limiting intervals
inside \(D\). A sufficiently
small round neighbourhood of
that compact hull remains in
\(D\). Include its forward
segments through \(u=e\)
in the hull: they lie in
the original closed truncated
cone, whose extension through
\(e+\eta\) was chosen
compactly in \(O\cap D\).
Small enough neighbourhoods
therefore retain the common
future-cap condition.
Use one such neighbourhood
for one sufficiently late closed
carrier and all its successors.
It may meet \(S\) elsewhere;
CP.F requires actual acyclicity
only over the changing closed
carrier.

Apply
§5.80, CP.F
to \(V\) and \(V\cap J(b)\),
with the original operator
carrier \(G_0\), cone field
\(\operatorname{int}G_b\)
and time function \(u\).
Every required negative covector
has nonzero time coefficient,
and every base point is outside
\(S\). The whole actual complex
is exact there. The resulting
supported Hom is zero in
every degree.

The increasing complements
\(V\cap J(b)\) give exactly
the derived inverse limit of
CS.9. The explicit
\(1-\mathrm{shift}\) sequence
CP.24 kills both its limit
and first derived limit,
since all the cohomology
groups are zero.
Thus every cohomology stalk
at \(y\) vanishes.
Off \(Y\) the supported
object is zero as well.
This proves FD.6.
All terms use the same
auxiliary module resolution
and coefficient domain before
the erosion limit.

#### FD.B. Support pairs with no inner closure buffer

Let \(I\subset\Omega\) be
\(G_0\)-opens and let \(W\)
be an ordinary transverse
open set. Assume
\[
\begin{gathered}
\overline{\Omega\setminus I}\Subset U,\\
S\cap F^{-1}(W)\subset\Omega\setminus I.
\end{gathered}
\tag{FD.11}
\]
There is no closure on
the inner open in the
second condition.
Set
\[
\begin{gathered}
\mathscr A_{\Omega,I}
 =R\mathcal Hom_R\bigl(P^\bullet,\\
 R\Gamma_{\Omega\setminus I}
                  (q_*\mathcal O|_\Omega)\bigr),\\
\mathscr L_{\Omega,I}
 =R(F|_\Omega)_*\mathscr A_{\Omega,I}|_W.
\end{gathered}
\tag{FD.12}
\]
This complex is canonically
independent of the pair
satisfying FD.11.
Every one of its cohomology
sheaves is locally constant
on \(W\).

For independence compare two
pairs over \(W'\Subset W\).
Use the common outer
intersection and the union
of the two inner opens
intersected with that outer
intersection, as in CS.12.
In the outer restriction
triangle the removed closed
carrier is contained in the
original compact carrier
and outside one of the
outer opens. It avoids
\(S\), since \(S\) is
inside both outer opens
over \(\overline{W'}\).
CS.A makes that term zero.

For inner enlargement
\(I\subset I'\subset\Omega\),
write \(j:I'\hookrightarrow\Omega\).
The third support-triangle
term, after finite Hom, is
\[
\begin{gathered}
Rj_*R\mathcal Hom_R\bigl(P^\bullet|_{I'},\\
 R\Gamma_{I'\setminus I}
                 (q_*\mathcal O|_{I'})\bigr).
\end{gathered}
\tag{FD.13}
\]
It is zero by FD.A, since
the outer domain \(I'\)
avoids \(S\).
The closure of the inner
difference may meet \(S\)
on the boundary of \(I'\);
FD.A is precisely the
local assertion needed here.

We verify FD.13 as a map
of actual complexes.
For \(I\subset I'\), the
three restriction maps
from \(\Omega\) to \(I'\)
to \(I\) give the three
relative cones in a finite
commuting square.
Its total cone is the
support triangle displayed
above. In the strict
module-derived models of
DL.6–DLG.7 these are the
same support cubes.
The exact inverse-image
functor \(j^{-1}\) has
right adjoint \(j_*\);
therefore \(j_*\) preserves
injectives. Each finite
free term of \(P^\bullet\)
gives only a finite power
of that injective complex.
Its matrix entries commute
with restriction.
Thus finite internal Hom
commutes with this derived
open image, term by term,
including its differential.
This proves FD.13 with its
actual comparison map.
No exactness over the
polynomial coefficient
ring is needed.

A third pair is compared
through the intersection
of all three outer opens
and the union of their
inner opens there.
The same finite cubes
commute, proving the
cocycle identity.

For local constancy,
shrink \(W\) relatively
compactly. The set
\(S\cap F^{-1}(\overline W)\)
is compact in \(U\), by
FD.11. Choose an ordinary
neighbourhood of it with
closure compact in \(U\),
independently of the old
inner boundary.
The order-diamond
construction CS.14–CS.15
gives a new thick-cone
pair whose closed
difference is compact
in \(U\), and for which
\(S\) is outside the
closure of its inner
open. The strong
CS.C–CS.F theorem applies
to that pair.
The pair independence
just proved compares it
canonically with the
original pair.
This proves the asserted
local constancy.

On a ball contained in
\(W\) there are natural
isomorphisms
\[
\begin{gathered}
H^k(B;\mathscr L_{\Omega,I})\\
 \simeq\mathcal H^k(\mathscr L_{\Omega,I})_x,\\
x\in B.
\end{gathered}
\tag{FD.14}
\]
Indeed the constant-vector
sheaf resolution and radial
homotopy CS.33 give zero
higher cohomology on balls,
even for arbitrary vector
fibres. In each fixed
degree the bounded-below
truncation argument CS.34
uses only a finite tower
of these sheaves.
Derived sections of the
remaining truncation start
in the next degree.
This proves FD.14 and its
restriction compatibility.

#### FD.C. Flat tangent geometry and cofinal supports

The whole projected support
has tangent cone in
\(\{t=0\}\).
Here is a proof including
its singular fibres.
The finite normalized
projection in CF.B and
CF.1 imply that, for
every \(\eta>0\), one
may shrink so that
\[
|\xi|\le\eta
\quad\text{on }A\cap\{\tau=1\}.
\tag{FD.15}
\]
Otherwise a compact
subsequence of finite
fibre lifts would limit
to a different normalized
covector above zero.

On every smooth graph
\(t=h_j(x)\) away from
the finite discriminant,
the Liouville form
vanishes on the conic
Lagrangian.
To see this, contract
the symplectic form
with the homogeneous
radial vector field,
which is tangent to
that Lagrangian.
Both tangent arguments
give zero. Its contraction
is the Liouville form.
Thus
\[
\begin{gathered}
dh_j=-\sum_i\xi_i\,dx_i,\\
|dh_j|\le\eta.
\end{gathered}
\tag{FD.16}
\]

Take a transverse \(x\)
whose complex radial line
is not contained in the
discriminant.
Its discriminant equation
has no zero near the
scalar origin other than
that origin, and only
finitely many zeros on
compact segments away
from it. Join \(rx\)
to \(x\) by a radial
segment with arbitrarily
small planar detours
around those zeros.
Its length tends to
\((1-r)|x|\).
Lift it backwards from
any prescribed root over
\(x\).
The root over \(rx\)
tends to zero, by the
monic root bound.
Integrating FD.16 and
letting the detours and
\(r\) tend to zero gives
\[
\begin{gathered}
|t|\le\eta|x|\\
\text{on every such fibre}.
\end{gathered}
\tag{FD.17}
\]
These line directions
are dense, since a
nonzero first homogeneous
part of the discriminant
does not vanish on a
dense open set.
Root multiset continuity,
proved in
analytic parametrization, §4,
extends FD.17 to every
fibre. This proves the
tangent assertion for
all branches and all
collisions.

Let \(G_+=-G_0\).
The closed positive
ray hull \(S+G_+\)
is an admissible
directional support
near zero.
If \(y=s+g\to0\),
its transverse coordinates
are those of \(s\).
The finite root bound
forces \(s\to0\),
and then \(g\to0\).
For every \(\eta>0\),
FD.17 gives
\[
\operatorname{Re}t_y\ge-\eta|x_y|.
\tag{FD.18}
\]
Every tangent limit
therefore has nonnegative
real time.
Local closedness follows
by truncating the ray
over a smaller closed
transverse ball; the
discarded far tail
cannot approach zero.

Start with any support
in CF.2. Enlarge it by
this positive hull.
Finite unions retain
the tangent condition,
by a subsequence in
one of the two sets.
Apply the convex-majorant
construction CF.A to
that union.
Consequently there is
a cofinal family of
convex-future supports
\(Z\) which contain
\(S\), have \(G_0\)-open
complement, and still
satisfy
\[
C_0(Z)\subset\{\operatorname{Re}t\ge0\}.
\tag{FD.19}
\]
They avoid each fixed
strict negative cone
after a sufficiently
small shrink.

#### FD.D. The actual directional Hom and a fixed transverse complex

Take a small product
\[
\begin{gathered}
B_{\epsilon,\delta}
 =\{|t|<\delta,\ |x|<\epsilon\},\\
\Omega=B_{\epsilon,\delta}+G_0.
\end{gathered}
\tag{FD.20}
\]
For a support \(Z\)
from FD.C put
\(I=\Omega\setminus Z\).
The product can be
chosen so that the
closed changing carrier
is compact in \(U\).
This is the thin-ray
case of the explicit
cap estimate CF.7:
the transverse coordinates
are bounded by the
initial product and
the future boundary
is bounded by its
convex modulus.
Ordinary excision compares
the group on \(\Omega\)
with its ordinary germ
neighbourhood.
Both \(\Omega\) and
\(I\) are convex, so
the support complex is
concentrated in degree
one, with quotient
\(\mathcal O(I)/\mathcal O(\Omega)\).
The exact filtered limit
of those quotients is
\(\mathcal C_{p_0}\),
by CF.A.

Fix one cone
\[
\begin{gathered}
W_a=\bigl\{(t,x):\\
\operatorname{Re}t<-a(|\operatorname{Im}t|+|x|)\bigr\}
\end{gathered}
\tag{FD.21}
\]
and the pair
\((\Omega,\Omega\cap W_a)\).
Its closed difference
is bounded: imaginary
time and \(x\) are
bounded by the product,
real time is bounded
on the right by
\(\delta\), and outside
\(W_a\) it is bounded
on the left by
\(-a(\delta+\epsilon)\).
Choose the radii small
enough to put this
whole closure inside
\(U\).
FD.17 puts every
point of \(S\) over
a smaller transverse
ball inside \(\Omega\)
and outside \(W_a\).
Both this fixed pair
and every sufficiently
refined support pair
satisfy FD.11.

The pair comparisons
in FD.B and their
cocycle identity identify
all their direct-image
complexes over a common
smaller base ball.
For varying radii use
their common outer
intersection and inner
union. For varying
transverse balls use
FD.14. Thus the
identifications commute
with every refinement.
There is one fixed
coefficient domain and
auxiliary module resolution
throughout these comparisons.

The finite actual
resolution of \(M\)
computes its Ext into
the directional quotient.
Finite powers commute
with the exact filtered
class limit, and a
finite system of kernel
or boundary equalities
holds on one common
refinement. The action
of each actual matrix
on a representative is
the calibrated canonical
action in CF.A.
Consequently, if
\(\mathscr L\) denotes
the fixed-pair complex
FD.12, then
\[
\begin{gathered}
\operatorname{Ext}_{\mathscr E_{p_0}}^k
 (M_{p_0},\mathcal C_{p_0})\\
\simeq\mathcal H^{k+1}(\mathscr L)_0.
\end{gathered}
\tag{FD.22}
\]
The shift is explicit:
\(R\Gamma_Z\mathcal O\)
has its quotient in
degree one, so its
finite Hom has that
same shift.
The comparison uses
identical finite matrices;
it does not require
the canonical complex
to be exact over its
polynomial coefficient
ring.

In particular this
identifies the entire
directional Hom space
with one fixed transverse
cohomology stalk.
A representative domain
depending on an individual
solution is not being
used to choose a different
generic fibre for that
solution.

#### FD.E. Euler solutions in the full directional module

For a finite constant
matrix \(A\), let
\(N_A\) be the actual
Euler module of
§5.54, EE.2.
It has the finite free
resolution EE.11.
Its solution complex
in \(\mathcal C_n\)
is the cochain Koszul
complex on
\[
\begin{gathered}
tD_tI-A,\\
D_{x_1}I,\ldots,D_{x_n}I.
\end{gathered}
\tag{FD.23}
\]

Use the cofinal quotient
representatives of CF.A
on
\[
\begin{gathered}
U_{\phi,r,\epsilon}
 =(\Delta_r\times B_\epsilon)\\
\cap\bigl\{(t,x):\\
\operatorname{Re}t<-\phi(|\operatorname{Im}t|+|x|)\bigr\}.
\end{gathered}
\tag{FD.24}
\]
Both this domain and
the full product are
preserved by
\((t,x)\mapsto(t,sx)\),
\(0\le s\le1\).
The exact radial
homotopy EE.14–EE.15
therefore descends to
their quotient.
It commutes with
\(tD_tI-A\).
On a total cochain
\(\alpha+e_0\wedge\beta\)
its homotopy is
\(H_x\alpha-e_0\wedge H_x\beta\),
so the mixed terms
cancel.
Evaluation at \(x=0\)
and extension independent
of \(x\) contract the
tangential complex onto
the one-variable quotient
\[
\begin{gathered}
Q_{\phi,r}
 =\mathcal O(U_{\phi,r})/\mathcal O(\Delta_r),\\
U_{\phi,r}=\Delta_r\cap\\
 \{\operatorname{Re}t<-\phi(|\operatorname{Im}t|)\}.
\end{gathered}
\tag{FD.25}
\]
The extension is
defined because FD.24
projects into \(U_{\phi,r}\).
All maps commute with
the support refinements.

On \(W=\mathcal O(U_{\phi,r})^q\)
the remaining operator
\(T=tD_tI-A\) is
surjective. The domain
is convex in the strict
left half-plane.
With its common logarithm
branch and an anchor
\(a\in U_{\phi,r}\), set
\[
v(t)=t^A\int_a^t s^{-A}g(s)\,\frac{ds}{s}.
\tag{FD.26}
\]
Differentiation gives
\(Tv=g\); its kernel
is \(t^A\mathbf C^q\).
No boundary growth
estimate is used.
On \(V=\mathcal O(\Delta_r)^q\),
Taylor expansion gives
the coefficient maps
\(mI-A\), \(m\ge0\).
For \(m>2\|A\|\)
their inverse norm is
at most \(2/m\);
the remaining invertible
maps have a finite
uniform bound.
Hence the coefficient
solutions converge on
each compact subdisc,
and
\[
\begin{gathered}
\ker_VT\\
=\bigoplus_{m\ge0}t^m\ker(mI-A),\\
\operatorname{coker}_VT\\
 =\bigoplus_{m\ge0}\operatorname{coker}(mI-A).
\end{gathered}
\tag{FD.27}
\]
Only the finitely
many noninvertible
summands are retained.
Their dimensions agree.
The elementary quotient
sequence
\[
\begin{gathered}
0\to\ker_VT\to\ker_WT\\
\to\ker_{W/V}T\\
\to\operatorname{coker}_VT\to0
\end{gathered}
\tag{FD.28}
\]
follows by subtracting
a \(V\)-preimage when
the obstruction vanishes;
surjectivity at its
last term is FD.26.
It proves quotient
kernel dimension \(q\)
and zero quotient
cokernel.

The \(q\) explicit
power/logarithm columns
\(F_A\) of EE.21 are
defined on the entire
left half-disc.
Their classes remain
independent on every
\(U_{\phi,r}\):
if a combination
extends as a holomorphic
germ there, the identity
theorem gives the same
extension on the smaller
left half-disc.
The pole, jump and
Jordan triangular argument
EE.21 makes all its
coefficients zero.
They consequently form
a basis of every
refined quotient kernel.
Every transition carries
these same columns
to themselves.
Taking the filtered
limit and the finite
resolution gives
\[
\begin{gathered}
\dim\operatorname{Hom}_E(N_A,\mathcal C_n)=q,\\
\operatorname{Ext}_E^k(N_A,\mathcal C_n)=0,\\
k>0.
\end{gathered}
\tag{FD.29}
\]

The right \(E\)-flatness
of the whole infinite
ring \(B\), proved in
§5.47,
and ordinary Hom
adjunction give the
identical complex for
\(B\otimes_E N_A\)
over \(B\).
CF.A provides the
full \(B\)-action
on \(\mathcal C_n\).
Thus FD.29 holds
there as well.

For clarity, the
halfspace module
of EE.3 injects in
\(\mathcal C_n\):
the closed support
\(\{\operatorname{Re}t\ge0\}\)
is one admissible
support in CF.2.
If its class becomes
a holomorphic germ
after a refinement,
the identity theorem
on the connected
left half-product
shows that its
original class was
already zero.
The inclusion respects
the full real kernel
action, since both
actions use the same
supported cup, excision
and trace maps.
In particular the
full-coefficient detector
EE.4 remains injective
with target \(\mathcal C_n^q\).

#### FD.F. A generic fibre of the entire complex

Choose a small transverse
patch outside the finite
discriminant. Its sheets
are holomorphic graphs
\(t=h_j(x)\).
The actual smooth-branch
construction
§5.69, GM.A–GM.B
gives a proper analytic
coefficient-minor exceptional
set on each sheet.
Avoid the finitely
many such sets.
At every remaining
point the entire
full-infinite module
is actually isomorphic
to a constant Euler
module of finite
rank \(r_j\).
This is the proved
inverse Gevrey gauge,
not a formal normal
form.

One may also arrange
that the sheet values
have distinct imaginary
parts.
No difference \(h_j-h_k\)
can be a nonzero
constant \(c\) on
an open set.
Otherwise the resultant
\[
\operatorname{Res}_t(p(t,x),p(t+c,x))
\tag{FD.30}
\]
vanishes there and
hence identically on
the connected original
base. At \(x=0\),
where \(p(t,0)=t^m\),
it is a nonzero
power of \(c\),
a contradiction.
A nonconstant holomorphic
function cannot have
zero imaginary part
on an open set:
the Cauchy–Riemann
equations would make
all its derivatives
zero there.
Thus these finitely
many imaginary-equality
sets have empty interior.
Choose \(x_*\) outside
them and the analytic
exceptional sets.

Over a smaller ball
about \(x_*\), choose
\(\rho>0\) with
\[
\begin{gathered}
|\operatorname{Im}(h_j-h_k)|>4\rho,\\
j\ne k.
\end{gathered}
\tag{FD.31}
\]
Put \(T_j=t-h_j(x)\).
The \(G_0\)-opens
\[
\begin{gathered}
\Omega_j=\bigl\{(t,x):x\in B_r(x_*),\\
 |T_j|<\rho\bigr\}+G_0,\\
I_j=\Omega_j\cap\\
\{\operatorname{Re}T_j<-\rho/2\}
\end{gathered}
\tag{FD.32}
\]
are pairwise disjoint
in \(j\).
Their closed changing
carriers are compact
in the fixed ordinary
coefficient neighbourhood,
after choosing the
base ball with closure
inside the graph patch.
They include every
sheet over a smaller
base ball, with a
strict inner buffer.
Their union satisfies
FD.11.

Pair independence
compares this union
with the fixed pair
of FD.D.
Disjointness gives
the direct sum of
its component supported
complexes.
Shrink the transverse
ball and each normal
cap. In the ordinary
graph coordinate
\(T_j\), apply the
local construction
of FD.C–FD.D at
the conormal point
\[
\begin{gathered}
p_j=(h_j(x_*),x_*;\\
1,-dh_j(x_*)).
\end{gathered}
\tag{FD.33}
\]
It compares the
component cap with
the local directional
module \(\mathcal C_{p_j}\).
All comparisons use
the same finite
actual matrices and
the same strict
supported models.
FD.14 makes every
transverse shrinking
map an isomorphism.
The weak pair comparison
makes every sufficiently
small normal-cap map
an isomorphism.
Consequently the cap
and its filtered
directional limit have
the same finite Hom
cohomology.

The graph time shift
has the actual finite,
infinite and real
coordinate maps proved
in
§5.56, BCT.5.
It also preserves
the thin carrier
exactly: on its
spatial diagonal the
two values of \(h_j(x)\)
are equal, so the
time difference is
unchanged. Its support
cube and trace maps
therefore identify
the preceding canonical
comparisons with the
actual local directional
actions.
We compare the Hom
complexes after these
maps; an unrestricted
cap function is not
being identified with
a directional germ.

At \(p_j\) the
actual \(B\)-module
gauge and finite
resolution identify
\[
\begin{gathered}
R\operatorname{Hom}_{E_{p_j}}
       (M_{p_j},\mathcal C_{p_j})\\
\simeq R\operatorname{Hom}_{B_{p_j}}\bigl(\\
 B_{p_j}\otimes_{E_{p_j}}M_{p_j},\\
 \mathcal C_{p_j}\bigr).
\end{gathered}
\tag{FD.34}
\]
Indeed tensor the
finite actual free
resolution with the
right-flat \(B_{p_j}\);
ordinary adjunction
identifies each Hom
term and differential.
Use the actual inverse
Euler gauge and FD.29.
Each summand is
concentrated in degree
zero and has dimension
\(r_j\).

FD.14 transports that
calculation back to
the stalk at zero.
Together with the
shift FD.22 it proves
FD.3 and FD.4 for
the entire original
module.
For \(n=0\) the
transverse base is
a point and this
is the same local
Euler calculation;
the finite discriminant
step is empty.
The zero module is
immediate.

Choose a basis of
the resulting finite
space \(V(M)\).
For any finite list
of module elements,
Theorem CF.1 gives
one common continuation
domain for the basis
representatives.
Their linear combinations
give representatives
for every element
of \(V(M)\) on
that same domain.
This last statement
does not yet identify
every continued branch
with another actual
module homomorphism.
Monodromy compatibility,
singular separation
and faithful full
infinite-order linearity
remain separate.

#### FD.G. Exact transverse geometry

![A real cusp slice, two disjoint complex-time caps, and the transverse solution comparison](assets/finite-directional-fibres.png)

The upper panel is only
the real slice of \(t^2=x^3\).
Its negative-time cone has
vertex on the cusp, while
its open interior misses
the cusp near zero, as
required by FD.11.
The middle panel takes
\(x_*=-1/4\), with
roots \(t=\pm i/8\)
and conormals
\((\tau,\xi)=(1,\pm3i/4)\).
For \(\rho=1/32\)
the two disc-plus-ray
opens of FD.32 are
disjoint: their imaginary
separation is \(1/4>4\rho\).
The shaded closed caps
start at \(\operatorname{Re}T=-\rho/2\).
The last panel records
the degree-one shift
FD.22 and the finite
Euler sum FD.29–FD.34.

Full figure, editable SVG, exact data and figure source.

Free human source
context:
[Kashiwara–Kawai, Holonomic systems III, IV.4.4 and IV.4.6](https://www.kurims.kyoto-u.ac.jp/~kenkyubu/kashiwara/HolIII.pdf).


### 5.84. Directional functionals separate the whole infinite extension

Keep the whole actual generic-position
holonomic module \(M\) at
\(p_0=(0,0;1,0)\), its projected
hypersurface \(S\), and its
directional function module
\(\mathcal C_{p_0}\) from
§5.82, CF.A.
Put
\[
\begin{gathered}
E=\mathscr E_{p_0},\quad
B=\mathscr E^\infty_{p_0},\\
N=B\otimes_E M,\\
V=\operatorname{Hom}_E(M,\mathcal C_{p_0}).
\end{gathered}
\tag{SE.1}
\]
All tensors are algebraic.
The complete actual kernel action
makes \(\mathcal C_{p_0}\) a
left \(B\)-module.
Every \(\Phi\in V\) consequently
has the unique \(B\)-linear extension
\[
\Phi^\infty(b\otimes m)=b\Phi(m).
\tag{SE.2}
\]
The formula respects the finite-ring
balancing relation because
\(\Phi(Pm)=P\Phi(m)\).

**Theorem SE.1.** If \(s\in N\) satisfies
\[
\Phi^\infty(s)=0
\quad\text{for every }\Phi\in V,
\tag{SE.3}
\]
then \(s=0\).
In particular, for any basis
\(\Phi_1,\ldots,\Phi_q\) of the
entire finite-dimensional space \(V\),
the actual \(B\)-linear map
\[
\begin{gathered}
N\longrightarrow\mathcal C_{p_0}^{\,q},\\
s\longmapsto
 \bigl(\Phi_1^\infty(s),\ldots,\Phi_q^\infty(s)\bigr)
\end{gathered}
\tag{SE.4}
\]
is injective.
Finite dimensionality is
§5.83, FD.1.
The assertion includes every
infinite coefficient tail, singular
characteristic point and principal
nilpotent of the whole module.

The target in SE.4 is the
directional function module.
A finite coherent D-type target
and the compatibility of continued
branches with all operators
require additional proofs.

#### SE.A. Identity for actual infinite sections on a smooth branch

First consider the whole constant
Euler extension \(T_A=B\otimes_E N_A\)
over a graph chart
\(\Lambda=\{t=z=0\}\),
where \(z=\xi/\tau\).
The full coefficient normal form
§5.52, AM.A
gives a unique row
\[
R=\sum_{j\in\mathbf Z}r_j(x)\tau^j
\tag{SE.5}
\]
for each locally represented section.
Every \(r_j\) is holomorphic
on one common parameter domain.
The positive every-epsilon and
negative factorial estimates
of AM.2 hold there.

We check the required restriction
property of this normal form.
Its spatial division AM.5 consists
of finite holomorphic differences
and divisions by spatial variables.
Its normal division AM.8–AM.10
consists of evaluation at \(t=0\),
the holomorphic quotient
\((f-f|_{t=0})/t\), constant
matrix products, and normally
convergent coefficient sums.
All these operations commute
with restriction in \(x\).
The estimates AM.11–AM.15
use a common normal radius
and compact parameter norms.
They apply uniformly on every
smaller parameter ball,
with the same radius for
all homogeneous indices.
Thus the normal form on
an overlap is the restriction
of SE.5, by its proved uniqueness.
This is an identity of actual
coefficient sequences, including
both tails.

Suppose a section of \(T_A\)
vanishes on a nonempty open
part of a connected parameter
ball on which that section
has one representative.
At each point of that open
part, uniqueness makes the
germ of every \(r_j\) zero.
The identity theorem makes
every \(r_j\) zero on the
whole parameter ball.
One elementary proof of that
identity theorem is to consider
the set where all Taylor
coefficients of a holomorphic
function vanish. It is closed
by continuity of its derivatives
and open by its convergent
Taylor expansion. If nonempty
on a connected domain it is
the entire domain.
This argument applies to
each of the countably many
coefficients separately; their
common domain was fixed first.
The actual zero normal form
makes the section zero.

Local representatives suffice.
The set of parameter points
where a sheaf section has zero
germ is open. At a boundary
point choose one representative
on a smaller connected parameter
ball. That ball meets the
open zero set, so the preceding
argument makes its entire
normal form zero.
The zero set is therefore
also relatively closed.
Consequently a section of the
whole Euler extension that is
zero on a nonempty open part
of a connected graph patch
is zero on that patch.

Now let the reduced support
of an arbitrary holonomic
\(M\) be a single smooth
graph on a small chart.
The graph time shift and its
actual finite and infinite
maps are
§5.56, BCT.5.
They put the graph into
\(\Lambda=\{t=z=0\}\).
The actual lattice construction
and finite minor argument in
§5.69, GM.A–GM.C
give a nonzero holomorphic
\(g(x)\) on the parameter
ball \(B_0\).
On \(\{g\ne0\}\), the
entire infinite extension is
locally isomorphic to a
constant Euler extension by
the two actual inverse
gauge matrices GM.9–GM.11.

The complement of \(\{g=0\}\)
in a complex ball is path
connected. For two points
outside it, restrict \(g\)
to their affine complex line.
It is not identically zero
because its value at the
first point is nonzero.
Its zeros on a compact
neighbourhood of the real
joining segment are finite.
Replace the segment by a
path with small planar detours
around those zeros.
The segment has a positive
margin inside the ball,
so the detours stay inside.
Isolated zeros and their
finiteness follow directly
by factoring out the first
nonzero Taylor power locally
and using a finite compact
cover.

If an infinite section is zero
on a nonempty open part of
the graph, that part meets
\(\{g\ne0\}\).
The Euler identity just proved
makes its zero set both open
and closed on the connected
set \(\{g\ne0\}\).
It is zero there.
Off \(\Lambda\) the source
module is zero, by the full
extension support equality
§5.49, ISR.13.
The section is thus supported
on the fixed closed analytic set
\[
Z_g=\Lambda\cap\{g(x)=0\}.
\tag{SE.6}
\]
If \(d=\dim_{\mathbf C}B_0\),
then its dimension is at most
\(d-1\), while the normalized
ambient chart has dimension
\(2d+1\).
Its codimension is at least
\(d+2\), strictly larger than
the base dimension \(d+1\).
The actual full-coefficient
support-purity theorem
§5.51, PG.1
kills this section.
No analyticity of its own
support has been assumed.
For \(g=1\) the exceptional
set is empty.
For \(d=0\), the parameter
base is a point and the
Euler identity is immediate.

We have proved the following
local assertion at every isolated
smooth characteristic branch
which is a graph over
the transverse base,
including its coefficient-minor
exceptional points:
an actual infinite section
which is zero on a nonempty
open part of a connected
smooth graph patch is zero
on that patch.
The proof used actual normal
coefficients and one local
analytic purity step.

#### SE.B. One canonical expression for an arbitrary infinite germ

Keep the actual whole-module
generators \(u_1,\ldots,u_b\)
and finite polynomial complex
of
§5.49, ISD.1 and ISR.1–ISR.2.
The full Späth division gives
for the arbitrary element
\(s\in N\) a finite expression
\[
s=\sum_{i=1}^b a_i u_i,
\qquad a_i\in A^\infty,
\tag{SE.7}
\]
where each \(a_i\) is
independent of the spatial
covariable \(z\).
All other relations of \(M\)
remain in force.
This generation is ISR.2;
it is neither a free-module
identification nor a formal
coefficient completion.
Finite sums and the actual
division estimates give
one common coefficient
domain for this expression.
The coefficients extend
over the finite normalized
covariable product, as
ISF.9 proves.
After shrinking, SE.7 is
an actual section expression
on the whole finite support
representative used below.
Indeed its germ equality
has an ordinary neighbourhood
witness, by the finite
balancing argument FEP.9.
The normalized support
projection is finite and
has only \(p_0\) over
the base origin.
The image of its closed
part outside that witness
is therefore closed and
misses zero, after choosing
a compact finite representative.
A smaller base ball
puts its entire support
inside the witness.

These particular coefficients
also have actual canonical
thin-ray kernel representatives.
Here is the analytic reason.
Write
\[
a=\sum_{j\in\mathbf Z}a_j(t,x)D_t^j.
\tag{SE.8}
\]
For \(j\ge0\), the positive
time kernel has coefficients
\(j!a_j(t,x)/(s-t)^{j+1}\),
with the common Cauchy
normalization of AM.B.
On any compact set with
\(|s-t|\ge\delta>0\),
choose the input epsilon
less than \(\delta/2\).
The every-epsilon bound
makes its series geometric.
For the negative part use
the Volterra kernel
\[
\begin{gathered}
K_a(t,x,s)=\\
\sum_{n\ge1}a_{-n}(t,x)
             \frac{(t-s)^{n-1}}{(n-1)!},\\
|K_a|\le
\frac{MC}{(1-C|t-s|)^2}.
\end{gathered}
\tag{SE.9}
\]
The bound follows from
\(|a_{-n}|\le MC^n n!\)
and the differentiated
geometric series.
It holds for \(C|t-s|<1\).
Spatial variables are diagonal
kernel variables, since no
spatial derivative occurs.
Every fixed parameter derivative
has the same assertion after
one reserved Cauchy margin.

The complete cochain,
primitive and composition
calibration in
§5.53, AM.B
and
§5.49, ISF.3–ISF.4
identifies these kernels
with the full \(B\)-actions.
It also identifies their
supported restrictions and
the action on the actual
directional module CF.A.
Use one coefficient domain
for the finite list \(a_i\)
and all presentation matrices.
The kernel support carrier
remains the original negative
time ray \(G_0\).

Let \(R=E(G_0;D)\) on this
coefficient domain, and let
\(P^\bullet_R\) be the
matched finite canonical
complex of FD.1, in
nonpositive degrees.
Its actual finite counterpart
resolves the whole \(M\)
on the nonzero-time chart.
Put
\[
\begin{gathered}
\mathfrak M=H^0(P_R^\bullet),\\
\widetilde s=\sum_i a_i\bar u_i\in\mathfrak M.
\end{gathered}
\tag{SE.10}
\]
This is a canonical coefficient
module used for evaluation.
We only compare the displayed
common coefficient expressions
and their genuine function
actions.
An arbitrary canonical kernel
is not sent to a finite symbol.
Exactness of \(P_R^\bullet\)
over the polynomial coefficient
ring is not needed.

#### SE.C. Finite solution dimension gives one vanishing cap

For a sufficiently small
product and a fixed \(a>0\),
write
\[
\begin{gathered}
\Omega=B_{\epsilon,\delta}+G_0,\\
W_a=\bigl\{(t,x):\\
\operatorname{Re}t<-a(|\operatorname{Im}t|+|x|)\bigr\},\\
I=\Omega\cap W_a,\\
Q_{\Omega,I}=\mathcal O(I)/\mathcal O(\Omega).
\end{gathered}
\tag{SE.11}
\]
The closed changing carrier
has compact closure in
the fixed coefficient
neighbourhood, as FD.D proves.
Both opens are convex.
Their supported holomorphic
complex has its quotient
in degree one.
Because \(P_R^\bullet\)
is nonpositive and finite free,
its degree-one supported
Hom is literally
\[
\operatorname{Hom}_R
       (\mathfrak M,Q_{\Omega,I}).
\tag{SE.12}
\]
The weak pair comparisons
FD.B and fixed-stalk
comparison FD.D identify
SE.12 canonically with \(V\),
compatibly with sufficiently
small cap and base refinements.

Choose a basis of the entire
finite space \(V\).
Represent its finitely many
generator images and their
finite relation equalities
on one common convex
directional support.
The hypothesis SE.3 says
that the finitely many
sums SE.7 evaluated on
this basis vanish as
directional classes.
The actual kernel action
and the definition of
the filtered class limit
put all these zero equalities
on one further common
support refinement.
Take a convex future
majorant again, by CF.A.
This preserves all equalities
and all finite relation maps.

That majorant avoids the fixed
strict cone \(W_a\) after
a small enough shrink.
Supported restriction therefore
gives representatives of the
basis in SE.12 satisfying
\[
\psi(\widetilde s)=0
\quad\text{in }Q_{\Omega,I}
\tag{SE.13}
\]
for every member of that
basis.
These restrictions still form
a basis of SE.12, by the
canonical isomorphisms in
FD.B–FD.D.
Linearity proves SE.13
for every cap homomorphism.
The common cap here is
chosen using a finite basis,
not a domain chosen
separately for each of
infinitely many solutions.

#### SE.D. Evaluation survives the comparison with disjoint normal caps

Choose the generic point
\(x_*\ne0\) and all sheets
\(t=h_j(x)\) as in FD.F,
avoiding the finite
discriminant and all
generic Euler-minor exceptions.
Shrink its base ball so
that its closure misses zero.
By the flat tangent estimate
FD.17 choose \(\eta<a\)
and arrange
\[
|h_j(x)|\le\eta|x|
\tag{SE.14}
\]
on these sheets.
Let \(b=\min_{\overline{B_r(x_*)}}|x|>0\).
Choose the common normal
radius \(\rho>0\) so small
that the opens
\[
\begin{gathered}
T_j=t-h_j(x),\\
\Omega_j=\bigl\{(t,x):x\in B_r(x_*),\\
|T_j|<\rho\bigr\}+G_0,\\
I_j=\Omega_j\cap\\
\{\operatorname{Re}T_j<-\rho/2\}
\end{gathered}
\tag{SE.15}
\]
are disjoint, lie inside
\(\Omega\) over that base
ball, and contain all sheets.
Use FD.31 for disjointness,
and also impose
\[
\rho/2<(a-\eta)b.
\tag{SE.16}
\]
For every point in \(W_a\)
over this ball,
\[
\begin{gathered}
\operatorname{Re}T_j
< -a|x|-\operatorname{Re}h_j(x)\\
\le -(a-\eta)|x|<-\rho/2.
\end{gathered}
\tag{SE.17}
\]
Consequently
\[
J_j:=W_a\cap\Omega_j\subset I_j.
\tag{SE.18}
\]
Each \(J_j\) is nonempty
within the common coefficient
completion, before its future
cutoff \(u=e\).
Indeed \(x_*\) can be
chosen arbitrarily near zero.
Let \(c=\max_{\overline{B_r(x_*)}}|x|\).
In addition choose it and
\(\rho\) so small that
\[
\begin{gathered}
a(\eta+1)c+2\eta c\\
{}+2\rho<e.
\end{gathered}
\tag{SE.18a}
\]
For fixed \(x\), the point
\[
\begin{gathered}
t=h_j(x)-L,\\
L=a(\eta+1)c+\eta c+\rho.
\end{gathered}
\tag{SE.18b}
\]
is on the negative real
ray from the graph point,
lies in \(W_a\), and
has \(u<e\).
Its ray segment is
in the common coefficient
domain by CS.3.
The strict inequalities
give a nonempty open
intersection there.

First restrict the base
of SE.11 to \(B_r(x_*)\).
FD.14 makes the Hom
restriction an isomorphism.
Next restrict the outer open
to the disjoint union of
the \(\Omega_j\), retaining
the old inner open there.
The removed closed carrier
is compact and misses \(S\),
since all its sheets are
inside the new outer opens
over the base-ball closure.
The outer comparison of
FD.B is therefore an
isomorphism after finite Hom.
Its support maps commute
with evaluation at the
fixed canonical element
\(\widetilde s\).
Thus all old-inner cap
homomorphisms on the
disjoint tubes annihilate
\(\widetilde s\).

For precision, no convexity
of \(J_j\) is needed in
this step. In graph
coordinates \((T_j,x)\),
\(\Omega_j\) is biholomorphic
to a product of a convex
disc-plus-ray domain and
a ball. Its holomorphic
higher cohomology vanishes
by the explicit convex
acyclicity proof DLH.1–DLH.4
and coordinate pullback.
It is connected, and
\(J_j\) is nonempty.
Hence its degree-zero
supported cohomology is
zero and its degree-one
supported cohomology is
the quotient
\[
Q_{\Omega_j,J_j}
=\mathcal O(J_j)/\mathcal O(\Omega_j).
\tag{SE.19}
\]
The support triangle proves
these two statements even
if \(J_j\) has higher
holomorphic cohomology.
For a nonpositive finite
free \(P_R^\bullet\), terms
of supported degree at
least two contribute total
degree at least two to
its Hom.
The degree-one truncation
therefore identifies the
degree-one Hom with
\(\operatorname{Hom}_R
(\mathfrak M,Q_{\Omega_j,J_j})\).
This proves the literal
degree-zero map assertion
used above without a
Stein claim for that
intersection.

The inner comparison
FD.A–FD.B for SE.18 gives
an isomorphism on these
Hom spaces from the
new-inner pair to the
old-inner pair.
The underlying quotient
restriction
\[
Q_{\Omega_j,I_j}\longrightarrow
Q_{\Omega_j,J_j}
\tag{SE.20}
\]
is itself injective.
Indeed \(I_j\) is connected
in graph coordinates.
If \(f\in\mathcal O(I_j)\)
agrees on \(J_j\) with a
function \(g\in\mathcal O(\Omega_j)\),
the identity theorem on
the connected \(I_j\)
makes \(f=g|_{I_j}\).
Its class was already zero.

Take any homomorphism for
the disjoint new-inner caps.
Its old-inner restriction
corresponds to a homomorphism
in SE.12, so its value on
\(\widetilde s\) is zero
there by SE.13.
Injectivity of SE.20 now
proves
\[
\psi_j(\widetilde s)=0
\quad\text{in }Q_{\Omega_j,I_j}
\tag{SE.21}
\]
for every new-inner cap
homomorphism, on every
component.
This uses an actual
injection of the evaluation
targets, in addition to
the isomorphism after Hom.

The same reasoning applies
to every sufficiently small
normal cap and smaller
base ball about \(x_*\).
Condition SE.16 persists.
Outer restriction preserves
a zero evaluation.
Inner enlargement again
has injective quotient
restriction by the same
connectedness argument.
Thus SE.21 holds throughout
the shrinking-cap comparisons.

#### SE.E. The local directional evaluation is zero

At
\[
\begin{gathered}
p_j=(h_j(x_*),x_*;\\
1,-dh_j(x_*)).
\end{gathered}
\tag{SE.22}
\]
use \(T_j\) as normal
time coordinate and \(x-x_*\)
as parameter.
The actual graph coordinate
maps preserve \(G_0\) on
the spatial diagonal,
as FD.F proves.
They send SE.8 to a
time-only operator with
coefficients
\(a_m(T_j+h_j(x),x)\).
Its complete estimates
hold on one smaller
parameter product.
Thus the finite canonical
matrix comparisons and
the evaluations SE.7 use
the same actual \(B_{p_j}\)
actions as the local
directional module.

We record exactly how
zero cap evaluation detects
a local directional class.
Represent such a class
on a cofinal convex
future support \(Z\),
with completed outer
product \(\Omega_Z\).
Its numerator domain
\(I_Z=\Omega_Z\setminus Z\)
is convex.
By ordinary excision
CF.A and FD.D, the
class is represented by
\[
f\in\mathcal O(I_Z)
\quad\bmod\mathcal O(\Omega_Z).
\tag{SE.23}
\]
Choose a sufficiently small
normal cap \(\Omega'\)
and its inner \(I'\)
of the form SE.15
in these local coordinates.
Choose its parameter
radius at most its
normal radius.
The inequality
\(\operatorname{Re}T<-\rho'/2\),
with \(|\operatorname{Im}T|<\rho'\)
and \(|x-x_*|<\rho'\),
places \(I'\) inside the
fixed strict negative cone
of aperture \(1/4\).
After a sufficiently small
shrink that cone misses \(Z\).
The completed tubes also
satisfy \(\Omega'\subset\Omega_Z\).
Thus \(I'\subset I_Z\).

If the image of SE.23
in \(Q_{\Omega',I'}\)
is zero, there is
\(g\in\mathcal O(\Omega')\)
agreeing with \(f\)
on \(I'\).
The intersection
\(I_Z\cap\Omega'\)
is convex, nonempty
and connected in these
local coordinates.
The identity theorem
extends the agreement
to that intersection.
The function \(g\)
therefore extends the
directional representative
through the origin,
so its original directional
class is zero.
Only this detection
statement is used.
An arbitrary unrestricted
cap function is not
identified with a
directional germ.

Now take any
\(\Psi\in\operatorname{Hom}_{E_{p_j}}
(M_{p_j},\mathcal C_{p_j})\).
Its finite generator images,
their relation equalities
and their SE.7 evaluation
have one common convex
future support representative.
For small enough caps,
FD.C–FD.F gives its
matched cap homomorphism,
with exactly the same
matrix and coefficient
actions.
SE.21 makes its
SE.7 evaluation zero
on that cap.
The detection argument
SE.23 consequently proves
\[
\Psi^\infty(s_{p_j})=0
\quad\text{for every }\Psi.
\tag{SE.24}
\]

At this generic point the
whole \(B_{p_j}\)-module
is actually Euler by GM.B.
The full-coefficient
detector EE.4 separates
that whole module into
powers of the halfspace
quotient.
The halfspace quotient
injects into \(\mathcal C_{p_j}\)
with its actual actions,
by FD.E.
The finite right-flat
adjunction FD.34 identifies
these \(B\)-linear functionals
with the displayed
\(E\)-linear \(\Psi\)'s.
SE.24 therefore gives
\[
\begin{gathered}
s_{p_j}=0\\
\text{for every point of the}\\
\text{chosen generic fibre}.
\end{gathered}
\tag{SE.25}
\]
A zero stalk of a sheaf
section has an open
neighbourhood on which
that section is zero.
Hence this gives vanishing
on an open part of every
sheet over that fibre.

#### SE.F. Every smooth sheet and the singular exceptional set

Choose one nonzero
discriminant equation \(h(x)\)
for the finite normalized
support projection of CF.B,
enlarged by the finitely
many component intersections
and projection singularities.
Over
\[
B_0\setminus H,\qquad H=\{h=0\},
\tag{SE.26}
\]
all reduced sheets are
smooth and disjoint
finite graphs locally.
The generic fibre above
was chosen outside \(H\).
The base complement is
path connected by the
complex-line detour proof
in SE.A.

From any point of a
sheet over that complement,
lift a base path back
to \(x_*\).
Finite proper projection
and its graph charts
give the lift along
the whole compact path:
in a local graph chart
the next part is unique,
and compactness gives
a finite subdivision
and prevents escape.
The endpoint is one
of the points in SE.25.
Its zero germ propagates
backwards along the lifted
path by the smooth-branch
identity SE.A.
More explicitly, the set
of path parameters with
zero germ is open.
At a boundary parameter,
one connected smooth graph
patch meets that open
zero set; SE.A makes
the whole patch zero.
The set is also closed,
and is therefore the
entire path interval.
No transitivity assumption
on the sheet monodromy
is required, since every
point of the ending
generic fibre is zero.

It follows that the
actual section SE.7 is
zero at every point
of the reduced normalized
support above \(B_0\setminus H\).
It is zero off that
support as well, by
ISR.13.
Its support is contained
in the fixed analytic set
\[
Z=A\cap F^{-1}(H).
\tag{SE.27}
\]
Here \(A\) is the
whole reduced normalized
characteristic support,
not only its smooth part.
The support projection
is finite.
The analytic dimension
calculation in CF.B
therefore gives in the
normalized ambient chart, with
\(n=\dim_{\mathbf C}B_0\),
\[
\begin{gathered}
\dim Z\le n-1,\\
\dim(\text{ambient})=2n+1,\\
\operatorname{codim}Z\ge n+2>n+1.
\end{gathered}
\tag{SE.28}
\]
The last quantity \(n+1\)
is the ordinary complex
base dimension of \(M\).
PG.1 applies to the
entire actual \(B\)-extension
and this fixed analytic
\(Z\).
It gives \(\mathcal H_Z^0(N)=0\),
and thus \(s=0\).
The section's own support
has not been declared
analytic or coherent.

When \(n=0\), the normalized
support near \(p_0\) is
an isolated smooth point.
There is no transverse
discriminant.
The generic lattice and
Euler gauge have no
parameter-minor exception,
so the local Euler
detector proves SE.1
directly.
The zero module is
immediate in every
dimension.
This finishes Theorem SE.1.

Finally the extensions SE.2
are \(B\)-linear by their
literal balancing formula.
Their finite product is
therefore \(B\)-linear,
and SE.3 proves its
injectivity, giving SE.4.
No tensor-flatness for the
real kernel ring over \(B\)
has been used.

Theorem SE.1 separates every
singular infinite-extension
germ by finitely many
actual directional functionals.
A monodromy representation
on their common continued
representatives and a
coherent D-type realization
still require the full
continuation/action and
infinite-order linearity
arguments.

#### SE.G. Geometry of section detection

![Smooth-branch identity, injective cap restriction and singular support purity](assets/singular-directional-separation.png)

The upper panel depicts
the identity hypothesis on
the unit parameter disc,
with \(g(x)=x\).
The initial zero patch
is \(D(-1/2,1/8)\).
The arc \(x=e^{i\theta}/2\),
from \(\theta=\pi\) to zero,
lies in the complement
of the exceptional point.
The normal-coefficient identity
propagates zero on that
complement; SE.A and PG.1
treat the center.
No numerical section is
being sampled.

The middle panel uses
the cusp \(t^2=x^3\)
at \(x_*=-1/4\), with
roots \(\pm i/8\),
\(a=2\) and \(\rho=1/32\).
Here \(W_2\) is exactly
\(\operatorname{Re}t<-1/2-2|\operatorname{Im}t|\).
The old inner intersections
\(J_j=W_2\cap\Omega_j\)
are shaded on the left.
The new \(I_j\) extend
up to \(\operatorname{Re}T_j=-1/64\).
The left picture edge is
only a plot crop; every
tube continues in negative
real time.
The notation \(Q(I_j)\)
means \(Q_{\Omega_j,I_j}\),
and similarly for \(J_j\).
Their displayed injection
is proved in SE.20.
The lower panel is the
complete section argument
SE.24–SE.28.

Full figure,
editable SVG,
exact data
and figure source.

Free human source context:
[Kashiwara–Kawai, Holonomic systems III, IV.4.6.2](https://www.kurims.kyoto-u.ac.jp/~kenkyubu/kashiwara/HolIII.pdf).


### 5.85. Monodromy of actual directional solution homomorphisms

Let \(M\) be the whole actual
generic-position holonomic module
at \(p_0=(0,0;1,0)\).
Keep the directional function
module \(\mathcal C_{p_0}\) and
all its actual actions from
§5.82, CF.A.
Write
\[
\begin{gathered}
V=\operatorname{Hom}_{\mathscr E_{p_0}}
             (M,\mathcal C_{p_0}),\\
q=\dim_{\mathbf C}V<\infty .
\end{gathered}
\tag{MC.1}
\]
Finite dimensionality is
§5.83, FD.1.
Choose one basis of \(V\)
and the finite actual generators
\(u_1,\ldots,u_b\) of the
matched presentation in FD.1.

**Theorem MC.1.** There is one
ordinary product neighbourhood \(X\)
of zero on which representatives
\(f_{ai}\) of all the finitely
many generator images
\(\Phi_a(u_i)\) continue on the
universal cover of \(X\setminus S\).
For every continuation loop, their
continued classes satisfy all
actual finite presentation relations.
They define another element of
the entire space \(V\).
This gives a genuine linear
representation
\[
\begin{gathered}
\rho:\Gamma\longrightarrow\operatorname{GL}(V),\\
\Gamma=\pi_1(X\setminus S).
\end{gathered}
\tag{MC.2}
\]
Changes of representatives which
continue on the same complement
give the same action.
Restriction to a common smaller
admissible product pulls back
this representation along the
induced map of groups.

Let \(m\) be the number of
distinct points in a generic
time fibre of the reduced
support hypersurface.
The group \(\Gamma\) has a
generating set of size \(d\le m\).
There is a two-sided ideal
\(\mathfrak a\subset\mathbf C[\Gamma]\)
which annihilates all functions
\(f_{ai}\), as functions, and
\[
\dim_{\mathbf C}\mathbf C[\Gamma]/\mathfrak a
\le 1+dq^2 .
\tag{MC.3}
\]
Thus the same finite determination
applies to every chosen generator
representative and all its branches.
This assertion concerns actual
functions and actual homomorphisms.
It does not yet identify their
finite D-type realization or
prove infinite-order linearity
of that realization.

#### MC.A. Finite fibres, covers and arbitrarily small meridians

The complete finite projection
and root bound are CF.B–CF.C.
After shrinking, take
\[
\begin{gathered}
X=\Delta_R\times B_\epsilon,\\
F(t,x)=x,\\
S=\{p(t,x)=0\},\\
p(t,0)=t^m .
\end{gathered}
\tag{MC.4}
\]
The polynomial is monic and
reduced. All its roots are
inside \(\Delta_{R/8}\) after
a smaller base ball.
Let \(H\) be a proper analytic
discriminant hypersurface containing
all root collisions, put
\[
\begin{gathered}
Y=B_\epsilon\setminus H,\\
D=F^{-1}(H),\\
Z=X\setminus(S\cup D).
\end{gathered}
\tag{MC.5}
\]
Empty exceptional sets are allowed.
The complex-line detour proof
SE.A
shows that \(Y\) is
path connected.

Choose a negative real
anchor \(A=-R/2\).
Make the base small enough
that its anchor product lies
inside a fixed strict negative
cone \(W\), disjoint from \(S\).
The flat root estimate FD.17
permits this choice.
The set \(W\cap X\) is
convex and hence contractible.
Choose the base point in
the anchor product over \(Y\).

The path-class construction in
§5.70, PM.A
gives universal covers and
unique path and homotopy lifting:
append straight paths inside
endpoint balls, and subdivide
a compact homotopy square
into such balls.
All complements here are
ordinary complex manifolds,
so these balls exist.

The map \(Z\to Y\) is
locally a punctured-disc
fibration with its fixed
anchor section.
For clarity its trivializations
are precisely the root-disc
homeomorphisms of CF.E.
At a base point put disjoint
closed discs around its simple
roots. On a smaller base
open, the roots move by
holomorphic functions. Translate
each root using a continuous
cutoff supported in its disc.
Take displacements so small
that the Lipschitz constant
of the displacement map is
less than one.
Then the map is injective;
the equation for its inverse
is a convergent contraction.
It fixes the outer annulus
and the anchor.
These are homeomorphisms of
the time disc, carrying
all punctures to their new
positions. They provide the
claimed trivializations.
Analytic formulas below use
fixed contour pieces instead
of differentiating these
topological maps.

We will use an explicit
lifting construction.
On any compact base path
or homotopy, the root
sets have a positive
minimum separation and
positive margins from the
anchor and the outer
boundary. Uniform continuity
permits a finite subdivision
of its time parameter
such that each new root
is uniquely matched to
the nearby old root.
On every step use the
cutoff translations above,
with discs smaller than
the common separation and
displacements small enough
to keep their Lipschitz
constant below one.
The sum of their displacements
is independent of a
labelling of the roots.
It therefore depends continuously
on all the path or
homotopy parameters.
Finite composition gives
homeomorphisms \(L_s\) along
a path, fixing the
anchor, and a continuous
family \(L_{s,u}\) along
a homotopy.
For a based homotopy
the initial maps are
the identity, and all
maps fix the anchor.
The final map of a
loop need not be
the identity on the
rest of the fibre.
This proves the required
homotopy lifting by an
actual finite construction.

For a loop \(\alpha(s)\)
in \(Z\), transport its
base path by these maps
and set \(v(s)=L_s^{-1}\alpha(s)\)
in the initial punctured
fibre. It is a loop
based at the anchor,
since all \(L_s\) fix
that anchor.
The square
\((s,w)\mapsto L_s(v(w))\)
lies in \(Z\).
Its diagonal is the
original loop; its bottom
edge is the anchor
base loop, and its
right edge is a loop
in the initial fibre.
The elementary diagonal-to-two-edges
homotopy in the unit
square proves that the
fibre loops and anchor
base loops generate
\(\pi_1(Z)\).

We also verify the
fibre assertion for the
covers. Over \(Y\), form
the space of endpoint-fixed
homotopy classes of paths
inside each punctured fibre,
starting at the anchor.
A fixed polygonal path
and a short final
endpoint segment give its
local charts over \(Z\).
Two equivalent paths remain
equivalent on a smaller
base open: their compact
fibre homotopy keeps a
positive distance from the
old roots, so it still
avoids all nearby roots.
The same statement for
inequivalent paths follows
by reversing the nearby
cutoff homeomorphisms.
Thus the charts form
a covering of \(Z\),
with its complex structure
inherited from the projection.
Pull it back to the
universal cover of \(Y\).
Each fibre is now,
by construction, the usual
path-class universal cover
of the punctured disc.
The total space is
simply connected as well.
A projected loop bounds
a based homotopy on
that universal base cover.
The continuous cutoff maps
\(L_{s,u}\) constructed above
lift to the fibre
path classes by applying
each map to every
point of a representative
path. They fix the
constant anchor path.
Write the original loop
as \(L_{s,0}(v(s))\).
The paths
\(L_{s,u}(v(s))\) are
based loops throughout
the contraction: both ends
are the constant anchor
path, even when the
final homeomorphism is
nontrivial.
At the contracted base
loop all root displacements
are zero, so \(L_{s,1}\)
is the identity.
The remaining loop \(v\)
is in one simply
connected fibre and
contracts there.
Consequently this is the
universal cover of \(Z\),
and its fibres over the
base cover are indeed
simply connected.
No assertion about the
second homotopy group of
the base is being
imported.

Inclusion \(Z\hookrightarrow X\setminus S\)
is surjective on fundamental
groups. To see this, cover
a loop in \(X\setminus S\)
by small convex balls there.
Perturb its subdivision points
off \(D\) inside the overlapping
balls. Join consecutive new
points by their complex
affine line, replacing the
finitely many zeros of
a defining equation of \(D\)
by small planar detours.
These paths avoid \(D\).
The replacements and endpoint
adjustments are homotopic to
the old segments inside
the same convex balls.
This gives the required
surjection.

Every anchor lift of a
base loop contracts inside
the anchor product in
\(X\setminus S\), since
that whole product includes
the base discriminant and
misses \(S\).
Consequently the image of
any one fibre group
generates \(\Gamma\).

The group of a disc
with \(m\) punctures is
generated by \(m\) meridians.
For the needed elementary
picture, choose a generic
fibre with distinct imaginary
parts, as FD.F permits.
Cut horizontally from each
root to the right boundary.
The cuts are disjoint.
Their complement is contractible:
its horizontal intervals deform
to the curve
\[
\operatorname{Re}t
=-\frac34\sqrt{R^2-(\operatorname{Im}t)^2}.
\tag{MC.6}
\]
At every cut height this
curve lies to the left
of all roots, since they
are in \(\Delta_{R/8}\).
Then contract along the
curve. A polygonal loop
transverse to the finitely
many cuts has finitely
many crossings; each crossing
is removed by the corresponding
meridian. The remaining loop
lies in the contractible cut
domain. This proves generation,
including after a small
perturbation of a general loop.
Thus \(d\le m\).

These generating meridians can
be taken arbitrarily close
to the origin.
Given a smaller product
\(\Delta_r\times B_\eta\),
choose a generic \(x'\)
in its base so small
that all the roots over
\(x'\) lie in \(\Delta_{r/8}\).
The full fibre disc
and this smaller root-containing
disc have the same fibre
group: a strictly increasing
radial map fixes the inner
root disc and retracts the
outer annulus, with its
radial interpolation giving
the two homotopies.
Transport the base point
to that fibre along the
anchor product, then along
the negative time segment
to \(-r/2\).
Both paths can be kept
inside \(W\).
The small meridians, conjugated
by this path, generate
the original \(\Gamma\).
No assertion that arbitrary
neighbourhood groups are
isomorphic is needed.

#### MC.B. A continued holomorphic error has trivial monodromy

Let \(h\) be holomorphic
on the universal cover of
\(X\setminus S\).
Suppose its reference branch
on \(W\) agrees near
zero with an ordinary
holomorphic germ \(h_0\).
Choose the small meridians
of MC.A inside a product
where \(h_0\) is holomorphic.
Along their conjugating paths
inside \(W\), continuation
is the fixed reference
branch. Inside the smaller
product it is \(h_0\),
by the identity theorem.
Each small meridian therefore
returns that branch unchanged,
and the inverse conjugating
path returns the same
germ at the original
base point.
These loops generate
\(\Gamma\), so every loop
returns \(h\) unchanged.
The equality of the two
functions then holds on
the whole connected universal
cover by the identity
theorem in path charts.

In particular a difference
of two globally continued
representatives which is
holomorphic at zero stays
that same holomorphic germ
under every loop.
This is stronger than
merely saying that the
initial difference has zero
directional class.

Use the left continuation
convention of PM.A:
traverse \(\delta\) before
\(\gamma\) in \(\gamma\delta\).
Equivalently use inverse
pullback of geometric deck
transformations for a left
action on functions.
Then
\[
T_{\gamma\delta}=T_\gamma T_\delta.
\tag{MC.7}
\]

#### MC.C. The special action without a finite-monodromy hypothesis

Let \(\Pi=\pi_1(Z)\), and let
\(\widetilde Z\) and \(\widetilde Y\)
be the covers just constructed.
The fibre map gives a
map from \(\widetilde Z\)
to \(\Delta_R\times\widetilde Y\)
away from the roots.
Its fibres are connected
universal covers of the
punctured time disc.
Let
\[
\begin{gathered}
\mathscr F=\underset{U}{\operatorname{colim}}\,
 \mathcal O\bigl(c^{-1}(U\cap Z)\bigr),\\
\mathscr G=\underset{U}{\operatorname{colim}}\,
 \mathcal O\bigl(\beta^{-1}(U\setminus D)\bigr).
\end{gathered}
\tag{MC.8}
\]
Here \(\beta:\Delta_R\times\widetilde Y
\to X\setminus D\), and
pullback identifies \(\mathscr G\)
with a subspace of
\(\mathscr F\).
Use smaller products containing
all the roots over their
smaller bases.
These products are cofinal,
by the finite root bound.
The covers are restrictions
of the fixed covers;
they are not replaced
by newly identified groups.
Each component of a
restricted base cover is
treated separately.
Its fibres remain the
same universal time covers:
the smaller time disc
contains all punctures,
and the radial annulus
homotopy in MC.A
identifies its fibre
path classes with the
old ones.
A new negative anchor
inside that disc has
a lift on every component,
obtained from the old
anchor along the fixed
root-free negative segment.
Thus no value of an
input outside its smaller
domain is required.

Use exactly the special
fibration ring
\(\widetilde{\mathscr E}^{\,\infty}\)
of
§5.49, ISF.1–ISF.4.
Write its pair as
\(\mu=P_0+V_J\), with
\(J=-K\) in ISF.4.
Its action has the positive
Volterra sign:
\[
\begin{gathered}
P_0=\sum_{\ell,\alpha}
 p_{\ell,\alpha}(t,x)D_t^\ell D_x^\alpha,\\
J=\sum_\alpha j_\alpha(t,s,x)D_x^\alpha,\\
\mathcal A_\lambda(\mu)f=P_0f+V_{\lambda,J}f,\\
V_{\lambda,J}f\\
=\int_{\lambda}^{\,p}(Jf)(t,s,x)\,ds .
\end{gathered}
\tag{MC.9}
\]
Here \(Jf\) means the
kernel \(J\) acting in
\(x\) on \(f(s,x)\).
The lower endpoint is a
lift of the anchor over
the base cover.
Paths are in its simply
connected lifted time fibre.
The coefficient hypotheses are
\[
\begin{gathered}
|p_{\ell,\alpha}|
\le M_\varepsilon
 \frac{\varepsilon^{\ell+|\alpha|}}{\ell!\alpha!},\\
\text{every }\varepsilon>0,\\
|j_\alpha|
\le M_\delta\frac{\delta^{|\alpha|}}{\alpha!},\\
\text{every }\delta>0 .
\end{gathered}
\tag{MC.10}
\]
The joint time and parameter
domain is fixed before
either epsilon is chosen.

The construction and estimates
PM.C–PM.D apply without
assuming finite determination
of \(f\). We give the
required extension explicitly.
Fix any compact finite
family of lifted paths
and slightly larger root-free
coordinate products around it.
For an arbitrary holomorphic
input \(f\), their finitely
many compact norms have
a finite maximum \(M_f\).
If the spatial Cauchy
margin is \(\eta>0\),
\[
\begin{gathered}
|\partial_x^\alpha f|
\le M_f\alpha!\eta^{-|\alpha|},\\
\sum_\alpha
 |j_\alpha\partial_x^\alpha f|\\
\le M_\delta M_f(1-\delta/\eta)^{-n},\\
0<\delta<\eta .
\end{gathered}
\tag{MC.11}
\]
The tail of total degree
at least \(N\), with
\(\theta=\delta/\eta<1\),
is bounded by
\[
M_\delta M_f
\theta^{N/2}(1-\sqrt\theta)^{-n}.
\tag{MC.12}
\]
These are the exact products
of geometric series, as
in PM.14–PM.15.
Integration multiplies the
bound by the finite total
path length.
The ordinary part has the
same estimate in \(n+1\)
variables, after choosing
\(\varepsilon\) below both
normal and spatial Cauchy
margins.
Every fixed further derivative
uses one additional reserved
margin.

There is no bound on
the number of branches
in this argument.
Each compact family uses
only its finite lifted
chart cover and its
finite path family.
All paths stay in the
same physical time disc.
The arbitrary spatial epsilon
does not shrink that disc.
Consequently a long contour
costs its length, while
the full joint kernel
still converges there.

On a small output chart,
choose fixed intermediate
contour pieces and one
short final segment to
the varying endpoint.
The same pieces avoid
the roots for nearby
base parameters.
Normal convergence proves
joint holomorphy by differentiation
under these finite integrals.
The polygonal Cauchy argument
on a lifted homotopy square
proves path independence.
It is the path argument
of PM.C with a single
arbitrary input in place
of finitely many coefficient
labels.

A change of lifted anchor
subtracts a fixed connecting
path integral.
Its output time appears
only in the kernel
coefficient. This integral
is therefore holomorphic
on the whole smaller
output time disc over
the base cover, including
the root points. It
belongs to \(\mathscr G\).
Both parts preserve
\(\mathscr G\): use local
Cauchy estimates for \(P_0\)
and a straight time segment
for \(V_J\), since an
input in \(\mathscr G\)
is nonsingular at the
fibre roots.

The four exact product
formulas ISF.11–ISF.17
remain valid on these
lifted paths.
For two negative parts,
Fubini on the ordered
path triangle gives the
convolution integral ISF.16.
Its inner integrand contains
only joint holomorphic
kernel coefficients, so
its inner path may be
replaced by a straight
segment, even across
the root positions.
The input function remains
on the outer root-avoiding
path.
All spatial products are
the absolutely convergent
ISF.11 sums with bound
ISF.12.
For the two mixed cases,
differentiation and integration
by parts give ISF.14
and ISF.15 on each
fixed contour family.
The complete infinite sums
have the fixed normal
and spatial margins in
that proof.
The lower endpoint series
ISF.17 is retained; as
a function of output
time it is holomorphic
on the full disc and
therefore belongs to
\(\mathscr G\).
The ordinary-ordinary product
is its full Cauchy product.
Thus
\[
\begin{gathered}
(\mu\nu)[f]=\mu(\nu[f]),\\
1[f]=[f]
\end{gathered}
\tag{MC.13}
\]
in \(\mathscr F/\mathscr G\),
with exactly the actual
special-ring product.

The action commutes with
the \(\Pi\)-action.
Every deck transformation
induces a base deck
transformation and fixes
the physical time coordinate.
It consequently preserves
\(\mathscr G\).
Indeed for a geometric
deck transformation \(d\),
the two expressions
\(d^*\mathcal A_\lambda(\mu)f\)
and \(\mathcal A_\lambda(\mu)d^*f\)
have the same upper endpoint.
Their difference is the
integral between the two
lifted anchor sections.
This is holomorphic on
the full output time
disc over the base cover,
by MC.11–MC.12.
It lies in \(\mathscr G\).
The ordinary infinite part
commutes exactly.
Hence
\[
\begin{gathered}
T_\gamma\bigl(\mu[f]\bigr)
=\mu\,T_\gamma[f]\\
\text{in }\mathscr F/\mathscr G .
\end{gathered}
\tag{MC.14}
\]
Only the chosen-fibration
special ring has been
used on this quotient.

#### MC.D. A cylinder error which is regular at an anchor

We need the following
holomorphic extension statement.
Suppose \(g(t,\widetilde x)\)
is holomorphic on a full
time disc times a connected
covering of \(B\setminus H\).
Suppose that near one
nonzero time \(A'\), it
is the pullback of a
holomorphic function on
a time neighbourhood of
\(A'\) times the whole
base ball \(B\).
Then, after a smaller
base ball, \(g\) descends
and extends holomorphically
through the base discriminant
for every time in the
given disc.

First fix any two points
of the cover above the
same \(x\). Their two
time functions agree near
\(A'\), by the assumed
anchor function. The identity
theorem in time makes
them agree on the
whole time disc.
If initially the equality
to the anchor function
is known on just one
open lift, the identity
theorem on the connected
base cover gives that
equality on its whole
anchor product first.
The descended function is
holomorphic on
\(\Delta\times(B\setminus H)\).

Here is the extension
through \(H\), with no
growth hypothesis.
Choose a complex coordinate
line on which the
first nonzero homogeneous
part of a defining
equation for \(H\)
does not vanish identically.
Write the coordinates
as \(x=(z,w)\).
Choose a small circle
\(|\zeta|=r\) on that
line which misses \(H\).
By compactness it still
misses \(H\) for all
small \(w\).
The finite-root preparation
and continuity used in
CF.B place all the
nearby intersections of
\(H\) strictly inside
that circle.
Write \(g_t(z,w)=g(t,z,w)\)
and define
\[
\begin{gathered}
\widehat g(t,z,w)\\
=\frac1{2\pi i}
 \int_{|\zeta|=r}
 \frac{g_t(\zeta,w)}{\zeta-z}\,d\zeta .
\end{gathered}
\tag{MC.15}
\]
The integrand has a
fixed compact nonsingular
contour, so this is
jointly holomorphic for
\(|z|<r\).
Near \(t=A'\), the
ordinary Cauchy formula
for the anchor function
gives \(\widehat g=g\).
For every fixed \(x\)
outside \(H\), both
sides are holomorphic on
the whole time disc.
The identity theorem in
time gives equality there
for every \(t\).
Thus MC.15 is the
required extension.
For \(n=1\) this is
the same one-circle proof
without \(w\); for \(n=0\)
there is no discriminant
to remove.

#### MC.E. Relating the function quotient to a directional relation

Take finitely many functions
\(f_i\) continuing on
the cover of \(X\setminus S\),
and finitely many special
operators \(\mu_i\).
Their directional classes
are actual classes of
CF.A. Namely, the cofinal
future-convex supports in
CF.A and FD.C contain
the whole \(S\).
Each convex complement
\(I_Z\) is simply connected
and avoids \(S\).
Its path-class lift, fixed
by the reference branch
on \(W\), gives a
holomorphic representative
on that whole complement.
The same construction applies
to every \(T_\gamma f_i\).
On the reference negative
cone form
\[
H_\lambda=\sum_i\mathcal A_\lambda(\mu_i)f_i .
\tag{MC.16}
\]
The actual calibration
ISF.13, PM.D and CF.A
identifies its directional
class with
\(\sum_i\mu_i[f_i]\)
using the full kernel
action, not a formal
normal-order sum.

Assume that this directional
class is zero.
On a smaller negative
cone, \(H_\lambda\) then
agrees with an ordinary
holomorphic germ \(h_0\).
Indeed the zero has a
cofinal support witness;
that support misses every
fixed strict negative cone
after shrinking. Restriction
and the identity theorem
give the asserted equality
of holomorphic representatives.

The anchor can be moved
closer to zero inside
the domain of \(h_0\).
The difference is a
fixed past-path integral,
with input paths in the
negative cone.
All these input functions
are holomorphic near that
compact path for the
whole smaller base ball.
MC.11–MC.12 proves that
its full spatial sum
is holomorphic on a
full output time disc
and that smaller base.
Thus it changes \(h_0\)
only by an ordinary
holomorphic function.
This is the two-anchor
normalization, with all
spatial indices retained.

Choose the new anchor
inside a product where
this new holomorphic
representative is defined.
Shrink the base until
all fibre roots are
inside its normal disc.
First verify that the
anchor values over
the base cover agree
with one ordinary anchor
function. To verify this,
the anchor lift and
its short time neighbourhood
are inside the reference
negative cone on
\(X\setminus S\).
Every \(f_i\) is single
valued there. Base loops
in the anchor product
contract in \(X\setminus S\),
so their lifts do not
change those values.
The local derivative and
integral formulas have
that same property.
Equivalently apply the
identity theorem on a
connected restricted base
cover to its anchor
product.
On the ordinary anchor
product this function
equals the new holomorphic
representative, since they
already agree on its
nonempty negative-cone open.
Thus they agree at the
anchor of every lifted
fibre, on every restricted
base component.

In each such fibre,
all meridians can be
drawn inside the ordinary
holomorphic product.
Continuation of \(H_\lambda\)
around them returns the
same holomorphic function.
Since they generate the
whole fibre group, the
function descends and
extends across every root.
These extensions are
jointly holomorphic: near
the roots they are the
given ordinary representative,
and off the roots they
are the descended holomorphic
function on the cover.
They agree on overlaps
by the identity theorem.
Consequently
\[
H_\lambda\in\mathscr G .
\tag{MC.17}
\]
Changing back to the
old anchor adds only
the full-time holomorphic
past-path integral, so
MC.17 holds for it
as well.

Now take a loop in
\(\Gamma\), and lift it
to a loop in \(\Pi\)
using MC.A.
The functions \(T_\gamma f_i\)
still continue on
\(X\setminus S\).
Equations MC.14 and MC.17
give
\[
\sum_i\mathcal A_\lambda(\mu_i)T_\gamma f_i
\in\mathscr G .
\tag{MC.18}
\]
Near any sufficiently small
negative anchor its left
side is holomorphic on
the whole ordinary base
ball: use the short
anchor paths and the
ordinary branches of the
\(T_\gamma f_i\) on \(W\).
If the representative of
MC.18 uses a smaller
output disc, move the
anchor inside that disc;
the fixed past-path integral
is again ordinary holomorphic.
The cylinder-extension lemma
MC.D applies. Thus the
left side of MC.18
is an ordinary holomorphic
germ at zero.
Its directional class
is zero:
\[
\sum_i\mu_i[T_\gamma f_i]=0
\quad\text{in }\mathcal C_{p_0}.
\tag{MC.19}
\]
This proves the precise
continuation rule needed
for every presentation row.
It does not assert an
action of arbitrary finite
or full microlocal symbols
on \(\mathscr F/\mathscr G\).

#### MC.F. All actual relations and the monodromy group law

The bounded actual presentation
from
§5.31, L.4–L.6
and
§5.49, ISF.3–ISF.4
has finite matrices over
the spatial-polynomial coefficient
ring. That ring is
contained in the special
ring by ISF.9.
Its actual finite counterpart
has degree-zero cokernel
equal to the whole
\(M\).
No exactness over the
canonical coefficient ring
is needed here.

Theorem CF.1 supplies
one common continuation
domain for the finitely
many basis-generator values.
For every presentation row
and each basis vector,
the initial directional
relation is zero.
There are finitely many
such relations.
The two-anchor construction
in MC.E and a common
shrink give the product
\(X\) used above for
all of them.
Every continued row
has zero directional class
by MC.19.
The continued generator
classes thus define an
actual \(\mathscr E_{p_0}\)-linear
homomorphism from the
finite presentation of \(M\).
This establishes the asserted
map \(T_\gamma\Phi\in V\)
on the whole module.
Finite-ring linearity follows
from that actual cokernel,
including coefficients outside
the special ring.

The action is linear.
To check its group law,
write the generator values
of \(T_\delta\Phi_a\)
as the corresponding linear
combination of the fixed
basis-generator functions.
Their difference from
the functions \(T_\delta f_{ai}\)
has zero directional class.
Both sets of functions
continue on the whole
cover of \(X\setminus S\),
so that difference agrees
near zero with an ordinary
holomorphic germ, by the
same representative identity
used in MC.E.
MC.B makes its every
continuation return that
same germ.
Continuing once more
therefore gives exactly
the classes prescribed
by MC.7. Hence
\[
\begin{gathered}
\rho(\gamma\delta)
=\rho(\gamma)\rho(\delta),\\
\rho(1)=I .
\end{gathered}
\tag{MC.20}
\]
The inverse loop gives
the inverse matrix.
This is a representation,
not only a family of
linear maps chosen separately
for different loops.

The same argument applies
to a change of continuing
representatives: their difference
is a continued holomorphic
germ, and MC.B makes
its directional class stay
zero. On a common
smaller admissible product,
the path-class restriction
and the same finite
presentation matrices commute
with all formulas. Its
representation is the pullback
along the inclusion-induced
map of fundamental groups.
No inverse for that
group map is presumed.

#### MC.G. One finite-determination ideal for the function representatives

Put \(A_\Gamma=\mathbf C[\Gamma]\),
let \(I\) be its augmentation
ideal, and set
\[
\begin{gathered}
\chi:A_\Gamma\longrightarrow\operatorname{End}_{\mathbf C}(V),\\
\mathfrak c=\ker\chi,\\
\mathfrak a=I\mathfrak c .
\end{gathered}
\tag{MC.21}
\]
Both ideals are two-sided.
The representation gives
\[
\dim_{\mathbf C}A_\Gamma/\mathfrak c
\le q^2 .
\tag{MC.22}
\]
For \(c\in\mathfrak c\),
every function \(c f_{ai}\)
has zero directional class
and is an ordinary
holomorphic germ near zero.
It continues on the whole
cover of \(X\setminus S\),
as a finite linear
combination of the fixed
continued functions.
MC.B therefore gives the
function equality
\[
\begin{gathered}
(\gamma-1)c f_{ai}=0\\
\text{for every }\gamma\in\Gamma.
\end{gathered}
\tag{MC.23}
\]
By the identity theorem
the equality holds on
the whole connected cover.
Thus \(\mathfrak a\) annihilates
every \(f_{ai}\) as a
function, not only as
a directional quotient class.

Let \(g_1,\ldots,g_d\)
generate \(\Gamma\).
The word identity
\[
\begin{gathered}
ab-1\\
=(a-1)b+(b-1)
\end{gathered}
\tag{MC.24}
\]
and
\(g_j^{-1}-1=-(g_j-1)g_j^{-1}\)
show that
\[
I=\sum_{j=1}^d(g_j-1)A_\Gamma .
\tag{MC.25}
\]
Since \(\mathfrak c\) is
two-sided,
\(I\mathfrak c=\sum_j(g_j-1)\mathfrak c\).
The linear map
\[
\begin{gathered}
(A_\Gamma/\mathfrak c)^d
\longrightarrow I/I\mathfrak c,\\
(\bar r_j)_j\longmapsto
 \sum_j(g_j-1)r_j
\end{gathered}
\tag{MC.26}
\]
is well defined and
surjective. The quotient
by \(I\) has dimension
one, so
\[
\begin{gathered}
\dim A_\Gamma/\mathfrak a\\
=1+\dim I/I\mathfrak c\\
\le 1+dq^2 .
\end{gathered}
\tag{MC.27}
\]
This proves MC.3.
Each individual function
orbit is an image of
\(A_\Gamma/\mathfrak a\);
the same ideal works
for all the finitely
many chosen functions.
Zero functions and the
zero module are immediate;
one may take the whole
group algebra as annihilator
when there are no
nonzero inputs.

The theorem supplies the
actual monodromy representation
and a common finite
function determination.
The conversion to a
finite coherent D-type
module, with the full
infinite-order evaluation map,
still requires the algebraic
tensor witnesses of III.4.9
and the actual comparison
maps proved earlier.

![Small fibre meridians, the anchor extension contour and the finite function envelope](assets/special-continuation-monodromy.png)

**Figure MC.1.** In the
first panel the curve is
\(t^2=x^3\), the fibre is
\(x_*=-1/16\), and the
display coordinate is
\(\zeta=16t\).
The roots are
\(\zeta=\pm i/4\);
the meridian circles have
radius \(1/8\) and
are joined to the
anchor \(-1/2\).
The cuts run to the
right boundary of
\(|\zeta|<1\).
This is the punctured-disc
generation in MC.A.
In the second panel,
\(H=\{z=0\}\) and the
fixed Cauchy circle has
radius \(1/2\).
Regularity near the negative
anchor and the identity
theorem in time make
MC.15 an extension for
the whole time disc,
as proved in MC.D–MC.E.
The last panel displays
the actual solution action
and the ideal construction
MC.21–MC.27.
Its dimension bound concerns
the function representatives.

Figure SVG ·
exact diagram data ·
reproducible figure source ·
figure PNG.

Free source:
[Kashiwara–Kawai, Holonomic systems III, III.3.4 and IV.4.6–4.7](https://www.kurims.kyoto-u.ac.jp/~kenkyubu/kashiwara/HolIII.pdf).


### 5.86. A faithful D-type realization of the whole infinite extension

Let \(M\) be a coherent actual holonomic microlocal module in
generic position at \(p_0=(0;dt)\), and put
\[
\begin{gathered}
E=\mathscr E_{p_0},\qquad B=\mathscr E^\infty_{p_0},\\
R=\mathcal D^\infty_{X,0},\\
N=B\otimes_E M .
\end{gathered}
\tag{EV.1}
\]
The action of \(R\) on \(N\) uses the actual inclusions
\(R\subset T\subset B\), where \(T\) is the chosen-fibration
special ring of ISF.1–ISF.5. Its ordinary part and its map to
the full microlocal ring are those actual kernel maps.
The hypersurface \(S\) is the ordinary projection of the whole
characteristic support. Keep the actual directional module \(\mathcal C\)
of §5.82, CF.A.
Write
\[
\begin{gathered}
V=\operatorname{Hom}_E(M,\mathcal C),\\
q=\dim_{\mathbf C}V .
\end{gathered}
\tag{EV.2}
\]
Its finite dimension and its whole-module interpretation are
§5.83, FD.1.

**Theorem EV.1.** There are a finite D-type module \(L\) along exactly
\(S\), an embedded trivial connection \(\mathcal O_X^q\), and a coherent
holonomic finite-operator quotient
\[
\begin{gathered}
K=L/\mathcal O_X^q,\\
\operatorname{Hom}_{\mathcal D_X}(\mathcal O_X,K)_0=0 .
\end{gathered}
\tag{EV.3}
\]
There is an actual \(R\)-linear injection \(e:N\to K^\infty_0\), where
\(K^\infty=R\otimes_{\mathcal D_X}K\), such that
\[
\begin{gathered}
\Psi:N\longrightarrow B\otimes_R K^\infty_0\\
\simeq B\otimes_{\mathcal D_{X,0}}K_0,\\
\Psi(s)=1\otimes e(s)
\end{gathered}
\tag{EV.4}
\]
is \(B\)-linear and injective. These are stalk maps of the actual
ordinary algebraic tensors. No finite-pole conclusion for \(M\) is
included in this assertion.

#### EV.A. Continuation of arbitrary infinite-extension sections

Choose the finitely many spanning ordinary fibrations \(F_j\) of
CF.C, with vertical vectors
\[
v_0,\ v_0-\lambda e_1,\ldots,v_0-\lambda e_n .
\tag{EV.5}
\]
They keep the time coordinate and its directional tangent condition.
The actual finite, infinite and real coordinate maps are
§5.56, BCT.5.3.
The entire transported module has a finite principal fibre in each
fibration, by CF.C. Thus the full Späth generation theorem
§5.49, ISR.2
applies in each of them.

For any \(s\in N\), it gives a finite expression
\[
\begin{gathered}
s=\sum_i a_{j,i}u_{j,i},\\
a_{j,i}\in A_j^\infty .
\end{gathered}
\tag{EV.6}
\]
The generators \(u_{j,i}\) belong to the actual finite module in that
fibration. The time-only ring \(A_j^\infty\) is contained in its actual
special ring by ISF.9. Each expression is obtained in its own
coordinates; it is not obtained by transporting a finite spatial degree.

Fix finitely many such sections and finitely many functionals in \(V\).
CF.1 gives common full-complement continuations for their finitely many
generator values. Apply every \(a_{j,i}\) by its full ordinary head and
joint Volterra kernel. The arbitrary-input construction
§5.85, MC.C
produces holomorphic functions on the covers of
\[
\begin{gathered}
U_j\setminus(S\cup D_j),\\
D_j=F_j^{-1}(H_j).
\end{gathered}
\tag{EV.7}
\]
All normal domains are chosen before the spatial epsilon; long lifted
contours cost their finite length. Both infinite coefficient tails and
all spatial indices are therefore retained.

On a common strict negative cone these functions represent the same
class \(\Phi^\infty(s)\). Their differences extend as ordinary
holomorphic germs by CF.8. Subtract these finitely many germs on a
common smaller ordinary ball. The functions then have exactly the same
initial scalar germ.

The bad cylinders have intersection of codimension at least two by
CF.13 and CF.F. That proof depends only on the spanning vertical
directions: a common hypersurface component would be tangent to all
of them at a regular point. The all-branch path argument of CF.F now
applies to EV.7. Its convex-ball detours arrange each overlap path
outside all cylinders, so the next fibration continues the branch
which has actually arrived there. CF.G then removes the remaining
codimension-two set: its small complement balls are connected and
simply connected, and the fixed-circle Cauchy extension identifies
the continued holomorphic functions. These arguments use no equation
for the resulting scalar function.

Consequently every prescribed finite list of values
\(\Phi^\infty(s)\), with \(s\) in the whole \(N\), has full multivalued
holomorphic representatives on one ordinary \(U\setminus S\).
The neighbourhood may depend on that finite list.

#### EV.B. A stable finite function algebra

Fix a product \(U_0\) carrying the basis-generator functions of MC.1
and its actual representation \(\rho_0\) on \(V\). Use a cofinal system
of smaller root-containing products \(U\), with the same reduced
Weierstrass degree \(m\). Base points and their comparison paths lie
in the fixed negative cone. Write
\[
\begin{gathered}
\Gamma_U=\pi_1(U\setminus S),\\
\rho_U:\Gamma_U\longrightarrow\operatorname{GL}(V).
\end{gathered}
\tag{EV.8}
\]
Here \(\rho_U\) is the pullback of \(\rho_0\).

Every inclusion of two such products induces a surjective group map.
Indeed MC.A shows that the larger group is generated by the meridians
in any one generic fibre. Choose that fibre sufficiently close to zero
that all punctures lie in the smaller time disc. The same meridians
then lie in the smaller product. Its own negative anchor paths and
the larger anchor paths differ by a homotopy in the common convex
negative cone. Thus their images are precisely a generating set of
the larger group. An isomorphism of the groups is unnecessary.

For \(s\in N\), choose a full-complement representative \(f_s\) of
EV.A on a common such \(U\), and use the original fibration expression
\(s=\sum_i a_i u_i\) in EV.6. The relation
\[
\sum_i a_i[f_i]-[f_s]=0\quad\text{in }\mathcal C
\tag{EV.9}
\]
is a finite relation between globally continued inputs with special
coefficients. MC.E applies to this exact relation, including the
coefficient \(-1\) of \(f_s\). For every \(\gamma\in\Gamma_U\) it gives
\[
\begin{gathered}
{}[T_\gamma f_s]\\
=\sum_i a_i[T_\gamma f_i]\\
=(\rho_U(\gamma)\Phi)^\infty(s).
\end{gathered}
\tag{EV.10}
\]
The last equality uses actual extension adjunction and ISR.2, not
continuation of an infinite normal-order sum.

Put
\[
\begin{gathered}
A_U=\mathbf C[\Gamma_U],\\
c_U=\ker(A_U\to\operatorname{End}_{\mathbf C}V),\\
a_U=I_Uc_U ,
\end{gathered}
\tag{EV.11}
\]
where \(I_U\) is the augmentation ideal. The proof MC.G applies with
EV.10. Each \(c f_s\), for \(c\in c_U\), is a globally continued
ordinary holomorphic germ at zero. MC.B makes all its continuations
unchanged. Hence \(a_U\) annihilates \(f_s\) as a function, on every
branch, and
\[
\dim_{\mathbf C}A_U/a_U\le 1+mq^2 .
\tag{EV.12}
\]
This ideal works for the whole finite list of sections and functionals.

There is one stabilized algebra for all further germs. For
\(U'\subset U\), the surjection \(h:A_{U'}\to A_U\) satisfies
\[
\begin{gathered}
c_{U'}=h^{-1}(c_U),\\
h(I_{U'})=I_U,\\
h(a_{U'})=a_U .
\end{gathered}
\tag{EV.13}
\]
It therefore gives a surjective algebra map
\[
A_{U'}/a_{U'}\longrightarrow A_U/a_U .
\tag{EV.14}
\]
The finite dimensions are nondecreasing under shrinking and are
bounded by EV.12. Choose \(U_*\) at which their maximum is attained.
Every map EV.14 to this product, from a smaller admissible product,
is an isomorphism: it is surjective between spaces of the same
dimension. In particular \(a_{U'}\) is the inverse image of \(a_{U_*}\).

Thus
\[
A_*=A_{U_*}/a_{U_*}
\tag{EV.15}
\]
controls every value of every whole infinite-extension section,
on its own sufficiently small representative domain. This proves
stability of finite algebra quotients; it makes no assertion that
the underlying fundamental groups stabilize.

#### EV.C. The finite D-type quotient and its directional injection

Let \(a_*=a_{U_*}\). The cyclic dual construction PD.39–PD.44 and
the prescribed-complement theorem
§5.79, FB.D–FB.E
give an actual finite D-type module \(L_*\) along \(S\).
Its ordinary infinite extension is, by the actual all-parameter map
FB.39,
\[
\begin{gathered}
R\otimes_{\mathcal D_X}L_*
\simeq\mathscr F_{a_*},\\
\operatorname{rank}(L_*|_{X\setminus S})
=\dim_{\mathbf C}A_* .
\end{gathered}
\tag{EV.16}
\]
Here \(\mathscr F_{a_*}\) consists of all jointly holomorphic branch
functions of that type, as germs on the fixed restricted covers.
EV.14 identifies this space with the corresponding local function
spaces on all further products.
Explicitly, if a small-complement loop maps to the identity in
\(\Gamma_{U_*}\), its difference from \(1\) lies in \(a_U\).
Every function of type \(a_U\) is invariant under that kernel,
so descends from the small universal cover to the restriction of
the fixed cover. The surjective group map makes this restricted
cover connected. Conversely, pullback of a function of type
\(a_*\) is annihilated by \(a_U\), since these ideals are exact
inverse images. The descents are holomorphic in the ordinary
covering charts.

Since \(a_*\subset I_{U_*}\), augmentation gives a nonzero invariant
functional on \(A_*\). Its invariant functional space has dimension
one:
\[
A_*/I_{U_*}A_*\simeq\mathbf C .
\tag{EV.17}
\]
Every horizontal germ in \(L_*\) restricts to an invariant vector
on a smaller complement. Its group surjects onto
\(\Gamma_{U_*}\), so this is the same one-dimensional invariant
space. A meromorphic section vanishing on that complement is
zero. Thus the horizontal germs of \(L_*\) are exactly the
constant multiples of this unit.
The corresponding horizontal section is the scalar function \(1\).
It embeds the trivial connection \(\mathcal O_X\) in \(L_*\).
Here is its finite-module extension. Apply FB.D also to the
trivial local system. Its normalized lattice is
\(\mathcal O_X\): this is the residue-zero logarithmic lattice on
the normalized locus, and the reflexive extension DT.E identifies
it everywhere. DT.21 therefore identifies that finite D-type
module with \(\mathcal O_X(*S)\).
The invariant local-system map extends uniquely to
\(\mathcal O_X(*S)\to L_*\) by DT.E.
Restrict to \(\mathcal O_X\).
The resulting map is injective, since its restriction off \(S\)
is injective and an ordinary holomorphic function vanishing on
that dense open is zero. Under FB.39 its map is precisely
the ordinary scalar-function inclusion.

Set \(K_*=L_*/\mathcal O_X\). It is coherent holonomic. It has no
horizontal germ. To see this directly, lift a horizontal germ of
\(K_*\) to \(l\in L_*\). In coordinates \(z_1,\ldots,z_{n+1}\),
write \(\partial_i l=b_i\in\mathcal O_X\).
Commuting derivatives gives \(\partial_i b_j=\partial_j b_i\).
The holomorphic function
\[
h(z)=\int_0^1\sum_i z_i b_i(\lambda z)\,d\lambda
\tag{EV.18}
\]
has \(\partial_i h=b_i\): differentiate the integrand and use the
closedness identity to obtain the derivative of
\(\lambda b_i(\lambda z)\). Thus \(l-h\) is horizontal in \(L_*\).
EV.17 makes it a constant multiple of the embedded unit, so its
quotient germ is zero.

Ordinary right flatness of \(R\) over \(\mathcal D_X\), proved in
§5.47, MF.6–MF.7,
and the empty-divisor case of SI give
\[
K_*^\infty\simeq\mathscr F_{a_*}/\mathcal O_X .
\tag{EV.19}
\]
Every function on the right gives an actual directional class by
the anchored lift on the convex complements of CF.A and FD.C.
This defines an \(R\)-linear map
\[
j:K_*^\infty\longrightarrow\mathcal C .
\tag{EV.20}
\]
Its ordinary infinite action is the actual action: FB.39 and ISF
use the same every-epsilon holomorphic heads on each larger
Cauchy product.

The map is injective. A class in its kernel has a branch agreeing
near zero with an ordinary holomorphic germ. MC.B fixes this
globally continued error under all meridians, and hence under
the whole complement group. Apply it on the universal cover
of a smaller representative product. Surjectivity to the fixed
group accounts for every branch of its restricted cover.
It is the same ordinary germ on
every branch of a smaller representative domain, so is zero
in EV.19.

For each \(\Phi\in V\), EV.A–EV.15 therefore define a unique map
\[
\begin{gathered}
e_\Phi:N\longrightarrow K_*^\infty,\\
j e_\Phi=\Phi^\infty .
\end{gathered}
\tag{EV.21}
\]
Uniqueness also proves its linearity and its \(R\)-linearity:
apply the injective \(j\), use the actual \(R\)-action, and use
the \(B\)-linearity of \(\Phi^\infty\). This does not yet assert
linearity of the tensor-unit map over \(B\).

#### EV.D. The actual normal-parameter module

One independent parameter \(u\) suffices. Let \(X'\) have
coordinates \((u,s,x)\), let \(p'_0=(0;0,ds)\), and use finite
and full rings \(E',B'\) there. The original rings embed by
operators independent of \(u,D_u\). Put
\[
\begin{gathered}
H=E'/E'D_u,\\
M^{[u]}=H\otimes_E M,\\
N^{[u]}=B'\otimes_{E'}M^{[u]} .
\end{gathered}
\tag{EV.22}
\]
These are actual algebraic quotients and tensors.

Here is the normal form needed for the first quotient. In the
chart \(\tau_s\ne0\), use \(\eta=\nu_u/\tau_s\).
For every homogeneous coefficient \(p_k(u,s,x,\xi/\tau_s,\eta)\),
Taylor division gives
\[
\begin{gathered}
p_k(\eta)-p_k(0)=\eta q_k(\eta),\\
q_k(\eta)=\int_0^1
                  \partial_\eta p_k(\lambda\eta)\,d\lambda .
\end{gathered}
\tag{EV.23}
\]
Choose a larger covariable disc before all indices.
Cauchy's formula bounds \(q_k\) by the old coefficient norm
times its fixed inverse margin. Multiplying by \(\tau_s^{-1}\)
shifts the homogeneous order by one. A fixed exponential
enlargement absorbs the resulting negative factorial ratio;
for the positive every-epsilon head choose the input epsilon
smaller first. The same fixed-margin argument bounds every
prescribed further derivative. Thus the coefficient sums give
an actual operator \(Q\), in the finite class for a finite
upper order and in the full class otherwise, with
\[
P=P|_{\nu_u=0}+QD_u .
\tag{EV.24}
\]
Right multiplication by \(D_u\) is exactly multiplication by
\(\nu_u\) on these symbols. The remainder is unique.
For several parameters repeat this division in their ordered
covariables.

A finite presentation of \(M\), together with \(D_u\) on its
generators, is a finite presentation of EV.22 over \(E'\).
Finite-ring coherence makes \(M^{[u]}\) coherent. Its actual
finite-module support is contained in
\[
\operatorname{supp}M\times\{(u;\nu_u=0)\}.
\tag{EV.25}
\]
Use the actual finite support convention of
§5.32, J.1–J.2.
If \(\nu_u\ne0\), \(D_u\) is an actual elliptic unit, so its
relations kill every generator. If \(\nu_u=0\) and the old
cotangent point is outside \(\operatorname{supp}M\), the old
finite presentation has zero cokernel. Its finitely many
unit rows express all generators in its relation rows.
Their actual coefficient identities, extended independently
of \(u,\nu_u\), express every new generator as zero too.
This proves the stated containment without a scalar-flatness
or a principal base-change assertion.
The reverse support containment follows from EV.24.
At any fixed \(u=a\), evaluation of its remainder is a right
\(E\)-linear map \(H\to E\), split by constant operators.
Right \(E\)-linearity holds because an old operator differentiates
only \(s,x\), and its coefficients are independent of \(u\).
Tensor these two maps with \(M\). Wherever \(M\) is nonzero,
the resulting split copy is nonzero in the new stalk.
Thus the reduced support is exactly the product in EV.25.
The set in EV.25 is Lagrangian of the new base dimension.
Thus \(M^{[u]}\) is holonomic, with the zero object allowed.
Its fibre over the new origin is contained in the one ray
through \(p'_0\), so it is in generic position.

The map \(N\to N^{[u]}\), sending a finite tensor
\(b\otimes m\) to the same operator and generator, respects
every \(E\)-balance relation. Its image is killed by \(D_u\).
No injectivity of a parameter tensor is required below.

Its whole directional Hom is canonically \(V\). Pullback of
representatives gives one direction. Conversely, the images
of the original generators under a functional for \(M^{[u]}\)
satisfy \(D_u f=0\) in the new directional module.
Choose a common convex representative domain from CF.A;
its isotropic future majorant is preserved by \(u\mapsto\lambda u\),
for \(0\le\lambda\le1\).
The derivative \(\partial_u f\) is an ordinary holomorphic germ.
Subtract its holomorphic \(u\)-primitive based at zero.
The resulting representative is independent of \(u\).
Restriction to \(u=0\) then gives the original directional class.
The old finite relations hold there, and pullback restores all
classes. The universal property of \(H\otimes_E M\) verifies
the two inverse Hom maps. For several parameters use the
closed-form radial primitive EV.18.

The actual actions agree under these maps. Symbols independent
of the parameter act by their original full kernels, and
the removed \(D_u\)-terms act by zero on constant representatives.
The finite quotient EV.24 and the common Cauchy estimates verify
this with all tails retained.

The characteristic projection in EV.25 is exactly
the product of the parameter disc with \(S\).
Apply EV.A–EV.B to this actual module, restricting functions
to the complement of that product. Its group is \(\Gamma_U\),
by contraction of the parameter disc, and its Hom representation
is \(\rho_U\). The same stabilized algebra \(A_*\) therefore
applies, with no new dimension bound or stabilization choice.

The all-parameter comparison FB.39, followed by the ordinary
exact quotient by the trivial connection, gives an injective
map of the parameter version of \(K_*^\infty\) into its
directional module. Hence EV.21 has a canonical parameter
version
\[
e_\Phi^{[u]}:N^{[u]}\longrightarrow K_*^{\infty,[u]} .
\tag{EV.26}
\]
It is linear over the full ordinary
\(D^\infty_{u,s,x}\), by the same injective-action argument.
On the embedded \(N\) it is the constant-parameter pullback
of \(e_\Phi\). These statements concern actual ordinary tensors:
the inverse in FB.39 has the finite row witnesses PD.35–PD.36.
The finite parameter quotient is exact as well. In ST.13 its
transfer is \(\mathcal O_{X'}\otimes_{\mathcal O_X}\mathcal D_X\);
the coefficient projection is flat by the completion proof of
GL06, Lemma3.0c.1.
Explicitly, let \(A_0=\mathcal O_{X,0}\), and localize
\(A_0[u]\) at its origin maximal ideal. This is flat over \(A_0\).
Its Noetherian completion and that of \(\mathcal O_{X',0}\)
are the same formal power-series ring. Flatness of the first
completion and faithful flatness of the second are the
Noetherian completion theorem
Completion, Theorems3.1–3.2.
A kernel after tensor with \(\mathcal O_{X',0}\) would survive
its faithful completion, where flatness over \(A_0\) makes it
zero. This proves the stated coefficient flatness.
Thus its tensor sends \(L_*/\mathcal O_X\) to
\(L_*^{[u]}/\mathcal O_{X'}\). The ordinary infinite extension
is exact by MF.6–MF.7 in the new dimension. This gives precisely
the target in EV.26.

#### EV.E. A finite tensor witness from the normal primitive

Use the exact special quotient and external transfer of
§5.71, ST.A–ST.D.
Thus
\[
\begin{gathered}
T=\widetilde{\mathscr E}^{\,\infty}_{X,0},\\
A=D^\infty_{u,s,x},\\
C_{\mathrm{tr}}=A/AD_u,\\
J=T/R\\
\simeq C_{\mathrm{tr}}/D_sC_{\mathrm{tr}} .
\end{gathered}
\tag{EV.27}
\]
Here \(C_{\mathrm{tr}}\) is the transfer quotient, distinct
from the directional function module \(\mathcal C\).
All variables \(u,s\) remain independent.

For the finite module \(K_*\), ST.13–ST.15 identifies
\[
C_{\mathrm{tr}}\otimes_R K_*^\infty
       \simeq K_*^{\infty,[u]} .
\tag{EV.28}
\]
There is also a direct map
\[
\begin{gathered}
\kappa:C_{\mathrm{tr}}\otimes_R N\longrightarrow N^{[u]},\\
c\otimes s\longmapsto c\,s^{[u]} .
\end{gathered}
\tag{EV.29}
\]
It is well defined: \(D_u s^{[u]}=0\), so \(AD_u\) acts
by zero, and the right \(R\)-balance is the literal operator
action. It commutes with \(D_s\).

Take any element \(v\) of the source of EV.29. It is one
finite sum of elementary tensors. Since \(\tau_s(p'_0)=1\),
the actual operator \(D_s^{-1}\) exists in \(B'\). Form
\[
w=e_\Phi^{[u]}(D_s^{-1}\kappa(v)).
\tag{EV.30}
\]
This is an actual element of the right side of EV.28.
Using EV.26 and \(D_sD_s^{-1}=1\) gives the exact module
identity
\[
D_sw=(C_{\mathrm{tr}}\otimes e_\Phi)(v).
\tag{EV.31}
\]
The equality uses the constant-parameter compatibility of
EV.26 on each of the finitely many entries of \(v\).
It is not an equality only after applying a function map.

ST.12 and EV.31 imply \(J\otimes_R e_\Phi=0\).
The equivalent finite tensor criterion ST.16–ST.19 now gives
\[
\begin{gathered}
\operatorname{im}(T\otimes_R e_\Phi)\\
\subseteq
\operatorname{im}(K_*^\infty\to T\otimes_R K_*^\infty).
\end{gathered}
\tag{EV.32}
\]
Specifically, EV.30 supplies the primitive in ST.18.
Represent it by the finite ordinary row supplied by FB.39.
Its equality EV.31 is a finite sum of tensor balancing
relations. Lift their finitely many coefficient factors
through ST.8, project to \(J\), and balance the remaining
ordinary factors into \(K_*^\infty\), exactly as in ST.19.
This produces a finite elementary-tensor equality for every
input in EV.32. No infinite sum of elementary tensors,
completed tensor, or closure of a relation space is involved.

#### EV.F. The full infinite-order linearity and faithfulness

For \(P\in T\) and \(s\in N\), EV.32 gives \(k\in K_*^\infty\)
with
\[
\begin{gathered}
P\otimes e_\Phi(s)=1\otimes k\\
\text{in }T\otimes_R K_*^\infty .
\end{gathered}
\tag{EV.33}
\]
Apply the balanced map to the actual directional module,
\[
\begin{gathered}
T\otimes_R K_*^\infty\longrightarrow\mathcal C,\\
P\otimes k\longmapsto Pj(k).
\end{gathered}
\tag{EV.34}
\]
Its right balance is precisely the \(R\)-linearity of \(j\).
The left side of EV.33 becomes
\(P\Phi^\infty(s)=\Phi^\infty(Ps)=j(e_\Phi(Ps))\).
The right side is \(j(k)\). Injectivity of \(j\) therefore
identifies \(k=e_\Phi(Ps)\). Consequently
\[
\begin{gathered}
s\longmapsto1\otimes e_\Phi(s)\\
N\longrightarrow T\otimes_R K_*^\infty
\end{gathered}
\tag{EV.35}
\]
is \(T\)-linear.

The actual special balancing ISR.3, equations ISR.8–ISR.10, gives
\[
B\otimes_TN\xrightarrow{\;\sim\;}N .
\tag{EV.36}
\]
For completeness, its earlier polynomial-coefficient tensor
is \(B\otimes_{R_{\mathrm{pol}}^\infty}N\).
The two inverse multiplication maps are ISR.4–ISR.7, and
\(R_{\mathrm{pol}}^\infty\subset T\) by ISF.9.
Quotient by the additional \(T\)-balance relations.
The inverse \(s\mapsto1\otimes s\) is \(B\)-linear before that
quotient, hence after it.
Multiplication and this map are inverse on every elementary
tensor. This proves EV.36 without an extension-flatness assertion.

Tensor EV.35 on the left by \(B\) over \(T\), then use EV.36
and finite tensor reassociation. The result
\[
\begin{gathered}
\Psi_\Phi(s)=1\otimes e_\Phi(s)\\
\text{in }B\otimes_R K_*^\infty
\end{gathered}
\tag{EV.37}
\]
is \(B\)-linear.

Choose a basis \(\Phi_1,\ldots,\Phi_q\) of \(V\), and take
\[
\begin{gathered}
L=L_*^{\,q},\\
K=K_*^{\,q},\\
e=(e_{\Phi_1},\ldots,e_{\Phi_q}) .
\end{gathered}
\tag{EV.38}
\]
The full \(B\)-action on \(\mathcal C\) gives a balanced contraction
\[
B\otimes_R K^\infty\longrightarrow\mathcal C^q .
\tag{EV.39}
\]
Its composite with \(\Psi=(\Psi_{\Phi_a})_a\) is exactly
\((\Phi_a^\infty)_a\).
This latter map is injective on the whole \(N\), including
its singular sections, by
§5.84, SE.1.
Thus both \(\Psi\) and \(e\) are injective.
The finite reassociation
\[
\begin{gathered}
B\otimes_R(R\otimes_{\mathcal D_X}K)\\
\simeq B\otimes_{\mathcal D_X}K,\\
b\otimes(r\otimes k)\\
\longmapsto br\otimes k
\end{gathered}
\tag{EV.40}
\]
has inverse \(b\otimes k\mapsto b\otimes(1\otimes k)\).
Both balance identities hold on each generator.
This proves every assertion of EV.1.
If \(q=0\), SE.1 gives \(N=0\), and the zero target proves
the theorem.

The theorem retains the original projected hypersurface,
the entire infinite extension and the actual finite D-type
quotient. Its finite-pole and intrinsic microlocal order
properties require the further regularity hypotheses and
arguments stated in Q.8.

![The stabilized finite algebra, actual parameter primitive and faithful full tensor map](assets/actual-infinite-evaluation.png)

**Figure EV.1.** Panel A is
the exact quotient-algebra argument
EV.13–EV.15. The group arrow
is surjective; the finite algebra
arrow becomes an isomorphism at
the maximum dimension.
Panel B keeps \(u\) and \(s\)
independent. The normal covector
is \(\nu_u=0,\tau_s=1\).
Its primitive identity EV.31 is
an equality in the actual ordinary
parameter tensor module.
Panel C is the commuting evaluation
diagram EV.37–EV.39. Its right
arrow is the full \(B\)-action
through the directional injection.
The reconstruction in EV.36 uses
the actual special ring \(T\).
These are algebraic diagrams with
no geometric scale.

Figure SVG ·
exact diagram data ·
reproducible figure source ·
figure PNG.

Free source:
[Kashiwara–Kawai, Holonomic systems III, IV.4.1 and IV.4.7](https://www.kurims.kyoto-u.ac.jp/~kenkyubu/kashiwara/HolIII.pdf).


### 5.87. Faithful finite-pole realization of regular actual modules

Work over a complex manifold \(X\) of dimension \(n\ge1\).
Let \(M\) be a coherent actual holonomic microlocal module,
regular on a dense smooth open of every reduced characteristic
component, and in generic position at \(p=(0;dt)\).
Here regularity means that on that open there is a coherent
finite order-zero lattice stable under all order-one operators
whose principal symbol vanishes on the smooth characteristic
Lagrangian. DP.R identifies this premise with Euler and
tangential stability in a standard smooth conormal chart.
An everywhere-stable lattice is not a premise.

Put
\[
\begin{gathered}
D=\mathcal D_{X,0},\quad E=\mathcal E_{X,p},\\
R=D^\infty_{X,0},\quad B=\mathcal E^\infty_{X,p},\\
N=B\otimes_E M_p,\\
S=\pi(\operatorname{supp}M).
\end{gathered}
\tag{RF.1}
\]
All tensors are ordinary algebraic tensors.
For the zero module every assertion holds with zero targets;
hence assume the module nonzero in the proof.
The support is the actual finite-module support of J.1–J.2.
Choose its closed conical representative \(\Lambda\) on a
sufficiently small base as in CF.C.

**Theorem RF.T.** The stalk \(M_p\), with its actual ordinary
operator action, is finite over \(D\). There are a finite
D-type module \(L\) along exactly \(S\), an embedded
trivial connection \(\mathcal O_X^q\), and a coherent
holonomic quotient
\[
\begin{gathered}
K=L/\mathcal O_X^q,\\
\operatorname{Hom}_{\mathcal D_X}(\mathcal O_X,K)_0=0,\\
e':M_p\hookrightarrow K_0 .
\end{gathered}
\tag{RF.2}
\]
The last map is \(D\)-linear. The actual tensor-unit map
to \(E\otimes_D K_0\) is \(E\)-linear and injective,
and multiplication gives
\[
\begin{gathered}
E\otimes_D M_p\xrightarrow{\sim}M_p,\\
R\otimes_D M_p\xrightarrow{\sim}N .
\end{gathered}
\tag{RF.3}
\]
There is a coherent holonomic ordinary module \(F\) on
one base neighbourhood, with regular finite
microlocalization and \(F_0\simeq M_p\), such that
\[
\begin{gathered}
\mathcal E_X\otimes_{\pi^{-1}\mathcal D_X}\pi^{-1}F\\
\simeq M\quad\text{near }p,\\
\operatorname{Char}(F)\subseteq\Lambda\cup T_X^*X .
\end{gathered}
\tag{RF.4}
\]
In particular its finite microlocalization at every
other nonzero ray over \(0\) is zero. The quotient
in RF.2 and the original projected hypersurface are
retained; all finite-source sections are included.

The exact earlier proofs used below are
actual normal completion, §§5.22.1–5.22.2,
commutative coefficient flatness, F.3,
ordinary operator coherence and analytic coefficient flatness, GL06 Lemmas3.0c.1–3.0c.3,
actual operator coherence, §5.24,
the microlocal and ordinary matching-ring faithful units, FEP.7,
ordinary infinite right flatness, MF.6–MF.7,
whole infinite generation, ISR.2,
actual finite Euler division, AM.A,
its full function action, AM.C,
the finite Euler normal form, DP.R,
whole point-coordinate transport, §5.56,
the normalized finite D-type lattice and its finite-kernel extension, DT.E,
finite cyclic duals and actual ordinary reconstruction, PD.35–PD.44,
the all-parameter function comparison, FB.39,
actual finite projection and directional classes, CF.A–CF.C,
and the entire infinite D-type tensor realization, EV.1.
The finite analytic dimension bound is proved by
successive local equations, AG-CA Theorem3.2
and the finite local-parametrization argument in CF.C.

#### RF.A. Finite microlocalization and characteristic support

Let \(D=\mathcal D_{X,q}\), \(E=\mathcal E_p\),
where \(p\ne0\) lies above \(q\). Choose a linear time coordinate
with \(\tau(p)\ne0\), normalize its ray, and put
\[
\begin{gathered}
H=\mathcal O_{X,q},\\
T=\mathcal O_{X\times\mathbf C_z,(q,z(p))},\\
h=\tau^{-1}.
\end{gathered}
\tag{RF.5}
\]
Write \(E_0\) for the actual order-zero ring. Its normal element,
actual coefficient shifts, Noetherianity, exact normal
Artin--Rees completion, tensor identity and zero detection are
the complete earlier proofs §§5.21, 5.22.1--5.22.2.
In particular
\[
\begin{gathered}
\widehat E_0=\varprojlim E_0/(h^r)\\
=\prod_{n\ge0}T h^n,\\
\widehat E=\widehat E_0[h^{-1}].
\end{gathered}
\tag{RF.6}
\]
These formal coefficients are individual holomorphic germs.
No common coefficient domain or factorial estimate is
asserted for an arbitrary element of \(\widehat E\).
The product has finitely many contributions in every
homogeneous order, so the ordinary \(D\)-map and multiplication
are well defined.

##### RF.A.1. Exact faithful formalization of finite actual E-modules

For a finite \(E\)-module \(V\), let \(V_0\subset V\) be the
\(E_0\)-span of finitely many \(E\)-generators. It is finite,
\(E V_0=V\), and \(h\) acts injectively on it.
For an exact sequence of finite \(E\)-modules, take \(V_0\)
in its middle term, intersect with the first term and take
the image in the last term. Noetherianity of \(E_0\) makes
the intersection finite. These three lattices generate the
three \(E\)-modules: multiply a finite localized expression
by a large enough left power of \(h\), using normality.
All three lattices embed in their respective modules, so
all are \(h\)-torsion free.

Exact completion §5.22.2 followed by exact normal localization
gives an exact sequence over \(\widehat E\).
Its terms are canonically
\(\widehat E\otimes_E V\), by the finite tensor
reassociation and localization identities. Thus
\[
\begin{gathered}
\widehat E\otimes_E-\\
\text{is exact on finite actual}\\
E\text{-modules}.
\end{gathered}
\tag{RF.7}
\]
It is faithful. A nonzero \(V_0\) has nonzero completion
by the actual normal Nakayama/zero-detection proof.
That completion is still \(h\)-torsion free:
apply exact completion to the injective left-linear map
\({}^{\sigma^{-1}}V_0\to V_0,\ v\mapsto hv\).
Here \(a\) acts on the twisted source by \(\sigma^{-1}(a)\);
then \(h\sigma^{-1}(a)v=ahv\).
Localizing a nonzero \(h\)-torsion-free module cannot kill
it. Hence \(\widehat E\otimes_E V\ne0\).
This argument uses actual faithful completion only at
order zero, where \(h\) is in the Jacobson radical; it does
not put a unit \(h\) in the radical of \(E\).

##### RF.A.2. Formal graded flatness over D

The increasing order filtrations have
\[
\begin{gathered}
\operatorname{gr}D=H[\tau,\xi],\\
\operatorname{gr}\widehat E=T[\tau,\tau^{-1}],\\
\xi_i\longmapsto\tau z_i .
\end{gathered}
\tag{RF.8}
\]
The graded target is flat over the graded source.
First invert \(\tau\), giving \(H[z,\tau,\tau^{-1}]\).
Then extend \(H[z]\to T\). This coefficient extension
is flat by the complete completion argument F.3
(equivalently GL06 Lemma3.0c.1); a translation of \(z(p)\)
reduces to its zero-coordinate proof. Laurent extension
and the two tensor reassociations give precisely RF.8.

Every finite \(D\)-module has a strict filtered free
resolution with finite shifted free terms.
Start with finitely many generators and their quotient
order filtration. For its kernel, the induced associated
graded module is a submodule of a finite graded free
module over the Noetherian \(H[\tau,\xi]\).
Lift a finite homogeneous generating list of that kernel
to actual relation rows. Leading-order cancellation
terminates, because ordinary differential-operator order
in each shifted free term is bounded below. It makes
this new map strict onto the whole kernel. Repeat.
This proves the needed resolution directly; an infinite
resolution is allowed, since every degree and every
cycle uses only adjacent finite free terms.

Tensor that resolution with \(\widehat E\).
Its graded complex is the tensor of the original
graded free resolution with RF.8 and is exact in
positive degrees.
For a cycle \(v\) of order \(m\), solve its leading
homogeneous equation by that graded exactness.
Lift the solution to a row of homogeneous formal
operators of order \(m\), subtract its boundary, and
repeat at orders \(m-1,m-2,\ldots\).
The sum of the correcting rows belongs to the finite
shifted free formal term: it has a finite upper order,
and every one of its homogeneous coefficients is a
finite sum of holomorphic germs. Coefficientwise
continuity gives its boundary exactly \(v\).
Consequently the formal tensor complex is exact in
positive degrees.

##### RF.A.3. Descent to actual operators

Tensor the same ordinary free resolution with actual \(E\).
Every term is finite free over the left Noetherian
actual \(E\), so its cycles, boundaries and homology
are finite actual \(E\)-modules.
RF.7 commutes with their kernels and images.
Its homology is the formal tensor homology just proved
zero. Faithfulness in RF.A.1 therefore makes the actual
homology zero. In particular tensoring an injection
of finite \(D\)-modules with \(E\) is injective:
\[
E\text{ is right flat over }D .
\tag{RF.9}
\]
For arbitrary modules the same conclusion follows
by finite witnesses. If a finite tensor from \(U\subset V\)
dies in \(E\otimes_DV\), its vanishing is a finite sum
of elementary additivity and balancing relations.
All entries lie in a finite \(D\)-submodule \(V_1\).
The intersection \(U\cap V_1\) is finite, since \(D\)
is left Noetherian. Finite-module exactness makes
the tensor zero already in \(E\otimes_D(U\cap V_1)\),
and hence in \(E\otimes_DU\).

No formal correcting series has been declared convergent.
The actual conclusion follows because its actual finite
homology module has zero faithful formalization.
The actual factorial-growth inverses and Noetherian
arguments establishing that faithfulness are §5.20--5.22,
applied on their stated common domains.
This proves finite microlocalization flatness, not
flatness of \(E^\infty\) over \(D^\infty\).

##### RF.A.4. Exact characteristic support

For a finite coherent ordinary \(D\)-module \(F\),
the induced formal quotient filtration is strict.
To check this at its presentation map, suppose a
boundary has order at most \(m\) but its chosen
preimage has order \(l>m\). The leading preimage
lies in the kernel of the graded differential.
Graded exactness supplies a leading second-boundary
which can be subtracted without changing the
boundary and which lowers that preimage's order.
After finitely many reductions its order is at
most \(m\). Thus
\[
\begin{gathered}
\operatorname{gr}(\widehat E\otimes_D F)\\
\simeq T[\tau,\tau^{-1}]\\
\otimes_{H[\tau,\xi]}\operatorname{gr}F .
\end{gathered}
\tag{RF.10}
\]
The formal quotient filtration is separated and
complete in decreasing order on a finite generating
lattice. Indeed the order-zero formal ring is
Noetherian and all its finite modules are complete
and separated by §5.22.2. Thus the formal tensor is
zero exactly when RF.10 is zero.

After inverting \(\tau\), a graded module is its
degree-zero slice extended by \(\tau^{\pm1}\).
This slice is a finite \(H[z]\)-module.
The map from \(H[z]\) localized at \((\mathfrak m_q,
z-z(p))\) to \(T\) is faithfully flat by F.3.
Consequently RF.10 is nonzero exactly at the
ordinary projective characteristic point \([p]\).
Faithfulness of RF.7 identifies that nonvanishing
with nonvanishing of \(E\otimes_DF\).
Hence, at every nonzero covector,
\[
\begin{gathered}
\operatorname{supp}\bigl(\mathcal E_X
\otimes_{\pi^{-1}\mathcal D_X}\pi^{-1}F\bigr)\\
=\operatorname{Char}(F)\setminus T_X^*X .
\end{gathered}
\tag{RF.11}
\]
Coherence of this actual \(E\)-module follows from
the finite ordinary presentation and actual operator
coherence §5.24. Coordinate translations and a
linear choice of nonzero \(\tau\) cover all such
points. Stalk exactness gives the sheaf assertion.

#### RF.B. Inhomogeneous Euler classes and resonances

Use a standard smooth conormal chart with coordinates \(t,x\), and the
actual directional quotient of CF.A. Let \(A\) be a constant complex
\(r\)-square matrix. Suppose a column of classes \([f]\) satisfies
\[
\begin{gathered}
(\theta I-A)[f]=0,\\
D_{x_i}[f]=0,\qquad \theta=tD_t .
\end{gathered}
\tag{RF.12}
\]
Choose representatives on one cofinal isotropic future-convex complement.
There are ordinary holomorphic germs \(b_i=D_{x_i}f\).
They satisfy \(\partial_i b_j=\partial_j b_i\).
The ordinary germ
\[
\begin{gathered}
h(t,x)\\
=\int_0^1\sum_i x_i b_i(t,\lambda x)\,d\lambda
\end{gathered}
\tag{RF.13}
\]
has \(D_{x_i}h=b_i\). Indeed differentiation and the closedness
identity make the integrand the derivative of
\(\lambda b_i(t,\lambda x)\). The spatial contraction preserves
the representative domain, so \(f-h=F(t)\) there. On restriction
to \(x=0\), RF.12 says
\[
\begin{gathered}
(\theta I-A)F=g(t),\\
g\in\mathcal O_{\Delta}^{\,r}.
\end{gathered}
\tag{RF.14}
\]

Write \(g=\sum_{n\ge0}g_nt^n\). If \(nI-A\) is invertible, set
\(c_n=(nI-A)^{-1}g_n\). For \(n>2\|A\|\), geometric inversion
gives \(\|(nI-A)^{-1}\|\le2/n\).
Thus all these terms form a convergent ordinary holomorphic series
on any smaller disc. There are only finitely many excluded
nonnegative integers.

On the generalized eigenspace with \(A=nI+J\), \(J^\ell=0\),
the excluded term has the particular solution
\[
t^n\sum_{k=0}^{\ell-1}
       \frac{(\log t)^{k+1}}{(k+1)!}J^kg_n .
\tag{RF.15}
\]
Applying \(\theta I-A\) differentiates the polynomial in \(\log t\)
and subtracts its product by \(J\); all terms cancel except \(g_n\).
On the other generalized eigenspaces the ordinary inverse applies.
The elementary primary decomposition and the nilpotent blocks are
the complete finite matrix arguments in AM.C, not an additional
classification premise.

Subtracting this particular solution from \(F\) leaves the homogeneous
solution \(e^{A\log t}c\). Its derivative after multiplication by
\(e^{-A\log t}\) is zero, so \(c\) is constant on the connected lifted
sector. On a block \(A=\lambda I+J\),
\[
\begin{gathered}
e^{A\log t}c\\
=t^\lambda\sum_{k=0}^{\ell-1}
\frac{(\log t)^k}{k!}J^kc .
\end{gathered}
\tag{RF.16}
\]
Every class in RF.12 therefore has a representative which is a finite
Nilsson sum, plus an ordinary holomorphic germ. This includes integer
resonances and classes represented by inhomogeneous holomorphic errors.
It does not discard such errors by presuming a homogeneous scalar
equation for the original representative.

#### RF.C. Finite-order rows and every negative coefficient

The actual Euler division AM.A reduces a finite-order row to
\[
\begin{gathered}
R(x,D_t)=\sum_{j\le m}r_j(x)D_t^j,\\
|r_{-n}|_K\le MC^n n!
\end{gathered}
\tag{RF.17}
\]
on one smaller tangential product. The upper bound \(m\) is finite.
DP.7.5 explicitly checks that division of a finite-order input retains
finite upper order; no full positive head enters RF.17.

For a noninteger exponent, AM.22 gives
\[
\begin{gathered}
D_t^{-n}t^\lambda\\
=\frac{t^{\lambda+n}}
{(\lambda+1)\cdots(\lambda+n)}\\
\text{modulo ordinary}\\
\text{holomorphic germs}.
\end{gathered}
\tag{RF.18}
\]
There are constants \(C_\lambda,K_\lambda,a_\lambda\) such that
the reciprocal denominator is bounded by
\(C_\lambda K_\lambda^n(n+1)^{a_\lambda}/n!\).
Split the product after \(2|\lambda|+2\); the initial nonzero factors
cost a fixed constant, and each remaining factor has modulus at
least half its index. A polynomial factor also absorbs any fixed
shift in the factorial index.

Here is one explicit treatment of every log power and integer
resonance, without a special-function normalization. Fix the
input exponent \(\lambda_0\), and put
\[
\begin{gathered}
a(\lambda)=\\
\begin{cases}
\lambda-\lambda_0,&\lambda_0\in-\mathbf Z_{>0},\\
1,&\text{otherwise},
\end{cases}\\
Q_n(\lambda,t)=
\frac{a(\lambda)t^{\lambda+n}}
{\prod_{s=1}^{n}(\lambda+s)} .
\end{gathered}
\tag{RF.19}
\]
Choose one small exponent disc about \(\lambda_0\) containing
no other negative integer. Every \(Q_n\) is analytic in that
disc: the only possible denominator zero there is canceled by
\(a\), and it occurs at most once. Also \(D_tQ_n=Q_{n-1}\).
If \(a=1\), its Taylor coefficient of degree \(\ell\) at
\(\lambda_0\) starts with
\(t^{\lambda_0}(\log t)^\ell/\ell!\) at \(n=0\).
If \(a=\lambda-\lambda_0\), use degree \(\ell+1\) instead.
Thus these coefficients are exact iterated primitives of every
input power/log monomial, up to the displayed fixed factor.

On the exponent-disc boundary, all finite initial denominator
factors have one positive lower bound; every sufficiently late
factor has modulus at least half its index. Uniformly in \(n\),
\(a(\lambda)/\prod_{s=1}^{n}(\lambda+s)\) is consequently
bounded by \(M K^n/n!\), enlarging \(M,K\) for the finitely
many early \(n\). The Cauchy formula in \(\lambda\) gives the
same bound, times a fixed radius factor, for every one of the
finitely many Taylor coefficients used above. Factoring
\(t^{\lambda_0+n}\) shows each primitive is a polynomial in
\(\log t\) of uniformly bounded degree with these coefficient
bounds. In the negative-integer case its degree may increase
by one upon crossing exponent zero; it does not keep increasing.
The finitely many primitives before this crossing are included.

Multiplying by RF.17 cancels \(n!\). After choosing one radius with
\(CK|t|<1/2\), the resulting coefficient series is normally convergent:
\[
\sum_{n\ge0}(CK|t|)^n<\infty .
\tag{RF.20}
\]
It is a holomorphic coefficient at zero after a single fixed
power of \(t\) has been removed. The finitely many nonnegative
derivatives \(j\le m\) have the same property. Thus the result
is a finite power/log expression with meromorphic coefficients.

These are the actual directional actions, not only formal
primitive symbols. The exact anchor corrections are AM.23b and
the Taylor coefficients of RF.19. At a negative anchor \(a_0\),
their \(n\)-th primitive values satisfy
\(M_{a_0}K^n|a_0|^n/n!\), with \(M_{a_0}\) allowed to depend
on that anchor. AM.23b therefore bounds their anchor
polynomial by
\(M_{a_0}(K|a_0|+r+|a_0|)^n/n!\).
Their gauges converge on the full time disc after the same
single radius choice as AM.23c. Fixed further
derivatives use one larger Cauchy product. In particular all
negative-order coefficients and tangential indices remain present.

Together with DP.R, this supplies the smooth generic calculation
for an actual module with a finite Euler/tangentially stable
order-zero lattice. Applying DP.R here uses precisely its proved
stable-lattice premise. It does not infer that premise for every
point of an arbitrary generically regular module.

#### RF.D. Propagating a scalar finite pole bound

Let \(S\) be a reduced hypersurface germ in a smooth complex
manifold. Let \(g\) be holomorphic on its complement. Suppose on a
nonempty smooth open of each irreducible component of \(S\),
\(g\) is meromorphic with a finite pole bound. Then \(g\) is a
meromorphic germ with poles only on \(S\).

Here is a uniform-bound proof. For one irreducible component
choose a reduced monic Weierstrass equation
\(p(t,x)\), with all roots in an inner time disc. Outside its
discriminant the roots form an unramified finite cover of a
connected base complement. This cover is connected. Otherwise
the elementary symmetric functions in the roots of one
monodromy orbit give a nontrivial monic factor of \(p\).
They are holomorphic off the discriminant and bounded by the
fixed root radius. A fixed transverse Cauchy circle extends
each coefficient across the discriminant; this is the same
bounded removable-singularity construction as MC.D/DT.E.
The resulting factorization of the irreducible germ is impossible.
The base complement is path connected by the complex-line
detours proved in MC.A and SE.A.

Near an unramified root use \(v=t-a(x)\). The negative Laurent
coefficients of \(g\) in \(v\) are holomorphic in \(x\), by their
fixed-circle integrals. If the pole order is at most \(N\) on
one nonempty open in this chart, every coefficient below
\(-N\) vanishes identically by the holomorphic identity theorem.
Along a path on the connected root cover, finitely many
overlapping such charts propagate the SAME \(N\).
Changing \(v\) by a holomorphic unit preserves that bound.
Thus the bound holds on every unramified smooth sheet.

Repeat this argument for the finitely many irreducible
components, avoiding their pairwise intersections. Choose a
single exponent \(N\) at least all the resulting bounds. If
\(f\) is a reduced equation for their union, \(f^Ng\) extends
holomorphically across all unramified sheets.
The remaining subset of \(S\) consists of the discriminant
critical loci, component intersections and singular loci.
Each has ambient codimension at least two; the rank and
Weierstrass dimension proofs are the ones in CF.C and DT.E.
The fixed-circle Hartogs proof DT.E extends \(f^Ng\) across
this subset. Hence \(g\in\mathcal O_X(*S)\).
In dimension one there is no such exceptional subset, and
the finitely many punctures give the assertion directly.

The use of bounded removal for a scalar holomorphic function
across a hypersurface is elementary as well: choose a
transverse line, use the uniform bound to kill its negative
Laurent coefficients, and use the same fixed-circle integral
in the other variables. Repeat on the smooth hypersurface
locus, then apply DT.E at its codimension-two singular locus.

#### RF.E. The test in a fixed finite D-type target

For the already constructed finite D-type \(L\) of EV.1,
its normalized reflexive lattice \(L_\Sigma\) exists by
DT.E, with \(L_\Sigma(*S)=L\).
This use of Section5.27 concerns the ALREADY finite
meromorphic target \(L\), not the source \(M\); it does not
assume the sought source finite-pole realization.

Let \(v\in L^\infty\), interpreted by FB.39 as a holomorphic
bundle section on \(X\setminus S\). Suppose it belongs to
the finite meromorphic bundle on a nonempty smooth open of
every local component of \(S\).
Choose a finite presentation of \(L_\Sigma^*\).
Dualizing and using reflexivity identifies \(L_\Sigma\)
with the kernel of a finite holomorphic matrix
\(\mathcal O^b\to\mathcal O^a\).
On \(X\setminus S\), evaluation of \(v\) against the finite
dual generators gives \(b\) scalar holomorphic functions.
They have finite poles on the stipulated generic opens.
By RF.D all \(b\) functions are meromorphic with poles on \(S\).
Take one power \(f^N\) clearing their denominators.
The finite kernel equations hold off \(S\), so hold
everywhere by density. Therefore \(f^Nv\in L_\Sigma\)
and \(v\in L\). No assertion about the support of
an arbitrary infinite quotient is needed for this test.

#### RF.F. The projective conormal-fibre exclusion

First establish this local fact. If a holomorphic ambient one-form
\(\alpha\) vanishes on the smooth locus of a reduced analytic germ
\(A\), then its pullback vanishes on a dense smooth open of any
irreducible analytic subgerm \(W\subset A\).
It suffices to take one component of \(A\) containing \(W\).
Write \(d=\dim A\), \(k=\dim W\); the case \(k=d\) follows by density.
At a generic smooth point of \(W\), choose a finite linear
Noether normalization of \(A\) whose restriction has rank \(k\)
on \(W\). These are simultaneously open conditions on the
linear projection: its kernel must avoid the tangent cone of
\(A\), and one \(k\)-minor on \(T W\) must be nonzero.
The finite projection construction is the earlier analytic
local-parametrization theorem used in CF.C.

Use \(w\in\mathbf C^k\) on the image of \(W\), and transverse
coordinates \(y\in\mathbf C^{d-k}\) on the normalized base.
After shrinking about this generic point and subtracting the
holomorphic graph of that image, it is \(y=0\).
The finite projection's discriminant and the image of
\(A_{\rm sing}\) are proper analytic subsets of the base.
For the latter assertion a finite map preserves dimension;
\(\dim A_{\rm sing}<d\). Choose a nonzero holomorphic germ
vanishing on their union. Choose a constant vector \(c\)
whose substitution \(y=\epsilon c\) makes this germ nonzero
as a germ in \((w,\epsilon)\). This is possible by taking
the first nonzero transverse homogeneous coefficient and
avoiding its zero polynomial in \(c\); if the germ does not
vanish identically on \(y=0\), no transverse condition is
needed. On a further dense \(w\)-open its substitution is
\(\epsilon^a h(w,\epsilon)\), with \(h(w,0)\ne0\).
After shrinking there, its only zero is \(\epsilon=0\).

The resulting finite family over \(B_w\times\Delta_\epsilon\)
is therefore unramified and lies in \(A_{\rm reg}\) for
\(\epsilon\ne0\). The point on the chosen local branch of
\(W\) lies on a component dominating this family.
Indeed the base family is cut out by \(d-k-1\) equations
in the \(d\)-dimensional normalization base. Their
pullbacks cut the pure \(d\)-dimensional \(A\). Iterating
the local one-equation dimension bound makes the cut
germ at that point have dimension at least \(k+1\).
Thus at least one component through that point has
dimension at least \(k+1\). A finite map preserves dimension, while its image
is in the irreducible \((k+1)\)-dimensional base family.
Properness and the analytic proper-subset dimension
drop make this component's image the entire family.
Its punctured sheets consequently limit to the chosen
central point. This dimension argument is essential:
no flatness of the arbitrary finite normalization map
has been presumed.

The punctured product has fundamental group \(\mathbf Z\):
contract the convex \(w\)-disc and use the elementary
logarithm cover of the punctured time disc. The sheet
permutation has finite order \(N\). Pulling back by
\(\epsilon=z^N\) trivializes that finite covering.
Each sheet's ambient coordinate functions are bounded
holomorphic functions on \(B_w\times\Delta_z^*\), because
the original finite map is proper over a smaller product.
Their negative Laurent coefficients vanish by boundedness;
fixed-circle integration extends them jointly across \(z=0\).
Choose the sheet limiting to \(W\). It gives a holomorphic map
\[
\begin{gathered}
\mu:B_w\times\Delta_z\longrightarrow A,\\
\mu(w,0)\in W,\\
\mu(w,z)\in A_{\rm reg}\quad(z\ne0).
\end{gathered}
\tag{RF.21}
\]
On \(z=0\) the \(w\)-projection is the identity. The pullback
\(\mu^*\alpha\) is a holomorphic one-form zero for \(z\ne0\);
each coefficient is zero everywhere by the identity theorem.
Its restriction to \(z=0\) proves the local fact.

Now let \(X\) have dimension \(n\), let \(\Lambda\) be a closed
homogeneous Lagrangian germ off the zero section, and suppose
\[
\Lambda\cap\pi^{-1}(q)=\mathbf C^*p .
\tag{RF.22}
\]
Its projectivization \(A=\mathbf P\Lambda\) is a closed Legendrian
germ, and its base projection is proper with a one-point fibre.
Choose one projective affine chart containing \([p]\).
The closed part of \(A\) outside this chart has proper
closed image avoiding \(q\). Shrink the base to avoid
that image. The entire \(A\) now lies in the affine
chart, and its fibre at \(q\) is the isolated point
\([p]\). The dimension-zero fibre case of CF.C's
finite preparation proof makes this projection finite
on a smaller base. No unproved assertion about an
infinite-coefficient support is used. Put \(Y=\pi(A)\).

For any irreducible analytic germ \(Z\subset Y\) through \(q\),
some irreducible component \(W\) of \(A\times_X Z\) maps
finitely onto \(Z\). Indeed its finitely many proper
component images cover the irreducible \(Z\), so one is
all of \(Z\). On a dense smooth open this map has rank
\(\dim Z\), by separability of its finite coordinate
equations in characteristic zero.

On each cotangent projective chart the tautological contact
form is a holomorphic one-form, up to a nowhere zero line
factor. Its pullback vanishes on \(A_{\rm reg}\).
For the conical lift this follows from
\(\theta=\iota_{\mathcal R}\omega\): the radial field
\(\mathcal R\) is tangent to the homogeneous Lagrangian,
and its symplectic form restricts to zero.
RF.21 therefore makes this contact form vanish on the
dense smooth open of \(W\) as well. At its points of
full projection rank, the corresponding covector
annihilates \(T Z\). Hence
\[
W\subset\mathbf P\overline{T^*_{Z_{\rm reg}}X}.
\tag{RF.23}
\]
Both sides are closed after taking the conormal closure;
the containment follows from the dense-open containment.
Properness of \(W\to Z\) gives a point of \(W\) over \(q\);
RF.22 says it is \([p]\). Consequently every such conormal
closure contains \([p]\) in its fibre over \(q\).

Finally let \(\Lambda'\) be another closed homogeneous
Lagrangian, with \(\pi(\Lambda')\subset Y\), and suppose
\(p\notin\Lambda'\). If it had a nonzero point over \(q\),
one projective irreducible component would have proper
image an irreducible germ \(Z\subset Y\) through \(q\).
On the dense open where this component and \(Z\) are
smooth and projection has maximal rank, the same
tautological-form argument places it in
\(\mathbf P T^*_{Z_{\rm reg}}X\). Both irreducible
varieties have dimension \(n-1\): the fibre dimension
of the conormal is \(n-\dim Z-1\).
The earlier analytic proper-subset dimension drop
therefore identifies the component with the full
projective conormal closure. RF.23 makes it contain
\([p]\), a contradiction. Thus
\[
\Lambda'\cap\pi^{-1}(q)=\varnothing .
\tag{RF.24}
\]

#### RF.G. The whole finite source in the finite quotient

Assume \(M\) is regular in the original microlocal sense:
on a dense smooth open of each reduced characteristic
component it has a coherent order-zero lattice stable
under the order-one principal-vanishing operators.
This gives DP.34 in a standard smooth conormal chart.
No everywhere-stable lattice or intrinsic-order theorem
is assumed.

Apply EV.1, retaining \(S=\pi(\operatorname{supp}M)\),
\[
\begin{gathered}
K=L/\mathcal O_X^q,\\
e:B\otimes_EM\hookrightarrow K^\infty_0,\\
\Psi(s)=1\otimes e(s).
\end{gathered}
\tag{RF.25}
\]
The latter is an injective \(B\)-linear map into
\(B\otimes_DK_0\).
Fix \(u\in M_p\). The matching-ring faithful unit FEP
identifies it with its image in \(B\otimes_E M_p\).
Ordinary right flatness MF.6--MF.7 and the equality
\(\mathcal O^\infty=\mathcal O\) let us lift \(e(u)\)
to \(v\in L^\infty_0\).

There is a common cotangent neighbourhood on which the
tensor map needed for this fixed calculation is defined.
Choose a finite actual \(E\)-presentation of \(M\).
Its scalar extension is a finite \(B\)-presentation.
Represent the finitely many values of \(\Psi\) and its
finitely many relation equalities on one neighbourhood.
Each equality is a finite algebraic tensor equality:
its additivity and balancing witnesses have only finitely
many section entries. Shrink once to represent all
witnesses and identities. This defines a \(B\)-linear
sheaf map there, without a coherence assumption on its
target. Represent \(u\) as a finite row in the chosen
generators and represent its equality with \(1\otimes e(u)\)
on a further common neighbourhood.

The projective characteristic support can be kept
proper and finite over a smaller base and within this
neighbourhood, by CF.C's finite normal-coordinate
preparation at the isolated central projective fibre.
Thus each component of the original \(S\) still has a
nonempty smooth open whose whole conormal lies in the
neighbourhood and in the regular-lattice locus of \(M\).
At a smooth projected point, the characteristic covector
annihilates the tangent to \(S\), by the tautological-form
calculation in RF.F. Since \(S\) is a hypersurface it is
the unique conormal ray. Remove the component crossings
and the proper rank/discriminant exceptional loci.

At such a point \(a\), choose any local branch observation
of the finite cyclic-dual function model of \(L^\infty\).
It sends a section to its scalar holomorphic branch.
Modulo ordinary holomorphic functions it is \(D_a\)-linear.
It kills the embedded \(\mathcal O^q\), so gives a map
\(K_a\to\mathcal C_{p_a}\). The actual full \(B_{p_a}\)
action on \(\mathcal C_{p_a}\) extends it by the balanced
rule
\[
\begin{gathered}
B_{p_a}\otimes_{D_a}K_a
\longrightarrow\mathcal C_{p_a},\\
b\otimes k\longmapsto b[k].
\end{gathered}
\tag{RF.26}
\]
Compose with the spread map RF.25 and restrict to finite
\(M_{p_a}\). This is an actual \(E_{p_a}\)-linear
directional functional. Its value on \(u\) is the
directional class of the observed branch of \(v\).

An ordinary chart making \(S=\{t=0\}\) transports the
finite stable lattice and the actual directional actions
by the point-coordinate kernel maps of §5.56.
DP.R identifies the whole finite module with a constant
Euler model. RF.B–RF.C prove that every finite section's
image under RF.26 is a finite power/log expression,
modulo an ordinary holomorphic germ. Therefore the
observed branch of \(v\) has moderate power growth near
this smooth divisor point, uniformly on smaller
tangential compacts and closed angular intervals.
This is asserted for all finite-order input sections,
not only the Euler basis.

Only finitely many branch observations are needed:
in PD.39--PD.44 the cyclic-dual model is the dual of
one finite quotient algebra \(A_*\). Choose finitely
many group words whose images span \(A_*\).
Their scalar observations are the coordinates of
the corresponding equivariant \(A_*^*\)-valued
holomorphic function. Every other observation is
a constant linear combination of these.
Apply the preceding local calculation to this finite
list, including the \(q\) target copies.

On a smooth-divisor product, the fixed finite D-type
target has the normalized logarithmic frame already
proved in §5.27.11 and DT.E. Its change from the
horizontal frame is a holomorphic invertible gauge
times \(t^C\), with \(C\) a constant finite matrix.
Both \(t^C\) and \(t^{-C}\) have fixed power/log
growth on closed angular intervals: on a generalized
block expand the nilpotent exponential as in RF.16.
Thus the single-valued holomorphic coefficients of
\(v\) in this logarithmic frame have a uniform
bound \(M|t|^{-N}\). A finite angular cover of one
circle supplies one bound on that circle; the
finite observations and their monodromy transition
matrices supply the same conclusion across its seam.
The coefficient Laurent integral makes all powers
below \(-N\) zero, by letting the circle radius
tend to zero. Hence these coefficients are
meromorphic at the smooth divisor point.

This proves the hypothesis of RF.E for a nonempty
smooth open of every original \(S\)-component.
The scalar propagation and reflexive finite-kernel
argument RF.D–RF.E give \(v\in L_0\), and consequently
\[
e(M_p)\subset K_0.
\tag{RF.27}
\]
The restriction
\(e':M_p\hookrightarrow K_0\) is \(D\)-linear and
injective. This source is therefore finite over
the left Noetherian ordinary \(D\): it is a
submodule of the finite \(K_0\).
The argument proves this for every element of
the entire finite \(E\)-stalk, including singular
sections, including a selected generator list.

#### RF.H. Finite reconstruction and the whole characteristic fibre

For the \(D\)-linear injection \(e'\), define
\[
\begin{gathered}
\phi:M_p\longrightarrow E_p\otimes_DK_0,\\
u\longmapsto1\otimes e'(u).
\end{gathered}
\tag{RF.28}
\]
It is \(E_p\)-linear. Indeed the matching-ring unit
of \(E_p\otimes_D K_0\) into its \(B_p/E_p\)
scalar extension is injective by FEP for arbitrary
modules. That scalar extension is \(B_p\otimes_DK_0\).
There RF.28 agrees with the \(B_p\)-linear \(\Psi\).
For every \(P\in E_p\), the difference
\(\phi(Pu)-P\phi(u)\) has zero image under this
injective unit, and is zero already in the finite
tensor target. No mixed infinite-order flatness
is used to prove this finite linearity.

Let
\[
\begin{gathered}
\mu:E_p\otimes_D M_p
\longrightarrow M_p,\\
P\otimes u\longmapsto Pu .
\end{gathered}
\tag{RF.29}
\]
It is surjective. Its composite with \(\phi\)
is \(E_p\otimes_D e'\), since both maps are
\(E_p\)-linear and agree on \(1\otimes u\).
RF.9 makes that latter map injective.
Therefore \(\mu\) is injective as well:
\[
E_p\otimes_D M_p\simeq M_p.
\tag{RF.30}
\]
Here ordinary finiteness of \(M_p\) was proved
in RF.G before finite flatness is applied.

Choose a finite \(D\)-generator list of
\(e'(M_p)\subset K_0\), and represent its elements
as actual sections of \(K\) on a common base
neighbourhood. Their \(D_X\)-span is a coherent
ordinary submodule \(F\subset K\), by ordinary
operator coherence. It has \(F_0=e'(M_p)\);
ordinary submodule holonomicity gives that
\(F\) is holonomic. The isomorphism RF.30
extends to
\[
\begin{gathered}
\mathcal E_X\otimes_{\pi^{-1}\mathcal D_X}\pi^{-1}F\\
\simeq M\\
\text{on a neighbourhood of }p.
\end{gathered}
\tag{RF.31}
\]
To verify the spread, represent the finitely many
generator images, relation equalities and inverse
identities on one neighbourhood. Both modules are
coherent over the actual finite \(E\); their finite
kernel and cokernel vanish as germs, hence vanish
on a smaller neighbourhood. This does not
assert an isomorphism merely from equal ranks.

Let \(\Lambda\) denote the closed conical
characteristic representative used in RF.F, with
isolated projective fibre \([p]\).
Since \(F\subset K\), its ordinary characteristic
set \(W\) is contained in
the zero section together with \(\pi^{-1}S\).
RF.11 and RF.31 identify \(W\) with \(\Lambda\)
near \(p\). Every nonzero irreducible
characteristic component is Lagrangian.
The projective components containing \([p]\)
therefore agree with the corresponding components
of \(\mathbf P\Lambda\): they agree on a
nonempty open, and these are closed irreducible
analytic varieties of the same dimension.
Every remaining component avoids \([p]\).
Its base image is contained in \(S\), so RF.24
shows it has no point at all over \(0\).
There are only finitely many components near the
compact central projective fibre. Their proper
projective images are closed and avoid \(0\);
shrink the base to exclude them. Consequently
\[
\begin{gathered}
\operatorname{Char}(F)
\subseteq\Lambda\cup T_X^*X\\
\text{over that base}\\
\text{neighbourhood}.
\end{gathered}
\tag{RF.32}
\]
In particular
\[
\begin{gathered}
E_{p'}\otimes_D M_p=0,\\
p'\in\pi^{-1}(0),\\
p'\notin\mathbf C^*p\cup T_X^*X .
\end{gathered}
\tag{RF.33}
\]
This is an assertion about the entire finite
ordinary module \(F_0\), not one source section.

On every nonzero characteristic component,
RF.31 transports the generic stable lattice
of \(M\) to that of the finite microlocalization
of \(F\). After RF.32 every such component
lies in this neighbourhood. Thus \(F\) has
regular singularities in the same generic
microlocal sense. This is not an assumed
everywhere-stable lattice or a claim of full C1.
The maps RF.27 and RF.30--RF.33 give the
finite ordinary realization and faithful finite-pole
quotient injection on the stated neighbourhood.

#### RF.I. The ordinary infinite extension

There is a further useful comparison at the
fixed origin. Put \(R=D^\infty_0\) and
\(N=B_p\otimes_{E_p}M_p\).
The natural map
\[
\mu_R:R\otimes_D M_p\longrightarrow N
\tag{RF.34}
\]
is injective: its composite with \(e\) is
\(R\otimes_D e'\), which is injective by
ordinary right flatness MF.6--MF.7.

It is surjective as well. ISR.2 writes any
section of \(N\) as a finite sum
\(\sum_i a_i u_i\), with \(u_i\in M_p\) and
\(a_i\) in the actual time-only \(A^\infty\).
Split each coefficient into its ordinary
nonnegative-order head and its negative tail.
The first is in \(R\), by the every-epsilon
ordinary head bound of ISF. The second has
finite upper order \(-1\) and the actual
negative factorial bound, so is in \(E_p\).
It acts on \(M_p\) itself. Hence every term
lies in the \(R\)-span of \(M_p\), proving
\[
\begin{gathered}
D^\infty_0\otimes_{D_0}M_p\\
\simeq E^\infty_p\otimes_{E_p}M_p.
\end{gathered}
\tag{RF.35}
\]
This uses all tails, finite algebraic tensors
and the already proved ordinary injection.
It is not a proof that \(B\otimes_R N=N\)
for an arbitrary \(B\)-module, and is not
mixed-ring flatness.

The theorem gives finite poles and both actual finite and ordinary-infinite reconstruction for a regular generic-position module. Section5.89, CK.A–CK.K proves intrinsic orders, the half-order criterion and coherent characteristic-stable cutoffs; Section5.90, GG.A–GG.G proves the initial analytic generators near compact sets. Arbitrary analytic proper/support-proper regularity requires its further argument.

![Actual finite flatness, a full negative-order Nilsson example and the finite tensor reconstruction](assets/regular-finite-pole.png)

**Figure RF.1.** Panel A is the formal and actual argument of
RF.A, equations RF.7–RF.9; no formal series is identified
with an actual operator merely by its coefficients.
Panel B is a positive real sample of the complex
multivalued input with exponent \(2/5\). For the indicated
finite-upper-order operator, the power-series coefficient
is \(a_n=n!/\prod_{s=1}^n(s+2/5)\), so
\(0<a_n\le1\). Consequently its sum through index \(N\)
has omitted tail at most \(t^{2/5}t^{N+1}/(1-t)\)
for \(0<t<1\). The plotted 200-term sum uses
\(N=199\) and \(0<t\le3/4\); the dashed curve is
the proved upper bound \(t^{2/5}/(1-t)\).
Every negative operator coefficient is included in the
analytic argument RF.C; the finite sum is the displayed
numerical sample. Panel C is the exact commuting
triangle RF.28–RF.30. Its upper arrow is injective by
finite microlocalization flatness, and multiplication
is surjective, hence an isomorphism. Panels A and C
are algebraic diagrams with no geometric scale.

Figure SVG ·
exact diagram and sample data ·
reproducible source ·
figure PNG.

Free source:
[Kashiwara–Kawai, Holonomic systems III, II.3.5 and V.1.1–V.1.4](https://www.kurims.kyoto-u.ac.jp/~kenkyubu/kashiwara/HolIII.pdf).


### 5.88. Indicial orders and ordinary finiteness of every order-zero submodule

Let \(X\) be a complex manifold of dimension \(n\ge1\).
Use the actual finite-order microlocal ring \(\mathcal E_X\),
its usual order filtration, and left modules. Write
\(E_j=\mathcal E_X(j)\); thus \(E_0\) is the order-zero
ring and \(E_{-1}\) is its ideal of negative-order operators.
All symbols below are normal symbols in the convention
\(\sigma([D_{x_i},x_j])=\delta_{ij}\).

Let \(M\) be coherent and holonomic, and suppose that on
a dense smooth open of each reduced characteristic component
it has a coherent \(E_0\)-lattice stable under all first-order
operators whose principal symbols vanish on that component.
The word lattice means a submodule generating \(M\) over
\(\mathcal E_X\). There is no lattice premise at the singular
points of the characteristic variety.

**Theorem IO.T.** Suppose that \(M\) is in generic position
at \(p\in T^*X\setminus T_X^*X\). For every coherent
\(E_0\)-submodule \(M_0\subset M\), the entire stalk
\(M_{0,p}\) is a finite \(\mathcal O_{X,\pi(p)}\)-module.
The submodule need not generate \(M\) or be stable under
first-order characteristic operators.

The proof also constructs indicial orders on the smooth
regular characteristic locus. They are invariant under
ordinary holomorphic changes of base coordinates, and
an elliptic operator of usual order \(a\in\mathbf Z\)
adds \(a\) to these orders. In a smooth hypersurface
conormal chart the indicial generator is
\[
\Theta=-tD_t-\frac12 .
\tag{IO.1}
\]
The half in this formula is part of the calculation.

The actual symbol product and commutator are proved in
§5.19;
the negative-order inverses and Jacobson property in
§5.20;
Noetherianity and normal Artin--Rees in
§§5.21–5.22;
and common-neighbourhood coherence in
§5.24.
The generating lattice and its exact analytic support are
J.1–J.2.
We use the actual Bernstein inequality
EC.D,
the smooth Euler form
DP.7,
and its full finite-order power/logarithm action
RF.B–RF.C.
The ordinary-coordinate symbol transformation is the finite-order
part of BCT.1–BCT.5.
The finite target, its normalized coefficient lattice, and
the scalar pole propagation are respectively
RF.T,
DT.E,
and RF.D–RF.E.
These links identify earlier proofs, with the same actual
rings and hypotheses.

#### IO.A. A canonical first-order residue

First let \(\Lambda\) be a smooth conic Lagrangian.
Let \(E_\Lambda\) be the subring generated by \(E_0\)
and the first-order operators with principal symbol zero
on \(\Lambda\). Work in a projective covector chart
\(\tau\ne0\), and put \(h=D_t^{-1}\).
The order ideal \(E_{-1}=hE_0=E_0h\) is independent
of the choice of this elliptic negative-order generator.

Choose finitely many homogeneous degree-one functions \(f_i\)
generating the ideal of \(\Lambda\), after multiplication
of degree-zero ideal generators by \(\tau\).
Choose actual operators \(P_i\) of order one with these
principal symbols. The ring \(E_\Lambda\) is generated
by \(E_0\) and these finitely many \(P_i\).
Indeed the principal symbol of any other permitted
first-order operator is a sum of the \(f_i\) with
degree-zero coefficients; subtract the corresponding
operator sum to leave order zero.

Set
\[
J_\Lambda=E_{-1}E_\Lambda .
\tag{IO.2}
\]
This is a two-sided ideal: the symbol product gives
\([P_i,E_{-1}]\subset E_{-1}\), and \(E_0\) preserves
\(E_{-1}\) on both sides. If \(L\) is an
\(E_\Lambda\)-stable \(E_0\)-module, then
\[
J_\Lambda L=E_{-1}L=hL .
\tag{IO.3}
\]
Equality here uses \(1\in E_\Lambda\).

For an operator \(P\) of order at most one, write its
first two homogeneous symbols as \(p_1,p_0\), allowing
\(p_1=0\), and define
\[
s(P)=p_0-\frac12\sum_i
             \partial_{x_i}\partial_{\xi_i}p_1 .
\tag{IO.4}
\]
Let \(H_f=\sum_i(f_{\xi_i}\partial_{x_i}
-f_{x_i}\partial_{\xi_i})\). The normal product,
retained through its first two orders, gives
\[
\begin{gathered}
s(AB)=a_m s(B)+s(A)b_\ell\\
\hspace{12mm}+\tfrac12\{a_m,b_\ell\},\\
s([P,Q])\\
=H_{p_1}s(Q)-H_{q_1}s(P).
\end{gathered}
\tag{IO.5}
\]
In the first line \(s(A)\) and \(s(B)\) mean the
corresponding subprincipal symbols of orders \(m\)
and \(\ell\). To check the second identity directly,
expand \(PQ-QP\) to order zero. The terms with two
symbol derivatives are
\(\frac12\sum_{ij}
(p_{\xi_i\xi_j}q_{x_ix_j}
-q_{\xi_i\xi_j}p_{x_ix_j})\).
For \(\Delta=\sum_i\partial_{x_i}\partial_{\xi_i}\),
the product rule gives
\[
\begin{gathered}
\Delta\{p,q\}-\{\Delta p,q\}
-\{p,\Delta q\}\\
=\sum_{ij}\bigl(
p_{\xi_i\xi_j}q_{x_ix_j}
-p_{x_ix_j}q_{\xi_i\xi_j}\bigr).
\end{gathered}
\tag{IO.6}
\]
Subtracting half of this identity proves IO.5.
The first identity follows by the same expansion of
the product to order \(m+\ell-1\).

The conormal identification furnished by the symplectic
form sends the class of a function in
\(I_\Lambda/I_\Lambda^2\) to its Hamiltonian field on
\(\Lambda\). It is an isomorphism because \(\Lambda\)
is Lagrangian: its kernel consists of functions whose
full ambient differential vanishes at points of
\(\Lambda\), and the two
bundles have the same rank. Let
\(\chi=\sum_i\xi_i\partial_{\xi_i}\) be the radial field.
It is tangent to \(\Lambda\) and homogeneous of degree
zero. Expressing it in the fields \(H_{f_i}|_\Lambda\)
and extending their degree-zero coefficients gives
a degree-one function \(q_1\in I_\Lambda\) with
\(H_{q_1}|_\Lambda=\chi\).
Quantize \(q_1\) and give it order-zero symbol
\(\frac12\Delta q_1\). This produces an actual
first-order operator \(\Theta\) with
\[
\begin{gathered}
q_1|_\Lambda=0,\qquad
H_{q_1}|_\Lambda=\chi,\\
s(\Theta)|_\Lambda=0 .
\end{gathered}
\tag{IO.7}
\]
Only finitely many homogeneous symbol terms are needed.

We shall need the following first-order kernel calculation.
If \(P\) has order at most one and
\[
H_{p_1}|_\Lambda=0,\qquad s(P)|_\Lambda=0,
\tag{IO.8}
\]
then \(P\in J_\Lambda\).
Indeed \(p_1\in I_\Lambda^2\). In our chart it is a
finite sum \(h\,a_{ij}f_if_j\), with \(a_{ij}\)
homogeneous of degree zero, where \(h\) in a symbol
denotes \(\tau^{-1}\). Lift this to a finite operator
sum \(Q=\sum h A_{ij}P_iP_j\), which lies in
\(J_\Lambda\). IO.5 and the Lagrangian identity
\(\{I_\Lambda,I_\Lambda\}\subset I_\Lambda\)
give \(s(Q)|_\Lambda=0\): every product term has a
factor \(f_i\) or \(f_j\), or their vanishing bracket.
Thus \(P-Q\) is of order zero with principal symbol
zero on \(\Lambda\). Express that symbol as a
sum of \(h\,a_i f_i\); subtract \(\sum h A_iP_i\).
The remainder has order at most minus one.
All these terms belong to \(J_\Lambda\), as required.

Two choices in IO.7 therefore differ by \(J_\Lambda\).
For each \(P_i\), IO.5 gives
\[
\begin{gathered}
H_{\sigma_1([\Theta,P_i])}|_\Lambda
=[\chi,H_{f_i}|_\Lambda]=0,\\
s([\Theta,P_i])|_\Lambda=0 .
\end{gathered}
\tag{IO.9}
\]
The first equality uses the degree-zero homogeneity of
\(H_{f_i}\); the second uses that \(s(P_i)\) has degree
zero, while \(s(\Theta)\) restricts to zero.
IO.8 applies. Also \([\Theta,A]\), for \(A\in E_0\),
has order zero with principal symbol
\(\chi(\sigma_0 A)|_\Lambda=0\), hence lies in
\(J_\Lambda\). Consequently \(\Theta\) acts centrally
on the residue \(L/hL\) of every stable lattice \(L\).

#### IO.B. Coordinates, shifts and the indicial polynomial

The residue operator is invariant under ordinary base
coordinates. Here is the chain-rule verification.
Put \(y=F(x)\), \(G=\det F'(x)\), and
\(\xi=F'(x)^T\eta\). For first-order normal symbols,
the first two terms of the coordinate substitution are
\[
\begin{gathered}
p'_1(y,\eta)=p_1(x,F'^T\eta),\\
p'_0=p_0\\
+\tfrac12
\sum_{ijk}p_{1,\xi_i\xi_j}
F_{k,x_ix_j}\eta_k .
\end{gathered}
\tag{IO.10}
\]
The terms without a prime on the right are evaluated
at the transformed point. They follow by expanding
the coordinate action on a test exponential through
one degree below its principal degree. Terms with
one second derivative of \(F\) give precisely the
displayed Hessian sum; higher derivatives contribute
only lower orders. This is also the first two terms
of the full coordinate formula in BCT.5.
Applying the chain rule once more gives
\[
\begin{gathered}
\sum_k\partial_{y_k}\partial_{\eta_k}p'_1\\
=\Delta p_1+H_{p_1}\log G\\
\hspace{10mm}+
\sum_{ijk}p_{1,\xi_i\xi_j}
F_{k,x_ix_j}\eta_k,\\
s_y(P)\\
=s_x(P)-\tfrac12H_{p_1}\log G .
\end{gathered}
\tag{IO.11}
\]
For IO.7 the last derivative restricts to
\(\chi\log G=0\), since \(G\) is a base function.
Thus the transformed operator again satisfies IO.7,
and the first-order kernel calculation proves equality
of the two residue actions.

If \(P\) is scalar elliptic of usual order
\(a\in\mathbf Z\), conjugation by \(P\) preserves
\(E_\Lambda\) and \(J_\Lambda\). For a first-order
characteristic generator its conjugate has the same
principal symbol; conjugation preserves the usual
filtration and its inverse gives equality of these
rings and ideals. Moreover
\[
\begin{gathered}
P^{-1}\Theta P=\Theta+a\\
\pmod{J_\Lambda}.
\end{gathered}
\tag{IO.12}
\]
Indeed \(P^{-1}[\Theta,P]\) is of order zero and
has principal symbol \(H_{q_1}p_a/p_a=a\) on
\(\Lambda\), by homogeneity; subtracting \(a\)
leaves an element of \(J_\Lambda\).

Let \(L\) now be a coherent \(E_0\)-module stable under
\(E_\Lambda\), contained in a coherent actual \(E\)-module.
The quotient \(V=L/hL\) is finite over the holomorphic
functions on \(\mathbf P\Lambda\). In fact the
principal-symbol quotient \(E_0/E_{-1}\) is the
holomorphic function ring on the projective chart.
An order-zero symbol vanishing on \(\Lambda\) lifts
to \(\sum h A_iP_i\) modulo \(E_{-1}\), so it acts
as zero on \(V\). This proves the asserted factorization
through \(\mathcal O_{\mathbf P\Lambda}\).

The fields \(H_{f_i}|_\Lambda\) span \(T\Lambda\).
Their projections therefore span the tangent bundle
of \(\mathbf P\Lambda\). The actions of the \(P_i\)
on \(V\), and local holomorphic combinations of them,
provide a connection in every coordinate direction:
the commutator with a degree-zero scalar is its
directional derivative. The connection need not be
integrable for the following argument.

Here is a proof that \(V\) is locally free. Present
it as a cokernel of a finite holomorphic matrix \(C\).
Lift one connection to the finite free module of
generators. Its preservation of the kernel gives
matrices \(A_i,B_i\) with
\[
\partial_i C+A_iC=CB_i .
\tag{IO.13}
\]
Differentiate each minor of \(C\). Row and column
replacement in IO.13 expresses its derivative in the
same ideal of minors. Every Fitting ideal is therefore
stable under all coordinate derivatives. A nonzero
such ideal in a holomorphic local ring is the whole
ring: take a nonzero Taylor coefficient of least
total degree and differentiate that many times to
obtain a unit. Thus each Fitting ideal is either zero
or the whole ring. Let \(r\) be the rank of \(C\) over the
fraction field. If \(r>0\), its nonzero ideal of
\(r\)-minors is the whole ring, so some such
minor is a unit. All \((r+1)\)-minors are zero.
Finite row and column elimination about the unit
minor reduces \(C\) to an identity block and
a zero block: the entries of the remaining
Schur complement are these vanishing larger
minors divided by the unit minor.
Its cokernel is free. If \(r=0\), all entries
of \(C\) vanish and the conclusion is immediate.
The unit minor persists on a neighbourhood,
proving local freeness.

By IO.9 the action of \(\Theta\) on \(V\) is linear
over the holomorphic scalars and commutes with these
connections. In a local frame, its matrix \(T\)
satisfies
\[
\partial_iT+[A_i,T]=0 .
\tag{IO.14}
\]
Along a coordinate segment solve \(U'=-A_iU\),
\(U(0)=I\). Successive integrals converge on a small
disc, bounded by the exponential majorant
\(\sum (C|z|)^k/k!\); the equation for the inverse
gives invertibility. Differentiation shows
\(U^{-1}TU\) constant. Coordinate paths consequently
conjugate all the matrices on a connected patch.
Their monic minimal polynomial is one constant
polynomial, denoted \(b_L(z)\). On a connected
regular characteristic component the polynomials
agree on overlaps and hence are constant there.
For \(L=0\) put \(b_L=1\).

Every local section \(u\) of \(M\) on this regular
locus has a coherent cyclic stable lattice
\[
L(u)=E_\Lambda u .
\tag{IO.15}
\]
To see this, choose a stable generating lattice \(L_*\).
Localization by the normal element \(h\) puts \(u\)
in some \(h^{-k}L_*\). This shift is still stable:
conjugating a first-order generator by \(h^k\)
changes it only by an order-zero term.
Noetherianity makes \(E_\Lambda u\) finite over \(E_0\).
Choose finite generators, express them as finite words
on \(u\), and express their products by the finitely
many \(P_i\) in those generators. All these finite
identities hold on one smaller neighbourhood.
Their \(E_0\)-span is a coherent stable sheaf there
by §5.24, proving IO.15 as a sheaf assertion.

Define \(\operatorname{ord}u\) to be the root set
of \(b_{L(u)}\), and define the order set of zero
to be empty. A nonzero cyclic lattice has nonzero
residue by the negative-order Jacobson property and
Nakayama, so its order set is nonempty.
It follows from IO.12 that
\[
\operatorname{ord}(Pu)
=\operatorname{ord}(u)+a .
\tag{IO.16}
\]
Indeed \(L(Pu)=P L(u)\) and
\(E_{-1}P L(u)=P E_{-1}L(u)\); pulling the residue
action back by \(P\) gives \(T+a\).
This also checks the direction of conjugation in IO.12.

For the standard conormal
\(\Lambda=\{t=0,\xi_x=0,\tau\ne0\}\),
take \(q_1=-t\tau\). Its Hamiltonian restricts to
\(\tau\partial_\tau\), and \(\Delta q_1=-1\).
IO.4–IO.7 give IO.1. Thus on a power/logarithm term,
\[
\begin{gathered}
\Theta\bigl(t^\lambda(\log t)^k\bigr)\\
=(-\lambda-\tfrac12)
       t^\lambda(\log t)^k\\
\hspace{13mm}-k\,t^\lambda(\log t)^{k-1}.
\end{gathered}
\tag{IO.17}
\]
In particular \(D_t^{-1}\Theta D_t=\Theta+1\).
The square root of
\(\Omega_\Lambda\otimes\pi^*\Omega_X^{-1}\)
has radial weight \(1/2\) in this chart: in
coordinates \((x,\tau)\) the top form contains
one factor \(d\tau\). This explains geometrically
the same half appearing in the subprincipal
calculation, without requiring a global choice of
that square root.

#### IO.C. Comparing lattices without changing the upper bound

Let \(L'\subset L\) be two finite stable \(E_0\)-modules
on the smooth regular locus. Normal Artin--Rees supplies
an integer \(N\ge0\) such that
\[
L'\cap h^{N+1}L\subset hL'.
\tag{IO.18}
\]
Filter \(L'/hL'\) by the finite decreasing filtration
\[
\begin{gathered}
F^j=\frac{L'\cap h^jL+hL'}{hL'},\\
0\le j\le N+1 .
\end{gathered}
\tag{IO.19}
\]
It begins with the whole residue and ends with zero.
Each successive quotient is a subquotient of
\(h^jL/h^{j+1}L\): first map the intersection into
this latter quotient, and then divide its image by
the image of its intersection with \(hL'\).
This is a subquotient statement and uses no map
from \(L'/hL'\) to \(L/h^{N+1}L\).

IO.12 with \(P=h^j\) identifies the action of
\(\Theta\) on \(h^jL/h^{j+1}L\) with that on
\(L/hL\) minus \(j\). All the filtrations are
\(\Theta\)-stable. Multiplying the annihilators
of their successive quotients gives
\[
b_{L'}(z)\ \mid\
             \prod_{j=0}^{N}b_L(z+j).
\tag{IO.20}
\]
Consequently the largest real part of any root
cannot increase on passing to a stable submodule.
If finitely many stable lattices are added, the
residue of their sum is a quotient of the direct
sum of their residues. Its root set is contained
in the union of their root sets.

If \(u_1,\ldots,u_s\) generate an \(E_0\)-submodule
and \(u=\sum_j A_ju_j\), with \(A_j\in E_0\), then
\(L(u)\subset\sum_j L(u_j)\). Therefore
\[
\begin{gathered}
\max\operatorname{Re}\operatorname{ord}u\\
\le\max_j\max\operatorname{Re}\operatorname{ord}u_j .
\end{gathered}
\tag{IO.21}
\]
with zero generators omitted. This is a bound for
every order-zero coefficient, irrespective of its
analytic norm or the size of its representative
neighbourhood.

#### IO.D. One bound on every smaller representative

We justify using IO.21 simultaneously on arbitrarily
small neighbourhoods of the original point.
First the failure of regularity on the smooth
characteristic locus is analytic. Choose the
generating coherent lattice \(G\) of J.1 on one
projective neighbourhood, and finite first-order
generators \(P_i\) for \(E_\Lambda\).
Here the reduced characteristic ideal is coherent
even where \(\Lambda\) is singular, so such
generators exist on this neighbourhood.
Let \(G_k\) be the \(E_0\)-span of the words of
length at most \(k\) in the \(P_i\), applied to
finite generators of \(G\).
Every \(G_k\) is coherent on this same neighbourhood:
there are finitely many words, their images lie
in a finite shift \(h^{-k}G\), and §5.24 gives
coherent images and kernels.
The commutators \([P_i,E_0]\subset E_0\) give
\[
G_{k+1}=G_k+\sum_i E_0P_iG_k .
\tag{IO.22}
\]

The quotient \(G_{k+1}/G_k\) is coherent over \(E_0\).
Its support is analytic: reduction modulo \(h\)
is a finite module over the holomorphic principal
symbol ring, and Nakayama detects its nonzero
stalks. Denote this closed analytic set by \(Y_k\).
They decrease. If one quotient vanishes on an open
set, IO.22 makes the lattice stable there, so all
subsequent quotients vanish there too.

A decreasing sequence of these analytic sets is
stationary on a neighbourhood of the point.
For completeness, take finitely many local
irreducible representatives of \(Y_0\).
For each representative either every subsequent
germ contains it, or some finite index fails to
contain it. In the first case the identity theorem
extends each containment over that fixed connected
irreducible representative. In the second case,
at its first failing index, the intersection has
strictly smaller dimension. There are finitely
many representatives, so take the largest of these
finitely many indices and shrink once.
Repeat on the finitely many lower dimensional
components. Dimension decreases each time; the
process ends, including the zero-dimensional case.
This proves stationarity on one common neighbourhood.

Call the stationary set \(Y\). Outside \(Y\)
one of the \(G_k\) is a finite stable lattice
generating \(M\). Conversely a stable generating
lattice contains \(G\) after a finite shift;
all the \(G_k\) lie in that stable finite shift,
and Noetherianity forces their ascending chain
to stabilize at the stalk. Thus on the smooth
characteristic locus \(Y\) is exactly the
failure set of the stable-lattice condition.
Our generic regularity premise makes it a
proper analytic subset of each component.

Shrink so that each projective irreducible
characteristic germ has one irreducible
representative. Remove \(Y\), the singular sets,
and the intersections of distinct components.
On each component the resulting smooth open
\(U_\alpha\) is connected.
Here is the local analytic reason for connectedness.
A finite Noether normalization and a separating
primitive holomorphic function identify a dense
open with the sheets of one irreducible monic
polynomial over the base, away from its discriminant.
The finite image of each additional proper analytic
excluded set may be put in an additional base
hypersurface. If the remaining cover were
disconnected, one monodromy orbit of its roots
would give a proper monic factor. Its coefficients
are bounded elementary symmetric functions of
the bounded roots. The bounded holomorphic
extension proof used in RF.D extends them
across the base hypersurface, giving a proper
factor over the base germ. This contradicts
irreducibility of the primitive polynomial.
The dense remaining cover is connected, and
adding its limit points within the smooth
complement preserves connectedness.
Finite normalization and the finite-image
statement are the same convergent preparation
arguments used in CF.C and RF.D.

Choose generators \(u_1,\ldots,u_s\) of the
given \(M_0\) on this fixed neighbourhood.
IO.B makes each cyclic indicial polynomial
constant on each \(U_\alpha\), including
the possibility that the cyclic module is zero.
There are finitely many polynomials, hence
an integer \(m\ge0\) strictly above the real
parts of all their roots. IO.21 proves
\[
\operatorname{Re}\beta<m
\quad
\bigl(\beta\in\operatorname{ord}u\bigr)
\tag{IO.23}
\]
on every \(U_\alpha\), for every local section
\(u\) of \(M_0\). In particular the bound applies
on the intersection with the representative
of any germ at \(p\).

If all these cyclic modules vanish, \(EM_0\)
is a coherent \(E\)-submodule of \(M\) whose
support is contained in a proper analytic
subset of every Lagrangian component.
Every such subset has dimension less than \(n\).
EC.D forbids any nonzero coherent module
with that support, so \(M_0=0\).
For zero-dimensional projective components the
same argument has no positive-dimensional
connectedness step: the generic regular locus
is the component itself.

#### IO.E. From the indicial bound to a uniform pole bound

At a generic point of a characteristic component,
the projective projection is finite in generic
position. Its generic rank is \(n-1\), so over
a dense smooth part of its image \(S_\alpha\)
that component is the conormal of a smooth
hypersurface. The Lagrangian tangent calculation
and equality of dimensions prove this, as in RF.F.
Choose ordinary base coordinates with that
hypersurface \(t=0\). IO.B preserves the
indicial action in these coordinates.

Consider an actual \(E\)-linear functional from
the source to directional function classes, as
constructed in RF.G, and let \([v]\) be the image
of \(u\). DP.7 and RF.B–RF.C show that this is
a finite Nilsson class: on a branch it is a
finite sum of powers and logarithms with
convergent holomorphic coefficients, modulo an
ordinary holomorphic germ. Integer resonances,
nilpotent logarithms and the entire negative
operator tail are retained in those proofs.
The map on cyclic stable modules and then on
their residues is surjective:
\[
\begin{gathered}
E_\Lambda u\longrightarrow E_\Lambda[v],\\
E_\Lambda u/hE_\Lambda u\\
\twoheadrightarrow\\
E_\Lambda[v]/hE_\Lambda[v].
\end{gathered}
\tag{IO.24}
\]

We spell out the exponent information supplied by
this residue. Combine exponent classes modulo
integers and remove all ordinary holomorphic
terms. Expand the remaining holomorphic
coefficients in nonnegative powers of \(t\).
If the class is nonzero, let \(\rho\) be the
smallest real part of its first nonzero
exponents. There are only finitely many
initial exponent classes and logarithmic
degrees, so this minimum exists.

Neither an order-zero operator nor a
characteristic first-order generator lowers
this minimum. For order zero, commute each
holomorphic coefficient to its right and
apply the RF.C normal remainder to the
constant power/logarithm terms. Its upper
\(D_t\)-order is at most zero. Each negative
power is a primitive and raises the exponent
by its positive integer index.
At a negative-integer resonance it can
increase the logarithmic degree by one;
it still raises that exponent. The factorial
bounds in RF.C make the full series converge
normally on a smaller disc, so the assertion
holds for the entire operator.
For order one, subtract order-zero multiples
of \(tD_t,D_{x_i}\); the remainder is of
order zero. These displayed generators
also preserve the lower exponent bound.
All finite words therefore preserve it.
The primitive \(h=D_t^{-1}\) raises it by
at least one, modulo ordinary holomorphic terms.

Take the coefficients at the exponents with
real part \(\rho\) in the identity
\(b_{L(u)}(\Theta)[v]\in hE_\Lambda[v]\).
The right side has no such coefficient.
The left side is a sum of distinct powers
times polynomials in \(\log t\). These are
linearly independent: after setting \(t=e^z\)
they are exponential polynomials, and
successive applications of the operators
\(\partial_z-\lambda\) isolate each
exponent and logarithmic coefficient.
IO.17 then shows that for at least one
leading exponent \(\mu\)
\[
\begin{gathered}
b_{L(u)}(-\mu-\tfrac12)=0,\\
\operatorname{Re}\mu=\rho .
\end{gathered}
\tag{IO.25}
\]
More explicitly, the coefficient of the highest
logarithmic power for any nonzero leading
exponent is multiplied by
\(b_{L(u)}(-\mu-\tfrac12)\).
One can restrict to a nonempty parameter
open where it is nonzero. The identity
there suffices, since the polynomial is
constant on \(U_\alpha\).
By IO.23 every observed branch therefore
has all its nonordinary exponents satisfying
\[
\operatorname{Re}\mu\ge -m-\tfrac12 .
\tag{IO.26}
\]
The weaker closed inequality is convenient;
the strict inequality also follows from
our choice of \(m\). A zero class contributes
only its holomorphic ambiguity, which obeys
the same growth bound because \(m\ge0\).

Use the finite target from RF.T:
\[
\begin{gathered}
e':M_p\hookrightarrow K_0,\\
K=L/\mathcal O_X^q,\\
L_\Sigma(*S)=L,\\
0\le\operatorname{Re}\lambda<1 .
\end{gathered}
\tag{IO.27}
\]
Here \(S=\pi(\operatorname{supp}M)\) is the
original reduced hypersurface; \(L_\Sigma\)
is the coherent normalized reflexive
coefficient lattice of DT.E. The embedded
trivial connection satisfies
\(\mathcal O_X^q\subset L_\Sigma\):
the normalized lattice of the trivial
connection is \(\mathcal O_X^q\), and the
normalized-map extension proved in DT.E
sends it into \(L_\Sigma\).

For \(u\in M_{0,p}\), choose a finite meromorphic
lift \(w\in L_0\) of \(e'u\).
RF.G realizes the finitely many required
branch observations of this lift as the
functionals in IO.24. A finite group-word
basis of the cyclic dual spans every
additional observation, so all branches
obey IO.26. On a full punctured normal
disc they grow at worst as
\(|t|^{-m-1/2}\) times a fixed power
of \(|\log|t||+1\); constants and logarithmic
degrees may depend on \(u\).

In a normalized logarithmic frame of \(L_\Sigma\)
on this smooth part, the inverse power/logarithm
matrix grows at worst as
\(|t|^{-1}\) times a logarithmic power.
This follows directly from its finite
Jordan matrices and the strip in IO.27.
The scalar, single-valued coefficients of
\(w\) consequently grow at worst as
\(|t|^{-m-3/2}\) times a logarithmic power.
For the one integer
\[
N=m+2
\tag{IO.28}
\]
their products with \(t^N\) are bounded and
tend to zero. The Laurent coefficient formula
on shrinking circles kills all negative
coefficients of these products. Hence
\(t^Nw\in L_\Sigma\) near a nonempty smooth
open of every relevant component of \(S\).
At a component with zero source cyclic
module the observed lift is holomorphic
modulo the trivial target, and the same
conclusion holds.

Let \(f\) be a reduced local equation for
the entire \(S\). At its generic smooth
points \(f/t\) is a unit, so the same
integer \(N\) gives generic membership
\(f^Nw\in L_\Sigma\).
This membership propagates with the same
integer. Indeed realize the reflexive
lattice as a kernel of a finite
holomorphic matrix, by the finite
presentation of its dual used in RF.E.
The coordinates of \(f^Nw\) are
meromorphic. They have no pole on a
nonempty open of each irreducible
divisor component, by what was just
proved. The connected finite-cover
Laurent argument RF.D propagates the
zero pole order, and its codimension-two
Cauchy argument extends the coordinates
everywhere. Their kernel identities hold
by the identity theorem. Thus
\[
f^Nw\in L_{\Sigma,0}
\tag{IO.29}
\]
on its whole germ, including singular
and intersecting components.
This step uses the finite meromorphic
target already supplied by RF.T.

Different lifts differ by
\(\mathcal O_{X,0}^q\), contained in
\(f^{-N}L_{\Sigma,0}\).
The integer in IO.28 is independent of
\(u\) and of the domain of its
order-zero coefficients. We have proved
an inclusion of entire ordinary modules
\[
e'(M_{0,p})\subset
f^{-N}L_{\Sigma,0}/\mathcal O_{X,0}^q .
\tag{IO.30}
\]
The module on the right is finite over
the Noetherian ring \(\mathcal O_{X,0}\).
The injection \(e'\) is ordinary
\(\mathcal D_{X,0}\)-linear, hence
\(\mathcal O_{X,0}\)-linear.
Every submodule of that finite
ordinary coefficient module is finite.
This proves IO.T.

#### IO.F. Examples and scope

For \(\lambda\notin\mathbf Z_{\ge0}\)
the nonzero class of \(t^\lambda\)
has order \(-\lambda-\frac12\).
For a logarithmic block the same
eigenvalue has the nilpotent action
shown in IO.17. Differentiation
adds one to its indicial order;
one primitive subtracts one.
For example \(t^{-1}\) has order
\(1/2\), while its primitive
\(\log t\) has order \(-1/2\).
These computations retain the
resonant logarithm.

The theorem concerns every coherent
order-zero submodule at a
generic-position point. The proof
constructs orders on the smooth
regular locus and the one-way
exponent bound needed for ordinary
finiteness. An intrinsic coherent
cutoff across singular characteristic
strata, the converse cutoff
membership criterion, its full
contact and half-density
compatibility, and the everywhere
characteristic-operator condition
require further arguments.

![Indicial half-order, Artin–Rees spectral shifts and the finite ordinary coefficient target](assets/indicial-order-zero.png)

**Figure IO.1.** Panel A is the resonant calculation IO.17:
one primitive sends \(t^{-1}\) to \(\log t\) and
subtracts one from its indicial order. Panel B
is the finite subquotient filtration IO.18–IO.20;
its layers have the old roots shifted by nonpositive
integers. Panel C is the inclusion of entire
modules IO.30 with the common integer IO.28.
The panels are exact algebraic diagrams with no
geometric scale.

Figure SVG ·
exact diagram data ·
reproducible source ·
figure PNG.

Free source:
[Kashiwara–Kawai, Holonomic systems III, I.5 and V.1.5](https://www.kurims.kyoto-u.ac.jp/~kenkyubu/kashiwara/HolIII.pdf).


### 5.89. Coherent indicial cutoffs and everywhere characteristic stability

#### CK.A. The generic order cutoff

Let \(M\) be a coherent actual holonomic \(E_X\)-module with generic
regular stable lattices on every reduced characteristic component.
Write \(\Lambda=\operatorname{supp}M\), \(n=\dim X\), \(E_0=E_X(0)\),
\(E_{-1}=E_X(-1)\). On the dense smooth regular locus let orders be
the root sets of the radial residue constructed in IO.A--IO.B.

For \(c\in\mathbf R\), define \(M_c\) by the generic condition that
every order of the section has real part at most \(c\), on the
smooth regular part of every characteristic component met by its
domain. This definition only uses points where the residue is
already constructed. Its independence of smaller dense regular
opens follows from IO.B and IO.D constant cyclic polynomials.

**Theorem CK.T.** For every real number \(c\),
\[
\begin{gathered}
M_c\text{ is coherent}\\
\text{over }E_0,\\
E M_c=M,\\
E_\Lambda M_c=M_c,
\end{gathered}
\tag{CK.1}
\]
and
\[
\begin{gathered}
\mathcal H^0_Z(M/M_c)=0\\
\text{if }\operatorname{codim}_{T^*X}Z\ge n+1.
\end{gathered}
\tag{CK.2}
\]
Consequently \(M\) has an everywhere coherent generating
characteristic-stable lattice. For every homogeneous involutive
\(V\supset\Lambda\), the ring \(E_V\) is a subring of \(E_\Lambda\),
so that same lattice proves regularity along \(V\).

The actual rings, normal completion and coherence are the earlier
proofs §§5.19–5.24.
The radial residue, its ordinary-coordinate invariance, cyclic
minimal polynomials, normal Artin--Rees comparison and connected
regular locus are IO.A–IO.D.
Whole-germ generic position and its actual quadratic quantization
are Q.1–Q.7.
The coherent generating lattice and principal support are
J.1–J.2.
We use EC.D's actual Bernstein inequality,
the actual Euler normal form DP.7,
and the full negative power/logarithm action
RF.B–RF.C.
The ordinary realization, finite microlocalization flatness,
exact support and whole characteristic fibre are the proofs
RF.T and RF.A–RF.I.
The finite meromorphic coefficient theorem, with generic divisorial
regularity and the reflexive normalized lattice, is
§5.27.1–§5.27.11.
Its extension by scalar Cauchy integrals and finite dual presentations
is DT.E.
The pole propagation argument is RF.D–RF.E.
Finite holomorphic matrix kernels are proved by
Oka coherence, Lemma2.2 and Theorem2.1;
the finite presentation argument of §5.27.1 gives their module
kernel and image consequences. All these are exact earlier
programme proofs.

The transformations used below are ordinary base-coordinate changes,
the actual quadratic quantizations for generic position, and
the explicit scalar order-shifting automorphism CK.D.
Their order, subprincipal and half-order effects are computed there.

#### CK.B. Smooth spectral cutoffs and their exact saturation

On a smooth regular Lagrangian let \(L\) be finite over \(E_0\),
stable under \(E_\Lambda\), and generating \(M\). Put
\(h=D_t^{-1}\), \(V=L/hL\), and let \(b_L\) be the constant minimal
polynomial of \(\Theta\) on \(V\).
IO.C gives the precise action on each layer:
\[
\begin{gathered}
\Theta\text{ on }h^jL/h^{j+1}L\\
\text{has polynomial }b_L(z+j).
\end{gathered}
\tag{CK.3}
\]
This also holds for negative \(j\), by IO.12.

**Saturation lemma.** If a polynomial \(a(z)\) is coprime to
every \(b_L(z-k)\), \(k=1,2,\ldots\), and \(a(\Theta)u\in L\),
then \(u\in L\). Indeed \(u\in h^{-k}L\) for some \(k\).
For \(k>0\), both \(a(\Theta)u\) and
\(b_L(\Theta-k)u\) belong to \(h^{-(k-1)}L\).
Bézout for these two scalar polynomials puts \(u\) there too.
Induct downward through the finite \(k\).

**Elementary lowering.** Let \(\lambda\) be a root and write
\(b_L(z)=(z-\lambda)^a d(z)\), \(d(\lambda)\ne0\).
The generalized eigenspace projection \(V\to V_\lambda\) is
a polynomial in \(\Theta\), by Bézout. Let
\[
K=\ker(L\longrightarrow V_\lambda).
\tag{CK.4}
\]
Then \(hL\subset K\subset L\); it is finite over \(E_0\),
stable, and generating. There is an exact sequence
\[
\begin{gathered}
0\longrightarrow hL/hK\\
\longrightarrow K/hK\\
\longrightarrow K/hL\longrightarrow0.
\end{gathered}
\tag{CK.5}
\]
The last term is \(V_{\ne\lambda}\).
Multiplication by \(h\) identifies the first with \(L/K\)
and shifts the \(\Theta\)-action by minus one:
the error in IO.12 lies in \(h^2L\subset hK\).
Hence
\[
b_K(z)\mid d(z)(z-\lambda+1)^a .
\tag{CK.6}
\]
This keeps every nilpotent multiplicity and allows nontrivial
extensions between the two displayed terms.

Let \(Z\subset\mathbf C\) be downward closed under subtraction
of nonnegative integers, and meet every integer orbit in a
nonempty proper initial segment. An example is
\(Z=\{\operatorname{Re}z\le c\}\).
Start with \(h^{-k}L\) for a sufficiently large integer \(k\)
so that every root \(\beta\) satisfies \(\beta+1\notin Z\).
Choose \(N\ge0\) with \(\beta-N\in Z\) for all its finitely many roots.
Repeatedly apply CK.4 to each root outside \(Z\).
CK.6 removes that root and moves its contribution down by one.
After the finitely many roots outside \(Z\) at this stage have
been treated, every remaining root satisfies
\(\beta-(N-1)\in Z\), while still \(\beta+1\notin Z\).
To verify the first assertion, each new root is one less than
a treated root, and every untreated surviving root was in \(Z\).
If a moved root coincides with an already treated one, it still
has this asserted lower shift from the initial stage.
The second assertion follows from downward closure: a root
just entering \(Z\) has its successor outside \(Z\).
Induction on \(N\) yields a finite stable generating lattice \(L_Z\) with
\[
\begin{gathered}
\beta\in Z,\qquad\beta+1\notin Z\\
(\beta\text{ a root of }b_{L_Z}).
\end{gathered}
\tag{CK.7}
\]

Its section orders lie in \(Z\), by IO.C.
Conversely suppose \(\operatorname{ord}u\subset Z\).
Let \(a=b_{E_\Lambda u}\).
For a sufficiently large \(\ell\), the layer polynomials give
\[
\begin{gathered}
\left(\prod_{j=0}^{\ell-1}a(\Theta+j)\right)u\\
\in h^\ell E_\Lambda u\subset L_Z .
\end{gathered}
\tag{CK.8}
\]
The last inclusion follows by placing the finite cyclic lattice
inside some \(h^{-k}L_Z\) and taking \(\ell\ge k\).
Every root of the polynomial in parentheses is in \(Z\).
Every root of \(b_{L_Z}(z-k)\), \(k\ge1\), is outside \(Z\),
because its predecessor by \(k-1\) is already outside \(Z\).
The saturation lemma proves \(u\in L_Z\). Thus
\[
L_Z=\{u:\operatorname{ord}u\subset Z\}.
\tag{CK.9}
\]
This proves the entire smooth cutoff criterion, including
coherence, generation and stability. It also proves the
top-strip characterization CK.7 for any supplied finite
stable generating lattice with that root condition.

#### CK.C. The exact strict-Nilsson comparison in a conormal chart

Take \(\Lambda=\{t=\xi_x=0,\tau\ne0\}\).
The radial generator is \(\Theta=-tD_t-\tfrac12\).
For a directional class \(v\) that is a finite Nilsson sum,
IO.E's proof works for any real upper order bound \(c\):
if every source order is at most \(c\), then every nonordinary
leading exponent \(\mu\) of any E-linear observation satisfies
\[
\operatorname{Re}\mu\ge-c-\tfrac12 .
\tag{CK.10}
\]
The same proof uses the highest logarithmic coefficient in
the residue polynomial identity, with all negative operator
terms converging as in RF.C. For \(c=-1/2\), the exponent
bound is precisely nonnegative real part.

For the converse we construct a coherent target lattice. Fix finitely
many normalized exponents \(\lambda\) in \(0\le\operatorname{Re}\lambda<1\)
and logarithmic degrees \(0\le k\le r_\lambda\).
In a finite direct sum of directional function modules let \(T\)
be the holomorphic coefficient span of all
\(t^\lambda(\log t)^k\), omitting the ordinary holomorphic
term \((\lambda,k)=(0,0)\).
It is finite over \(\mathcal O_{t,x}\), hence over \(E_0\),
once its actual order-zero stability is proved.

Distinct exponent classes modulo integers, and distinct logarithmic
degrees within them, are independent over meromorphic coefficients:
after analytic continuation around \(t=0\), apply successively the
generalized eigenspace polynomials of monodromy, then finite
differences in \(\log t\). The last coefficient must vanish;
descend in logarithmic degree. For the zero exponent, apply these differences to an actual
holomorphic representative of a putative zero class; its
monodromy difference is zero as a function. This also
kills the coefficient of the first logarithm.
The sole ordinary holomorphic class is the omitted term. Thus the displayed scalar generators
form a finite free holomorphic module in the function quotient.

The full RF.C negative-tail estimate proves \(E_0T\subset T\):
the normal remainder of an order-zero operator has upper
\(D_t\)-order zero. Every negative term is a primitive.
For normalized \(\lambda\), its denominators \(\lambda+s\),
\(s\ge1\), never vanish, so no additional logarithmic degree
appears. Holomorphic coefficient composition preserves order zero.
All negative terms converge normally, with the same finite list
of powers and logs and convergent holomorphic coefficients.
The operators \(tD_t,D_{x_i}\) preserve \(T\), so it is
\(E_\Lambda\)-stable.
To justify actual \(E_0\)-coherence, including its neighbourhood
relations, use the standard coherent stable lattice in the
actual finite Euler module. For a noninteger \(\lambda\),
one Jordan block of size \(r+1\) is evaluated on
\(t^\lambda(\log t)^k/k!\), \(0\le k\le r\).
For \(\lambda=0\), use the size-\(r\) zero Euler block
evaluated on \((\log t)^k/k!\), \(1\le k\le r\);
the last derivative is ordinary holomorphic and is zero
in the directional quotient.
These evaluation maps are injective on the actual finite
Euler modules. Indeed AM.A/DP.7 gives unique normal rows
with coefficients in the tangential variables and finite
upper \(D_t\)-order. For noninteger \(\lambda\), at each
integer shift the falling-factorial (or primitive)
coefficient is nonzero, and the logarithmic matrix is
triangular with that nonzero diagonal.
For the zero block, a positive derivative of order \(j\)
of \((\log t)^k\) has leading term
\[
\begin{gathered}
k(-1)^{j-1}(j-1)!\\
{}\cdot t^{-j}(\log t)^{k-1}.
\end{gathered}
\tag{CK.11}
\]
Its distinct \(k\)'s give degrees \(0,\ldots,r-1\).
An \(n\)-fold primitive has exponent \(n\) and top
logarithmic degree \(k\), with nonzero coefficient \(1/n!\).
Normal convergence of every negative tail makes these
holomorphic coefficient series; independence of their
Taylor and logarithmic coefficients kills every row
when the evaluated class is zero.
Thus the maps are actual injections on a common neighbourhood.
The \(E_0\)-span of the Euler generator family is a coherent
stable generating lattice. Its coherence follows by putting
the finite family in one finite shift of J.1's lattice
and applying Section5.24. Its stability follows directly from
\([tD_t,E_0],[D_{x_i},E_0]\subset E_0\) and the constant
Euler/tangential generator relations. Thus no classification
assumption is needed to supply this lattice.
Its cyclic \(E_\Lambda\)-closure is the same lattice.
Its image equals \(T\):
stability gives containment in \(T\), and ordinary
coefficients and powers of \(tD_t-\lambda\) give all
the displayed generators. This proves coherence of \(T\)
without assuming coherence of an arbitrary kernel
with target the infinite directional module.

Moreover
\[
hT=tT.
\tag{CK.12}
\]
The primitive formulas give the forward inclusion.
For the reverse, \(D_t(tT)\subset T\) and the actual inverse
identity \(hD_t=1\) holds in the directional module.
The order-zero connection terms do not change this identity.
On \(T/hT=T/tT\), the eigenvalues of \(\Theta\) are
\(-\lambda-\tfrac12\), with the full logarithmic nilpotents.
They have real part at most \(-1/2\).

Suppose a finite tuple of E-linear directional observations
is injective on the source and takes \(u\) to strict Nilsson
classes with these exponents. Their \(E_\Lambda\)-cyclic
image is a stable coherent \(E_0\)-submodule of \(T\).
IO.C then bounds every source order by \(-1/2\).
This proves the converse of CK.10 for a faithful finite tuple.

For a meromorphic regular connection \(A\) in a smooth divisor
chart, its normalized constant logarithmic model has exponent
matrix \(C\), \(0\le\operatorname{Re}\operatorname{spec}C<1\).
It is generated as a D-module by \(t^{-1}\) times its normalized
frame: successive derivatives have invertible leading matrices
\(C-kI\) for \(k\ge1\), generating all poles.
There is consequently a natural surjection from the
constant Euler module \(N_{C-I}\) to its finite
microlocalization. The flat dual observations below,
composed with this surjection, are injective by their
normal rows. This proves that the surjection is also
injective, and hence identifies the entire actual
microlocalization with \(N_{C-I}\).
Inhomogeneous holomorphic Euler errors
are removable, because \(nI-(C-I)\) is invertible for \(n\ge0\);
RF.B's convergent Taylor estimate applies. A flat dual
basis gives homogeneous power/logarithm observations.
They are jointly injective on that microlocalization.
For a noninteger eigenvalue the normal-row argument above applies.
For the eigenvalue \(-1\), a positive derivative of order \(j\)
has leading logarithmic coefficient
\((-1)^j j!t^{-j-1}\), which is nonzero.
An \(n\)-fold primitive, \(n\ge1\), of
\(t^{-1}(\log t)^k\) has exponent \(n-1\) and highest
logarithmic term
\[
\frac{t^{n-1}(\log t)^{k+1}}
     {(k+1)(n-1)!}.
\tag{CK.13}
\]
The normal coefficients are determined successively by
these distinct exponent and triangular logarithmic
coefficients, with actual convergence from RF.C.
A flat dual basis separates different Jordan blocks.
These observations are exactly the full flat dual
basis of the meromorphic connection.

If \(F\subset A\) is any coherent D-submodule, actual E/D flatness
injects \(E\otimes_DF\) into \(E\otimes_DA\).
Thus those flat dual observations are faithful on its source.
For \(w\in F\), the normalized coefficient lattice criterion is
therefore
\[
\begin{gathered}
w\in A_\Sigma\\
\Longleftrightarrow\\
\operatorname{Re}\operatorname{ord}(1\otimes w)\le-\tfrac12
\end{gathered}
\tag{CK.14}
\]
on the smooth conormal.
For the forward direction use \(T\) and IO.C.
For the reverse, a negative pole of the coefficient vector in
the normalized frame supplies a nonzero leading vector
polynomial in \(\log t\), with real exponent less than zero.
At least one flat dual component detects it, contradicting CK.10.
Ordinary holomorphic ambiguities cannot cancel a negative real
exponent. This handles every nilpotent and resonant block.

#### CK.D. Actual contact and scalar-order changes used in the reduction

For the quadratic quantization of Section5.28 put
\[
\begin{gathered}
Q=\tfrac12 D_x^T C D_xD_t^{-1},\\
C=C^T.
\end{gathered}
\tag{CK.15}
\]
Its degree-one symbol is independent of the base variables
and its subprincipal symbol is zero.
The actual common-domain quantization there is
\(\gamma_s=\exp(s\,\mathrm{ad}Q)\). Let \(F_s\) be the
Hamiltonian flow of \(H_q\); normal symbols transform by
pullback along \(F_s\), so their zero sets transform to
\(F_{-s}\Lambda\). Differentiate the
convergent coefficient formula on compact parameter discs;
the actual bounds justify this differentiation.
The first two symbol equations from IO.5 are
\[
\begin{gathered}
\frac{d}{ds}p_s=\{q,p_s\},\\
\frac{d}{ds}s(P_s)=H_qs(P_s).
\end{gathered}
\tag{CK.16}
\]
If \(P_0=\Theta_\Lambda\), the first Hamiltonian restricts
to the transported radial field on \(F_{-s}\Lambda\),
because \(H_q\) commutes with the radial field.
The second symbol vanishes on the transported Lagrangian.
IO.A's first-order kernel assertion therefore identifies
the transported radial generator modulo \(J_{F_{-s}\Lambda}\).
The quantization preserves \(E_0,E_{-1}\) and sends the
characteristic subring and its ideal to those of the image.
It preserves the cyclic residue polynomial exactly.
Ordinary-coordinate invariance was proved in IO.B.
Consequently the actual transformations used for whole-germ
generic position preserve the cutoff and its half-order sign.

For a scalar \(s\in\mathbf C\), define an actual automorphism
in a \(\tau\ne0\) chart by its full normal symbol:
\[
\begin{gathered}
\Phi_s(a)\\
=\sum_{k\ge0}\binom{-s}{k}
       (\partial_t^k a)\tau^{-k}.
\end{gathered}
\tag{CK.17}
\]
This is a definition by actual symbols; a noninteger
\(D_t^s\) is not asserted to be an element of \(E\).
The notation \(D_t^{-s}aD_t^s\) is only a mnemonic.

To prove actuality, bound the coefficient of input order
\(m-j\) by \(MC^j j!\) on one common domain, including
the finitely many nonnegative orders after changing \(M,C\).
Cauchy bounds its \(k\)-th time derivative by
\(MC^j j!k!\delta^{-k}\) on a smaller fixed domain.
The product formula
\[
\begin{gathered}
\left|\binom{-s}{k}\right|\\
\le C_s(k+1)^{|s-1|+1}
\end{gathered}
\tag{CK.18}
\]
follows by bounding the factors \(|1+(s-1)/\ell|\)
by \(\exp(|s-1|/\ell)\), absorbing initial factors.
At output order \(m-n\) there are exactly the finite
terms \(j+k=n\). Since \(j!k!\le n!\), their sum is
bounded by \(M'C'^n n!\). The same estimates on every
smaller domain give the actual derivative bounds.
Thus CK.17 has finite upper order and one negative
factorial bound on a common domain.

At each homogeneous order all the composition identities
are finite coefficient identities. Binomial convolution
and the normal Leibniz product give
\[
\begin{gathered}
\Phi_s\Phi_r=\Phi_{s+r},\\
\Phi_s(ab)=\Phi_s(a)\Phi_s(b).
\end{gathered}
\tag{CK.19}
\]
One may verify the product identity first for an indeterminate
integer \(s\): it is conjugation by the ordinary invertible
integer power of \(D_t\); at any fixed output order both
sides are polynomial in \(s\), hence the identity holds
for every complex \(s\). CK.18 makes both resulting
symbols actual. The inverse is \(\Phi_{-s}\).

It preserves the principal symbol, every usual order,
\(E_0,E_{-1},E_\Lambda,J_\Lambda\).
For a radial first-order generator,
\(\partial_t q_1|_\Lambda=-\tau\), so
\[
\begin{gathered}
\Phi_s(\Theta)=\Theta+s\\
\pmod{J_\Lambda}.
\end{gathered}
\tag{CK.20}
\]
Pulling the module action back by \(\Phi_s\) adds \(s\)
to every indicial order. It changes no characteristic
support or generic stable-lattice property.

On one small neighbourhood, all section orders lie in
\(B+\mathbf Z\) for a finite set \(B\).
Choose finitely many E-generators, take their stable cyclic
lattices on each connected dense regular component, and
use their constant polynomials. Every section is in a
finite shift of their sum; IO.C proves the assertion.
It applies to every coherent E-quotient as well, by
using the image of the same stable generating lattice.

For a requested \(c\), choose \(\epsilon>0\) so that
\((c,c+\epsilon]\) contains no real part from \(B+\mathbf Z\).
Then \(M_c=M_{c+\epsilon}\), and the real scalar
\(s=-\tfrac12-c-\epsilon\) reduces the desired cutoff
to \(-1/2\), with no spectrum real part in
\(\tfrac12+\mathbf Z\).
This finite spectral gap also excludes every half-integer
order of a hypersurface-supported module in the next step.

#### CK.E. Ordinary modules supported on a smooth hypersurface

For a D-module on which \(t\) is locally nilpotent, put
\[
\Pi u=\sum_{k\ge0}\frac{D_t^kt^ku}{k!}.
\tag{CK.21}
\]
The sum is finite for each section. The identity
\(tD_t^k=D_t^kt-kD_t^{k-1}\) gives \(t\Pi=0\)
by telescoping. Direct substitution gives the finite
unique normal decomposition
\[
u=\sum_{j\ge0}\frac{(-1)^j}{j!}
                 D_t^j\Pi(t^ju).
\tag{CK.22}
\]
Uniqueness follows by applying the largest possible power
of \(t\) and then descending. The tangential operators
commute with \(t,D_t,\Pi\).
Finite D-generators, with their finite \(t\)-nilpotence
orders, therefore give finitely many generators of
\(\ker t\) over the tangential D-ring.
The ordered normal filtration identifies its characteristic
set with the tangential part of the original set:
normal \(D_t\) is a free polynomial variable in CK.22.
If the original characteristic lies in the conormal
of \(t=0\), the tangential characteristic is the zero
section. A local good filtration then makes \(\ker t\)
finite over the tangential holomorphic ring.
Its connection is locally free by the differential
Fitting-ideal argument IO.B. A horizontal frame exists
by convergent coordinate ODEs and their flat compatibility.
Thus locally the module is a finite direct sum of the
normal delta module, tensored with a flat tangential bundle.

For its normal generator \(\delta\),
\[
\begin{gathered}
t\delta=0,\qquad tD_t\delta=-\delta,\\
\Theta\delta=\tfrac12\delta.
\end{gathered}
\tag{CK.23}
\]
All section orders of its finite microlocalization lie
in \(\tfrac12+\mathbf Z\), by IO.C and integer shifts.
If this microlocalization is nonzero, some section has
nonempty order set by the actual Jacobson/Nakayama proof.

#### CK.F. A lower-dimensional ordinary vanishing argument

Suppose \(F\) is coherent over D, its characteristic is
contained in \(\Lambda\cup T_X^*X\), and \(\mathbf P\Lambda\to S\)
is finite near the origin, as in RF.T. Let \(G\) be a coherent
D-subquotient of \(F\), supported on \(S\), whose finite
microlocalization vanishes over a dense smooth part of
every component of \(S\).
Its remaining nonzero characteristic support has dimension
at most \(n-1\): over \(S_{\rm sing}\) this follows from
the finite projective projection, and any other excluded
proper analytic subset is treated by the same dimension bound.
RF.A identifies that support with the actual finite E-support.
EC.D makes each such finite E-module zero. Thus
\(\operatorname{Char}G\subset T_X^*X\).

A local good filtration of \(G\) then has associated graded
annihilated by a sufficiently large power of the cotangent
variables: use its finite presentation and the analytic
radical theorem. It is finite over the holomorphic ring,
and has only finitely many nonzero graded degrees.
Lifting those finitely many generators proves \(G\)
finite over the holomorphic ring. Its D-action is a
connection, hence its differential Fitting ideals make
it locally free. A locally free module supported on the
proper hypersurface \(S\) is zero.
This proves \(G=0\) without assuming a general proper-image
theorem or a source cutoff purity assertion.

#### CK.G. The faithful ordinary module in a finite meromorphic target

After CK.D's reductions assume generic position, cutoff
\(-1/2\), and no generic section order with real part
in \(\tfrac12+\mathbf Z\).
RF.T supplies \(F\) on one base neighbourhood with
\[
\begin{gathered}
F_0=M_p,\\
E\otimes_DF=M\text{ near }p,\\
\operatorname{Char}F\subset\Lambda\cup T_X^*X,
\end{gathered}
\tag{CK.24}
\]
and \(F\subset K=L/\mathcal O_X^q\), with \(L\) the
finite D-type target along the original \(S\).

First \(F\) has no S-torsion. Any putative torsion section
generates one coherent cyclic D-submodule \(G\subset F\)
supported on \(S\); derivative commutators preserve
finite \(f\)-torsion for individual sections.
At a generic smooth point of \(S\), CK.E shows its
microlocalization has half-integer orders.
It is an E-submodule of \(M\), whose orders all belong
to the disjoint set \(B+s+\mathbf Z\). Hence it vanishes
there, and CK.F's argument makes the entire \(G\) zero.
This excludes every torsion section without assuming
coherence of an infinite torsion union.

Take the finite meromorphic quotient connection
\[
A=L/\mathcal O_X(*S)^q.
\tag{CK.25}
\]
This notation is a quotient over the meromorphic
coefficient ring; no D-coherence of this quotient
is needed here. A finite presentation is obtained
by localizing the finite coefficient lattice of \(L\)
and the finite map from the trivial bundle.
Away from \(S\) that map is a constant-rank flat
injection. The quotient is a finite meromorphic
connection and is generically regular on every
component: in a normalized logarithmic chart a
quotient of the finite monodromy representation
has its constant logarithmic model.
Section5.27 therefore supplies the coherent
reflexive normalized lattice \(A_\Sigma\), with
\(A_\Sigma(*S)=A\). Its hypotheses are coherence
over \(\mathcal O(*S)\) and generic divisorial
regularity, not assumed D-coherence.

The kernel of \(F\to A\) is S-torsion: a lift
in \(L\) lies in \(\mathcal O(*S)^q\), and a
finite power of \(f\) puts it in \(\mathcal O^q\).
Thus \(F\subset A\) by the preceding paragraph.

#### CK.H. Coherence of the ordinary intersection

Set \(H=F\cap A_\Sigma\). This is an ordinary holomorphic
coefficient subsheaf; its coherence needs a neighbourhood
argument, not just finite generation of each stalk.

We first prove that an ascending chain of coherent submodules
of a coherent holomorphic module stabilizes locally.
At a point its stalk stabilizes, by Noetherianity.
Let \(H_N\) be the stabilized term and put \(Q=A_\Sigma/H_N\).
A finite prime filtration of \(Q_0\) spreads on a smaller
neighbourhood to a filtration with factors
\(\mathcal O_{Z_i}\), where each \(Z_i\) is a reduced
irreducible representative through the point.
Here is the algebra and spread: choose a maximal
annihilator of a nonzero element; it is prime, since
a product in it and a factor outside it would otherwise
give a strictly larger annihilator. Quotient by that
cyclic prime factor and repeat. The ascending sequence
of submodules stops by Noetherianity.
There are finitely many generators, relations and
exactness identities, so coherence spreads the finite
filtration; choose irreducible representatives of the
prime germs and shrink to remove extra components.

Any coherent submodule \(B\subset Q\) with \(B_0=0\)
is zero on this same neighbourhood. Intersect it with
the finite filtration. The successive image is a
coherent ideal in \(\mathcal O_{Z_i}\), zero near the
original point. A nonzero such ideal has support all
the irreducible \(Z_i\): its analytic support contains
a nonempty open, hence all of \(Z_i\) by the identity
theorem. That would contain the original point.
Thus every image is zero, and \(B=0\).
Apply this to every later chain term modulo \(H_N\).
This proves local stationarity on one common neighbourhood.

Choose a good D-filtration \(F_k\) of \(F\) on one
fixed base neighbourhood, from finitely many D-generators.
Every \(F_k\) is coherent holomorphic.
All their pole bounds hold on this same neighbourhood.
In fact, put the finitely many initial D-generators in
\(f^{-a}A_\Sigma\) and the derivatives of the finitely many
generators of \(A_\Sigma\) in \(f^{-b}A_\Sigma\).
Both statements have finite witnesses and hence hold
on one neighbourhood. Leibniz then gives
\[
F_k\subset f^{-a-(b+1)k}A_\Sigma
\tag{CK.26}
\]
there, with no new shrink depending on \(k\).
Clearing that bound places both
\(F_k\) and \(A_\Sigma\) inside one finite
holomorphic lattice in \(A\), so
\(F_k\cap A_\Sigma\) is coherent holomorphic, by the
finite kernel theorem. These intersections ascend
and their union is \(H\).
The proved local stationarity makes \(H\) coherent.

By CK.14 and the scalar/reflexive extension RF.D--RF.E,
\[
\begin{gathered}
H_0=\\
\left\{\begin{gathered}
u\in M_p:\\
\operatorname{Re}\operatorname{ord}u\le-1/2\\
\text{on every generic}\\
\text{regular component}
\end{gathered}\right\}.
\end{gathered}
\tag{CK.27}
\]
For this germ comparison use the finite prepared
projective projection: every sufficiently small
neighbourhood of \(p\) contains the entire
characteristic representative over a smaller base.
RF.T identifies \(F_0\) with the whole \(M_p\).
Membership in \(A_\Sigma\) is the generic strict
Nilsson criterion on each component and propagates
across singularities with the same zero pole order.
Thus CK.27 is for all source germs, not just
sections represented on one fixed generic chart.

#### CK.I. A coherent generating candidate and its smooth identification

Let \(L'=E_0H\subset M\). It is coherent over \(E_0\):
finite ordinary generators have images in one finite
shift of J.1's coherent lattice, and Section5.24
gives their coherent image. IO.C shows CK.27 is
stable under \(E_0\), so
\[
L'_p=H_0=(M_{-1/2})_p .
\tag{CK.28}
\]

On the smooth divisor chart, \(H\) is stable under
\(tD_t,D_{x_i}\), since both \(F\) and \(A_\Sigma\)
are. Thus \(L'\) is stable under the whole local
\(E_\Lambda\), by DP.7's generator proof.
Let \(b(\theta)\) be the characteristic polynomial
of the constant normalized residue matrix \(C\).
It satisfies
\[
D_t b(tD_t) A_\Sigma\subset A_\Sigma .
\tag{CK.29}
\]
Indeed \(b(tD_t)\) kills the residue modulo \(t\);
then \(D_t(tA_\Sigma)\subset A_\Sigma\).
Both operators preserve \(F\), so CK.29 also holds
with \(H\). Consequently
\[
b(-\Theta-\tfrac12)L'\subset hL'.
\tag{CK.30}
\]
To pass from generators to \(E_0H\), commute
the polynomial in \(\Theta\) past an \(E_0\)
coefficient; IO.A puts its commutator in
\(J_\Lambda\), which acts into \(hL'\).
The roots in CK.30 satisfy
\[
-\tfrac32<\operatorname{Re}\beta\le-\tfrac12 .
\tag{CK.31}
\]
The previously selected spectral gap excludes the boundary.

We must also prove generation. \(DH\) is coherent over D.
The quotient \(F/DH\) is supported on \(S\): multiply
each of the finitely many ordinary D-generators
of \(F\) by a sufficient power of \(f\) to put it
in the finite lattice \(A_\Sigma\), hence in \(H\).
At generic smooth points, the finite E-quotient
of \(M\) has no half-integer order, by CK.D's quotient
spectral observation. CK.E shows its supported
ordinary module must have such orders if nonzero.
It vanishes generically, so CK.F's lower-dimensional
argument gives \(F/DH=0\).
Therefore \(EL'=M\).

Now CK.31 and the smooth top-strip characterization
CK.7--CK.9 give
\[
\begin{gathered}
L'=M_{-1/2}\\
\text{away from}\\
W=\Lambda\cap\pi^{-1}(S_{\rm sing}).
\end{gathered}
\tag{CK.32}
\]
This includes any points where initial smooth regularity
was unknown: the lattice \(L'\) has just supplied
smooth characteristic stability there.
For the comparison of generic definitions, its
generating quotient argument is independent of
that newly proved regularity.

Every section of \(L'\) obeys the generic cutoff
on all characteristic components, by CK.14 and
IO.C. Thus \(L'\subset M_{-1/2}\) everywhere.

#### CK.J. The single adjacent quotient and extension across W

The quotient
\[
Q=h^{-1}L'/L'
\tag{CK.33}
\]
is coherent over \(E_0\) and is killed by \(h\).
It is a finite holomorphic principal-symbol module,
supported on \(\Lambda\).
At \(p\), a germ of \(Q\) supported on \(W\)
lifts to \(u\in h^{-1}L'\) whose restriction
off \(W\) is in \(L'\). It therefore satisfies
the generic cutoff condition. CK.28 gives
\(u\in L'_p\). Hence
\[
\mathcal H^0_W(Q)_p=0.
\tag{CK.34}
\]
This supported submodule is coherent holomorphic.
Indeed for finitely generated ideal \(I_W\)
the annihilators \(\ker I_W^k\) form an ascending
chain of coherent submodules of the finite
principal-symbol module, so CK.H's local stationarity
applies. Thus CK.34 holds on one neighbourhood.

Multiplication by \(h^{-k+1}\) identifies, up to
the normal coefficient automorphism,
\[
h^{-k}L'/h^{-k+1}L'\quad\text{with }Q.
\tag{CK.35}
\]
That automorphism is the identity on the principal
symbol ring and hence preserves \(W\).
All these quotient sheaves consequently have
no W-supported sections on the same neighbourhood,
with no additional shrink depending on \(k\).
Finite extensions preserve absence of supported
sections. Since \(EL'=M\), every section of
\(M/L'\) lies locally in one finite negative
shift. Hence
\[
\mathcal H^0_W(M/L')=0.
\tag{CK.36}
\]
By CK.32 the subquotient \(M_{-1/2}/L'\) is
supported on \(W\); by CK.36 it is zero.
This proves \(M_{-1/2}=L'\) as sheaves near \(p\),
and in particular actual \(E_0\)-coherence at
every singular characteristic point.

#### CK.K. Every characteristic operator and supported-section purity

Undo CK.D's actual transformations. They preserve the
usual order-zero and characteristic rings, with the
explicit scalar shift accounted for. This proves
coherence of the original \(M_c\) for every real \(c\).
On the dense regular locus it generates \(M\);
the coherent E-quotient away from that locus has
support in a proper subset of every Lagrangian
component and is zero by EC.D. Equivalently the
generation in CK.I already transports back.

For a characteristic first-order operator \(P\),
\(Pu\in E_\Lambda u\) on every dense smooth regular
component, so IO.C proves its orders have no larger
real parts than those of \(u\). The generic definition
therefore makes \(M_c\) stable under \(P\) everywhere.
This proves CK.1, and the inclusion \(E_V\subset E_\Lambda\)
gives full regularity along every involutive \(V\)
containing the support.

Finally let a germ of \(M/M_c\) be supported on \(Z\)
of ambient codimension at least \(n+1\).
Lift it locally to \(u\in M\).
Each Lagrangian component has dimension \(n\), so its
dense smooth regular locus has a nonempty open
outside \(Z\). There the orders of \(u\) satisfy the
cutoff. IO.B--IO.D constant cyclic polynomials on
each connected dense regular representative extend
the same bound throughout it. The generic definition
puts \(u\) in \(M_c\), proving CK.2.
The assertion concerns sections of the quotient by the canonical cutoff lattice.


The theorem proves coherent source cutoffs across every singular characteristic stratum and the everywhere characteristic-stable generating lattice. Section5.90, GG.A–GG.G proves initial coherent ordinary generators near every compact set. Arbitrary analytic proper/support-proper regularity requires its further argument.

![Spectral lowering, the strict half-order boundary and the adjacent quotient across singular strata](assets/canonical-cutoff-C1.png)

**Figure CK.1.** Panel A gives two exact spectral examples at
\(c=-1/2\): the roots \(-1/5\) and \(7/10\) move to
\(-6/5\) and \(-13/10\) in the top strip. Each step
is the elementary lowering of CK.B, with its full nilpotent
multiplicity. Panel B shows the power/logarithm example
\(t^{2/5}(\log t)^2\), whose indicial eigenvalue is
\(-9/10\), at the boundary criterion of CK.C.
Panel C is CK.J's single adjacent quotient: absence of
sections supported on \(W\) propagates to every negative
shift on one neighbourhood and identifies the coherent
candidate with the intrinsic cutoff. The panels are
algebraic diagrams with no geometric scale.

Figure SVG ·
exact diagram data ·
reproducible source ·
figure PNG.

Free source:
[Kashiwara–Kawai, Holonomic systems III, I.5, II.3.7–II.3.8 and V.1.6–V.1.7](https://www.kurims.kyoto-u.ac.jp/~kenkyubu/kashiwara/HolIII.pdf).


### 5.90. Analytic good filtrations from the dummy-variable cutoff

#### GG.A. The global filtration and the compact-neighbourhood generator

Let \(X\) be a complex manifold of dimension \(d\), and let \(M\)
be a coherent analytic left \(\mathcal D_X\)-module. In this
section regular holonomicity means holonomicity together with
generic regular stable lattices for its actual finite
microlocalization on every nonzero characteristic component.
The zero module is allowed.

Set \(Y=\mathbf C_t\times X\), \(i(x)=(0,x)\), and
\[
\begin{gathered}
\Omega=\{\tau\ne0\}\subset T^*Y,\\
h=D_t^{-1},\\ j_0(x)=(0,x;1,0),\\
N=\mathcal E_Y\otimes_{\mathcal D_Y}i_+M .
\end{gathered}
\tag{GG.1}
\]
The last line is restricted to \(\Omega\), and the ordinary
sheaves there are pulled back by \(\pi\).
The normal line of \(i\) is trivialized by \(t\). Thus the
left-module transfer has the global normal generator
\(u_0=1\bmod\mathcal E_Yt\), and \(N\) is exactly
\(\Phi(M)\) of §5.25.3.
The ordinary embedding is \(m\mapsto u_0\otimes m\).

**Theorem GG.T.** If \(M\) is regular holonomic, \(N\) is an actual
regular holonomic \(\mathcal E_Y\)-module on \(\Omega\). Fix any
real number \(c\), and let \(L=N_c\) be the intrinsic cutoff of
CK.T.
Then
\[
\begin{gathered}
L\text{ is coherent over }\mathcal E_Y(0),\\
\mathcal E_YL=N,\qquad tL\subset hL .
\end{gathered}
\tag{GG.2}
\]
The subsheaves of the actual ordinary module
\[
\begin{gathered}
m\in B_kM\\
\Longleftrightarrow h^k(u_0\otimes m)\in j_0^{-1}L,\\
k\in\mathbf Z,
\end{gathered}
\tag{GG.3}
\]
are globally defined coherent \(\mathcal O_X\)-modules and form
an exhaustive, locally bounded-below, locally good order
filtration. Every point has one neighbourhood on which a finite
list \(m_\nu\) and integer shifts \(a_\nu\) give, simultaneously
for all integers \(k\),
\[
\begin{gathered}
B_kM=\sum_\nu\mathcal D_{X,\le k-a_\nu}m_\nu,\\
\mathcal D_{X,\le b}=0\quad(b<0).
\end{gathered}
\tag{GG.4}
\]
For every compact subset \(K\subset X\), there are one open
neighbourhood \(U\) of \(K\) and an integer \(q\) such that
\[
\begin{gathered}
G=B_qM|_U,\\
G\text{ is coherent over }\mathcal O_U,\\
\mathcal D_UG=M|_U,\\
B_{q+r}M|_U=\mathcal D_{U,\le r}G\\
(r\ge0).
\end{gathered}
\tag{GG.5}
\]
Thus the initial coherent generator exists on a neighbourhood
of every compact set. The integer \(q\) may depend on that
compact set.

We use actual finite operators throughout. Their coherent
kernels and the faithful dummy completion are
§§5.19–5.24.
The actual ambient comparison, ordinary injection and lattice
bounds are §5.25.
The complete actual descent statement is
Proposition5.26.1.
Finite microlocalization is right flat, coherent, and has exact
characteristic support by
RF.A.1–RF.A.4.
The residue and ordinary-coordinate calculation are
IO.A–IO.B.
Finite actual generating lattices are
J.1.
The analytic radical theorem and the dimension and empty-interior
properties of proper analytic subsets are proved in
Analytic germs, Theorems5.1,5.4 and Proposition5.5.
The horizontal-frame construction is
GL02, Lemma4.1.
These earlier proofs supply the indicated inputs.

#### GG.B. The ordinary normal transfer and its exact characteristic set

Work first on an ordinary coordinate neighbourhood with any
local good filtration \(F_aM\), zero for sufficiently negative
\(a\). This local filtration follows from a finite ordinary
operator presentation; it is not a global generator hypothesis.
The normal derivative description of closed direct image is
\[
\begin{gathered}
i_+M=\bigoplus_{j\ge0}D_t^j(u_0\otimes M),\\
tD_t^j(u_0\otimes m)\\
=-jD_t^{j-1}(u_0\otimes m).
\end{gathered}
\tag{GG.6}
\]
Every displayed sum is finite. To verify the presentation,
start with a finite \(\mathcal D_X\)-presentation of \(M\), extend
its relation matrix by the same tangential operators on \(Y\),
and impose \(t(u_0\otimes m_\nu)=0\) on its generators.
Move \(t\) to the right by \([D_t,t]=1\) and move tangential
operators through \(D_t\). The remaining rows are precisely
finite sums in (GG.6), with their original relation rows
coefficient by coefficient. A holomorphic coefficient on \(Y\)
acts on a row with largest normal derivative \(j\) through its
Taylor polynomial in \(t\) of degree \(j\); the remainder kills
that row. Consequently the normal form is valid for arbitrary
holomorphic coefficients, and its inverse respects their action.
It proves the finite presentation, coherence, and exactness of
this normal transfer.

Put
\[
\begin{gathered}
F_ai_+M\\
=\bigoplus_{j\ge0}D_t^j(u_0\otimes F_{a-j}M).
\end{gathered}
\tag{GG.7}
\]
For each fixed \(a\) this is a finite sum of coherent
holomorphic modules. Taylor expansion just used and the
ordinary order compatibility of \(F\) prove that it is a
filtered \(\mathcal D_Y\)-module. A finite homogeneous
generating list for \(\operatorname{gr}_FM\), with the same
shifts, generates its normal extension, giving
\[
\begin{gathered}
\operatorname{gr}_Fi_+M\\
\simeq i_*(\operatorname{gr}_FM)[\tau],\qquad t=0,\\
\operatorname{Char}(i_+M)\\
=\bigl\{(0,x;\tau,\xi):\\
(x,\xi)\in\operatorname{Char}M\bigr\}.
\end{gathered}
\tag{GG.8}
\]
The first identity is coefficientwise in normal PBW rows;
in particular it is an equality of the actual associated
graded presentation. Thus this filtration is good.
Each component in the second identity has dimension \(d+1\)
when \(M\) is holonomic.

RF.A.3–RF.A.4 applied on \(Y\) now give coherence and
\[
\begin{gathered}
\Lambda'=\operatorname{supp}N\\
=\bigl\{(0,x;\tau,\xi)\in\Omega:\\
(x,\xi)\in\operatorname{Char}M\bigr\}.
\end{gathered}
\tag{GG.9}
\]
Hence \(N\) is holonomic. Tensor reassociation of the finite
transfer presentation gives the actual identification
\[
N=(\mathcal E_Y/\mathcal E_Yt)\otimes_{\mathcal D_X}M.
\tag{GG.10}
\]
Along \(j_0(X)\), §5.25.3 proves the injection of \(M\) into
this module using exact faithful completion and the
\(h\)-torsion-free ambient module. Thus the embedding used
in (GG.3) retains every ordinary section.

#### GG.C. Regularity on every lifted component

First suppose \(\xi\ne0\). The projection from \(t=0\) to
\(T^*X\) then meets the ordinary finite microlocalization
\(M^\mu=\mathcal E_X\otimes_{\mathcal D_X}M\).
In a chart with \(\xi_\ell\ne0\), a tangential actual symbol
on \(X\), independent of \(t,\tau\), defines an actual symbol
on \(Y\). Homogeneous order is unchanged. Indeed the ratio
\(\tau/\xi_\ell\) stays bounded on a smaller product chart,
and the original common coefficient domain and factorial
bound remain a common domain and factorial bound there.
More explicitly, the source term
\(\xi_\ell^{-n}p_n(x,\xi/\xi_\ell)\) becomes a
\(\tau^{-n}\)-term whose coefficient is multiplied by
\((\tau/\xi_\ell)^n\). If that ratio is bounded by \(R\),
the bound \(AC^n n!\) becomes \(A(CR)^n n!\).
The normal symbol multiplication has no additional
derivatives for these \(t,\tau\)-independent symbols.

Accordingly \(\mathcal E_Y/\mathcal E_Yt\) has a right
\(\mathcal E_X\)-action on this chart: the tangential
operators commute with \(t\). Tensor associativity gives
\[
\begin{gathered}
N\simeq(\mathcal E_Y/\mathcal E_Yt)
\otimes_{\mathcal E_X}M^\mu .
\end{gathered}
\tag{GG.11}
\]
This identity requires only the right action and tensor
reassociation.

Take a smooth generic regular point of a nonzero
characteristic component \(\Lambda\) of \(M^\mu\).
Let \(A\subset M^\mu\) be a coherent generating
\(\mathcal E_X(0)\)-lattice stable under
\(\mathcal E_{X,\Lambda}\). Define \(A'\) as the image in
(GG.11) of
\[
\begin{gathered}
(\mathcal E_Y(0)/\mathcal E_Y(0)t)
\otimes_{\mathcal E_X(0)}A .
\end{gathered}
\tag{GG.12}
\]
A finite generating list for \(A\) gives a finite
\(\mathcal E_Y(0)\)-generating list for \(A'\).
By J.1 and a finite order shift, its finitely many
generators lie in one coherent order-zero ambient
lattice in \(N\). The coherent submodule theorem
therefore makes their span \(A'\) coherent. Since
\(\mathcal E_XA=M^\mu\), (GG.11) gives
\(\mathcal E_YA'=N\) here.

We verify characteristic stability directly. On this
smooth locus choose finitely many homogeneous degree-one
equations \(g_\alpha(x,\xi)\) for \(\Lambda\), and
tangential first-order lifts \(H_\alpha\). Such equations
are obtained in the chart \(\xi_\ell\ne0\) by multiplying
local degree-zero defining equations by \(\xi_\ell\).
Let \(P\) have order at most one on \(Y\), with degree-one
symbol vanishing on \(p^{-1}\Lambda=\{t=0,(x,\xi)\in\Lambda\}\).
The smooth defining-coordinate Taylor formula gives
\[
\begin{gathered}
\sigma_1(P)=f\,t+\sum_\alpha b_\alpha g_\alpha,\\
\deg f=1,\\ \deg b_\alpha=0.
\end{gathered}
\tag{GG.13}
\]
Lift \(f,b_\alpha\) to actual \(F,B_\alpha\) of the indicated
orders. The finite sum \(P-Ft-\sum_\alpha B_\alpha H_\alpha\)
has order zero. Acting on \(u_0\otimes a\), its \(Ft\)-term
is zero, and its other terms lie in \(A'\) by stability of
\(A\). Put \(v=u_0\otimes a\). For any order-zero \(Q\),
\[
\begin{gathered}
P Qv=Q Pv+[P,Q]v,
\end{gathered}
\tag{GG.14}
\]
and \([P,Q]\) has order at most zero by §5.19's actual
symbol product. Therefore \(P A'\subset A'\).
This proves regularity on the dense lifted part of every
component with generic \(\xi\ne0\).

The original zero section requires a separate argument.
If \(T_X^*X\) is a characteristic component, choose a
generic point of it outside its intersections with the
other characteristic components. Locally there are
finitely many components; distinct \(d\)-dimensional
components have proper intersections. The characteristic
set is a cone. A nonzero covector over a nearby point
can be scaled into any fixed neighbourhood of the zero
covector, and its limit lies in the same closed conical
component. Hence, after shrinking an ordinary neighbourhood
\(W\) of the chosen point,
\[
\operatorname{Char}(M|_W)\subset T_W^*W .
\tag{GG.15}
\]
No characteristic component over that neighbourhood can
be discarded in this deduction.

Here is the ordinary consequence of (GG.15). In a finite
homogeneous good presentation the radical of the
annihilator of the associated graded module contains
every cotangent variable. The analytic radical theorem
therefore puts a power of each variable in that
annihilator. One may first obtain this at the analytic
germ at the zero covector: clearing its denominator and
taking the homogeneous component of the indicated degree
leaves its invertible degree-zero coefficient. This gives
the same relation over the polynomial symbol ring.
There are finitely many such identities, so they hold on
one ordinary neighbourhood. A sufficiently large power
of the whole cotangent ideal then kills the associated
graded module. It has finitely many generators and
therefore only finitely many nonzero degrees. Lifting
them makes \(M|_W\) coherent over \(\mathcal O_W\).

Its operator action is an integrable connection. The
Fitting-ideal calculation in IO.B shows that it is
locally free: differentiating the presentation minors
preserves their ideals, and a nonzero differential
ideal in a holomorphic local ring contains a unit.
The unit-minor row reduction there proves local
freeness. GL02 Lemma4.1 now gives a horizontal frame
\(e_1,\ldots,e_r\). In this frame the lifted module has
presentation
\[
\begin{gathered}
\mathcal J=\mathcal E_Yt\\
{}+\sum_{\ell=1}^d\mathcal E_YD_{x_\ell},\\
N|_W\simeq(\mathcal E_Y/\mathcal J)^r,\\
t(u_0\otimes e_\nu)=0,\\
D_{x_\ell}(u_0\otimes e_\nu)=0 .
\end{gathered}
\tag{GG.16}
\]
The order-zero span of these \(r\) generators is coherent
by the same J.1 ambient argument and generates.
Its characteristic set is
\(\{t=0,\xi=0,\tau\ne0\}\).
For every first-order operator whose principal symbol
vanishes on this conormal, (GG.13) holds with the defining
equations \(\xi_1,\ldots,\xi_d\). The \(Ft\)-term and all
tangential \(D_{x_\ell}\)-terms kill the generators.
The commutator calculation (GG.14) again proves stability.
Thus this entire lifted zero-section component is
generically regular as well.

Every irreducible component of (GG.9) is covered by one
of these two cases. \(N\) has the generic regularity
required by CK.T on the whole dummy chart, including
near \(j_0(X)\).

#### GG.D. The intrinsic lattice lowers the normal coordinate

Apply CK.T to \(N\). Its cutoff \(L=N_c\) is coherent,
generates \(N\), and is stable under every first-order
operator with principal symbol zero on \(\Lambda'\).
Since \(\Lambda'\subset\{t=0\}\), the actual operator
\(D_tt\) is such an operator:
\[
\begin{gathered}
D_tt=tD_t+1,\\
\sigma_1(D_tt)=\tau t,\\
D_tt\,L\subset L,\\
tL=hD_tt\,L\subset hL.
\end{gathered}
\tag{GG.17}
\]
The identity in the last line is an identity of finite
actual operators. It includes the order-zero \(+1\).

The construction is global on \(\Omega\). The closed
embedding, the coordinate \(t\), and the normal
trivialization are global. On overlaps in \(X\), (GG.10)
is the same intrinsic transfer module, and the residue
is invariant under ordinary coordinate changes by IO.B.
The generic root condition defining \(N_c\) therefore
agrees on overlaps. CK.T supplies coherence through every
singular characteristic stratum, so these local cutoffs
are the restrictions of one \(L\).

The filtration (GG.3) is consequently a filtration of
globally defined subsheaves of \(M\). It depends on the
fixed real bound \(c\); it uses no local choice of an
initial ordinary generator.

#### GG.E. All actual ordinary pieces on one neighbourhood

We give the exact application of §§5.23–5.26. At any
\(x\), choose a local ordinary good filtration and its
actual coherent ambient lattice \(V\) from §5.25. The
two finite actual denominator bounds give
\[
h^C V\subset L\subset h^{-B}V
\tag{GG.18}
\]
near \(j_0(x)\), for some nonnegative \(B,C\). Exact
faithful completion realizes its middle term in
\(h^{-B}\prod_a h^a F_aM\). By (GG.17) the preserving
operator \(D_tt=tD_t+1\) acts there as
\[
\begin{gathered}
(tD_t+1)(h^a(u_0\otimes m))\\
=a h^a(u_0\otimes m).
\end{gathered}
\tag{GG.19}
\]
The bounds imply equality after finite \(h\)-localization.
The closed-submodule and finite-node Euler projections
of §5.23 therefore give finitely many homogeneous
generators \(h^{a_\nu}(u_0\otimes m_\nu)\) of that
completed lattice, with actual ordinary germs \(m_\nu\).

Represent these finitely many germs on one ordinary
neighbourhood. Their actual order-zero span \(L'\)
and \(L\) lie in one coherent shifted ambient \(V\).
Their completions have the same image: the Ore
calculation of §5.23.4 reduces every formal
\(t\)-coefficient to its homogeneous ordinary action.
Exact completion makes both coherent quotients
\[
\begin{gathered}
(L+L')/L,\\ (L+L')/L',
\end{gathered}
\tag{GG.20}
\]
have zero completion. Faithfulness makes their stalks
zero, and finite local generation makes both sheaves
zero on one neighbourhood. Thus \(L=L'\) as actual
sheaves there, as in Proposition5.26.1.

Shrink the ordinary neighbourhood only once so that
its \(j_0\)-image lies there. At every point of it the
coefficientwise degree-\(k\) calculation of §5.26
identifies (GG.3) with (GG.4), for every \(k\).
The right side is coherent for each \(k\): put its
finite operator images in one local ordinary coherent
order piece and use the coherent kernel/image theorem.
The integer bounding that piece may depend on \(k\);
the neighbourhood and the finite list \(m_\nu,a_\nu\)
do not. This proves coherence of all the actual
intersections on a common neighbourhood.

The finite list in (GG.4) proves the order-filtration
relations and the local lower bound. The lattice
generation in (GG.2) proves exhaustion, since a finite
operator expression has one finite upper order.
These are sheaf statements at every ordinary point,
so the globally defined \(B_kM\) are coherent and
locally good on \(X\).

#### GG.F. A uniform generator near a compact set or proper support

Cover \(K\) by finitely many of the neighbourhoods in
GG.E, with finite generating lists and shifts
\(a_{\nu,\alpha}\). Let \(U\) be their union, and choose
an integer \(q\) not smaller than every such shift.
On the \(\alpha\)-th neighbourhood all its generators
belong to \(B_qM\), so
\[
\mathcal D_U(B_qM|_U)=M|_U .
\tag{GG.21}
\]
Moreover, for every \(r\ge0\), ordinary coordinate
operators and PBW give
\[
\begin{gathered}
\mathcal D_{\le r}\mathcal D_{\le q-a_{\nu,\alpha}}\\
=\mathcal D_{\le q+r-a_{\nu,\alpha}},\\
B_{q+r}M|_U\\
=\mathcal D_{U,\le r}(B_qM|_U).
\end{gathered}
\tag{GG.22}
\]
The first equality follows by splitting each ordered
derivative monomial of order at most the sum into
two monomials with the indicated bounds; its
holomorphic coefficient can be placed in the first
factor. Both bounds on the first line are
nonnegative. Thus it applies to every summand of
(GG.4). The finitely many local lower bounds also
give one lower bound on \(U\). This proves (GG.5).

In particular the filtration with negative pieces
zero and nonnegative pieces \(\mathcal D_{U,\le r}G\)
is an exhaustive good filtration generated in degree
zero by one coherent holomorphic subsheaf \(G\).
The same \(q\) can be chosen for any finite list of
regular holonomic modules by taking the maximum of
the finitely many shift bounds.

**Corollary GG.P.** Let \(f:X\to Z\) be a holomorphic
map of complex manifolds and let \(M\) be regular holonomic with closed
support \(S\). If \(f|_S\) is proper, every \(z\in Z\)
has an open neighbourhood \(V_z\) on whose inverse
image \(M\) has a coherent holomorphic generator
and a global good filtration generated by it.

Indeed \(K=S\cap f^{-1}(z)\) is compact. Apply
(GG.5) on \(U\). Proper maps of locally compact
Hausdorff spaces are closed: for a point outside
the image of a closed set, choose a neighbourhood
with compact closure. The inverse image of that
closure is compact; its intersection with the
closed set is compact and has closed image.
Remove that image inside the neighbourhood.
Apply this to \(S\setminus U\)
and shrink \(V_z\) to avoid its image. Then
\[
S\cap f^{-1}(V_z)\subset U.
\tag{GG.23}
\]
The globally defined coherent sheaf \(B_qM\)
restricts to \(G\) on \(U\) and is zero off \(S\).
Thus its restriction to \(f^{-1}(V_z)\) generates
\(M\) there. Formula (GG.22) holds on \(U\) and
holds trivially off \(S\), proving the asserted
global good filtration on that inverse image.
For a bounded complex, the same construction
supplies such generators for its finitely many
regular holonomic cohomology modules.

This corollary supplies the initial-generator
hypothesis in the actual locally projective
operator-image coherence and characteristic
comparison of
§§5.9–5.12.
Their remaining hypotheses and conclusions apply
as stated. A proof of regularity of the full
arbitrary proper analytic direct image requires
the additional direct-image argument.

#### GG.G. Normal signs and an exact local example

Let \(M=\mathcal O_X\) with its trivial connection.
Its lift is the rank-one conormal module in
(GG.16). Put \(\delta=u_0\otimes1\).
The exact normal signs, valid for every integer
\(a\), are
\[
\begin{gathered}
t\delta=0,\\ tD_t\delta=-\delta,\\
th^a\delta=a h^{a+1}\delta,\\
(tD_t+1)h^a\delta=a h^a\delta .
\end{gathered}
\tag{GG.24}
\]
For \(a\ge0\) use \([t,h]=h^2\); for negative
\(a\) use \([t,D_t^j]=-jD_t^{j-1}\).
With IO.B's radial operator
\(\Theta=-tD_t-\tfrac12\), this gives
\[
\Theta(h^a\delta)=(\tfrac12-a)h^a\delta .
\tag{GG.25}
\]
CK.B's smooth top-strip description therefore
identifies the cutoff at \(c=0\) with
\(L=\mathcal E_Y(0)h\delta\). The generating
residue has eigenvalue \(-1/2\) in \((-1,0]\).
Along \(j_0\), completion of this lattice consists
of the series with degrees at least one and
holomorphic coefficients. The exact actual
intersection result of GG.E yields
\[
\begin{gathered}
B_k\mathcal O_X=0\quad(k<1),\\
B_k\mathcal O_X=\mathcal O_X\quad(k\ge1).
\end{gathered}
\tag{GG.26}
\]
The shift records the delta half-order. Replacing
\(D_tt\) by \(tD_t\) in the Euler projection would
give the wrong degree even in this example.

**Exercise.** For this flat example and any real
\(c\), determine the first nonzero ordinary
filtration piece.

**Solution.** The least allowed integral \(a\)
satisfies \(1/2-a\le c\), so it is
\[
a_0=\lceil\tfrac12-c\rceil .
\tag{GG.27}
\]
The smooth cutoff is
\(\mathcal E_Y(0)h^{a_0}\delta\).
Its formal homogeneous degrees start at \(a_0\);
GG.E gives zero pieces below \(a_0\) and
\(\mathcal O_X\) at and above it. In particular
the endpoint at an integral value is included.

![The lifted characteristic set, the exact Euler identity and ordinary generators near a compact set](assets/initial-analytic-generators.png)

**Figure GG.1.** The first panel separates the dense
nonzero-covector part from the original zero-section
component. GG.C proves regularity in both cases.
The second panel is the actual operator identity
(GG.17) with its \(+1\). The third panel shows
the finite shifted presentations (GG.4), from
which taking the maximum of the shifts gives
the coherent generator in (GG.5). These are
algebraic diagrams, with no geometric scale.

Figure SVG ·
exact diagram data ·
reproducible source ·
figure PNG.

Free source:
[Kashiwara–Kawai, Holonomic systems III, Lemma5.1.9,
Sublemma5.1.10, Corollary5.1.11 and AppendixA.8](https://www.kurims.kyoto-u.ac.jp/~kenkyubu/kashiwara/HolIII.pdf).


### 5.91. Finite cokernels of relative nuclear perturbations

Restrictions between analytic neighbourhoods enter proper-image
finiteness through their small infinite tails. The coefficient
algebra must be retained in the finite-rank argument: finiteness
over that algebra is stronger than a bound on any one fibre.
The theorem below keeps the full algebra and permits
non-Hausdorff algebraic quotients.

#### NP.A. The coefficient algebra and the finiteness statement

Let \(B\) be a nonzero unital complex Banach algebra. Its norm is
submultiplicative. A Fréchet left \(B\)-module is a complete
Hausdorff locally convex complex vector space with a countable
seminorm topology and a jointly continuous unital left
\(B\)-action. The scalar action of \(\mathbf C\) agrees with
the action through the centre of \(B\). The algebra \(B\)
need not be commutative.

A continuous \(B\)-linear map \(u:E\to F\) is **\(B\)-nuclear**
if it has a representation
\[
\begin{gathered}
u(x)=\sum_{j\ge1}\lambda_j\xi_j(x)y_j,\\
\sum_j|\lambda_j|<\infty,\\
\xi_j:E\longrightarrow B,\\
\xi_j\text{ continuous and }B\text{-linear},
\end{gathered}
\tag{NP.1}
\]
where the \(\xi_j\) are equicontinuous and \((y_j)\) is
bounded in \(F\). Equicontinuity here means that one
continuous seminorm \(p\) on \(E\), after a fixed rescaling,
satisfies
\[
\begin{gathered}
\|\xi_j(x)\|_B\le p(x)\\
(j\ge1,\ x\in E).
\end{gathered}
\tag{NP.2}
\]
Boundedness means boundedness for every continuous
seminorm on \(F\).

The map \(u\) is **\(B\)-subnuclear** if there is a Fréchet
left \(B\)-module \(G\) and a continuous \(B\)-linear
surjection \(p_G:G\to E\) such that \(u p_G\) is
\(B\)-nuclear.

**Theorem NP.T.** Let \(E,F\) be Fréchet left \(B\)-modules,
let \(f:E\to F\) be continuous, \(B\)-linear and surjective,
and let \(u:E\to F\) be \(B\)-subnuclear. Then
\[
\begin{gathered}
\operatorname{Coker}_{B}(f-u)\\
\text{is finitely generated}\\
\text{over }B.
\end{gathered}
\tag{NP.3}
\]
The cokernel in this statement is the algebraic quotient.
Neither a closed range nor a Hausdorff quotient is asserted.
NP.G gives a rank-one example where the quotient is not
Hausdorff.
For the zero coefficient algebra every unital module is zero,
so the finite-cokernel assertion holds there as well.

We first prove all the completeness, lifting and finite-rank
steps used in the theorem. NP.H supplies explicit relative
nuclear restrictions between multivariable analytic
power-series modules. These are coefficient-space
statements; no proper-image theorem is a premise.

#### NP.B. Convergent relative series

For every continuous seminorm \(q\) on a Fréchet \(B\)-module
\(F\), joint continuity of the action gives a continuous
seminorm \(q'\) on \(F\) such that
\[
q(by)\le\|b\|_B q'(y).
\tag{NP.4}
\]
Indeed continuity at \((0,0)\) gives a product of
neighbourhoods on which \(q(by)<1\). The neighbourhood
in \(B\) contains a norm ball; the other contains a
finite intersection of seminorm balls. Take their
rescaled maximum and then scale the two variables
separately. If that maximum is zero, scale its argument
by arbitrarily large positive constants to obtain the
same inequality.

For a bounded sequence \((y_j)\), put
\(M_q=\sup_jq'(y_j)<\infty\). The tails in (NP.1) satisfy
\[
\begin{gathered}
q\!\left(\sum_{j=m}^n\lambda_j\xi_j(x)y_j\right)\\
\le M_q p(x)\sum_{j=m}^n|\lambda_j|.
\end{gathered}
\tag{NP.5}
\]
They are Cauchy for every seminorm, so completeness gives
a limit. The same estimate proves continuity of the
resulting map. Passing to the limit in the finite sums
proves \(B\)-linearity.

Composition before or after a \(B\)-nuclear map by a
continuous \(B\)-linear map preserves nuclearity.
For precomposition, compose the \(\xi_j\) and pull back
the one seminorm in (NP.2). For postcomposition, map
the bounded sequence \((y_j)\); its image is bounded.
Finite sums and finite block matrices of nuclear maps
are nuclear: concatenate their finitely many summable
coefficient sequences, use the maximum of the finitely
many controlling seminorms and the finite union of
their bounded sequences.

We will also use a normalization with \(y_j\to0\).
There are numbers \(c_j\ge1\), tending to infinity,
such that
\[
\sum_j c_j|\lambda_j|<\infty.
\tag{NP.6}
\]
To construct them, choose strictly increasing integers
\(N_k\) so that the tail starting at \(N_k\) has sum
at most \(2^{-2k}\), and put \(c_j=2^k\) on
\([N_k,N_{k+1})\); handle the finite initial segment
with \(c_j=1\). The weighted sum of the later blocks
is bounded by \(\sum_k2^{-k}\).
Replace \(\lambda_j,y_j\) by
\(c_j\lambda_j,c_j^{-1}y_j\). Formula (NP.1) is
unchanged, its coefficients remain summable, and
the boundedness of \((y_j)\) proves that the new
\(y_j\) tend to zero for every seminorm.

#### NP.C. Open surjections and null-sequence lifting

We give the needed open-mapping argument.
Increasing seminorms \(p_k\) defining a Fréchet
topology give the complete translation-invariant metric
\[
\begin{gathered}
d(x,y)=\sum_{k\ge1}2^{-k}\\
{}\cdot\min\{1,p_k(x-y)\}.
\end{gathered}
\tag{NP.7}
\]
A sequence is Cauchy for this metric exactly when it is
Cauchy for every \(p_k\); its limit exists by
completeness of the locally convex space.

A complete metric space cannot be a countable union
of closed sets with empty interior. Otherwise choose
successive nonempty closed balls, each contained in
the interior of its predecessor, avoiding the next
closed set, and with radii tending to zero at least
as fast as \(2^{-k}\). Their centres form a Cauchy
sequence. Its limit belongs to every ball and
therefore to none of the alleged covering sets,
a contradiction.

Let \(f:E\to F\) be a continuous linear surjection
between Fréchet spaces. If \(V\) is an absolutely
convex neighbourhood of zero in \(E\), its positive
integer dilates cover \(E\). Thus the closures of
the images \(f(nV)\) cover \(F\). The complete-metric
argument gives interior to one of these closures.
Taking differences puts a neighbourhood of zero
in the closure of \(f(2nV)\). Rescaling shows that
\(\overline{f(V)}\) contains a neighbourhood of zero.

We remove the closure. Given a neighbourhood \(U\)
of zero, choose \(m,\varepsilon>0\) so that
\(\{p_m<\varepsilon\}\subset U\), and set, for
\(n\ge0\),
\[
\begin{gathered}
V_n=\bigl\{x:\\
p_{m+n}(x)<\varepsilon 2^{-n-2}\bigr\}.
\end{gathered}
\tag{NP.8}
\]
Every series with \(x_n\in V_n\) is Cauchy in each
seminorm, and its sum has \(p_m\)-value at most
\(\varepsilon/2<\varepsilon\), hence belongs to \(U\).
Choose open neighbourhoods \(W_n\) of zero in \(F\)
such that
\[
\begin{gathered}
W_n\subset\overline{f(V_n)},\\
W_n\subset\bigl\{y:\\
q_j(y)<2^{-n}\ (1\le j\le n)\bigr\},
\end{gathered}
\tag{NP.9}
\]
where the \(q_j\) define the topology of \(F\).
For \(y_0\in W_0\), density gives \(x_n\in V_n\)
recursively with
\[
\begin{gathered}
y_{n+1}=y_n-f(x_n),\\
y_{n+1}\in W_{n+1}.
\end{gathered}
\tag{NP.10}
\]
The sum \(x=\sum_nx_n\) lies in \(U\), while
\(y_n\to0\) and continuity give \(f(x)=y_0\).
Thus \(W_0\subset f(U)\), proving that \(f\) is open.

**Null-sequence lifting.** If \(y_j\to0\) in \(F\),
there are \(x_j\to0\) in \(E\) with \(f(x_j)=y_j\).
Choose a decreasing neighbourhood basis \(U_k\)
of zero in \(E\). Openness and convergence give
strictly increasing integers \(J_k\ge k\) such that
\(y_j\in f(U_k)\) for all \(j\ge J_k\).
For \(J_k\le j<J_{k+1}\), choose a preimage
\(x_j\in U_k\); choose arbitrary preimages for the
finite initial segment. These preimages tend to zero.
No linear or continuous right inverse has been used.

#### NP.D. One Banach module in place of a Fréchet module

Let
\[
\begin{gathered}
\ell^1(B)=\bigl\{(b_j):\\
\sum_j\|b_j\|_B<\infty\bigr\},\\
\|(b_j)\|_1=\sum_j\|b_j\|_B .
\end{gathered}
\tag{NP.11}
\]
It is a Banach left \(B\)-module with componentwise
action. To check completeness, a Cauchy sequence has
a limit in every coordinate. Bounds on each finite
partial sum pass to these limits. Taking their
supremum gives a summable limiting row, and the
Cauchy tail bound proves convergence in \(\ell^1\).
The module inequality follows by summing the
submultiplicative inequality in \(B\).

Suppose \(v:S\to S\) is \(B\)-nuclear and \(S\) is
Fréchet. Using its representation (NP.1), define
\[
\begin{gathered}
\alpha:S\longrightarrow\ell^1(B),\\
\alpha(x)=(\lambda_j\xi_j(x))_j,\\
\beta:\ell^1(B)\longrightarrow S,\\
\beta((b_j))=\sum_j b_jy_j .
\end{gathered}
\tag{NP.12}
\]
The map \(\alpha\) is continuous by (NP.2).
Inequality (NP.4) and boundedness give
\(q(\beta(b))\le M_q\|b\|_1\) for each \(q\);
completeness gives the series in \(S\).
Thus \(\beta\) is continuous. Both maps are
\(B\)-linear, and
\[
\begin{gathered}
v=\beta\alpha,\\ w:=\alpha\beta,\\
w\in\operatorname{End}_B(\ell^1(B)).
\end{gathered}
\tag{NP.13}
\]
The map \(w\) is \(B\)-nuclear. If \(e_j\) is the
row with \(1_B\) in coordinate \(j\) and zero
elsewhere, then
\[
w(b)=\sum_j\lambda_j
(\xi_j\beta)(b)e_j .
\tag{NP.14}
\]
The \(e_j\) have uniformly bounded norm. Continuity
of \(\beta\) for the single seminorm \(p\) in
(NP.2) gives one constant controlling the norms
of every \(\xi_j\beta\).

There are algebraic inverse isomorphisms
\[
\begin{gathered}
S/(1-v)S\\
\xrightarrow{\ \bar\alpha\ }
\ell^1(B)/(1-w)\ell^1(B),\\
\bar\beta\bar\alpha=1,\qquad
\bar\alpha\bar\beta=1 .
\end{gathered}
\tag{NP.15}
\]
Indeed \(\alpha(1-v)=(1-w)\alpha\) and
\(\beta(1-w)=(1-v)\beta\), so the induced maps
are well defined. Their composites are \(v\)
and \(w\), which equal the identity in the
respective quotients. No closure is taken in
either quotient.

#### NP.E. The finite head and the invertible tail

First let \(S\) be a Banach \(B\)-module. Joint
continuity gives an action bound
\(\|bx\|\le C\|b\|_B\|x\|\).
In a nuclear representation the \(\xi_j\) have
uniformly bounded operator norms and the \(y_j\)
uniformly bounded norms. Consequently
\[
\begin{gathered}
v=v_{\le N}+v_{>N},\\
K_\xi=\sup_j\|\xi_j\|,\\ K_y=\sup_j\|y_j\|,\\
\|v_{>N}\|\\
\le C K_\xi K_y\sum_{j>N}|\lambda_j|.
\end{gathered}
\tag{NP.16}
\]
Choose \(N\) so that this last bound is less
than one. The operator
\[
\begin{gathered}
A=1-v_{>N},\\
A^{-1}=\sum_{k\ge0}v_{>N}^{\,k}
\end{gathered}
\tag{NP.17}
\]
is a continuous \(B\)-linear automorphism.
The series converges in operator norm by the
geometric bound. A norm-Cauchy sequence of
bounded operators has a pointwise limit on
the complete space \(S\), with the same
uniform bounds, and that limit is bounded
and linear. Thus the series defines an
operator; multiplying its finite partial
sums by \(1-v_{>N}\) and letting the final
power tend to zero proves both inverse
identities.

For any \(x\in S\), write \(x=Az\). Then
\[
x=(1-v)z+v_{\le N}z .
\tag{NP.18}
\]
The second term belongs to
\(\sum_{j=1}^N B y_j\). Hence the classes
of \(y_1,\ldots,y_N\) generate the
algebraic cokernel of \(1-v\).

For a Fréchet \(S\), apply this Banach result
to \(w\) on \(\ell^1(B)\) from NP.D.
Formula (NP.14) supplies its nuclear
representation; its finite head has range
in the span of finitely many \(e_j\).
The isomorphism \(\bar\beta\) in (NP.15)
sends their classes to the classes of
the corresponding \(y_j\). We have proved
\[
\begin{gathered}
v:S\to S\text{ is }B\text{-nuclear}\\
\Longrightarrow S/(1-v)S\\
\text{is finite over }B .
\end{gathered}
\tag{NP.19}
\]
Here and below “finite over \(B\)” means
finitely generated as an algebraic module.

#### NP.F. Proof for a surjection and a subnuclear perturbation

First assume that \(u:E\to F\) is \(B\)-nuclear.
Normalize its representation as in NP.B
so that \(y_j\to0\). NP.C lifts this sequence
through \(f\) to a sequence \(x_j\to0\) in \(E\).
In particular \((x_j)\) is bounded. The
convergent relative series
\[
v(x)=\sum_j\lambda_j\xi_j(x)x_j
\tag{NP.20}
\]
is \(B\)-nuclear by NP.B. Continuity gives
\[
\begin{gathered}
fv=u,\\ f-u=f(1-v).
\end{gathered}
\tag{NP.21}
\]
Since \(f\) is onto, it induces a surjection
\[
\begin{gathered}
E/(1-v)E\\
\twoheadrightarrow F/(f-u)E .
\end{gathered}
\tag{NP.22}
\]
Its source is finite over \(B\) by NP.E;
the images of a finite generating list
generate its target.

For a \(B\)-subnuclear \(u\), choose
\(p_G:G\to E\) as in NP.A. Then \(fp_G\)
is surjective and \(up_G\) is nuclear.
The result just proved applies to their
difference, while
\[
\begin{gathered}
\operatorname{im}((f-u)p_G)\\
=\operatorname{im}(f-u),\\
\operatorname{Coker}(fp_G-up_G)\\
=\operatorname{Coker}(f-u).
\end{gathered}
\tag{NP.23}
\]
This proves NP.T.

Subnuclear maps admit the finite-diagram
operations used in this argument. For a
continuous map \(a:E'\to E\) and a
subnuclear presentation \(p_G:G\to E\),
form the closed submodule
\[
\begin{gathered}
G'=\bigl\{(x',g)\in E'\times G:\\
a(x')=p_G(g)\bigr\},\\
p':G'\longrightarrow E',\\
p'(x',g)=x'.
\end{gathered}
\tag{NP.24}
\]
It is Fréchet because it is the kernel
of a continuous map into the Hausdorff
space \(E\); completeness follows by
taking the limit in its defining equality.
The map \(p'\) is onto. The composite
\((ua)p'\) is \(up_G\) composed with
the projection to \(G\), and is nuclear
by NP.B. Thus precomposition preserves
subnuclearity. Postcomposition does so
by the same nuclear composition rule.
For a finite sum, take the fibre product
over the common domain of the finitely
many surjective presentations. It is
again a closed Fréchet submodule with
surjective projection. The lifted maps
are nuclear and their finite sum is
nuclear. Finite block matrices follow
by precomposing with the component
projections and postcomposing with the
component inclusions, then adding the
finitely many entries.

#### NP.G. A finite cokernel need not be Hausdorff

Take
\[
\begin{gathered}
B=C([0,1],\mathbf C),\\
E=F=B,\qquad f=1,\\
s(x)=x,\\ u(b)=(1-s)b .
\end{gathered}
\tag{NP.25}
\]
Use the supremum norm and pointwise
multiplication. The map \(u\) is
\(B\)-nuclear with a single term:
\(\lambda_1=1\), \(\xi_1=\operatorname{id}_B\) as
the identity map of \(B\), and
\(y_1=1-s\).
Its perturbed map \(f-u\) is multiplication
by \(s\). The algebraic cokernel \(B/sB\)
is cyclic, generated by the class of one.

The ideal \(sB\) is not closed. Indeed
\[
g_n(x)=\frac{x}{\sqrt{x+1/n}}\in sB
\tag{NP.26}
\]
converges uniformly to \(\sqrt{x}\).
For \(x\le1/n\) the difference is at most
\(1/\sqrt n\). For \(x\ge1/n\), rationalizing
the difference bounds it by
\(1/(2n\sqrt x)\le1/(2\sqrt n)\).
Thus the uniform difference tends to zero.
But \(\sqrt{x}\notin sB\), since the
necessary quotient \(1/\sqrt{x}\) on
\(x>0\) has no continuous extension
to zero. The zero class of the quotient
therefore is not closed. This is exactly
why NP.T asserts algebraic finite
generation without a Hausdorff conclusion.

#### NP.H. Actual power-series restrictions and their finite tails

For a positive polyradius
\(R=(R_1,\ldots,R_n)\), define
\[
\begin{gathered}
W_R(B)=\\
\biggl\{\sum_{\alpha\in\mathbf N^n}b_\alpha z^\alpha:\\
\sum_\alpha\|b_\alpha\|_B R^\alpha<\infty\biggr\},\\
\|b\|_R=\sum_\alpha\|b_\alpha\|_B R^\alpha .
\end{gathered}
\tag{NP.27}
\]
It is a Banach left \(B\)-module: rescaling
each coordinate by \(R^\alpha\) identifies
its normed coefficient space with
\(\ell^1(\mathbf N^n,B)\), whose completeness
was proved in NP.D. The series converges
absolutely and uniformly on the closed
polydisc of polyradius \(R\). On every
smaller polydisc its derivatives converge
normally, because a fixed derivative
costs a polynomial in \(\alpha\), absorbed
by the strict geometric radius margin.
Thus these are actual analytic series.

Let \(0<r_i<R_i\) for all \(i\).
For the restriction
\(\rho_{R,r}:W_R(B)\to W_r(B)\), put
\[
\begin{gathered}
\lambda_\alpha=(r/R)^\alpha,\\
\xi_\alpha(b)=R^\alpha b_\alpha,\\
y_\alpha(z)=r^{-\alpha}z^\alpha1_B .
\end{gathered}
\tag{NP.28}
\]
These are \(B\)-linear coefficient maps
with norm at most one and a bounded
sequence of target monomials. Coordinate
by coordinate,
\[
\begin{gathered}
\rho_{R,r}(b)\\
=\sum_\alpha\lambda_\alpha\xi_\alpha(b)y_\alpha,\\
\sum_\alpha|\lambda_\alpha|\\
=\prod_{i=1}^n(1-r_i/R_i)^{-1}.
\end{gathered}
\tag{NP.29}
\]
Hence the restriction is \(B\)-nuclear.
For \(n=0\) there is one coefficient
and the statement is the identity
map on the rank-one module \(B\).

The finite head can use the box
\(0\le\alpha_i\le N\). With
\(a_i=r_i/R_i\), its remaining
coefficient weight is exactly
\[
\begin{gathered}
T_N=\prod_i(1-a_i)^{-1}\\
{}-\prod_i\frac{1-a_i^{N+1}}{1-a_i}.
\end{gathered}
\tag{NP.30}
\]
It tends to zero by the finite product
formula. This supplies the explicit
small tail used in NP.E after continuous
finite-matrix precomposition and
postcomposition; the finitely many
operator bounds multiply \(T_N\).

For the connection with ordinary
holomorphic charts, take \(B\) to be
the Banach algebra of scalar functions
continuous on a closed base polydisc
and holomorphic in its interior.
A uniform limit stays holomorphic
by the scalar Cauchy integral formula
on interior coordinate circles, so
this is a Banach algebra.
If a scalar function is holomorphic
on a neighbourhood of a larger product
of that closed base polydisc and a
closed fibre polydisc of radius \(R'\),
its fibre Taylor coefficients belong
to \(B\). Cauchy's coefficient bound
gives, for \(R_i<R'_i\),
\[
\begin{gathered}
\|b_\alpha\|_B\le M(R')^{-\alpha},\\
\sum_\alpha\|b_\alpha\|_B R^\alpha\\
\le M\prod_i(1-R_i/R'_i)^{-1}.
\end{gathered}
\tag{NP.31}
\]
Thus a reserved Cauchy margin places
the actual holomorphic function in
the power-series module (NP.27).
The coefficient estimate is the
ordinary multivariable Cauchy estimate
proved in
Holomorphic functions, Theorems1.2 and2.1;
it is applied uniformly on the compact
closed base polydisc.

For example, suppose \(p:G\to E\) is
a continuous \(B\)-linear surjection,
\(G\) is a finite sum of \(W_R(B)\),
and \(q\) is a continuous \(B\)-linear
map from the corresponding finite
sum of \(W_r(B)\) to \(F\).
Any map \(u:E\to F\) satisfying
\(up=q\rho_{R,r}\) is subnuclear:
the restriction is nuclear by
(NP.28)–(NP.29), and postcomposition
preserves that property. The same
conclusion holds with a continuous
finite matrix before the restriction.
NP.T therefore applies to every
surjection perturbed by such an
induced finite-diagram restriction.

**Exercise.** If \(v\) is a nuclear
endomorphism of a Banach \(B\)-module,
why do the original head vectors
\(y_1,\ldots,y_N\), rather than
only their images under \(A^{-1}\),
generate the cokernel?

**Solution.** For arbitrary \(x\),
take \(z=A^{-1}x\) in (NP.18).
Its class is the class of
\(\sum_{j\le N}\lambda_j\xi_j(z)y_j\).
Those coefficients belong to \(B\),
so the stated original head
vectors generate.

For an application to analytic Čech complexes, the perturbation
map into the cycle module must itself satisfy the nuclear or
subnuclear hypothesis. A displayed nuclear expansion in an
ambient term supplies that hypothesis for the cycle module
only after its coefficient maps and vectors have been
constructed there. The compatible analytic coefficient
base changes are separate hypotheses of such an application.

#### NP.I. A diagonal model and a multivariable tail

On \(\ell^1(B)\), consider the actual bounded operator
\[
\begin{gathered}
v(b_1,b_2,b_3,\ldots)\\
=(b_1,\tfrac12b_2,\tfrac14b_3,\ldots).
\end{gathered}
\tag{NP.32}
\]
It is \(B\)-nuclear: use the coordinate maps,
the unit rows \(e_j\), and the summable weights
\(\lambda_j=2^{1-j}\). Its tail after the first
two coordinates has operator norm \(1/4\).
The sum of the tail weights is \(1/2\), so
the finite-head criterion in NP.E also applies
using its summable bound.

The range of \(1-v\) consists exactly of the
rows whose first coordinate is zero. For the
reverse inclusion, divide coordinate \(j\ge2\)
by \(1-2^{1-j}\); these denominators are at
least \(1/2\), so the resulting row remains
in \(\ell^1(B)\). Hence projection onto the
first coordinate gives the algebraic isomorphism
\[
\ell^1(B)/(1-v)\ell^1(B)\simeq B.
\tag{NP.33}
\]
The finite-head proof supplies two generators;
the explicit diagonal calculation shows that
only the first is necessary.

For the two-variable analytic restriction,
take \(R=(2,3)\), \(r=(1,1)\). Formula (NP.29)
has total coefficient weight \(3\). Its head
with \(0\le\alpha_1,\alpha_2\le2\) has weight
\[
\begin{gathered}
(1+\tfrac12+\tfrac14)(1+\tfrac13+\tfrac19)\\
=\tfrac{91}{36},\\
T_2=3-\tfrac{91}{36}=\tfrac{17}{36}.
\end{gathered}
\tag{NP.34}
\]
Both fibre radii enter the product. A
one-variable tail bound would lose the
second coefficient direction.

![Relative nuclear factorization, a diagonal finite cokernel and the exact two-variable restriction tail](assets/nuclear-perturbation.png)

**Figure NP.1.** Panel A is the pair of maps
in (NP.12) and their inverse algebraic
cokernel maps (NP.15). Panel B is the
diagonal example (NP.32)–(NP.33), with its
exact tail norm and its actual cokernel.
Panel C is the two-variable coefficient
weight in (NP.34). These are algebraic
diagrams, with no geometric scale.

Figure SVG ·
exact diagram data ·
reproducible source ·
figure PNG.

Free source:
[Demailly, Complex Analytic and Differential Geometry,
ChapterIX, §5, Theorem5.42 and Lemma5.43,
author's freely accessible manuscript](https://www-fourier.univ-grenoble-alpes.fr/~demailly/manuscripts/agbook.pdf).


### 5.92. Nuclear tensor exactness and closed coefficient targets

A cycle space is a closed subspace of a term of a complex.
Its topology matters when a restriction map is expressed by a
convergent nuclear series. This section proves the tensor and
closed-target statements with that topology retained.

All vector spaces are complex. All tensor products without a
coefficient subscript are over \(\mathbf C\). A Fréchet space
has its complete Hausdorff countable seminorm topology.
We use the complete open mapping proof in
NP.C
and the absolutely convergent series and Banach factorization
in NP.B--NP.D.
The scalar holomorphic input is the
polydisc Cauchy and Taylor proof.

#### NT.A. Separation and complete quotients

We first give the separation argument used below. If \(p\) is
a real sublinear function and a real linear functional \(l\)
on a subspace satisfies \(l\le p\), it extends to the whole
space with the same bound. Indeed, to add a vector \(x\), its
value \(c\) can be chosen in the interval
\[
\begin{gathered}
\sup_{m}\{l(m)-p(m-x)\}\le c,\\
c\le\inf_{m}\{p(m+x)-l(m)\}.
\end{gathered}
\tag{NT.1}
\]
The interval is nonempty: for \(m,n\) in the old subspace,
\(l(m)+l(n)\le p(m+n)\), while
\(p(m+n)\le p(m-x)+p(n+x)\).
The inequalities in NT.1 imply domination on every
\(m+tx\), by positive rescaling when \(t>0\) and negative
rescaling when \(t<0\). The unions of chains of dominated
extensions are dominated extensions. The maximal-element
principle therefore gives an extension to the whole space.

For a complex linear \(l\) satisfying \(|l|\le p\), apply
this argument to its real part, with \(p\) a seminorm.
If \(a\) is the real extension, set
\(L(x)=a(x)-i a(ix)\). Then \(L\) is complex linear,
extends \(l\), and is dominated by \(p\): choose a unit scalar
\(\theta\) with \(\theta L(x)=|L(x)|\), and use
\(|L(x)|=a(\theta x)\le p(x)\).
In particular,
\[
p(x)=\sup_{\substack{L\text{ complex linear}\\ |L|\le p}}|L(x)|.
\tag{NT.2}
\]
For \(p(x)>0\), the functional \(cx\mapsto c p(x)\) on
the line through \(x\) supplies the lower bound; the case
\(p(x)=0\) is immediate. This proves both norm recovery and
the extension of a functional bounded by a restricted
seminorm on a subspace.

Here are the completeness details for quotients. If \(S\)
is a closed subspace of a Fréchet space \(E\), with increasing
defining seminorms \(p_j\), give \(E/S\) the seminorms
\[
\bar p_j([x])=\inf_{s\in S}p_j(x-s).
\tag{NT.3}
\]
Their open balls are the images of the corresponding open
balls in \(E\); hence this is the quotient topology. It is
Hausdorff because \(S\) is closed. Given a Cauchy sequence
of classes, choose a subsequence \(u_j\) with
\(\bar p_j(u_{j+1}-u_j)<2^{-j}\), and representatives
\(v_j\) of these differences with \(p_j(v_j)<2^{-j}\).
The series \(\sum_j v_j\) converges in every seminorm and
therefore in \(E\). Its class, added to \(u_1\), is the
limit of the subsequence. The original Cauchy sequence has
the same limit. Thus \(E/S\) is Fréchet. The metric
\(\sum_j2^{-j}\min(1,\bar p_j(u-v))\) expresses this
completeness in the countable seminorm topology.
Closed subspaces are complete by taking their limits in
\(E\).

Write \(E_p\) for the Banach completion of \(E/\ker p\).
The completion of \(S/\ker(p|_S)\) identifies isometrically
with the closure \(S_p\) of its image in \(E_p\).
Distance to \(S\) and distance to \(S_p\) are equal.
Consequently
\[
(E/S)_{\bar p}=E_p/S_p.
\tag{NT.4}
\]
Indeed the map is isometric on the dense image of \(E/S\),
and the right side is complete by the preceding series
argument with one norm.

#### NT.B. The two tensor topologies

For continuous seminorms \(p,q\) on \(E,F\), define on their
algebraic tensor product
\[
\begin{gathered}
(p\otimes_\pi q)(t)\\
 =\inf\sum_{\nu=1}^N p(x_\nu)q(y_\nu),\\
(p\otimes_\varepsilon q)(t)\\
 =\sup_{\substack{|l|\le p\\ |m|\le q}}
       |(l\otimes m)(t)|.
\end{gathered}
\tag{NT.5}
\]
The infimum runs over all finite representations
\(t=\sum_{\nu=1}^N x_\nu\otimes y_\nu\).
The first seminorm is projective; the second is injective.
Both are seminorms by combining finite representations or
using the supremum, and
\(p\otimes_\varepsilon q\le p\otimes_\pi q\).
Their Hausdorff completions are denoted
\(E\widehat\otimes_\pi F\) and
\(E\widehat\otimes_\varepsilon F\).
For Fréchet factors they are Fréchet, since paired increasing
seminorms are cofinal and their metric completion is complete.

These topologies are Hausdorff already on \(E\otimes F\).
To see this, continuous functionals separate points by NT.2.
Their restrictions span the full dual of any finite-dimensional
subspace: otherwise a nonzero vector in that subspace would
annihilate every restriction, contradicting separation.
For a nonzero finite tensor, use such coordinate functionals
on the spans of its factors to obtain a nonzero
\((l\otimes m)(t)\).

Continuous linear maps in both factors give continuous maps
of either completed tensor product. This follows directly
from NT.5: pull back the bounding seminorms and their
dominated functionals. The projective topology has the
following universal property. A jointly continuous bilinear
map \(b:E\times F\to G\), with \(G\) complete, extends
uniquely to \(E\widehat\otimes_\pi F\). For every seminorm
\(r\) on \(G\), continuity and rescaling give
\(r(b(x,y))\le C p(x)q(y)\); summing a finite
representation gives the required bound, and completeness
extends the map. Conversely the tensor map
\((x,y)\mapsto x\otimes y\) is jointly continuous by NT.5.
The same reasoning identifies continuous multilinear maps
and proves associativity and symmetry of completed
projective tensor products: the maps on elementary tensors
and their inverse parenthesizations are bounded for each
product seminorm, and agree on a dense subspace.

The Banach completion for \(p\otimes_\pi q\) is
\(E_p\widehat\otimes_\pi F_q\). One can first divide out
the zero seminorm factors. Bounded bilinear forms on the
resulting normed factors extend uniquely to their Banach
completions, with the same bound, by successive norm limits.
NT.2, applied to the projective normed tensor space,
recovers its norm from these forms. Its norm therefore
agrees with the projective norm after completing the factors;
finite tensors in the dense factors are dense in the latter
space by NT.5.

Two facts about closed embeddings and quotients will be
needed. First, if \(S\subset E\) has the subspace topology,
NT.2 extends every functional bounded by \(p|_S\) to one
bounded by \(p\) on \(E\). Thus
\[
\begin{gathered}
(p|_S\otimes_\varepsilon q)(t)
 =(p\otimes_\varepsilon q)(t),\\
t\in S\otimes F.
\end{gathered}
\tag{NT.6}
\]
If \(S,E,F\) are Fréchet and \(S\) is closed, completion
gives a topological embedding for \(\widehat\otimes_\varepsilon\)
with closed image. The image is closed because a convergent
sequence in it is Cauchy in the identical source seminorms
and the source is complete.

Second, for \(Q=E/S\), the algebraic quotient
\((E\otimes F)/(S\otimes F)\) is \(Q\otimes F\).
For example, write a finite tensor using linearly independent
second factors; its image is zero exactly when every first
coefficient belongs to \(S\). Moreover the quotient
seminorm of \(p\otimes_\pi q\) is
\(\bar p\otimes_\pi q\). One inequality follows by mapping a
representation to \(Q\). For the other, lift the finitely
many first coefficients of a representation in \(Q\otimes F\),
with \(p(x_\nu)\) arbitrarily close to
\(\bar p([x_\nu])\). This makes its cost arbitrarily close
to the quotient cost, also when a coefficient has zero
quotient seminorm.

Let \(T\) be the closure of \(S\otimes F\) in
\(E\widehat\otimes_\pi F\). Taking the infimum over this
closure gives the same quotient seminorm as taking it over
\(S\otimes F\), by continuity. NT.A makes the quotient by
\(T\) complete. Its dense subspace \(Q\otimes F\) has
exactly the preceding projective topology, so uniqueness
of completion gives
\[
\begin{gathered}
(E\widehat\otimes_\pi F)/T
 \simeq Q\widehat\otimes_\pi F,\\
u:E\widehat\otimes_\pi F\longrightarrow
           Q\widehat\otimes_\pi F,\\
\ker u=T.
\end{gathered}
\tag{NT.7}
\]
In particular the last map is onto and open. No nuclearity
is needed in NT.7; the kernel is a closure at this stage.

#### NT.C. Nuclear maps, Hilbert factors and closed targets

A scalar nuclear map \(v:E\to F\) between Fréchet spaces
means a series
\[
\begin{gathered}
v(x)=\sum_j\lambda_j l_j(x)y_j,\\
\sum_j|\lambda_j|<\infty,\\
|l_j(x)|\le p(x),\\
\sup_j q(y_j)<\infty.
\end{gathered}
\tag{NT.8}
\]
Here \(q\) ranges over all continuous target seminorms.
This is NP.1 for scalar coefficients. NP.B proves convergence,
continuity and stability under continuous precomposition and
postcomposition. NP.D factors it through \(\ell^1\).
Conversely, a norm-convergent nuclear representation on
Banach spaces has this form after absorbing the functional
and vector norms into \(\lambda_j\).
No identification of a tensor space with a space of
operators is required here.

A Banach nuclear map also factors through a Hilbert space.
Normalize its representation with
\(\|l_j\|,\|y_j\|\le1\), put
\(a_j=|\lambda_j|^{1/2}\), and write
\(\lambda_j=\theta_j a_j^2\), with \(|\theta_j|\le1\).
The maps
\[
\begin{gathered}
u(x)=(a_j l_j(x))_j\in\ell^2,\\
w(c)=\sum_j\theta_j a_j c_j y_j,\\
v=wu,\\
\|u\|,\|w\|\le\Big(\sum_j|\lambda_j|\Big)^{1/2}
\end{gathered}
\tag{NT.9}
\]
are bounded by the sum of squares and Cauchy--Schwarz.
For completeness, the latter inequality follows by
minimizing the nonnegative quadratic
\(\sum|a_j-tb_j|^2\) first for finite sequences and then
taking limits. The same finite partial-sum bounds show
that a Cauchy sequence in \(\ell^2\) has coordinatewise
limits with square-summable norm and converges in norm.
Thus the intermediate space in NT.9 is complete.

Every closed subspace \(H_1\) of a Hilbert space has a
norm-one orthogonal projection (norm zero for the zero
subspace). To prove this, minimize \(\|x-h\|\) over \(h\in H_1\).
The parallelogram identity makes a minimizing sequence
Cauchy, so it has a minimizer. Varying the minimizer by
each real and imaginary multiple of a vector in \(H_1\)
shows orthogonality. Uniqueness, linearity and the norm
bound follow by the orthogonal decomposition.

**Binuclear lemma.** If \(f:E\to F\) and \(g:F\to G\)
are Banach nuclear maps and \(gf(E)\subset T\) for a
closed subspace \(T\subset G\), then \(gf:E\to T\)
is nuclear. If \(gf\) vanishes on a closed subspace
\(S\subset E\), its induced map \(E/S\to G\) is nuclear.

For the first assertion factor \(g=wu\) through a Hilbert
space \(H\), put \(H_1=w^{-1}(T)\), and insert its orthogonal
projection \(P\):
\[
gf=w|_{H_1}\,Puf:E\longrightarrow T.
\tag{NT.10}
\]
Here \(uf(E)\subset H_1\), and \(Puf\) is nuclear because
\(f\) is nuclear. For the second assertion factor \(f=wu\)
through \(H\), put \(H_1=\overline{u(S)}\), and let \(P\)
project to \(H_1^\perp\). Then \(gw\) vanishes on \(H_1\),
so \(gf=gwPu\). The map \(Pu\) vanishes on \(S\), and
its induced map \(E/S\to H_1^\perp\) is bounded by the
quotient norm. Postcompose it with \(gw|_{H_1^\perp}\),
which is nuclear by nuclearity of \(g\). This proves the
second assertion. Both uses require two nuclear arrows;
no assertion about a single arbitrary Banach nuclear
arrow into a closed subspace is being made.

#### NT.D. Nuclear Fréchet spaces and their subquotients

A Fréchet space \(E\) is **nuclear** if it has increasing
defining seminorms \(p_j\) such that every canonical map
\[
\begin{gathered}
E_{p_{j+1}}\longrightarrow E_{p_j}\\
\text{is Banach nuclear.}
\end{gathered}
\tag{NT.11}
\]
Passing to a cofinal subsequence retains this property,
since compositions with bounded maps retain nuclearity.
The space itself is the inverse limit of its Banach
completions. In fact, for compatible \(z_j\in E_{p_j}\)
choose \(x_j\in E\) approximating \(z_j\) in \(p_j\)
within \(1/j\). For each fixed \(k\), their images in
\(E_{p_k}\) converge to \(z_k\). They are Cauchy in
every defining seminorm; completeness supplies the
unique element of \(E\) with these images.

If \(E\) is nuclear, every continuous map from \(E\)
to a Banach space is nuclear: it factors through one
\(E_{p_j}\), and inserting \(E_{p_{j+1}}\) gives a
nuclear arrow. It follows that for a Fréchet target \(F\),
\[
\begin{gathered}
v:E\to F\text{ is nuclear}\\
\Longleftrightarrow\
v=w u,\\
E\xrightarrow{u}M\xrightarrow{w}F,\\
M\text{ Banach},\\
u,w\text{ continuous}.
\end{gathered}
\tag{NT.12}
\]
The forward implication is NP.D; the reverse implication
is the preceding observation followed by continuous
postcomposition.

Closed subspaces and Hausdorff quotients of nuclear
Fréchet spaces are nuclear. Indeed, with \(S_j\subset E_{p_j}\)
as in NT.A, the map
\(S_{j+2}\to E_{p_{j+1}}\to E_{p_j}\) is the composite
of two nuclear maps (restrict the first to \(S_{j+2}\)).
Its image lies in \(S_j\). NT.C makes
\(S_{j+2}\to S_j\) nuclear. Similarly the binuclear map
\[
\begin{gathered}
E_{p_{j+2}}\longrightarrow E_{p_{j+1}}\\
\longrightarrow E_{p_j}/S_j
\end{gathered}
\tag{NT.13}
\]
vanishes on \(S_{j+2}\); NT.C and NT.4 make the induced
map between the corresponding quotient completions
nuclear. The even-indexed seminorms now satisfy NT.11
on both \(S\) and \(E/S\).

A countable product of nuclear Fréchet spaces is nuclear.
For factors \(E^k\) with seminorms \(p_j^k\), use
\[
q_j((x_k))=\max_{k\le j}p_j^k(x_k).
\tag{NT.14}
\]
Its Banach completion is the finite product of the indicated
Banach completions. The connecting map is the finite sum
of their nuclear connecting maps, composed with coordinate
projections and inclusions; the extra coordinate maps to
zero. Finite sums are nuclear by concatenating their
absolutely summable representations. Countable inverse
limits with continuous bonding maps are closed subspaces
of these products, since the compatibility equations are
closed. They too are nuclear.

Here is the precise closed-target consequence used later.
If \(E\) is nuclear and \(v:E\to F\) is nuclear, then
\(v:E\to T\) is nuclear whenever \(T\subset F\) is closed
and contains its image. Also \(v:E/S\to F\) is nuclear
if \(S\subset\ker v\) is closed. To prove these assertions,
factor \(v=wu:E\to M\to F\) through a Banach space.
In the first case use the Banach subspace \(w^{-1}(T)\).
In the second use \(M/\overline{u(S)}\). The induced
maps are continuous for the subspace or quotient
topologies. Both \(E\) and \(E/S\) are nuclear, so NT.12
applies. These are scalar nuclear assertions.

#### NT.E. Equality of the tensor topologies

Let \(a:E_P\to E_p\) be a nuclear Banach connecting map,
with a representation
\(a=\sum_j\lambda_j l_j\otimes x_j\) and
\(\|l_j\|,\|x_j\|\le1\). For a finite tensor \(t\),
NT.2 gives
\[
\begin{gathered}
q((l_j\otimes1)t)\\
 =\sup_{|m|\le q}|(l_j\otimes m)(t)|\\
\le (P\otimes_\varepsilon q)(t).
\end{gathered}
\tag{NT.15}
\]
The series for \((a\otimes1)t\) consequently converges
in \(E_p\widehat\otimes_\pi F_q\), with bound
\[
\begin{gathered}
(p\otimes_\pi q)(t)\\
 \le\Big(\sum_j|\lambda_j|\Big)
          (P\otimes_\varepsilon q)(t).
\end{gathered}
\tag{NT.16}
\]
To check that its sum is the image of \(t\), expand each
of the finitely many first factors of \(t\) by the nuclear
series for \(a\). The equality holds in the Banach
projective completion, whose norm on these finite
tensors is the original seminorm by NT.B.

If \(E\) is nuclear, each \(p=p_j\) has such a
\(P=p_{j+1}\). NT.16 and the reverse elementary
inequality from NT.5 prove equality of the projective
and injective locally convex topologies. Thus, if either
factor is nuclear,
\[
E\widehat\otimes_\pi F
 \simeq E\widehat\otimes_\varepsilon F.
\tag{NT.17}
\]
We then write \(\widehat\otimes\) without a decoration.
This proof uses series and seminorm bounds; it assumes
no injectivity theorem for a map from tensors to operators.

If both factors are nuclear, their completed tensor
product is nuclear. On Banach completions, the tensor
of two nuclear maps has the double representation
\[
\begin{gathered}
(a\otimes b)(t)\\ =
 \sum_{j,k}\lambda_j\mu_k u_{jk}(t),\\
u_{jk}(t)=c_{jk}(t)(x_j\otimes y_k),\\
c_{jk}(t)=(l_j\otimes m_k)(t),\\
\sum_{j,k}|\lambda_j\mu_k|
\\ =\Big(\sum_j|\lambda_j|\Big)
  \Big(\sum_k|\mu_k|\Big).
\end{gathered}
\tag{NT.18}
\]
The functionals have norm at most one for the projective
norm, as do the elementary target vectors. Absolute
convergence proves the formula first on finite tensors
and then on the completion. Applying it to paired
defining seminorms proves NT.11 for the tensor product.

#### NT.F. Strict tensor exactness

**Theorem NT.T.** Let
\(0\to E_1\to E_2\to E_3\to0\) be an exact sequence
of Fréchet spaces and continuous maps. If \(E_2\)
or \(F\) is nuclear, then
\[
\begin{gathered}
0\longrightarrow E_1\widehat\otimes F
 \longrightarrow E_2\widehat\otimes F\\
\longrightarrow E_3\widehat\otimes F
 \longrightarrow0
\end{gathered}
\tag{NT.19}
\]
is exact, its first map is a closed topological embedding,
and its last map is an open surjection.

By NP.C, \(E_1\) has exactly the closed subspace topology
of the kernel in \(E_2\), and \(E_3\) is its topological
quotient. If \(E_2\) is nuclear, so are its subspace
\(E_1\) and quotient \(E_3\), by NT.D. If \(F\) is
nuclear, every tensor here has a nuclear factor directly.
Thus NT.17 applies to every term of NT.19.
NT.6 makes its first map a closed embedding. NT.7 says
that the last map is an open surjection whose kernel is
the closure of \(E_1\otimes F\). The closed image of the
first map is precisely that closure, since its algebraic
tensors are dense in its complete source. This proves
all assertions.

The tensor product with a nuclear factor commutes with
countable products of Fréchet spaces, whenever either
that fixed factor is nuclear or every product factor is
nuclear. For the injective topology, a defining seminorm
on a product involves only finitely many coordinates.
For a finite product with its maximum seminorm, its dual
unit ball consists of coordinate functionals whose norms
have sum at most one: restriction gives these functionals;
the reverse bound and equality follow by choosing each
coordinate vector almost maximizing its functional,
with phases chosen to align. NT.5 then makes the
injective tensor seminorm the maximum of the coordinate
tensor seminorms. Finite-coordinate tensors are dense
in the product of completed tensors, by approximating
each of the finitely many relevant coordinates.
Completion therefore gives
\[
E\widehat\otimes\prod_k F_k
 \simeq\prod_k(E\widehat\otimes F_k).
\tag{NT.20}
\]
NT.D makes the product nuclear in the second alternative;
NT.17 translates the preceding injective proof into the
stated tensor notation.

#### NT.G. Analytic coefficient spaces

On a polydisc \(D_R\subset\mathbf C^n\), let \(A_r\)
be the Banach space of holomorphic functions on \(D_r\)
continuous on \(\overline D_r\), with its supremum norm.
Here \(0<r_i<R_i\) in every coordinate. These spaces are
complete: a uniformly Cauchy sequence has a continuous
limit on the closed polydisc, and the Cauchy integral
formula on every smaller polydisc makes its limit
holomorphic inside.

The completion of \(\mathcal O(D_R)\) for the
\(\overline D_r\) supremum seminorm is \(A_r\).
For density, replace \(g(z)\in A_r\) by \(g(sz)\) with
\(s<1\) tending to one. Uniform continuity on the closed
polydisc gives uniform convergence; the Taylor series
of \(g(sz)\) converges uniformly there and its polynomials
belong to \(\mathcal O(D_R)\).

For \(r_i<s_i<R_i\), the Cauchy coefficient bound gives
the Banach nuclear restriction
\[
\begin{gathered}
A_s\longrightarrow A_r,\\
g\longmapsto\sum_\alpha\lambda_\alpha\xi_\alpha(g)y_\alpha,\\
\lambda_\alpha=(r/s)^\alpha,\\
\xi_\alpha(g)=s^\alpha a_\alpha(g),\\
y_\alpha=r^{-\alpha}z^\alpha,\\
|\xi_\alpha(g)|\le\|g\|_{A_s},\\
\|y_\alpha\|_{A_r}=1,\\
\sum_\alpha(r/s)^\alpha\\
 =\prod_{i=1}^n(1-r_i/s_i)^{-1}.
\end{gathered}
\tag{NT.21}
\]
For coefficients on the boundary of \(D_s\), first apply
Cauchy on strictly smaller radii and then let the radii
increase to \(s\). A sequence of closed polydiscs exhausting
\(D_R\) and NT.21 prove that \(\mathcal O(D_R)\), with
its compact-open topology, is nuclear.

For every open \(U\subset\mathbf C^n\),
\(\mathcal O(U)\) is likewise nuclear Fréchet.
Completeness for its countable compact-open seminorms
follows from the same local Cauchy argument; a countable
compact exhaustion is obtained from rational boxes with
closures in \(U\). Cover \(U\) by countably many relatively
compact polydiscs \(D_j\). Restriction identifies
\(\mathcal O(U)\) with the closed subspace given by
\[
\begin{gathered}
(g_j)\in\prod_j\mathcal O(D_j),\\
g_j|_{D_j\cap D_k}=g_k|_{D_j\cap D_k}.
\end{gathered}
\tag{NT.22}
\]
These equations hold for every pair \(j,k\).
They are closed because restriction to each
overlap is continuous into the Hausdorff compact-open
space on that overlap. Compatible functions glue
pointwise, and are holomorphic locally on the cover.
The subspace topology is the compact-open topology:
each compact subset of \(U\) has a finite cover by
smaller compact subsets contained in the \(D_j\).
NT.D now proves the assertion.

We also need a nuclear map between whole Fréchet
coefficient algebras. If \(D_r\Subset D_R\), choose
vectors of radii \(r<s<t<R\). The restriction
\(\rho:\mathcal O(D_R)\to\mathcal O(D_r)\) has the
scalar nuclear representation
\[
\begin{gathered}
\rho(g)=\sum_\alpha\lambda_\alpha\xi_\alpha(g)y_\alpha,\\
\lambda_\alpha=(s/t)^\alpha,\\
\xi_\alpha(g)=t^\alpha a_\alpha(g),\\
y_\alpha=s^{-\alpha}z^\alpha.
\end{gathered}
\tag{NT.23}
\]
The functionals are bounded by the single compact
seminorm \(\sup_{\overline D_t}|g|\). The target
monomials form a bounded family for every compact
seminorm on \(D_r\); in fact their suprema on the
whole \(D_r\) are at most one. The weights sum to
\(\prod_i(1-s_i/t_i)^{-1}\). Thus this is nuclear
as a map of Fréchet spaces, not only separately for
each target seminorm.

#### NT.H. Closed cycles and cohomology

Let \(C^\bullet\) be a complex of nuclear Fréchet spaces
with continuous differentials. Suppose every image is
closed, or equivalently every cohomology quotient is
Hausdorff. Equivalence holds because
\(Z^q=\ker d^q\) is closed and
\(B^q=\operatorname{im}d^{q-1}\subset Z^q\):
being closed in this closed subspace is the same as
being closed in \(C^q\).
The spaces \(Z^q,B^q,H^q\) are nuclear Fréchet by NT.D.
The exact sequences
\[
\begin{gathered}
0\to Z^q\to C^q\\
\to B^{q+1}\to0,\\
0\to B^q\to Z^q\to H^q\to0
\end{gathered}
\tag{NT.24}
\]
are strict by NP.C. For any Fréchet space \(V\),
NT.T applied to both sequences shows that the cycles
and boundaries of \(V\widehat\otimes C^\bullet\) are
the embedded spaces \(V\widehat\otimes Z^q\) and
\(V\widehat\otimes B^q\). In particular its images
are closed, and
\[
H^q(V\widehat\otimes C^\bullet)
 \simeq V\widehat\otimes H^q(C^\bullet)
\tag{NT.25}
\]
topologically. To check the cycle assertion explicitly,
factor \(d^q\) as the surjection onto \(B^{q+1}\)
followed by its closed inclusion into \(C^{q+1}\);
both maps retain exactly these properties after tensoring.
Then use the second sequence of NT.24 for the quotient.
No boundedness of the complex or passage to a fibre is
needed for this degreewise assertion.

#### NT.I. Relative nuclearity into a closed coefficient target

Let \(A,B\) be commutative unital nuclear Fréchet
algebras, with jointly continuous multiplication, and
let \(\rho:A\to B\) be a continuous unital homomorphism.
Modules have jointly continuous unital actions.
For nuclear Fréchet \(A\)-modules \(E,F\), define
their balanced tensor by the actual cokernel
\[
\begin{gathered}
E\widehat\otimes_A F
 =(E\widehat\otimes F)/\operatorname{im}d,\\
d:E\widehat\otimes A\widehat\otimes F\\
       \longrightarrow E\widehat\otimes F,\\
d(x\otimes a\otimes y)\\
 =ax\otimes y-x\otimes ay.
\end{gathered}
\tag{NT.26}
\]
The quotient may be non-Hausdorff: no closure is inserted
in its definition. If it is Hausdorff, \(\operatorname{im}d\)
is closed, and NT.A and NT.D make it nuclear Fréchet.
All actions and maps in NT.26 extend by the projective
universal property and associativity in NT.B.

For a Fréchet algebra \(A\), an \(A\)-nuclear map
means a series as in NP.1, now with continuous
\(A\)-linear \(\xi_j:E\to A\). Equicontinuity means
that for every seminorm \(p\) on \(A\) there is a
seminorm \(q\) on \(E\), independent of \(j\), with
\(p(\xi_j(x))\le q(x)\). The target vectors are bounded
and the scalar weights are absolutely summable.
Joint continuity of the action gives, for each target
seminorm \(r\), a bound
\(r(ay)\le C p(a)q(y)\). This proves convergence and
continuity of the series and stability under continuous
module maps by exactly the seminorm summation in NP.B.
Here **\(A\)-subnuclear** means that precomposition
with a continuous \(A\)-linear surjection from a
nuclear Fréchet \(A\)-module makes the map \(A\)-nuclear.
The requirement that this auxiliary module be nuclear
is part of the definition in this paragraph.

**Closed-target base-change theorem NT.CB.**
Let \(E,F\) be nuclear Fréchet \(A\)-modules,
\(F_1\subset F\) a closed \(A\)-submodule, and
\(f:E\to F\) an \(A\)-subnuclear map whose image lies
in \(F_1\). Denote the induced map by \(f_1:E\to F_1\).
Assume the following precise coefficient hypotheses:
\(\rho:A\to B\) is scalar nuclear;
\(E_B=B\widehat\otimes_A E\) and
\(F_B=B\widehat\otimes_A F\) are Hausdorff; and
the natural map
\[
\begin{gathered}
T=B\widehat\otimes_A F_1
 \longrightarrow F_B\\
\text{is a closed embedding.}
\end{gathered}
\tag{NT.27}
\]
Then \(1_B\widehat\otimes_A f_1:E_B\to T\)
is \(B\)-subnuclear, with a nuclear Fréchet auxiliary
module. In particular NT.CB applies to a cycle
submodule when NT.27 has been proved for that module.

First suppose \(f\) itself is \(A\)-nuclear.
Write
\[
\begin{gathered}
f(x)=\sum_j\lambda_j\xi_j(x)y_j,\\
\rho(a)=\sum_k\mu_k\eta_k(a)b_k.
\end{gathered}
\tag{NT.28}
\]
The second line is a scalar nuclear representation.
The natural map \(h_F:E\to F_B\), \(x\mapsto[1\otimes f(x)]\),
has the representation
\[
\begin{gathered}
h_F(x)=\sum_{j,k}\lambda_j\mu_k v_{jk}(x),\\
v_{jk}(x)=c_{jk}(x)[b_k\otimes y_j],\\
c_{jk}(x)=(\eta_k\circ\xi_j)(x).
\end{gathered}
\tag{NT.29}
\]
Indeed the balancing relation replaces
\([1\otimes\xi_j(x)y_j]\) by
\([\rho(\xi_j(x))\otimes y_j]\).
The double weights are summable. The composed
functionals are equicontinuous: the \(\eta_k\) have one
seminorm bound on \(A\), and the \(\xi_j\) have a uniform
bound for that seminorm. The target vectors are bounded,
since the tensor map and quotient map are continuous
and the two factor families are bounded. Absolute
seminorm convergence justifies the substitutions and
the double sum. Thus \(h_F\) is scalar nuclear.

The map \(h:E\to T\), \(x\mapsto[1\otimes f_1(x)]\),
is continuous. Under NT.27 its composition with the
closed embedding is \(h_F\). NT.D therefore makes
\(h\) scalar nuclear. For a subnuclear \(f\), take its
specified nuclear auxiliary module \(G\) and surjection
\(p:G\to E\). The same argument makes \(hp:G\to T\)
scalar nuclear. It vanishes on the closed kernel of
\(p\); NT.D descends its nuclearity to \(G/\ker p\),
which is topologically \(E\) by NP.C. This proves the
same scalar nuclear assertion for \(h\).

Finally write \(h(x)=\sum_j\lambda_j l_j(x)t_j\).
On \(P=B\widehat\otimes E\), the maps
\[
\begin{gathered}
\zeta_j(b\otimes x)=b\,l_j(x)\in B,\\
k(b\otimes x)=b\,h(x)\\
 =\sum_j\lambda_j\zeta_j(b\otimes x)t_j
\end{gathered}
\tag{NT.30}
\]
are \(B\)-linear. If \(|l_j|\le p\), then for every
seminorm \(q\) on \(B\),
\(q(\zeta_j(z))\le(q\otimes_\pi p)(z)\), first
on finite tensors and then on the completion.
Thus the \(\zeta_j\) are equicontinuous into \(B\);
NT.30 makes \(k:P\to T\) \(B\)-nuclear.
The natural map \(P\to E_B\) is a continuous
\(B\)-linear surjection by the definition of the
balanced quotient. It is a map between nuclear Fréchet
spaces by the Hausdorff assumption. On elementary
tensors, and hence by continuity on \(P\),
\[
\begin{gathered}
f_B=1_B\widehat\otimes_A f_1,\\
p:P\to E_B,\qquad k=f_Bp.
\end{gathered}
\tag{NT.31}
\]
This is the required \(B\)-subnuclear presentation.

The coefficient hypothesis NT.27 is essential to this
argument. Scalar tensor exactness NT.T concerns
\(\widehat\otimes_{\mathbf C}\); it does not assert
exactness of an arbitrary balanced tensor
\(\widehat\otimes_A\). For geometric applications,
closed-target coefficient exactness and the required
analytic restriction identifications must be proved
for the modules in question.

#### NT.J. A strict coefficient example

For a nuclear Fréchet \(A\)-module \(M\),
\(A\widehat\otimes_A M\simeq M\), with its actual
topology. The action has section \(m\mapsto1\otimes m\).
The continuous map \(a\otimes m\mapsto1\otimes a\otimes m\)
to \(A\widehat\otimes A\widehat\otimes M\) has
boundary \(a\otimes m-1\otimes am\) under NT.26.
Thus the kernel of the action is exactly the image of
the relation map, on completed tensors as well.
For a coefficient homomorphism \(A\to B\), the same
calculation with
\(b\otimes a\mapsto b\otimes a\otimes1\) proves
\[
B\widehat\otimes_A A^m\simeq B^m.
\tag{NT.32}
\]
In this calculation the boundary is
\(b\rho(a)\otimes1-b\otimes a\), so its negative is
the identity minus the section followed by the action.

Take \(A=\mathcal O(D_R)\), \(B=\mathcal O(D_r)\),
and \(\rho\) restriction, with \(D_r\Subset D_R\).
These are nuclear algebras by NT.G, their multiplication
is jointly continuous by the supremum seminorm bounds,
and \(\rho\) is scalar nuclear by NT.23.
Let \(L:\mathbf C^m\to\mathbf C^n\) be a constant
linear map. Choosing a basis of its kernel and extending
it to a basis of \(\mathbf C^m\) splits its kernel over
both \(A\) and \(B\). Therefore
\[
\begin{gathered}
F_1=\ker(L:A^m\to A^n),\\
F_1\simeq A^{\dim\ker L},\\
B\widehat\otimes_A F_1\\
 =\ker(L:B^m\to B^n)
\end{gathered}
\tag{NT.33}
\]
as closed subspaces with their exact topologies.
This verifies NT.27, including its closedness, in this
example. Any \(A\)-linear matrix map from \(A^s\) to
\(A^m\) with image in \(F_1\) is \(A\)-nuclear:
use its finitely many coordinate functionals and column
vectors in NT.28. NT.CB then applies to its restriction
into the displayed actual kernel over \(B\).

![Scalar nuclear tensor exactness, closed-target coefficient descent, and the nested radii in the analytic restriction.](assets/nuclear-tensor-closed-target.png)

The upper panel depicts NT.19: the left image is closed
and equals the full kernel. The middle panel depicts
NT.27--NT.31: the coefficient target \(T\) has its closed
subspace topology, and the nuclear presentation is made
on \(P=B\widehat\otimes E\). The lower panel records
the strict inequalities \(r<s<t<R\) used in NT.23;
the concentric circles depict the one-coordinate example
\((r,s,t,R)=(1,2,3,4)\).
They do not depict general fibres or characteristic
supports.

The freely accessible source is
[Demailly's author manuscript, Chapter IX, §5](https://www-fourier.univ-grenoble-alpes.fr/~demailly/manuscripts/agbook.pdf),
especially the tensor constructions, binuclear lemma,
nuclear exactness and closed-target coefficient argument
on printed pages 429--435 and 443--444.


### 5.93. Coefficient transversality and analytic tensor products

Tensoring over a coefficient algebra requires more than
scalar nuclear exactness. The bar complex records the
additional obstruction. Its vanishing, together with a
Hausdorff degree-zero quotient, controls closed cycle
spaces and changes of analytic coefficients.

Throughout, \(A\) is a commutative unital nuclear Fréchet
algebra with jointly continuous multiplication. Its
modules are nuclear Fréchet spaces with jointly continuous
unital actions; all module maps are continuous and linear
over the stated algebra. Scalar completed tensors have
the topology proved in
NT.E--NT.F.
Balanced tensors are the actual cokernels of NT.26,
with no closure added to their defining relation images.
Their homology quotients may be non-Hausdorff.

We use the proved
Fréchet subquotient and nuclear stability statements,
the open mapping theorem NP.C,
and the closed-target theorem NT.CB.
The analytic example uses the
holomorphic coefficient spaces of NT.G
and the polydisc Cauchy and Taylor proof.

#### CT.A. Nuclearly free modules and the augmented bar resolution

A nuclearly free module means \(A\widehat\otimes V\),
where \(V\) is a nuclear Fréchet vector space and \(A\)
acts on its first factor. Its balanced tensor with a
module \(E\) has the topological identification
\[
\begin{gathered}
E\widehat\otimes_A(A\widehat\otimes V)
 \simeq E\widehat\otimes V,\\
[x\otimes(a\otimes v)]\longmapsto ax\otimes v.
\end{gathered}
\tag{CT.1}
\]
Here is the actual relation calculation on completed
tensors. The action map \(\mu:E\widehat\otimes A
\widehat\otimes V\to E\widehat\otimes V\) has the
continuous section \(s(x\otimes v)=x\otimes1\otimes v\).
Insert a further unit after \(a\) by
\(r(x\otimes a\otimes v)=x\otimes a\otimes1\otimes v\).
For the relation map \(d\) of the balanced tensor,
\(dr=s\mu-1\). Also \(\mu d=0\). These identities hold
on elementary tensors and extend by continuity and
density. They show that \(\ker\mu=\operatorname{im}d\)
exactly. This kernel is closed, and the displayed
section makes CT.1 a topological isomorphism.
In particular its balanced quotient is Hausdorff.

For a module \(F\), put
\[
\begin{gathered}
L_{-1}(F)=F,\\
L_q(F)=A^{\widehat\otimes(q+1)}
             \widehat\otimes F,\\
q\ge0.
\end{gathered}
\tag{CT.2}
\]
The \(A\)-action is on the first factor. A finite
elementary tensor is written \((a_0,\ldots,a_q;y)\).
There are \(q+1\) face maps from \(L_q\) to \(L_{q-1}\):
for \(i<q\) multiply \(a_i\) and \(a_{i+1}\);
for \(i=q\) replace the last \(A\)-factor and \(y\)
by \(a_qy\). When \(q=0\), this is the augmentation
\(d_0(a_0;y)=a_0y\). For \(q\ge1\), define
\[
d_q=\sum_{i=0}^q(-1)^i\partial_i.
\tag{CT.3}
\]
Multiplication and action are jointly continuous, so
the projective tensor universal property extends each
face map to the completed tensors. Associativity of
multiplication and action gives
\(\partial_i\partial_j=\partial_{j-1}\partial_i\)
when \(i<j\). In the double sum for \(d_{q-1}d_q\),
the term indexed by \(i<j\) cancels the term with
the same composite indexed by \(j-1,i\), whose sign
is opposite. Thus \(d_{q-1}d_q=0\), including the
augmentation.

Define \(h_{-1}(y)=(1;y)\) and let \(h_q\) insert
\(1\) as the first factor for \(q\ge0\). Directly,
the first face of \(h_q\) is the identity and every
other face is \(h_{q-1}\) followed by the preceding
face. With their alternating signs this gives
\[
\begin{gathered}
d_{q+1}h_q+h_{q-1}d_q=1
       \quad\\
q\ge0,\qquad
d_0h_{-1}=1_F.
\end{gathered}
\tag{CT.4}
\]
These are continuous scalar linear homotopies;
\(h_q\) need not be \(A\)-linear.
They prove exactness of the entire augmented resolution.
The image of each \(d_{q+1}\) is the closed kernel of
\(d_q\), and \(d_{q+1}h_q\) is a continuous projection
onto that kernel: it equals the identity there and
has image there. Thus the resolution is split as a
complex of vector spaces with their topologies.
Every \(L_q\) is nuclearly free and nuclear by NT.E.

#### CT.B. Completed Tor and its symmetry

Tensor CT.2 over \(A\) with \(E\). By CT.1 the terms
are the nuclear Fréchet spaces
\[
C_q(E,F)=E\widehat\otimes
          A^{\widehat\otimes q}\widehat\otimes F.
\tag{CT.5}
\]
Use \([x|a_1|\cdots|a_q|y]\) for an elementary tensor.
For \(q\ge1\), face zero acts by \(a_1\) on \(x\),
faces \(1,\ldots,q-1\) multiply adjacent \(A\)-factors,
and face \(q\) acts by \(a_q\) on \(y\).
Their alternating sum is the differential.
For example
\[
\begin{gathered}
d_1[x|a|y]\\
 =ax\otimes y-x\otimes ay,\\
d_2[x|a|b|y]\\
 =[ax|b|y]-[x|ab|y]\\
{}+[x|a|by].
\end{gathered}
\tag{CT.6}
\]
This is the differential induced from CT.3 under CT.1,
so its square is zero. Define
\[
\begin{gathered}
\widehat{\operatorname{Tor}}^A_q(E,F)
\\ =\ker d_q/\operatorname{im}d_{q+1},\\
d_0:C_0(E,F)\to0.
\end{gathered}
\tag{CT.7}
\]
The quotients in positive degree are taken as vector
spaces with their quotient topology, without inserting
closures. In degree zero this definition gives exactly
\(E\widehat\otimes_A F\), topologically as well as
algebraically, by CT.6 and NT.26.

Reversal of the factors is the continuous isomorphism
\[
\begin{gathered}
R_q[x|a_1|\cdots|a_q|y]\\
 =[y|a_q|\cdots|a_1|x].
\end{gathered}
\tag{CT.8}
\]
Commutativity of \(A\) identifies its face \(i\)
with face \(q-i\) on the reversed tensor. Hence
\(dR_q=(-1)^qR_{q-1}d\). With
\[
S_q=(-1)^{q(q+1)/2}R_q
\tag{CT.9}
\]
we have \(dS_q=S_{q-1}d\), because the ratio of
successive signs is \((-1)^q\).
Reversal twice gives the identity, including its
scalar signs. It follows that
\(\widehat{\operatorname{Tor}}^A_q(E,F)\) and
\(\widehat{\operatorname{Tor}}^A_q(F,E)\) are naturally
isomorphic with their quotient topologies.

If \(E=A\widehat\otimes V\) is nuclearly free,
CT.1 identifies \(E\widehat\otimes_A L_\bullet(F)\)
with \(V\widehat\otimes L_\bullet(F)\).
Tensor the scalar homotopy CT.4 with \(1_V\);
all maps remain continuous. Thus
\[
\widehat{\operatorname{Tor}}^A_q(E,F)=0
 \quad(q\ge1)
\tag{CT.10}
\]
in this case, and also when \(F\) is nuclearly free
by CT.8--9. The degree-zero balanced tensor is then
Hausdorff by CT.1.

#### CT.C. The exact sequence and its topology

For an exact sequence of nuclear Fréchet modules
\(0\to E_1\to E_2\to E_3\to0\), scalar tensor exactness
NT.T applied to CT.5 gives a short exact sequence
of complexes \(C_\bullet(E_i,F)\), strict in each
degree. Its homology has the algebraic exact sequence
\[
\begin{gathered}
\cdots\to T_q(E_1)\to T_q(E_2)\\
\to T_q(E_3)\\
\xrightarrow{\delta}T_{q-1}(E_1)\to\cdots,\\
T_0(E_1)\to T_0(E_2)\\
\to T_0(E_3)\to0,
\end{gathered}
\tag{CT.11}
\]
where \(T_q(E_i)=\widehat{\operatorname{Tor}}^A_q(E_i,F)\).

We give the cycle argument for every assertion of
exactness. Lift a cycle \(z_3\) in degree \(q\) to
\(z_2\) in the middle complex. Its boundary is the
image of a unique \(z_1\) in degree \(q-1\) of the
first complex. This \(z_1\) is a cycle, since its
image has boundary zero and the first inclusion is
injective. Define \(\delta[z_3]=[z_1]\).
A different lift changes \(z_1\) by a boundary.
Changing \(z_3\) by a boundary allows a lift in
degree \(q+1\), and again changes the result by a
boundary. The map is therefore well defined,
linear and natural.

At the middle homology, a cycle whose image is a
boundary can be adjusted by the boundary of a lift
of that primitive. The adjusted cycle lies in the
first complex. At the third homology, \(\delta[z_3]=0\)
means that the boundary of its lift is a boundary in
the first complex; subtract its primitive to obtain a
cycle lift in the middle complex. At the next first
homology, a cycle becoming a boundary in the middle
complex is the boundary of an element whose projection
is a cycle in the third complex; its connecting image
is the given class. These three verifications prove
exactness throughout CT.11. In degree zero every
element is a cycle, so the last map is onto.
No closedness of an arbitrary homology denominator
is used. Symmetry gives the same sequence in the
second module variable.

In particular balanced tensoring is algebraically
right exact in this category. This statement uses
actual cokernels. When all relevant quotients are
Hausdorff, the maps in a short exact resulting
sequence are strict by NP.C.

#### CT.D. Any nuclearly free resolution computes the groups

Let \(L_\bullet\to F\) now be any exact augmented
resolution by nuclearly free modules, with continuous
maps. It need not admit a scalar splitting.
Put \(B_{-1}=F\) and
\(B_q=\ker(L_q\to L_{q-1})\) for \(q\ge0\),
where \(L_0\to L_{-1}\) is the augmentation.
Exactness makes these the preceding images, and
they are closed nuclear Fréchet submodules. Thus
\[
\begin{gathered}
0\to B_q\to L_q\\
\to B_{q-1}\to0
\end{gathered}
\tag{CT.12}
\]
is strict. Use CT.11 in the second variable and
the vanishing CT.10. For \(j>1\) it gives
\[
\begin{gathered}
\widehat{\operatorname{Tor}}^A_j(E,B_{q-1})
\\ \simeq
\widehat{\operatorname{Tor}}^A_{j-1}(E,B_q),\\
\widehat{\operatorname{Tor}}^A_1(E,B_{q-1})\\
 \simeq \ker(K_q\to G_q).
\end{gathered}
\tag{CT.13}
\]
Here \(G_q=E\widehat\otimes_A L_q\) and
\(K_q=E\widehat\otimes_A B_q\).
The first identification is the connecting map;
the second is its injection followed by exactness.
They are natural algebraic identifications.

To identify the homology directly, use these abbreviations.
Right exactness gives an onto map \(G_q\to K_{q-1}\)
whose kernel is the image of \(K_q\to G_q\).
Also \(G_{q+1}\to K_q\) is onto. The differential of
\(G_\bullet\) factors through
\(G_q\to K_{q-1}\to G_{q-1}\).
It follows, by taking the preimage of the last
kernel and dividing by the first kernel, that
\[
\begin{gathered}
H_q(G_\bullet)\\
 \simeq\ker(K_{q-1}\to G_{q-1}),\\
q\ge1.
\end{gathered}
\tag{CT.14}
\]
Successively apply the first line of CT.13, starting
at \(B_{-1}=F\), and then its second line with
index \(q-1\). This identifies CT.14 with
\(\widehat{\operatorname{Tor}}^A_q(E,F)\).
For \(q=1\) the second line alone does so.
For \(q=0\), right exactness gives the actual
cokernel \(E\widehat\otimes_A F\).

This degree-zero identification is topological:
the scalar map
\(E\widehat\otimes L_0\to E\widehat\otimes F\)
is an open surjection by NT.T. Composing it with
the balanced quotient gives an open surjection onto
\(E\widehat\otimes_A F\). Its factor through
\(G_0\) is still onto and open, because the preceding
balanced quotient map is onto. CT.14 in degree zero
identifies its kernel with the image of \(G_1\).
The quotient topology is therefore exactly the
homology quotient topology. This does not require
the degree-zero quotient to be Hausdorff.
In positive degrees we use the algebraic
identifications just proved and make no additional
topological assertion about non-Hausdorff homology.

#### CT.E. Transverse coefficients and actual closed kernels

Call \(E\) and \(F\) **transverse over \(A\)** when
\[
\begin{gathered}
E\widehat\otimes_A F\text{ is Hausdorff},\\
\widehat{\operatorname{Tor}}^A_q(E,F)=0
       \quad(q\ge1).
\end{gathered}
\tag{CT.15}
\]
The condition is symmetric by CT.B. Nuclearly free
modules are transverse to every module.
Finite sums of transverse modules remain transverse:
CT.5 is the finite direct sum of the individual
complexes, so its kernels, images and homology are
their finite direct sums, including degree zero
and its topology.

**Kernel theorem.** If CT.12 is replaced by an exact
sequence \(0\to E_1\to E_2\to E_3\to0\), and
\(E_2,E_3\) are transverse to \(F\), then \(E_1\)
is transverse to \(F\), and
\[
\begin{gathered}
0\to E_1\widehat\otimes_A F
 \to E_2\widehat\otimes_A F\\
 \to E_3\widehat\otimes_A F\to0
\end{gathered}
\tag{CT.16}
\]
is strict. The positive Tor vanishing follows at
every index from CT.11. Its degree-zero segment
makes the first arrow injective, with image the
kernel of the second arrow, and the last arrow
onto. A continuous injection into a Hausdorff
space makes its domain Hausdorff: two distinct
points have distinct images and disjoint inverse
images of separating neighbourhoods. Thus the
first balanced quotient is Hausdorff and hence
nuclear Fréchet by NT.A and NT.D. Its image is
closed because it is a kernel with Hausdorff
target. NP.C now makes the injection a
topological embedding onto that kernel, and
makes the final surjection open.

This gives the coefficient embedding needed in NT.CB.
Let \(\rho:A\to B\) be a scalar nuclear homomorphism
of commutative unital nuclear Fréchet algebras.
Let \(F_1\subset F\) be closed, put \(Q=F/F_1\),
and suppose \(B\) is transverse over \(A\) to
\(E,F,Q\). Then
\[
\begin{gathered}
0\to B\widehat\otimes_A F_1
 \to B\widehat\otimes_A F\\
 \to B\widehat\otimes_A Q\to0
\end{gathered}
\tag{CT.17}
\]
is strict by CT.16. In particular NT.27 holds
with its actual closed subspace topology, and
\(E_B,F_B\) are Hausdorff. If an \(A\)-subnuclear
map \(f:E\to F\), with a nuclear auxiliary module,
has image in \(F_1\), NT.CB therefore proves that
\[
\begin{gathered}
B\widehat\otimes_A E
 \longrightarrow B\widehat\otimes_A F_1
\\ \text{is }B\text{-subnuclear}.
\end{gathered}
\tag{CT.18}
\]
This is a statement into the closed target itself.
The transversality assumptions in CT.17 are the
coefficient conditions that supply its topology.

#### CT.F. Successive changes of coefficients

Let \(A\to B\to D\) be continuous unital homomorphisms
of commutative nuclear Fréchet algebras. If \(B\)
is transverse to \(E\) over \(A\), then
\[
\begin{gathered}
\widehat{\operatorname{Tor}}^B_q
 (D,B\widehat\otimes_A E)
\\ \simeq\widehat{\operatorname{Tor}}^A_q(D,E).
\end{gathered}
\tag{CT.19}
\]
To prove this, take the standard resolution
\(L_q=A\widehat\otimes V_q\) of \(E\).
After tensoring with \(B\), CT.1 gives the terms
\(B\widehat\otimes V_q\).
Its positive homology vanishes by transversality,
and its degree-zero cokernel is
\(E_B=B\widehat\otimes_A E\).
This is Hausdorff, so the augmented complex is
an actual exact resolution of \(E_B\) by
nuclearly free \(B\)-modules.

Use CT.D for this resolution and for the original
resolution over \(A\). In both cases the complex
computing the groups in CT.19 is exactly
\(D\widehat\otimes V_\bullet\), with the same
differentials. This proves CT.19 in positive
degrees algebraically. In degree zero both
objects have the topology of the quotient of
\(D\widehat\otimes V_0\) by the same image
of \(D\widehat\otimes V_1\), by the last
paragraph of CT.D. Hence the degree-zero
identification is topological even before
assuming it Hausdorff. On elementary tensors
it is the canonical coefficient map
\[
\begin{gathered}
{}[d\otimes[b\otimes x]]\\
 \longmapsto[d\,\rho_{BD}(b)\otimes x].
\end{gathered}
\tag{CT.20}
\]
Thus, if \(D\) is also transverse to \(E\)
over \(A\), it is transverse to \(E_B\)
over \(B\). No general flatness of \(B\)
or \(D\) over \(A\) is assumed.

#### CT.G. Bounded complexes and cycle spaces

Let \(K^\bullet\) be a complex of nuclear Fréchet
\(A\)-modules, zero in degrees \(q>N\).
Let \(M\) be transverse to every term, and
suppose \(H^q(K^\bullet)=0\) for \(q\ge k\).
These are algebraic cohomology vanishings;
no initial Hausdorff assumption on lower
cohomology is required.

Put \(Z^q=\ker d^q\). For \(q\ge k-1\),
exactness at the next degree gives
\[
\begin{gathered}
0\to Z^q\to K^q\\
\to Z^{q+1}\to0.
\end{gathered}
\tag{CT.21}
\]
These are sequences of nuclear Fréchet modules:
the cycle spaces are closed kernels. Starting
with \(Z^{N+1}=0\), descend through CT.21.
The kernel theorem CT.E makes \(M\) transverse
to \(Z^q\) for every \(q\ge k-1\), and
makes every tensor of CT.21 strict.
Consequently
\[
\begin{gathered}
Z^q(M\widehat\otimes_A K^\bullet)
\\ =M\widehat\otimes_A Z^q,\\
B^q(M\widehat\otimes_A K^\bullet)
\\ =M\widehat\otimes_A Z^q,\\
q\ge k.
\end{gathered}
\tag{CT.22}
\]
For the first equality, factor the differential
through \(Z^{q+1}\). Its inclusion into \(K^{q+1}\)
remains injective because it is the first arrow
of the tensored sequence at \(q+1\). At the
last zero term the assertion is immediate.
For the second equality use the onto map to
\(Z^q\) at \(q-1\). This proves that the
tensored complex is acyclic in degrees \(q\ge k\).
It also proves all the closed cycle embeddings
displayed in CT.22.

Here is the precise statement for comparison maps.
Let \(f:K^\bullet\to J^\bullet\) be a continuous
chain map, with both complexes bounded above
and with \(M\) transverse to all their terms.
Define
\[
\begin{gathered}
C(f)^q=K^q\oplus J^{q-1},\\
D^q(x,y)\\
 =(d_K^q x,\ f^q x-d_J^{q-1}y).
\end{gathered}
\tag{CT.23}
\]
The mixed term of \(D^{q+1}D^q\) is
\(f^{q+1}d_K^q x-d_J^q f^q x=0\);
the two square terms vanish as well. Thus
this is a complex. Its inclusion of
\(J[-1]\), whose differential is \(-d_J\),
and projection to \(K\) give a short exact
sequence of complexes. Lift a cycle \(x\)
by \((x,0)\); its boundary is \((0,f(x))\).
The same lifting proof as CT.C, with degrees
reversed, gives the connecting map
\(+H^q(f)\). Its exact sequence shows
\[
\begin{gathered}
H^q(C(f))=0\\
\Longleftrightarrow\
H^q(f)\text{ injective and}\\
H^{q-1}(f)\text{ surjective}.
\end{gathered}
\tag{CT.24}
\]
Indeed the adjacent exact segment starts with
\(H^{q-1}(K)\to H^{q-1}(J)\), passes through
\(H^q(C(f))\), and ends with \(H^q(K)\to H^q(J)\);
exactness gives both implications.

If \(H^q(f)\) is an isomorphism for \(q\ge k\)
and a surjection for \(q=k-1\), then CT.24
makes \(C(f)\) acyclic in degrees \(q\ge k\).
Its terms are finite sums of transverse modules.
Apply CT.21--22, and note that finite sums and
the differential CT.23 commute with balanced
tensoring. CT.24 applied to the tensor comparison
then gives
\[
\begin{gathered}
H^q(1_M\widehat\otimes_A f)
\\ \text{is an isomorphism},\\
q\ge k,\\
H^{k-1}(1_M\widehat\otimes_A f)
\\ \text{is surjective}.
\end{gathered}
\tag{CT.25}
\]
This includes the degree adjacent to the
comparison range and the sign of its
connecting map.

#### CT.H. Faithful evaluations and arbitrary kernels

We next establish the scalar analytic product
identifications without requiring a closed image
for a gluing differential. Let \(U,V\) be complete
Fréchet spaces. Every continuous \(l:U\to\mathbf C\)
gives a continuous evaluation
\(l\widehat\otimes1:U\widehat\otimes_\varepsilon V\to V\).
If \(|l|\le Cp\), NT.2 and NT.5 give
\[
\begin{gathered}
q((l\otimes1)t)\\
 \le C(p\otimes_\varepsilon q)(t).
\end{gathered}
\tag{CT.26}
\]
The map extends by completeness. These evaluations
separate tensors in the completion. In fact, for
each pair \(p,q\), the injective seminorm is the
supremum of all scalar evaluations with
\(|l|\le p, |m|\le q\). This equality passes to
the completion because a Cauchy sequence for
that seminorm converges uniformly on these
two families of functionals. If every
\((l\widehat\otimes1)t\) is zero, every defining
seminorm of \(t\) is therefore zero.

It follows that a continuous injective map
\(a:V\to W\) between Fréchet spaces remains
injective after \(U\widehat\otimes_\varepsilon-\).
Indeed a tensor in the kernel has all its
evaluations in \(\ker a=0\); the preceding
separation proves it zero. This does not
assert a topological embedding for an
arbitrary injective \(a\).

If \(U\) is nuclear and \(a:V\to W\) is
any continuous map, put \(S=\ker a\).
NT.T gives the strict sequence
\[
\begin{gathered}
0\to U\widehat\otimes S
 \to U\widehat\otimes V\\
 \to U\widehat\otimes(V/S)\to0.
\end{gathered}
\tag{CT.27}
\]
The induced \(V/S\to W\) is continuous and
injective. NT.17 and the preceding evaluation
argument make its tensor with \(U\) injective.
Hence
\[
\ker(1_U\widehat\otimes a)
 =U\widehat\otimes\ker a
\tag{CT.28}
\]
with its closed subspace topology. No
closedness of \(\operatorname{im}a\) is needed.

#### CT.I. Holomorphic products on arbitrary open sets

Let \(U\subset\mathbf C^n\) and
\(V\subset\mathbf C^m\) be open. Then the
pointwise product map induces a topological
isomorphism
\[
\begin{gathered}
\mathcal O(U)\widehat\otimes\mathcal O(V)\\
 \simeq\mathcal O(U\times V).
\end{gathered}
\tag{CT.29}
\]
All spaces have their compact-open topologies
and are nuclear Fréchet by NT.G.

First take \(V=D\) to be a polydisc about zero,
of radii \(R\). For a finite sum
\(h(z,w)=\sum_\nu f_\nu(z)g_\nu(w)\),
compact \(K\subset U\), and
\(\overline D_r\Subset D\), the injective
seminorm is exactly
\[
(p_K\otimes_\varepsilon p_r)(h)
 =\sup_{K\times\overline D_r}|h|.
\tag{CT.30}
\]
Point evaluations give the inequality from
right to left. Conversely, apply a functional
\(l\) bounded by \(p_K\) to the \(z\)-variable.
For each fixed \(w\), the resulting function
\(\sum_\nu l(f_\nu)g_\nu(w)\) has absolute
value at most the right side. Applying a
functional bounded by \(p_r\) proves the
other inequality, using precisely NT.5.
The argument works for any compact \(K\),
without a representation of functionals by
measures.

Every \(g\in\mathcal O(U\times D)\) has a
Taylor expansion in \(w\), with coefficients
\(a_\alpha\in\mathcal O(U)\). Cauchy's formula
on an intermediate polydisc gives, uniformly
for \(z\in K\) and \(w\in\overline D_r\),
\[
\begin{gathered}
g(z,w)=\sum_{\alpha\in\mathbf N^m}
                  a_\alpha(z)w^\alpha,\\
\sup_K|a_\alpha|
 \le M_{K,s}s^{-\alpha},\\
r<s<R,\\
M_{K,s}=\sup_{K\times\overline D_s}|g|.
\end{gathered}
\tag{CT.31}
\]
The coefficient integrals are holomorphic in
\(z\): on each smaller \(z\)-polydisc, Cauchy's
formula in \(z\) and uniform convergence allow
the parameter integrals to be interchanged;
the resulting Cauchy formula gives their
local holomorphic Taylor series. The bound
in CT.31 makes the series uniformly
absolutely convergent on the stated compact
product. Its rectangular polynomial partial
sums in \(w\) are finite tensors. Compact
products of this form are cofinal among
compact subsets of \(U\times D\).
CT.30, density and completeness prove CT.29
when the second open set is a polydisc,
also after translating its centre.

For general \(V\), take a countable polydisc
cover \((D_j)\). In every nonempty
\(D_j\cap D_k\), choose a countable dense
set of points \(w_{jk\ell}\). Set
\[
\begin{gathered}
P=\prod_j\mathcal O(D_j),\\
Q=\prod_{j,k,\ell}\mathbf C,\\
a((g_j))_{jk\ell}
\\ =g_j(w_{jk\ell})-g_k(w_{jk\ell}).
\end{gathered}
\tag{CT.32}
\]
The products are countable, and \(a\) is
continuous. Equality on the chosen dense
sets is equality on each overlap by
continuity. The gluing and topology
argument NT.22 therefore gives
\(\mathcal O(V)=\ker a\), as a closed
topological subspace of \(P\).

Apply CT.28 with \(U\) there replaced
by \(\mathcal O(U)\), and use NT.20
to commute its tensor with the two
countable products. The polydisc case
already proved identifies the resulting
kernel with the tuples
\[
\begin{gathered}
(h_j)\in\prod_j\mathcal O(U\times D_j),\\
h_j(z,w_{jk\ell})=h_k(z,w_{jk\ell}).
\end{gathered}
\tag{CT.33}
\]
For each fixed \(z\), density of the chosen
points gives equality on the whole overlap.
Thus these functions glue to a holomorphic
function on \(U\times V\). Conversely every
such function gives the tuple.
The induced topology is exactly compact-open:
cover a compact subset by finitely many
compact subsets of \(U\times D_j\), as in
NT.22. This proves CT.29 for all open sets.
The proof neither requires a closed image
of \(a\) nor a Stein hypothesis on \(U,V\).
For empty open sets the function spaces
are zero and the same formula holds.

#### CT.J. A coefficient obstruction in every positive degree

Let \(A=\mathbf C[\epsilon]/(\epsilon^2)\) with its
finite-dimensional topology and let \(B=A/(\epsilon)\).
Both are nuclear Fréchet algebras; any linear map
between finite-dimensional normed spaces is a finite
nuclear series by choosing coordinate functionals
and basis vectors. The periodic augmented resolution
\[
\begin{gathered}
\cdots\xrightarrow{\epsilon} A
 \xrightarrow{\epsilon}A
 \xrightarrow{\epsilon}A\\
 \to B\to0
\end{gathered}
\tag{CT.34}
\]
is exact: the image and kernel of multiplication by
\(\epsilon\) are both \(\mathbf C\epsilon\), and
the augmentation has that same kernel.
Its terms are nuclearly free. Tensoring with \(B\)
over \(A\) makes every displayed multiplication
map zero. CT.D gives
\[
\begin{gathered}
\widehat{\operatorname{Tor}}^A_q(B,B)
 \simeq B,\\
q\ge0.
\end{gathered}
\tag{CT.35}
\]
For \(I=\epsilon A\subset A\), the sequence
\(0\to I\to A\to B\to0\) is strict, but
\[
\begin{gathered}
B\widehat\otimes_A I\simeq B,\\
B\widehat\otimes_A A\simeq B,\\
B\widehat\otimes_A(I\hookrightarrow A)=0.
\end{gathered}
\tag{CT.36}
\]
Indeed \(I\simeq B\) as an \(A\)-module,
and \(1\otimes\epsilon=\epsilon\cdot1\otimes1=0\)
in \(B\widehat\otimes_A A\).
Thus a closed submodule need not stay embedded
under a coefficient change, even though the
algebras and all scalar tensor spaces are nuclear.
The nonzero group CT.35 is exactly the obstruction
to the injective arrow in CT.16. This is a
finite-dimensional calculation, so every quotient
topology in this example is Hausdorff.

The same example illustrates the target distinction.
Multiplication by \(\epsilon\) is an \(A\)-nuclear
map \(A\to A\) with image in \(I\), and the quotient
map \(A\to B\) is scalar nuclear. The balanced
target \(B\widehat\otimes_A I\) is nevertheless
sent to zero in \(B\widehat\otimes_A A\).
An ambient coefficient calculation therefore
cannot identify that closed target; CT.17
requires its transversality premises.

![The split bar resolution, the signed comparison complex, and the dual-number obstruction to coefficient exactness.](assets/coefficient-transversality.png)

The first panel records the homotopy identity CT.4.
The second records the mixed-term cancellation in
CT.23 and the degree range CT.25. The third records
CT.35--36: its zero arrow has a nonzero domain,
whereas every positive Tor group is \(B\).
These are algebraic diagrams; no geometric scale
or representation of a proper fibre is intended.

Free source:
[Demailly's author manuscript, Chapter IX, §5.C](https://www-fourier.univ-grenoble-alpes.fr/~demailly/manuscripts/agbook.pdf),
printed pages 439--442, for nuclear coefficient
resolutions and transversality. The signed
complex and all connecting maps used here are
defined and proved in CT.C and CT.G.


### 5.94. Weighted Dolbeault solvability and holomorphic approximation

A strictly plurisubharmonic exhaustion supplies both a
complete metric and weights that separate two compact
sublevel sets. The complete metric permits integration
by parts without a boundary condition. The weights
then give a weak solution estimate, vanishing for the
structure sheaf, and approximation of holomorphic
functions with all their derivatives.

Here \(\Omega\) is an open subset of \(\mathbb C^n\),
\(n\ge1\), with a smooth nonnegative proper strictly
plurisubharmonic function \(\psi\). Proper means that
every set \(\{\psi\le c\}\) is compact in \(\Omega\).
No regularity of its boundary is assumed. All Hilbert
inner products are linear in the first variable.

The measure operations used below are proved in the
earlier programme: monotone and dominated convergence,
product integration and linear changes of variables,
Hölder and convolution estimates,
[\(L^p\) completeness and smooth density](../elliptic-boundary-reduction/banach-foundation-bridges.html#completeness-and-compact-smooth-density),
scaled mollifiers,
weak derivatives and local approximation,
and polar integration.
We also use Hilbert projection and Riesz representation,
Hahn–Banach, NT.A,
the polydisc Cauchy and Taylor proof,
and the explicit programme proofs of
distributional holomorphic regularity, R.3
and acyclicity of smooth-module sheaves, R.5.

#### WD.A. A complete metric with controlled cutoffs

Put
\[
\begin{gathered}
F(t)=t+t^2,\\
\omega=i\partial\bar\partial F(\psi),\\
\omega=(1+2\psi)i\partial\bar\partial\psi
       \\
       {}+2i\partial\psi\wedge\bar\partial\psi .
\end{gathered}
\tag{WD.1}
\]
Strict plurisubharmonicity makes this a positive
Kähler form. Our convention is
\(\omega=i\sum g_{j\bar k}dz_j\wedge d\bar z_k\);
the associated real metric is \(2\operatorname{Re}g\).
The rank-one inequality in WD.1 gives
\[
\begin{gathered}
|\partial\psi|_g^2\le\tfrac12,\\
|d\psi|_\omega^2=2|\partial\psi|_g^2\le1 .
\end{gathered}
\tag{WD.2}
\]
Indeed, for a positive matrix \(g\), evaluate
\(g\ge2aa^*\) on \(g^{-1}a\). If
\(c=a^*g^{-1}a\), the result is \(c\ge2c^2\);
as \(c\ge0\), it follows that \(c\le1/2\).
This is precisely WD.2.

Along every piecewise smooth path the change in
\(\psi\) is bounded by its metric length. A metric
Cauchy sequence therefore has bounded \(\psi\). It
lies in a compact sublevel set, has a convergent
subsequence there, and converges in the metric,
because the smooth positive metric and Euclidean
metric are locally equivalent. Thus \(\omega\) is
complete. This argument works on each connected
component; bounded sublevels meet only finitely many
components.

Choose a real smooth function \(\eta\), equal to one
on \((-\infty,1]\), zero on \([2,\infty)\), with
\(0\le\eta\le1\), and set
\[
\chi_R=\eta(\psi/R),\qquad R\ge1 .
\tag{WD.3}
\]
These functions have compact support, converge
pointwise to one, and satisfy
\(|d\chi_R|_\omega\le\|\eta'\|_\infty/R\).
Such a smooth \(\eta\) is obtained by integrating
and normalizing a nonnegative smooth bump on
\((1,2)\); smooth bumps and locally finite
partitions are constructed in R.5.

#### WD.B. Maximal operators and their common graph core

Let \(\varphi\) be a smooth real function. Write
\(H_q=L^2_{n,q}(\Omega,\omega,e^{-\varphi})\) for
the Hilbert space of measurable \((n,q)\)-forms
with norm
\[
\|u\|_\varphi^2
 =\int_\Omega |u|_\omega^2 e^{-\varphi}
                       \frac{\omega^n}{n!}.
\tag{WD.4}
\]
These spaces are complete: in a measurable
orthonormal frame they are finite sums of scalar
\(L^2\) spaces, whose completeness was proved at
the linked earlier result. Equivalently one may
use a countable coordinate partition. Positive
smooth weights and bundle metrics are boundedly
equivalent to Euclidean ones on each compact.

The maximal operator \(D''=\bar\partial\) has domain
the forms \(u\in H_q\) whose distributional derivative
belongs to \(H_{q+1}\). It is closed: an \(L^2\)
limit and an \(L^2\) derivative limit satisfy the
same test-form identities, by Cauchy–Schwarz on
each compact. Compactly supported smooth forms
are dense by the earlier mollifier proof, so this
domain is dense. Distributional derivatives
commute, hence \(D''D''=0\), with the full domain
assertion that \(D''u\) belongs to the next domain
and has derivative zero.

Let \(P\) denote the formal weighted adjoint of
\(D''\), defined by integration by parts on compact
test forms. Its maximal distributional domain is
defined in the same way. We prove
\[
\begin{gathered}
C_c^\infty(\Omega,\Lambda^{n,q}),\\
\text{dense in}\\
\operatorname{Dom}D''\cap\operatorname{Dom}P,\\
\text{with norm}\\
\|u\|_\varphi+\|D''u\|_\varphi\\
{}+\|Pu\|_\varphi .
\end{gathered}
\tag{WD.5}
\]
The same assertion holds with either operator
alone. First multiply by WD.3. The commutators
are exterior multiplication by
\(\bar\partial\chi_R\), and its negative adjoint
contraction, respectively. Exterior multiplication
by a covector and its contraction have operator
norm at most the covector norm: choose a unitary
basis with that covector in its first direction.
Consequently each commutator norm is bounded by
\(\|d\chi_R\|_\infty\|u\|_\varphi\).
Dominated convergence for the three original
terms and WD.3 show convergence in the joint
graph norm. It remains to smooth a compactly
supported form.

Take a finite smooth partition on a neighbourhood
of its support. Each piece has support strictly
inside a coordinate box, so may be extended by
zero before convolution. All operators in such
a box are matrix first-order operators with smooth
coefficients. For a scalar summand \(a\partial_j\),
integration by parts gives the exact commutator
\[
\begin{aligned}
&a(x)\partial_j(u*\rho_\epsilon)(x)\\
&\quad-((a\partial_j u)*\rho_\epsilon)(x)\\
&=\int(a(x)-a(y))u(y)\\
&\qquad\cdot(\partial_j\rho_\epsilon)(x-y)\,dy\\
&\quad+\int(\partial_j a)(y)\\
&\qquad\cdot\rho_\epsilon(x-y)u(y)\,dy .
\end{aligned}
\tag{WD.6}
\]
On a fixed slightly larger box the coefficients
are Lipschitz. The first kernel is bounded by
a constant times
\(|x-y||\partial_j\rho_\epsilon(x-y)|\);
its integral is independent of \(\epsilon\).
The second kernel has bounded integral as well.
Young's inequality thus bounds WD.6 uniformly
on \(L^2\). For smooth \(u\) the commutator tends
to zero in \(L^2\), since both differentiated
convolutions tend to \(a\partial_j u\).
Smooth \(L^2\) density and the uniform bound
give the same conclusion for every \(u\in L^2\).
Zeroth-order commutators tend to zero by uniform
continuity of the coefficient on that box and
the convolution bound. Apply this argument to
all coefficients of \(D''\) and \(P\). The
convolution of each distributional derivative
converges to its \(L^2\) value, so the smoothed
partition pieces converge in both graph norms.
Their finite sum proves WD.5. Supports remain
inside the coordinate boxes for small
\(\epsilon\); no boundary extension is applied
to a piece touching its box boundary.

The Hilbert adjoint \((D'')^*\) is precisely
the maximal operator \(P\). One inclusion follows
by testing the Hilbert adjoint against compact
smooth forms. Conversely, if \(Pu\in L^2\),
the compact integration-by-parts identity
extends to every vector in \(\operatorname{Dom}D''\)
using its single-operator graph core. This is
the defining adjoint identity. In particular
WD.5 applies to the actual Hilbert adjoint,
not merely to a formal expression.

#### WD.C. The weighted Kähler estimate

On the trivial holomorphic line bundle with metric
\(e^{-\varphi}\), define
\[
\begin{gathered}
D'=\partial-\partial\varphi\wedge,\\
D''=\bar\partial,\\
L=\omega\wedge,\qquad\Lambda=L^* .
\end{gathered}
\tag{WD.7}
\]
We establish the needed Kähler identities directly.
At any chosen point take holomorphic coordinates
with \(g\) the identity and its first derivatives
zero. To construct them, first use a complex
linear change to make \(g\) the identity. The
Kähler equations give
\(\partial_\ell g_{j\bar k}=\partial_jg_{\ell\bar k}\).
A holomorphic quadratic coordinate change with
these symmetric coefficients cancels the
holomorphic first derivatives; their conjugates
cancel the antiholomorphic ones. A holomorphic
change of line frame cancels the first derivatives
of \(\varphi\): in the new frame its weight is
\(\varphi-2\operatorname{Re}h\), and the linear
holomorphic function \(h\) can have any specified
holomorphic first derivatives. Thus the connection,
volume and metric first-order terms all vanish
at this point.

Let \(e_j\) and \(\bar e_j\) denote exterior
multiplication by \(dz_j\) and \(d\bar z_j\);
let \(\iota_j,\bar\iota_j\) be their adjoints.
The only nonzero anticommutators of exterior
and contraction operators are
\(\iota_je_k+e_k\iota_j=\delta_{jk}\) and
\(\bar\iota_j\bar e_k+\bar e_k\bar\iota_j=\delta_{jk}\).
All mixed ones vanish. Therefore
\[
\begin{gathered}
L=i\sum_j e_j\bar e_j,\\
\Lambda=-i\sum_j\bar\iota_j\iota_j,\\
{}[\Lambda,\bar e_k]=i\iota_k,\\
{}[\Lambda,e_k]=-i\bar\iota_k .
\end{gathered}
\tag{WD.8}
\]
For example, moving \(\iota_j\) past
\(\bar e_k\) changes sign, and moving
\(\bar\iota_j\) past \(\bar e_k\) leaves the
term \(\delta_{jk}\iota_j\); this gives the
first commutator. Moving \(\iota_j\) past
\(e_k\) gives the second. At the normal point,
the formal adjoints of
\(\sum e_j\partial_j\) and
\(\sum\bar e_j\partial_{\bar j}\) are
\(-\sum\iota_j\partial_{\bar j}\) and
\(-\sum\bar\iota_j\partial_j\).
WD.8 hence proves the global first-order identities
\[
\begin{gathered}
{}[\Lambda,D'']=-i(D')^*,\\
{}[\Lambda,D']=i(D'')^* .
\end{gathered}
\tag{WD.9}
\]
They are frame-invariant identities, and their
coefficients and first-order terms have been
checked at an arbitrary point.

Write \(\{A,B\}=AB+BA\) when both operators
have odd degree. Direct differentiation of
WD.7 gives
\(\{D',D''\}=\partial\bar\partial\varphi\wedge\).
Expanding all four products verifies
\[
\begin{gathered}
\{D'',[\,\Lambda,D'\,]\}\\
=[\,\Lambda,\{D',D''\}\,]\\
       {}+\{D',[\,D'',\Lambda\,]\},\\
\Delta''=\Delta'
       +[\,i\partial\bar\partial\varphi\wedge,\Lambda\,],\\
\Delta'=\{D',(D')^*\},\\
\Delta''=\{D'',(D'')^*\}.
\end{gathered}
\tag{WD.10}
\]
For the second line, multiply the first by
\(-i\), then use WD.9. The second term becomes
\(\Delta'\); the first becomes the displayed
curvature commutator with the indicated order
and sign.

Suppose now
\[
i\partial\bar\partial\varphi\ge\omega .
\tag{WD.11}
\]
Diagonalize this Hermitian form relative to
\(\omega\) at the point, with eigenvalues
\(\lambda_j\ge1\). The finite-dimensional
diagonalization used here follows by maximizing
the real quadratic form on the unit sphere:
differentiation in every tangent direction
gives a unit eigenvector; its orthogonal
complement is invariant by Hermitian symmetry,
so induction gives an orthonormal eigenbasis.
For one exterior pair,
the commutator
\([i\lambda_je_j\bar e_j,-i\bar\iota_j\iota_j]\)
acts on
\(1,dz_j,d\bar z_j,dz_j\wedge d\bar z_j\)
by \(-\lambda_j,0,0,\lambda_j\), respectively.
Thus on
\(dz_1\wedge\cdots\wedge dz_n\wedge d\bar z_J\)
it acts by \(\sum_{j\in J}\lambda_j\).
For an \((n,q)\)-form this is at least \(q\).
Compact integration by parts in WD.10 now yields
\[
\begin{gathered}
\|D''u\|_\varphi^2+\|(D'')^*u\|_\varphi^2\\
\ge q\|u\|_\varphi^2 .
\end{gathered}
\tag{WD.12}
\]
Indeed \(D'u=0\) in holomorphic top degree, and
the remaining contribution
\(\|(D')^*u\|_\varphi^2\) is nonnegative.
WD.5 passes WD.12 to every vector in the joint
maximal domain. This passage uses only the lower
bound \(q\|u\|^2\); it requires no boundedness of
the full curvature operator.

#### WD.D. The weak solution theorem

For \(1\le q\le n\), let
\[
\begin{gathered}
T=D'':H_{q-1}\longrightarrow H_q,\\
S=D'':H_q\longrightarrow H_{q+1},
\end{gathered}
\tag{WD.13}
\]
with maximal domains; set \(H_{n+1}=0\) and
let \(S\) have full domain when \(q=n\).
Under WD.11, every \(g\in H_q\) with \(Sg=0\)
has a solution \(f\in\operatorname{Dom}T\) satisfying
\[
\begin{gathered}
Tf=g,\\
\|f\|_\varphi^2\le q^{-1}\|g\|_\varphi^2 .
\end{gathered}
\tag{WD.14}
\]

Here are the domain and functional-analytic details.
The kernel \(K=\ker S\) is closed, and
\(\operatorname{im}T\subset K\) by the full
distributional square-zero assertion in WD.B.
For \(v\in\operatorname{Dom}T^*\), use Hilbert
projection to write \(v=v_1+v_2\), with
\(v_1\in K\) and \(v_2\in K^\perp\).
Since \(v_2\perp\operatorname{im}T\), the
adjoint definition gives \(v_2\in\operatorname{Dom}T^*\)
and \(T^*v_2=0\). Hence
\(v_1\in\operatorname{Dom}T^*\cap\operatorname{Dom}S\),
and \(T^*v_1=T^*v\). Applying WD.12 to \(v_1\)
gives
\[
\begin{gathered}
|\langle v,g\rangle|=|\langle v_1,g\rangle|\\
 \le q^{-1/2}\|g\|_\varphi\|T^*v\|_\varphi .
\end{gathered}
\tag{WD.15}
\]
The linear functional
\(T^*v\mapsto\langle v,g\rangle\) on
\(\operatorname{im}T^*\) is therefore well-defined
and bounded. Hahn–Banach extends it to \(H_{q-1}\)
with the same bound. Riesz representation gives
\(f\in H_{q-1}\) such that
\(\langle T^*v,f\rangle=\langle v,g\rangle\)
for every \(v\in\operatorname{Dom}T^*\), and
the norm bound in WD.14 holds. In particular
the identity against compact smooth \(v\)
states precisely \(D''f=g\) distributionally.
This identifies \(f\) in the maximal domain
of \(T\) and proves WD.14.

Projection of \(f\) off the closed space \(\ker T\)
retains its derivative and decreases its norm.
The resulting solution is unique among solutions
orthogonal to \(\ker T\): the difference of any
two is in that kernel. No smoothness in positive
form degree is asserted by this weak solution
argument.

#### WD.E. Arbitrary locally square-integrable data

Every distributionally closed locally \(L^2\)
\((n,q)\)-form \(g\), \(1\le q\le n\), has a
locally \(L^2\) primitive on \(\Omega\).
To see this with an actual admissible weight,
keep the metric WD.1 and let
\[
\begin{gathered}
A_j=\{j\le\psi\le j+1\},\\
M_j=\int_{A_j}
                    |g|_\omega^2\,\frac{\omega^n}{n!},\\
b_j=\log\!\bigl(2^j(1+M_j)\bigr),\\
j\ge0 .
\end{gathered}
\tag{WD.16}
\]
Each integral is finite because its region is
compact. Overlaps of these regions cause no
problem: they only increase the following
upper bound. Construct a smooth convex
nondecreasing \(\kappa\), zero on
\((-\infty,0]\), positive at one, by
\[
\begin{gathered}
\kappa(t)=\int_0^t(t-s)e^{-1/s}\,ds\\
(t>0),\\
\kappa(t)=0\qquad(t\le0).
\end{gathered}
\tag{WD.17}
\]
Smoothness at zero follows because every
derivative of \(e^{-1/t}\) is a polynomial
in \(1/t\) times \(e^{-1/t}\), and each
of these tends to zero. Its first derivative
is nonnegative, and its second derivative
is \(e^{-1/t}\) for \(t>0\).
Choose \(a_j\ge0\) inductively so that
\[
\begin{gathered}
\chi(t)=\\
\sum_{j=0}^{\infty}a_j\kappa(t-j+1),\\
\chi(j)\ge b_j .
\end{gathered}
\tag{WD.18}
\]
At integer \(j\), the newly available summand
has value \(a_j\kappa(1)>0\) if \(a_j>0\);
all future summands vanish there. Thus such
coefficients exist. On every bounded interval
only finitely many summands are nonzero,
including after differentiation. The sum is
smooth, convex and nondecreasing.

Set \(\varphi=F(\psi)+\chi(\psi)\).
The chain rule and nonnegative first and
second derivatives of \(\chi\) imply WD.11.
Also \(\varphi\ge\chi(\psi)\), so
\[
\begin{gathered}
\|g\|_\varphi^2
 \le\sum_{j\ge0}M_j e^{-\chi(j)}\\
 \le\sum_{j\ge0}2^{-j}<\infty .
\end{gathered}
\tag{WD.19}
\]
WD.14 gives a globally weighted \(L^2\)
primitive, which is locally \(L^2\) since
the weight and metric are smooth and positive.
Multiplication or division by the nowhere
zero holomorphic form
\(dz_1\wedge\cdots\wedge dz_n\) transfers
the assertion to \((0,q)\)-forms, with the
fixed sign \((-1)^n\) if this top form is
written before the antiholomorphic factors.
Explicitly,
\(\bar\partial(\zeta\wedge u)=(-1)^n
\zeta\wedge\bar\partial u\) for
\(\zeta=dz_1\wedge\cdots\wedge dz_n\).
Thus solving for \((-1)^n\zeta\wedge g\)
and dividing its primitive by \(\zeta\)
solves \(\bar\partial u=g\).

#### WD.F. Scalar holomorphic vanishing

On every open subset of \(\Omega\), let
\(\mathcal A^q\) be the sheaf of locally
\(L^2\) \((0,q)\)-forms whose distributional
\(\bar\partial\) is locally \(L^2\).
For \(q=n\) the derivative is zero.
These are sheaves because membership and
the weak identities are local and locally
square-integrable forms glue uniquely up
to almost-everywhere equality. They form
a complex, since the distributional square
is zero. Its degree-zero kernel is
\(\mathcal O_\Omega\), by R.3: a
distribution killed by every
\(\partial/\partial\bar z_j\) is an
ordinary holomorphic function. Conversely,
holomorphic functions are locally \(L^2\)
and have zero derivative.

Near any point use a coordinate ball whose
closure lies in the given open set. That
ball has a nonnegative proper strictly
plurisubharmonic exhaustion, for example
\((r^2-|z-z_0|^2)^{-1}\). WD.E on this ball
gives a locally \(L^2\) primitive of every
closed form of positive degree. The
primitive itself lies in the graph sheaf,
because its derivative is the given
locally \(L^2\) form. Hence the sequence
\[
\begin{gathered}
0\longrightarrow\mathcal O_\Omega
\longrightarrow\mathcal A^0,\\
\mathcal A^0\xrightarrow{\bar\partial}\mathcal A^1
\longrightarrow\cdots,\\
\cdots
\xrightarrow{\bar\partial}\mathcal A^n
\longrightarrow0 .
\end{gathered}
\tag{WD.20}
\]
is exact as a sequence of sheaves.

Each \(\mathcal A^q\) is a module over smooth
functions. Indeed, for smooth \(a\) the weak
Leibniz rule gives
\(\bar\partial(au)=\bar\partial a\wedge u
+a\bar\partial u\), and both terms are
locally \(L^2\). The differential need not
be linear over this smooth-function ring;
acyclicity concerns the individual sheaves.
The full injective-resolution and
partition-of-unity proof R.5 applies to
each \(\mathcal A^q\), so each is acyclic
for ordinary global sections.

For completeness, an acyclic resolution
such as WD.20 computes the cohomology of
its initial sheaf. Put \(Z^0=\mathcal O_\Omega\)
and \(Z^{j+1}=\operatorname{im}
(\mathcal A^j\to\mathcal A^{j+1})\), with
\(Z^{n+1}=0\). The short exact sequences
\(0\to Z^j\to\mathcal A^j\to Z^{j+1}\to0\)
give the usual long exact cohomology
sequence as follows. Use the functorial
injective resolution \(I^r=GQ^r\) constructed
in R.5. Both \(G\) and \(Q\) there are
exact, so a short exact sequence gives
a short exact sequence in every resolution
degree. It splits in that degree because
its first term is injective. Applying
global sections therefore gives a short
exact sequence of complexes of vector
spaces. For a cocycle in the quotient
complex, lift it to the middle complex.
Its differential has a unique preimage
in the first complex, and that preimage
is a cocycle. Different lifts change it
by a boundary. This defines the connecting
map. Its kernel consists of the classes
whose lifts can be corrected to cocycles;
the kernel of the middle-to-quotient map
consists of classes that can be corrected
by a lifted boundary into the first
complex; and a class of the first complex
that becomes a boundary in the middle
is obtained by the connecting construction
from the image of that primitive. These
three statements prove exactness at
every position.

Acyclicity of
\(\mathcal A^j\) identifies positive
cohomology in adjacent degrees, and
identifies the degree-one group with the
cokernel of the section map. Iterating
these identities gives exactly the
cohomology of the section complex in
WD.20; after the last term all higher
groups vanish. This is also the elementary
dimension-shifting proof of the
acyclic-resolution assertion.

Finally WD.E solves every globally closed
positive-degree section of that complex.
Consequently
\[
\begin{gathered}
H^q(\Omega,\mathcal O_\Omega)=0,\\
q\ge1 .
\end{gathered}
\tag{WD.21}
\]
The same proof works for any fixed finite
direct sum of the structure sheaf, by
solving each component. WD.21 is the
structure-sheaf assertion; extension to an
arbitrary coherent sheaf requires additional
coherent-section arguments.

#### WD.G. Holomorphic \(L^2\) bounds for every jet

Let \(D_R\) be a polydisc with coordinate
radii \(R_1,\ldots,R_n>0\), centred at zero.
If \(h(z)=\sum_\alpha c_\alpha z^\alpha\)
is holomorphic there, then
\[
\begin{gathered}
A_\alpha(R)=\prod_{j=1}^n
        \frac{\pi R_j^{2\alpha_j+2}}{\alpha_j+1},\\
\int_{D_R}|h|^2\,dV\\
 =\sum_{\alpha\in\mathbb N^n}|c_\alpha|^2 A_\alpha(R).
\end{gathered}
\tag{WD.22}
\]
To prove it, first integrate on a smaller
closed polydisc. The Cauchy–Taylor proof
gives a uniformly absolutely convergent
series there. Finite polynomial partial
sums converge uniformly, so their squared
absolute values converge in integral.
Polar integration and Fubini show that
different monomials are orthogonal:
the angular integral of
\(e^{i(\alpha_j-\beta_j)\theta_j}\)
is zero unless \(\alpha_j=\beta_j\);
the radial integral is
\(R_j^{2\alpha_j+2}/(2\alpha_j+2)\).
Combining it with \(2\pi\) gives WD.22
at smaller radii. Increase those radii
to \(R_j\); monotone convergence on
both sides proves WD.22, allowing
infinite values.

In particular,
\[
\begin{gathered}
C_\alpha(R)=\alpha!\\
\quad\cdot\prod_{j=1}^n
    \left(\frac{\alpha_j+1}
      {\pi R_j^{2\alpha_j+2}}\right)^{1/2},\\
|\partial^\alpha h(0)|\\
 \le C_\alpha(R)\|h\|_{L^2(D_R)} .
\end{gathered}
\tag{WD.23}
\]
If \(K\Subset V\) and \(h\) is holomorphic
on \(V\), choose a fixed positive
polydisc radius so that the translated
polydisc about every point of \(K\)
is contained in a relatively compact
open subset \(V_0\) of \(V\). Applying
WD.23 at each such centre gives
\[
\begin{gathered}
\sup_K|\partial^\alpha h|\\
 \le C_{K,V_0,\alpha}\|h\|_{L^2(V_0)} .
\end{gathered}
\tag{WD.24}
\]
Replacing Euclidean measure and norm
on \(\overline{V_0}\) by any smooth
positive weighted bundle norm changes
only the constant. In particular this
applies to \(h\zeta\) with the metric
and weights used above. There is no
assertion that a general positive-degree
weak solution has these holomorphic
estimates.

#### WD.H. Approximation across an exhaustion gap

Let \(U\subset\Omega\) be open. Assume that
for each compact \(K\subset U\) there is
a nonnegative smooth proper strictly
plurisubharmonic function \(\psi\) on
\(\Omega\) and numbers \(b_0<b\) such that
\[
\begin{gathered}
K\subset\{\psi<b_0\},\\
\{\psi<b_0\}\Subset\{\psi<b\},\\
\{\psi<b\}\Subset U .
\end{gathered}
\tag{WD.25}
\]
The notation \(\Subset\) means that the
closure of the set on the left is
compact and contained in the set on
the right. Under this exact hypothesis
the restriction
\(\mathcal O(\Omega)\to\mathcal O(U)\)
has dense image for uniform convergence
on compact sets, and approximation
holds simultaneously for every
holomorphic derivative.

Fix such \(K,\psi,b_0,b\), and
\(h\in\mathcal O(U)\).

We may assume that the closed outer sublevel is compactly
contained in the given open set. Indeed, decrease the outer
level to any value strictly between the two original levels;
its closed sublevel lies in the original open outer sublevel.
Keep the notation for this decreased level.

Choose a real
smooth compactly supported function
\(\eta\) in \(U\), equal to one on a
neighbourhood of \(\{\psi\le b\}\).
Its extension by zero is smooth on
\(\Omega\). With
\(\zeta=dz_1\wedge\cdots\wedge dz_n\),
set
\[
g=\bar\partial(\eta h\zeta).
\tag{WD.26}
\]
This is a smooth compactly supported
closed \((n,1)\)-form. Its support lies
in \(\{\psi>b\}\), since the derivative
of \(\eta\) is zero near the whole
closed sublevel \(\{\psi\le b\}\).

Use the function WD.17 to put
\[
\begin{gathered}
\theta(t)=\frac{\kappa(t-b_0)}{\kappa(b-b_0)},\\
\varphi_\nu=F(\psi)+\nu\theta(\psi),
\\
\nu\ge0,\\
\omega=i\partial\bar\partial F(\psi).
\end{gathered}
\tag{WD.27}
\]
The denominator is positive.
The function \(\theta\) is smooth,
convex and nondecreasing, zero for
\(t\le b_0\), and at least one for
\(t\ge b\). Thus WD.11 holds for
every \(\nu\), with one fixed complete
metric. Since \(g\) is compactly
supported, \(C=\|g\|_{F(\psi)}^2\)
is finite, and
\[
\|g\|_{\varphi_\nu}^2\le Ce^{-\nu}.
\tag{WD.28}
\]
WD.D supplies an \((n,0)\)-form \(u_\nu\)
with
\[
\begin{gathered}
\bar\partial u_\nu=g,\\
\|u_\nu\|_{\varphi_\nu}^2\le Ce^{-\nu}.
\end{gathered}
\tag{WD.29}
\]
The global form
\(\eta h\zeta-u_\nu\) is
distributionally closed. Its coefficient
is therefore a holomorphic function
\(H_\nu\in\mathcal O(\Omega)\), by R.3.
On \(\{\psi<b_0\}\), the form \(u_\nu\)
is holomorphic and
\((H_\nu-h)\zeta=-u_\nu\).
Choose \(V_0\) with
\(K\subset V_0\Subset\{\psi<b_0\}\).
There the weights \(\varphi_\nu\)
all equal \(F(\psi)\), so WD.29
and WD.24 give, for every multi-index,
\[
\begin{gathered}
\sup_K|\partial^\alpha(H_\nu-h)|\\
 \le C_{K,\alpha}e^{-\nu/2}.
\end{gathered}
\tag{WD.30}
\]
All constants on the right are
independent of \(\nu\).

To obtain a single approximating
sequence on \(U\), choose an increasing
compact exhaustion \(K_j\) of \(U\)
whose interiors cover \(U\). Such sets
are obtained, for example, from
Euclidean bounds and a positive
distance from the closed complement.
Apply the preceding construction to
\(K_j\), and choose its parameter
\(\nu_j\) large enough that the
error is at most \(1/j\) for all
\(|\alpha|\le j\) on \(K_j\).
There are only finitely many such
indices. Every compact subset of \(U\)
is contained in some \(K_j\), and
each fixed derivative is eventually
among these indices. Thus the
resulting global holomorphic functions
converge to \(h\), with all derivatives,
uniformly on every compact subset
of \(U\). This proves the claimed
density without a boundary regularity
or a smoothness assertion for
positive-degree weak solutions.

#### WD.I. A concrete exhaustion and the estimate mechanism

For \(\Omega=\mathbb C\) and
\(\psi(z)=|z|^2\), the metric is
\[
\omega=i(1+4|z|^2)\,dz\wedge d\bar z .
\tag{WD.31}
\]
Take \(K=\{|z|\le1/2\}\),
\(b_0=1\), \(b=9/4\), and
\(U=\{|z|<5/2\}\).
A cutoff may be one on
\(\{|z|\le7/4\}\) and zero on
\(\{|z|\ge2\}\).
Its derivative is supported beyond
the radius \(3/2\) sublevel, where
the weight penalty is at least
\(\nu\). On the unit disc the
penalty is zero. WD.28–WD.30
therefore apply to every holomorphic
function on \(U\), with the indicated
compact \(K\).

![Concentric sublevels for the concrete one-variable example, and the weighted solution estimate leading to holomorphic jet approximation.](assets/weighted-dbar.png)

*The left panel uses the exact radii in WD.I;
the cutoff derivative is confined to the
shaded annulus. The right panel shows the
three implications proved in WD.D and WD.H.
The disc example illustrates the exhaustion
gap; the proofs WD.A–WD.H apply in every
dimension under their stated hypotheses.*

The weighted identity and complete-manifold
solution method can also be found in
[Demailly's freely available manuscript,
Chapters VII.1 and VIII.3–VIII.5](https://www-fourier.univ-grenoble-alpes.fr/~demailly/manuscripts/agbook.pdf).
The proof above includes the common graph
core, the operator domains, the Hilbert
solution argument, scalar cohomology
vanishing and the holomorphic approximation
estimate.


### 5.95. Coherent analytic sections and support-proper cohomology

Coherent sections carry a natural nuclear Fréchet
topology, even on a singular or nonreduced analytic
space. Locally this topology is an actual quotient
of holomorphic coefficient spaces. Finite jets detect
its closed relations and make the local quotient
topologies compatible. A finite cohomology bound
supplies the local vanishing needed for these
quotients and a uniform upper bound near a proper
fibre.

All analytic spaces here are Hausdorff and second
countable, locally given by a coherent ideal in an
open subset of a finite-dimensional complex affine
space. Nilpotent elements of the defining ideal
quotient are retained. Sheaf cohomology is derived
ordinary global-section cohomology. A sheaf without
a specified ring means a sheaf of complex vector
spaces.
These analytic spaces are locally compact:
in a chart, intersect the closed analytic model
with a small closed Euclidean ball inside
the ambient open set.

We use the complete earlier programme proofs of
weighted scalar vanishing, WD.F,
holomorphic jet estimates, WD.G,
the injective construction, R.5,
open mapping, NP.C,
nuclear subquotients, products and scalar closed-target descent, NT.D,
and nuclear holomorphic restrictions, NT.G.
The local analytic algebra inputs are
Oka coherence, Theorem 2.1,
Noetherian analytic local rings, Lemma 2.1,
and Krull intersection, Theorem 6.1.
The volume calculation in the first argument uses
the proved product integration and linear Jacobian
formula.

#### SC.A. Covers with a finite multiplicity bound

Let \(V\subset\mathbb R^m\) be open, \(m\ge1\),
and let \(\mathcal U\) be any open cover of \(V\).
The empty open set has zero section cohomology,
so assume \(V\) is nonempty.
Its members are open in \(\mathbb R^m\).
For an empty complement use distance \(+\infty\).
Put
\[
\begin{gathered}
E_U=\mathbb R^m\setminus U,\\
d_U(x)=\operatorname{dist}(x,E_U),\\
u(x)=\sup_{U\in\mathcal U}\min\{1,d_U(x)\},\\
\delta(x)=u(x)/100,\qquad x\in V .
\end{gathered}
\tag{SC.1}
\]
For a nonempty closed set the triangle inequality
gives \(d(x,E)\le|x-y|+d(y,E)\).
Exchanging the two points proves that distance
is Lipschitz with constant one. Truncation by
one preserves that bound; an empty complement
gives the constant function one.
The supremum of functions with Lipschitz constant
one has that same constant: compare each value
at two points and then take suprema in both
directions. Thus \(u\) is continuous, positive
on \(V\), and at most one. The function \(\delta\)
has Lipschitz constant \(1/100\). For each \(x\)
some member \(U_x\) satisfies
\[
\overline B(x,50\delta(x))\subset U_x .
\tag{SC.2}
\]
Indeed its distance value may be chosen strictly
larger than \(u(x)/2\). Also
\(\delta(x)\le\operatorname{dist}(x,\mathbb R^m\setminus V)/100\).

Choose a maximal pairwise disjoint family of balls
\(B(x_j,\delta_j/10)\), where
\(\delta_j=\delta(x_j)\). Such a family exists
by taking unions of chains of disjoint families.
It is countable because each of its disjoint
nonempty balls contains a distinct rational point.
Set \(V_j=B(x_j,\delta_j)\).
These larger balls cover \(V\). For the small
ball about an arbitrary \(x\) must meet a selected
small ball, by maximality. The Lipschitz constant,
bounded also by one, then gives
\[
\begin{gathered}
|x-x_j|<(\delta(x)+\delta_j)/10,\\
\delta(x)\le(11/9)\delta_j,\\
|x-x_j|<(2/9)\delta_j<\delta_j .
\end{gathered}
\tag{SC.3}
\]
The second inequality follows by moving the
\(\delta(x)/10\) term to the other side of
\(\delta(x)-\delta_j\le|x-x_j|\).
Every \(\overline V_j\) lies in the member \(U_{x_j}\)
of SC.2, so this is a shrinking refinement.

The family is locally finite. Fix \(y\in V\)
and consider the larger balls meeting
\(B(y,\delta(y)/10)\). Their centres satisfy
\[
|x_j-y|<\delta_j+\delta(y)/10 .
\tag{SC.4}
\]
The Lipschitz bound implies
\(\delta(y)/2<\delta_j<2\delta(y)\).
Consequently their disjoint small balls have
radii larger than \(\delta(y)/20\), and all
lie in \(B(y,4\delta(y))\). Only finitely
many such disjoint balls fit there. To justify
this last assertion and the next bound, the
volume of a ball of radius \(r\) is
\(c_mr^m\), with \(0<c_m<\infty\): the unit
ball contains and is contained in boxes of
positive finite volume, and the linear
Jacobian formula gives the scaling. Add
the volumes of any finite subfamily of
disjoint balls. This bounds its cardinality
uniformly, and hence bounds the entire
family's cardinality.

If \(y\in V_j\), the sharper comparison is
\[
\begin{gathered}
\frac{\delta(y)}{1+1/100}<\delta_j,\\
\delta_j<\frac{\delta(y)}{1-1/100}.
\end{gathered}
\tag{SC.5}
\]
The associated disjoint small balls lie
in the ball about \(y\) of radius
\((11/10)\delta(y)/(1-1/100)\).
Their radii exceed
\(\delta(y)/(10(1+1/100))\).
The volume comparison therefore bounds
the number of indices with \(y\in V_j\) by
\[
\left(\frac{1111}{99}\right)^m<12^m .
\tag{SC.6}
\]
In particular the multiplicity is at most
\(D_m=12^m\). This deliberately nonoptimal
integer is a uniform finite bound.

The same cover is a star refinement of
\(\mathcal U\). If \(V_k\cap V_j\ne\varnothing\),
the Lipschitz estimate at their centres gives
\(\delta_k/\delta_j<101/99\). Every point
of \(V_k\) is consequently at distance less
than \((1+2\cdot101/99)\delta_j<4\delta_j\)
from \(x_j\). Thus
\[
\begin{gathered}
\bigcup_{V_k\cap V_j\ne\varnothing}V_k
 \subset B(x_j,4\delta_j),\\
B(x_j,4\delta_j)\subset U_{x_j}.
\end{gathered}
\tag{SC.7}
\]
Applying the construction to an intersection
cover makes it subordinate to several prescribed
covers at once. We have proved, for every open
cover, a countable locally finite shrinking
star refinement of multiplicity at most \(D_m\).

#### SC.B. Čech cochains and derived sections

For a cover \(\mathcal U=(U_i)\), use alternating
Čech cochains
\[
\begin{gathered}
I=(i_0,\ldots,i_p),\\
U_I=U_{i_0}\cap\cdots\cap U_{i_p},\\
C^p(\mathcal U,F)
 =\prod_{i_0<\cdots<i_p}F(U_I),\\
p\ge0 .
\end{gathered}
\tag{SC.8}
\]
Extend a cochain antisymmetrically to all ordered
tuples, with repeated indices giving zero.
The differential is given below; deleting one
entry keeps the order of the others.
\[
\begin{gathered}
I=(i_0,\ldots,i_{p+1}),\\
(\partial c)_I
 =\sum_{r=0}^{p+1}(-1)^r
    c_{I\setminus i_r}\big|_{U_I}.
\end{gathered}
\tag{SC.9}
\]
Every pair of omitted indices occurs twice in
\(\partial^2\), with opposite signs, so its
square is zero.

We record the comparison with derived sections
in full. For an open inclusion \(j:W\hookrightarrow V\),
extension by zero \(j_!\) has stalk \(F_x\) on
\(W\) and zero off \(W\); its sections are
sections on the intersection that vanish near
the complement. This description proves both
its exactness, by stalks, and the adjunction
\(\operatorname{Hom}(j_!F,G)=
\operatorname{Hom}(F,G|_W)\).
Restriction to an open set therefore sends
injective sheaves to injective sheaves.

There is an augmented complex of sheaves
\[
\begin{gathered}
\cdots\longrightarrow
\bigoplus_{i_0<i_1}j_{i_0i_1!}\mathbb C
\\
\longrightarrow
\bigoplus_i j_{i!}\mathbb C
\\
\longrightarrow\mathbb C_V\longrightarrow0 .
\end{gathered}
\tag{SC.10}
\]
Its arrows are the signed inclusions omitting
one index. At a point \(x\), it is the
augmented simplicial chain complex of the
set of indices whose opens contain \(x\).
Choose one such index and insert it at the
front of a chain, interpreting tuples
alternatingly. The two composites with
the boundary add to the identity: terms
omitting an old index cancel in pairs,
and the term omitting the inserted index
is the original chain. This also gives
the augmentation identity in degree zero.
Hence SC.10 is exact at every stalk.

Applying \(\operatorname{Hom}(-,I)\) for an
injective sheaf \(I\) preserves exactness
and gives the augmented Čech complex of
sections of \(I\). Here a morphism
\(\mathbb C_W\to I|_W\) is determined by the
image of the constant section one, so the
adjunction gives exactly \(I(W)\), including
disconnected \(W\). Thus
\[
\begin{gathered}
H^p(C^\bullet(\mathcal U,I))=0,\\
p>0,\\
H^0(C^\bullet(\mathcal U,I))=I(V).
\end{gathered}
\tag{SC.11}
\]

Take an injective resolution \(F\to I^\bullet\);
the explicit construction in R.5 supplies one.
The double complex
\[
\begin{gathered}
B^{p,q}=C^p(\mathcal U,I^q),\\
d_{\mathrm{tot}}=\partial+(-1)^p d_I
\end{gathered}
\tag{SC.12}
\]
has square-zero differential, because restriction
commutes with \(d_I\). Only finitely many
bidegrees occur in any fixed total degree.
Its total cohomology is \(H^\bullet(V,F)\).
Indeed, for a total cocycle first eliminate its
largest Čech-degree component by SC.11.
That component is a horizontal cocycle;
when its degree is positive it is a horizontal
boundary. Subtract a total boundary with that
primitive. The replacement has smaller
Čech degree and larger resolution degree.
After finitely many steps only Čech degree
zero remains, where the horizontal kernel
is exactly \(\Gamma(V,I^\bullet)\).
The same elimination on a total primitive
shows that two resulting global cocycles
are cohomologous exactly when the original
total cocycles are. This proves the
cohomology identification, including
injectivity as well as surjectivity.

We need one local refinement assertion. Suppose
\(q>0\) and \(z\in C^p(\mathcal U,I^q)\)
satisfies \(d_Iz=0\). After a refinement,
\(z\) has a primitive for \(d_I\) as a
Čech cochain. First take a countable
locally finite shrinking refinement
\((W_j)\) with
\(\overline W_j\subset U_{\sigma(j)}\).
At a point \(x\), only finitely many
\(\overline W_j\) contain \(x\).
There is a neighbourhood meeting no
other \(W_j\): start with one meeting
only finitely many, then avoid the
finitely many closures not containing
\(x\). For every tuple of the remaining
indices, the corresponding component
of \(z\) is defined near \(x\), since
all \(U_{\sigma(j)}\) contain \(x\).
Exactness of \(I^\bullet\) in positive
degree gives a primitive near \(x\).
There are finitely many such tuples,
so choose one neighbourhood \(N_x\)
on which all these primitives exist.

Take a star refinement of \((N_x)\)
which also refines \((W_j)\), using
SC.A. For an intersecting tuple of
its opens, their union lies in some
\(N_x\). Every old \(W_j\) labelling
that tuple meets \(N_x\), so belongs
to the finite list just considered.
Restrict its chosen primitive to
the tuple intersection. Make these
choices for increasing tuples and
extend alternatingly. They form
a cochain primitive of the refined
\(z\), as claimed. No compatibility
between different tuples is required
for a primitive in the resolution
direction.

It follows that every derived
cohomology class is represented by
an ordinary Čech cocycle after
refinement. To see all signs, start
with a total cocycle of degree \(k\).
Its component \((0,k)\) is vertically
closed. If \(k>0\), apply the preceding
assertion, choose a vertical primitive
and subtract its total differential.
Then the first component has been
eliminated. At component
\((p,k-p)\), which is now vertically
closed, use the primitive multiplied
by \((-1)^p\) to cancel it.
Proceed while \(k-p>0\).
Only \((k,0)\) remains. Its vertical
kernel is \(C^k(\mathcal U,F)\), and
its horizontal differential is zero.
For injectivity, start with a total
primitive of an ordinary Čech cocycle.
Apply the same elimination to the
positive-resolution-degree components
of that primitive. Each is vertically
closed because its total differential
has only resolution degree zero.
After refinement the primitive lies
in \(C^{k-1}(\mathcal U,F)\).
Thus a class becomes zero precisely
when its Čech representative becomes
a boundary after refinement.
In degree zero this assertion is
the sheaf gluing axiom directly.

Refinement choices cause no ambiguity.
For two refinement maps \(\alpha,\beta\)
from a cover \(\mathcal V\) to
\(\mathcal U\), set, on a degree-\(p\)
cochain with \(p>0\),
\[
\begin{gathered}
J=(j_0,\ldots,j_{p-1}),\\
a_r=(\alpha j_0,\ldots,\alpha j_r),\\
b_r=(\beta j_r,\ldots,\beta j_{p-1}),\\
(hc)_J=\sum_{r=0}^{p-1}(-1)^r
 c_{a_rb_r}\big|_{V_J}.
\end{gathered}
\tag{SC.13}
\]
Here \(V_J\) is the intersection
of the opens labelled by \(J\);
the two lists in \(c_{a_rb_r}\)
are concatenated in the indicated
order.
Omitting any index away from the
change between the two lists cancels
with the corresponding term of
\(h\partial\). At the change the
adjacent terms cancel telescopically;
the two remaining endpoint lists
are the all-\(\beta\) list and the
negative all-\(\alpha\) list.
Consequently
\[
\partial h+h\partial=\beta^*-\alpha^* .
\tag{SC.14}
\]
In degree zero this is just the
difference of the two restrictions,
obtained from \(h\partial\).
Any two covers have a common
refinement by intersecting them and
using SC.A. We have therefore proved
the canonical comparison
\[
\begin{gathered}
H^k(V,F)\\
 \simeq\underset{\mathcal U}{\operatorname{colim}}\,
       H^k(C^\bullet(\mathcal U,F)).
\end{gathered}
\tag{SC.15}
\]

If every finite nonempty intersection
of a fixed cover is acyclic for \(F\),
no refinement is needed in the vertical
elimination: restriction of the injective
resolution computes cohomology there,
and each closed positive-degree
component has a section primitive.
Thus that fixed Čech complex already
computes \(H^\bullet(V,F)\). This is
the acyclic-cover assertion, with its
comparison proved by SC.12.

#### SC.C. Finite cohomological dimension

Let \(D_m=12^m\) as in SC.A. A cover
of multiplicity at most \(D_m\) has
no nonempty intersection of \(D_m+1\)
distinct members, so its alternating
Čech groups vanish in degrees at
least \(D_m\). Such covers are cofinal
by SC.A. Applying SC.15 gives
\[
\begin{gathered}
H^k(V,F)=0,\\
k\ge D_m .
\end{gathered}
\tag{SC.16}
\]
for every sheaf \(F\) on every open
\(V\subset\mathbb R^m\).
For \(m=0\), an open subset of
\(\mathbb R^0\) is empty or a point;
sections are exact, and the same
statement holds with \(D_0=1\).

For a closed inclusion \(i:A\hookrightarrow V\),
direct image has stalk \(F_x\) on
\(A\) and zero off \(A\), so it is
exact. The inverse-image functor
is exact by its stalk description.
Its adjunction with \(i_*\) implies
that \(i_*\) preserves injectives.
Since \(\Gamma(V,i_*F)=\Gamma(A,F)\),
injective resolutions give
\[
H^k(A,F)=H^k(V,i_*F).
\tag{SC.17}
\]
The bound SC.16 holds on \(A\) as
well. It holds also on any open
subset of \(A\), because that subset
is closed in a suitable open subset
of \(V\). No analytic structure or
smoothness of the closed subset
is needed for this conclusion.

#### SC.D. Coherent algebra and finite free prefixes

We make explicit the module algebra
used with Oka's theorem. A coherent
module is locally finitely generated,
and the kernel of each map from a
finite free module to it is locally
finitely generated.

First, a finite-type submodule of
a coherent module is coherent:
the relation sheaf of any finite
list is unchanged when the list
is regarded in the larger module.
Second, a direct sum of two coherent
modules is coherent. Given a map
from a finite free module to the
sum, first impose the relations
in the second summand. Choose
finitely many generators of that
kernel near the point. Their
images in the first summand have
a finite-type relation sheaf.
The images of those relations
generate the kernel of the
original map. This proves the
assertion and, by induction, the
coherence of every finite free
module over a coherent ring sheaf.

A finite presentation over a coherent
ring sheaf defines a coherent module.
Indeed, write the presentation as
\[
\mathcal A^a\xrightarrow{P}\mathcal A^b
 \longrightarrow F\longrightarrow0 .
\tag{SC.18}
\]
Locally lift any map
\(\mathcal A^q\to F\) to a matrix
\(Q:\mathcal A^q\to\mathcal A^b\).
There are only finitely many columns,
so one common neighbourhood suffices.
The kernel of \((Q,-P)\) in
\(\mathcal A^{q+a}\) is finite type
by coherence of \(\mathcal A^b\).
Its image under projection to
\(\mathcal A^q\) is exactly the
relation sheaf of the original
map to \(F\). Thus that relation
sheaf is finite type.
Conversely, a coherent module has
a finite presentation by choosing
generators and then generators
of their relation sheaf.

For a map of coherent modules,
its image is finite type and hence
coherent as a submodule of its
coherent target. Its kernel is
finite type by pulling back along
a finite free presentation of
the source: the kernel of the
composite free-to-target map is
finite type and maps onto the
desired kernel. That kernel is
therefore a finite-type submodule
of the coherent source. The cokernel
has a finite presentation: add
lifted generators of the image to
the relations in a presentation
of the target. It too is coherent.
These arguments prove all coherence
stability statements used below.

Let \(A\subset W\subset\mathbb C^N\)
be an analytic patch, with ideal
sheaf \(J\), and let \(S\) be coherent
over \(\mathcal O_A=\mathcal O_W/J\).
The ambient module \(i_*S\) is coherent
over \(\mathcal O_W\). Near a point,
choose a presentation of \(S\) by
finite free \(\mathcal O_A\)-modules.
Lift its finite matrix to holomorphic
functions on an ambient neighbourhood,
and choose finitely many generators
\(g_1,\ldots,g_t\) of \(J\) there.
If the presentation has \(b\) target
generators, an ambient presentation
matrix is
\[
\bigl(\widetilde P,\ g_1I_b,\ldots,g_tI_b\bigr).
\tag{SC.19}
\]
Its cokernel is the prescribed
presentation modulo \(J\) on \(A\).
Off \(A\), one \(g_j\) is a unit
at each point, so that cokernel
is zero. This proves the assertion
as a statement about the actual
sheaf. Oka's theorem and SC.18
give coherence. The ideal \(J\)
has not been replaced by its
radical.

For any integer \(L\ge1\), coherence
now supplies a neighbourhood of
the chosen point and a finite free
prefix with \(L\) free terms:
\[
\begin{gathered}
P_{L-1}\longrightarrow\cdots
\longrightarrow P_0\\
\longrightarrow i_*S\longrightarrow0,\\
P_j=\mathcal O^{p_j}.
\end{gathered}
\tag{SC.20}
\]
Choose a finite generating list,
then a finite list for its coherent
kernel, and repeat \(L\) times,
shrinking the neighbourhood at
each step. The final neighbourhood
is common to this finite list
of sheaf maps. If \(Z_{-1}=i_*S\)
and \(Z_j=\ker(P_j\to Z_{j-1})\),
the precise exact sequences are
\[
\begin{gathered}
0\longrightarrow Z_j\longrightarrow P_j\\
\longrightarrow Z_{j-1}\longrightarrow0,\\
0\le j<L .
\end{gathered}
\tag{SC.21}
\]
The last kernel is retained;
no terminating free resolution
or bound on projective dimension
is assumed.

#### SC.E. Local coherent vanishing and actual section quotients

In SC.D take \(L=D_{2N}=12^{2N}\),
and let \(V\) be any open subset
of the resulting ambient neighbourhood
with a smooth nonnegative proper
strictly plurisubharmonic exhaustion.
For \(N=0\), use an open point and
the exactness of sections directly.
Write \(U=A\cap V\).
Every \(P_j|_V\) is a finite direct
sum of \(\mathcal O_V\), so WD.F
gives its vanishing in every
positive cohomology degree.
The long exact sequences of SC.21
therefore give, for \(q\ge1\),
\[
\begin{gathered}
H^q(U,S)=H^q(V,Z_{-1}),\\
H^q(V,Z_{-1})\\
 \simeq H^{q+1}(V,Z_0)\\
 \simeq\cdots\\
 \simeq H^{q+L}(V,Z_{L-1})=0 .
\end{gathered}
\tag{SC.22}
\]
The first equality is SC.17.
The last degree is at least \(L\),
so SC.16 proves its vanishing.
The same dimension shifting,
starting with \(Z_0\), gives
\[
\begin{gathered}
H^1(V,Z_0)\\
 \simeq H^L(V,Z_{L-1})=0 .
\end{gathered}
\tag{SC.23}
\]
For \(L=1\) this is the direct
bound \(H^1(V,Z_0)=0\).
Consequently the free presentation
is also surjective on sections:
\[
\begin{gathered}
0\longrightarrow Z_0(V)
\longrightarrow\mathcal O(V)^{p_0}
\\
\longrightarrow S(U)\longrightarrow0 .
\end{gathered}
\tag{SC.24}
\]
This is an exact sequence of actual
vector spaces, with its full
relation space, before any topology
is put on the last term.

Every point has arbitrarily small
patches of this form: choose a
polydisc compactly contained in
the common ambient neighbourhood.
The sum of
\((r_j^2-|z_j-a_j|^2)^{-1}\)
is a nonnegative smooth proper
strictly plurisubharmonic exhaustion
of that polydisc. Its positivity
follows by differentiating each
one-variable term, and properness
follows because approaching its
boundary makes one summand tend
to infinity. Thus SC.22 applies
at every point, with all singular
and nonreduced coefficient data.

#### SC.F. Finite jets and closed analytic relation spaces

Let \(x\) be a point of an analytic
space and \(\mathfrak m_x\) the
maximal ideal of its local ring.
For a coherent module \(S\), put
\[
\begin{gathered}
J_x^r(S)=S_x/\mathfrak m_x^rS_x,\\
r\ge1 .
\end{gathered}
\tag{SC.25}
\]
This is a finite-dimensional complex
vector space. In an ambient chart
the maximal ideal is generated
by the coordinate differences.
Taylor expansion modulo their
\(r\)-th power is spanned by
the finitely many monomials of
total degree less than \(r\).
Their products with a finite
generating list of \(S_x\)
span the displayed quotient.

The analytic local ring is a quotient
of the Noetherian holomorphic local
ring. It is Noetherian: take the
preimage of an ideal and descend
a finite generating list.
Krull intersection applies because
the analytic local ring is Noetherian
and \(S_x\) is finite. It gives
\[
\bigcap_{r\ge1}\mathfrak m_x^rS_x=0 .
\tag{SC.26}
\]
The earlier proof includes Artin–Rees
and the determinant argument, so
SC.26 applies also to quotient
modules and to nonreduced local
rings. Hence a section is zero
exactly when its image in every
\(J_x^r(S)\), at every point,
is zero.

On a patch in SC.24, each map
from \(\mathcal O(V)^{p_0}\) to
\(J_x^r(S)\) is continuous for
uniform convergence on compact
sets. In fact the map factors
through the finite Taylor jets
of its holomorphic coefficient
functions at the ambient point.
Those jets depend continuously
on the coefficients by WD.G,
or directly by the polydisc
Cauchy formula. Multiplication
by the fixed generator germs,
followed by the finite-dimensional
quotient, is a fixed linear map
between finite-dimensional spaces.
SC.26 therefore identifies
\[
\begin{gathered}
e_x^r:\mathcal O(V)^{p_0}
                     \longrightarrow J_x^r(S),\\
Z_0(V)=\bigcap_{x\in U,\ r\ge1}\ker e_x^r .
\end{gathered}
\tag{SC.27}
\]
This is a closed subspace. The
intersection may have uncountably
many conditions; each kernel is
closed, and arbitrary intersections
of closed sets are closed.

Give \(S(U)\) the quotient topology
in SC.24. The source is nuclear
Fréchet by NT.G. The relation
space is closed by SC.27, so
NT.D makes this actual quotient
a Hausdorff nuclear Fréchet space.
Every section-to-jet map
\[
j_x^r:S(U)\longrightarrow J_x^r(S)
\tag{SC.28}
\]
is continuous, by the quotient
definition and the preceding
coefficient calculation.

#### SC.G. The topology on sections of any coherent sheaf

Let \(U\) be any open subset of
an analytic space \(X\), and let
\(S\) be coherent. Choose a countable
cover \((U_i)\) by patches of SC.24.
Such a subcover exists because
second countability implies the
countable-subcover property: for
each basis member contained in
a cover member, select one such
member. Set \(E_i=S(U_i)\), with
the quotient topology of SC.F.

Inside the nuclear Fréchet product
\(\prod_iE_i\), impose the equations
\[
\begin{gathered}
j_x^r(s_i)=j_x^r(s_j),\\
x\in U_i\cap U_j,\qquad r\ge1 .
\end{gathered}
\tag{SC.29}
\]
Each equation is closed by SC.28.
By SC.26 it means equality of
the two germs. Equality of all
germs means equality of sections
on the overlap, by the sheaf
axiom. Thus the solution space
is exactly the compatible-tuple
space, canonically \(S(U)\).
It is closed in a countable
nuclear Fréchet product. NT.D
therefore supplies a Hausdorff
nuclear Fréchet topology on
the full space of sections.
All global section-to-jet maps
are continuous: select any
patch of the cover containing
the point and use its coordinate
projection and SC.28.

This topology is independent of
every presentation, ambient chart
and cover. Indeed, for two choices
the graph of the identity consists
of pairs whose jets agree at all
points and all orders. It is closed
in the product of the two Fréchet
spaces, and SC.26 shows that it is
exactly the identity graph.
The closed graph argument here
is a consequence of NP.C:
a closed graph is Fréchet;
its projection onto the source
is a continuous linear bijection;
open mapping makes the inverse
continuous; projection onto the
target then gives continuity
of the map. Apply this in both
directions. In particular, on
a patch of SC.24 the global
construction agrees with its
original quotient topology.

Every restriction and every
coherent sheaf morphism induces
a continuous map on sections.
For a morphism \(S\to T\) over
an open subset, the maps on
finite jets are linear maps
\[
J_x^r(S)\longrightarrow J_x^r(T),
\tag{SC.30}
\]
because the morphism preserves
the powers of \(\mathfrak m_x\).
The graph of the section map
is specified by equality with
these finite-jet images. It is
closed by SC.28–SC.29, and
SC.26 makes these equations
sufficient. The same closed
graph argument proves continuity.
Restriction is the case of
the identity sheaf morphism
with a smaller open domain.

If \(T\subset S\) is a coherent
subsheaf, the quotient is coherent
by SC.D. The space \(T(U)\) is
the kernel of the continuous map
\(S(U)\to(S/T)(U)\), hence is
closed in \(S(U)\). Its own
topology agrees with the induced
topology by open mapping onto
that closed image. We have proved
the strict left-exact section
sequence
\[
\begin{gathered}
0\longrightarrow T(U)
\longrightarrow S(U)\\
\longrightarrow(S/T)(U).
\end{gathered}
\tag{SC.31}
\]
Surjectivity of the last arrow
requires its cohomological
hypothesis. On the patches
of SC.24 it has already been
proved for the displayed free
presentation. No global vanishing
for arbitrary coherent sheaves
on a general open set is inferred.

#### SC.H. Relatively compact restrictions are nuclear

Suppose \(V\Subset U\) are open
in \(X\), meaning that
\(\overline V\) is compact and
contained in \(U\). Then
\[
\begin{gathered}
S(U)\longrightarrow S(V),\\
\text{a scalar nuclear map.}
\end{gathered}
\tag{SC.32}
\]
To prove this, first take nested
concentric polydiscs \(W'\Subset W\) inside
one common SC.D ambient neighbourhood.
Let \(U_0=A\cap W\) and
\(U_0'=A\cap W'\).
The same finite presentation gives
surjective quotient maps from
\(\mathcal O(W)^p\) and
\(\mathcal O(W')^p\) onto
\(S(U_0)\) and \(S(U_0')\).
The holomorphic restriction between
these finite free coefficient
spaces is nuclear by NT.G.
Postcompose with the target
quotient map. It vanishes on
the closed source relation space.
The scalar closed-quotient
conclusion of NT.D therefore
gives a nuclear induced map
\(S(U_0)\to S(U_0')\).
Both the coefficient source
and its Hausdorff quotient
are nuclear, as required
in that conclusion.

Choose finitely many such pairs
\(U_i'\Subset U_i\Subset U\)
whose smaller members cover
\(\overline V\).
Restriction from \(S(U)\) to
\(\prod_iS(U_i)\) is continuous
by SC.G. The following map is
nuclear, by the local assertion
and its continuous postcomposition:
\[
\begin{gathered}
\prod_iS(U_i)\\
\longrightarrow
\prod_iS(U_i'\cap V).
\end{gathered}
\tag{SC.33}
\]
Finite sums of nuclear maps
are nuclear by concatenating
their absolutely summable
representations. The composite
from \(S(U)\) has image in
the compatible tuples on the
finite cover \((U_i'\cap V)\)
of \(V\).

That compatible-tuple space is
closed in the displayed product:
use the continuous restriction
maps to pairwise overlaps and
take the kernels of their
differences. Its topology is
the canonical topology on
\(S(V)\). For both are nuclear
Fréchet spaces, their bijective
restriction map is continuous,
and open mapping makes its
inverse continuous. NT.D's
scalar closed-target conclusion
now makes the composite nuclear
as a map into this actual
closed space \(S(V)\).
Its domain \(S(U)\) is nuclear
by SC.G. This proves SC.32.
Only scalar nuclear descent
has been used here; a balanced
coefficient tensor requires
its separate transversality
argument.

#### SC.I. A finite-cover cohomology bound

Let \(Z\) have a finite open
cover of \(r\) members.
Suppose every nonempty finite
intersection has sheaf cohomology
zero in degrees at least \(B\),
for the restricted sheaf in
question. Then
\[
\begin{gathered}
H^k(Z,F)=0,\\
k\ge r+B-1 .
\end{gathered}
\tag{SC.34}
\]
Use the injective double complex
SC.12 for this finite cover.
Its Čech columns have degrees
zero through \(r-1\).
For a total cocycle of degree
\(k\ge r+B-1\), the component
of Čech degree zero is vertically
closed and has resolution degree
at least \(B\). It has a section
primitive on each cover member.
Subtract its signed total
differential. At Čech degree
\(p\), the remaining component
is vertically closed, of degree
\(k-p\ge B\), so has a primitive
on every tuple intersection.
Multiply that primitive by
\((-1)^p\) and subtract its
total differential. After
\(r\) steps every component
has been removed. This is
a total boundary, and SC.12
proves SC.34.

#### SC.J. Uniform bounds near every proper-support fibre

Let \(f:X\to Y\) be an analytic
map and let \(C\subset X\) be
closed with \(f|_C\) proper.
For every \(y\in Y\), there
are a neighbourhood \(T\) and
an integer \(M\) such that,
for every sheaf \(F\) supported
in \(C\) and every open
\(W\subset T\),
\[
\begin{gathered}
H^k(f^{-1}W,F)=0,\\
k\ge M .
\end{gathered}
\tag{SC.35}
\]
The integer and neighbourhood
depend on the chosen fibre
and its finite chart cover,
and work for all such sheaves.

Cover the compact fibre \(C_y\)
by finitely many analytic chart
opens \(U_1,\ldots,U_r\), with
\(U_i\) a closed analytic model
in an open subset of
\(\mathbb C^{N_i}\).
The part \(C\setminus\bigcup_iU_i\)
is closed in \(C\), and its
proper image is closed in \(Y\).
Here proper maps are closed
because \(Y\) is locally compact
Hausdorff: intersect with a
compact neighbourhood of a
point outside the image;
the inverse image is compact,
and the image of its closed
part is compact and closed.
Its complement inside the
neighbourhood avoids the
whole closed part's image.
Shrink \(T\) to avoid that
image. Then
\[
C\cap f^{-1}T\subset\bigcup_{i=1}^r U_i .
\tag{SC.36}
\]
If \(C_y\) is empty, the same
argument makes \(C\cap f^{-1}T\)
empty, and SC.35 is immediate.

For the nonempty case put
\[
\begin{gathered}
B=\max_i12^{2N_i},\\
M=r+B-1 .
\end{gathered}
\tag{SC.37}
\]
For every open \(W\subset T\),
cover \(C\cap f^{-1}W\) by
its intersections with the
\(U_i\). Any multiple
intersection is open in one
closed subset of an ambient
open subset of
\(\mathbb R^{2N_i}\).
SC.C gives its cohomology
vanishing in degrees at
least \(12^{2N_i}\), hence
at least \(B\), for every
sheaf on that intersection.
SC.I gives vanishing in
degrees at least \(M\).

A sheaf supported in the
closed set \(C\cap f^{-1}W\)
is the direct image of its
restriction to that closed
set: the natural adjunction
map is an isomorphism on
all stalks, including the
zero stalks outside the
support. SC.17 now gives
SC.35 on \(f^{-1}W\).
The closed support need
not be reduced, smooth
or itself given an analytic
structure for this
cohomology argument.

The derived direct image
therefore satisfies
\[
R^k f_*F|_T=0\qquad(k\ge M).
\tag{SC.38}
\]
In fact its stalk at any
point is the filtered
colimit of
\(H^k(f^{-1}W,F)\) over
neighbourhoods of that
point. To verify this,
compute with an injective
resolution. A stalk is
the filtered colimit of
the section complexes.
Filtered colimits of vector
spaces commute with their
cohomology: a represented
cocycle becomes a cocycle
at a sufficiently late
stage, and a represented
boundary has its primitive
at a sufficiently late
stage. These two
representative checks give
both directions of the
cohomology comparison.
SC.35 makes every indicated
cohomology group zero.

The same conclusion holds
in the category of modules
over a sheaf of complex
algebras. Forgetting that
module structure sends
injectives to injective
complex-vector sheaves:
its left adjoint is tensor
with the algebra over
\(\mathbb C\), which is
exact on stalks. Thus
the preceding cohomology
bound computes the
underlying vector sheaves
of those derived sections
as well.

For a bounded complex
with cohomology supported
in \(C\) and concentrated
in degrees \([a,b]\),
the resulting upper
bound is
\[
\begin{gathered}
\mathcal H^k(Rf_*K)|_T=0,\\
k<a\quad\text{or}\\
k>b+M-1 .
\end{gathered}
\tag{SC.39}
\]
For one sheaf this is
SC.38 and the nonnegative
degrees of its injective
resolution. For a finite
cohomology interval,
successively separate the
top cohomology sheaf by
the canonical truncation
triangle. The long exact
cohomology sequence gives
the claimed bound by
induction. Truncation
can be constructed on
the complex by replacing
the terminal term by
the kernel or cokernel
of its differential;
the corresponding
quotient has just the
single top cohomology
sheaf. Thus only the
finite list of cohomology
sheaves enters the
induction. The lower
bound also follows from
a bounded-below injective
resolution starting in
degree \(a\).

#### SC.K. Proper supports and the two section functors

For a sheaf \(F\) supported
in \(C\), the natural
comparison is an isomorphism
\[
Rf_!F\simeq Rf_*F .
\tag{SC.40}
\]
Use the closed inclusion
\(i:C\hookrightarrow X\).
An injective resolution on
\(C\), followed by the
exact injective-preserving
functor \(i_*\), resolves
\(F\) by injectives on
\(X\) supported in \(C\).
For any open \(W\subset Y\),
every section of one of
these terms over \(f^{-1}W\)
has support closed in
\(C\cap f^{-1}W\).
Its map to \(W\) is proper,
being a closed part of
the proper map \(C\to Y\).
Thus \(f_!\) and \(f_*\)
are identical on this
resolution, including
their natural comparison
map. This proves SC.40.
For a bounded complex
with cohomology supported
in \(C\), the same
isomorphism follows by
the finite truncation
induction just used in
SC.J. Hence SC.39
also holds for \(Rf_!\).

SC.39 bounds derived sections
of coefficient sheaves.
For an operator direct image,
the transfer tensor contributes
an additional homological
amplitude. The total amplitude
depends on that tensor as
well as on the section bound.

![A planar pair of cover balls with their disjoint small balls, and the quotient, gluing and proper-support bounds.](assets/coherent-section-topology.png)

*The left panel measures lengths in units of
the first cover radius. The normalized
radii are \(1\) and \(101/100\),
their centres have distance \(3/2\),
and the disjoint small balls have
one tenth those radii. The circle of
radius \(4\) shows the star margin
in SC.7 in the first ball's units.
The margin \(50\) in SC.2 is outside
the drawn scale. This pair illustrates
the estimates; SC.A proves the
countable cover and multiplicity
bound in every dimension. The right
panel records the actual quotients,
closed compatibility space and
cohomology bound proved in
SC.E–SC.K.*

The local coherent vanishing
and section-topology method
are discussed in
[Demailly's free author manuscript,
Chapter IX, §4.B–§4.C](https://www-fourier.univ-grenoble-alpes.fr/~demailly/manuscripts/agbook.pdf).
SC.A–SC.C give the finite
cohomology bound used here;
SC.D–SC.H give the actual
coherent section spaces,
their closed relations and
nuclear restrictions;
SC.I–SC.K give the
support-proper upper bound.


### 5.96. Compact maps and cohomology of inverse limits

The cohomology of a complex of Fréchet spaces carries its quotient
topology, which can fail to be Hausdorff. Compact restrictions and
dense approximation have different consequences for this topology.
We prove the finite-dimensionality and inverse-limit statements
with the actual spaces of cycles and boundaries retained.

All vector spaces are complex. A Fréchet space is Hausdorff,
locally convex, complete and defined by countably many seminorms.
We use the complete open mapping argument in
NP.C
and the separation and complete-quotient argument in
NT.A.
For the sheaf application we use the full
derived/Čech comparison, SC.B
and the
coherent-section topology, SC.G.

#### ML.A. Quotient topologies and finite codimension

Let a complex and its cohomology be
\[
\begin{gathered}
d^q:E^q\longrightarrow E^{q+1},\\
d^{q+1}d^q=0,\\
Z^q(E)=\ker d^q,\\
B^q(E)=d^{q-1}E^{q-1},\\
H^q(E)=Z^q(E)/B^q(E).
\end{gathered}
\tag{ML.1}
\]
The differentials are continuous, so the cycle spaces are closed
Fréchet subspaces. Give cohomology its actual quotient topology;
do not close the boundary space in this definition. A quotient map
of topological vector spaces is open: the preimage of the image
of an open set is the union of its translates by the kernel.
The quotient is Hausdorff exactly when its kernel is closed.
Indeed, a closed kernel makes distinct cosets separable by
translates of sufficiently small zero neighbourhoods; conversely
the inverse image of the closed zero coset is the kernel.

A finite-dimensional subspace of a Fréchet space is closed. To see
this directly, fix an independent basis and suppose a sequence in
its span converges in the ambient space. If its coefficient
vectors were unbounded, division by their Euclidean norms and
passage to a convergent subsequence on the unit sphere would
give a nonzero coefficient vector representing zero. This
contradicts independence. The coefficient vectors are therefore
bounded, a subsequence converges, and its limit represents the
ambient limit. Since Fréchet spaces are metrizable, this proves
closedness. The same argument gives the usual topology on that
finite-dimensional span.

**Finite-codimension lemma.** If a continuous linear map
\[
\begin{gathered}
t:P\longrightarrow Q,\\
\dim_{\mathbf C}Q/t(P)<\infty
\end{gathered}
\tag{ML.2}
\]
has Fréchet domain and target, then its image is closed.
Choose a finite-dimensional algebraic complement \(S\) to the
image. The continuous map
\[
\begin{gathered}
a:P\oplus S\longrightarrow Q,\\
a(x,s)=t(x)+s
\end{gathered}
\tag{ML.3}
\]
is surjective and is open by NP.C. Projection onto \(S\) vanishes
on its kernel, because \(S\cap t(P)=0\). It descends through this
open quotient to a continuous map \(Q\to S\). Its kernel is
exactly \(t(P)\), proving the lemma. Consequently the quotient in
ML.2 is Hausdorff with its ordinary finite-dimensional topology.
This argument uses finite dimension over the scalar field; it
does not assert that every finite module over a Banach algebra
has a closed image.

#### ML.B. Compact perturbations of surjections

A continuous linear map is **compact** here if it maps some
zero neighbourhood to a relatively compact set. A map merely
sending bounded sets to relatively compact sets is not being
assumed compact in this definition.

**Compact perturbation theorem.** Let \(P,Q\) be Fréchet spaces,
let \(g:P\to Q\) be a continuous linear surjection, and let
\(h:P\to Q\) be compact. Then
\[
\begin{gathered}
f=g+h,\\
f(P)\text{ is closed in }Q,\\
\dim_{\mathbf C}Q/f(P)<\infty.
\end{gathered}
\tag{ML.4}
\]
Choose an open absolutely convex zero neighbourhood \(U\) of
\(P\) such that
\[
K=\overline{h(U)}
\tag{ML.5}
\]
is compact. Its image \(V=g(U)\) is open by NP.C. Compactness
gives finitely many elements of \(K\) such that
\[
K\subset\bigcup_{j=1}^s(b_j+\tfrac12 V).
\tag{ML.6}
\]
Let \(S\) be their finite-dimensional span. Quotient the target
by \(S\), a closed subspace by ML.A. The target quotient is
Fréchet by NT.A, and the quotient of \(g\) is still an open
surjection. The closure of the quotient of \(h(U)\) is the
image of \(K\): the image is compact and hence closed, and
every point in it is a limit of images of points in \(h(U)\).
In the quotient, ML.6 gives
\[
K\subset\tfrac12 g(U).
\tag{ML.7}
\]
For the next argument use the quotient target and the same
letters. We will prove that \(f\) is surjective there.

Fix a decreasing absolutely convex open zero-neighbourhood
basis in \(P\) satisfying
\[
U_{p+1}\subset\tfrac12 U_p,\qquad p\ge1.
\tag{ML.8}
\]
For fixed \(p\), the increasing open sets
\[
g(2^N U_p\cap\tfrac12 U),\qquad N\ge0,
\tag{ML.9}
\]
cover \(K\). Indeed, ML.7 supplies a lift in
\(\tfrac12 U\), and the absorbing neighbourhood \(U_p\)
contains that lift after multiplication by a sufficiently
small power of two. A finite subcover and monotonicity
give an integer \(N(p)\) with
\[
K\subset g(2^{N(p)}U_p\cap\tfrac12 U).
\tag{ML.10}
\]
Increase these integers so that they are strictly increasing.

For \(y_0\in g(U)\), choose \(x_0\in U\) with
\(g x_0=y_0\), and set \(y_1=-h x_0\). Inductively choose
\[
\begin{gathered}
g x_n=y_n,\\
x_n\in2^{-n}U,\\
y_{n+1}=y_n-fx_n,\\
y_{n+1}=-h x_n\in2^{-n}K.
\end{gathered}
\tag{ML.11}
\]
These choices are possible because ML.7 gives
\(2^{1-n}K\subset g(2^{-n}U)\). For indices in a block,
ML.10 permits the additional choice
\[
\begin{gathered}
N(p)<n\le N(p+1),\\
x_n\in2^{N(p)+1-n}U_p,\\
x_n\in2^{-n}U.
\end{gathered}
\tag{ML.12}
\]
For the finitely many initial indices no additional condition
is needed. Any partial sum inside the \(p\)-th block belongs
to \(2U_p\), because the sum of its displayed coefficients
is at most two. The sum of all block tails starting at
block \(p\), including an incomplete first or last block,
belongs to \(4U_p\) by ML.8 and absolute convexity.
These sets tend to zero. Thus
\[
x=\sum_{n\ge0}x_n
\tag{ML.13}
\]
exists by completeness. Since the compact set \(K\) is bounded,
ML.11 gives \(y_n\to0\). Telescoping ML.11 and continuity
show \(fx=y_0\). The image of \(f\) therefore contains the
open neighbourhood \(g(U)\) and, being a linear subspace,
is the whole quotient target.

Before taking that target quotient, the conclusion is
\[
Q=f(P)+S.
\tag{ML.14}
\]
The scalar cokernel is finite dimensional. ML.A gives its
Hausdorff property and closedness of \(f(P)\), proving ML.4.
No finite-rank approximation of the compact map was required.

Compactness in this proof passes to a closed target subspace.
More precisely, if a compact map takes its values in a closed
subspace, the compact closure of the image of the chosen
neighbourhood is still contained in that subspace and is compact
for its induced topology. This is a statement about the
neighbourhood definition of compactness.

#### ML.C. Finite and Hausdorff cohomology

Let \(E^\bullet,F^\bullet\) be Fréchet complexes, and let
\(\rho^\bullet:E^\bullet\to F^\bullet\) be a continuous chain
map. Suppose in one degree that
\[
\begin{gathered}
\rho^q:E^q\longrightarrow F^q,\\
\rho^q\text{ is compact},\\
H^q(\rho):H^q(E)\\
\longrightarrow H^q(F),\\
H^q(\rho)\text{ is surjective}.
\end{gathered}
\tag{ML.15}
\]
Then \(H^q(F)\), with the topology in ML.1, is finite dimensional
and Hausdorff. Define maps between actual Fréchet spaces by
\[
\begin{gathered}
P=Z^q(E)\oplus F^{q-1},\\
Q=Z^q(F),\\
g(x,y)=\rho^q x+d_F^{q-1}y,\\
h(x,y)=-\rho^q x,\\
(g+h)(x,y)=d_F^{q-1}y.
\end{gathered}
\tag{ML.16}
\]
Surjectivity on cohomology says exactly that \(g\) is
surjective. The map \(h\) is compact into the closed cycle
target by the final paragraph of ML.B. ML.4 therefore gives
a closed finite-codimensional boundary space in \(Q\).
This proves the claim without first assuming closed boundaries.

There is also a useful topological statement without compactness:
every surjective \(H^q(\rho)\) is open. Indeed \(g\) in ML.16
is an open map by NP.C. If \(O\) is an open set in \(H^q(E)\),
let \(\widetilde O\) be its open inverse image in \(Z^q(E)\).
Then
\[
g(\widetilde O\times F^{q-1})
\tag{ML.17}
\]
is the inverse image of \(H^q(\rho)(O)\) in \(Z^q(F)\).
It is open, so the image in cohomology is open. The induced
cohomology map is continuous by the definition of the two
quotient topologies. Consequently a bijective induced map is
a homeomorphism even when its cohomology spaces are not
Hausdorff.

#### ML.D. Metrics for a tower of complexes

Consider a tower of Fréchet complexes with continuous chain maps
\[
\begin{gathered}
r_n:E_{n+1}^\bullet\longrightarrow E_n^\bullet,\\
n\ge0,\qquad r_{nn}=1,\\
r_{nk}=r_k\cdots r_{n-1},\\
k<n,\\
E^\bullet=\varprojlim_n E_n^\bullet.
\end{gathered}
\tag{ML.18}
\]
For the inverse limit take compatible elements in the
countable product in each degree. Compatibility is a closed
condition, so each limit term is a Fréchet space. Products
are complete because a Cauchy sequence converges in every
coordinate, and the product topology makes these coordinate
limits its limit. Closed subspaces inherit completeness.
The differentials on the limit act coordinatewise.

Throughout ML.D–ML.G assume that
\[
\begin{gathered}
r_n:E_{n+1}^j\longrightarrow E_n^j\\
\text{has dense image}\\
\text{for every }n,j.
\end{gathered}
\tag{ML.19}
\]
There is no assumption that the maps on terms or cycles are
surjective.

Here are explicit metrics used to control the limit
constructions. A countable defining family of continuous
seminorms gives a complete translation-invariant metric by
summing their truncations with positive summable weights.
For example
\[
\begin{gathered}
a_m(v)=\min\{1,p_m(v)\},\\
a(v)=\sum_{m\ge1}2^{-m}a_m(v),\\
\alpha(x,y)=a(x-y).
\end{gathered}
\tag{ML.20}
\]
The triangle inequality follows from subadditivity of the
seminorms and of truncation at one. This metric defines the
seminorm topology. A Cauchy sequence for it is Cauchy for
each seminorm, and completeness of the Fréchet space gives
a limit to which the sequence also converges in the metric.

Fix a degree \(q\). Choose such metrics \(\alpha_n\) in degree
\(q\) and \(\beta_n\) in degree \(q-1\). In the next display,
a metric with one argument means distance to zero, all maxima
over \(k\) range from zero to \(n\), and a maximum of functions
is pointwise. Put
\[
\begin{gathered}
a_n(u)=\max_k\alpha_k(r_{nk}u),\\
b_n^0(v)=\max_k\beta_k(r_{nk}v),\\
b_n^1(v)=a_n(d_nv),\\
b_n=\max\{b_n^0,b_n^1\},\\
A_n(x,x')=a_n(x-x'),\\
B_n(y,y')=b_n(y-y').
\end{gathered}
\tag{ML.21}
\]
Both are complete metrics with the original topologies:
they contain the original \(k=n\) metric, and add only finitely
many continuous terms. A sequence Cauchy for either metric
has an original limit, and continuity gives convergence for
the additional terms. The chain maps imply
\[
\begin{gathered}
A_k(r_{nk}x,0)\le A_n(x,0),\\
B_k(r_{nk}y,0)\le B_n(y,0),\\
A_n(d_ny,0)\le B_n(y,0).
\end{gathered}
\tag{ML.22}
\]
Thus all errors measured at a later stage control the earlier
stages, including errors after taking a differential.

The same finite-maximum construction in a single degree
shows that each projection
\[
\varprojlim_n E_n^j\longrightarrow E_N^j
\tag{ML.23}
\]
has dense image. To prove this, start with an element at
stage \(N\), and use ML.19 to choose successive lifts whose
images differ from the preceding choice by less than
\(\varepsilon2^{-n-1}\) in the metric at stage \(n\).
At each fixed earlier stage, the successive images are Cauchy.
Their limits are compatible by continuity and have image at
stage \(N\) at distance less than \(\varepsilon2^{-N}\)
from the chosen element. The finite-maximum metric includes
every prescribed finite set of earlier neighbourhood
conditions. This proves density for the entire Fréchet topology.

#### ML.E. Lifting compatible cohomology classes

There is a natural linear map
\[
\begin{gathered}
\Theta^q:H^q(E^\bullet)\\
\longrightarrow\varprojlim_n H^q(E_n^\bullet),
\end{gathered}
\tag{ML.24}
\]
where the limit on the right consists of compatible algebraic
cohomology classes. Under ML.19 it is surjective, without any
closed-boundary assumption.

Take a compatible family of classes \(c_n\). Choose a cycle
\(x_0\) representing \(c_0\). Having chosen a representative
\(x_n\), choose any representative \(\widetilde x_{n+1}\)
of \(c_{n+1}\). Compatibility gives
\[
\begin{gathered}
r_n\widetilde x_{n+1}-x_n=d_n v_n,\\
v_n\in E_n^{q-1}.
\end{gathered}
\tag{ML.25}
\]
By ML.19 choose \(w_{n+1}\in E_{n+1}^{q-1}\) so close
to \(v_n\) after restriction that, on defining
\[
\begin{gathered}
x_{n+1}=\widetilde x_{n+1}-d_{n+1}w_{n+1},\\
y_n=v_n-r_nw_{n+1},\\
r_nx_{n+1}-x_n=d_ny_n,
\end{gathered}
\tag{ML.26}
\]
we have
\[
B_n(y_n,0)<2^{-n-1}.
\tag{ML.27}
\]
The class of \(x_{n+1}\) is still \(c_{n+1}\).
For each fixed \(k\), ML.22 and ML.27 show that the sequences
of restricted representatives and the series of restricted
primitives converge. Explicitly set
\[
\begin{gathered}
Y_k=\sum_{n\ge k}r_{nk}y_n,\\
Y_k\in E_k^{q-1},\\
\xi_k=x_k+d_kY_k,\\
\xi_k=\lim_{n\to\infty}r_{nk}x_n.
\end{gathered}
\tag{ML.28}
\]
The metric tail estimates prove that the partial sums are
Cauchy; completeness gives the sums. Taking the differential
of their limits is legitimate by continuity. The last equality
is the telescoping identity from ML.26. Each \(\xi_k\) is a
cycle representing \(c_k\), and continuity of restriction
gives
\[
r_k\xi_{k+1}=\xi_k.
\tag{ML.29}
\]
The family is a cycle in the inverse limit and proves
surjectivity of ML.24. The series of primitives in ML.28 is
essential: convergence of the representatives alone would
only place their differences in the closure of the boundaries.

If every transition on degree-\(q\) cohomology is surjective,
any class at stage zero extends recursively to a compatible
family. Hence the map
\[
H^q(E^\bullet)\longrightarrow H^q(E_0^\bullet)
\tag{ML.30}
\]
is surjective under that additional assumption.

#### ML.F. Density, primitives and injectivity

First we prove a density assertion that respects actual
quotient cohomology. Let \(s:P^\bullet\to Q^\bullet\) be a
continuous chain map of Fréchet complexes. If
\[
\begin{gathered}
s^{q-1}:P^{q-1}\\
\longrightarrow Q^{q-1},\\
H^q(s):H^q(P)\\
\longrightarrow H^q(Q),\\
\text{both have dense image},
\end{gathered}
\tag{ML.31}
\]
then \(s^q:Z^q(P)\to Z^q(Q)\) has dense image.
Indeed, for \(z\in Z^q(Q)\) and any small cycle
neighbourhood, density on the quotient supplies a cycle
\(a\in Z^q(P)\) and \(b\in Q^{q-1}\) for which
\[
z-s^q a-d_Qb
\tag{ML.32}
\]
is as small as prescribed in the cycle topology. This follows
because the quotient image of every open cycle neighbourhood
is open. Approximate \(b\) by \(s^{q-1}v\). Continuity of
the differential then makes the cycle
\[
s^q(a+d_Pv)
\tag{ML.33}
\]
arbitrarily close to \(z\). No separation property of the
cohomology quotient was used.

Suppose every \(H^q(r_n)\) has dense image. Applying ML.31
shows that the maps on degree-\(q\) cycle spaces have dense
image. These are closed Fréchet spaces. The construction for
ML.23 applied to their tower proves
\[
\begin{gathered}
Z^q(E^\bullet)\longrightarrow Z^q(E_N^\bullet)\\
\text{has dense image}
\end{gathered}
\tag{ML.34}
\]
for every \(N\). The induced cohomology projection also has
dense image, by continuity of the open quotient map.

For injectivity of ML.24 assume instead that every
\[
\begin{gathered}
H^{q-1}(r_n):H^{q-1}(E_{n+1})\\
\longrightarrow H^{q-1}(E_n)\\
\text{has dense image}.
\end{gathered}
\tag{ML.35}
\]
Apply ML.31 one degree lower, using ML.19 in degree
\(q-2\). The transitions on \(Z^{q-1}\) have dense image.
Let a cycle \(\xi\) in the limit map to zero at every stage.
Choose a primitive \(y_0\) for \(\xi_0\). Given a primitive
\(y_n\), choose any primitive \(\widetilde y_{n+1}\)
for \(\xi_{n+1}\). Their discrepancy
\[
\begin{gathered}
z_n=r_n\widetilde y_{n+1}-y_n,\\
z_n\in Z^{q-1}(E_n)
\end{gathered}
\tag{ML.36}
\]
can be approximated by \(r_n w_{n+1}\), where
\(w_{n+1}\in Z^{q-1}(E_{n+1})\). Make the approximation
so that, after replacing the primitive by
\[
\begin{gathered}
y_{n+1}=\widetilde y_{n+1}-w_{n+1},\\
e_n=r_ny_{n+1}-y_n,\\
B_n(e_n,0)<2^{-n-1},
\end{gathered}
\tag{ML.37}
\]
the displayed estimate holds. Taking limits at every earlier
stage as in ML.28 gives a compatible family \(y\). Every
approximate primitive still has differential \(\xi_n\), so
continuity gives
\[
dy=\xi.
\tag{ML.38}
\]
Thus \(\Theta^q\) is injective, and together with ML.E it is
an algebraic isomorphism under ML.35.

If, in addition, every \(H^q(r_n)\) is injective, then the
map ML.30 is injective. A class zero at stage zero is zero at
every stage by those injective transition maps; ML.38 then
shows it was zero in the limit. No topological isomorphism
with a limit of non-Hausdorff quotient spaces is asserted
merely from these algebraic conclusions.

#### ML.G. A common approximation source

Let \(P^\bullet\) be another Fréchet complex and let
\(\varphi^\bullet:P^\bullet\to E^\bullet\) be a continuous
chain map. Write \(\varphi_n\) for its composite with
the \(n\)-th projection. Suppose for every \(n\) that
\[
\begin{gathered}
\varphi_n^{q-1}:P^{q-1}\longrightarrow E_n^{q-1},\\
H^q(\varphi_n):H^q(P)\\
\longrightarrow H^q(E_n),\\
\text{both have dense image}.
\end{gathered}
\tag{ML.39}
\]
Then
\[
\begin{gathered}
H^q(\varphi):H^q(P)\\
\longrightarrow H^q(E^\bullet)\\
\text{has dense image}.
\end{gathered}
\tag{ML.40}
\]
ML.31 shows that \(\varphi_n(Z^q(P))\) is dense in
\(Z^q(E_n)\). For a limit cycle \(\xi\), choose cycles
\(a_n\in Z^q(P)\) with
\[
A_n(\varphi_n a_n,\xi_n)<2^{-n}.
\tag{ML.41}
\]
ML.22 implies that the images \(\varphi a_n\) converge to
\(\xi\) in every coordinate and hence in the limit topology.
Cycle density implies cohomology density, proving ML.40.

One sufficient hypothesis for the first line of ML.39 is
density of \(P^{q-1}\to E^{q-1}\) itself. Its composite
with a projection is then dense in the projection image,
and ML.23 makes that image dense in \(E_n^{q-1}\).
This explains precisely how an approximation source for
the entire tower can be used.

#### ML.H. Exhaustions and a vanishing criterion

Here is a sheaf formulation with the cover conditions explicit.
Let \(X_0\subset X_1\subset\cdots\) be open subsets with
union \(X\), and let \(\mathscr S\) be a sheaf of complex
vector spaces. Suppose a single countable cover
\(\mathcal W=(W_i)_{i\in I}\) of \(X\) has increasing
subfamilies indexed by
\[
\begin{gathered}
I_0\subset I_1\subset\cdots,\\
\bigcup_n I_n=I,
\end{gathered}
\tag{ML.42}
\]
each of which is a cover of \(X_n\) by sets contained in \(X_n\).
Suppose every nonempty finite intersection in this cover is
\(\mathscr S\)-acyclic in positive degrees. Suppose its
section spaces on those intersections are Fréchet and the
restriction maps are continuous. For a coherent analytic
sheaf, SC.G supplies these section topologies and continuity;
acyclicity is a separate hypothesis here.

For an increasing tuple \(J\), let \(W_J\) be the intersection
of its members. Let \(\mathcal I_n^{p+1}\) and
\(\mathcal I^{p+1}\) denote the increasing tuples of length
\(p+1\) in \(I_n\) and \(I\), respectively. Take alternating
Čech complexes with their product topologies:
\[
\begin{gathered}
C_n^p=\prod_{J\in\mathcal I_n^{p+1}}\mathscr S(W_J),\\
C^p=\prod_{J\in\mathcal I^{p+1}}\mathscr S(W_J).
\end{gathered}
\tag{ML.43}
\]
An empty intersection contributes the zero space. The
differential in any target coordinate is a finite sum of
continuous restrictions, hence is continuous. Transition
maps are coordinate projections, so they are surjective:
extend an arbitrary cochain by zero on the additional tuples.
This extension need not commute with the differential and
is used only to prove surjectivity on terms.

Every finite tuple of indices appears in some \(I_n\).
Consequently compatible cochains are exactly all cochains:
\[
C^\bullet=\varprojlim_n C_n^\bullet
\tag{ML.44}
\]
as topological complexes. The topology agrees because every
finite coordinate condition occurs at some stage. SC.B
identifies their cohomology with derived sheaf cohomology
on \(X\) and \(X_n\), and identifies the induced maps with
ordinary restriction. In degree zero this is just the
sheaf gluing isomorphism. Give these cohomology groups the
quotient topologies of these particular complexes.

Assume now that
\[
\begin{gathered}
H^q(X_n,\mathscr S)=0,\\
q\ge1,\quad n\ge0,\\
\mathscr S(X_{n+1})\longrightarrow\mathscr S(X_n)\\
\text{has dense image}.
\end{gathered}
\tag{ML.45}
\]
For the second line the section topology must agree with the
degree-zero cycle topology of ML.43. In the coherent analytic
case this follows from SC.G: the compatible-section subspace
of the product of local sections is the canonical section
space. The topological isomorphism in degree zero can also
be seen from continuity of the section restrictions, a closed
compatibility subspace, and NP.C.

ML.E and ML.F now prove
\[
\begin{gathered}
H^q(X,\mathscr S)=0\\
(q\ge1).
\end{gathered}
\tag{ML.46}
\]
In degree one, the preceding cohomology is the section
space and its transition density is the second line of
ML.45. In every higher degree, the preceding cohomology
groups vanish, so their transition maps have dense image.
Thus ML.24 is injective and has zero target in every
positive degree, proving ML.46.

The density clause cannot be dropped merely because all
positive cohomology groups at the individual stages vanish.
The proof of ML.38 needs approximation of differences of
primitives in the preceding cycle spaces; ML.J gives an
explicit tower for which this density fails and the limit
has nonzero positive cohomology.

#### ML.I. Compact restrictions in a Čech model

Let two countable acyclic covers of a space carry the
Fréchet section topologies above. Suppose a refinement
chain map between their Čech complexes has a compact map
in degree \(q\), and induces a surjection on \(H^q\).
SC.B identifies its cohomology map with the corresponding
sheaf-cohomology map. ML.C proves that the target group is
finite dimensional and Hausdorff in its actual Čech quotient
topology. If the induced map is bijective, ML.C also identifies
the two topologies.

For a finite cover, a sufficient compactness condition is
that each cochain restriction factors through a relatively
compact restriction of coherent analytic sections as in
SC.H.
Such a restriction is scalar nuclear and hence compact:
in a nuclear representation, choose an absolutely convex
neighbourhood on which all coefficient functionals have
absolute value at most one. For each seminorm in the target,
boundedness of the coefficient vectors and summability of
the scalar weights bound the tail uniformly. Set
\(t_j(x)=\ell_j(x)y_j\); then
\[
\begin{gathered}
T_N(x)=\sum_{j>N}\lambda_jt_j(x),\\
M_p=\sup_jp(y_j),\\
s_N=\sum_{j>N}|\lambda_j|,\\
\sup_{x\in U}p(T_N(x))
\le M_p s_N.
\end{gathered}
\tag{ML.47}
\]
The finite head ranges in a bounded finite-dimensional
set. For any target zero neighbourhood, finitely many
seminorm conditions and ML.47 make a tail smaller than
half that neighbourhood; a finite cover of the head gives
a finite cover of the full image. The image is therefore
totally bounded. Its closure in the complete metric target
is compact: successively choose subsequences in finite
covers of diameters tending to zero; the diagonal sequence
is Cauchy and hence converges. Finite sums, finite products,
and continuous pre- or postcomposition preserve compactness
by this same neighbourhood definition.

This supplies the topological implication when the required
acyclic covers and refinement have been constructed. It does
not identify an arbitrary intersection of analytic charts
with a single ambient polydisc, or infer coherent vanishing
from scalar vanishing without a resolution.

#### ML.J. A tower with vanishing stages and a nonzero limit

Let \(\ell^2\) be the space of square-summable complex sequences
indexed by nonnegative integers, with norm
\[
\|u\|_2^2=\sum_{j\ge0}|u_j|^2.
\tag{ML.48}
\]
It is complete. A norm-Cauchy sequence has a limit in every
coordinate. For any sufficiently late fixed term, pass to
the coordinate limit in each finite partial sum of the
squared distance to that term. Its supremum is at most
the squared Cauchy bound. Thus the coordinate limit differs
from that term by an \(\ell^2\) sequence with arbitrarily small
norm, and belongs to \(\ell^2\) and is the norm limit.

At stage \(n\), take the two-term complex in degrees zero
and one
\[
\begin{gathered}
E_n^0=\ell^2,\qquad E_n^1=\mathbf C^n,\\
d_nu=(u_0,\ldots,u_{n-1}).
\end{gathered}
\tag{ML.49}
\]
The differential is continuous and surjective: append zeros
to any finite row. All other terms are zero. Transition maps
are the identity in degree zero and truncation of the last
coordinate in degree one. They commute with the differentials
and are surjective on all terms. Nevertheless
\[
\begin{gathered}
H^1(E_n)=0,\\
H^0(E_n)=\ker d_n,\\
u\in\ker d_n\ \Longleftrightarrow\\
u_j=0\quad(0\le j<n).
\end{gathered}
\tag{ML.50}
\]
The image of \(H^0(E_{n+1})\) in \(H^0(E_n)\) is the
kernel of its \(n\)-th coordinate functional. That functional
is continuous and surjective, as witnessed by the sequence
with only coordinate \(n\) equal to one. Thus the image is
a proper closed subspace and is not dense.

The inverse limit complex is
\[
\begin{gathered}
E^0=\ell^2,\qquad E^1=\mathbf C^{\mathbf N},\\
du=(u_j)_{j\ge0},
\end{gathered}
\tag{ML.51}
\]
where the target has its product topology. Compatible
finite rows are exactly all infinite rows; their finite
coordinate conditions are precisely the product topology.
The degree-zero identity tower gives the original norm
topology. The displayed inclusion is continuous because
each coordinate is bounded by the norm. Its cohomology is
\[
\begin{gathered}
H^1(E)=\mathbf C^{\mathbf N}/\ell^2\ne0,\\
[(1,1,1,\ldots)]\ne0.
\end{gathered}
\tag{ML.52}
\]
The all-ones row is not square summable, proving the second
line and hence the first. Finite-support rows belong to
\(\ell^2\) and can match every finite coordinate condition.
The boundary space is therefore dense and proper in the
target. The cohomology in ML.52 is non-Hausdorff. Its
nonzero class maps to zero at every stage, exactly as allowed
when the preceding-degree density hypothesis ML.35 fails.

![Compact perturbation on cycle spaces and convergent primitive corrections.](assets/cohomology-inverse-limits.png)

*The left panel gives the actual maps in ML.16 and their
closed finite-codimensional image. The right panel gives
ML.26–ML.29 and the first six terms of the error-bound
envelope. Those terms sum to \(63/64\), with remaining
envelope \(1/64\); no particular errors are asserted to
attain these bounds. The reproducible drawing source
and exact envelope data
specify the diagram.*

The compact-cohomology and inverse-limit arguments are also
developed in
[Demailly's free author manuscript, Chapter IX, §1.B–§1.C](https://www-fourier.univ-grenoble-alpes.fr/~demailly/manuscripts/agbook.pdf).
ML.A–ML.J give the quotient, finite-codimension, convergent
primitive and cover arguments used here.


### 5.97. Coherent vanishing and approximation on analytic exhaustions

Let \(X\) be a Hausdorff second-countable analytic space, possibly
singular or nonreduced. Suppose it has a smooth strictly
plurisubharmonic exhaustion. For every coherent analytic sheaf
on \(X\), we prove vanishing of positive cohomology, generation
by global sections at every stalk, finite global generation
near every compact subset, and approximation on the Runge
opens specified below.

The geometry first gives genuine ambient neighbourhoods and
acyclic intersections. Compact cohomology and convergent
inverse limits then supply the global passage. The coherent
ideal and every coherent coefficient module retain their
nonreduced structures throughout.

We use the earlier proofs of
coherent algebra and finite free prefixes, SC.D,
local coherent vanishing and actual section quotients, SC.E,
coherent section topology, SC.G,
nuclear restrictions, SC.H,
and derived/Čech comparison, SC.B.
The functional-analytic inputs are the complete
open mapping and null-sequence lifting proof, NP.C,
compact-cohomology theorem, ML.C,
and inverse-limit arguments, ML.D–ML.H.
Holomorphic approximation uses the full weighted solution
and jet argument in
WD.H.
The convolution integrals use the complete
product-integration and linear-change-of-variables proof.

#### CX.A. Strict positivity and changes of embedding

A smooth real function on an analytic space means one which
is locally the restriction of a smooth function in an ambient
complex affine space. It is **strictly plurisubharmonic** if
such extensions can be chosen with positive-definite Levi
form. For an ambient smooth function the convention is
\[
\begin{gathered}
L_\phi(z;v)=
\sum_{a,b}\frac{\partial^2\phi}
 {\partial z_a\,\partial\overline z_b}
 v_a\overline v_b,\\
L_\phi(z;v)>0\quad(v\ne0).
\end{gathered}
\tag{CX.1}
\]
These are conditions on real functions on the underlying
space; they do not replace its structure sheaf by a reduced
one. An **exhaustion** here is bounded below and has compact
closed sublevel sets. A space with a smooth strictly
plurisubharmonic exhaustion is called **strongly
one-complete**.

Strict positivity is independent of the chosen local
embedding. Suppose two analytic charts for the same space
are identified near a point. Their coordinate maps have
holomorphic ambient lifts \(H,K\) in the two directions:
each coordinate germ is an element of the local analytic
quotient ring and therefore has a holomorphic representative.
Shrink so that their composites are defined. If \(\phi\)
is a strict ambient extension in the first chart, put in the
second chart
\[
\begin{gathered}
\phi'(z')=\phi(K(z'))\\
              +|z'-H(K(z'))|^2.
\end{gathered}
\tag{CX.2}
\]
On the analytic space the second term is zero and the first
has the required value. The holomorphic chain rule gives
\[
\begin{gathered}
a=DK(z')v,\\
b=v-DH(K(z'))a,\\
P=L_\phi(K(z');a),\\
L_{\phi'}(z';v)=P+|b|^2.
\end{gathered}
\tag{CX.3}
\]
If the first vector is nonzero the first term is positive.
If it is zero, the second term is \(|v|^2\).
Thus the extension is strict in either case. The same
argument applies to a closed analytic subspace inside a
chart of a larger analytic space: its own local strict
extension can be transferred to that chart. All coordinate
lifts are taken in the actual analytic quotient rings.

Finite sums of strict functions are strict: transfer their
finitely many local extensions into one common chart using
CX.2 and add them. The chain rule also gives, for a smooth
real function \(F\) of finitely many real variables,
\[
\begin{gathered}
u_i=\partial\phi_i(v),\\
L_{F(\phi)}(v)\\
=\sum_i F_i(\phi)L_{\phi_i}(v)\\
 +\sum_{i,j}F_{ij}(\phi)u_i\overline u_j.
\end{gathered}
\tag{CX.4}
\]
If \(F\) is convex and coordinatewise nondecreasing, its
Hessian term is nonnegative. Indeed a real positive
semidefinite matrix is also nonnegative on complex vectors
by separating their real and imaginary parts. In particular,
composition with a convex increasing function preserves
strictness if its first derivative is positive.

We will use uniform positive Levi margins on compact
ambient sets. Continuity and compactness give these margins
on the product with the unit sphere. The elementary
finite-dimensional compactness fact follows by repeatedly
bisecting bounded intervals in each real coordinate,
retaining a half containing infinitely many terms.
The resulting nested intervals have lengths tending to
zero and yield a convergent subsequence, successively in
all finitely many coordinates. A closed bounded set is
therefore sequentially compact. Such a metric space is
compact: failure of a finite net of some fixed radius
would give a sequence of pairwise separated points.
Also every open cover has a positive Lebesgue radius;
otherwise choose points whose balls of radii tending
to zero lie in no cover member, take a convergent
subsequence, and use an open cover member at its limit
to contradict that choice. A finite net of smaller radius
then gives a finite subcover. This proves compactness
in the Euclidean metric and the finite-dimensional
compactness used for the Levi margins.

#### CX.B. Nested chart balls and smooth maxima

There is a compact exhaustion
\[
\begin{gathered}
K_n\subset\operatorname{int}K_{n+1},\\
X=\bigcup_n\operatorname{int}K_n.
\end{gathered}
\tag{CX.5}
\]
For completeness, small closed ambient balls inside an
analytic chart intersect its closed analytic model in
compact sets. Their smaller open balls form a basis of
relatively compact neighbourhoods. Second countability
selects a countable cover of this kind: for each member
of a countable basis contained in one such neighbourhood,
select one containing neighbourhood. Inductively cover
the preceding compact set and the first finitely many
chosen compact closures by finitely many relatively
compact open neighbourhoods. Their compact closures give
the next \(K_n\) and contain the preceding set in their
interior. This proves CX.5.

Set \(K_0=K_{-1}=\varnothing\) for the following
annulus construction. If \(Y\) is closed, any prescribed family of chart
neighbourhoods of its points has a refinement by nested
chart balls
\[
\begin{gathered}
V_i\Subset U_i,\\
Y\subset\bigcup_i V_i,\\
\overline U_i\text{ lies in}\\
\text{a prescribed chart domain},
\end{gathered}
\tag{CX.6}
\]
with the \(U_i\) locally finite in \(X\).
To construct it, cover each compact annulus
\(Y\cap(K_n\setminus\operatorname{int}K_{n-1})\)
by finitely many smaller chart balls. Choose their
slightly larger closed balls inside
\(\operatorname{int}K_{n+1}\setminus K_{n-2}\)
and inside the prescribed domains, taking the first
two excluded compact sets to be empty. This is possible
because \(K_{n-2}\) is inside the interior of \(K_{n-1}\).
Every compact subset meets only finitely many of these
larger balls. Smaller concentric balls still cover each
annulus, proving CX.6.

There are smooth cutoffs \(\chi_i\) equal to one on
\(V_i\), between zero and one, and zero near the boundary
of \(U_i\). Take an ordinary radial ambient cutoff
between the two ball radii and restrict it to the
analytic model. Extending by zero outside the chart is
smooth because its support is compactly contained there.
One explicit smooth scalar bump is obtained from
\[
\begin{gathered}
e(t)=
\begin{cases}e^{-1/t},&t>0,\\0,&t\le0,\end{cases}\\
s(t)=\frac{e(t)}{e(t)+e(1-t)}.
\end{gathered}
\tag{CX.7}
\]
Then \(s=0\) for \(t\le0\), \(s=1\) for \(t\ge1\).
Its denominator never vanishes. All derivatives of
\(e\) at zero vanish: each right derivative is an
exponential times a polynomial in \(1/t\), and
\(e^u\ge u^m/m!\) for every integer \(m\) makes this
product tend to zero. This proves smoothness and
justifies the cutoffs.

For positive numbers \(\eta_i\), choose even nonnegative
smooth functions of integral one supported in
\((-\eta_i,\eta_i)\). They are rescalings of the bump
\(e(1-t^2)\), divided by its positive finite integral.
For a finite nonempty index set define
\[
\begin{gathered}
m(t)=\max_i t_i,\\
k_\eta(s)=\prod_i\kappa_{\eta_i}(s_i),\\
M_\eta(t)\\
=\int m(t+s)k_\eta(s)\,ds.
\end{gathered}
\tag{CX.8}
\]
The integral is an ordinary finite-dimensional integral
with compact support. Moving derivatives to the smooth
kernel after the change of variables \(u_i=t_i+s_i\)
shows that this function is smooth: on any compact
set of \(t\)'s, all differentiated integrands have
common compact support and uniform integrable bounds,
so their difference quotients converge under the
integral. The product-integration argument in the
earlier programme justifies these iterated integrals.

Convexity and monotonicity follow by integrating the
corresponding inequalities for the maximum. Evenness
gives zero first moments, so
\[
\begin{gathered}
\max_i t_i\le M_\eta(t),\\
M_\eta(t)\le\max_i(t_i+\eta_i),\\
M_\eta(t+c{\bf1})\\
=M_\eta(t)+c,\\
(M_\eta)_i\ge0,\\
\sum_i(M_\eta)_i=1.
\end{gathered}
\tag{CX.9}
\]
For the lower bound integrate
\(\max_j(t_j+s_j)\ge t_i+s_i\) for each \(i\).
The upper bound is pointwise, and differentiation
of translation covariance gives the final sum.
There is an exact deletion rule:
\[
\begin{gathered}
t_i+\eta_i<t_j-\eta_j\\
\Longrightarrow\\
M_\eta(t)\text{ ignores }t_i.
\end{gathered}
\tag{CX.10}
\]
That variable can never achieve the maximum for any
point of the support of the kernel; its kernel factor
then integrates to one. This proves consistency when
an irrelevant variable is removed.

By CX.4 and CX.9, the smooth maximum of finitely many
strictly plurisubharmonic functions is strict. The
weighted sum of their positive Levi forms has weights
with sum one, and the Hessian term is nonnegative.
If their Levi forms merely have a common lower bound
\(-C|v|^2\), their smooth maximum has that lower bound
as well.

#### CX.C. Approximate extension from a closed analytic subspace

Let \(Y\) be a closed analytic subspace of \(X\), let
\(\psi\) be smooth and strictly plurisubharmonic on \(Y\),
and let \(\delta>0\) be continuous on \(Y\). There is
an open neighbourhood \(N\) of \(Y\) and a smooth
strictly plurisubharmonic function \(\phi\) on \(N\)
such that
\[
\psi\le\phi|_Y<\psi+\delta.
\tag{CX.11}
\]
Choose the locally finite balls of CX.6 so that each
has a strict ambient extension \(\phi_i\) of \(\psi\)
on a neighbourhood of its closure. This uses CX.A
to work in a chart of \(X\). Let \(\chi_i\) be the
cutoff of CX.B. Choose positive constants so small that
\[
\begin{gathered}
\rho_i=\phi_i+\varepsilon_i(2\chi_i-1),\\
\rho_i\text{ is strict near }\overline U_i,\\
0<\eta_i<\varepsilon_i/4,\\
\varepsilon_i<
\tfrac14\min_{Y\cap\overline U_i}\delta.
\end{gathered}
\tag{CX.12}
\]
The compact Levi margin and the bounded second
derivatives of the cutoff permit the first choice.
The final minimum is positive on the nonempty compact
set in question. Balls disjoint from \(Y\) are unnecessary.

Here is a precise neighbourhood on which to glue
these functions. Extend each \(\rho_i\) continuously
to a neighbourhood of \(\overline U_i\). On the boundary
of \(U_i\), let \(B_i\) be the set where
\[
\rho_i+\eta_i\ge
\sup_{j:\,x\in U_j}(\rho_j-\eta_j).
\tag{CX.13}
\]
The supremum is lower semicontinuous: each term,
extended by minus infinity outside its open domain,
is lower semicontinuous. Thus \(B_i\) is closed in
the compact boundary. The family of boundaries is
locally finite, so the union of the \(B_i\) is closed.
It misses \(Y\). In fact, at a point of
\(Y\cap\partial U_i\), the cutoff \(\chi_i\) is zero,
whereas some \(V_j\) contains that point and
\(\chi_j=1\). Hence
\[
\begin{gathered}
\rho_i+\eta_i=\psi-\varepsilon_i+\eta_i,\\
\rho_j-\eta_j=\psi+\varepsilon_j-\eta_j,\\
\rho_i+\eta_i<\rho_j-\eta_j.
\end{gathered}
\tag{CX.14}
\]
Remove the closed union of the \(B_i\) from
\(\bigcup_iU_i\), and call the resulting neighbourhood
\(N\). At a boundary point of any \(U_i\) in \(N\),
an interior term strictly dominates the boundary
term as in CX.10. The inequality persists locally.
Only finitely many boundaries meet a sufficiently
small neighbourhood. All their terms can therefore
be deleted there, and the remaining finite smooth
maximum defines
\[
\phi(x)=M_\eta\bigl((\rho_i(x))_{x\in U_i}\bigr).
\tag{CX.15}
\]
The deletion rule proves that these expressions agree
across all boundaries. They are smooth and strict
by CX.B and the common-chart argument of CX.A.
At a point of \(Y\), one of the terms is
\(\psi+\varepsilon_j\). Every term plus its smoothing
radius is less than \(\psi+\delta\), by CX.12.
The two bounds in CX.9 give CX.11.

#### CX.D. A logarithmic barrier with a controlled negative Levi part

Write \(\mathscr J\) for the coherent ideal defining
\(Y\) in \(X\). There is a function \(v\), defined
near \(Y\), which is smooth and real on the complement
of \(Y\), tends to minus infinity at \(Y\), and has
locally uniformly bounded negative Levi part. More
precisely, near every point, in a fixed ambient chart
its smooth local representatives off \(Y\) satisfy
\[
L_v(z;w)\ge-C|w|^2
\tag{CX.16}
\]
with one constant on a smaller neighbourhood, including
when points of the complement approach \(Y\).

Choose locally finite chart balls \(U_i\) with closures
inside domains on which a finite tuple \(g_i\) generates
the same ideal \(\mathscr J\). Lift the tuple to
holomorphic functions in the ambient chart. Shrinking
the balls first ensures that these lifts are defined
on neighbourhoods of their closed balls. If their
centres and radii are \(a_i,r_i\), put on \(U_i\setminus Y\)
\[
\begin{gathered}
b_i(z)=\frac1{r_i^2-|z-a_i|^2},\\
v_i=\log|g_i|^2-b_i.
\end{gathered}
\tag{CX.17}
\]
The tuple is nonzero outside \(Y\). Direct
differentiation gives, for a holomorphic tuple \(g\)
where it is nonzero,
\[
\begin{gathered}
a=Dg(w),\\
Q=|g|^2|a|^2-|\langle a,g\rangle|^2,\\
L_{\log|g|^2}(w)=Q/|g|^4\ge0.
\end{gathered}
\tag{CX.18}
\]
The inequality is Cauchy–Schwarz, obtained by expanding
the nonnegative squared norm of the component of
\(Dg(w)\) perpendicular to \(g\).
On compact subsets inside a chart ball, the smooth
function \(b_i\) has bounded second derivatives.
Thus each \(v_i\) has the required lower Levi bound
there.

Use a smooth maximum with all smoothing radii equal
to one. A term approaching its chart boundary
becomes irrelevant. To check this even at a point
of \(Y\), select another ball \(U_j\) containing the
boundary point in its interior. Since \(g_j\)
generates \(\mathscr J\), locally
\[
\begin{gathered}
g_i=A_{ij}g_j,\\
|g_i|\le C_{ij}|g_j|
\end{gathered}
\tag{CX.19}
\]
for a finite holomorphic coefficient matrix, bounded
on a smaller neighbourhood. Its entries are actual
sections of \(\mathscr O_X\) and admit local ambient
representatives. There
\[
\begin{gathered}
c_{ij}=2\log C_{ij},\\
v_i-v_j\le c_{ij}-b_i+b_j,\\
c_{ij}-b_i+b_j\longrightarrow-\infty
\end{gathered}
\tag{CX.20}
\]
as the boundary of \(U_i\) is approached from within.
The function \(b_j\) stays bounded there. At a boundary
point outside \(Y\), the same conclusion follows
directly from continuity and \(b_i\to+\infty\).
CX.10 therefore deletes every boundary term locally.
The resulting locally finite smooth maximum defines
\(v\) smoothly off \(Y\).

At a point of \(Y\), only finitely many chart closures
meet a sufficiently small neighbourhood. Every
corresponding tuple tends to zero there, while each
\(b_i\) is nonnegative. The upper bound in CX.9
therefore shows \(v\to-\infty\). For CX.16, discard
the boundary terms using CX.20. The remaining terms
are in the interiors of finitely many charts.
Transfer their holomorphic coordinate functions and
their ideal generators into one common ambient
chart. Their barrier functions have smooth extensions
with bounded second derivatives on a smaller
neighbourhood. Formula CX.18 and the last assertion
of CX.B give one uniform lower bound for their smooth
maximum. This proves CX.16. Only smooth representatives
off \(Y\) and this uniform bound are used below;
no distributional extension across \(Y\) is required.

#### CX.E. Complete neighbourhoods preserving exhaustion values

Suppose now that \(\psi\) is a strict smooth exhaustion
on \(Y\). By a common additive shift we can assume
\(\psi\ge0\); restore that shift to the constructed
exhaustion at the end.
For every neighbourhood \(O\) of \(Y\) and every
continuous positive \(\delta\) on \(Y\), there is
a strongly one-complete open neighbourhood \(V\)
inside \(O\), with a strict smooth exhaustion \(\Psi\)
such that
\[
\psi\le\Psi|_Y<\psi+\delta.
\tag{CX.21}
\]
Apply CX.C to obtain \(\phi\), and intersect its domain
with the domain of the barrier in CX.D. Shrink also
to \(\{\phi>-1\}\), which still contains \(Y\).

First make \(\phi\) proper on a closed neighbourhood
of \(Y\). Choose the compact exhaustion in CX.5
large enough that
\[
\begin{gathered}
Y\cap\{\phi\le n\}\subset\operatorname{int}K_n\\
(n\ge1).
\end{gathered}
\tag{CX.22}
\]
These sets on \(Y\) are compact: they are closed in
the compact set \(\{\psi\le n\}\), by CX.11.
In the domain of \(\phi\), remove the closed sets
\(\{\phi\le n\}\setminus\operatorname{int}K_n\).
Their family is locally finite, since its \(n\)-th
member lies outside \(\operatorname{int}K_n\).
Their union misses \(Y\). Apply CX.6 once more,
with all closed balls inside the resulting open
neighbourhood, and let \(W_1\) be their union.
Local finiteness implies that \(\overline W_1\)
lies in that open neighbourhood. For each real \(c\),
\[
\overline W_1\cap\{\phi\le c\}
\text{ is compact}.
\tag{CX.23}
\]
It is closed and lies in one \(K_n\) with \(n>c\).
All closures here are taken in \(X\).

Choose an open \(W\) containing \(Y\) with
\(\overline W\subset W_1\cap O\), using the same
locally finite shrinking construction. Its boundary
is disjoint from \(Y\). On each compact \(\phi\)-band
in \(\overline W_1\), CX.16 and a finite collection
of strict local extensions of \(\phi\) bound the
negative Levi part of \(v\) by a fixed multiple
of \(L_\phi\). On the intersection of that band
with \(\partial W\), the finite continuous function
\(v\) has a finite lower bound.

There is consequently a smooth convex increasing
function \(\chi\), with positive first derivative,
so large that
\[
\begin{gathered}
L_{v+\chi(\phi)}>0\quad\text{on }W_1\setminus Y,\\
v+\chi(\phi)>0\quad\text{on }\partial W.
\end{gathered}
\tag{CX.24}
\]
Here is an explicit domination construction. Define
the smooth convex function
\[
\begin{gathered}
\kappa(t)=0\quad(t\le0),\\
\kappa(t)=\int_0^t(t-s)e(s)\,ds\\
(t>0).
\end{gathered}
\tag{CX.25}
\]
It is flat at zero by CX.7, with positive first
derivative for positive arguments and
\(\kappa''(t)=e^{-1/t}>0\) there. A function
\[
\begin{gathered}
u_n(t)=a_n\kappa(t-n+1),\\
R(t)=\sum_{n\ge0}u_n(t),\\
\chi(t)=A+Bt+R(t).
\end{gathered}
\tag{CX.26}
\]
has a locally finite sum on \(t>-1\). Choose \(B>0\)
and all \(a_n\ge0\). At \(t=n\), the \(n\)-th term
has positive value and positive first derivative.
Choose \(a_n\) successively to make both the value
and first derivative at \(n\) exceed the required
bounds for the band \(n\le\phi\le n+1\).
The constant and slope handle the initial band
\(-1<\phi\le0\), whose closed part in
\(\overline W_1\) is compact by CX.23.
Monotonicity then handles each whole band.
Convexity and CX.4 give the Levi conclusion in
CX.24; the value bounds give its boundary conclusion.

Give \(v\) the value minus infinity on \(Y\), and put
\[
\begin{gathered}
u=v+\chi(\phi),\\
V=\{x\in W:u(x)<0\},\\
\theta(t)=\kappa(-1/t-1),\\
t<0,\\
\Psi=\phi+\theta(u)\quad\text{on }V.
\end{gathered}
\tag{CX.27}
\]
The function \(-1/t\) has positive first and second
derivatives for \(t<0\). Thus \(\theta\) is convex,
nondecreasing, zero for \(t\le-1\), and tends to
positive infinity as \(t\) tends to zero from below.
The last assertion follows, for example, by bounding
\(\kappa''\) below by \(e^{-1}\) for arguments at
least one and integrating twice. Define the value
of \(\theta\) at minus infinity to be zero.
Since \(v\to-\infty\), the term \(\theta(u)\)
is identically zero on a neighbourhood of every
point of \(Y\). Thus \(\Psi\) is smooth there
and equals \(\phi\); elsewhere it is smooth and
strict by CX.4, CX.24 and the strict \(\phi\) term.
The set \(V\) is an open neighbourhood of \(Y\).

It remains to prove exhaustion. No point of
\(\partial W\) is in \(\overline V\), because
\(u>0\) there and \(u\) is continuous off \(Y\).
If a sequence in \(V\) has bounded \(\Psi\),
then \(\theta\ge0\) bounds \(\phi\) above.
CX.23 gives a convergent subsequence in
\(\overline W_1\). Its limit lies in \(W\).
A limit point in \(Y\) lies in \(V\).
At any other limit point, \(u\le0\).
Equality would force \(\theta(u)\to+\infty\),
contradicting the bound and \(\phi>-1\).
Thus the limit again lies in \(V\).
Closed sublevels of \(\Psi\) are therefore compact,
and \(\Psi>-1\). Its equality to \(\phi\) near
\(Y\) proves CX.21.

#### CX.F. Intersections and compatible ambient sublevels

Finite intersections of strongly one-complete opens
are strongly one-complete. Add a constant to make each
chosen strict exhaustion nonnegative, restrict them
to the intersection, and add them. The sum is strict
by CX.A. Its closed sublevel lies in the intersection
of finitely many compact closed sublevels. Within
that compact intersection it is closed, so it is
compact. This proves the assertion, including
intersections not represented by a single ambient
polydisc.

An open subset \(U'\) of a strongly one-complete
space \(U\) is **Runge complete in \(U\)** if, for
every compact \(L\subset U'\), there is a smooth
strict exhaustion \(\psi\) on \(U\) and a number \(b\)
such that
\[
L\subset\{\psi<b\}\Subset U'.
\tag{CX.28}
\]
Every sublevel of a strict exhaustion has this
property: move its level slightly down above the
maximum on the given compact. Its own strict
exhaustion is \(1/(b-\psi)\), after restricting
to that sublevel.

Suppose \(A\) is closed analytic in an ambient
open \(\Omega\), \(U\) is open in \(A\), and
the exhaustion \(\psi\) in CX.28 is given on
\(U\). Choose an open \(\Omega'\subset\Omega\)
with \(A\cap\Omega'=U\); this is precisely
the relative openness of \(U\). Now \(U\) is
a closed analytic subspace of \(\Omega'\).
For any positive constant \(\varepsilon\), CX.E
gives an ambient neighbourhood \(V\subset\Omega'\)
and strict exhaustion \(\Psi\) with
\[
\begin{gathered}
A\cap V=U,\\
\psi\le\Psi|_U<\psi+\varepsilon.
\end{gathered}
\tag{CX.29}
\]
In particular \(V\) is a neighbourhood of \(U\),
with that exact intersection with \(A\).
It is not required to contain points of \(A\setminus U\).

Choose \(\varepsilon\) and levels so that
\[
\sup_L\psi+\varepsilon<a<b'<b.
\tag{CX.30}
\]
Then
\[
\begin{gathered}
L\subset A\cap V_a,\\
A\cap\overline V_{b'}\Subset U',\\
V_t=\{\Psi<t\}.
\end{gathered}
\tag{CX.31}
\]
Indeed \(\Psi|_U\ge\psi\) makes the second set
a compact subset of \(\{\psi<b\}\Subset U'\);
compactness also follows from exhaustion on \(V\).
This proves a compatible ambient Runge construction
with all neighbourhoods and domains specified.

#### CX.G. Coherent vanishing and approximation inside a chart

Fix a coherent analytic sheaf \(\mathscr S\). Throughout CX.G–CX.O, write
\(\Gamma(T)=\mathscr S(T)\) and
\(H^q(T)=H^q(T,\mathscr S)\).
A **distinguished patch** will mean a closed analytic
model \(A\subset\Omega\subset\mathbf C^N\) on which
the ambient coherent module \(i_*\mathscr S\)
has the finite free prefix of SC.D–SC.E, of length
\[
L=12^{2N}.
\tag{CX.32}
\]
All maps and its last coherent kernel are defined
on the same \(\Omega\). SC.D supplies such a patch
around every point, after shrinking once. No finite
projective dimension or local freeness of \(\mathscr S\)
is assumed.

If \(U\subset A\) is strongly one-complete, CX.E–CX.F
produce an ambient strongly one-complete neighbourhood
\(V\subset\Omega\) with \(A\cap V=U\). Restrict the
whole prefix to \(V\) and apply SC.E. The result is
\[
\begin{gathered}
H^q(U)=0\quad(q\ge1),\\
\mathscr O(V)^{p_0}\longrightarrow
             \Gamma(U)\longrightarrow0.
\end{gathered}
\tag{CX.33}
\]
Here \(H^q\) means derived sheaf cohomology. The
ambient closed inclusion gives the same cohomology
by SC.C–SC.E. The section map in CX.33 is an
actual surjection. It is continuous for the
canonical section topologies by SC.G, and is
therefore open by NP.C. The closed-inclusion
identification has the same topology: its graph
is closed because all finite jets agree, and SC.G
and NP.C give the resulting Fréchet isomorphism.
The entire defining ideal of \(A\), including
nilpotents, is present in this prefix and quotient.

There is a countable basis of strongly one-complete
distinguished chart balls. The radial function
\[
\frac1{r^2-|z-a|^2}
\tag{CX.34}
\]
is a strict exhaustion on a ball, and remains one
on its closed analytic model. Its closed sublevels
are closed subsets of smaller compact balls.
Choose these balls arbitrarily small inside
distinguished patches. For each member of a
countable topological basis, a countable selection
covers that member; the union of these selections
is the required countable basis.
Every finite intersection of its members is
strongly one-complete by CX.F and lies in one
distinguished patch. CX.33 therefore proves
acyclicity of all these intersections.

**Local coherent approximation.** If \(U\subset A\)
is strongly one-complete and \(U'\subset U\)
is Runge complete in \(U\), then
\[
\begin{gathered}
\Gamma(U)\longrightarrow\Gamma(U')\\
\text{has dense image}.
\end{gathered}
\tag{CX.35}
\]
To prove density in the actual section topology,
fix \(h\in\Gamma(U')\) and a zero neighbourhood
there. By SC.G, take a countable cover of \(U'\)
by relatively compact local section domains.
A prescribed zero neighbourhood contains the
conditions imposed by finitely many neighbourhoods
in these section spaces. Let \(L\) be the union
of the closures of those finitely many domains.
It is a compact subset of \(U'\).

Use CX.28–CX.31 for that \(L\), retaining the
same distinguished ambient prefix. In particular,
choose \(a<b'<b\) as there. The restriction of
\(h\) to \(A\cap V_{b'}\) lifts, by CX.33, to
a tuple in \(\mathscr O(V_{b'})^{p_0}\).
This use of CX.33 is legitimate because
\(V_{b'}\) has the strict exhaustion
\(1/(b'-\Psi)\) and is inside the fixed prefix
domain. The full holomorphic approximation
theorem WD.H supplies tuples in
\(\mathscr O(V)^{p_0}\) converging to this lift
on every compact subset of \(V_a\), with all
holomorphic derivatives.

The exhaustion gap required in WD.H is explicit:
choose two more levels strictly between \(a\)
and \(b'\). The closed sublevel at the larger
of them is compactly contained in \(V_{b'}\),
where the lifted tuple is defined. The cutoff
and its outer closed sublevel therefore have
their required common domain.

The continuous quotient and restriction maps
in SC.G send these tuples to sections of
\(\Gamma(U)\) whose restrictions converge
to \(h\) on \(\Gamma(A\cap V_a)\).
Each of the finitely many section domains
chosen above lies in \(A\cap V_a\).
Continuity of restriction makes their prescribed
errors as small as desired. This proves CX.35.
Since the target is metrizable, density also
permits approximation of any fixed section
by a sequence: choose successively a countable
zero-neighbourhood basis.

#### CX.H. The topology and exact sequence of cohomology

On any open set choose a countable cover from the
acyclic basis in CX.G. Its alternating Čech
complex is a complex of Fréchet spaces, with
continuous differential and the actual quotient
topology on cohomology. SC.B identifies this
cohomology with derived sheaf cohomology.
Degree-zero cycles have precisely the SC.G
section topology: restriction gives a continuous
bijection from the section space to the closed
compatible subspace of the product, and NP.C
makes it a homeomorphism.
This topology is independent of the chosen
acyclic basis cover. A common countable basic
refinement has acyclic intersections. Its
refinement cochain maps are continuous, since
each target coordinate is a restriction of
one source coordinate, with the alternating
sign and repeated-index zero convention.
They induce the natural cohomology isomorphisms
by SC.B. ML.C makes these bijective induced
maps homeomorphisms, even before cohomology
is known to be Hausdorff. The same argument
shows that restriction to any open subset
induces a continuous cohomology map.

Let \(G,U\) be strongly one-complete opens,
write \(W=G\cap U\), and suppose \(V=G\cup U\).
There is the Mayer–Vietoris sequence
\[
\begin{gathered}
\cdots\longrightarrow H^q(V)\\
\longrightarrow H^q(G)\oplus H^q(U)\\
\longrightarrow H^q(W)\\
\xrightarrow{\ \partial\ }H^{q+1}(V)\\
\longrightarrow\cdots.
\end{gathered}
\tag{CX.36}
\]
The difference map is the restriction from \(U\)
minus the restriction from \(G\).
Algebraically, take an injective resolution and
apply the two-open augmented sheaf chain in
SC.B. Its stalk contraction and injectivity
give an exact sequence on sections in every
resolution degree, which gives CX.36.

Here is the full complex argument for that
last implication. For a short exact sequence
of complexes, lift a cycle in the quotient
complex to the middle complex. Its differential
is the image of a unique cycle in the first
complex; take that class as the connecting
class. Changing the lift changes it by a
boundary, and lifting a primitive in the
quotient shows that its boundary class has
zero connecting class. If the connecting
class is zero, subtract the corresponding
element of the first complex from the chosen
lift to make the lift a cycle. This proves
exactness at the quotient cohomology.
If a cycle from the first complex becomes
a boundary in the middle, the image of its
primitive is a quotient cycle whose connecting
class is the original one. This proves
exactness at the first cohomology. Finally,
if a middle cycle maps to a quotient boundary,
subtract the differential of a lift of its
primitive; what remains is a cycle from the
first complex. This proves exactness at the
middle cohomology. These arguments in every
degree give the entire sequence and its
naturality.

All connecting maps in CX.36 are continuous
for the topology just defined. To prove this,
choose one countable basic cover \(T_i\) of
\(V\). Every intersection \(T_I\), and its
intersections with \(G,U,W\), is strongly
one-complete and lies in the distinguished
patch of one of its members. They are all
acyclic by CX.33. The algebraic sequence
on the two-open cover of \(T_I\), together
with \(H^1(T_I)=0\), gives the
surjection on the right of the following sequence,
where \(P_I=\Gamma(T_I\cap G)\oplus\Gamma(T_I\cap U)\):
\[
\begin{gathered}
0\longrightarrow\Gamma(T_I)\\
\longrightarrow P_I\\
\longrightarrow\Gamma(T_I\cap W)
\longrightarrow0.
\end{gathered}
\tag{CX.37}
\]
Take products over all increasing tuples.
The resulting short exact sequence of
Čech complexes has continuous maps between
Fréchet spaces, and is onto in each degree:
choose a preimage in every coordinate.
All its three cohomologies are the ones
in CX.36 by SC.B.

More generally, write such a short exact
sequence as \(0\to A\overset{i}\to B
\overset{p}\to C\to0\).
For complexes, \(H^q\) here means ordinary
complex cohomology.
The map \(i\) is a homeomorphism onto
the closed kernel of \(p\), by NP.C.
For each \(q\), the space
\[
\begin{gathered}
R\subset Z^q(C)\oplus B^q,\\
R=\{(z,b):p(b)=z\}.
\end{gathered}
\tag{CX.38}
\]
is closed and Fréchet. Here \(B^q\) denotes the degree-\(q\)
term of the middle complex. The projection of
\(R\) onto
\(Z^q(C)\) is onto and hence open by NP.C.
The map
\[
\begin{gathered}
(z,b)\longmapsto[i^{-1}(db)]\\
\text{in }H^{q+1}(A).
\end{gathered}
\tag{CX.39}
\]
is continuous and constant on the fibres
of that projection. It therefore descends
continuously to cycles in \(C\), and then
through their open quotient to \(H^q(C)\).
This is precisely the connecting map.
No continuous linear choice of preimages
and no closed-boundary assumption has been
used.

#### CX.I. Finite bumps of a strict exhaustion

Let \(\psi\ge0\) be a smooth strict exhaustion
on \(X\), and write
\[
X_t=\{x:\psi(x)<t\}.
\tag{CX.40}
\]
Fix \(c<d\). The closed band
\(\{c\le\psi\le d\}\) is compact.
Cover it by finitely many distinguished
chart balls \(A_j\), with closures in their
chart and prefix domains. There are
nonnegative smooth functions of compact
support in these balls such that
\[
\begin{gathered}
0\le\sum_{j<s}\theta_j\le1,\\
\sum_{j<s}\theta_j=1\\
\text{near the closed band}.
\end{gathered}
\tag{CX.41}
\]
For an explicit construction, choose finitely
many chart cutoffs \(\beta_j\) whose smaller
balls cover the band. Their sum is positive
on a neighbourhood of the band, and has a
positive minimum on the band. Choose a
smooth scalar cutoff \(\zeta\), zero near
zero and one above a smaller positive
threshold. Set
\(\theta_j=\zeta(\sum\beta_k)\beta_j/
\sum\beta_k\), with value zero where the
denominator is zero. CX.7 makes these
smooth, with the stated supports and sum.

There is one \(\varepsilon_0>0\) such that
all partial perturbations
\[
\begin{gathered}
\Theta_j=\sum_{k<j}\theta_k,\\
\psi_j=\psi-\varepsilon\Theta_j,\\
0\le j\le s,\quad0<\varepsilon\le\varepsilon_0,
\end{gathered}
\tag{CX.42}
\]
are strict smooth exhaustions. Their
interpolations adding any fraction of the
next \(\theta_j\) are strict as well.
To verify this uniformly, cover the compact
union of the supports by finitely many
charts carrying a strict extension of
\(\psi\). The finitely many cutoff functions
have smooth local extensions with bounded
second derivatives on smaller compact
charts. The minimum positive Levi margin
of \(\psi\) dominates their total perturbation
when \(\varepsilon_0\) is sufficiently small.
Outside those supports the function is
unchanged. The bound
\(|\psi_j-\psi|\le\varepsilon_0\)
proves exhaustion and a common lower bound.

For any \(t\in[c,d)\) and
\(0<\varepsilon\le\min(\varepsilon_0,d-t)\),
put
\[
\begin{gathered}
G_j=\{\psi_j<t\},\\
U_j=G_{j+1}\cap A_j,\\
W_j=G_j\cap A_j.
\end{gathered}
\tag{CX.43}
\]
Then
\[
\begin{gathered}
G_0=X_t,\qquad G_s=X_{t+\varepsilon},\\
G_{j+1}=G_j\cup U_j,\\
G_j\cap U_j=W_j.
\end{gathered}
\tag{CX.44}
\]
The only change at step \(j\) is supported
inside \(A_j\), proving the second line.
For the first line, if
\(t\le\psi<t+\varepsilon\), the sum in
CX.41 is one. If \(\psi\ge t+\varepsilon\),
the inequality that sum is at most one
keeps the point outside \(G_s\); points
with \(\psi<t\) stay inside. This proves
the exact equality, including its boundary.

The sets \(G_j\) are strongly one-complete
sublevels of global strict exhaustions.
The sets \(U_j,W_j\) have explicit strict
exhaustions on \(U_j,W_j\), respectively, in the chart ball of radius
\(r_j\) and centre \(a_j\):
\[
\begin{gathered}
b_j=\frac1{r_j^2-|z-a_j|^2},\\
\varphi_j=\frac1{t-\psi_{j+1}}+b_j,\\
\widetilde\varphi_j=\frac1{t-\psi_j}+b_j.
\end{gathered}
\tag{CX.45}
\]
Each summand is strict by CX.4. The
denominators force divergence when either
defining boundary is approached. The
closed chart balls and global sublevels
are compact, so bounded sublevels of these
functions are compact within their domains.

Moreover \(W_j\) is Runge complete in \(U_j\).
For a compact \(L\subset W_j\), choose
\[
\sup_L\psi_j<b<t.
\tag{CX.46}
\]
On \(U_j\), the function
\[
\psi_j+\eta\varphi_j,\qquad\eta>0,
\tag{CX.47}
\]
is strict and exhausting. Strictness is
the sum of strict functions, and exhaustion
follows from a lower bound for \(\psi_j\)
and properness of \(\varphi_j\).
For sufficiently small \(\eta\), its
sublevel at \(b\) contains \(L\).
Its closed sublevel is compact in \(U_j\)
and lies in \(\{\psi_j\le b\}\subset W_j\),
because \(\varphi_j>0\).
This is exactly CX.28.

Finally partition \([c,d]\) into finitely
many intervals of length at most
\(\varepsilon_0\), and perform these
constructions at each left endpoint.
The same original band and cutoffs work
at every step; each block starts with
the original \(\psi\) and its new threshold.
CX.44 identifies successive endpoints.
We obtain a finite chain of these bumps
from \(X_c\) to \(X_d\).

#### CX.J. Hausdorff finiteness before using density

Local acyclicity CX.33 and CX.36 first
give, for every one of the bumps,
\[
H^q(G_{j+1})\longrightarrow
H^q(G_j)
\tag{CX.48}
\]
surjective for \(q\ge1\) and bijective
for \(q\ge2\). Both \(U_j\) and \(W_j\)
are acyclic; the only preceding term
that can obstruct injectivity in degree
one is the connecting map from
\(\Gamma(W_j)\).
In particular the finite chain gives
surjections
\[
\begin{gathered}
H^q(X_d)\longrightarrow H^q(X_c)\\
(q\ge1).
\end{gathered}
\tag{CX.49}
\]
No density or separation assertion has
entered this conclusion.

Choose finitely many nested distinguished
chart balls
\[
P_i'\Subset P_i\subset X_d
\tag{CX.50}
\]
whose smaller balls cover \(\overline X_c\).
Their existence follows from compactness
of that closure in \(X_d\). Include the
larger \(P_i\) in a countable basic
acyclic cover of \(X_d\). The finite
sets \(P_i'\cap X_c\) form an acyclic
cover of \(X_c\): all their intersections
are strongly one-complete by CX.F and
lie in a distinguished patch.
The refinement map on Čech complexes
has in each degree only finitely many
target coordinates. Each nonzero coordinate
is a restriction of coherent sections
from an intersection of larger balls
to a relatively compact intersection
of the smaller ones with \(X_c\).
The target closure is compact and lies
inside every relevant larger ball.

SC.H makes every such restriction scalar
nuclear. It is compact by the uniform
tail proof ML.I. Precomposing with a
coordinate projection and taking finitely
many components preserves compactness,
again by ML.I. Thus the refinement is
a compact chain map in every degree.
It induces CX.49 by SC.B. ML.C proves
\[
\begin{gathered}
H^q(X_c)\\
\text{is finite dimensional}\\
\text{and Hausdorff}\quad(q\ge1).
\end{gathered}
\tag{CX.51}
\]
The statement holds for every sublevel
of every smooth strict exhaustion.
In particular it applies to every \(G_j\)
in CX.43 by using its own global
perturbed exhaustion. This argument
establishes separation before separation
is used to take limits in cohomology.

#### CX.K. Injectivity and approximation across a bump

The connecting map
\[
\partial:\Gamma(W_j)\longrightarrow
                H^1(G_{j+1})
\tag{CX.52}
\]
is continuous by CX.H. Its target is
Hausdorff by CX.J. Its restriction to
the image of \(\Gamma(U_j)\) is
zero by exactness. That image is dense
by CX.35, since \(W_j\) is Runge
complete in \(U_j\), both inside
the same distinguished patch \(A_j\).
Continuity into the Hausdorff target
therefore forces CX.52 to vanish
identically. CX.36 now proves that
CX.48 is bijective also in degree one.
It is a homeomorphism by ML.C.

In degree zero, exactness gives the
continuous surjection of Fréchet spaces
\[
\begin{gathered}
\Gamma(G_j)\oplus\Gamma(U_j)\\
  \longrightarrow\Gamma(W_j),\\
(g,u)\longmapsto u|_{W_j}-g|_{W_j}.
\end{gathered}
\tag{CX.53}
\]
It is open by NP.C. For fixed
\(g\in\Gamma(G_j)\), choose
\(u_\nu\in\Gamma(U_j)\) with
\[
w_\nu=u_\nu|_{W_j}-g|_{W_j}\longrightarrow0.
\tag{CX.54}
\]
Null-sequence lifting in NP.C gives
\((g_\nu',u_\nu')\to0\) mapped to
\(w_\nu\) by CX.53. Thus
\[
g-g_\nu'\quad\text{and}\quad u_\nu-u_\nu'
\tag{CX.55}
\]
agree on \(W_j\) and glue to a section
on \(G_{j+1}\). Its restriction to
\(G_j\) tends to \(g\).
This proves dense restriction on sections
across each bump. Finite composition
preserves density. If the image of
a map \(f\) is dense in an intermediate
Fréchet space and \(g\) is continuous,
approximate each intermediate point
by a sequence from that image.
Its image under \(g\) converges to
the image of the point. Thus the
closure of the composite image
contains the image of \(g\), and
is the whole target whenever that
image is dense.
We have proved, for every \(c<d\),
\[
\begin{gathered}
H^q(X_d)\simeq H^q(X_c),\\
q\ge1,\\
\Gamma(X_d)\longrightarrow\Gamma(X_c)\\
\text{has dense image}.
\end{gathered}
\tag{CX.56}
\]

#### CX.L. Global coherent vanishing and Runge approximation

Choose \(c\) below the lower bound of
\(\psi\), so \(X_c\) is empty.
CX.56 gives
\[
\begin{gathered}
H^q(X_d)=0\\
(q\ge1,\ d\in\mathbf R).
\end{gathered}
\tag{CX.57}
\]
To pass to \(X\), take the countable
basic acyclic cover of CX.G and let
its \(n\)-th subfamily consist of the
members whose closures lie in \(X_n\).
It covers \(X_n\): around any point
one can choose a relatively compact
neighbourhood in \(X_n\), then a basic
member inside it. Its closure is
still in that neighbourhood's compact
closure inside \(X_n\).
Every basic member appears at a finite
stage, because its closure is compact
and \(\psi\) is continuous.
The subfamilies are increasing.

All finite intersections are acyclic
by CX.G. Their section spaces have
the canonical Fréchet topology SC.G.
CX.57 gives vanishing at every stage,
and CX.56 gives dense section
restrictions. The exact Čech
exhaustion criterion ML.H therefore
proves the full coherent vanishing
theorem
\[
H^q(X)=0\qquad(q\ge1).
\tag{CX.58}
\]
The convergent primitive argument in
ML.F, not merely convergence of
cocycles, is what makes this passage
valid for actual cohomology.

ML.F in degree zero also shows that
\[
\begin{gathered}
\Gamma(X)\longrightarrow\Gamma(X_c)\\
\text{has dense image}
\end{gathered}
\tag{CX.59}
\]
for every sublevel. To see the stated
level directly, choose the exhaustion
subfamilies starting with that level
and then increasing by integers.
Their term maps are surjective
coordinate projections, and their
degree-zero cohomology maps are dense
by CX.56. The density conclusion
ML.34 is precisely CX.59.

More generally, if an open \(U\subset X\)
is Runge complete in \(X\) in the
sense of CX.28, then
\[
\begin{gathered}
\Gamma(X)\longrightarrow\Gamma(U)\\
\text{has dense image}.
\end{gathered}
\tag{CX.60}
\]
For a desired section and zero
neighbourhood in \(\Gamma(U)\),
choose the finite collection of
relatively compact local section
domains controlling that neighbourhood,
as in the proof of CX.35. Their
compact closures form \(L\).
The Runge condition chooses an
exhaustion, and a sublevel compactly
contained in \(U\), containing \(L\).
Use CX.59 for that exhaustion and
that sublevel. Continuous restriction
to the finitely many local domains
gives the prescribed approximation.
This proves CX.60 in the canonical
topology, for arbitrary coherent
coefficients.

#### CX.M. Global generation at stalks and near compact sets

Fix \(x\in X\), and let \(\mathscr I_x\)
be the ideal sheaf of functions
vanishing at \(x\). It is coherent.
Near \(x\), ambient coordinate
differences generate its stalk and
its ideal sheaf: a holomorphic
function vanishing at the point
has the identity
\[
\begin{gathered}
F_t=a+t(z-a),\\
h_j(z)=\int_0^1\partial_j f(F_t)\,dt,\\
q_j(z)=(z_j-a_j)h_j(z),\\
f(z)-f(a)=\sum_jq_j(z).
\end{gathered}
\tag{CX.61}
\]
on a sufficiently small convex
ambient neighbourhood. The integrals
are holomorphic, as follows by
differentiation under their compact
parameter integral. This proves
generation before passing to the
analytic quotient ring.
At a nearby point distinct from
\(x\), one coordinate difference
is a unit on a smaller neighbourhood.
Thus the generated ideal has
exactly the claimed point ideal
stalks; away from \(x\) it is
\(\mathscr O_X\).
The coherent module algebra SC.D
now proves coherence of that finite
image. It also proves coherence of
\(\mathscr I_x\mathscr S\).

The quotient
\(\mathscr S/\mathscr I_x\mathscr S\)
is the finite-dimensional skyscraper
with fibre
\[
\mathscr S_x/\mathfrak m_x\mathscr S_x.
\tag{CX.62}
\]
Finiteness follows by applying the
finite stalk generating list of
\(\mathscr S\) modulo the maximal
ideal. Theorem CX.58 applied to
\(\mathscr I_x\mathscr S\) makes
the section map onto CX.62.
Lift a finite vector-space generating
list to global sections of
\(\mathscr S\). Their germs generate
\(\mathscr S_x\).

Here is the entire local-ring step.
The quotient \(M\) of that finite
stalk module by the lifted germs
is finite and satisfies
\(M=\mathfrak m_x M\).
For finite generators \(m_i\), write
\[
\begin{gathered}
m_i=\sum_j a_{ij}m_j,\\
a_{ij}\in\mathfrak m_x,\\
\det(1-A)m_i=0.
\end{gathered}
\tag{CX.63}
\]
The second identity follows by
multiplying the matrix equation by
the adjugate. Its determinant belongs
to \(1+\mathfrak m_x\), so is a unit
in the local ring. Hence all \(m_i\)
vanish and \(M=0\).

The coherent cokernel of the map
from the finite free sheaf defined
by these global sections has zero
stalk at \(x\). A finite local
generating list of that cokernel
then vanishes on one common
neighbourhood, so the cokernel
vanishes there. Consequently
\[
\begin{gathered}
\mathscr O_X^{r_x}\longrightarrow\mathscr S\\
\text{is onto near }x
\end{gathered}
\tag{CX.64}
\]
for a tuple of global sections.
For a compact subset choose finitely
many of these neighbourhoods and
take the union of their finite
global section lists. They generate
on an open neighbourhood of the
compact set. This proves global
generation at every stalk and finite
generation near every compact;
it does not assert a single finite
list on a general noncompact \(X\).

#### CX.N. Discrete interpolation and holomorphic convexity

If \(Z\subset X\) is closed and
discrete, its point ideal sheaf is
coherent by the same local argument:
each point has a neighbourhood
meeting at most one point of \(Z\),
and off \(Z\) the ideal is the whole
structure sheaf. Its quotient has
one copy of \(\mathbf C\) at each
point of \(Z\). On the discrete
closed set its sections are
arbitrary collections of scalars.
CX.58 therefore gives
\[
\begin{gathered}
\mathscr O_X(X)\longrightarrow
\prod_{z\in Z}\mathbf C\\
\text{surjective}.
\end{gathered}
\tag{CX.65}
\]
This includes separation of two
distinct points by global holomorphic
functions.

For a compact \(K\subset X\), define
its holomorphic hull by
\[
\begin{gathered}
\mathcal H=\mathscr O_X(X),\\
M_f=\sup_K|f|,\\
A_f=\{x:|f(x)|\le M_f\},\\
\widehat K=\bigcap_f A_f.
\end{gathered}
\tag{CX.66}
\]
The intersection is over \(f\in\mathcal H\).
It is closed as an intersection
of closed sets. If it were
noncompact, use CX.5 to select
distinct points in it escaping
every compact set: inductively
choose the next point outside a
larger compact exhaustion member
containing the preceding points.
The resulting set \(Z\) is closed
and discrete, because each compact
set meets only finitely many terms.
By CX.65 choose \(f\) with its
values on that sequence equal to
the positive integers. This
contradicts the finite bound
\(\sup_K|f|\) required by CX.66.
Thus every holomorphic hull of
a compact set is compact.

The theorem concerns every analytic
space with the stated strict smooth
exhaustion and every coherent sheaf
on it. No smoothness of the space,
reducedness, coefficient flatness,
or global finite free resolution
has been assumed.

#### CX.O. A normal section of the smooth maximum

In \(\mathbf C^2\), with coordinates
\((z,w)\), the two functions
\[
\begin{gathered}
a=|z|^2+|w|^2,\\
b=\tfrac12\operatorname{Re}w,\\
\phi_\pm=a\pm b,\\
\eta_+=\eta_-=\tfrac15
\end{gathered}
\tag{CX.67}
\]
have identity Levi matrix and agree
on the analytic subspace \(Y=\{w=0\}\).
Their smooth maximum from CX.8 has
identity as a lower Levi bound:
CX.9 gives total derivative weight
one, and the Hessian contribution
in CX.4 is nonnegative.
On the real normal section
\(z=0,\ w=t\in\mathbf R\), put
\(m(t)=M_\eta(\phi_+(0,t),\phi_-(0,t))\).
The exact bounds and deletion rule are
\[
\begin{gathered}
t^2+\tfrac12|t|\le m(t)\\
\le t^2+\tfrac12|t|+\tfrac15,\\
|t|>\tfrac25\ \Longrightarrow\\
m(t)=t^2+\tfrac12|t|.
\end{gathered}
\tag{CX.68}
\]
The last implication follows because
the difference of the two inputs
has absolute value \(|t|\), exceeding
the sum of their smoothing radii.
With only one active variable, the
even kernel makes its smooth
maximum equal to that variable.

![A real normal section of the smooth maximum and the order of the coherent vanishing argument.](assets/coherent-cartan.png)

*The left panel is the real normal
section specified in CX.67. The
shaded envelope and the deletion
thresholds are the exact bounds
CX.68. The smooth curve is evaluated
by the stated finite quadrature of
CX.8; this numerical evaluation is
an illustration, while CX.A–CX.C
prove positivity and locality.
The right panel records the actual
implications CX.G–CX.M, including
the Hausdorff conclusion before
the density argument. The
drawing source
and constants and quadrature data
specify the figure.*

#### CX.P. Coherent cohomology on an arbitrary compact analytic space

The local arguments also give the
compact finiteness theorem without
a global strict exhaustion. Let
\(Z\) be any compact Hausdorff
analytic space and \(\mathscr T\)
any coherent analytic sheaf on it.
Compactness gives second countability:
finitely many analytic charts cover
\(Z\), and each has a countable basis.
Choose finitely many pairs of
nested distinguished chart balls
\(P_i'\Subset P_i\) such that the
smaller balls cover \(Z\).
Both families are finite acyclic
covers by CX.F–CX.G; all their
finite intersections are acyclic
for the same reason.

The refinement cochain map from
the larger cover to the smaller
one has compact maps in every
degree. Each target intersection
is relatively compact in the
corresponding source intersection;
SC.H and ML.I make its section
restriction compact. There are
only finitely many coordinates.
SC.B identifies the induced map
with the identity on sheaf
cohomology. ML.C therefore proves
\[
\begin{gathered}
H^q(Z,\mathscr T)\\
\text{is finite dimensional}\\
\text{and Hausdorff}\quad(q\ge0).
\end{gathered}
\tag{CX.69}
\]
In degree zero this includes the
ordinary section topology SC.G,
by CX.H.
The proof applies to singular
and nonreduced \(Z\) and arbitrary
coherent \(\mathscr T\).

The exhaustion, neighbourhood and
bump method is also developed in
[Demailly's free author manuscript, Chapter IX, §2 and §4](https://www-fourier.univ-grenoble-alpes.fr/~demailly/manuscripts/agbook.pdf).
The arguments CX.A–CX.P include the
ambient construction, coherent
quotients, exact-sequence topology
and convergent global passage used
here.


### 5.98. Coherent products and analytic coefficient restriction

All analytic spaces in this section are Hausdorff and
second countable; their structure sheaves may have
nilpotents. Section spaces carry the canonical nuclear
Fréchet topology of
SC.G.
Scalar completed tensor products use
NT.E--NT.F.
Balanced tensors and completed Tor mean the actual
quotients and bar homology of
CT.A--CT.D;
their defining relation images are not replaced by closures.

The analytic hypotheses below use a smooth strictly
plurisubharmonic exhaustion, in the precise sense of
CX.A.
We use the complete coherent vanishing and generation
proofs
CX.L
and
CX.M.
The coherent algebra and finite local presentations are
those proved in
SC.D.

#### AT.A. Analytic products and coherent exterior products

In local models
\[
\begin{gathered}
A\subset D\subset\mathbf C^n,\\
B\subset D'\subset\mathbf C^m,\\
\mathcal O_A=\mathcal O_D/\mathcal J,\\
\mathcal O_B=\mathcal O_{D'}/\mathcal K,
\end{gathered}
\tag{AT.1}
\]
the analytic product is the closed model in \(D\times D'\)
with ideal generated by \(\mathcal J(z)\) and
\(\mathcal K(w)\). These are the entire ideals, including
their nilpotent information. To check compatibility,
a holomorphic map into this model is a pair of coordinate
maps into \(D,D'\) that kills the respective ideals.
This is exactly a pair of maps into \(A,B\).
Holomorphic coordinate changes in either factor therefore
induce inverse coordinate changes of the product models,
and these agree on every triple overlap by this same
pair-of-maps property. They glue to the product space.
The underlying topology is the product topology:
this is so in each local model, and the model products
form an open cover. Products of countable bases give a
countable base; distinct points are separated in at
least one Hausdorff factor.

For coherent sheaves \(\mathcal F,\mathcal G\) on \(X,Y\),
respectively, define
\[
\begin{gathered}
\mathcal F\boxtimes\mathcal G
=p_X^*\mathcal F
 \otimes_{\mathcal O_{X\times Y}}p_Y^*\mathcal G.
\end{gathered}
\tag{AT.2}
\]
Here pullback means the inverse-image sheaf tensored
with the structure sheaf of the product. If presentations
in the two ambient charts have matrices \(M,N\) and
numbers of generators \(p,q\), the corresponding
presentation of the closed pushforward of AT.2 has
generators \(e_a\otimes f_b\) and relations
\[
\begin{gathered}
M(z)e\otimes f,\\
e\otimes N(w)f,\\
\mathcal J(z)(e\otimes f),\\
\mathcal K(w)(e\otimes f).
\end{gathered}
\tag{AT.3}
\]
One can include the ideal relations in \(M,N\) from
the start by presenting the closed pushforwards.
This is a finite presentation locally, so SC.D proves
coherence. The presentation is independent of the
chosen generators because its maps to any module
are precisely the bilinear maps subject to the
stated relations.

This also proves right exactness of the exterior
product in each variable. Indeed, replacing a
module by a cokernel adds exactly the images of its
defining relations to AT.3. No exactness assertion
about the kernel of an arbitrary pullback is involved.
For a closed inclusion \(j:B\hookrightarrow D'\),
the same presentation gives
\[
\begin{gathered}
\mathcal F\boxtimes j_*\mathcal G\\
\simeq(1_X\times j)_*
       (\mathcal F\boxtimes\mathcal G).
\end{gathered}
\tag{AT.4}
\]
Its underlying sheaf is supported on \(X\times B\);
the quotient structure and all its relations are
the same on both sides.

There is a canonical jointly continuous section product.
To check continuity locally, choose distinguished
chart patches on which the closed pushforwards have
the finite ambient presentations of SC.D. Their section
maps from finite tuples of ambient holomorphic functions
are open surjections by SC.E and the open mapping
theorem NP.C. Multiplying two such tuples in separate
variables is continuous for the compact-open seminorms.
Its composite with the coherent target section map is
continuous by SC.G and descends to the required
bilinear product. A product of two open surjections
is open: images of basic product neighbourhoods are
product neighbourhoods. Thus the descended bilinear
map is jointly continuous. On arbitrary opens the
same conclusion follows by restriction to these
patches, since SC.G gives the closed compatible-product
topology. A compact subset of a product has a finite
cover by smaller chart rectangles; hence the same
local argument controls every target seminorm.
The completed projective tensor universal property
and completeness now give a continuous map
\[
\begin{gathered}
\Phi_{X,Y}:
\mathcal F(X)\widehat\otimes\mathcal G(Y)\\
\longrightarrow
(\mathcal F\boxtimes\mathcal G)(X\times Y),\\
s\otimes t\longmapsto p_X^*s\otimes p_Y^*t.
\end{gathered}
\tag{AT.5}
\]
The defining formula proves naturality for coherent
maps and open restrictions on elementary tensors;
continuity and density extend each equality to
the completions.

#### AT.B. Complete product domains

If \(X,Y\) are strongly one-complete, choose their
strict smooth exhaustions \(\psi_X,\psi_Y\) nonnegative.
Then
\[
\Psi(x,y)=\psi_X(x)+\psi_Y(y)
\tag{AT.6}
\]
is a strict smooth exhaustion on \(X\times Y\).
In a product ambient chart its Levi form is the sum
of the two positive definite forms, on the two
coordinate components of a tangent vector. It is
therefore positive on every nonzero ambient vector.
A closed sublevel of AT.6 is a closed subset of the
product of the two compact closed sublevels at that
same level. That product is compact: replace an open
cover locally by product rectangles. For each fixed
point of the first factor, finitely many rectangles
cover its compact vertical fibre. Intersect their
first-factor neighbourhoods; over that intersection
the same finite rectangles cover every fibre.
Finitely many such intersections cover the compact
first factor, giving a finite subcover of the product.
Thus AT.6 is proper and
bounded below. This proves the product assertion
with the full analytic structures.

Polydiscs have the explicit strict exhaustion
\(\sum_j(R_j^2-|z_j-a_j|^2)^{-1}\).
Each summand has positive Levi coefficient
\((R_j^2+|z_j-a_j|^2)/(R_j^2-|z_j-a_j|^2)^3\)
in its own coordinate. Its closed sublevels stay
inside a smaller closed polydisc, and are compact
by the finite-dimensional compactness proof CX.A.
Restriction to a closed analytic model retains
strictness in ambient charts and compactness of
closed sublevels. Thus those closed polydisc models
are strongly one-complete as well.

In particular CX.L applies to every coherent sheaf
on such a product. If
\(0\to\mathcal R\to\mathcal E\to\mathcal Q\to0\)
is an exact sequence of coherent sheaves there,
its sequence of sections is exact because
\(H^1(\mathcal R)=0\). Its maps are continuous by
SC.G. Its kernel is closed, and its surjection is
open by NP.C. We shall use this strict section
exactness on products and on polydiscs.

#### AT.C. The section product in distinguished patches

We prove that AT.5 is a topological isomorphism.
First let \(A\) be a distinguished closed model
in a polydisc \(D\), and let \(D'\) be a polydisc.
The ambient presentation of its coherent sheaf
has a right exact sequence
\[
\mathcal O_D^{p_1}\longrightarrow
\mathcal O_D^{p_0}\longrightarrow i_*\mathcal F
\longrightarrow0.
\tag{AT.7}
\]
The kernel and image sheaves are coherent.
Coherent vanishing on \(D\), or directly SC.E for
this prefix, gives the actual right exact section
sequence. In particular the first image is the
closed kernel of the last surjection.
Scalar strict tensor exactness NT.F therefore
gives the right exact sequence
\[
\begin{gathered}
\mathcal O(D)^{p_1}\widehat\otimes\mathcal O(D')\\
\longrightarrow
\mathcal O(D)^{p_0}\widehat\otimes\mathcal O(D')\\
\longrightarrow
\mathcal F(A)\widehat\otimes\mathcal O(D')
\longrightarrow0.
\end{gathered}
\tag{AT.8}
\]
For completeness, factor the first arrow of the
section sequence through its closed image.
Tensor the resulting short exact sequence and
the open surjection onto that image. Their
composite has precisely the kernel claimed in
AT.8. Thus no strictness of a nonclosed image
has been assumed.

Right exactness in AT.A pulls AT.7 to \(D\times D'\).
This product is strongly one-complete by AT.B.
Taking sections gives the second right exact row
with terms
\[
\begin{gathered}
\mathcal O(D\times D')^{p_1},\\
\mathcal O(D\times D')^{p_0},\\
(\mathcal F\boxtimes\mathcal O_{D'})(A\times D').
\end{gathered}
\tag{AT.9}
\]
The scalar product isomorphism CT.I identifies the
first two terms of AT.8 with those of AT.9,
including their maps: both maps are multiplication
by the matrix in AT.7. Their actual cokernels,
with quotient topologies, are therefore identical.
Using AT.4 proves
\[
\begin{gathered}
\mathcal F(A)\widehat\otimes\mathcal O(D')\\
\simeq
(\mathcal F\boxtimes\mathcal O_{D'})(A\times D').
\end{gathered}
\tag{AT.10}
\]

Now let \(B\subset D'\) also be a distinguished
closed model, with presentation
\[
\begin{gathered}
\mathcal O_{D'}^{q_1}\longrightarrow
\mathcal O_{D'}^{q_0}\\
\longrightarrow j_*\mathcal G
\longrightarrow0.
\end{gathered}
\tag{AT.11}
\]
Tensor its strict section cokernel with the
nuclear space \(\mathcal F(A)\).
By the same factor-through-image argument as
in AT.8, the result is a right exact row ending
in \(\mathcal F(A)\widehat\otimes\mathcal G(B)\).
Exterior product of AT.11 with \(\mathcal F\)
gives a right exact coherent sequence on
\(A\times D'\). This product is strongly
one-complete, since its factors are closed
polydisc models. Taking sections is again right
exact. Its first two terms agree topologically
with those of the tensor row by AT.10.
Taking the same cokernel and using AT.4 gives
\[
\begin{gathered}
\Phi_{A,B}:\\
\mathcal F(A)\widehat\otimes\mathcal G(B)\\
\simeq(\mathcal F\boxtimes\mathcal G)(A\times B).
\end{gathered}
\tag{AT.12}
\]
This is the map AT.5, as its formula on elementary
tensors shows. Every step used the actual coherent
presentations; the factors need not be smooth,
reduced or locally free.

#### AT.D. Gluing the product isomorphism

Every coherent sheaf has a countable cover by
the distinguished patches just used: start
with the local prefixes in SC.D, shrink their
ambient domains to polydiscs, and use second
countability. For a cover \((V_i)\) of \(Y\),
choose a countable cover
\((T_{ij\ell})_\ell\) of every nonempty
\(V_i\cap V_j\) by further distinguished patches.
These patches use the restricted sheaf
\(\mathcal G\); their presentations may be
different from those on \(V_i\).
Set
\[
\begin{gathered}
P=\prod_i\mathcal G(V_i),\\
Q=\prod_{i,j,\ell}\mathcal G(T_{ij\ell}),\\
a((t_i))_{ij\ell}\\
=t_j|_{T_{ij\ell}}-t_i|_{T_{ij\ell}}.
\end{gathered}
\tag{AT.13}
\]
Both products are countable nuclear Fréchet
spaces. The map \(a\) is continuous. The sheaf
axiom gives
\(\mathcal G(Y)=\ker a\); its topology is the
closed compatible-product topology SC.G.
One may also identify it by the continuous
bijection onto that closed kernel and NP.C.

Fix a distinguished \(A\subset X\).
By CT.H, tensor with the nuclear space
\(\mathcal F(A)\) preserves this kernel even
if the image of \(a\) is not closed.
It commutes with the countable products by
NT.20, proved in NT.F. Consequently
\[
\begin{gathered}
P_A=\prod_i
 \mathcal F(A)\widehat\otimes\mathcal G(V_i),\\
Q_A=\prod_{i,j,\ell}
 \mathcal F(A)\widehat\otimes\mathcal G(T_{ij\ell}),\\
\mathcal F(A)\widehat\otimes\mathcal G(Y)\\
\simeq\ker(P_A\to Q_A).
\end{gathered}
\tag{AT.14}
\]
Use AT.12 on each product factor. Naturality of
AT.5 makes the resulting map precisely the
difference of restrictions of the exterior
product sheaf. Equality on all
\(A\times T_{ij\ell}\) is equality on
\(A\times(V_i\cap V_j)\), since those rectangles
cover that overlap. Thus the kernel is
\((\mathcal F\boxtimes\mathcal G)(A\times Y)\),
with its SC.G topology.

Next cover arbitrary \(X\) by distinguished
patches \(A_i\), and cover each \(A_i\cap A_j\)
by countably many further distinguished patches.
Apply exactly the kernel construction AT.13
to \(\mathcal F\). Tensor it with
\(\mathcal G(Y)\), use CT.H and NT.20, and use
the case \(A\times Y\) just proved in each
factor. The kernel is the sections of the
exterior product on \(X\times Y\).
We obtain the natural topological isomorphism
\[
\begin{gathered}
\mathcal F(X)\widehat\otimes\mathcal G(Y)\\
\simeq
(\mathcal F\boxtimes\mathcal G)(X\times Y).
\end{gathered}
\tag{AT.15}
\]
The empty-space cases are zero on both sides.
No finite global presentation on an overlap
was needed: only its further distinguished
cover entered AT.13.

For the structure sheaves this includes
\[
\begin{gathered}
\mathcal O(X)\widehat\otimes\mathcal O(Y)\\
\simeq\mathcal O(X\times Y)
\end{gathered}
\tag{AT.16}
\]
on arbitrary analytic spaces under the stated
topological hypotheses. This is an isomorphism
of topological algebras: multiplication agrees
on elementary tensors, then on the completion
by joint continuity.
The same local open-quotient proof in AT.A,
with multiplication in one chart, proves
joint continuity of every coherent section
module action. Thus all coefficient modules
used below meet the action hypothesis in CT.

#### AT.E. One product domain for an entire free resolution

Let \(X,Y\) be strongly one-complete and let
\(\mathcal H\) be coherent on \(X\times Y\).
Let \(U\Subset X\), \(V\Subset Y\) be strongly
one-complete open subsets. The set
\[
K=\overline U\times\overline V
\tag{AT.17}
\]
is compact in \(X\times Y\).
We construct a resolution on the same
\(U\times V\) in every degree.

Apply CX.M to \(\mathcal H\) and \(K\).
Finitely many sections defined on all
\(X\times Y\) give a map
\(\mathcal L_0=\mathcal O_{X\times Y}^{r_0}
\to\mathcal H\) that is onto near \(K\).
Its global kernel \(\mathcal S_0\) is coherent
by SC.D, although the map need not be onto
on all \(X\times Y\). Apply CX.M to
\(\mathcal S_0\) and the same \(K\). This
gives \(\mathcal L_1\to\mathcal S_0\) onto
near \(K\); compose with its inclusion in
\(\mathcal L_0\). Repeat with each global
coherent kernel. With
\(\mathcal S_{-1}=\mathcal H\), each step is
\[
\begin{gathered}
\mathcal L_q=\mathcal O_{X\times Y}^{r_q},\\
\mathcal L_q\longrightarrow\mathcal S_{q-1},\\
\mathcal S_q
=\ker(\mathcal L_q\to\mathcal S_{q-1}).
\end{gathered}
\tag{AT.18}
\]
Each \(r_q\) is finite and each kernel is
coherent on the whole product. Consecutive
differentials compose to zero because the
intermediate map lands in the preceding
kernel. Every surjectivity neighbourhood
contains \(K\), so every map is onto on
\(U\times V\). The resulting sequence
\[
\begin{gathered}
\cdots\longrightarrow\mathcal L_1
\longrightarrow\mathcal L_0\\
\longrightarrow\mathcal H\longrightarrow0
\end{gathered}
\tag{AT.19}
\]
is therefore exact on that fixed domain
in all degrees. There is no need to take
an open intersection of infinitely many
different neighbourhoods.

Let \(U'\subset U\) be any strongly
one-complete open subset. Restriction to
\(U'\times V\) preserves exactness at each
stalk. The products \(U\times V\) and
\(U'\times V\) are strongly one-complete
by AT.B. Apply CX.L to each coherent
kernel in AT.19. The augmented complexes
of sections on both products are exact.
Their maps are continuous and all images
are closed kernels, including the
augmentation kernel. NP.C gives strict
short exact sequences at every syzygy.

In particular AT.19 supplies an actual
nuclearly free resolution over the algebra
\[
A=\mathcal O(U).
\tag{AT.20}
\]
Indeed AT.16 identifies its terms with
\[
\begin{gathered}
\mathcal L_q(U\times V)\\
\simeq A\widehat\otimes\mathcal O(V)^{r_q}.
\end{gathered}
\tag{AT.21}
\]
The coefficient spaces are nuclear by
SC.G, and multiplication is continuous
by AT.D. Finite rank in AT.21 concerns
each individual degree; no uniform bound
on all the \(r_q\) is asserted.

#### AT.F. Coherent coefficient restriction and its topology

Put
\[
\begin{gathered}
B=\mathcal O(U'),\\
E=\mathcal H(U\times V),\\
E'=\mathcal H(U'\times V).
\end{gathered}
\tag{AT.22}
\]
There is a continuous \(A\)-algebra map
\(A\to B\) by restriction. CT.1 and AT.16
give in every degree
\[
\begin{gathered}
B\widehat\otimes_A\mathcal L_q(U\times V)\\
\simeq\mathcal L_q(U'\times V).
\end{gathered}
\tag{AT.23}
\]
This identification respects the
differentials and augmentation:
on elementary tensors a holomorphic
matrix coefficient is restricted in the
first variable and then multiplied.
These are exactly the restricted matrices
of AT.19. Density and continuity prove
the same identities on completed tensors.
CT.D, applied to the actual resolution
AT.21, and the exactness on \(U'\times V\)
therefore prove
\[
\begin{gathered}
\widehat{\operatorname{Tor}}^A_q(B,E)=0
\quad(q\ge1),\\
B\widehat\otimes_A E\simeq E',\\
[b\otimes s]\longmapsto b\,s|_{U'\times V}.
\end{gathered}
\tag{AT.24}
\]
The degree-zero isomorphism is topological,
by the degree-zero cokernel proof in CT.D.
Its target has the intrinsic SC.G topology
and is Hausdorff. Thus \(B,E\) are transverse
over \(A\) in the full sense CT.15, with
the actual balanced relation quotient.
Neither local freeness of \(\mathcal H\)
nor flatness over the first factor was
assumed.

All restriction comparisons in AT.24
are canonical. If \(U''\subset U'\) is
strongly one-complete, apply the same
resolution AT.19 on \(U''\times V\).
Its degreewise formula identifies both
successive coefficient restriction and
direct restriction with the same section
complex. Equivalently CT.F gives
\[
\begin{gathered}
\mathcal O(U'')\widehat\otimes_B E'\\
\simeq \mathcal H(U''\times V),
\end{gathered}
\tag{AT.25}
\]
because both \(B\) and \(\mathcal O(U'')\)
are transverse to \(E\) over \(A\) by
AT.24. The isomorphism sends
\([c\otimes[b\otimes s]]\) to
\(c\,b|_{U''}\,s|_{U''\times V}\).
This proves compatibility with each
arbitrary smaller strongly one-complete
base, not merely with a sequence of radii.
The same formula proves commutation with
restriction to a smaller strongly
one-complete subset of \(V\); apply AT.24
there and compare on elementary tensors.

#### AT.G. Coherent kernels and actual cycle embeddings

For an exact coherent sequence on \(X\times Y\),
\[
\begin{gathered}
0\to\mathcal H_1\to\mathcal H_2\\
\to\mathcal H_3\to0,
\end{gathered}
\tag{AT.26}
\]
set \(E_i=\mathcal H_i(U\times V)\).
CX.L gives a strict short exact sequence
of these section modules. AT.24 makes
each \(E_i\) transverse to \(B\).
The kernel theorem CT.E gives
\[
\begin{gathered}
0\to B\widehat\otimes_A E_1
\to B\widehat\otimes_A E_2\\
\to B\widehat\otimes_A E_3\to0
\end{gathered}
\tag{AT.27}
\]
strictly. AT.24 identifies it, arrow by
arrow, with AT.26 on \(U'\times V\).
This supplies a closed embedding into the
actual smaller-base section module.

There is also a useful statement for
complexes whose terms come from different
product charts. Let \(C^\bullet\) be
bounded above, with continuous
\(A\)-linear differentials. Suppose each
term is a finite sum of modules of the
form AT.22, possibly for different
strongly one-complete relatively compact
fibre domains and different coherent
sheaves on their stated outer products.
Include finite free \(A\)-modules if
needed. Suppose the same \(U'\) meets
the AT.24 hypotheses for every summand.
Then \(B\) is transverse to every term,
by AT.24 and finite direct sums CT.E.

If \(H^q(C^\bullet)=0\) for \(q\ge k\),
the actual cycle modules \(Z^q=\ker d^q\)
are transverse to \(B\) for \(q\ge k-1\).
More precisely CT.G proves the strict
sequences
\[
\begin{gathered}
0\to B\widehat\otimes_A Z^q
\to B\widehat\otimes_A C^q\\
\to B\widehat\otimes_A Z^{q+1}\to0,\\
q\ge k-1.
\end{gathered}
\tag{AT.28}
\]
The first arrow has its actual closed
subspace topology. Starting at the last
zero degree and applying CT.E downwards
to
\(0\to Z^q\to C^q\to Z^{q+1}\to0\)
proves every transversality premise in
this statement. In degrees \(q\ge k\)
the middle differential has kernel
\[
\begin{gathered}
Z^q(B\widehat\otimes_A C^\bullet)
=B\widehat\otimes_A Z^q,
\end{gathered}
\tag{AT.29}
\]
and this is its preceding boundary
module, by the surjection in AT.28
at \(q-1\). Thus the base-changed complex
is acyclic in those degrees.
The stated range of exactness is a
hypothesis on the original complex;
termwise nuclearity alone does not
provide it.

For comparison maps between such
bounded-above complexes, CT.23--CT.25
apply with their signed complex
\(C(f)^q=C^q\oplus D^{q-1}\).
An isomorphism in degrees \(q\ge k\)
and a surjection at \(k-1\) retains
exactly that range after coefficient
restriction. This supplies both the
actual cycle embedding and its adjacent
comparison degree whenever that
comparison range has been proved.

#### AT.H. Relative nuclearity into the closed target

Let \(S\Subset X\) and
\(W'\Subset W\Subset Y\) be strongly
one-complete opens in strongly
one-complete \(X,Y\), and let
\(\mathcal H\) be coherent on \(X\times Y\).
Set \(A=\mathcal O(S)\).
The coherent restriction
\[
\begin{gathered}
\mathcal H(S\times W)\\
\longrightarrow\mathcal H(S\times W')
\end{gathered}
\tag{AT.30}
\]
is \(A\)-subnuclear with a nuclear
auxiliary module in the sense NT.I.

Here is a complete factorization.
AT.18 supplies a finite free map onto
\(\mathcal H\) on \(S\times W\), and
CX.L makes the corresponding section
map an open surjection. Its source is
\(A\widehat\otimes\mathcal O(W)^r\).
The scalar restriction
\(\mathcal O(W)\to\mathcal O(W')\)
is nuclear by SC.H. For a nuclear
series
\[
u(v)=\sum_j\lambda_j\,\xi_j(v)y_j
\tag{AT.31}
\]
therefore, the map \(1_A\widehat\otimes u\)
has the \(A\)-nuclear series with
coefficient maps \(1_A\widehat\otimes\xi_j\)
and target vectors \(1\otimes y_j\).
These coefficient maps are equicontinuous:
if \(|\xi_j(v)|\le p(v)\), then for each
continuous seminorm \(a\) of \(A\),
\[
a((1_A\widehat\otimes\xi_j)t)
\le(a\widehat\otimes_\pi p)(t).
\tag{AT.32}
\]
This holds for finite tensors by the
projective seminorm infimum, and then
for completions by continuity. The target
vectors are bounded for every tensor
seminorm, and \(\sum_j|\lambda_j|<\infty\).
The series converges by the same
seminorm estimate. Finite sums preserve
this representation. Composition with
the smaller-domain coherent section
map gives the factorization
\[
\begin{gathered}
p:A\widehat\otimes\mathcal O(W)^r\\
\twoheadrightarrow\mathcal H(S\times W),\\
A\widehat\otimes\mathcal O(W)^r\\
\longrightarrow
A\widehat\otimes\mathcal O(W')^r\\
\longrightarrow\mathcal H(S\times W').
\end{gathered}
\tag{AT.33}
\]
The second line is \(A\)-nuclear and
equals AT.30 composed with the first
line. Its auxiliary space is nuclear.
This proves the asserted subnuclearity.

Suppose now that a closed coherent
subsheaf \(\mathcal T\subset\mathcal H\)
on the same outer product contains
the image of a continuous \(A\)-subnuclear
section map
\[
\begin{gathered}
E=\mathcal H_0(S\times W),\\
F=\mathcal H(S\times W'),\\
f:E\to F.
\end{gathered}
\tag{AT.34}
\]
Assume \(\mathcal H_0\) is coherent on
the outer product and that the auxiliary
module for \(f\) is nuclear.
Let \(T=\mathcal T(S\times W')\).
Coherent vanishing identifies \(F/T\)
topologically with the sections of
\(\mathcal H/\mathcal T\), by the open
mapping theorem and the exact section
sequence. For every strongly one-complete
\(S'\Subset S\), the restriction
\(A\to B=\mathcal O(S')\) is scalar
nuclear by SC.H. AT.24 makes \(B\)
transverse to \(E,F,F/T\).
CT.17 supplies the closed embedding,
and the closed-target theorem CT.18
therefore gives the \(B\)-subnuclear map
\[
\begin{gathered}
B\widehat\otimes_A E
\longrightarrow B\widehat\otimes_A T\\
\simeq\mathcal T(S'\times W').
\end{gathered}
\tag{AT.35}
\]
This conclusion is into the submodule
itself, with its canonical section
topology.

For a cycle target in AT.28 the same
conclusion holds in its stated exact
range, provided the source is transverse
and the original map is \(A\)-subnuclear
with nuclear auxiliary module. Indeed
the quotient \(C^q/Z^q\) is
topologically \(Z^{q+1}\), by exactness
and NP.C, and all three required
transversality statements follow from
AT.28. CT.18 then applies. For a cycle
outside that range those premises
must be established separately.

#### AT.I. A nonflat thickened coefficient module

Take discs \(U\Subset X\subset\mathbf C_z\)
and \(V\Subset Y\subset\mathbf C_w\),
with \(U,X\) centred at zero. For an
integer \(m\ge1\), put
\[
\mathcal H=\mathcal O_{X\times Y}/(z^m).
\tag{AT.36}
\]
It is coherent by its one-relation
presentation. Taylor division in \(z\)
gives the topological identification
\[
\begin{gathered}
E=\mathcal H(U\times V)\\
\simeq\bigoplus_{j=0}^{m-1}\mathcal O(V)e_j,\\
e_j=[z^j],\\
ze_j=e_{j+1}\ (j<m-1),\\
ze_{m-1}=0.
\end{gathered}
\tag{AT.37}
\]
For an actual holomorphic representative
the coefficient of \(e_j\) is
\(\partial_z^j f(0,w)/j!\).
Its remainder is holomorphic and divisible
by \(z^m\), by the convergent Taylor series.
CX.L lifts every quotient section to a
holomorphic representative. The coefficient
maps are continuous by Cauchy's estimates;
the inverse is the finite polynomial sum.
They annihilate precisely the ideal
sections, so AT.37 has the intrinsic
quotient topology SC.G.

As a module over \(A=\mathcal O(U)\),
\(E\) is not algebraically flat.
Multiplication by \(z^m\) is injective
on \(A\): a holomorphic function with
zero product is zero away from zero
and hence zero everywhere by continuity.
Tensoring this injection algebraically
with \(E\) gives the zero map on the
nonzero module \(E\), because \(z^mE=0\).
That violates the injectivity required
of a flat module.

Nevertheless for every disc \(U'\subset U\),
AT.24 proves all positive completed
Tor groups vanish for \(B=\mathcal O(U')\).
The actual coefficient quotient is
\[
\begin{gathered}
B\widehat\otimes_A E\\
\simeq
\begin{cases}
\mathcal O(V)^m,&0\in U',\\
0,&0\notin U'.
\end{cases}
\end{gathered}
\tag{AT.38}
\]
If \(0\in U'\), Taylor division gives
the same finite jet module as AT.37.
If \(0\notin U'\), the function \(z^m\)
is an invertible holomorphic function
on \(U'\), so the quotient sheaf is zero.
Thus the vanishing and Hausdorff
identification of AT.24 cover nonflat
coefficients and retain every nilpotent
layer of the thickening.

![A common-domain coherent resolution, its actual coefficient restriction, and the nilpotent three-jet module.](assets/analytic-coefficient-transversality.png)

The left panel records AT.18--AT.24 on the
same product domain in every degree.
The right panel gives the exact
\(m=3\) action in AT.37 and the two
restriction cases AT.38. It is an
algebraic module diagram; the nilpotent
layers are basis vectors, not separate
points of the analytic support.
The drawing source
and exact matrix data
specify the diagram.

The product, coherent restriction and
relative nuclearity statements are also
developed in
[Demailly's free author manuscript, Chapter IX, §5.B--§5.D](https://www-fourier.univ-grenoble-alpes.fr/~demailly/manuscripts/agbook.pdf).
AT.A--AT.I give the local presentations,
gluing kernels, common-domain resolution,
balanced quotient topology and closed-target
premises used here.


### 5.99. Finite free models and proper analytic coherent images

Let \(f:X\to Y\) be a proper holomorphic map of
Hausdorff second-countable complex analytic spaces.
For every coherent \(\mathcal O_X\)-module \(\mathcal S\),
we prove that every \(R^qf_*\mathcal S\) is coherent.
Neither the spaces nor the coefficient module are
required to be smooth, reduced or flat over the base.
The same conclusion holds when \(f\) is proper on
the closed support of \(\mathcal S\).

The proof constructs a coefficient complex with finite free terms
and compares it with actual coherent Čech sections on
every smaller base domain. We use the complete
Banach-coefficient perturbation proof NP.D--NP.F,
the open mapping and null-lift theorem NP.C,
scalar tensor products NT.B--NT.F,
the bar resolution and actual balanced quotients CT.A--CT.D,
and the closed-kernel and comparison theorems CT.E--CT.G.
The analytic section and restriction inputs are
AT.A--AT.I,
with coherent vanishing CX.L,
coherent algebra SC.D,
and the full Čech/derived-section comparison SC.B.

#### GI.A. Nuclear coefficient scales

Let \(A\) be a commutative unital nuclear Fréchet algebra.
Suppose its topology is given by increasing continuous
submultiplicative seminorms \(p_r\), \(0<r<1\), such that
the Banach connecting map
\[
\begin{gathered}
P_s\longrightarrow P_r,\qquad r<s,\\
P_r=\widehat{A/\ker p_r}
\end{gathered}
\tag{GI.1}
\]
is scalar nuclear. Here \(P_r\) has the completed
\(p_r\)-norm. Define \(A_t\), for \(0<t<1\), as the
Hausdorff completion of \(A\) for the seminorms
\(p_r\), \(r<t\), and set \(A_1=A\).
Increasing sequences of radii tending to \(t\)
are cofinal, so these completions are Fréchet.
Multiplication extends by the submultiplicative
estimates. The Banach completion of \(A_t\) for
its \(p_r\)-seminorm is \(P_r\): both contain
the same dense image of \(A\), with the same norm.
Thus GI.1 and the defining nuclear-space criterion
NT.11 prove that every \(A_t\) is nuclear.

There are continuous algebra maps
\[
\begin{gathered}
A_t\longrightarrow P_v
\longrightarrow P_u\longrightarrow A_s,\\
0<s\le u<v<t\le1.
\end{gathered}
\tag{GI.2}
\]
The first map is defined by the \(p_v\)-seminorm.
The last exists because every \(p_r\), \(r<s\),
is bounded by \(p_u\); completeness extends
the map from \(A\) to \(P_u\).
The middle map is nuclear by hypothesis.
Composing its nuclear series with the two
continuous maps proves that \(A_t\to A_s\)
is scalar nuclear whenever \(s<t\).
The same seminorm argument gives
\[
P_t\longrightarrow A_t.
\tag{GI.3}
\]
No injectivity of this map is required.
All compositions agree on the dense image
of \(A\), hence agree everywhere.
The zero-algebra case has only zero unital
modules and causes no exception to the conclusions.

A nuclear Fréchet \(A\)-module \(E\) will be called
fully transverse when \(A_t\) is transverse to
it, in the exact sense CT.15, for every \(t<1\).
Write
\[
E_t=A_t\widehat\otimes_A E.
\tag{GI.4}
\]
This is then a Hausdorff nuclear Fréchet module.
CT.F identifies \(A_s\widehat\otimes_{A_t}E_t\)
with \(E_s\) for \(s<t\). In particular \(E_t\)
is fully transverse for the rescaled family
\(p_{\lambda t}\), \(0<\lambda<1\).

For the analytic application take \(A=\mathcal O(D)\),
where \(D=D(a,R)\) is a polydisc, and take \(p_r\)
to be the supremum on \(\overline{D(a,rR)}\).
The Taylor and nuclear restriction proof NT.G
gives GI.1. It also gives
\[
A_t=\mathcal O(D(a,tR)).
\tag{GI.5}
\]
Indeed the completion topology is compact-open
on that smaller polydisc, and Taylor polynomials
from \(A\) are dense there.

#### GI.B. A free balanced quotient over a Banach algebra

We shall use one calculation with a coefficient
algebra that need not be nuclear. Let \(R\to C\)
be a continuous homomorphism of commutative unital
Fréchet algebras, let \(M\) be Fréchet, and use
completed projective scalar tensors throughout
this paragraph. Define the balanced quotient
by the actual image of its relation map.
Then
\[
\begin{gathered}
C\widehat\otimes_R(R\widehat\otimes_\pi M)
\simeq C\widehat\otimes_\pi M,\\
[c\otimes(b\otimes m)]\longmapsto c\rho(b)\otimes m.
\end{gathered}
\tag{GI.6}
\]
Here is the full relation argument. On the
unbalanced space put
\[
\begin{gathered}
\mu(c\otimes b\otimes m)\\
=c\rho(b)\otimes m,\\
s(c\otimes m)=c\otimes1\otimes m,\\
r(c\otimes b\otimes m)\\
=c\otimes b\otimes1\otimes m.
\end{gathered}
\tag{GI.7}
\]
All maps are continuous by the projective
multilinear universal property. With
\[
\begin{gathered}
d(c\otimes b\otimes b'\otimes m)\\
=c\rho(b)\otimes b'\otimes m\\
-c\otimes bb'\otimes m
\end{gathered}
\tag{GI.8}
\]
one has \(\mu d=0\), \(\mu s=1\) and
\(dr=s\mu-1\). Verification on elementary
tensors gives these identities; density
extends them. Hence \(\ker\mu=\operatorname{im}d\)
exactly. This kernel is closed and the
section \(s\) identifies its quotient
topologically with the right side of GI.6.
Nuclearity of \(R\) was not used.

If \(Q:H\oplus R^b\to H\) is a continuous
surjective \(R\)-linear map of Fréchet modules,
then
\[
\begin{gathered}
(C\widehat\otimes_R H)\oplus C^b
\\
\longrightarrow C\widehat\otimes_R H\\
\quad\text{is onto}.
\end{gathered}
\tag{GI.9}
\]
To see this, first apply \(C\widehat\otimes_\pi-\)
to \(Q\). It is onto by the open-surjection
statement NT.7, which has no nuclearity
hypothesis. Compose with the balanced
quotient on the target. The resulting onto
map kills the balanced relations on its
source because \(Q\) is \(R\)-linear.
It therefore factors onto through its
actual balanced quotient. Finite direct
sums commute with this quotient, and
GI.6 with a finite-dimensional \(M\)
identifies the second summand with \(C^b\).
This proves GI.9 without tensoring an
algebraic cokernel that might be non-Hausdorff.

#### GI.C. Relative subnuclear maps under restriction

An \(A\)-nuclear series and an \(A\)-subnuclear
map have the meaning NT.I, with a nuclear
auxiliary module for a subnuclear map.
The seminorm convergence proof NP.B works
for a Fréchet algebra as well: for each
target seminorm, joint continuity of the
action bounds it by one coefficient
seminorm times one module seminorm.
Equicontinuity controls the former on
all series coefficient maps, and boundedness
controls the latter on all target vectors.
The summable scalar weights then give
absolute convergence in every seminorm.

Precomposition preserves subnuclearity.
Explicitly, for a presentation \(p: M\twoheadrightarrow E\)
and a continuous module map \(a:E'\to E\), use
\[
\begin{gathered}
M'\subset E'\oplus M,\\
(x,m)\in M'\\
\Longleftrightarrow a(x)=p(m),\\
M'\twoheadrightarrow E',\\
(x,m)\longmapsto x.
\end{gathered}
\tag{GI.10}
\]
The equality defines a closed submodule;
it is nuclear Fréchet when \(E',M\) are.
The projection is onto and hence open
by NP.C. The lifted composite is the
original nuclear map composed with
the projection to \(M\), so remains
nuclear. Postcomposition preserves its
series directly. Finite sums use a
finite fibre product of the surjective
presentations and concatenate the series.
This also proves the finite-block
matrix version.

For a presentation of \(u:E\to F\), replace
the auxiliary module \(M\) by the nuclearly
free module \(A\widehat\otimes M\). Its
action map to \(M\) is onto, with scalar
section \(m\mapsto1\otimes m\). We obtain
\[
\begin{gathered}
Q=A\widehat\otimes M\twoheadrightarrow E,\\
uQ\text{ is }A\text{-nuclear}.
\end{gathered}
\tag{GI.11}
\]
Here and subsequently a letter denoting
a surjection may also denote its map.

If \(E,F\) are transverse to a nuclear
coefficient algebra \(B\), the induced
\(u_B:E_B\to F_B\) is \(B\)-subnuclear.
Indeed \(Q_B=B\widehat\otimes M\) is
nuclear and still maps onto \(E_B\)
by CT.C. For a coefficient map
\(\xi_j:Q\to A\), put
\(\ell_j(m)=\xi_j(1\otimes m)\).
After the coefficient change its
extension on \(Q_B\) is
\[
b\otimes m\longmapsto b\,\rho(\ell_j(m)).
\tag{GI.12}
\]
For each target seminorm in \(B\),
joint continuity of multiplication,
continuity of \(\rho\), and the original
equicontinuity bound give a uniform
projective tensor seminorm bound on
all maps GI.12. The images in \(F_B\)
of the bounded series vectors remain
bounded. The same summable weights
therefore give a \(B\)-nuclear series
for \(u_BQ_B\). This proves the assertion.
In particular every radius restriction
of a subnuclear map between fully
transverse modules remains subnuclear.
Identity on a finite free coefficient
module is nuclear: use its finitely
many coordinate maps and basis vectors.

#### GI.D. Finite coefficient cokernels after radius decrease

**Perturbation theorem.** Let \(E,F\) be fully
transverse nuclear \(A\)-modules, let
\(g: E\twoheadrightarrow F\) be continuous
and \(A\)-linear, and let \(u:E\to F\)
be \(A\)-subnuclear with nuclear auxiliary
module. For every \(t<1\),
\[
\begin{gathered}
\operatorname{Coker}(g_t-u_t)\\
\text{ is finite over }A_t.
\end{gathered}
\tag{GI.13}
\]
“Finite” means algebraically finitely
generated; the cokernel is not assumed
Hausdorff.

Use GI.11 to replace the source by
\(Q=A\widehat\otimes M\), so the
perturbation is nuclear. Its coefficient
restriction still maps onto \(E_t\)
by CT.C; composing with this onto map
does not change the image of \(g_t-u_t\).
It is therefore enough to treat the
nuclear perturbation with this free source.
Normalize its nuclear series so that
the target vectors tend to zero.
The weighted-block construction NP.6
does this while keeping the scalar
weights summable; it only rescales
the bounded vectors and works for
the present Fréchet coefficient algebra.
Lift those null vectors through
\(gQ:Q\twoheadrightarrow F\) by NP.C.
The lifted vectors \(x_j\) tend to
zero in \(Q\). The series
\[
\begin{gathered}
v(x)=\sum_j\lambda_j\xi_j(x)x_j,\\
(gQ)v=uQ,\\
(g-u)Q=(gQ)(1-v)
\end{gathered}
\tag{GI.14}
\]
defines an \(A\)-nuclear endomorphism
of \(Q\). Thus finite generation of
\(Q_t/(1-v_t)Q_t\) suffices: the
surjection \((gQ)_t\) maps that
quotient onto the cokernel in GI.13.

Pass first to the Banach coefficient
algebra \(P_t\) of GI.1. Set
\[
Q_P=P_t\widehat\otimes_\pi M.
\tag{GI.15}
\]
For \(\ell_j(m)=\xi_j(1\otimes m)\),
equicontinuity gives one continuous
seminorm \(q\) on \(M\) with
\[
\begin{gathered}
\|\ell_j(m)\|_{P_t}\le q(m)\\
\quad\text{for every }j.
\end{gathered}
\tag{GI.16}
\]
The coefficient maps
\(b\otimes m\mapsto b\,\ell_j(m)\)
extend equicontinuously to \(Q_P\):
their norm is bounded by the
projective seminorm \(\|\cdot\|_{P_t}\otimes_\pi q\).
The images of the \(x_j\) in \(Q_P\)
are bounded. Hence GI.14 extends
to a \(P_t\)-nuclear endomorphism \(v_P\).
Its restriction agrees with \(v\)
on \(A\otimes M\).

NP.19, including its Fréchet-module
proof, gives finitely many vectors
\(w_1,\ldots,w_b\) in \(Q_P\) for which
\[
\begin{gathered}
Q_P\oplus P_t^b\twoheadrightarrow Q_P,\\
(x,c)\longmapsto\\
(1-v_P)x+\sum_{j=1}^b c_jw_j
\end{gathered}
\tag{GI.17}
\]
is onto. It is continuous and hence
open by NP.C. Use GI.9 with
\(R=P_t\) and \(C=A_t\), and GI.6
to identify both free balanced
products with \(Q_t=A_t\widehat\otimes M\).
We obtain the onto map
\[
\begin{gathered}
Q_t\oplus A_t^b\twoheadrightarrow Q_t,\\
(x,c)\longmapsto\\
(1-v_t)x+\sum_jc_jw'_j .
\end{gathered}
\tag{GI.18}
\]
The endomorphism here is \(v_t\):
both its constructions agree on
elementary tensors from \(A\otimes M\).
Those tensors have dense image in
\(A_t\widehat\otimes M\), by the
definition of \(A_t\) and the
projective seminorm estimates.
Continuity proves equality everywhere.
GI.18 proves the required finite
generation and hence GI.13.
Every step took a quotient by its
actual image; no separation of the
perturbation cokernel was imposed.

#### GI.E. Cycles at the boundary of an exact range

Let \(C^\bullet\) be a bounded-above
complex of fully transverse nuclear
modules, with \(H^j(C)=0\) for \(j>q\).
All cycles \(Z^j=\ker d^j\) are
closed nuclear Fréchet submodules.
Descending from the last zero term,
CT.E gives full transversality of
\(Z^j\) for \(j\ge q\) from
\[
\begin{gathered}
0\to Z^j\to C^j\\
\to Z^{j+1}\to0,\\
j\ge q.
\end{gathered}
\tag{GI.19}
\]
These sequences are exact since
the next cohomology group vanishes.
For every smaller coefficient radius
they remain strict by CT.E.
The inclusion of the tensor of
\(Z^{q+1}\) into that of \(C^{q+1}\)
is therefore a closed embedding.
Factor the outgoing differential at
degree \(q\) through this inclusion.
The exact sequence at \(q\) then proves
\[
(Z^q(C))_s=Z^q(C_s)
\tag{GI.20}
\]
with the actual closed subspace
topology. The same argument applies
at every higher degree. This includes
the boundary degree \(q\), although
acyclicity was assumed only above it.

A quasi-isomorphism between bounded-above
complexes of fully transverse modules
remains a quasi-isomorphism at every
smaller radius. Its signed comparison
complex CT.23 is bounded above,
termwise transverse, and acyclic in
all degrees. Apply CT.G at each
integer cutoff; the tensored comparison
complex is acyclic in all degrees.
CT.24 gives the asserted isomorphisms.
This argument does not assume
Hausdorff lower cohomology.

#### GI.F. Finite top-degree cohomology

Let \(f^\bullet:E^\bullet\to F^\bullet\)
be a quasi-isomorphism of bounded-above
complexes of fully transverse nuclear
modules. Suppose every \(f^j\) is
subnuclear with nuclear auxiliary
module. If both complexes are acyclic
in degrees greater than \(q\), then
for every \(s<1\),
\[
\begin{gathered}
H^q(E_s)\simeq H^q(F_s)\\
\text{ is finite over }A_s .
\end{gathered}
\tag{GI.21}
\]

Choose an intermediate radius \(s<r<1\).
By GI.E the cycle modules \(Z^q(E)\),
\(Z^q(F)\) and \(Z^{q+1}(F)\) are
fully transverse, and
\[
F^q/Z^q(F)\simeq Z^{q+1}(F)
\tag{GI.22}
\]
topologically, by NP.C and exactness
at the next degree. Restrict \(f^q\)
to the closed source \(Z^q(E)\).
GI.10 proves that this map into
\(F^q\) is still subnuclear with
nuclear auxiliary module.
Its image lies in \(Z^q(F)\).
The scalar nuclear coefficient map
\(A\to A_r\) and the transversality
of the source, target and quotient
GI.22 supply exactly CT.17--CT.18.
Thus after this first decrease it
is subnuclear into the actual target
\(Z^q(F_r)\); GI.20 identifies both
cycle tensors used here.

GI.E also preserves the cohomology
isomorphism. Consequently
\[
\begin{gathered}
\Delta:F_r^{q-1}\oplus Z^q(E_r)\\
\twoheadrightarrow Z^q(F_r),\\
\Delta(a,b)=d_Fa+f_rb
\end{gathered}
\tag{GI.23}
\]
is onto: subtract from any target
cycle the image of a cycle representing
its cohomology class, then lift the
remaining boundary. Its terms are
fully transverse modules over \(A_r\).
The summand \(u(a,b)=f_rb\) is
subnuclear into the stated closed
target. Apply GI.13 over the rescaled
coefficient algebra \(A_r\), decreasing
from \(r\) to \(s\). The difference
\(\Delta_s-u_s\) is \(d_F\oplus0\).
Its cokernel is
\(Z^q(F_s)/d_F(F_s^{q-1})\), by GI.20
and the canonical coefficient composition
CT.F. This is precisely \(H^q(F_s)\).
It proves GI.21 with both strict
radius inequalities and all closed-target
premises supplied.

#### GI.G. A finite free model in every degree

**Finite-model theorem.** Under the hypotheses
on \(f^\bullet\) in GI.F, for every
\(0<t<1\) there are finite integers
\(b_j\), a bounded-above complex
\[
\begin{gathered}
L^j=A_t^{b_j},\\
h^\bullet:L^\bullet\to E_t^\bullet,
\end{gathered}
\tag{GI.24}
\]
and a continuous coefficient-linear
quasi-isomorphism \(h^\bullet\).
There may be infinitely many nonzero
degrees to the left; each degree
has finite rank.

Choose \(N\) above which both initial
complexes vanish, and put
\[
\begin{gathered}
t_j=t+(1-t)2^{\,j-N-1},\\
j\le N.
\end{gathered}
\tag{GI.25}
\]
Then \(t<t_j<1\) and \(t_j<t_{j+1}\)
for \(j<N\), and the
radii decrease to \(t\) towards the
left. GI.21 makes \(H^N(E_{t_N})\)
finite. Choose cycle representatives
of finitely many generators and use
them as columns of
\(h^N:A_{t_N}^{b_N}\to Z^N(E_{t_N})\).
With this single term \(L^N\), the
signed comparison complex
\[
\begin{gathered}
M^j=L^j\oplus E^{j-1},\\
D(x,y)=(d_Lx,hx-d_Ey)
\end{gathered}
\tag{GI.26}
\]
is acyclic in every degree greater
than \(N\), including \(N+1\).
Indeed its cohomology at \(N+1\)
is the cokernel of the chosen
surjection onto \(H^N(E)\);
higher terms vanish.

Suppose terms and maps have been
constructed in degrees at least \(q\),
over their respective larger radii,
and restrict them to \(A_{t_q}\).
Let \(L_{\ge q}\) denote this truncated
complex. Form
\[
\begin{gathered}
M=C(h_{\ge q}),\\
P=C(f_{t_q}h_{\ge q}),\\
\Psi^j(x,y)=(x,f_{t_q}^{j-1}y).
\end{gathered}
\tag{GI.27}
\]
The chain condition for \(\Psi\) is
the equality \(f d_E=d_F f\), with
the signs of GI.26. It is termwise
subnuclear: the first component is
identity on a finite free module
and the second is subnuclear by GI.C.

The map \(\Psi\) is a quasi-isomorphism.
Here is a direct proof using the
exact cone sequences CT.C and CT.23.
Given a class in \(H^j(P)\), its
projection to \(H^j(L_{\ge q})\)
is killed by \(H^j(fh)\).
Since \(H(f)\) is an isomorphism,
it is killed by \(H^j(h)\), so it
lifts to \(H^j(M)\). Subtract the
image of that lift. The remainder
comes from \(H^{j-1}(F)\), which
comes from \(H^{j-1}(E)\) under
the isomorphism \(H(f)\).
Adding its cone image gives the
desired lift. For injectivity,
a class of \(H^j(M)\) with zero
image has zero projection to
\(H^j(L_{\ge q})\). It comes from
a class \(e\) in \(H^{j-1}(E)\).
Its \(f\)-image is in the image
of \(H^{j-1}(fh)\), by exactness
for \(P\). Injectivity of \(H(f)\)
makes \(e\) an \(h\)-image, whose
cone image is zero. This proves
both implications without a
separation assertion about cohomology.

The induction hypothesis makes
\(M\) acyclic above \(q\), hence
so is \(P\). Their terms are
fully transverse. Apply GI.21
to \(\Psi\) at radius \(t_{q-1}\).
Choose cycle representatives of
finitely many generators of
\(H^q(M_{t_{q-1}})\):
\[
\begin{gathered}
(a_\nu,e_\nu)\in
L^q_{t_{q-1}}\oplus E^{q-1}_{t_{q-1}},\\
d_La_\nu=0,\qquad
h a_\nu=d_Ee_\nu .
\end{gathered}
\tag{GI.28}
\]
Introduce a finite free \(L^{q-1}\).
Its differential columns are \(a_\nu\);
the columns of \(h^{q-1}\) are
\(+e_\nu\). The two equations in
GI.28 say precisely that the
new differential squares to zero
and that \(h\) is a chain map.
In the new comparison complex
these pairs are the boundaries
of the new basis vectors.
They kill \(H^q(M)\). Higher
terms and the boundaries entering
higher degrees are unchanged.
Thus the new comparison complex
is acyclic above \(q-1\), the
required next invariant.

Perform this induction for every
integer \(q\le N\), then restrict
all terms and matrices to \(A_t\).
At each finite stage GI.E preserves
the invariant under that restriction.
For any fixed degree \(j\), choose
a stage \(q<j\). The final comparison
complex agrees with that stage
in degrees \(j-1,j,j+1\); hence its
degree-\(j\) cohomology is zero.
It is therefore acyclic in all
degrees. CT.24 proves that the
final \(h\) is a quasi-isomorphism.
This proves GI.24 on one coefficient
domain in all degrees.

#### GI.H. Closed graphs of every chart overlap

Return to the proper map in the
opening statement. Fix \(y_0\in Y\).
Choose a local embedding of a
neighbourhood of \(y_0\) as a
closed analytic subspace of an
ambient open set \(W\subset\mathbf C^n\).
Take concentric polydiscs
\[
\begin{gathered}
y_0\in D\Subset D_0\Subset W,\\
K=f^{-1}(Y\cap\overline D).
\end{gathered}
\tag{GI.29}
\]
The set \(Y\cap\overline D\) is
compact, and properness makes
\(K\) compact. Work in
\(f^{-1}(Y\cap D_0)\).
Choose finitely many source chart
opens \(U_\alpha^0\), each a
closed analytic model
\[
i_\alpha:U_\alpha^0\hookrightarrow
\Omega_\alpha^0
\tag{GI.30}
\]
in an ambient polydisc, and nested
concentric smaller polydiscs
\(\Omega'_\alpha\Subset\Omega_\alpha
\Subset\Omega_\alpha^0\), such that
both corresponding smaller source
opens cover \(K\).
For existence, choose such a
triple around every point of \(K\)
with its smallest source open
containing the point, then select
a finite subcover. Initial chart
balls can be chosen inside
\(f^{-1}(Y\cap D_0)\), since it
is open in \(X\).

For an increasing nonempty index
tuple \(I\), put
\[
\begin{gathered}
T_I=\bigcap_{\alpha\in I}U_\alpha^0,\\
\Omega_I^0=\prod_{\alpha\in I}\Omega_\alpha^0,\\
j_I:T_I\longrightarrow D_0\times\Omega_I^0,\\
x\longmapsto(f(x),(i_\alpha(x))_{\alpha\in I}).
\end{gathered}
\tag{GI.31}
\]
This is a closed analytic embedding,
with the full structure of \(T_I\).
For closedness first take the product
of the closed chart models.
The tuples representing the same
point of \(X\) form the inverse image
of its closed diagonal, since \(X\)
is Hausdorff. Adding the base
coordinate is the graph of the
continuous map \(f\), also closed
in the product with the Hausdorff
space \(D_0\).
For the analytic structure, near
an image point use one chart
coordinate as the independent one.
The other chart coordinates and
the base coordinates have actual
holomorphic ambient representatives
after shrinking. Impose their
differences from these representatives,
together with the entire defining
ideal of the independent chart.
The resulting local quotient is
exactly \(\mathcal O_{T_I}\).
The kernels of the canonical
restriction maps to this sheaf
agree on overlaps, so these ideals
glue. Off the closed image the
ideal is the whole structure sheaf.
Their local finite presentations
and coherence follow from SC.D.
This proves the assertion without
passing to a reduced diagonal.

The sheaf
\[
\mathcal S_I=(j_I)_*(\mathcal S|_{T_I})
\tag{GI.32}
\]
is coherent on the whole
\(D_0\times\Omega_I^0\), by the
closed-embedding coherent algebra
in SC.D. Its finite local presentation
is obtained by adding the graph
ideal relations to a presentation
of \(\mathcal S\).

Let \(\Omega_I=\prod_{\alpha\in I}\Omega_\alpha\)
and similarly define \(\Omega'_I\).
For every strongly one-complete
open \(V\subset D\) the actual
chart intersection over \(V\) is
the closed analytic model
\[
\begin{gathered}
T_I\cap f^{-1}(V)
\cap\bigcap_{\alpha\in I}i_\alpha^{-1}(\Omega_\alpha)\\
\simeq j_I(T_I)\cap(V\times\Omega_I).
\end{gathered}
\tag{GI.33}
\]
Both ambient factors are strongly
one-complete; the product is so
by AT.B. A closed analytic model
inherits the strict ambient
exhaustion and compact closed
sublevels. Hence this intersection
is strongly one-complete, and
CX.L makes every coherent sheaf
on it acyclic in positive degrees.
The same statements hold for
the primed polydiscs.

#### GI.I. The actual finite Čech comparison

Denote the finite source opens
from \(\Omega_\alpha,\Omega'_\alpha\)
by \(U_\alpha,U'_\alpha\), respectively,
and their number by \(a\).
They cover \(f^{-1}(Y\cap D)\),
since they cover \(K\).
With \(A=\mathcal O(D)\), define
the finite alternating Čech complexes
\[
\begin{gathered}
E^q=\prod_{|I|=q+1}\mathcal S_I(D\times\Omega_I),\\
F^q=\prod_{|I|=q+1}\mathcal S_I(D\times\Omega'_I),\\
0\le q\le a-1.
\end{gathered}
\tag{GI.34}
\]
Their terms in other degrees are
zero. Closed pushforward identifies
these sections with the actual
sections of \(\mathcal S\) on the
respective intersections. This is
also a topological identification:
the maps are continuous for SC.G,
and the resulting continuous
bijection of nuclear Fréchet
section spaces is an isomorphism
by NP.C.

The differential on an increasing
tuple \(I=(i_0,\ldots,i_{q+1})\) is
\[
(dc)_I=\sum_{\nu=0}^{q+1}
(-1)^\nu c_{I\setminus i_\nu}|_{U_I}.
\tag{GI.35}
\]
Each double deletion occurs twice
with opposite sign, so \(d^2=0\).
The maps are continuous and
\(A\)-linear section restrictions.
Restriction from each unprimed
intersection to its primed one
defines a chain map \(E\to F\):
both orders in GI.35 are the
same restriction with the same
alternating sign.

AT.24 makes every term fully
transverse over \(A\), with
\[
\begin{gathered}
\mathcal O(V)\widehat\otimes_A E^\bullet
\simeq E^\bullet(V),\\
\mathcal O(V)\widehat\otimes_A F^\bullet
\simeq F^\bullet(V)
\end{gathered}
\tag{GI.36}
\]
for every strongly one-complete
\(V\subset D\). The complexes
on the right are exactly the same
Čech complexes restricted over \(V\).
The coefficient isomorphisms
respect GI.35 by the naturality
proved in AT.F; on elementary
tensors both maps are ordinary
section restriction and multiplication
by the base function.
Fibre restriction AT.30 makes each
component \(E^q\to F^q\)
\(A\)-subnuclear with nuclear
auxiliary module. Finite products
preserve this property by GI.C.

All finite intersections in both
covers are acyclic by GI.33.
SC.B therefore gives
\[
\begin{gathered}
H^q(E^\bullet(V))\\
\simeq H^q(f^{-1}(Y\cap V),\mathcal S),\\
H^q(F^\bullet(V))\\
\simeq H^q(f^{-1}(Y\cap V),\mathcal S).
\end{gathered}
\tag{GI.37}
\]
The nested-cover restriction induces
the identity of this derived-section
cohomology, by the refinement
comparison in SC.B. In particular
\(E\to F\) is a quasi-isomorphism
of bounded-above fully transverse
complexes with termwise subnuclear
maps. Every premise of GI.G has
now been verified on the actual
source sections.

#### GI.J. Sheafification of the finite model

Choose \(0<t<1\) and apply GI.G.
Let \(D_t=D(y_0,tR)\), where \(R\)
is the polyradius of \(D\).
Sheafify its finite free terms
and holomorphic differential
matrices to a complex
\[
\mathcal L^q=\mathcal O_{D_t}^{b_q}.
\tag{GI.38}
\]
The matrices square to zero
because their product is zero
as a matrix of sections on \(D_t\).
For a strongly one-complete
\(V\subset D_t\), the columns
of \(h\), restricted to \(V\),
define a natural chain map
\[
\mathcal L^\bullet(V)\longrightarrow E^\bullet(V).
\tag{GI.39}
\]
The coefficient module \(\mathcal O(V)\)
is transverse to every \(E_t^q\)
by AT.24 and coefficient composition
CT.F. It is transverse to each
finite free \(L^q\) by CT.1--CT.10.
Apply CT.G to the acyclic bounded-above
comparison complex of \(h\), at
every cutoff as in GI.E. It proves
that GI.39 is a quasi-isomorphism.
Thus for every such \(V\),
\[
\begin{gathered}
H^q(\mathcal L^\bullet(V))\\
\simeq H^q(f^{-1}(Y\cap V),\mathcal S),
\end{gathered}
\tag{GI.40}
\]
naturally for all smaller such opens.

We spell out the stalk passage.
Polydiscs around any point of \(D_t\)
form a strongly one-complete
neighbourhood basis. A germ in the
kernel of a differential can be
represented on a neighbourhood
where that differential is zero.
Equality of two germs or a boundary
identity likewise holds on a smaller
neighbourhood. Hence
\[
\begin{gathered}
\mathcal H^q(\mathcal L)_y
=\varinjlim_{V\ni y}
H^q(\mathcal L^\bullet(V)).
\end{gathered}
\tag{GI.41}
\]
This uses only the three adjacent
complex degrees; a left bound is
unnecessary.
For the right side of GI.40 take
an injective resolution of \(\mathcal S\).
Its direct-image section complex
on \(V\) is its section complex
on \(f^{-1}(Y\cap V)\).
It computes the indicated sheaf
cohomology by the injective
derived-section argument SC.B.
Taking germs commutes with its
cohomology by the same representative,
kernel and boundary reasoning.
Writing \(j:Y\cap D_t\hookrightarrow D_t\),
we obtain
\[
\mathcal H^q(\mathcal L)
\simeq j_*R^qf_*\mathcal S
\quad\text{on }D_t.
\tag{GI.42}
\]
Closed pushforward \(j_*\) is exact:
at a point of the closed subspace
its stalk is the original stalk,
and off it a neighbourhood misses
the subspace. Thus it commutes
with cohomology of the direct-image
resolution, justifying its placement
in GI.42. The natural section maps
in GI.39--GI.40 give the displayed
sheaf isomorphism, not merely an
unrelated isomorphism of stalk dimensions.

Every \(\mathcal H^q(\mathcal L)\)
is coherent by SC.D: it is the
quotient of the kernel of a map
between finite free sheaves by
the coherent image of the preceding
finite free sheaf. This remains
true for the bounded-above complex
with no left bound.

The full defining ideal of \(Y\)
annihilates the right side of GI.42.
Its pullback along \(f\) is zero
in \(\mathcal O_X\), so it acts
as zero on the section resolution
and its cohomology. A coherent
ambient module \(P\) annihilated
by that ideal is coherent over
\(\mathcal O_Y\): locally choose
finite ambient generators. The
kernel of their map from
\(\mathcal O_{D_t}^b\) is coherent
and contains the ideal times
\(\mathcal O_{D_t}^b\). Its finite
local generators, reduced modulo
that ideal, generate the kernel
over \(\mathcal O_Y\). This is a
finite presentation, and SC.D
gives coherence on the analytic
quotient. Therefore every
\(R^qf_*\mathcal S\) is coherent
near \(y_0\), and hence on \(Y\).
This proves the general analytic
proper coherent-image theorem.

#### GI.K. Proper support and cohomological bounds

Let \(f:X\to Y\) now be any
holomorphic map and suppose it
is proper on the closed support
of the coherent module \(\mathcal S\).
Its annihilator \(\mathcal J\)
is coherent. Locally, if
\(s_1,\ldots,s_b\) generate
\(\mathcal S\), it is the kernel
of \(a\mapsto(as_1,\ldots,as_b)\);
SC.D proves coherence and these
kernels give the same intrinsic
ideal on overlaps.
Give \(Z=V(\mathcal J)\) its
entire quotient analytic structure.
Its underlying set is precisely
the support: a zero stalk gives
the unit annihilator locally,
whereas a nonzero stalk has no
unit in its annihilator.
The module \(\mathcal S\) is the
closed pushforward of a coherent
\(\mathcal O_Z\)-module \(\mathcal S_Z\),
by the same finite-presentation
argument as at the end of GI.J.
The restricted map \(f_Z:Z\to Y\)
is proper by hypothesis. Exact
closed pushforward and its
derived-section identification
SC.B--SC.C give
\[
Rf_*\mathcal S\simeq R(f_Z)_*\mathcal S_Z.
\tag{GI.43}
\]
GI.J therefore proves coherent
higher images also for proper
support.

For a bounded complex \(K\) with
coherent cohomology supported on
a closed set proper over \(Y\),
successive finite cohomological
truncations express \(Rf_*K\)
by finitely many triangles with
terms \(Rf_*\mathcal H^j(K)[-j]\).
GI.43 makes all their cohomology
sheaves coherent; kernels, images
and extensions remain coherent
by SC.D and the exact cohomology
sequences. The uniform support-proper
ordinary section bound
SC.J--SC.K
supplies a target neighbourhood and
one integer \(M\) on which
\[
\begin{gathered}
\mathcal H^j(K)=0\\
j\notin[a,b]\\
\Longrightarrow
\mathcal H^n(Rf_*K)=0\\
n<a\text{ or }n>b+M-1.
\end{gathered}
\tag{GI.44}
\]
Thus the direct image has coherent
cohomology and is bounded on a
neighbourhood of every target point.
These are local bounds on \(Y\).
If the integers \(M\) can be chosen
uniformly, GI.44 gives global
boundedness. In particular this
holds when \(Y\) is compact:
choose finitely many of the target
neighbourhoods, with bounds \(M_i\),
and use \(M=\max_i M_i\).
Every degree outside that single
range vanishes on the finite cover
and hence on all of \(Y\).
The natural proper-support
comparison \(Rf_!K\simeq Rf_*K\)
is the one proved in SC.K.
It is \(\mathcal O_Y\)-linear:
the defining comparison of sections
commutes with multiplication by
each pulled-back base function,
and the same is true of the
resolution maps and truncations.
Consequently the coherent-image
and boundedness statements apply
to that proper-support functor as well.

![Decreasing coefficient radii, signed cone columns, and the actual graph and Čech construction of coherent proper images.](assets/general-analytic-coherent-images.png)

The left panel records GI.25--GI.28
for \(t=1/3\) and \(N=2\), with
the displayed exact sample radii.
The right panel gives the maps
GI.31--GI.42; its boxes denote
analytic spaces, coefficient
complexes or sheaves as labelled.
It is a diagram of the proof,
not a metric picture of a fibre.
The drawing source
and exact radius and sign data
specify the diagram.

The nuclear coefficient method
and its application to proper
coherent images are also developed
in [Demailly's free author manuscript,
Chapter IX, §5.D--§5.E](https://www-fourier.univ-grenoble-alpes.fr/~demailly/manuscripts/agbook.pdf).
GI.A--GI.K give the coefficient
quotient, closed-cycle, finite
model, overlap and stalk arguments
used here.


### 5.100. Induced analytic modules and proper operator coherence

Let \(f:X\to Y\) be a holomorphic map of Hausdorff second-countable complex manifolds, with \(\dim_{\mathbf C}X=n\). Fix a closed subset \(S\subset X\) such that \(f|_S\) is proper. A right operator module will have a **coherent ordinary generator near the fibres** if, for every \(y\in Y\), there is an open neighbourhood \(V\) and a coherent right \(\mathcal O_{f^{-1}V}\)-submodule \(F\subset M|_{f^{-1}V}\) such that \(F\mathcal D_{f^{-1}V}=M|_{f^{-1}V}\).

We prove that every coherent right \(\mathcal D_X\)-module supported in \(S\) with this generator property has bounded coherent proper direct image. The property holds for regular holonomic modules by GG.P, after the exact density side change. More generally the theorem applies to any supplied global good filtration near each proper fibre. The finite-dimensional hypothesis on \(X\) gives a single amplitude bound, independent of the number of charts covering a fibre.

The ordinary coherent-image theorem used here is the full GI.J–GI.K, including nonreduced coefficient support and nonflat coefficients. Analytic operator coherence, uniform filtered images and relations are Theorem3.0c and Lemma3.0c.4. We also use the actual proper filtered-coefficient comparison, Lemma3.1b, and the complete finite c-soft resolution, D8–D10. The sheaf results allow arbitrary complex coefficients.

#### AP.A. Coherent coefficients induce actual good modules

Let \(F\) be any coherent right \(\mathcal O_X\)-module. Define
\[
\begin{gathered}
I(F)=F\otimes_{\mathcal O_X}\mathcal D_X,\\
G_pI(F)=F\otimes_{\mathcal O_X}\mathcal D_{X,\le p},\\
p\ge0,\\
G_pI(F)=0\quad(p<0).
\end{gathered}
\tag{AP.1}
\]
The tensor uses the left ordinary action on the operator factor; the resulting module uses its right operator action. The order inclusions are split as left ordinary modules in local PBW coordinates. Consequently every indicated tensor is a subsheaf of \(I(F)\), even when \(F\) is not flat. PBW gives exact sequences of right ordinary modules
\[
\begin{gathered}
0\to G_{p-1}I(F)\to G_pI(F)\\
\to F\otimes_{\mathcal O_X}\operatorname{Sym}^pT_X\to0.
\end{gathered}
\tag{AP.2}
\]
Right multiplication by a function differs from its multiplication on the coefficient by a term of lower operator order. Thus the last term has exactly the displayed ordinary action. Each is coherent by SC.D, and induction in AP.2 proves coherence of every actual right ordinary piece. This argument does not identify right multiplication with left coefficient multiplication before passing to the symbol.

A finite local ordinary presentation of \(F\), tensored with the left ordinary-flat sheaf \(\mathcal D_X\), presents \(I(F)\) by finite free right operator modules. Operator coherence therefore makes it coherent. Its symbol is
\[
\operatorname{gr}_GI(F)
=F\otimes_{\mathcal O_X}\operatorname{Sym}T_X.
\tag{AP.3}
\]
It is locally finitely generated by degree zero, so AP.1 is a global good filtration. In PBW coordinates the underlying coefficient sheaf is the sheaf direct sum of copies of \(F\), with one copy for each derivative monomial. In particular
\[
\operatorname{supp}I(F)=\operatorname{supp}F.
\tag{AP.4}
\]
The zero-order inclusion exhibits a copy of every nonzero coefficient stalk; conversely a zero coefficient stalk kills all summands. These statements concern actual finite-order operators.

#### AP.B. Derived cancellation for nonflat coefficients

Put
\[
\begin{gathered}
T=\mathcal D_{X\to Y}\\
=\mathcal O_X\otimes_{f^{-1}\mathcal O_Y}f^{-1}\mathcal D_Y.
\end{gathered}
\tag{AP.5}
\]
The chain-rule construction gives its commuting left \(\mathcal D_X\)- and right \(f^{-1}\mathcal D_Y\)-actions. In target coordinates, left ordinary PBW identifies it with a direct sum of copies of \(\mathcal O_X\). Hence it is flat as a left \(\mathcal O_X\)-module.

For clarity, an ordinary flat resolution \(P^\bullet\to F\) exists without a global vector-bundle hypothesis. For every local section on an open \(U\), use the corresponding map from the open extension by zero of \(\mathcal O_U\) to \(F\). The direct sum of these maps is surjective on stalks. Each source is ordinary-flat, since its stalk is either \(\mathcal O_{X,x}\) or zero. Repeat on its kernel. This constructs a resolution in nonpositive degrees by ordinary-flat sheaves.

The complex \(P^\bullet\otimes\mathcal D_X\) resolves \(I(F)\), because \(\mathcal D_X\) is left ordinary-flat. Its terms are right operator-flat: for any left operator module \(N\), the balancing maps give
\[
\begin{gathered}
(P^j\otimes_{\mathcal O_X}\mathcal D_X)
\otimes_{\mathcal D_X}N\\
\simeq P^j\otimes_{\mathcal O_X}N,
\end{gathered}
\tag{AP.6}
\]
and the right side preserves exact sequences. Tensoring this resolution with \(T\) gives \(P^\bullet\otimes_{\mathcal O_X}T\). Its only cohomology is the ordinary tensor in degree zero, by the ordinary-flatness of \(T\). Thus the actual natural reassociation isomorphism is
\[
\begin{gathered}
I(F)\otimes^L_{\mathcal D_X}T
\simeq F\otimes_{\mathcal O_X}T\\
=F\otimes_{f^{-1}\mathcal O_Y}f^{-1}\mathcal D_Y.
\end{gathered}
\tag{AP.7}
\]
All maps retain the right target-operator action. This proves vanishing of every positive transfer Tor of this induced source. It makes no flatness assertion about the transfer as a left operator module.

#### AP.C. Finite order and the proper filtered union

Suppose \(\operatorname{supp}F\subset S\). Write \(D_p=\mathcal D_{Y,\le p}\). The finite-order projection map is
\[
\begin{gathered}
Rf_*F\otimes_{\mathcal O_Y}D_p\\
\longrightarrow Rf_*\bigl(
F\otimes_{f^{-1}\mathcal O_Y}f^{-1}D_p\bigr).
\end{gathered}
\tag{AP.8}
\]
It is an isomorphism. Indeed, on a target coordinate open, \(D_p\) is finite free for its left ordinary action. In that basis AP.8 is the identity between the same finite direct sum of \(Rf_*F\). A change of basis is a finite matrix of holomorphic functions; multiplication by their pullbacks gives the same matrix on the source. Thus these identifications glue to the canonical projection map. They are compatible with the order inclusions.

For this passage to the union it is essential to use proper support. Lemma3.1b applies to the filtered system
\[
\begin{gathered}
B_p=F\otimes_{f^{-1}\mathcal O_Y}f^{-1}D_p,\\
\operatorname{supp}B_p\subset S.
\end{gathered}
\tag{AP.9}
\]
It proves, for every \(q\),
\[
\begin{gathered}
\underset{p}{\operatorname{colim}}R^qf_*B_p\\
\simeq R^qf_*\bigl(\underset{p}{\operatorname{colim}}B_p\bigr).
\end{gathered}
\tag{AP.10}
\]
Here is the content of its proof that matters for this system. A section of a filtered colimit on a compact set is represented at one stage: finitely many local representatives, and finitely many compact overlaps, admit one common later index witnessing their agreement. The compact-neighbourhood gluing argument then glues them. Equality of two such representatives is witnessed at one later stage by the same finite-cover argument. Filtered colimits of c-soft sheaves are therefore c-soft. Take the functorial flabby resolution formed by products of stalks and successive cokernels for the diagram \(B_p\). Its termwise colimit is an exact resolution, has c-soft terms, and is proper-image acyclic. Proper underived image commutes with this colimit by the compact-fibre section formula. The two derived computations give AP.10. On the fixed closed support one uses its proper restricted map; closed pushforward is exact. These are sheaf statements, rather than statements about all sections on an arbitrary noncompact inverse image.

Taking the colimit of AP.8 and using AP.10 gives the natural derived isomorphism
\[
\begin{gathered}
Rf_*F\otimes_{\mathcal O_Y}\mathcal D_Y\\
\xrightarrow{\sim}Rf_*(F\otimes_{\mathcal O_X}T).
\end{gathered}
\tag{AP.11}
\]
One may represent all derived terms by bounded-below functorial resolutions in the preceding argument. Exact filtered colimits commute with their cohomology in each degree, so AP.10 proves that the resulting derived map is a quasi-isomorphism. The left ordinary-flatness of \(\mathcal D_Y\) also identifies its cohomology with \((R^qf_*F)\otimes\mathcal D_Y\).

We check the right action before using AP.11 as an operator identity. Right multiplication by an operator of order \(r\) defines a left ordinary-linear map \(D_p\to D_{p+r}\). AP.8 is natural for these maps. In coordinates, multiplication by a target derivative raises the corresponding PBW exponent, and multiplication by a target holomorphic function is the finite formula
\[
\begin{gathered}
\partial^\alpha a\\
=\sum_{\beta\le\alpha}
\binom\alpha\beta(\partial^\beta a)
\partial^{\alpha-\beta}.
\end{gathered}
\tag{AP.12}
\]
Its source coefficients are the pullbacks of precisely those derivatives. The finite-order maps, their derived comparisons and their filtered colimit therefore commute with both kinds of multiplication. Functions and coordinate derivatives generate all finite-order operators, so AP.11 is right \(\mathcal D_Y\)-linear.

Define the right direct image by
\[
f_+N=Rf_*\bigl(N\otimes^L_{\mathcal D_X}T\bigr)
\tag{AP.13}
\]
on objects supported in \(S\). The proper-support comparison identifies the star and shriek sheaf images here. Combining AP.7 and AP.11 proves
\[
\begin{gathered}
f_+I(F)\\
\simeq Rf_*F\otimes_{\mathcal O_Y}\mathcal D_Y,\\
\mathcal H^q(f_+I(F))\\
\simeq I_Y(R^qf_*F).
\end{gathered}
\tag{AP.14}
\]
GI.J–GI.K makes every coefficient on the last line coherent. AP.A consequently proves that every induced-term image cohomology is a coherent right operator module with its actual good induced filtration.

#### AP.D. A uniform bound for the actual functor

The bound is independent of the induced calculation. The augmented absolute Spencer resolution of \(T\) has terms
\[
\begin{gathered}
Q^{-r}=\mathcal D_X\otimes_{\mathcal O_X}E_r,\\
E_r=\bigwedge^rT_X\otimes_{\mathcal O_X}T,\\
0\le r\le n,
\end{gathered}
\tag{AP.15}
\]
and augmentation \(P\otimes t\mapsto Pt\). Its differential and signs are the complete Spencer differential of §5.9, finite transfer bounds: the alternating terms are \(P\xi_a\otimes t-P\otimes\xi_at\), together with the alternating Lie-bracket terms. The chain-rule action makes them right target-operator linear. Filter the first operator factor by order plus wedge degree. The associated graded augmented complex is the polynomial Koszul resolution in the source cotangent variables, with coefficient \(T\). Its exactness and the terminating leading-symbol reduction are proved in that section. Every term is left operator-flat, since \(\bigwedge^rT_X\otimes T\) is ordinary-flat. Thus AP.15 computes the actual transfer tensor of any right operator module \(N\), with amplitude \([-n,0]\).

The real manifold underlying \(X\) has dimension \(2n\). The earlier finite c-soft resolution D8–D10, followed by its proper-image acyclicity F6, proves
\[
R^qf_!A=0\qquad(q>2n)
\tag{AP.16}
\]
for every complex sheaf \(A\). In particular this uses one integer on the whole manifold; it is not a maximum taken over infinitely many unrelated local fibre covers. The result also applies on every open inverse image with the same integer. For sheaves supported in \(S\), the proper-support comparison gives the same bound for \(Rf_*\).

These bounds apply to the functor with its operator action. Forgetting a sheaf of \(f^{-1}\mathcal D_Y\)-modules to complex sheaves preserves injectives: its left adjoint is tensor with \(f^{-1}\mathcal D_Y\) over the constant complex sheaf, and is exact on stalks. Hence the underlying sheaf of the derived operator image is its ordinary sheaf-derived image. The proper-support comparison is operator-linear, since the section inclusion and all resolution maps commute with the target action. Applying AP.16 to AP.15 therefore yields, for every \(N\) supported in \(S\),
\[
f_+N\in D^{[-n,\,2n]}(\mathcal D_Y^{\mathrm{op}}).
\tag{AP.17}
\]
For a bounded input with cohomology in \([a,b]\), its finite cohomology truncation triangles give the uniform interval
\[
f_+K\in D^{[a-n,\,b+2n]}(\mathcal D_Y^{\mathrm{op}}).
\tag{AP.18}
\]
Neither coherence of an output nor a regularity theorem is used to obtain these bounds. In particular the local-only qualification of GI.K for general analytic spaces has not been replaced by an unproved global bound: the fixed smooth dimension supplies the independent uniform bound AP.16.

#### AP.E. Supported induced segments near a fibre

Assume the generator property in the opening paragraph and fix \(y\). After restricting to its neighbourhood, let \(F_0\subset M\) be the given coherent ordinary generator. Its action gives
\[
I(F_0)\twoheadrightarrow M.
\tag{AP.19}
\]
The source is coherent with the global good filtration AP.1, and its support lies in \(S\). The kernel \(N_1\) is coherent by operator coherence and has the induced global good filtration.

We recall the exact filtered-submodule argument, which also applies to these nonfree coefficients. For a coherent submodule \(H\) of any good filtered coherent module \(B\), locally choose a strict filtered epimorphism from a finite shifted free operator module \(L\) to \(B\). Lift finite homogeneous symbol generators and lower order to obtain this epimorphism and its strictness. The inverse image \(H'\subset L\) is coherent. Lemma3.0c.4 gives it a good induced filtration on one neighbourhood for every degree. Strictness identifies
\[
\begin{gathered}
H\cap G_pB\\
=\operatorname{im}(H'\cap G_pL\to G_pB).
\end{gathered}
\tag{AP.20}
\]
Both ordinary sheaves in this image map are coherent. Their image is coherent by SC.D. The associated graded image is a quotient of the finite graded module of \(H'\), and is therefore finite. This proves goodness of the induced filtration on \(H\), including coherence of all its actual ordinary pieces on the same neighbourhood. It is the general filtered-submodule proof already given in §5.9; local freeness of \(B\) is unnecessary.

At every point of the compact set \(S\cap f^{-1}(y)\), finite local operator generators of \(N_1\) belong to some induced degree. Membership in that degree holds on a neighbourhood, by the definition of the subsheaf. A finite subcover and the maximum of its degree bounds give one integer \(p\) such that \(G_pN_1\) generates on a neighbourhood \(U\) of this compact set. Properness on \(S\) supplies a smaller target neighbourhood \(V\) with
\[
S\cap f^{-1}(V)\subset U.
\tag{AP.21}
\]
Indeed \(f|_S\) is closed and the image of the closed set \(S\setminus U\) misses \(y\); remove that image. The globally defined coherent sheaf \(G_pN_1\), restricted to \(f^{-1}V\), generates \(N_1\) everywhere: it does so on \(U\), and both sheaves vanish off \(S\).

Set \(F_1=G_pN_1\) there and use \(I(F_1)\twoheadrightarrow N_1\). AP.A and AP.20 apply to its kernel. Repeating any specified finite number of times constructs, after only finitely many target shrinkings,
\[
\begin{gathered}
0\longrightarrow L\longrightarrow I(F_s)\longrightarrow\cdots\\
\longrightarrow I(F_0)\longrightarrow M\longrightarrow0.
\end{gathered}
\tag{AP.22}
\]
Every \(F_j\) is ordinary coherent and supported in the same \(S\). Every syzygy is operator-coherent and is supported there as well. These are actual action maps, not merely symbol maps. Their differentials need not be induced by ordinary coefficient maps \(F_j\to F_{j-1}\).

A global good filtration in the opening hypothesis supplies its coherent ordinary generator by the same compact-degree argument. Thus the stated alternate hypothesis is sufficient. GG.P supplies it for each regular holonomic cohomology module, and density conversion supplies the right-module version.

#### AP.F. Removing the finite resolution tail

Put \(A_j=I(F_j)\). Take
\[
\begin{gathered}
s=3n+1,\\
C=[A_s\to\cdots\to A_0],\\
\text{in degrees }[-s,0].
\end{gathered}
\tag{AP.23}
\]
The augmentation in AP.22 has
\[
\begin{gathered}
\operatorname{Cone}(C\to M)\\
\simeq L[s+1].
\end{gathered}
\tag{AP.24}
\]
This follows directly on cohomology: \(C\) has only the additional kernel \(L\) in degree \(-s\); in the cone it is in degree \(-s-1\). All other augmented degrees are exact. The image functor is triangulated. Its independent bound AP.17 gives
\[
\begin{gathered}
f_+L[s+1]\in\\
D^{[-n-s-1,\,2n-s-1]}\\
=D^{[-4n-2,\,-n-2]}.
\end{gathered}
\tag{AP.25}
\]
For every \(q\ge-n\), both cone degrees \(q-1\) and \(q\) vanish. The long exact sequence gives the natural right-operator isomorphism
\[
\begin{gathered}
\mathcal H^q(f_+C)\xrightarrow{\sim}\mathcal H^q(f_+M),\\
q\ge-n.
\end{gathered}
\tag{AP.26}
\]
AP.17 kills the remaining degrees \(q<-n\) of the actual output. Thus AP.26 covers every possibly nonzero output degree; no convergence of an infinite resolution is being assumed.

Each \(f_+I(F_j)\) has coherent cohomology by AP.14. It is bounded by AP.17. The finitely many brutal filtration triangles of \(C\) express \(f_+C\) as finitely many cones of shifts of these objects. Kernels, cokernels and finite extensions of coherent analytic operator modules are coherent by Theorem3.0c. The long exact sequences of those triangles therefore prove coherence of every \(\mathcal H^q(f_+C)\). Formula AP.26 proves the same for the actual output. The target neighbourhoods cover \(Y\), so this is coherence on all of \(Y\). AP.17 already gives a single global finite amplitude. We have proved
\[
f_+M\in D^b_{\mathrm{coh}}(\mathcal D_Y^{\mathrm{op}})
\tag{AP.27}
\]
for every module in the stated generator and proper-support scope, for an arbitrary holomorphic map.

For a bounded complex whose coherent cohomology modules have that property, use its finite cohomology truncation triangles. Each attached cohomology module has coherent image by AP.27; the long exact sequences preserve coherence. Their degree intervals give AP.18. There is no need for a global induced resolution of an entire complex, or an intersection of infinitely many shrinking target neighbourhoods.

#### AP.G. Left modules and regular holonomic inputs

For a coherent left module \(M\), density side change gives the right module \(\omega_X\otimes_{\mathcal O_X}M\). Its generator property follows from the one for \(M\): in a local volume frame the transpose of a vector field is minus that vector field minus its holomorphic divergence. This order-one identity and induction on order show that the density-twisted coherent ordinary generator generates on the other side. Tensoring by the density line and its inverse is exact. Thus the left direct image is
\[
\begin{gathered}
f_*^{\mathcal D}M\simeq\\
\omega_Y^{-1}\otimes_{\mathcal O_Y}
f_+(\omega_X\otimes_{\mathcal O_X}M),\\
f_*^{\mathcal D}K\in
D^{[a-n,\,b+2n]}_{\mathrm{coh}}(\mathcal D_Y)
\end{gathered}
\tag{AP.28}
\]
when the cohomology of \(K\) lies in \([a,b]\) and satisfies the generator hypothesis. The formula has the programme's transfer normalization; side change adds no cohomological shift.

In particular let \(K\) have regular holonomic cohomology and closed support proper over \(Y\). GG.P supplies the generator on a neighbourhood of every target fibre for each of its finitely many cohomology modules. A finite intersection of these target neighbourhoods suffices at each point. All hypotheses above therefore hold. This proves bounded coherent analytic operator image for every such \(K\), on every support allowed here, with no projectivity or coefficient-flatness assumption.

#### AP.H. A nonflat supported coefficient and all its layers

Let \(X=\mathbf C_z\), let \(f:X\to\{*\}\), and take, for any integer \(m\ge1\),
\[
\begin{gathered}
F=\mathcal O_X/(z^m),\\
I(F)=\mathcal D_X/z^m\mathcal D_X.
\end{gathered}
\tag{AP.29}
\]
Its closed support is the single point, so the support restriction of \(f\) is proper. It is not ordinary-flat: in the tensor of the injection \(\mathcal O_X\xrightarrow{z}\mathcal O_X\) with \(F\), the nonzero class of \(z^{m-1}\) is killed. AP.2 retains every layer of this nonflat coefficient in each derivative degree.

Writing \(e_j\) for the class of \(z^j\), with \(e_m=0\), the exact right actions are
\[
\begin{gathered}
(e_j\otimes\partial^k)\partial=e_j\otimes\partial^{k+1},\\
(e_j\otimes\partial^k)z\\
=e_{j+1}\otimes\partial^k+k e_j\otimes\partial^{k-1}.
\end{gathered}
\tag{AP.30}
\]
The second summand is zero when \(k=0\). These formulas follow from \(\partial^kz=z\partial^k+k\partial^{k-1}\); they verify explicitly the lower-order term used in AP.2.

Here \(T=\mathcal O_X\). Apply AP.7, or tensor the ordinary flat resolution \(0\to\mathcal O_X\xrightarrow{z^m}\mathcal O_X\to F\to0\) with the ordinary-flat \(T\), to get
\[
I(F)\otimes^L_{\mathcal D_X}\mathcal O_X\simeq F.
\tag{AP.31}
\]
Closed pushforward from the point is exact. Its sections are the full vector space of nilpotent jets, and its higher cohomology is zero. Therefore
\[
\begin{gathered}
f_+I(F)\simeq\mathbf C[z]/(z^m),\\
\text{in degree }0,\\
\dim_{\mathbf C}\mathcal H^0(f_+I(F))=m.
\end{gathered}
\tag{AP.32}
\]
This example retains all \(m\) layers. The induced cancellation is compatible with nonflat coefficients and with a nonreduced ordinary support structure.

**Exercise.** For \(n=1\), compute the source and cone degree intervals in AP.23–AP.25. Check directly that they separate every possibly nonzero output degree, including the lower endpoint.

**Solution.** Here \(s=4\), the finite induced complex lies in \([-4,0]\), and AP.17 gives the actual output interval \([-1,2]\). The shifted kernel image in AP.25 lies in \([-6,-3]\). For \(q=-1\), its two relevant degrees are \(-2,-1\), both outside that interval. Increasing \(q\) keeps both outside it. AP.26 therefore includes the lower endpoint and every larger output degree. These are sufficient bounds; AP.32 has its actual cohomology only in degree zero.

![Actual induced transfer and proper image, supported finite resolution and degree separation](assets/actual-induced-analytic-proper.png)

The upper panel gives the actual maps AP.7 and AP.11 and the supported exact segment AP.22. The lower left plots the exact sufficient intervals in the exercise, with their actual integer endpoints; the dotted line is degree \(-2\), which lies strictly above the cone interval. The lower right is AP.32 for \(m=3\); its three boxes represent nilpotent coefficient basis layers at one support point. Drawing source, SVG and exact interval and layer data specify the diagram.

The induced-module route to proper coherence is also developed in [Schapira's freely accessible D-module notes, Theorem2.5.4, pp.43–45](https://perso.imj-prg.fr/pierre-schapira/wp-content/uploads/schapira-pub/lectnotes/Dmod.pdf). AP.A–AP.H supply the nonflat induced terms, actual right-action comparison, supported finite segment and uniform degree separation used here; the ordinary analytic coherence input is proved in GI.A–GI.K.


### 5.101. General analytic characteristic transport

Let \(f: X\to Y\) be a holomorphic map of Hausdorff second-countable complex manifolds, of fixed dimensions \(n\) and \(m\). Let \(S\subset X\) be closed and proper over \(Y\). Suppose the coherent right \(\mathcal D_X\)-module \(M\) is supported in \(S\) and has a coherent ordinary generator on the whole inverse image of a neighbourhood of each target point, as defined in AP.E. This includes every regular holonomic module by GG.P.

The cotangent correspondence is
\[
\begin{gathered}
W=X\times_YT^*Y,\\
u: W\to X,\\
v: W\to T^*Y,\\
\rho: W\to T^*X,\\
\rho(x,\eta)=(x,(df_x)^*\eta).
\end{gathered}
\tag{GP.1}
\]
We prove the actual characteristic containment
\[
\begin{gathered}
\operatorname{Ch}\mathcal H^b(f_+M)\\
\subset v\bigl(\rho^{-1}\operatorname{Ch}M\bigr)
\end{gathered}
\tag{GP.2}
\]
in every degree. The proof constructs good filtrations of the actual output, retains derived Rees torsion until both specializations have been computed, and includes all singular pieces of the source characteristic support. If \(M\) is holonomic, every output cohomology module is holonomic. The same conclusions hold for bounded complexes with the stated cohomology hypotheses, and for left modules under density side change.

The complete ordinary analytic coherent-image input is GI.A–GI.K. The actual unfiltered operator image and its uniform amplitude are AP.A–AP.G. The analytic polynomial-extension and uniform operator arguments are Theorem3.0c. The finite graded Rees coherence proof and the signed Rees Spencer resolution are the full arguments in §5.12; their ring and resolution statements do not require a projective map. The new proper cotangent comparison uses the analytic parameter theorem below.

#### GP.A. Adding analytic parameters is flat

For any complex manifold \(D\) and polydisc \(B\), the projection \(D\times B\to D\) is ordinary-flat. Work at a point and translate it to the origin. Put
\[
\begin{gathered}
H=\mathbf C\{y\},\\
K=\mathbf C\{y,z\},\\
P=H[z]_{(y,z)},\\
\widehat P=\widehat K=\mathbf C[[y,z]].
\end{gathered}
\tag{GP.3}
\]
The Noetherianity, regular local presentations and the exact faithfully flat completion theorem are the earlier analytic local algebra proof and commutative completion, Theorem3.2. The equality of completions follows by finite Taylor quotients. The ring \(P\) is flat over \(H\), because polynomial extension and localization are flat: polynomials are free as coefficient modules, and localization preserves an injection by the fraction criterion. Its completion is flat over \(P\), hence over \(H\). Completion of \(K\) is faithfully flat over \(K\). For an \(H\)-module injection, a kernel after tensoring with \(K\) would remain nonzero after tensoring with \(\widehat K\), contradicting its flatness over \(H\). Thus \(K\) is flat over \(H\), for arbitrary modules. This proves the projection statement at every stalk, including nonzero parameter points by translation. The same conclusion holds for any analytic space: a local model has ring \(H/J\), its product model has ring \(K/JK\), and the latter is flat over \(H/J\) by base change of the proved flat map. Tensoring an injection of quotient-ring modules with it is exactly tensoring that injection over \(H\) with \(K\). This retains the full nonreduced ideal and needs no reducedness assumption.

We shall also use exact analytic extension from polynomial fibre variables on a manifold. The proof at an arbitrary fibre point is the same completion descent: localize the coefficient polynomial ring at that point and compare its completion with the convergent local ring. The argument just recalled from Theorem3.0c proves exactness on arbitrary modules. Write
\[
\begin{gathered}
S_Y=\operatorname{Sym}T_Y,\\
\pi_Y: T^*Y\to Y,\\
C_Y=\pi_Y^{-1}S_Y,\\
\mathfrak a_Y(N)\\
=\mathcal O_{T^*Y}\otimes_{C_Y}\pi_Y^{-1}N.
\end{gathered}
\tag{GP.4}
\]
The analogous notation will be used on \(X\). These are exact coefficient-extension functors; no assertion that a general holomorphic map is flat is involved.

#### GP.B. Proper images and independent analytic parameters

Let \(g: Z\to D\) be any holomorphic map of Hausdorff second-countable analytic spaces, with \(D\) a manifold, and let \(F\) be coherent with closed support proper over \(D\). For a polydisc \(B\), let \(p: Z\times B\to Z\), \(\beta: D\times B\to D\), and \(g_B=g\times1\). Then the canonical comparison is
\[
\beta^*Rg_*F\xrightarrow{\sim}R(g_B)_*p^*F.
\tag{GP.5}
\]
The assertion is local on \(D\times B\); a neighbourhood of any chosen target point suffices. We prove the actual comparison, not merely an equality of fibre dimensions.

First replace the source by the closed analytic support defined by the full annihilator of \(F\). This is the reduction of GI.K, which retains all nilpotents. Its restricted map is proper, closed pushforward is exact, and products with \(B\) preserve that reduction. We may therefore use the finite graph covers and actual coherent Čech models of GI.H–GI.J for \(g\) itself near a given target point. On a polydisc \(D_0\) they give bounded Čech complexes \(E\to E'\) and, on a smaller concentric \(D_t\), a finite-rank free model \(L\to E_t\) in every degree. This model is bounded above, with no left boundedness required. On every smaller polydisc \(V\subset D_t\), the map
\[
L(V)\longrightarrow E(V)
\tag{GP.6}
\]
is a quasi-isomorphism of nuclear Fréchet complexes. These are the actual complexes of sections and restriction maps in GI.I–GI.J.

Every nonempty chart intersection over \(V\) is a closed analytic graph in a strongly one-complete product. Its product with any smaller polydisc \(B'\subset B\) has the same property, by AT.B. Coherent acyclicity follows from CX.L. The full coherent exterior-product section theorem AT.A–AT.D identifies the Čech terms on these products with
\[
E(V)\widehat\otimes_{\mathbf C}\mathcal O(B').
\tag{GP.7}
\]
All restriction differentials agree under that identity. The same assertion holds for \(L(V)\), whose free terms become \(\mathcal O(V\times B')^{b_j}\).

Completed scalar tensoring in GP.7 preserves the quasi-isomorphism GP.6. Indeed its bounded-above signed cone is acyclic, each cycle is a closed nuclear Fréchet subspace, and each short cycle sequence is strict by the open-mapping theorem. The exact nuclear tensor theorem NT.F preserves those sequences. The tensor cone is therefore acyclic in every degree. Only adjacent degrees enter this argument; it takes no inverse limit over the left tail.

Sheafify the finite free terms of \(L\) on \(D_t\), obtaining the complex \(\mathcal L\) of GI.J. On the rectangle basis \(V\times B'\), the tensor maps just constructed give
\[
\beta^*\mathcal L\longrightarrow R(g_B)_*p^*F
\tag{GP.8}
\]
as a quasi-isomorphism. More explicitly, its columns are the restricted Čech cycle representatives of the original map \(L\to E_t\), tensored with holomorphic parameter functions. Finite Čech/derived-section comparison and the natural stalk passage of GI.J identify its cohomology with the indicated higher images. A section in a kernel, or a boundary equality, is represented on one smaller rectangle, so stalks commute with these cohomologies even for the left-unbounded free complex.

On the other hand \(\beta\) is flat by GP.A. Hence \(\beta^*\mathcal L\) has cohomology \(\beta^*\mathcal H^q(\mathcal L)\), and GI.J identifies the latter with \(\beta^*R^qg_*F\). The resulting isomorphism is GP.5 itself: the natural comparison is defined by pulling a coefficient section to the product and multiplying by the parameter function. Its composite with \(\mathcal L\to Rg_*F\) is exactly the column map in GP.8. Equivalently its adjoint is the pulled-back counit to \(F\); tensor reassociation gives the same coefficient multiplication. Thus the proof identifies the canonical map and its naturality in \(F\), restrictions and coefficient maps. The local comparisons agree on overlaps and prove GP.5 everywhere. Bounded coherent complexes follow from finite cohomology truncation triangles.

#### GP.C. The analytic cotangent candidate

Return to \(f\) and choose an actual global good filtration \(F_\bullet M\) on the inverse image of a sufficiently small target neighbourhood. A coherent ordinary generator supplies it: give its induced module the filtration AP.1, use the good induced kernel of AP.20, and take the quotient filtration on \(M\). Its actual pieces are coherent and its graded module is finite. Write
\[
\begin{gathered}
G=\mathfrak a_X(\operatorname{gr}_FM),\\
\mathcal K=Rv_*L\rho^*G.
\end{gathered}
\tag{GP.9}
\]
This is a bounded coherent complex. Here are the amplitude and support details, including at critical points of \(f\).

In source and target coordinates, put \(a_{ij}=\partial f_j/\partial x_i\). The symbol-transfer module is
\[
\begin{gathered}
B_f=\mathcal O_X\otimes_{f^{-1}\mathcal O_Y}f^{-1}S_Y,\\
\xi_i\cdot b=\Bigl(\sum_j a_{ij}\eta_j\Bigr)b.
\end{gathered}
\tag{GP.10}
\]
Over the ring \(S_X\otimes_{\mathcal O_X}B_f\), its Koszul resolution uses the \(n\) elements \(\xi_i-\sum_j a_{ij}\eta_j\). These form a regular sequence: the polynomial change of variables replacing \(\xi_i\) by those differences is invertible, and successive polynomial variables are injective with quotient deleting that variable. The same two-term cone induction as the earlier Spencer/Koszul proof gives exactness. Its terms are flat over \(S_X\), because \(B_f\) is free over \(\mathcal O_X\). It follows that symbol transfer has amplitude \([-n,0]\), without a non-characteristic or submersion assumption.

Analytically, consider \(T^*X\times_YT^*Y\). Its projection to \(T^*X\) is locally the addition of \(m\) independent parameters, so it is flat by GP.A. The graph of \(\rho\) in this space is the closed embedding defined by the same \(n\) differences. Analytic extension of the polynomial Koszul resolution therefore computes \(L\rho^*G\). Its terms and cohomology are coherent, and
\[
\begin{gathered}
L\rho^*G\in D^{[-n,0]}_{\mathrm{coh}}(\mathcal O_W),\\
\operatorname{supp}L\rho^*G
\subset \rho^{-1}\operatorname{Ch}M.
\end{gathered}
\tag{GP.11}
\]
The support assertion follows by restricting to an open where \(G=0\).

The map \(v\) is proper on the closed set \(u^{-1}S=S\times_YT^*Y\): it is a base change of the proper map \(S\to Y\). The cohomology in GP.11 is supported there. GI.K gives coherent higher images. The real dimension of the manifold \(W\) is \(2(n+m)\), so the earlier uniform all-sheaf bound used in AP.D supplies a single upper image bound. Consequently
\[
\begin{gathered}
\mathcal K\in D^{[-n,\,2(n+m)]}_{\mathrm{coh}}(\mathcal O_{T^*Y}),\\
\operatorname{supp}\mathcal K
\subset v(\rho^{-1}\operatorname{Ch}M).
\end{gathered}
\tag{GP.12}
\]
For the latter statement, the set on the right is closed by properness on that inverse image. Over its complement the entire source coefficient complex vanishes. These are sufficient bounds, not a claim of optimal amplitude.

#### GP.D. Strict finite segments with coherent coefficients

The actual graded Rees rings and module are
\[
\begin{gathered}
\mathscr R_X=\\
\bigoplus_{p\ge0}\mathcal D_{X,\le p}h^p,\\
\mathscr M=\bigoplus_{p\in\mathbf Z}F_pM\,h^p,\\
\vartheta_\xi=h\xi,\\
\deg h=\deg\vartheta_\xi=1.
\end{gathered}
\tag{GP.13}
\]
The finite graded Rees coherence theorem proved in §5.12 applies to these analytic sheaves. Its auxiliary filtration is least power of \(h\), with commutative associated graded \(\mathcal O_X[h,\xi]\). At fixed internal degree, raising that least power terminates. The proof keeps both the internal and auxiliary shifts in the replacement finite free modules, uses uniform homogeneous polynomial kernels in \(n+1\) variables, and then presents finite graded kernels, images and extensions on one neighbourhood. Thus every fixed degree piece is ordinary coherent. We use that full uniform theorem, rather than stalk Noetherianity as a substitute for neighbourhood coherence.

A finite strict induced segment can be constructed without projective twists or free ordinary coefficients. On a neighbourhood of the compact set \(S\cap f^{-1}(y)\), finitely many local good presentations give a common lower filtration bound and finitely many degrees \(a\) generating the symbol module. Proper closedness shrinks the target so these assertions hold on the entire inverse image. For each such degree take the whole coherent sheaf \(F_aM\), with its actual map to \(M\). Give \(I(F_aM)\) generator degree \(a\). The direct sum of these finitely many maps is strict and surjective: its symbol map is onto, and lifting the leading symbol and lowering degree terminates at the common lower bound.

Its kernel has the induced global good filtration by AP.20. Repeat for that kernel a specified finite number of times, shrinking the target finitely many times. With shifts defined by \(V(-a)_p=V_{p-a}\), this gives
\[
\begin{gathered}
0\to N\to A_s\to\cdots\\
\to A_0\to M\to0,\\
A_i=\bigoplus_a I(F_{i,a}),\\
\text{with generator degree }a,\\
\mathscr A_i=\\
\bigoplus_a F_{i,a}\otimes_{\mathcal O_X}\mathscr R_X(-a).
\end{gathered}
\tag{GP.14}
\]
Every \(F_{i,a}\) is ordinary coherent and supported in \(S\). Every differential is strict for the indicated filtrations. Therefore Rees formation and symbol formation both give exact sequences. The kernel \(N\) has its induced good filtration as well. No infinite set of target shrinkings has been intersected.

#### GP.E. Actual Rees transfer and its bounds

Define the forward Rees transfer by
\[
\begin{gathered}
\mathscr T=\\
\mathcal O_X[h]\otimes_{f^{-1}\mathcal O_Y[h]}f^{-1}\mathscr R_Y.
\end{gathered}
\tag{GP.15}
\]
For a source vector field \(\xi\), its left action in target coordinates is
\[
\begin{gathered}
\vartheta_\xi(a\otimes Q)\\
=h\xi(a)\otimes Q\\
{}+\sum_j a\,\xi(f_j)\otimes\vartheta_{y_j}Q.
\end{gathered}
\tag{GP.16}
\]
Balancing and the bracket identities follow by the chain rule, exactly as for the ordinary transfer, now with \([\vartheta_\xi,a]=h\xi(a)\) and \([\vartheta_\xi,\vartheta_\zeta]=h\vartheta_{[\xi,\zeta]}\). The intrinsic expression uses \(df\), so these formulas glue across coordinates. They commute with the right target Rees action. Target PBW makes \(\mathscr T\) free over \(\mathcal O_X[h]\).

Put \(A_X=\mathcal O_X[h]\). The absolute Rees Spencer resolution has length \(n\), with cochain degree \(-r\) term
\[
\begin{gathered}
E_r=A_X\otimes_{\mathcal O_X}\bigwedge^rT_X,\\
\mathscr R_X\otimes_{A_X}E_r(-r)\otimes_{A_X}\mathscr T.
\end{gathered}
\tag{GP.17}
\]
Its action-difference differential and \(h\)-multiplied bracket terms are the exact signed formulas proved in §5.12. The relations just checked make that proof apply to GP.16 for this arbitrary map. Every term is left Rees-flat, because its ordinary-polynomial coefficient is flat. The leading first-factor operator-order differential is the Koszul differential in \(n\) independent source variables. Its exactness and terminating order reduction prove the augmented resolution exact. The proof needs no vertical-coordinate splitting of \(f\).

Set
\[
\mathcal C=Rf_*\bigl(\mathscr M\otimes^L_{\mathscr R_X}\mathscr T\bigr).
\tag{GP.18}
\]
The Spencer bound and the uniform real-manifold image bound give amplitude \([-n,2n]\) for any Rees module supported in \(S\). These are bounds in the graded category: evaluation at an internal degree preserves injectives because its left adjoint is the free graded Rees module in that degree tensored over \(\mathbf C\), which is exact. Forgetting the grading is also compatible with proper image, by the compact-fibre coproduct and c-soft acyclicity proof already used in §5.12 and AP.C. Thus every homogeneous target action and every ordinary specialization is retained.

For a nonflat ordinary coefficient \(F\), an ordinary-flat resolution of \(F\), induced to \(\mathscr R_X\), is a right Rees-flat resolution. Both \(\mathscr R_X\) and \(\mathscr T\) are left ordinary-flat. The cancellation proof of AP.B therefore gives
\[
\begin{gathered}
(F\otimes_{\mathcal O_X}\mathscr R_X)
\otimes^L_{\mathscr R_X}\mathscr T\\
\simeq F\otimes_{\mathcal O_X}\mathscr T.
\end{gathered}
\tag{GP.19}
\]
Finite-order coefficient projection and the proper filtered-union proof of AP.C identify its image with \(Rf_*F\otimes_{\mathcal O_Y}\mathscr R_Y\). In each internal degree there are only finitely many target PBW monomials. The comparison commutes with all right actions, including the finite product rule
\[
\begin{gathered}
\vartheta^\alpha b\\
=\sum_{\beta\le\alpha}\binom\alpha\beta
h^{|\beta|}(\partial^\beta b)\vartheta^{\alpha-\beta}.
\end{gathered}
\tag{GP.20}
\]
Thus the image of each summand in GP.14 has coherent graded Rees cohomology, by GI and finite ordinary presentation of \(R^qf_*F_{i,a}\).

Choose
\[
s=3n+2m+2.
\tag{GP.21}
\]
Let \(A=[A_s\to\cdots\to A_0]\), in degrees \([-s,0]\). The strict Rees augmentation has cone \(\mathscr N[s+1]\). Its image lies in
\[
\begin{gathered}
{}[\ell_R,u_R],\\
\ell_R=-n-s-1\\
=-4n-2m-3,\\
u_R=2n-s-1\\
=-n-2m-3.
\end{gathered}
\tag{GP.22}
\]
Both adjacent cone degrees vanish in every possible output degree of \(\mathcal C\). The finite segment therefore computes those degrees. Finitely many cones of its induced-term images, and the coherent graded category, prove that every \(H^b\mathcal C\) is a coherent graded Rees module. This does not presume strictness of an image filtration.

#### GP.F. Both central specializations

Write \(\mathbf C_c=\mathbf C[h]/(h-c)\) for evaluation at a complex number \(c\). The input \(\mathscr M\subset M[h,h^{-1}]\) is torsion-free over \(\mathbf C[h]\), hence flat, by the complete PID/torsion-free argument of §5.12. Tensor GP.17 with it. Locally the resulting terms are direct sums of copies of \(\mathscr M\) with finite tangent factors, so they remain flat over \(\mathbf C[h]\). This bounded complex computes derived specialization by its ordinary specialization. At \(h=1\) it is the actual ordinary Spencer transfer complex. At zero it is the symbol Koszul transfer complex of GP.10. Derived image commutes with either specialization, because the two-term free resolution of \(\mathbf C[h]/(h-c)\) expresses it as a finite cone of \(h-c\). Consequently
\[
\begin{gathered}
\mathcal C\otimes^L_{\mathbf C[h]}\mathbf C_1\simeq f_+M,\\
\mathcal C_0:=\mathcal C\otimes^L_{\mathbf C[h]}\mathbf C_0\\
\simeq Rf_*\bigl(\operatorname{gr}_FM\otimes^L_{S_X}B_f\bigr).
\end{gathered}
\tag{GP.23}
\]
The first comparison retains the full right operator action, and the second the full target symbol action. At one the internal grading is forgotten; at zero it is retained. The sufficient interval for \(\mathcal C_0\) is \([-n-1,2n]\).

#### GP.G. The actual analytic cotangent comparison

There is a natural coefficient comparison
\[
\mathfrak a_Y(\mathcal C_0)\longrightarrow\mathcal K.
\tag{GP.24}
\]
It is obtained by pulling a polynomial-coefficient image section to the cotangent parameter space and multiplying by its analytic coefficient, then applying the image adjunction. The same map on resolutions gives the derived comparison. Exact analytic extension, the graph Koszul model of GP.C and tensor reassociation identify its source coefficient complex with the analytic cotangent pullback on the right. This construction is natural in the source symbol complex.

For each induced symbol term in GP.14, derived symbol transfer is ordinary transfer, since \(B_f\) is ordinary-flat and the same induced ordinary-flat resolution applies. Analytically, its coefficient is simply \(u^*F_{i,a}\): the projection \(u\) is flat by GP.A. On a coordinate open in \(Y\), the cotangent bundle is a product with a polydisc locally in its fibre coordinates. The parameter theorem GP.5 gives precisely
\[
\begin{gathered}
\mathfrak a_Y(Rf_*F_{i,a}\otimes S_Y)\\
\xrightarrow{\sim}Rv_*u^*F_{i,a}.
\end{gathered}
\tag{GP.25}
\]
This is the map GP.24 on that term, including its internal shift. Naturality makes it commute with the symbol differentials of the finite segment. Those differentials may contain fibre variables and need not arise from maps between the ordinary coefficient sheaves; it is the functorial comparison, rather than an independently chosen contraction, that gives commutation. Finite totalization proves GP.24 for the finite segment.

The finite segment computes the entire comparison. On the Rees side, specialize the image cone GP.22 at zero. Its upper bound stays \(2n-s-1\), and its lower bound can decrease by one. Analytic extension is exact. On the cotangent side the strict symbol cone is \(\operatorname{gr}N[s+1]\). GP.11 and the real dimension of \(W\) put its image in
\[
\begin{gathered}
{}[\ell_R,u_A],\\
u_A=2(n+m)-s-1\\
=-n-3.
\end{gathered}
\tag{GP.26}
\]
The Rees-side upper bound is no larger. Thus both cone degrees \(q-1,q\) vanish for every \(q\ge-n-1\). The finite comparison therefore gives an isomorphism in that entire range. Below it both full objects already vanish: the left has the interval following GP.23, and the right has GP.12. We have proved the actual derived isomorphism
\[
\mathfrak a_Y(\mathcal C_0)\xrightarrow{\sim}Rv_*L\rho^*G.
\tag{GP.27}
\]
No unrestricted analytic base-change theorem or infinite-width resolution is hidden in this comparison.

#### GP.H. Torsion and the actual output filtration

Put \(H_b=\mathcal H^b\mathcal C\). It is coherent graded Rees by GP.E. Multiplication by \(h-1\) is injective on any graded module considered as an ungraded module: the lowest nonzero homogeneous component of a finite sum contributes its negative and cannot cancel. The two-term specialization sequence in GP.23 therefore gives
\[
\begin{gathered}
\mathcal H^b(f_+M)\\
=H_b/(h-1)H_b.
\end{gathered}
\tag{GP.28}
\]
Keep the \(h\)-power torsion until this point. Let \(T_b\) be the union of the kernels of powers of \(h\) on \(H_b\). It is coherent and locally killed by one power. Indeed at a stalk the graded ascending-chain condition gives a power \(h^N\) killing all torsion. The coherent quotient by \(\ker h^N\) has zero \(h\)-kernel at that stalk. Its coherent \(h\)-kernel therefore vanishes on one smaller neighbourhood, so the quotient is \(h\)-torsion-free there. This proves the same bound \(N\) on that neighbourhood, without intersecting infinitely many opens.

Write \(\overline H_b=H_b/T_b\). Torsion specializes to zero at one, since \(h\) is invertible there, so its specialization is still GP.28. Every homogeneous piece of \(\overline H_b\) embeds in that specialization. If a nonzero homogeneous element were \((h-1)a\), the lowest degree of \(a\) would be the degree of that element; injectivity of \(h\) then makes the highest component of \(ha\) a nonzero higher-degree term, a contradiction. Let \(F'_p\) be the image of degree \(p\). Multiplication by \(h\) makes these pieces increasing. Raising finitely many homogeneous representatives to their largest degree proves exhaustion. Coherence of every piece and a local lower bound follow from a finite graded presentation. Finite homogeneous Rees generation proves that this is an actual good order filtration of the actual output, with
\[
\begin{gathered}
\mathscr R_{F'}\mathcal H^b(f_+M)\\
=\overline H_b,\\
\operatorname{gr}_{F'}\mathcal H^b(f_+M)\\
=\overline H_b/h\overline H_b.
\end{gathered}
\tag{GP.29}
\]
The zero-specialization sequence gives an injection of \(H_b/hH_b\) into \(\mathcal H^b\mathcal C_0\); its cokernel is the \(h\)-kernel of \(H_{b+1}\), with its internal shift. The map onto \(\overline H_b/h\overline H_b\) is surjective. Exact analytic extension and GP.27 therefore make the actual output symbol sheaf a quotient of a submodule of \(\mathcal H^b\mathcal K\). Its support is contained in GP.12, proving GP.2. No equality with the entire cotangent candidate, or torsion-free image before taking this quotient, has been asserted.

These constructions are local on the target but apply to every target point. Characteristic support is independent of the chosen local good filtration by the finite Rees-lattice proof in the earlier characteristic-variety lesson; that proof uses finite good generation and Noetherian symbols, which hold at the analytic stalks here. Thus the containments glue to the intrinsic statement GP.2 on all of \(T^*Y\).

#### GP.I. Singular-inclusive isotropic transport

Let \(\Lambda\subset T^*X\) be a closed reduced conic analytic Lagrangian set whose base support is contained in \(S\). Put
\[
A=\rho^{-1}\Lambda,\qquad B=v(A),
\tag{GP.30}
\]
with reduced analytic structures. The set \(A\) is closed analytic and proper over \(T^*Y\), being closed in \(u^{-1}S\). Its coherent structure sheaf has coherent direct image by GI.K. The support of that image is exactly \(B\): outside the closed image it is zero, while at an image point the unit cannot have zero direct-image germ, since its restriction would then vanish on an inverse image containing a point of the nonempty fibre. Coherent support is analytic by a finite presentation. Consequently \(B\) is closed analytic; scaling of \(\eta\) makes it conic.

The exact singular restriction lemma Lemma5.11.1 says that an ambient holomorphic form vanishing on the regular locus of a reduced analytic set also vanishes on the regular locus of every locally closed analytic subset, even one wholly inside its singular locus. Its proof uses the finite ramified transverse family, bounded holomorphic removal and dimension drop; it assumes no stratification or resolution. It has the following map version. If \(q: T\to\Lambda\) is holomorphic with \(T\) smooth, then \(q^*\theta_X=0\), where \(\theta_X\) is the tautological one-form. Indeed conicity and Lagrangianity give \(\theta_X=\iota_Ed\theta_X=0\) on \(\Lambda_{\rm reg}\). On each maximum-rank open of \(q\), the constant-rank theorem identifies its local image with a smooth locally closed analytic subset of \(\Lambda\). Lemma5.11.1 makes the form zero there. The pullback is holomorphic and vanishes on these dense maximum-rank opens in every connected source chart, hence vanishes everywhere. The rank-zero case is automatic for a one-form.

The intrinsic cotangent identity for the general correspondence is
\[
\rho^*\theta_X=v^*\theta_Y.
\tag{GP.31}
\]
At \((x,\eta)\), both sides evaluate a tangent vector as \(\eta(df_x(du(\cdot)))\). This proves the identity for all tangent directions and fixes its sign. Apply the map version to \(\rho\) on every smooth locally closed analytic piece of \(A\). It follows that \(v^*\theta_Y\) vanishes on every such piece, including pieces mapped wholly into \(\Lambda_{\rm sing}\).

The complete countable smooth/rank-cover argument of §5.11 applies to \(v: A\to B\). Its hypotheses are exactly a second-countable analytic source, an analytic image and the form vanishing just proved; its rank argument has no projectivity hypothesis. To state its essential steps: local finite parametrization gives dense smooth opens with lower-dimensional analytic complements. Recursing through dimension drop and a countable chart cover gives countably many smooth locally closed pieces. Apply the same recursion to maximum-rank minors so the map has constant rank on each piece. If all ranks over a smooth target component of dimension \(r\) were below \(r\), compact exhaustions of the rank charts would cover a target ball by countably many closed nowhere-dense images. The nested-ball proof of Baire in that section rules this out. Some piece therefore has full rank, and the differential surjects onto a target open. Form vanishing then forces \(\theta_Y=0\) there. If it were nonzero anywhere in the smooth target patch, repeat over the open where a coefficient is nonzero; no full-rank piece could exist there, giving the same contradiction. Thus it vanishes throughout every smooth component patch.

Taking its exterior derivative makes every such component generically isotropic. In the \(2m\)-dimensional symplectic tangent space, an isotropic subspace has dimension at most \(m\), because it is contained in its symplectic orthogonal and their dimensions sum to \(2m\). Therefore
\[
\dim v(\rho^{-1}\Lambda)\le m.
\tag{GP.32}
\]
Every singular source piece was included in the form and rank arguments. Dimension-zero and empty sets cause no exception.

#### GP.J. Holonomicity, complexes and density conversion

The earlier analytic Gabber and componentwise Bernstein argument in §5.12 proves that the reduced characteristic set of a holonomic analytic module is pure conic Lagrangian. Its proof applies the full Noetherian rational-symbol Gabber theorem to analytic PBW stalks, then uses exact homogeneous analytic extension, radical detection and the analytic Nullstellensatz. No map theorem enters that source statement. The same componentwise Bernstein bound applies to any nonzero coherent target module.

For holonomic \(M\), use \(\Lambda=\operatorname{Ch}M\) in GP.32. Formula GP.2 bounds every output characteristic dimension by \(m\). AP supplies coherence and global boundedness; the analytic componentwise Bernstein inequality gives the opposite dimension inequality for any nonzero component. Hence every output cohomology module is holonomic, with the zero module included.

For a bounded complex, its finite cohomology truncation triangles attach its cohomology modules in their original degrees. Their generator hypotheses hold on a common smaller target neighbourhood by a finite intersection. The image triangles and the exact-sequence characteristic-support theorem put each output characteristic support in the union of the corresponding module images. AP gives the uniform amplitude on the whole target. This proves bounded holonomic image and characteristic containment for every such bounded complex, without assuming a filtration on an arbitrary complex representative.

The density equivalences of AP.G convert right input and output to the programme's left-module direct image. Tensoring by the density line is exact and changes the local action only by transpose and its order-zero divergence term. It leaves reduced characteristic support unchanged. Thus GP.2 and holonomicity hold in those conventions, with the same degree bounds. GG.P supplies every required generator for regular holonomic inputs; this conclusion establishes holonomicity of their full arbitrary proper-support analytic images, while preservation of regularity is a further mathematical statement.

#### GP.K. A critical finite map

Let \(f:\mathbf C_x\to\mathbf C_y\) be \(y=x^e\), with \(e\ge1\). It is proper: the inverse image of a compact set is closed and bounded by \(|x|^e=|y|\), hence compact by the finite-dimensional compactness proof in CX.A. A fibre has at most \(e\) points, since each distinct root removes a linear factor from its degree-e defining polynomial. Take the zero-section source Lagrangian. The correspondence gives
\[
\begin{gathered}
y=x^e,\qquad \xi=e x^{e-1}\eta,\\
A=\{(x,\eta):e x^{e-1}\eta=0\}.
\end{gathered}
\tag{GP.33}
\]
For \(e=1\), its image is exactly the zero section. For \(e>1\), it is the union of the zero section and the full cotangent fibre over the critical value:
\[
\begin{gathered}
B=\{\eta=0\}\cup\{y=0\}\\
{}\subset T^*\mathbf C_y.
\end{gathered}
\tag{GP.34}
\]
Indeed \(x\ne0\) forces \(\eta=0\), and every complex \(y\) has an \(e\)-th root: if \(y=r e^{i\theta}\ne0\), take \(x=r^{1/e}e^{i\theta/e}\), and take zero for the root of zero. At \(x=0\) every \(\eta\) occurs. Both components have complex dimension one. The actual one-form identity is \(\xi\,dx=\eta\,dy=e x^{e-1}\eta\,dx\), which is zero on both source components, including the critical one. The inequality predicts a possible critical-value component; it does not assert that every candidate component occurs in every operator image.

**Exercise.** For \(n=m=1\), compute the two upper bounds on the finite comparison cones in GP.22 and GP.26, and check the lowest possible comparison degree.

**Solution.** Formula GP.21 gives \(s=7\). The Rees image cone has interval \([-9,-6]\); zero specialization may lower its first endpoint to \(-10\) and leaves its upper endpoint \(-6\). The analytic cone has interval \([-9,-4]\). The lowest comparison degree is \(q=-2\), whose two cone degrees are \(-3,-2\). Both lie strictly above both upper endpoints. This proves the claimed separation at the endpoint; all larger degrees are separated as well. These sufficient bounds are not a claim that any cone cohomology in those intervals is nonzero.

![Derived Rees specializations, the actual symbol subquotient and the cotangent image of a critical map](assets/general-analytic-characteristic.png)

The upper left diagram records GP.23 and GP.27. Its dashed connections indicate the actual filtration and the subquotient relation GP.29; they do not assert a map from the whole cotangent candidate onto the output symbol. The right panel is the exact real coordinate section of GP.34 for \(e=2\), with \(\operatorname{Im}y=\operatorname{Im}\eta=0\). The zero section and critical cotangent fibre are distinct complex one-dimensional components; their displayed real axes are a section of those components. Drawing source, SVG and exact map and interval data specify the diagram.

The analytic proper characteristic theorem is also stated in [Schapira's freely accessible D-module notes, Theorem2.5.4](https://perso.imj-prg.fr/pierre-schapira/wp-content/uploads/schapira-pub/lectnotes/Dmod.pdf). GP.A–GP.K give the parameter comparison, actual finite Rees and cotangent comparison, torsion quotient and singular-inclusive isotropic argument used here. Every ordinary coherent-image and functional-analytic input is proved in the exact earlier programme lessons cited above.


### 5.102. Analytic de Rham transport with nonflat coefficients

Let \(f: X\to Y\) be a holomorphic map of Hausdorff second-countable complex manifolds of fixed dimensions \(n\) and \(m\). Use right operator modules first. Put
\[
\begin{gathered}
T_f=\mathcal O_X\otimes_{f^{-1}\mathcal O_Y}f^{-1}\mathcal D_Y,\\
f_+M=Rf_*(M\otimes^L_{\mathcal D_X}T_f),\\
\operatorname{DR}^{\rm r}_X(M)=M\otimes^L_{\mathcal D_X}\mathcal O_X.
\end{gathered}
\tag{DRP.1}
\]
The left source action on \(T_f\) is the actual chain-rule action of AP.B. There is a canonical natural isomorphism
\[
\begin{gathered}
\eta_f:\operatorname{DR}^{\rm r}_Y(f_+M)\\
\xrightarrow{\sim}Rf_*\operatorname{DR}^{\rm r}_X(M)
\end{gathered}
\tag{DRP.2}
\]
for every bounded complex of right operator modules. Coherence, regularity and flatness over the source or target structure sheaf are not hypotheses for this comparison. When the support is proper and the cohomology modules have the near-fibre generator of AP.E, AP.A–AP.G supplies bounded coherent operator images. GP.A–GP.J additionally supplies characteristic containment and holonomicity for holonomic inputs. The de Rham comparison itself will be proved without using either regularity of the output or an analytic Riemann–Hilbert equivalence.

#### DRP.A. The finite target Spencer complex

On a target coordinate open, with commuting vector fields \(e_j=\partial_{y_j}\), write \(e_J=e_{j_1}\wedge\cdots\wedge e_{j_r}\). Resolve the left module \(\mathcal O_Y\) by
\[
\begin{gathered}
P_Y^{-r}=\\
\mathcal D_Y\otimes_{\mathcal O_Y}\bigwedge^rT_Y,\\
0\le r\le m,\\
b_i=(-1)^{i-1}Q\partial_{y_{j_i}},\\
d(Q\otimes e_J)\\
=\sum_{i=1}^r b_i\otimes e_{J\setminus j_i}.
\end{gathered}
\tag{DRP.3}
\]
The augmentation is \(Q\mapsto Q(1)\). Each term is finite free as a left \(\mathcal D_Y\)-module. In arbitrary vector fields the bracket terms have exactly the signs of the earlier absolute Spencer proof, AP.D; the intrinsic formula makes this an augmented complex independent of coordinates.

Here is a split exactness argument over the ordinary structure sheaf, needed when the map is nonflat. Left-coefficient PBW identifies the underlying complex in DRP.3 with the polynomial Koszul complex
\[
\begin{gathered}
\mathcal O_Y[t_1,\ldots,t_m]\otimes\bigwedge^r\mathbf C^m,\\
d=\sum_j t_j\iota_j.
\end{gathered}
\tag{DRP.4}
\]
On polynomial degree \(k\) and exterior degree \(r\), define
\[
\begin{gathered}
H=\frac{1}{k+r}\sum_j\partial_{t_j}\varepsilon_j,\\
k+r>0,
\end{gathered}
\tag{DRP.5}
\]
where \(\varepsilon_j\) is exterior multiplication by \(e_j\), and \(\iota_j\) contracts its dual. Set \(H=0\) on the constant term of exterior degree zero. Contraction and exterior multiplication satisfy
\[
\iota_i\varepsilon_j+\varepsilon_j\iota_i=\delta_{ij}.
\tag{DRP.6}
\]
Expanding the two compositions, the cross terms cancel and their sum before division is
\[
\sum_jt_j\partial_{t_j}+\sum_j\varepsilon_j\iota_j.
\tag{DRP.7}
\]
It acts by \(k+r\). Thus \(dH+Hd=1-ip\), where \(p\) takes the constant term and \(i\) includes it. These operators are \(\mathcal O_Y\)-linear. They use finite polynomials only. In particular this resolution remains exact after tensoring with any ordinary coefficient module. The contraction is used to establish ordinary split exactness; no operator-linear contraction is needed.

#### DRP.B. Cancellation of the target Spencer factor

The target resolution is bounded and left operator-flat. It therefore computes the actual derived tensor
\[
\begin{gathered}
T_f\otimes^L_{f^{-1}\mathcal D_Y}f^{-1}\mathcal O_Y\\
=T_f\otimes_{f^{-1}\mathcal D_Y}f^{-1}P_Y.
\end{gathered}
\tag{DRP.8}
\]
In source and target coordinates its underlying ordinary complex is DRP.4 with coefficients in \(\mathcal O_X\). Indeed the finite term in degree \(-r\) is
\[
\mathcal O_X\otimes_{f^{-1}\mathcal O_Y}f^{-1}P_Y^{-r},
\tag{DRP.9}
\]
and its differential is right multiplication by the constant target derivatives. The split contraction DRP.5 tensors to these terms. This proves exactness in negative degrees for an arbitrary holomorphic \(f\), including its critical and nonflat points.

The augmentation of DRP.8 is the evaluation map
\[
\begin{gathered}
(a\otimes Q)\otimes g\\
\longmapsto a\,f^*(Qg),\\
T_f\otimes^L_{f^{-1}\mathcal D_Y}f^{-1}\mathcal O_Y\\
\xrightarrow{\sim}\mathcal O_X.
\end{gathered}
\tag{DRP.10}
\]
Balancing over the target operator ring is precisely composition of the operators applied to \(g\). The map is left \(\mathcal D_X\)-linear: apply the source chain-rule action on \(T_f\), and the two terms are the derivative of \(a\) and the chain-rule derivative of \(f^*(Qg)\). Their sum is the usual derivative of the product. Holomorphic functions and vector fields generate the source ring, so this verifies its entire action. The evaluation map is intrinsic. The local exactness just established proves the displayed isomorphism on the whole source, with the left source action retained. It has no dimension shift.

#### DRP.C. A finite perfect projection comparison

Let \(A\) be a sheaf of complex algebras on \(Y\), let \(L\) be a bounded complex of right \(f^{-1}A\)-modules on \(X\), and let \(P\) be a left \(A\)-complex locally represented by a bounded complex of finite free modules. There is a natural map
\[
\begin{gathered}
(Rf_*L)\otimes_A^LP\\
\longrightarrow Rf_*V,\\
V=L\otimes^L_{f^{-1}A}f^{-1}P.
\end{gathered}
\tag{DRP.11}
\]
It is an isomorphism. Its adjoint is tensor reassociation followed by the counit \(f^{-1}Rf_*L\to L\). The inverse-image functor on complex sheaves is exact, so this defines the derived map by flat resolutions and the pushforward adjunction.

To prove the assertion, work on an open where the finite free complex for \(P\) is fixed. For a single term \(A^b\) in any degree, the map is the identity identification with \(b\) copies of \(Rf_*L\), with that degree shift. A finite direct sum is also a finite product; pushforward and its derived functor preserve these products. Filter the bounded free complex by its finite degree truncations. The comparison is natural in their differentials and respects their triangles. The cone of each comparison is zero by induction, because in a morphism of triangles two isomorphisms imply the third by the two long exact cohomology sequences. This proves DRP.11. The local maps are restrictions of the same counit construction and glue. The proof works for noncommutative rings and needs no ordinary flatness of \(f\), no properness and no commutation with infinite sums.

When \(A=\mathcal D_Y\) and \(P=\mathcal O_Y\), DRP.A provides exactly this finite perfect presentation. All derived pushforwards can be computed in sheaves of complex vector spaces while retaining their operator action. The forgetful functor from right \(f^{-1}A\)-modules to complex sheaves preserves injectives: its left adjoint is the free right module obtained by tensoring with \(f^{-1}A\) over \(\mathbf C\), which is exact on stalks. Thus its injective resolutions are also acyclic for the underlying pushforward. This justifies the comparison in the same derived category on both sides.

#### DRP.D. The actual analytic comparison

Apply DRP.11 to
\[
\begin{gathered}
L=M\otimes^L_{\mathcal D_X}T_f,\\
P=\mathcal O_Y.
\end{gathered}
\tag{DRP.12}
\]
Derived tensor reassociation and DRP.10 give
\[
\begin{gathered}
\operatorname{DR}^{\rm r}_Y(f_+M)\\
\simeq Rf_*V\\
\simeq Rf_*\operatorname{DR}^{\rm r}_X(M).
\end{gathered}
\tag{DRP.13}
\]
These are the maps in DRP.2. Reassociation may be checked without assuming a transfer flatness that fails for a general map. Resolve \(M\) by a right operator-flat complex and use the bounded target free resolution DRP.3. The latter computes tensoring on the target side. Its tensor with \(T_f\) has the quasi-isomorphic augmentation DRP.10. Tensoring that augmentation with the flat source resolution preserves it. This is the stated reassociation and cancellation, including all differentials. The same calculation is available with bounded-above flat resolutions; only the finite target Spencer direction is totalized in the comparison.

The actual source transfer Spencer complex has length \(n\), by AP.D. The bound for ordinary direct image on arbitrary complex sheaves is \(2n\), by the complete ordinary-cohomology proof, B1–B3. That proof first lifts ordinary sections through a quotient with c-soft kernel by compatible compact-exhaustion corrections. It then applies the finite c-soft resolution of length \(2n\) on every open subset of \(X\). Each stalk of \(R^qf_*A\) is the filtered colimit of \(H^q(f^{-1}V,A)\), which vanishes for \(q>2n\). Thus this is an ordinary-image bound even without proper support. These bounds put \(L\) and \(f_+M\) in bounded derived categories for bounded \(M\), without coherence or properness assumptions. They also apply to the finite complexes just used. Hence no convergence of an unbounded totalization is being assumed in DRP.13. The maps commute with morphisms of complexes and target open restriction because their counit, action and augmentation do. Identity maps give the identity derived coefficient comparison.

#### DRP.E. Density conversion and the normalization sign

For a left complex \(N\) on \(X\), put \(M=\omega_X\otimes_{\mathcal O_X}N\) with its right action. Its left-module direct image is the density conversion of \(f_+M\), as in AP.G. Write
\[
\begin{gathered}
\operatorname{DR}_X(N)\\
= (\Omega_X^\bullet\otimes_{\mathcal O_X}N)[n].
\end{gathered}
\tag{DRP.14}
\]
Forms precede coefficient degrees in the totalization. Its unshifted differential is the connection differential on forms and \((-1)^k d_N\) on a form of degree \(k\); the shift multiplies the total differential by \((-1)^n\).

Here is an explicit signed identification with \(\operatorname{DR}^{\rm r}_X(M)\). In a source coordinate chart put \(\omega=dx_1\wedge\cdots\wedge dx_n\). For \(u\in N^a\) and \(w=v_1\wedge\cdots\wedge v_r\), write \(\iota_w=\iota_{v_r}\cdots\iota_{v_1}\). Send
\[
\begin{gathered}
(\omega\otimes u)\otimes w\\
\longmapsto (-1)^\sigma(\iota_w\omega)\otimes u,\\
\sigma=ar+nr\\
{}+r(r+1)/2.
\end{gathered}
\tag{DRP.15}
\]
Both sides have degree \(a-r\). Determinant contraction makes the map independent of the coordinate frame; its scalar sign depends only on the displayed degrees.

We check the differentials in a constant coordinate frame, which suffices locally. The right action on \(\omega\otimes u\) is \(-\omega\otimes\nabla_vu\). Put \(u_i=-\nabla_{v_i}u\) and \(\epsilon_i=(-1)^{i-1}\), and let \(w_i\) omit \(v_i\) from \(w\), in its original order. The source tensor Spencer differential in exterior degree \(r\) therefore has the coefficients
\[
(-1)^a\sum_{i=1}^r\epsilon_i u_i\otimes w_i.
\tag{DRP.16}
\]
The exterior-contraction identity is
\[
\begin{gathered}
\alpha\wedge\iota_w\omega=(-1)^{r-1}\\
{}\cdot\sum_{i=1}^r(-1)^{i-1}\alpha(v_i)\iota_{w_i}\omega.
\end{gathered}
\tag{DRP.17}
\]
It follows by moving exterior multiplication through the \(r\) contractions and using that \(\alpha\wedge\omega=0\). If \(c_r=(-1)^{nr+r(r+1)/2}\), then
\[
c_r=(-1)^{n+r}c_{r-1}.
\tag{DRP.18}
\]
The connection part of the shifted forms differential applied to DRP.15 has coefficient
\[
\begin{gathered}
(-1)^{ar+n+r-1}c_r\\
=-(-1)^{ar}c_{r-1}.
\end{gathered}
\tag{DRP.19}
\]
This equals DRP.16 after applying DRP.15 in exterior degree \(r-1\). For the coefficient differential, its forms degree is \(n-r\), so the shifted totalization gives sign \((-1)^r\). The two coefficient-degree signs agree because
\[
\begin{gathered}
(-1)^{(a+1)r}\\
=(-1)^r(-1)^{ar}.
\end{gathered}
\tag{DRP.20}
\]
This proves the full complex identification, including the shifts, for arbitrary bounded left coefficient complexes. The intrinsic Spencer formula and determinant contraction glue the calculation. It is the analytic version of the normalization in de Rham Proposition1.1.

Converting DRP.2 on both sides gives the normalized left-module comparison
\[
\begin{gathered}
\operatorname{DR}_Y(f_+N)\\
\xrightarrow{\sim}Rf_*\operatorname{DR}_X(N).
\end{gathered}
\tag{DRP.21}
\]
There is no additional relative-dimension shift in DRP.21. Source and target dimensions already enter their density and de Rham normalizations.

#### DRP.F. Proper support and every bounded attaching map

Suppose the cohomology of \(N\) is coherent, supported in one closed set proper over \(Y\), and has the generator hypothesis in AP.E. Then its operator image in DRP.21 is bounded coherent by AP. Its characteristic estimate and holonomicity, when the input is holonomic, are GP.2 and GP.J. If the input is regular holonomic, GG.P supplies the generator premise. These are the full proper-support scopes of those earlier proofs.

The comparison DRP.21 is defined on the original complex through its actual derived tensor and pushforward. It therefore retains every attaching morphism and every map between complexes. Applying cohomology truncation triangles is another way to verify its isomorphism property on bounded inputs: it does not replace the constructed map by separately chosen isomorphisms on the cohomology modules. Preservation of regularity, constructibility for arbitrary holonomic analytic modules, full analytic Riemann–Hilbert and compatibility with its duality evaluations require their respective proofs.

#### DRP.G. Compact complex manifolds and finite local systems

Let \(X\) be any compact complex manifold of dimension \(n\), and let \(L\) be a finite-rank complex local system. The ordinary bundle
\[
E=\mathcal O_X\otimes_{\mathbf C_X}L
\tag{DRP.22}
\]
has constant transition matrices in locally constant frames. Differentiating only its holomorphic coefficient defines a flat connection, and its horizontal sections are precisely \(L\). It is coherent and locally free. The right module \(\omega_X\otimes E\) is its own coherent ordinary generator, so AP applies to the map \(a:X\to\mathrm{pt}\). The holomorphic Poincaré contraction of de Rham Proposition1.2 identifies \(\operatorname{DR}_X(E)=L[n]\). DRP.21 gives
\[
\begin{gathered}
a_+E\simeq R\Gamma(X,L)[n],\\
H^q(a_+E)\\
=H^{q+n}(X,L).
\end{gathered}
\tag{DRP.23}
\]
The coherent target ring is \(\mathcal D_{\mathrm{pt}}=\mathbf C\), so AP proves that every one of these groups is finite-dimensional. The fixed real-manifold bound also gives \(H^j(X,L)=0\) for \(j>2n\), and negative sheaf cohomology is zero. Thus the normalized complex in DRP.23 has degrees only in \([-n,n]\). This consequence includes compact nonprojective complex manifolds; its argument uses the proved analytic coherent-image theorem rather than an algebraic model.

**Exercise.** Compute DRP.15 for a module in degree zero when \(n=1\), and for the exterior degrees \(0,1,2\) when \(n=2\). Check the source tensor differential against the shifted de Rham differential.

**Solution.** For \(n=1\), the signs in degrees \(r=0,1\) are \(+,+\). The right action is \(-\nabla_{\partial_x}\), and the shift \([1]\) also changes the forms differential to its negative. For \(n=2\), the signs are \(+,-,-\). In exterior degree two the element \((dx_1\wedge dx_2\otimes u)\otimes(\partial_1\wedge\partial_2)\) maps to \(-u\). Its boundary maps to \(-dx_1\otimes\nabla_1u-dx_2\otimes\nabla_2u\), which is the de Rham differential of \(-u\). In exterior degree one, \(\partial_1\) contracts the volume form to \(dx_2\), whereas \(\partial_2\) contracts it to \(-dx_1\); the sign in DRP.15 gives the same top-form boundary. The coefficient-degree sign in DRP.20 supplies the calculation for complexes in any original degree.

![Finite target Spencer cancellation, the actual de Rham comparison and its density signs](assets/analytic-de-rham-transport.png)

The left panel records the canonical maps DRP.10–DRP.13. The right panel gives the exact signs and form degrees of DRP.15 for coefficient degree zero and dimension two. Drawing source, SVG and degree and sign data specify the figure. Its finite sign samples illustrate the general identities DRP.18–DRP.20.

The finite Spencer construction and operator projection framework occur in [Schapira's freely accessible notes, §§1.4 and2.5](https://perso.imj-prg.fr/pierre-schapira/wp-content/uploads/schapira-pub/lectnotes/Dmod.pdf). DRP.A–DRP.G give the split nonflat cancellation, finite perfect comparison and signed density calculation used here; the exact earlier programme proofs cited above supply the other inputs.


### 5.103. Compact-support duality for coherent analytic complexes

Let \(X\) be a Hausdorff second-countable complex manifold of fixed dimension \(n\), and put \(\omega_X=\Omega_X^n\). For an ordinary coherent complex define
\[
\begin{gathered}
\mathbb S_XF\\
=R\mathcal Hom_{\mathcal O_X}(F,\omega_X)[n].
\end{gathered}
\tag{SDC.1}
\]
For every bounded complex with coherent cohomology supported in a compact subset of \(X\), evaluation and integration give an isomorphism
\[
\begin{gathered}
R\Gamma_c(X,\mathbb S_XF)\\
\xrightarrow{\sim}\\
R\operatorname{Hom}_{\mathbf C}(R\Gamma(X,F),\mathbf C).
\end{gathered}
\tag{SDC.2}
\]
Both sides have finite-dimensional cohomology. In particular the induced pairing between \(H^j(X,F)\) and \(H_c^{-j}(X,\mathbb S_XF)\) is perfect. The support may be singular or nonreduced. No global resolution by holomorphic vector bundles is a hypothesis.

The exact earlier inputs are the actual smooth and distributional Dolbeault resolutions, R.1–R.8, coherent Cartan vanishing, CX.L, coherent section topology, SC.G–SC.I, and compact coherent-image finiteness, GI.K. The extension theorem for bounded functionals is proved in NT.A; the open-mapping and finite-codimension arguments are NP.C and ML.A. The finite c-soft resolutions apply to arbitrary complex sheaves, with the uniform bound \(2n\).

#### SDC.A. The density and its trace

Give \(X\) its complex orientation. Resolve \(\omega_X\) by smooth or distributional Dolbeault forms. In coordinates the full exterior realization is antiholomorphic-first, as in R.2: a coefficient with antiholomorphic multiindex \(I\) represents \(d\bar z_I\wedge dz_1\wedge\cdots\wedge dz_n\). The resolution computes compact cohomology because its terms are c-soft. This is proved in R.5, rather than inferred merely from local exactness.

After the shift \([n]\), a Dolbeault form of antiholomorphic degree \(q\) has cochain degree \(q-n\), and the differential is \((-1)^n\bar\partial\). The trace is
\[
\begin{gathered}
\operatorname{tr}_X:R\Gamma_c(X,\omega_X[n])\\
\longrightarrow\mathbf C,\\
u\longmapsto(2\pi i)^{-n}\int_Xu,\\
q=n,
\end{gathered}
\tag{SDC.3}
\]
and is zero in other degrees. The integral of a compactly supported \(\bar\partial\)-boundary of type \((n,n)\) is zero. On a coordinate box, the integral of each compact derivative is zero by the one-real-variable fundamental theorem. A finite smooth partition on the compact support, constructed in R.5, reduces the manifold assertion to these boxes; the cutoff derivatives cancel because their sum is zero. The same assertion for currents is their defining distributional integration by parts. Thus SDC.3 is a chain map with the indicated shifted differential. Coordinate changes preserve it: their real Jacobian is the positive quantity \(|\det(\partial z'/\partial z)|^2\), and differential forms transform by the same determinant rule. Extension by zero from an open subset preserves the integral. The evaluation on complexes and its derived tensor–Hom signs are the explicit evaluation calculation; the same local formula applies over the sheaf ring \(\mathcal O_X\). Evaluation followed by this trace defines the map SDC.2 before any assertion that it is invertible.

In dimension zero a compact support is a finite subset of a discrete manifold. Coherent stalks are finite-dimensional complex vector spaces. SDC.2 is then the ordinary finite direct-sum duality. In the remaining proof assume \(n>0\).

#### SDC.B. A compact Cauchy contraction on a polydisc

Let \(P\subset\mathbf C^n\) be a polydisc centred at zero. A form or current used here has compact support in a smaller closed product \(K\Subset P\). Choose separate radial cutoffs \(\chi_j\), equal to one on a neighbourhood of the \(j\)-th projection of \(K\), with compact support in the \(j\)-th disc. Their derivatives are supported in annuli separated from that projection. Write
\[
\begin{gathered}
E(z)=\frac1{\pi z},\\
T_ju=E*_ju.
\end{gathered}
\tag{SDC.4}
\]
Extend by zero only in the integrated coordinate. The input has compact support there. The distributional identity \(\bar\partial E=\delta\), its normalization and the partial convolution with arbitrary distributional parameters are proved in R.2. They give
\[
\begin{gathered}
\bar\partial_jT_ju=u,\\
T_j\bar\partial_ju=u.
\end{gathered}
\tag{SDC.5}
\]
The second equality uses the compact support of the input: integration by parts moves its derivative to the convolution kernel. These equalities concern the whole integrated complex line.

Let \(\varepsilon_j\) multiply by \(d\bar z_j\), and let \(\iota_j\) contract that coordinate. On forms with the chosen input support put
\[
\begin{gathered}
h_j=\chi_jT_j\iota_j,\\
p_j=-\varepsilon_j(\bar\partial_j\chi_j)T_j\iota_j.
\end{gathered}
\tag{SDC.6}
\]
All outputs have compact support in a fixed larger product still contained in \(P\). To verify the homotopy, write a form as \(a+\varepsilon_jb\), with neither coefficient containing \(d\bar z_j\). In degree zero for that coordinate, \(h_j\bar\partial_j a=\chi_ja=a\). On its degree-one part,
\[
\begin{gathered}
\bar\partial_j(\chi_jT_jb)\\
=b+(\bar\partial_j\chi_j)T_jb.
\end{gathered}
\tag{SDC.7}
\]
The other coordinate differentials cancel by their exterior anticommutation and commute with the partial kernels. Hence
\[
\bar\partial h_j+h_j\bar\partial=1-p_j.
\tag{SDC.8}
\]
The maps \(p_j\) commute with \(\bar\partial\) and with one another on these inputs. Operations in another coordinate leave the required \(j\)-th input support unchanged. Thus all successive uses of SDC.8 are valid with the original cutoffs. Telescoping gives
\[
\begin{gathered}
H=h_1+p_1h_2+\cdots\\
{}+p_1\cdots p_{n-1}h_n,\\
\Pi=p_1\cdots p_n,\\
\bar\partial H+H\bar\partial=1-\Pi.
\end{gathered}
\tag{SDC.9}
\]
This is a compact-support homotopy. The cutoffs are chosen for the given compact input; no assertion that one fixed cutoff works for every compact subset is needed.

The operator \(\Pi\) kills every form of degree less than \(n\), since each surviving component must contain every \(d\bar z_j\). In top degree its coefficient is
\[
\begin{gathered}
(-1)^n\Bigl(\prod_j\bar\partial_j\chi_j\Bigr)\\
{}\cdot T_1\cdots T_nu.
\end{gathered}
\tag{SDC.10}
\]
On that product of annuli, each kernel variable is separated from the corresponding input support. The full transposed kernel is therefore a smooth compact test near the input support, depending holomorphically on every output variable before multiplying by the cutoffs. It follows, even for a current input, that \(\Pi u\) is a smooth compactly supported top form. Fixed derivatives are obtained by differentiating this separated test. This is the same distributional parameter argument as R.2–R.3, with all integrated variables separated from the support.

#### SDC.C. Scalar compact duality, including surjectivity

For a smooth compact top form define a continuous functional on \(\mathcal O(P)\) by
\[
\Phi_u(g)=(2\pi i)^{-n}\int_Pg\,u.
\tag{SDC.11}
\]
Its continuity follows from a supremum norm on the compact support. If \(u\) is a boundary, integration by parts and \(\bar\partial g=0\) make this functional zero.

Conversely suppose it is zero. In particular all the moments against the monomials \(\zeta^\alpha\) vanish. On the product of cutoff annuli, the kernel of the full Cauchy transform expands as
\[
\begin{gathered}
\prod_{j=1}^n\frac1{z_j-\zeta_j}\\
=\sum_{\alpha\in\mathbf N^n}\zeta^\alpha z^{-\alpha-\mathbf1}.
\end{gathered}
\tag{SDC.12}
\]
The series and every fixed input derivative converge uniformly near the compact input support, because each ratio \(|\zeta_j/z_j|\) is strictly below one there. Its moments therefore show \(T_1\cdots T_nu=0\) on those annuli. Formula SDC.10 gives \(\Pi u=0\), and SDC.9 supplies a compact smooth primitive. For a closed form of degree below \(n\), the same primitive exists immediately because \(\Pi\) is zero in that degree.

Every continuous functional \(\ell\) on \(\mathcal O(P)\) occurs in SDC.11. Its continuity bounds it by a constant times a supremum on one smaller closed polydisc \(K\). NT.A extends it to a functional on smooth functions with that same supremum bound. This extension is a compactly supported distribution; no measure-representation theorem is required. Regard \((2\pi i)^n\) times this extension as a top-degree current \(u\). Its normalized pairing with a holomorphic function is \(\ell\). Apply SDC.9 to this current. The result \(\Pi u\) is smooth and compact by SDC.10. The difference \(u-\Pi u=\bar\partial Hu\) pairs to zero with every holomorphic function by distributional integration by parts. Thus \(\Phi_{\Pi u}=\ell\).

We have proved the actual vector-space identities
\[
\begin{gathered}
H_c^q(P,\omega_P)=0,\\
q\ne n,\\
H_c^n(P,\omega_P)\\
\xrightarrow{\sim}\mathcal O(P)',
\end{gathered}
\tag{SDC.13}
\]
where the prime means continuous linear functionals for the compact-open Fréchet topology. Degrees greater than \(n\) vanish from the finite Dolbeault resolution. Only this continuous dual, as a vector space with its actual functional maps, is needed below. No assertion about an unproved inductive-limit tensor topology is involved.

The identification is natural for open inclusions between polydiscs: extension of a compact form by zero is adjoint to restriction of a holomorphic function, by their identical integral. It is also natural for holomorphic multiplication and finite matrices of holomorphic coefficients.

#### SDC.D. Coherent coefficients on an adapted polydisc

Every coherent ordinary sheaf on a complex \(n\)-manifold has a finite free resolution on a neighbourhood of each point. Here are the exact prerequisites and the neighbourhood passage. The convergent local ring is regular of dimension \(n\), by Theorem3.0c, analytic local algebra. The regular-local finite resolution theorem gives a finite free stalk resolution of length at most \(n\). Its finitely many generators, matrices and relations extend on one neighbourhood. Coherence makes the kernels and cokernels of that finite complex coherent; since its augmented homology vanishes at the chosen stalk, it vanishes on a smaller neighbourhood. This gives a genuine finite free sheaf resolution there. It does not assert a global holomorphic bundle resolution on \(X\).

Choose a smaller polydisc \(U\) in that neighbourhood and write the resolution as \(P^\bullet\to F|_U\), in degrees \([-n,0]\). Its finitely many coherent kernels have no higher cohomology on \(U\), by CX.L. Taking sections is therefore exact. All its maps are continuous for the coherent Fréchet section topology SC.G–SC.I. Their images are the closed kernels of the following maps. The map onto \(\Gamma(U,F)\) is also strict by NP.C. Thus the continuous dual of this finite section resolution is exact except for \(\Gamma(U,F)'\) in degree zero. Explicitly, a functional on a closed subspace extends by NT.A; a functional that annihilates a kernel descends continuously through the open quotient and then extends from the closed image. These two operations prove the asserted exactness.

Compute \(\mathbb S_UF\) by the finite complex \(\mathcal Hom(P^\bullet,\omega_U)[n]\), and use its Dolbeault resolutions for compact sections. Formula SDC.13 on each finite free term identifies its compact cohomology with the continuous dual of the corresponding section term. Holomorphic coefficient matrices transpose under their evaluation-integral pairing. The signs can be checked term by term. If \(P^{-a}\) is the source term, the differential of \(\mathcal Hom(P,\omega)[n]\) on its dual has coefficient \((-1)^{n+a+1}\), whereas the differential of the continuous section dual has coefficient \((-1)^{a+1}\). After compact Dolbeault cohomology, the local pairing on this term includes the factor
\[
(-1)^{na}.
\tag{SDC.14}
\]
It is the Koszul sign for moving a coefficient of degree \(-a\) past the top Dolbeault degree \(n\) before evaluation. The two differential coefficients agree because the extra factor in the next term is \((-1)^{n(a+1)}\): multiplying it by \((-1)^{n+a+1}\) gives \((-1)^{na+a+1}\). This checks the actual Hom, shift and evaluation signs. Finite totalization therefore gives
\[
\begin{gathered}
H_c^a(U,\mathbb S_UF)=0,\\
a\ne0,\\
H_c^0(U,\mathbb S_UF)\\
\xrightarrow{\sim}\Gamma(U,F)'.
\end{gathered}
\tag{SDC.15}
\]
This is the map from actual evaluation and SDC.3. Its construction is independent of the chosen free resolution: comparison maps between free resolutions commute with evaluation, and the usual lifting of a map through a surjection gives their comparison homotopies. Each lift is made in a free term; taking sections of the required coherent kernel surjection is exact by CX.L on the polydisc. Subtracting its boundary leaves a cycle and allows the next lift. Only finitely many terms occur. The same evaluation description proves that SDC.15 commutes with restriction, its transposed compact extension by zero, and coefficient maps. In particular the maps for two different adapted polydiscs agree under these operations.

#### SDC.E. A countable basis resolution

For a coherent sheaf \(F\) on \(X\), choose a countable basis \(\mathcal B\) of coordinate polydiscs on which the preceding finite free resolutions exist. This is possible by second countability: first take a countable cover by such resolution neighbourhoods and then rational smaller polydiscs in their charts. Two basis neighbourhoods containing a point have a smaller basis neighbourhood in their intersection.

For an open \(U\), let \(\mathbf C_U\) mean the constant sheaf on \(U\), extended by zero to \(X\). Form the augmented complex
\[
\begin{gathered}
B^{-p}=\bigoplus_{U_0\subsetneq\cdots\subsetneq U_p}\mathbf C_{U_0},\\
p\ge0,\qquad B^0\longrightarrow\mathbf C_X.
\end{gathered}
\tag{SDC.16}
\]
Its differential is the alternating sum of deletions. Deleting \(U_0\) uses \(\mathbf C_{U_0}\to\mathbf C_{U_1}\); the other deletions keep the same coefficient. The face identities give \(d^2=0\). The augmentation is exact. At a stalk, a cycle involves finitely many chains of basis neighbourhoods of that point. Choose one smaller neighbourhood contained in all their vertices and insert it at the beginning. Its cone boundary gives that cycle as a boundary; a repeated vertex is treated as zero in the normalized complex. The same cone proves exactness in augmentation degree zero.

The complex
\[
\begin{gathered}
E^p=\\
\prod_{U_0\subsetneq\cdots\subsetneq U_p}\Gamma(U_0,F),\\
d_E^p=(-1)^{p+1}(d_B^{-p-1})^*
\end{gathered}
\tag{SDC.17}
\]
computes \(R\Gamma(X,F)\). To verify this, apply the global complex \(\operatorname{Hom}_{\mathbf C_X}(B,I)\) to an injective complex resolving \(F\). Exactness of the augmented basis resolution and injectivity give the ordinary global section complex in one direction. In the other direction the terms compute \(R\Gamma(U_0,F)\), which has no positive cohomology by CX.L. It therefore gives SDC.17. The double complex is in the first quadrant, so each total degree has only finitely many complex-direction entries. There is no infinite diagonal convergence assumption.

Each product in SDC.17 is countable and is a Fréchet space. The section spaces and their restriction maps have the topology proved in SC.G–SC.I; countable products are complete with the product seminorms. Every differential is continuous, since each output component is a finite alternating sum of restrictions.

#### SDC.F. Compact sections of the dual basis resolution

We give the compact-section argument, including the boundedness needed for the unbounded basis direction. A bounded complex of complex sheaves on \(X\) has a bounded c-soft representative. For clarity, the functor
\[
G(A)=\prod_{x\in X}i_{x*}A_x
\tag{SDC.18}
\]
is exact and injective-valued over \(\mathbf C\), as proved in R.5: maps into a point stalk extend by extending a vector-space basis, and products preserve injectivity. Its natural germ map is injective. The successive quotient construction gives a functorial Godement resolution. Its \(2n\)-th cycle is c-soft by D9–D10 of the earlier uniform-manifold proof. Truncate there. Applying these functorial finite resolutions termwise to a bounded complex and totalizing gives the asserted bounded c-soft representative.

For a coherent sheaf \(F\), local finite free resolutions put \(\mathbb S_XF\) in cohomology degrees \([-n,0]\). Take a bounded underlying sheaf representative in those degrees and apply the preceding construction. We obtain a bounded c-soft complex \(Q\) representing it, with terms only in \([-n,2n]\). Its restriction to every open is still c-soft. Thus \(\Gamma_c(U,Q)\) computes \(R\Gamma_c(U,\mathbb S_XF)\).

Tensor SDC.16 with \(Q\) over \(\mathbf C\) and augment to \(Q\). Exactness is stalkwise, since complex vector-space tensoring is exact. The terms are sums of open extensions of c-soft restrictions, hence are c-soft by the earlier compact-extension proof. Taking compact sections preserves this quasi-isomorphism even though the basis direction is unbounded to the left. Indeed, in an exact complex of c-soft sheaves, write its cycles as \(Z^a\). For \(q>0\), the short cycle sequences and c-soft acyclicity give
\[
\begin{gathered}
H_c^q(X,Z^a)\\
\simeq H_c^{q+1}(X,Z^{a-1})\\
\simeq H_c^{q+2n}(X,Z^{a-2n})\\
=0.
\end{gathered}
\tag{SDC.19}
\]
The last equality is the uniform bound for every complex sheaf. Thus every cycle is compact-section acyclic, and the short cycle sequences remain exact on compact sections. Apply this to the augmented cone. It proves
\[
\begin{gathered}
\Gamma_c(X,B\otimes Q)\\
\xrightarrow{\sim}\Gamma_c(X,Q).
\end{gathered}
\tag{SDC.20}
\]
There are only finitely many \(Q\)-degrees on each diagonal of this tensor totalization.

Compact sections of a sheaf coproduct involve only finitely many summands: a finite cover of their compact support supplies one finite index set. Consequently the basis terms on the left of SDC.20 are
\[
\bigoplus_{U_0\subsetneq\cdots\subsetneq U_p}\Gamma_c(U_0,Q).
\tag{SDC.21}
\]
By SDC.15, each column has cohomology only in degree zero, where it is \(\Gamma(U_0,F)'\). These identifications commute with every face map. Functorial good truncations of the column complexes give natural quasi-isomorphisms to those degree-zero spaces. They remain quasi-isomorphisms after totalization: the vertical degrees are bounded in one fixed interval, so a finite filtration checks every total cohomology degree. Thus \(R\Gamma_c(X,\mathbb S_XF)\) is computed by the complex with degree \(-p\) term
\[
\bigoplus_{U_0\subsetneq\cdots\subsetneq U_p}\Gamma(U_0,F)',
\tag{SDC.22}
\]
and the alternating extension-by-zero face maps.

#### SDC.G. The continuous dual and its sign

A continuous functional on a countable product of locally convex spaces depends on only finitely many factors. A bounding basic seminorm involves a finite set of coordinates; the functional vanishes when those coordinates vanish. Hence the direct sum in SDC.22 is exactly the continuous dual of the product \(E^p\).

Use the cochain-dual convention
\[
\begin{gathered}
(E^\vee)^q=(E^{-q})',\\
d_{E^\vee}^q=(-1)^{q+1}(d_E^{-q-1})'.
\end{gathered}
\tag{SDC.23}
\]
At degree \(-p\), SDC.17 makes this differential the negative of the extension face differential in SDC.22. Multiplication by \((-1)^p\) therefore identifies \(E^\vee\) with that compact basis complex: its two boundary coefficients agree because
\[
(-1)^{p-1}(-1)=(-1)^p.
\tag{SDC.24}
\]
This sign is also the evaluation sign. To evaluate a degree-\(p\) element of \(\operatorname{Hom}(B,F)\) on a degree-\(-p\) basis element, exchanging their order contributes \((-1)^{p(-p)}=(-1)^p\). The local identification in SDC.15 is evaluation followed by the same normalized integral. Thus the resulting comparison with \(E^\vee\) is precisely the canonical evaluation-trace map, rather than an independently chosen isomorphism. Its naturality under the basis augmentations also proves independence of the chosen countable basis.

#### SDC.H. Finite cohomology and the ordinary derived dual

Assume now that \(F\) has compact support. GI.K, applied to the point map and the full coherent analytic support, proves that every \(H^j(X,F)\) is finite-dimensional; the real-manifold bound makes their range bounded. This finiteness theorem was proved without coherent duality.

The cycle spaces of the Fréchet complex \(E\) are closed. Each boundary has finite codimension in the next cycle space, so ML.A makes it closed as well. The topology on cohomology is its usual finite-dimensional topology. The continuous dual complex consequently has cohomology
\[
H^{-j}(E^\vee)\xrightarrow{\sim}H^j(E)^*.
\tag{SDC.25}
\]
Here is the full functional argument. A continuous functional on a cycle space which kills its boundaries extends to the ambient term by NT.A. It then represents a dual cocycle. Conversely a functional on a term which vanishes on its cycle space descends through the open Fréchet quotient onto the closed image of its differential. It extends from that closed image to the following term by NT.A, and hence is a dual boundary, with the sign in SDC.23. These two assertions give exactly SDC.25. Every functional on the finite-dimensional cohomology space is continuous, so no algebraic functional is lost.

The inclusion of continuous functionals into all algebraic functionals therefore induces a quasi-isomorphism from \(E^\vee\) to the ordinary vector-space dual complex. Over \(\mathbf C\), that latter complex computes \(R\operatorname{Hom}_{\mathbf C}(R\Gamma(X,F),\mathbf C)\): linear functionals extend from any vector subspace by a basis extension, so the scalar dual functor is exact. Combining this with SDC.20–SDC.24 proves SDC.2 for every coherent sheaf with compact support, including its actual map and all natural coefficient morphisms.

#### SDC.I. Bounded complexes and the coherent cohomology bound

The map SDC.2 is defined for the original derived object by evaluation and trace. Both its functors take cohomology truncation triangles to triangles, with the contravariant reversal on the dual side. A bounded complex has finitely many such triangles. Applying the sheaf result to its coherent cohomology modules and using two out of three proves SDC.2 on the entire bounded coherent complex. This verifies an already constructed natural map, and retains every attaching morphism; no global coherent-term or vector-bundle representative has been assumed. A finite union of the compact cohomology supports is still compact.

The dual complex has coherent cohomology by local finite free resolutions, and has the same compact support. Compact and ordinary derived sections agree on such an object by the earlier proper-support comparison. Thus the theorem also gives ordinary compact-support Serre duality. For a sheaf in degree zero it implies the sharper coherent bound
\[
\begin{gathered}
H^j(X,F)=0,\quad j>n,\\
H^a(X,\mathbb S_XF)=0,\\
a\notin[-n,0].
\end{gathered}
\tag{SDC.26}
\]
Indeed \(\mathbb S_XF\) has cohomology sheaves only in \([-n,0]\), so its compact hypercohomology is zero below \(-n\). SDC.2 gives the first assertion. Its perfect pairings and the fact that ordinary sheaf cohomology of \(F\) is zero in negative degrees give the second. Finiteness of all groups follows from the theorem and GI.K, not from an assumption that a Dolbeault range is closed.

#### SDC.J. A nonreduced point and its residue pairing

On \(\mathbf C_z\), let \(F=\mathcal O/(z^d)\), with \(d\ge1\). Its two-term free resolution and SDC.1 give
\[
\begin{gathered}
0\to\mathcal O\xrightarrow{z^d}\mathcal O\\
\to F\to0,\\
\mathbb S_{\mathbf C}F\simeq\omega_{\mathbf C}/z^d\omega_{\mathbf C}
\end{gathered}
\tag{SDC.27}
\]
in degree zero. With the usual Hom differential and shift, the dual free complex is \([\omega\xrightarrow{z^d}\omega]\) in degrees \(-1,0\). Choose a cutoff \(\chi=1\) near zero. A class represented by \(b(z)\,dz\) has the compact Dolbeault representative with components
\[
\begin{gathered}
\gamma^{0,0}=\chi b\,dz,\\
\gamma^{-1,1}=-\frac{b}{z^d}\bar\partial\chi\wedge dz.
\end{gathered}
\tag{SDC.28}
\]
The second component is smooth because the cutoff derivative is away from zero. The coherent differential of that component cancels the Dolbeault differential of the first. Its remaining differential is zero. Evaluation on a polynomial representative \(a(z)\) and SDC.3 give
\[
\begin{gathered}
\langle a,b\,dz\rangle\\
=\operatorname{Res}_{z=0}\frac{a(z)b(z)}{z^d}\,dz\\
=\operatorname{coeff}_{z^{d-1}}(ab).
\end{gathered}
\tag{SDC.29}
\]
The sign follows directly from Stokes on the cutoff annulus: its inner boundary is clockwise, so the integral of \(\bar\partial\chi\,ab/z^d\wedge dz\) is minus the positive small-circle integral. The minus in SDC.28 cancels it. The positive circle integral of \(z^k dz\) is zero unless \(k=-1\), when it is \(2\pi i\), by its parametrization. This proves the last equality and its normalization.

For the bases \(1,z,\ldots,z^{d-1}\) and \(dz,z\,dz,\ldots,z^{d-1}dz\), the matrix is
\[
\begin{gathered}
\langle z^i,z^jdz\rangle\\
=\begin{cases}1&i+j=d-1,\\0&i+j\ne d-1.\end{cases}
\end{gathered}
\tag{SDC.30}
\]
It is invertible because reversing the basis order makes it the identity. All \(d\) nilpotent coefficient layers are retained.

**Exercise.** For \(X=\mathbf P^1\) and \(F=\mathcal O_X\), determine \(\mathbb S_XF\) and the degree of its global sections complex.

**Solution.** The canonical bundle is \(\mathcal O(-2)\), by the derivative of the coordinate transition \(w=1/z\). Thus \(\mathbb S_XF=\mathcal O(-2)[1]\). The earlier projective-line coefficient calculation in §5.8 gives \(H^0(\mathcal O)=\mathbf C\), \(H^1(\mathcal O)=0\), \(H^0(\mathcal O(-2))=0\) and \(H^1(\mathcal O(-2))=\mathbf C\). Consequently
\[
R\Gamma(\mathbf P^1,\mathbb S_X\mathcal O_X)\simeq\mathbf C[0].
\tag{SDC.31}
\]
The canonical line and the shift both enter this equality.

![Compact Cauchy support collar, oriented residue trace and the complete four-layer nilpotent pairing](assets/compact-coherent-serre.png)

The left panel is one integrated complex coordinate of the product construction in SDC.B–SDC.C; its exact radii and smooth cutoff specify the compact input and the separated derivative annulus. The inner boundary is clockwise. The right panel is the exact matrix of SDC.30 for \(d=4\). It illustrates the coherent, nonreduced pairing; the general coherent-complex theorem is SDC.2. Drawing source, SVG and cutoff, matrix and sign data specify both panels.

The classical integral pairing for holomorphic vector bundles on compact complex manifolds is described in [Demailly's freely accessible manuscript, ChapterVI, §7](https://www-fourier.univ-grenoble-alpes.fr/~demailly/manuscripts/agbook.pdf). SDC.A–SDC.J prove the compact-support coherent-complex statement here through the explicit Cauchy contraction, continuous functionals, basis resolution and evaluation signs. Relative proper analytic coherent duality and proper operator duality require their further comparisons.


### 5.104. Proper coherent analytic duality and its current trace

Let \(f:X\to Y\) be holomorphic between Hausdorff second-countable complex manifolds of fixed dimensions \(n\) and \(m\). Let \(F\) be a bounded ordinary coherent analytic complex whose cohomology sheaves have support in one closed subset \(S\subset X\), with \(f|_S\) proper. Put
\[
\begin{gathered}
D_XF\\
=R\mathcal Hom_{\mathcal O_X}(F,\omega_X)[n],\\
D_YA\\
=R\mathcal Hom_{\mathcal O_Y}(A,\omega_Y)[m].
\end{gathered}
\tag{RCD.1}
\]
There is a canonical evaluation-trace isomorphism
\[
\begin{gathered}
\alpha_f(F):Rf_*D_XF\\
\xrightarrow{\sim}D_Y(Rf_*F).
\end{gathered}
\tag{RCD.2}
\]
The map includes singular and nonreduced supports, arbitrary critical points and every bounded-complex attaching morphism. Neither flatness of \(f\) nor a global bundle resolution of \(F\) is a hypothesis.

The exact inputs are compact-support coherent evaluation duality, SDC.A–SDC.J, ordinary coherent proper images, GI.K, finite perfect projection, DRP.C, and the full current Dolbeault resolution, R.1–R.5. The supported sheaf constructions, their c-soft acyclicity and their composition are proved in the earlier proper-image fibre argument and composition argument. Tensor–Hom signs use the explicit evaluation calculation.

#### RCD.A. Current pushforward with proper support

Set \(r=n-m\); it may be negative. A degree-\(k\) current on the real \(2n\)-manifold \(X\) is a continuous functional on compact test forms of complementary degree. Use the differential convention
\[
(du)(\eta)=(-1)^{k+1}u(d\eta).
\tag{RCD.3}
\]
Suppose its support is proper over an open subset of \(Y\). For a compact test form \(\eta\) there, define
\[
(f_*u)(\eta)=u(\chi f^*\eta),
\tag{RCD.4}
\]
where \(\chi\) is a compact smooth cutoff equal to one near the intersection of the current support with \(f^{-1}\operatorname{supp}\eta\). That intersection is compact by properness of the support. Two such cutoffs have difference zero near it, so RCD.4 is independent of the cutoff. On any fixed compact test support, one cutoff works on a neighbourhood of that support. The usual test seminorm estimates for smooth pullback and multiplication prove continuity. The construction commutes with restriction to smaller target opens.

The resulting current has degree \(k-2r\). Since \(2r\) is even, the two differential signs in RCD.3 agree. Pullback commutes with \(d\); the terms involving \(d\chi\) vanish near the relevant current support. Hence
\[
df_*u=f_*du.
\tag{RCD.5}
\]
This proof works for an arbitrary smooth map between these oriented manifolds; it does not require a submersion or a fibre-volume form.

For a holomorphic map, pullback preserves complex type. A current of type \((n,q)\) therefore pushes to type \((m,q-r)\). Indeed the latter pairs with tests of type \((0,m-q+r)=(0,n-q)\), exactly the type required before pullback. It is zero if that type is outside the possible degrees. Formula RCD.5 then gives the same commutation for \(\bar\partial\). For a holomorphic coefficient \(h\) on the target,
\[
f_*((f^*h)u)=h f_*u.
\tag{RCD.6}
\]
Thus these are maps of target ordinary structure-sheaf modules.

The current Dolbeault resolution of \(\omega_X\) has terms of type \((n,q)\), with \(0\le q\le n\). Its terms are c-soft: they are modules over smooth functions, and a smooth cutoff equal to one near a given compact set extends its neighbourhood section with compact support. Their restrictions to every open have the same property. The exact earlier proper-image fibre proof makes these terms \(f_!\)-acyclic. Consequently applying \(f_!\) to this finite resolution computes the supported derived image.

This calculation agrees with the derived image in ordinary module sheaves. Here is the relevant comparison. An injective ordinary module sheaf is flabby: for an open inclusion \(j:U\hookrightarrow X\), the monomorphism \(j_!\mathcal O_U\to\mathcal O_X\) and injectivity extend any map into it, so sections over \(U\) lift to sections over \(X\). A section along a compact set extends to a neighbourhood by the earlier finite compact-germ gluing argument. Flabbiness then extends it globally, proving c-softness. Thus underlying complex sheaves of injective module sheaves are \(f_!\)-acyclic by the same fibre argument. An injective-resolution comparison identifies the underlying derived supported image with the complex-sheaf calculation. In particular the current calculation defines the stated ordinary-module derived map.

#### RCD.B. The density shift and the trace

In the shifted current resolution of \(\omega_X[n]\), an antiholomorphic degree-\(q\) term has cochain degree \(q-n\) and differential \((-1)^n\bar\partial\). Its image on the target has degree \((q-r)-m=q-n\). Define
\[
\begin{gathered}
\tau_f^q(u)=\\
(-1)^{r(q-n)}(2\pi i)^{-r}f_*u.
\end{gathered}
\tag{RCD.7}
\]
This gives a chain map
\[
\begin{gathered}
\tau_f:Rf_!(\omega_X[n])\\
\longrightarrow\omega_Y[m].
\end{gathered}
\tag{RCD.8}
\]
To verify the differential, the scalar in RCD.7 changes by \((-1)^r\) when \(q\) increases by one. Therefore
\[
(-1)^n(-1)^r=(-1)^m,
\tag{RCD.9}
\]
which is exactly the target shift sign. The current Dolbeault inclusion identifies its target resolution with \(\omega_Y[m]\). The pushforward construction is intrinsic, so the map is independent of coordinates and cutoffs.

The trace for a map to a point is SDC.3: only \(q=n\) survives, its sign is one, and its scalar is \((2\pi i)^{-n}\). The trace of the identity is the identity. The construction also composes with its full signs. If \(g:Y\to Z\) has relative dimension \(s=m-\dim Z\), its input degree after the first pushforward is \(q-r\), and
\[
\begin{gathered}
r(q-n)\\
{}+s(q-r-m)\\
=(r+s)(q-n).
\end{gathered}
\tag{RCD.10}
\]
The powers of \(2\pi i\) multiply, and the two current pushforwards compose by their definition on tests. With the supported derived-composition identification this proves
\[
\tau_{gf}=\tau_g\circ Rg_!\tau_f.
\tag{RCD.11}
\]

We shall use the following compact trace identity. A compact current on \(X\) pushes to a compact current on \(Y\). On its top degree the sign in RCD.7 is one. Testing the pushed current against a cutoff equal to one near its compact support gives the same integral as on \(X\). The normalization is
\[
\begin{gathered}
(2\pi i)^{-m}(2\pi i)^{-r}\\
=(2\pi i)^{-n}.
\end{gathered}
\tag{RCD.12}
\]
Thus, under compact supported composition,
\[
\operatorname{tr}_Y\circ R\Gamma_c(\tau_f)=\operatorname{tr}_X.
\tag{RCD.13}
\]
This equality concerns the actual current maps and normalized integrals.

#### RCD.C. Constructing the coefficient comparison

The local coherent dual \(D_XF\) is bounded coherent and vanishes off \(S\). Indeed an ordinary coherent sheaf has a finite free resolution on a neighbourhood of every point, by the exact local regular-ring and resolution proofs in SDC.D; its length is at most \(n\). Bounded-complex truncations then give a uniform bounded range for its dual. This argument is local and makes no global bundle-resolution assertion.

Put
\[
\begin{gathered}
A=Rf_*F,\\
B=Rf_*D_XF.
\end{gathered}
\tag{RCD.14}
\]
Both are bounded coherent by GI.K and the fixed-dimensional uniform image bound. For every complex whose cohomology support is contained in \(S\), the natural map from supported to ordinary image is an isomorphism: restriction to the full closed proper support gives the proper-support comparison. We use that canonical identification, including for the coefficient tensor product below.

The ordinary derived cup map, evaluation and RCD.8 give a pairing
\[
\begin{gathered}
B\otimes^L A\\
\longrightarrow Rf_*(D_XF\otimes^L F)\\
\simeq Rf_!(D_XF\otimes^L F)\\
\longrightarrow Rf_!(\omega_X[n])\\
\xrightarrow{\tau_f}\omega_Y[m].
\end{gathered}
\tag{RCD.15}
\]
Its tensor–Hom adjoint is \(\alpha_f(F)\) in RCD.2. In particular this map is defined before proving it invertible. It is contravariantly natural in \(F\), because every arrow in RCD.15 is. Truncation triangles are retained by the derived construction.

#### RCD.D. Testing with the full target Koszul complex

Fix \(y\in Y\) and restrict to a coordinate polydisc \(V\) centred there; replace \(X\) by \(f^{-1}V\). Let \(P=k_y\) on \(V\), regarded as a coherent sheaf. Its actual finite free resolution is the Koszul complex on the coordinate functions \(t_1,\ldots,t_m\), in degrees \([-m,0]\). This resolves the residue sheaf because successive coordinate quotients are convergent power-series rings and multiplication by the next coordinate is injective. The usual one-coordinate exact complex, tensored successively with these injective multiplications, proves exactness inductively. In dimension zero it is just the residue sheaf in degree zero.

Write
\[
\begin{gathered}
P^\vee\\
=R\mathcal Hom_{\mathcal O_V}(P,\mathcal O_V),\\
Q=Lf^*P.
\end{gathered}
\tag{RCD.16}
\]
The pullback is the same full finite Koszul complex with differentials \(f^*t_j\) on \(X\). Its terms remain finite free even when those pulled functions are zero divisors or all zero. We never replace it by the underived fibre. The complex
\[
E=F\otimes^L Q
\tag{RCD.17}
\]
is bounded coherent. Its cohomology support is contained in \(S\cap f^{-1}(y)\), which is compact by properness of \(f|_S\). The complex \(A\otimes^L P\) is likewise bounded coherent with compact support at \(y\).

Finite perfect projection gives the actual isomorphism
\[
\begin{gathered}
\pi_P:A\otimes^L P\\
\xrightarrow{\sim}Rf_*(F\otimes^L Q).
\end{gathered}
\tag{RCD.18}
\]
DRP.C proves this directly on a bounded finite free target complex, using finite totalizations and the coefficient matrices of its differentials. Here that complex is precisely the target Koszul resolution. Thus RCD.18 has no flatness or unproved underived base-change premise. The same argument applies to \(P^\vee\).

Finite tensor–Hom evaluation also gives
\[
\begin{gathered}
D_XE\\
\simeq D_XF\otimes^L Lf^*P^\vee,\\
D_Y(A\otimes^L P)\\
\simeq D_YA\otimes^L P^\vee.
\end{gathered}
\tag{RCD.19}
\]
These identities can be checked on the finite Koszul terms. For homogeneous local dual elements \(\alpha,\lambda\) and coefficient elements \(v,u\), the first pairing is
\[
\begin{gathered}
(\alpha\otimes\lambda)(v\otimes u)\\
=(-1)^{|\lambda||v|}\alpha(v)\lambda(u).
\end{gathered}
\tag{RCD.20}
\]
The sign moves \(\lambda\) past \(v\). Applying the Leibniz rule to both sides proves compatibility with their Hom differentials. The same formula gives the target identity. Finite free summands, shifts and finite cones establish the identities for the entire Koszul complex; the derived evaluation calculation ensures independence of the chosen local representatives.

#### RCD.E. The compact pairing proves the comparison on every point test

Apply the derived global Hom functor from \(P\) to RCD.2 over \(V\). Before asserting invertibility, its source is canonically
\[
\begin{gathered}
R\operatorname{Hom}_{\mathcal O_V}(P,B)\\
\simeq R\Gamma(V,B\otimes^L P^\vee)\\
\simeq R\Gamma\bigl(X,\\
D_XF\otimes^L Lf^*P^\vee\bigr)\\
\simeq R\Gamma(X,D_XE)\\
\simeq R\operatorname{Hom}_{\mathbf C}\\
\bigl(R\Gamma(X,E),\mathbf C\bigr).
\end{gathered}
\tag{RCD.21}
\]
The second step is finite perfect projection. The third is RCD.19. The last is the actual compact-support duality SDC.2, since \(E\) has compact support; ordinary and compact sections of its dual agree. No coherent duality relative to \(f\) is used in this calculation.

Its target is canonically
\[
\begin{gathered}
R\operatorname{Hom}_{\mathcal O_V}(P,D_YA)\\
\simeq R\Gamma\bigl(V,\\
D_Y(A\otimes^L P)\bigr)\\
\simeq R\operatorname{Hom}_{\mathbf C}\\
\bigl(R\Gamma(V,A\otimes^L P),\mathbf C\bigr).
\end{gathered}
\tag{RCD.22}
\]
Here the last step is SDC.2 on \(V\), applied to the compactly supported coherent complex \(A\otimes^L P\). RCD.18 identifies its ordinary sections with \(R\Gamma(X,E)\).

Under RCD.21 and RCD.22, the map induced by \(\alpha_f(F)\) is exactly the scalar dual of the sections map RCD.18. We verify that assertion at the level of the pairings defining the maps. Tensoring RCD.15 with the finite Koszul dual evaluates an element \(b\otimes\lambda\) on \(a\otimes u\) as the evaluation of \(b\) on \(a\), multiplied by \(\lambda(u)\) and the sign \((-1)^{|\lambda||a|}\). This is the same interchange sign as RCD.20. The finite perfect projection is defined by pullback, the counit and reassociation, so this evaluation becomes the identical coefficient evaluation on \(X\). Finally the two compact integrations agree by RCD.13. Every operation is a chain map, and the Koszul direction is finite, so the verification retains all its differentials. It proves commutation for the derived maps themselves, not just an equality of dimensions or a pairing on an underived fibre. Consequently
\[
\begin{gathered}
R\operatorname{Hom}_{\mathcal O_V}(k_y,\alpha_f(F))\\
\hbox{is an isomorphism}.
\end{gathered}
\tag{RCD.23}
\]

#### RCD.F. Coherent point tests detect the cone

Let \(C\) be the cone of \(\alpha_f(F)\). It is bounded coherent, by RCD.14 and local finite coherent duality. Formula RCD.23 says its derived Hom from \(k_y\) is zero for every \(y\). The sheaf derived Hom from \(k_y\) has support at that point. Global sections of a sheaf supported at one closed point equal its stalk and are exact: a section on a neighbourhood glues with zero on the complement, and all compatibility conditions are automatic. A finite cohomology truncation therefore identifies the derived global Hom with its stalk. Thus its vanishing implies
\[
\begin{gathered}
R\operatorname{Hom}_{\mathcal O_{Y,y}}(k_y,C_y)\\
=0.
\end{gathered}
\tag{RCD.24}
\]

For completeness, here is the full finite Koszul duality used in this test. Write \(K^{-a}=\bigwedge^a L\), where \(L\) has the coordinate basis and \(d_K=\iota_t\). For a complementary wedge \(v\in\bigwedge^{m-a}L\) and a determinant functional \(\epsilon\), define \(J_a(v\otimes\epsilon)(u)=\epsilon(u\wedge v)\). If \(u'\) has wedge degree \(a+1\), contraction of the zero wedge \(u'\wedge v\) gives \(\iota_tu'\wedge v=(-1)^a u'\wedge\iota_tv\). The Hom differential in degree \(a\) is \((-1)^{a+1}\) times precomposition with \(\iota_t\). Hence \(d_{K^\vee}J_a=-J_{a+1}\iota_t\). The degree-\(a\) map \((-1)^{(m+1)a}J_a\) consequently intertwines this with the shifted differential \((-1)^m\iota_t\) on \(K[-m]\). Each complementary-wedge pairing is nondegenerate. We have proved an isomorphism of the full finite free complexes, not merely their cohomology. Tensoring it with \(C_y\) gives
\[
\begin{gathered}
R\operatorname{Hom}_{\mathcal O_{Y,y}}(k_y,C_y)\\
\simeq\bigl((C_y\otimes^L k_y)\\
{}\otimes\det(T_yY)\bigr)[-m].
\end{gathered}
\tag{RCD.25}
\]
The determinant factor is the dual of the coordinate cotangent determinant. Moving the shift from the Koszul factor to the total tensor complex multiplies an original coefficient of degree \(b\) by \((-1)^{mb}\); the Leibniz rule checks this final shift sign as well.

It follows that \(C_y\otimes^L k_y=0\). If \(C_y\ne0\), let \(j\) be its highest nonzero cohomology degree. Derived tensor is right t-exact; the truncation below \(j\) therefore gives
\[
\begin{gathered}
H^j(C_y\otimes^L k_y)\\
=H^j(C_y)\otimes k_y.
\end{gathered}
\tag{RCD.26}
\]
The right side is nonzero by Nakayama. Here is the elementary finite-module argument: if a finitely generated local module \(M\) satisfies \(M=\mathfrak mM\), express each member of a finite generating list as an \(\mathfrak m\)-linear combination of that list. The adjugate of its matrix \(I-A\) shows that \(\det(I-A)M=0\). This determinant is one modulo \(\mathfrak m\), hence a unit, so \(M=0\). Apply the contrapositive to the nonzero finite module \(H^j(C_y)\). This contradicts the derived-fibre vanishing. Therefore \(C_y=0\) for every point. RCD.2 is an isomorphism.

#### RCD.G. Naturality, composition and the scope of the theorem

The proof applies to the original bounded coherent complex throughout. Its duals, the finite target Koszul tensor, supported images and evaluation are derived constructions. There is no replacement of the original complex by the direct sum of its cohomology sheaves. In particular all connecting maps and every nilpotent coefficient layer are retained.

For composable maps whose indicated support remains proper, the two evaluation-trace comparisons agree. The supported cup maps compose by their counits; the reassociation and evaluation signs are those in RCD.20. Their traces agree by RCD.11. Taking the tensor–Hom adjoint therefore proves the equality of the corresponding comparison morphisms. Identity maps have identity comparison. Restriction to target opens commutes with all constructions.

For a map to a point the theorem is exactly SDC.2 with its canonical density and degree. For a closed point in a complex \(m\)-manifold, RCD.7 sends its scalar current to \((2\pi i)^m\) times the point current of type \((m,m)\). Its compact integral against \((2\pi i)^{-m}\) is one. This gives the same normalized residue and shift as the nonreduced-point calculation SDC.J.

RCD.2 is ordinary coherent analytic duality. A proper differential-operator duality comparison still requires its transfer bimodules, the actual nonflat induced coefficient terms and their uniform truncation argument. Proper-image regularity and the full analytic Riemann–Hilbert equivalence also require their corresponding arguments.

![Current trace degrees and the two compact coherent Koszul point tests](assets/relative-coherent-duality.png)

The upper row gives the exact type, degree, shift sign and normalization of RCD.7. The lower square is RCD.21–RCD.23: both vertical arrows are compact-support duality, and the bottom arrow is the scalar dual of the actual finite perfect projection. The full target Koszul complex retains every nonflat fibre degree. RCD.24–RCD.26 then detect the cone. Drawing source, SVG and degree and composition data specify the diagram. Finite data illustrate the general identities RCD.9–RCD.12.


### 5.105. Analytic operator traces and the natural dual comparison

Let \(f:X\to Y\) be holomorphic between Hausdorff second-countable complex manifolds of fixed dimensions \(n\) and \(m\), and put \(r=n-m\). Use right operator modules. Write
\[
\begin{gathered}
T_f=\mathcal O_X\otimes_{f^{-1}\mathcal O_Y}f^{-1}\mathcal D_Y,\\
f_!^{\mathcal D}M=Rf_!(M\otimes^L_{\mathcal D_X}T_f).
\end{gathered}
\tag{OT.1}
\]
The source action is the full chain-rule action. The supported operator trace is a natural map
\[
\begin{gathered}
t_f:f_!^{\mathcal D}(\omega_X[n])\\
\longrightarrow\omega_Y[m].
\end{gathered}
\tag{OT.2}
\]
We construct it from currents and finite Spencer complexes, together with a trace having two target operator actions. The latter defines the natural comparison
\[
\begin{gathered}
\beta_f(M):f_+\mathbb D_X^{\rm r}M\\
\longrightarrow\mathbb D_Y^{\rm r}(f_+M).
\end{gathered}
\tag{OT.3}
\]
Here \(f_+\) uses ordinary derived image. For OT.3 assume that \(M\) is bounded coherent with cohomology support in a closed subset \(S\) proper over \(Y\), and that \(f_+M\) is bounded coherent. The proper coherence theorem AP supplies the last condition when the cohomology modules have coherent ordinary generators near the fibres, including all regular holonomic inputs. The construction is natural for every operator morphism and for all bounded-complex attaching maps. No invertibility assertion is included in OT.3.

The precise other prerequisites are density side change, the finite right Spencer resolution, Proposition1.1, analytic local operator resolutions, the complete current resolutions R, and proper-support current pushforward, RCD.A. We use the proved c-soft proper-image calculation, coproduct compatibility, and derived evaluation and currying.

#### OT.A. The dualizing kernel and both actions

Put
\[
\mathscr K_X=\omega_X\otimes_{\mathcal O_X}\mathcal D_X.
\tag{OT.4}
\]
It has two commuting right operator actions. Label them \(1\) and \(2\). The first is ordinary right multiplication on the operator factor. For a holomorphic function \(h\) and vector field \(v\), the second is
\[
\begin{gathered}
(\nu\otimes P)\cdot_1Q=\nu\otimes PQ,\\
(\nu\otimes P)\cdot_2h=h\nu\otimes P,\\
(\nu\otimes P)\cdot_2v\\
=-\mathcal L_v\nu\otimes P-\nu\otimes vP.
\end{gathered}
\tag{OT.5}
\]
The second is exactly density side change of the left regular operator action. The Lie derivative Leibniz identity verifies its balancing over \(\mathcal O_X\). Composition on the left commutes with multiplication on the right, so the two actions commute, including when \(Q\) has nonconstant coefficients. Operator relations on functions and vector fields prove the formulas for all finite-order operators.

There is a canonical involution \(J_X\) interchanging these actions. In a nonvanishing volume frame \(\nu\), let \(P^{t,\nu}\) be the formal transpose determined by
\[
\begin{gathered}
h^{t,\nu}=h,\\
v^{t,\nu}=-v-\operatorname{div}_\nu(v),\\
(PQ)^{t,\nu}=Q^{t,\nu}P^{t,\nu}.
\end{gathered}
\tag{OT.6}
\]
The function–field and field–field operator relations are preserved by this anti-map: the first is the ordinary product rule, and the second is the identity \(\operatorname{div}_\nu[v,w]=v(\operatorname{div}_\nu w)-w(\operatorname{div}_\nu v)\). That identity follows by applying \([\mathcal L_v,\mathcal L_w]=\mathcal L_{[v,w]}\) to \(\nu\). Thus OT.6 defines the anti-map on the operator ring. Its square is the identity on functions and vector fields, hence on every operator. Set
\[
J_X(\nu\otimes P)=\nu\otimes P^{t,\nu}.
\tag{OT.7}
\]
If \(\nu'=h\nu\), with \(h\) invertible, then \(P^{t,\nu'}=h^{-1}P^{t,\nu}h\). This follows first for the generators from the divergence formula and then for products. It makes OT.7 agree in the two frames. Directly on the same generators, OT.5 gives
\[
\begin{gathered}
J_X(k\cdot_1Q)=J_X(k)\cdot_2Q,\\
J_X(k\cdot_2Q)=J_X(k)\cdot_1Q,\\
J_X^2=1.
\end{gathered}
\tag{OT.8}
\]
This is a map exchanging two actions, not an ordinary module map that keeps their labels.

The right dual is consequently
\[
\begin{gathered}
\mathbb D_X^{\rm r}M\\
=R\mathcal Hom_{\mathcal D_X^{\rm op},1}
 (M,\mathscr K_X)[n],
\end{gathered}
\tag{OT.9}
\]
with its remaining right action \(2\). This is precisely the usual left operator Hom followed by density side change. The analytic local resolution theorem makes it bounded coherent; for a module in degree zero its degrees lie in \([-n,0]\).

#### OT.B. Transfer of the second action

The sheaf
\[
L_f=\omega_X\otimes_{\mathcal O_X}T_f
\tag{OT.10}
\]
has its density right source action and its commuting ordinary right target action. Explicitly the source field action is \((\nu\otimes t)v=-\mathcal L_v\nu\otimes t-\nu\otimes vt\), where the last \(v\) is the chain-rule left action on \(T_f\). There is a canonical isomorphism
\[
\begin{gathered}
\Psi_f:\mathscr K_X\otimes_{\mathcal D_X,2}T_f\\
\xrightarrow{\sim}L_f,\\
(\nu\otimes P)\otimes t
\longmapsto(\nu\otimes t)P.
\end{gathered}
\tag{OT.11}
\]
The last action on the right is the density right source action. To check balancing for a field \(v\), the image of \(((\nu\otimes P)\cdot_2v)\otimes t\) is
\[
\begin{gathered}
-\bigl(\mathcal L_v\nu\otimes t\bigr)P\\
{}-\bigl((\nu\otimes t)v\bigr)P\\
=(\nu\otimes vt)P.
\end{gathered}
\tag{OT.12}
\]
It is the image of \((\nu\otimes P)\otimes vt\). Function balancing is immediate from OT.5. The formula respects action \(1\) as the right source action on \(L_f\), and respects the right target action.

In a volume frame, \(\mathscr K_X\) is free of rank one for action \(2\): applying an operator to \(\nu\otimes1\) gives its transpose, and transposition is bijective. Formula OT.11 therefore has inverse determined by the same generator. It also proves that the displayed tensor is its derived tensor; no flatness of the holomorphic map is involved.

For bounded coherent \(M\), finite-projective evaluation gives a canonical isomorphism
\[
\begin{gathered}
\delta_f:
\mathbb D_X^{\rm r}M\otimes^L_{\mathcal D_X,2}T_f\\
\xrightarrow{\sim}
R\mathcal Hom_{\mathcal D_X^{\rm op}}
 (M,L_f)[n].
\end{gathered}
\tag{OT.13}
\]
On a finite projective source resolution this is the usual Hom–tensor map followed by OT.11. Each Hom term is a finite direct summand of copies of \(\mathscr K_X\); it is flat for action \(2\), and the termwise comparison is an isomorphism. Finite totalization proves the assertion. Local finite resolutions exist with a uniform length bound; their comparison maps and homotopies commute with the original evaluation, so the local comparisons glue. For a homogeneous tensor \(\varphi\otimes t\) and coefficient \(a\), the Hom–tensor map uses \((-1)^{|t||a|}\varphi(a)\otimes t\). This records the original complex degree; it is not replaced by a degree-zero convention.

#### OT.C. A Spencer resolution with flat connection coefficients

Here is the finite resolution needed for the trace. Let \(V\) be a left operator module flat over the ordinary structure sheaf. Its density side change \(\omega_X\otimes V\) has the right Spencer resolution
\[
\begin{gathered}
S_X(V)^{p-n}\\
=\Omega_X^p\otimes_{\mathcal O_X}V\\
{}\otimes_{\mathcal O_X}\mathcal D_X,\\
0\le p\le n.
\end{gathered}
\tag{OT.14}
\]
Its differential is \((-1)^n\) times the exterior covariant differential for the tensor of the two left connections. Write \(w=v\otimes P\). In coordinates that covariant differential is
\[
\begin{gathered}
d_\nabla(\alpha\otimes w)\\
=d\alpha\otimes w\\
{}+\sum_j dx_j\wedge\alpha\otimes\nabla_jw,\\
\nabla_j(v\otimes P)\\
=(\partial_jv)\otimes P+v\otimes\partial_jP.
\end{gathered}
\tag{OT.15}
\]
Its square is zero by the commuting flat connections. The product rule makes it balanced over ordinary coefficients. Its augmentation is the right action on the top density: \(\nu\otimes v\otimes P\mapsto(\nu\otimes v)P\). The field formula in OT.10 makes the augmented differential zero.

For exactness, filter only the last operator factor by order, shifting the filtration in degree \(p\) by \(p\). The connection of \(V\) does not raise this order. The leading differential is exterior multiplication by \(\sum_jdx_j\xi_j\) on the polynomial coefficient module \(V[\xi_1,\ldots,\xi_n]\). Each new variable acts injectively on any such polynomial module, with quotient the polynomial module in the remaining variables. Successive two-term Koszul cones give exactness outside the top augmented degree. Lowering finite operator order then proves exactness of the original augmented complex. Each term in OT.14 is right operator-flat because \(V\) is ordinary-flat: tensoring it over the right operator factor is the exact ordinary tensor with \(\Omega_X^p\otimes V\). Thus it computes actual derived tensors.

For \(V=\mathcal O_X\), it is the right density resolution. For \(V=T_f\), it resolves \(L_f\). The chain-rule connection is flat and left ordinary PBW makes \(T_f\) locally free, possibly of infinite rank, over \(\mathcal O_X\). Both uses therefore satisfy the proved flatness premise.

#### OT.D. The full current complex and the supported trace

Resolve each form coefficient in OT.14 by its distributional Dolbeault resolution. Tensoring with the ordinary-flat PBW coefficient modules preserves that resolution. Use the full exterior realization of currents; write \(\mathcal C_X^{p,q}\) for currents of type \((p,q)\). For the density resolution, tensor with \(T_f\) and shift by \([n]\). The resulting model for \(\omega_X[n]\otimes^L_{\mathcal D_X}T_f\) has terms
\[
\begin{gathered}
\mathcal C_X^{p,q}
 \otimes_{f^{-1}\mathcal O_Y}f^{-1}\mathcal D_Y,\\
\text{in degree }p+q-2n.
\end{gathered}
\tag{OT.16}
\]
Its full differential, including the target operator connection, is
\[
\begin{gathered}
D_f(u\otimes P)\\
=du\otimes P\\
{}+\sum_j f^*dy_j\wedge u\otimes\partial_{y_j}P.
\end{gathered}
\tag{OT.17}
\]
The two shifts by \(n\) explain the even shift \([2n]\), so the current differential here has no extra shift sign. In holomorphic-first form coordinates the vertical Dolbeault sign is exactly the sign obtained by moving its differential past the holomorphic degree; equivalently it is the ordinary full exterior differential in OT.17. This also fixes the signs in every original bidegree.

The residue normalization can be fixed within these same complexes. Let \(Z\) have dimension \(d\). On the ordinary current resolution of \(\omega_Z[d]\), the differential is \((-1)^d\bar\partial\). The full Spencer–current complex instead has the ordinary full exterior differential. Its augmentation to that shifted density resolution is zero on \(p<d\), and on its top holomorphic column is
\[
\begin{gathered}
A_Z^q(u\otimes P)\\
=\begin{cases}
(-1)^{dq}uP,&p=d,\\
0,&p\ne d.
\end{cases}
\end{gathered}
\tag{OT.18}
\]
Here \(uP\) is the density right action on currents. The horizontal augmented differential is zero by the same right-action product rule as OT.15, now on currents. The vertical chain identity is exactly \((-1)^{d(q+1)}=(-1)^d(-1)^{dq}\). For each fixed current degree, the finite-order polynomial Koszul argument in OT.C proves horizontal exactness with arbitrary current coefficients; it does not require those coefficients to be ordinary-flat. Thus OT.18 is a quasi-isomorphism. Its degree-zero current part is the original positive holomorphic Spencer augmentation. This also specifies the current comparison without choosing an inverse to an unspecified zigzag.

The corresponding ordinary density edge into the top Spencer column is
\[
 i_Z^q(u)=(-1)^{dq}u\otimes1.
\tag{OT.19}
\]
The same vertical identity proves it is a chain map of ordinary coefficient complexes. Its current-degree-zero map is the positive inclusion of top holomorphic forms. If \(u\) has source type \((n,q)\), pushforward has target type \((m,q-r)\). The two edge factors are therefore \((-1)^{nq}\) and \((-1)^{m(q-r)}\). To agree with the exact shifted coherent trace RCD.7, the raw full-current coefficient must be
\[
\begin{gathered}
\kappa_r=(-1)^r(2\pi i)^{-r},\\
(-1)^{r+nq+m(q-r)}\\
=(-1)^{r(q-n)}.
\end{gathered}
\tag{OT.20}
\]
Indeed the exponents differ by \(r(r+1)+2mq\), an even integer. Thus the ordinary density edge followed by the full-current trace and the target augmentation is precisely RCD's normalized coherent trace. The argument works also for negative \(r\).

These terms are c-soft. In target PBW coordinates they are sheaf coproducts of current sheaves. Compact sections of a coproduct involve only finitely many summands, and the earlier compact-extension proof gives c-softness. The same is true on every open. The proved c-soft fibre calculation therefore computes \(Rf_!\) on this finite double complex.

RCD.A defines current pushforward with proper support. It changes the bidegree to \((p-r,q-r)\): a complementary target test has type \((m-(p-r),m-(q-r))=(n-p,n-q)\), exactly the complementary source type, and holomorphic pullback preserves it. Hence the target total degree is still \(p+q-2n\). Define on OT.16
\[
\begin{gathered}
u\otimes P\\
\longmapsto\kappa_r f_*u\otimes P.
\end{gathered}
\tag{OT.21}
\]
This is balanced over the target ordinary structure sheaf and right target operator-linear. Locally a proper support has compact fibres. A section of the sheaf coproduct has a common finite PBW set near such a fibre by a finite cover of that compact set; properness of its support permits shrinking the target open so that the same finite set suffices near its support. Equivalently, this is the proved commutation of supported image with coproducts. Thus OT.21 lands in actual finite-order operators rather than a product or completion.

Pushforward commutes with \(d\). It also satisfies \(f_*(f^*dy_j\wedge u)=dy_j\wedge f_*u\): this follows directly by testing on a compact form, with no sign since the real dimension difference is \(2r\). These two identities prove compatibility with OT.17. The target complex, with shift \([2m]\), is the current model of \(\omega_Y[m]\), with the explicit augmentation OT.18. Composing with that augmentation defines OT.2. It proves right operator-linearity for arbitrary finite operators, not just for functions.

For the identity map, OT.21 is the identity. For a map to a point, only the top current type \((n,n)\) contributes. The raw current map is \((-1)^n(2\pi i)^{-n}\int_X\); its source edge OT.19 contributes the same sign \((-1)^n\), so the resulting coherent period is \((2\pi i)^{-n}\int_X\). In one variable the compact cutoff representative of a simple pole has clockwise inner boundary; Stokes gives its positive residue after this normalization, exactly as in SDC.J. The trace therefore has the prescribed residue normalization.

#### OT.E. Two target factors and the transposed kernel trace

Take two copies \(T_1,T_2\) of \(T_f\), and set
\[
E_f=L_f\otimes^L_{\mathcal D_X}T_2.
\tag{OT.22}
\]
Its two right target actions come respectively from \(T_1\) in \(L_f\) and from \(T_2\). Apply OT.14 with \(V=T_1\), tensor with \(T_2\), shift by \([n]\), and use the same current resolution. Put \(\mathcal A=f^{-1}\mathcal D_Y\) and \(\mathcal R=f^{-1}\mathcal O_Y\). The terms of \(E_f[n]\) are
\[
\begin{gathered}
\mathcal C_X^{p,q}\\
{}\otimes_{\mathcal R}
 (\mathcal A\otimes_{\mathcal R}\mathcal A),\\
\text{in degree }p+q-2n.
\end{gathered}
\tag{OT.23}
\]
The balanced tensor in parentheses uses left ordinary coefficients in both operator factors. Write \(w=P_1\otimes P_2\). Its covariant differential is
\[
\begin{gathered}
D_f(u\otimes w)\\
=du\otimes w\\
{}+\sum_j f^*dy_j\wedge u\otimes\Delta_jw,\\
\Delta_j(P_1\otimes P_2)\\
=\partial_{y_j}P_1\otimes P_2\\
{}+P_1\otimes\partial_{y_j}P_2.
\end{gathered}
\tag{OT.24}
\]
Moving a holomorphic coefficient between the two factors gives its derivative once on either side, so this formula respects their ordinary balancing. Left composition in either factor commutes with its ordinary right target action. The two right actions therefore commute with this entire differential.

The same normalized current pushforward defines a chain map to OT.23 on \(Y\). For \(f=\operatorname{id}_Y\), the complex resolves
\[
E_{\operatorname{id}_Y}\simeq\mathscr K_Y.
\tag{OT.25}
\]
Indeed tensoring \(L_{\operatorname{id}_Y}=\omega_Y\otimes\mathcal D_Y\) with the left regular transfer identifies \((\nu\otimes P_1)\otimes P_2\) with \((\nu\otimes P_1)P_2\), where the last action is density side change. Under this identification, the first target action is action \(1\) of OT.5, and the second is action \(2\). This is precisely the augmentation of the finite coefficient Spencer resolution, so includes its full differential and shift. In current degree \(q\), this augmentation includes the factor \((-1)^{mq}\) of OT.18; its right action on the second operator factor is the density side-change action on the entire coefficient \(u\otimes P_1\). The same horizontal and vertical calculations prove this augmented current comparison.

For the dual comparison, the second target factor is the input action, while the first is the output action. Append the involution \(J_Y\) to put those labels in the order of OT.9. We obtain
\[
\begin{gathered}
\epsilon_f:Rf_!(E_f[n])\\
\longrightarrow\mathscr K_Y[m],\\
T_2\text{ acts by }1,\\
T_1\text{ acts by }2.
\end{gathered}
\tag{OT.26}
\]
Every chain map before the final involution preserves the two existing right actions. OT.8 exchanges them at that last step. All tensor balancing, density divergence and positive-order operator terms are therefore retained.

#### OT.F. The actual natural dual comparison

Put \(C_M=M\otimes^L_{\mathcal D_X}T_2\), and put \(H_M=R\mathcal Hom_{\mathcal D_X^{\rm op}}(M,L_f)\). Tensoring a homogeneous map with the identity of a transfer resolution gives the natural map
\[
\begin{gathered}
H_M\longrightarrow\\
R\mathcal Hom_{f^{-1}\mathcal D_Y^{\rm op},2}\\
(C_M,E_f).
\end{gathered}
\tag{OT.27}
\]
On homogeneous cochains it sends \(\varphi\) to the map \(a\otimes t\mapsto\varphi(a)\otimes t\); the Leibniz rule verifies its Hom differential. This is a map with the remaining right action \(1\) from \(T_1\). Resolutions, their comparison maps and homotopies give the same derived map, by the proved tensor–Hom construction. It is natural for every operator morphism of \(M\).

Apply supported image to OT.27. The derived evaluation and cup map over \(\mathbf C\), followed by currying in the second operator action, gives
\[
\begin{gathered}
Rf_!H_M[n]\\
\longrightarrow
R\mathcal Hom_{\mathcal D_Y^{\rm op},2}\\
\bigl(Rf_!C_M,Rf_!(E_f[n])\bigr)\\
\xrightarrow{\epsilon_f}
R\mathcal Hom_{\mathcal D_Y^{\rm op},1}\\
\bigl(Rf_!C_M,\mathscr K_Y\bigr)[m].
\end{gathered}
\tag{OT.28}
\]
The two Hom subscripts before and after the last arrow record exactly the exchanged labels in OT.26. One may construct the first arrow by evaluating the two supported inputs into \(Rf_!E_f\); no balancing that identifies its two ordinary function actions is made. Those actions on a differential-operator kernel need not be equal. For the derived cup map, use the finite functorial c-soft models from the earlier proper-image proof. Tensoring complex sheaves over \(\mathbf C\) is exact on stalks, so these replacements preserve the source tensor. The underived supported cup maps their sections into the supported sections of that tensor, followed by its natural map to derived supported image. This construction does not require the tensor itself to be c-soft. Functorial models retain all the commuting operator actions; the usual derived comparison maps give the same evaluation. Evaluation is linear in the input target operator action and in the independent output action, so currying gives the displayed morphism of right output modules.

Use OT.13 for its source. The complexes \(H_M\) and \(C_M\) vanish off \(S\), so their supported and ordinary images agree. The target of OT.28 is OT.9 on \(f_+M\), with the output action in its correct label. This defines OT.3. All maps were constructed from the full complexes, so contravariant naturality, homotopies and every finite truncation triangle are respected. For \(f=\operatorname{id}\), OT.11 induces the kernel flip and OT.26 contributes the same flip once more. OT.8 makes their composite the identity; hence \(\beta_{\operatorname{id}}\) is the identity comparison.

#### OT.G. A uniform finite reduction for invertibility

The construction allows a precise finite reduction. Suppose the following additional mathematical assertion holds for the comparison just constructed: \(\beta_f(I_X(F))\) is an isomorphism for every ordinary coherent \(F\) supported in \(S\), where \(I_X(F)=F\otimes_{\mathcal O_X}\mathcal D_X\). Then OT.3 is an isomorphism for every coherent module in the proper-support, near-fibre generator scope of AP, and for every bounded complex whose cohomology modules have that property. Here is a finite proof of that implication.

Fix one target fibre and use AP.E to construct a finite induced segment
\[
\begin{gathered}
C=[I_X(F_s)\to\cdots\\
{}\to I_X(F_0)],\\
\operatorname{Cone}(C\to M)\\
\simeq L[s+1].
\end{gathered}
\tag{OT.29}
\]
The \(F_j\) are ordinary coherent supported in \(S\). The kernel \(L\) is coherent and has the same finite near-fibre generator construction. Only finitely many target shrinkings are involved for a fixed \(s\). AP proves coherent images for all these modules. The two natural comparison functors take the finite term filtration of \(C\) to triangles. The additional induced assertion therefore makes \(\beta_f(C)\) an isomorphism, including every operator differential in this complex.

For a coherent source module \(N\), analytic local operator resolutions give \(\mathbb D_X^{\rm r}N\in D^{[-n,0]}\). The independent AP amplitude is \(f_+:D^{[a,b]}\to D^{[a-n,b+2n]}\). Consequently
\[
\begin{gathered}
f_+\mathbb D_X^{\rm r}M\in D^{[-2n,2n]},\\
\mathbb D_Y^{\rm r}f_+M\in D^{[-2n-m,n]}.
\end{gathered}
\tag{OT.30}
\]
The second bound uses AP coherence of \(f_+M\), then the target analytic dual bound. Both objects for \(M\) vanish outside \([-2n-m,2n]\).

Dualizing the triangle in OT.29 makes its first term \(\mathbb D_X^{\rm r}L[-s-1]\). After image, this first term has no cohomology below \(s+1-2n\). The first term of the corresponding target-dual triangle is \(\mathbb D_Y^{\rm r}(f_+L[s+1])\); it has no cohomology below \(s+1-2n-m\). Choose
\[
s=4n+m+2.
\tag{OT.31}
\]
The two lower bounds are then \(2n+m+3\) and \(2n+3\). For every \(q\le2n\), both degree \(q\) and degree \(q+1\) of these first terms vanish. The long exact sequences show that the maps from the two functors on \(M\) to their values on \(C\) are isomorphisms throughout the full interval \([-2n-m,2n]\). Naturality of OT.3 makes a commutative comparison square. Its other three sides are isomorphisms there, so \(\beta_f(M)\) is too. OT.30 kills all remaining degrees. This also proves coherence of the source image from its finite comparison, without presupposing a separate generator theorem for the dual module.

The argument is local on the target and covers it. For bounded complexes, finite cohomology truncation triangles apply to the originally constructed natural comparison and retain all attaching maps. Thus the stated implication is proved without convergence of an infinite induced resolution.

The extra induced-coefficient assertion in OT.G remains a separate assertion: the natural trace and finite reduction alone do not prove it. In particular they do not prove general analytic proper duality, proper-image regularity or the full analytic Riemann–Hilbert equivalence. The full current trace construction is also described in [Schapira's freely accessible notes, §2.6](https://perso.imj-prg.fr/pierre-schapira/wp-content/uploads/schapira-pub/lectnotes/Dmod.pdf); the two-action kernel, transposition, derived maps and bounds used here are given explicitly above.

![Full current trace degrees, the two target operator actions and the exact finite tail interval](assets/analytic-operator-trace.png)

The first row is the complete Spencer–current trace OT.16–OT.21. The middle row tracks the kernel input and output through the actual transposition OT.26. The last panel is the exact interval example \(n=2,m=1\) of the general reduction OT.29–OT.31; the two first triangle terms start strictly beyond both required adjacent degrees. The induced-coefficient invertibility premise remains explicit. Drawing source, SVG and current-degree, transpose and interval data specify all panels. The finite data illustrate the general arguments; they are not substitutes for those arguments.


### 5.106. Induced coefficients and proper operator duality

Let \(f:X\to Y\) be holomorphic between Hausdorff second-countable complex manifolds of fixed dimensions \(n\) and \(m\). Fix a closed subset \(S\subset X\) such that \(f|_S\) is proper. Use right operator modules, and write
\[
\begin{gathered}
I_Z(F)=F\otimes_{\mathcal O_Z}\mathcal D_Z,\\
D_ZF\\
=R\mathcal Hom_{\mathcal O_Z}(F,\omega_Z)[\dim Z].
\end{gathered}
\tag{ID.1}
\]
The tensor defining induction uses the left ordinary action on the operator factor. The operator dual \(\mathbb D_Z^{\rm r}\) and the comparison \(\beta_f\) are exactly OT.3 and OT.9. The induced image is the actual AP.B–AP.C comparison, including its finite-order union and right action.

We prove that \(\beta_f\) is an isomorphism on every induced ordinary coherent coefficient complex supported in \(S\). It follows that it is an isomorphism on every bounded coherent operator complex whose cohomology modules have coherent ordinary generators near each fibre. In particular this includes every regular holonomic bounded complex with the indicated proper support. The generator condition is the exact one in AP; it is not an assumption that every analytic coherent operator module has a global good filtration.

The additional inputs are the complete ordinary coherent comparison RCD, finite ordinary perfection SDC.D, the two-action kernel OT.A–OT.B, and the current trace OT.D–OT.F. These are proved prerequisites.

#### ID.A. The induced dual with both ordinary actions

Put \(d=\dim Z\) and \(\mathscr K_Z=\omega_Z\otimes_{\mathcal O_Z}\mathcal D_Z\), with the two right actions of OT.5. Its intrinsic involution \(J_Z\) exchanges actions \(1\) and \(2\). There is a canonical isomorphism
\[
\begin{gathered}
\lambda_Z(F):I_Z(D_ZF)\\
\xrightarrow{\sim}\mathbb D_Z^{\rm r}I_Z(F).
\end{gathered}
\tag{ID.2}
\]
Here \(F\) may be any bounded ordinary coherent complex, including a nonflat one. To prove this and specify the map, use the induction–restriction adjunction in the input action:
\[
\begin{gathered}
R\mathcal Hom_{\mathcal D_Z^{\rm op},1}
 (I_Z(F),\mathscr K_Z)\\
\simeq R\mathcal Hom_{\mathcal O_Z,1}(F,\mathscr K_Z).
\end{gathered}
\tag{ID.3}
\]
The subscript on the last Hom uses right multiplication by functions in the operator factor. On a neighbourhood where \(F\) has a finite ordinary free representative \(P^\bullet\), tensoring that representative with the left ordinary-flat operator sheaf gives a finite right operator-free representative of \(I_Z(F)\). Evaluation at \(p\otimes1\) proves ID.3 term by term. Finite totalization proves it for the complex. Ordinary perfection is local, so no global bundle representative is asserted.

Apply \(J_Z\) to the values of the Hom on the right. Its input ordinary action becomes action \(2\), which is multiplication on the density coefficient. Its remaining output action becomes the regular action \(1\). Consequently finite ordinary perfection gives
\[
\begin{gathered}
R\mathcal Hom_{\mathcal O_Z,1}(F,\mathscr K_Z)\\
\xrightarrow{J_Z}
R\mathcal Hom_{\mathcal O_Z,2}(F,\mathscr K_Z)\\
\simeq R\mathcal Hom_{\mathcal O_Z}(F,\omega_Z)
 \otimes_{\mathcal O_Z}\mathcal D_Z.
\end{gathered}
\tag{ID.4}
\]
The last Hom–tensor map is an isomorphism on each finite free term, hence on its finite totalization. In particular it does not commute an infinite product with a tensor. Shift by \([d]\) and invert ID.4 to obtain ID.2.

Explicitly, for a homogeneous ordinary dual map \(g\) on \(P^\bullet\), its value is
\[
\begin{gathered}
\lambda_Z(g\otimes Q)(p\otimes R)\\
=J_Z\bigl(g(p)\otimes Q\bigr)\cdot_1R.
\end{gathered}
\tag{ID.5}
\]
The operators have cochain degree zero. The usual complex Hom evaluation is used for \(g(p)\), retaining its original degree. The equality \(g(ph)=g(p)h\), followed by the exchange of the two function actions under \(J_Z\), verifies input ordinary balancing. Extending by \(R\) proves input operator linearity. The exchange identity OT.8 proves that regular right multiplication by \(Q\) on the source becomes the output action \(2\) of the operator dual. Thus this is an isomorphism of right operator complexes.

All three maps in ID.3–ID.4 are their natural evaluation or adjunction maps. Comparison morphisms between finite free representatives and their homotopies commute with them. The local isomorphisms therefore specify a global derived isomorphism, natural for ordinary coefficient maps and bounded-complex attaching maps. The intrinsic formula for \(J_Z\) proves independence of the volume frame. If \(\nu'=h\nu\), the formal transpose is \(Q^{t,\nu'}=h^{-1}Q^{t,\nu}h\); the derivatives in this conjugation are retained. Neither moving a function through an operator nor dropping the divergence is used.

#### ID.B. Identifying the actual comparison on induced inputs

Let \(F\) be bounded ordinary coherent with support in \(S\), and set
\[
\begin{gathered}
A=Rf_*F,\\
B=Rf_*D_XF.
\end{gathered}
\tag{ID.6}
\]
RCD proves that these are bounded ordinary coherent. The actual AP induced-image isomorphism, extended through the finite cohomology truncations of a bounded coefficient complex, is
\[
\begin{gathered}
\theta_f(F):I_Y(A)\\
\xrightarrow{\sim}f_+I_X(F).
\end{gathered}
\tag{ID.7}
\]
This extension is the same derived projection map as AP.11, not a choice of a decomposition into cohomology sheaves. AP.7 computes transfer on an ordinary-flat representative, AP.8 computes each finite order, and compact proper-support colimits prove the union. Thus ID.7 retains all attaching maps. Apply it also to \(D_XF\).

Define the two identifications
\[
\begin{gathered}
\chi_L:f_+\mathbb D_X^{\rm r}I_X(F)\\
\xrightarrow{(f_+\lambda_X)^{-1}}
 f_+I_X(D_XF)\\
\xrightarrow{\theta_f(D_XF)^{-1}}I_Y(B),\\[2pt]
\chi_R:\mathbb D_Y^{\rm r}(f_+I_X(F))\\
\xrightarrow{\mathbb D_Y^{\rm r}\theta_f(F)}
 \mathbb D_Y^{\rm r}I_Y(A)\\
\xrightarrow{\lambda_Y(A)^{-1}}I_Y(D_YA).
\end{gathered}
\tag{ID.8}
\]
The precise identity is
\[
\begin{gathered}
\chi_R\,\beta_f(I_X(F))\,\chi_L^{-1}\\
=I_Y\bigl(\alpha_f(F)\bigr),
\end{gathered}
\tag{ID.9}
\]
where \(\alpha_f\) is the evaluation-trace map RCD.15. We prove the identity for the maps themselves.

First examine source evaluation on finite ordinary free representatives. A zero-order coefficient of the source is represented by an ordinary dual map \(g\) and the first transfer coefficient \(t_1\). In ID.5 the dual value contains \(J_X\). The transfer map \(\Psi_f\) of OT.11 satisfies
\[
\begin{gathered}
\Psi_f\bigl(J_X(\nu\otimes Q)\otimes t_1\bigr)\\
=\nu\otimes Qt_1.
\end{gathered}
\tag{ID.10}
\]
Indeed \(J_X(\nu\otimes Q)=(\nu\otimes1)\cdot_2Q\), and OT.11 is balanced for that action. Thus after source transfer the corresponding map on an induced input is
\[
\begin{gathered}
p\otimes R\\
\longmapsto\bigl(g(p)\otimes t_1\bigr)\cdot_{\mathcal D_X}R.
\end{gathered}
\tag{ID.11}
\]
The last action is the density source action on \(L_f\). In particular evaluation after cancelling the second induced transfer at \(p\otimes t_2\) is exactly the class of
\[
\bigl(g(p)\otimes t_1\bigr)\otimes t_2
 \quad\hbox{in }E_f[n].
\tag{ID.12}
\]
This is the evaluation used in OT.27, with the original complex Hom degrees. It has no residual source transpose.

Next evaluate the supported derived cup on the zero-order coefficients of the two induced target objects. Under ID.7 these are the zero-order inclusions of \(A\) and \(B\) into their actual PBW coefficient complexes. Their operator factors in ID.12 are both \(1\). The density-current edge of OT.19 has factor \((-1)^{nq}\) on an evaluated current \(u\) of type \((n,q)\). The full two-target current trace then multiplies its pushforward by
\[
\begin{gathered}
\kappa_r=(-1)^r(2\pi i)^{-r},\\
r=n-m.
\end{gathered}
\tag{ID.13}
\]
On the target, before its final transpose, the augmentation of OT.25 contributes \((-1)^{m(q-r)}\). Thus the resulting evaluated density is
\[
\begin{gathered}
(-1)^{nq}\kappa_r(-1)^{m(q-r)}f_*u\\
=(-1)^{r(q-n)}(2\pi i)^{-r}f_*u\\
=\tau_f^q(u).
\end{gathered}
\tag{ID.14}
\]
The parity calculation is OT.20. If the target type is outside its possible range, both current maps are zero, so the identity still applies.

The involution here also acts on the current density resolution. This extension can be described without multiplying distributions together. In a holomorphic volume frame, write one distributional coefficient as \(a\) and one finite holomorphic operator as \(P\). Transpose the expression \(aP\) as \(P^{t,\nu}a\), then expand it with the finite Leibniz rule. This uses derivatives of that single distribution and multiplication only by holomorphic functions. For an operator of order \(k\), all resulting operators have order at most \(k\), so the extension preserves the actual finite PBW union. The product rule proves that it exchanges the same two actions. Applying it twice gives the identity: in each positive derivative order the intervening coefficient is an alternating binomial sum, zero by expanding \((1-1)^j\); the zero-order coefficient is unchanged. Holomorphic derivatives and coefficients commute with the distributional Dolbeault differential. Thus this is an involutive chain map on the full density current resolution. The same volume conjugation as OT.7 makes it intrinsic.

The operator trace ends with \(J_Y\). Target identification by \(\lambda_Y^{-1}\) applies that same involution to the evaluated kernel value. These two involutions cancel by \(J_Y^2=1\). With both operator factors initially \(1\), the remaining operator coefficient is exactly \(1\). Hence this zero-order evaluation is ordinary coefficient evaluation followed by \(\tau_f\), the pairing RCD.15.

Here are the complex and resolution details of this verification. In a finite ordinary free representative, write a homogeneous ordinary dual coefficient as \(\varphi\otimes u\), with \(\varphi\) in the dual free complex and \(u\) in the shifted density resolution. Evaluating on a coefficient \(p\) has sign \((-1)^{|u||p|}\). It moves the density coefficient past \(p\); the Leibniz rule gives exactly the complex Hom differential. Both sides of ID.14 retain that same sign: the density degree is \(q-n\) on the source and \((q-r)-m=q-n\) on the target. All operator factors in this verification have degree zero. Consequently neither the shifts nor an original coefficient differential introduces an unrecorded interchange sign. For coefficients resolved by ordinary-flat complexes, use the same tensor–Hom evaluation maps; the local finite perfect representative proves their comparison is an isomorphism.

The derived cup in this argument is the functorial supported cup used in OT.28 and RCD.15. Coefficient evaluation is a morphism of derived tensors; it is not a multiplication of arbitrary distributional currents. On a finite free representative it can, for example, be computed with a smooth Dolbeault representative in the evaluated coefficient factor and a current representative in the density factor. Multiplication is then by smooth forms. Its comparison with the supported c-soft construction follows by their maps from the same ordinary evaluation and the c-soft acyclicity proved earlier. All equalities above concern those natural maps before taking the supported image, followed by the same current pushforward. They therefore pass through the supported cup and its counits. Changing a local finite free representative commutes with evaluation, transfer, \(J_X\), and the two density edges. Their comparison maps and homotopies give the identical derived pairing. This proves the verification globally without requiring a bundle resolution on the inverse image of an entire target open. The comparison diagrams are formed from the globally defined evaluation, adjunction, and Hom–tensor maps before their quasi-isomorphisms are inverted. The displayed balancing identities are identities of those chain maps on ordinary-flat and functorial supported representatives; local finite perfection is used to establish their invertibility. Thus the global equality follows from this natural diagram, rather than from an inference that local equality alone detects an arbitrary global derived morphism.

Finally, this zero-order calculation determines the entire right-operator map. Induction–restriction adjunction says that a morphism out of \(I_Y(B)\) is determined by its restriction to \(B\otimes1\); the derived assertion is checked on an ordinary-flat representative, as in ID.3. For its value in \(I_Y(D_YA)\), ID.4 and the same adjunction identify evaluation against \(A\otimes1\) with the ordinary Hom having values in \(\omega_Y\otimes\mathcal D_Y\). Ordinary perfection of \(A\) makes that Hom–tensor identification exact. The preceding calculation gives the coefficient \(\alpha_f(F)\otimes1\) in that Hom. It detects the whole operator-valued Hom, rather than only its symbol or a scalar quotient. Right operator linearity supplies all other input and output operators. At every finite order these assertions are identities of finite PBW matrices, including the lower-order derivatives of a holomorphic coefficient; AP.12 gives those matrices. Proper-support colimits pass their compatible identities to the actual operator union. No product completion occurs. This proves ID.9.

#### ID.C. Invertibility and the finite operator segment

RCD.2 proves that \(\alpha_f(F)\) is an isomorphism for every bounded ordinary coherent \(F\) supported in \(S\). Induction is exact, since the operator sheaf is left ordinary-flat. Formulas ID.8–ID.9 therefore prove
\[
\begin{gathered}
\beta_f(I_X(F))\\
\text{is an isomorphism}.
\end{gathered}
\tag{ID.15}
\]
In particular this supplies the premise for ordinary coherent sheaves in OT.G. That argument applies to the same comparison \(\beta_f\), which OT.F constructed naturally for all operator morphisms. There is no requirement that an operator differential between two induced terms come from an ordinary coefficient map.

For precision, let \(M\) be one coherent right operator module supported in \(S\), with a coherent ordinary generator near every proper fibre. AP.E gives, after finitely many target shrinkings, an induced segment \(C\to M\) whose cone is \(L[s+1]\). Choose
\[
\begin{gathered}
s=4n+m+2,\\
\mathcal I=[-2n-m,2n].
\end{gathered}
\tag{ID.16}
\]
Each term of \(C\) satisfies ID.15. The finite term filtration and operator naturality of \(\beta_f\) make \(\beta_f(C)\) an isomorphism, including all of its operator attaching maps. OT.30 puts both actual comparison objects for \(M\) inside \(\mathcal I\). The first terms in the two dualized triangles have lower cohomology bounds
\[
\begin{gathered}
s+1-2n=2n+m+3,\\
s+1-2n-m=2n+3.
\end{gathered}
\tag{ID.17}
\]
Thus their degrees \(q\) and \(q+1\) both vanish for every \(q\in\mathcal I\). The two long exact sequences identify the comparison on \(M\) with the comparison on \(C\) in that whole interval. Naturality gives the commutative square. Its other three sides are isomorphisms, so \(\beta_f(M)\) is an isomorphism there; the independent bounds kill all other degrees. This is a finite argument. It does not assert convergence of an infinite induced resolution or assume the desired image coherence for the dual object. The resulting comparison proves that coherence from its target.

The target neighbourhoods cover \(Y\). For any bounded coherent operator complex \(K\) whose cohomology modules have the same generator and support property, apply its finite cohomology truncation triangles to the original natural map. Induction on their number proves
\[
\begin{gathered}
f_+\mathbb D_X^{\rm r}K\\
\xrightarrow[\beta_f(K)]{\sim}
\mathbb D_Y^{\rm r}(f_+K).
\end{gathered}
\tag{ID.18}
\]
AP gives bounded coherent \(f_+K\); the target dual is bounded coherent by analytic local operator resolutions. Consequently the source in ID.18 is bounded coherent as well. All connecting maps of \(K\) are retained. GG supplies the generator property for all regular holonomic cohomology modules; no projectivity or flatness of \(f\) is used.

#### ID.D. Nilpotent coefficient layers and the scope of the comparison

For \(X=\mathbf C_z\), \(Y=\{*\}\), and \(F=\mathcal O_X/(z^\ell)\) with \(\ell\ge1\), AP.H gives
\[
\begin{gathered}
f_+I_X(F)=\mathbf C[z]/(z^\ell),\\
\mathbb D_X^{\rm r}I_X(F)\simeq I_X(D_XF).
\end{gathered}
\tag{ID.19}
\]
The finite ordinary resolution is multiplication by \(z^\ell\) on two copies of \(\mathcal O_X\). Its ordinary dual has its only cohomology in the shifted degree zero and identifies with \(F\,dz\). The complete residue pairing is the SDC.J pairing:
\[
\begin{gathered}
\langle z^i dz,z^j\rangle\\
=\operatorname{Res}_{z=0}\frac{z^{i+j}dz}{z^\ell}\\
=\begin{cases}1,&i+j=\ell-1,\\0,&i+j\ne\ell-1,\end{cases}\\
0\le i,j<\ell.
\end{gathered}
\tag{ID.20}
\]
The matrix is antidiagonal with all antidiagonal entries equal to one, hence has nonzero determinant \((-1)^{\ell(\ell-1)/2}\). This checks nondegeneracy on every nilpotent layer. Here the source edge has sign \(-1\), the raw operator trace has sign \(-1\), and the target augmentation has sign \(1\); their product is positive. ID.9 says that ID.18 gives precisely this residue pairing.

Formula ID.18 is proper operator duality in the stated near-fibre generator scope, including every regular holonomic bounded input with proper support. It identifies the already defined evaluation-trace comparison. It does not by itself prove preservation of regularity, constructibility, the analytic Riemann–Hilbert equivalence, or the full composition and adjunction identities for operator direct images. Those assertions require their corresponding proofs. The analytic operator trace and induced approach also appear in [Schapira's freely accessible notes, §2.6](https://perso.imj-prg.fr/pierre-schapira/wp-content/uploads/schapira-pub/lectnotes/Dmod.pdf); the exact density, two-action, coefficient and finite-segment arguments used here are given above.

![The induced duality comparison square, its density calibration and all nilpotent residue layers](assets/induced-operator-duality.png)

The square is ID.8–ID.9, with both vertical arrows the displayed canonical identifications. Its top arrow is induced ordinary coherent duality. The middle row shows the complete density sign calculation ID.14. The last panel is ID.20 for four nilpotent layers, with rows indexed by the dual monomials and columns by the original monomials. Each entry is the actual residue. Drawing source, SVG and exact sign and residue data specify these panels. The data illustrate the proofs given above.


### 5.107. Composition of analytic operator images and duality traces

Let \(X\xrightarrow fY\xrightarrow gZ\) be holomorphic maps of Hausdorff second-countable complex manifolds of fixed dimensions \(n,m,k\). Put \(r=n-m\) and \(s=m-k\). Use right operator modules and write
\[
\begin{gathered}
T_f=\mathcal O_X\otimes_{f^{-1}\mathcal O_Y}f^{-1}\mathcal D_Y,\\
f_!^{\mathcal D}M=Rf_!(M\otimes^L_{\mathcal D_X}T_f).
\end{gathered}
\tag{CP.1}
\]
The left source action on a transfer is its full chain-rule action. There are canonical associative isomorphisms on bounded operator complexes
\[
\begin{gathered}
g_!^{\mathcal D}f_!^{\mathcal D}M\\
\xrightarrow{\sim}(gf)_!^{\mathcal D}M.
\end{gathered}
\tag{CP.2}
\]
We prove these for analytic supported image, together with composition of the normalized operator trace, composition of its two-action kernel trace, and compatibility of the canonical dual comparison. No flatness, submersion or projectivity is a hypothesis.

The exact prerequisites are derived transfer composition, inverse images Theorem2.1, the finite left transfer Spencer resolution and amplitude, AP.D, proper-support coproduct compatibility, supported derived composition, the finite c-soft resolution and uniform dimension bound, and the actual operator trace and natural comparison, OT.A–OT.G. The induced comparison and proper duality in the generator scope are ID.A–ID.D. Complex Hom evaluation and currying use the explicit derived evaluation calculation.

#### CP.A. Supported projection for the whole transfer

Let \(N\) be a bounded complex of right \(f^{-1}\mathcal D_Y\)-modules on \(X\). There is a natural comparison
\[
\begin{gathered}
Rf_!N\otimes^L_{\mathcal D_Y}T_g\\
\longrightarrow Rf_!\bigl(N\otimes^L_{f^{-1}\mathcal D_Y}f^{-1}T_g\bigr).
\end{gathered}
\tag{CP.3}
\]
It is an isomorphism. Here is a finite calculation, including the infinite PBW coefficient union.

The finite left Spencer resolution of \(T_g\) has terms
\[
\begin{gathered}
Q^{-j}=\mathcal D_Y\otimes_{\mathcal O_Y}E_j,\\
E_j=\bigwedge^jT_Y\otimes_{\mathcal O_Y}T_g,\\
0\le j\le m.
\end{gathered}
\tag{CP.4}
\]
AP.D proves its exactness, full differential and left operator-flatness. On a coordinate open of \(Y\) mapping into a coordinate open of \(Z\), left ordinary PBW makes \(T_g\) a sheaf coproduct of copies of \(\mathcal O_Y\). Thus each \(E_j\) is a sheaf coproduct of finite free ordinary modules. The termwise projection map for CP.4 is therefore the finite-free projection map followed by the exact coproduct comparison for \(Rf_!\). The earlier c-soft proof establishes that comparison for arbitrary coefficient sheaves. On bounded complexes the finite functorial c-soft resolution, with length bounded by the real dimension \(2n\), gives the same comparison on finite total complexes. No section statement on an arbitrary noncompact inverse image is substituted for this sheaf calculation.

Tensoring with \(Q^{-j}\) over the regular operator factor is exactly ordinary tensor with \(E_j\). This computes both sides of CP.3. All Spencer differentials commute with the projection map, because it is defined from the coefficient tensor map and the supported-image counit. Their terms involving the connection of \(T_g\) and the Lie brackets are retained. The same naturality applies to right multiplication by a target operator. On PBW pieces its functions act by the complete Leibniz matrix AP.12 and its derivatives raise the corresponding exponent; these are finite maps between finite-order pieces. The compatible coproduct comparison retains them on the whole union. Consequently CP.3 is a right \(g^{-1}\mathcal D_Z\)-linear isomorphism. Local finite-coordinate calculations identify this globally defined comparison with an isomorphism, so the isomorphism is independent of charts.

There is no hypothesis that \(T_g\) is flat over \(\mathcal D_Y\). The finite operator-flat resolution CP.4 is what computes its derived tensor. Its ordinary-flat coefficient modules explain the coproduct projection calculation.

The derived transfer identity is
\[
\begin{gathered}
T_f\otimes^L_{f^{-1}\mathcal D_Y}f^{-1}T_g\\
\xrightarrow{\sim}T_{gf}.
\end{gathered}
\tag{CP.5}
\]
Its exact proof in the inverse-images prerequisite cancels the regular middle operator factor. The remaining ordinary derived tensor has no positive Tor, because \(f^{-1}T_g\) is ordinary-flat over \(f^{-1}\mathcal O_Y\). The two chain-rule terms give \(d(gf)\), and right multiplication is unchanged. This proof is valid in holomorphic coordinates, including a nonflat \(\mathcal O_X\) over \(f^{-1}\mathcal O_Y\).

Apply CP.3 to \(N=M\otimes^L_{\mathcal D_X}T_f\), use CP.5, and then use supported sheaf composition. This proves CP.2. The complexes remain bounded: the source transfer has amplitude \([-n,0]\), the next has amplitude \([-m,0]\), and the two supported sheaf images have the proved bounds \(2n\) and \(2m\). No coherence premise is needed for these bounds or for CP.2.

The isomorphism is associative for three maps. Before inversion, the projection comparisons and transfer comparisons are induced by the same reassociation, coefficient tensor map and supported counits. On flat representatives, both parenthesizations send the same homogeneous simple tensor to the same ordered tensor. All transfers themselves have degree zero. For their resolutions, reassociation retains the Koszul signs, and \((a+b)c=ac+bc\) identifies the two interchange signs. The supported-image counits compose by the proved sheaf composition argument. This proves the actual associativity diagram, including arbitrary coefficient differentials.

#### CP.B. Ordinary images and the support hypotheses

Suppose \(S\subset X\) is closed and \((gf)|_S\) is proper. Then \(f|_S\) is proper. Indeed, for compact \(K\subset Y\), its inverse image in \(S\) is a closed subset of the compact set
\[
S\cap(gf)^{-1}(g(K)).
\tag{CP.6}
\]
Here \(g(K)\) is compact. A proper map of these locally compact Hausdorff spaces is closed: near a target point take a compact neighbourhood; the image of the inverse image of that neighbourhood intersected with any closed source set is compact, hence closed, and determines the image on the neighbourhood's interior. Therefore \(f(S)\) is closed. For compact \(L\subset Z\),
\[
\begin{gathered}
f(S)\cap g^{-1}(L)\\
=f\bigl(S\cap(gf)^{-1}(L)\bigr).
\end{gathered}
\tag{CP.7}
\]
is compact. Thus \(g|_{f(S)}\) is proper as well.

For a complex whose cohomology support is in \(S\), restriction to its closed proper support identifies supported and ordinary sheaf image, as proved in OT.F. The transfer tensor does not enlarge this support. The cohomology support of its image is in the closed set \(f(S)\), by restriction to the complement. Consequently CP.2 gives the canonical ordinary operator comparison
\[
g_+f_+M\xrightarrow{\sim}(gf)_+M.
\tag{CP.8}
\]
It holds for every bounded operator complex with the stated support; no image coherence has been used.

#### CP.C. One common current model for the trace

For a complex manifold \(W\) of dimension \(d\), let \(\mathcal C_W^{p,q}\) denote full distributional currents. The one-factor Spencer–current resolution of \(\omega_W[d]\) has terms
\[
\begin{gathered}
\mathcal P_W^{p,q}
=\mathcal C_W^{p,q}\otimes_{\mathcal O_W}\mathcal D_W,\\
\deg=p+q-2d.
\end{gathered}
\tag{CP.9}
\]
Its differential is the full exterior current differential plus the left regular operator connection. Its augmentation to the shifted density current resolution is
\[
\begin{gathered}
A_W^q(u\otimes P)\\
=\begin{cases}(-1)^{dq}uP,&p=d,\\0,&p\ne d.\end{cases}
\end{gathered}
\tag{CP.10}
\]
These are precisely OT.16–OT.19 for the identity. Their differential, horizontal finite-order Koszul exactness and vertical sign prove the augmented quasi-isomorphism. No flatness of a current coefficient is required for that Koszul argument.

For a transfer whose source is \(W\), the terms of CP.9 are acyclic for its derived tensor. More generally, for any ordinary coefficient sheaf \(H\) and any left operator module \(V\) which is ordinary-flat, the ordinary-flat resolution argument AP.B gives
\[
\begin{gathered}
(H\otimes_{\mathcal O_W}\mathcal D_W)
 \otimes^L_{\mathcal D_W}V\\
\simeq H\otimes_{\mathcal O_W}V.
\end{gathered}
\tag{CP.11}
\]
That proof uses a flat resolution of \(H\); its ordinary tensor with \(V\) is exact because \(V\) is ordinary-flat. It does not require \(H\) coherent or flat. Thus transferring CP.9 computes the actual derived transfer of \(\omega_W[d]\).

For \(f\), the transferred terms are
\[
\begin{gathered}
\mathcal C_X^{p,q}\otimes_{f^{-1}\mathcal O_Y}f^{-1}\mathcal D_Y,\\
\deg=p+q-2n.
\end{gathered}
\tag{CP.12}
\]
They are c-soft sheaf coproducts of current sheaves. Applying \(f_!\) computes their derived supported image. Current pushforward defines the unaugmented chain map \(F_f^{(1)}\) from that image to \(\mathcal P_Y\):
\[
\begin{gathered}
u\otimes P\longmapsto\kappa_r f_*u\otimes P,\\
\kappa_r=(-1)^r(2\pi i)^{-r}.
\end{gathered}
\tag{CP.13}
\]
The source current type becomes \((p-r,q-r)\), and
\[
\begin{gathered}
(p-r)+(q-r)-2m\\
=p+q-2n.
\end{gathered}
\tag{CP.14}
\]
Pushforward commutes with \(d\) and with wedging a pulled-back target one-form, as proved on compact tests in OT.D. These identities include the whole target operator connection. They prove the chain-map identity and right operator linearity of CP.13.

There is a direct c-soft computation after the first supported image as well. For a current section with proper support over an open neighbourhood of a compact target set, the part of its support over that compact set is compact. Choose a compact smooth cutoff on the source, equal to one near that part and with support over the given target neighbourhood. Multiplying the current by this cutoff and extending by zero gives a compactly supported source current. Properness of the original section's support permits shrinking the target neighbourhood so that it agrees with the original section there. Thus \(f_!\mathcal C_X^{p,q}\) is c-soft on \(Y\). The same argument applies to a finite PBW set of coefficients, and compact-germ gluing supplies the sheaf section statement. This proves c-softness of the coproduct coefficient terms used below; it does not assert softness of arbitrary underived direct images.

Apply the transfer for \(g\) to CP.13. Write \(\mathcal A=(gf)^{-1}\mathcal D_Z\) and \(\mathcal R=(gf)^{-1}\mathcal O_Z\). By CP.11 and proper-support coproduct compatibility, its source terms become
\[
\begin{gathered}
f_!\mathcal C_X^{p,q}\otimes_{\mathcal O_Y}T_g\\
\simeq f_!\bigl(\mathcal C_X^{p,q}\otimes_{\mathcal R}\mathcal A\bigr).
\end{gathered}
\tag{CP.15}
\]
These are coproducts of the c-soft sheaves just considered. Their next supported image therefore computes the derived image. The target terms are the corresponding current model for \(g\). Under these identifications, the map is still \(\kappa_r f_*\) on currents and the identity on the final operator factor. The transfer identity CP.5 supplies the full composite chain-rule differential.

At this unaugmented level, the two current pushes compose on compact tests. Their constants satisfy
\[
\begin{gathered}
\kappa_s\kappa_r\\
=(-1)^{r+s}(2\pi i)^{-(r+s)}\\
=\kappa_{r+s}.
\end{gathered}
\tag{CP.16}
\]
The two maps therefore agree with \(F_{gf}^{(1)}\) as chain maps on the common model, including every original bidegree and operator factor.

The operator trace \(t_f\) is represented by \(A_YF_f^{(1)}\). In computing \(t_g\), use CP.9 as the resolution of its density input. Its defining roof precomposes \(A_ZF_g^{(1)}\) with the inverse of the derived transfer of \(A_Y\). In the composite with the transferred \(t_f\), that inverse cancels the transferred \(A_Y\). Thus the equality just proved gives
\[
\begin{gathered}
t_{gf}
=t_g\circ g_!^{\mathcal D}(t_f),\\
t_f:f_!^{\mathcal D}(\omega_X[n])\to\omega_Y[m].
\end{gathered}
\tag{CP.17}
\]
This is cancellation of a specified quasi-isomorphism in the derived category. It does not choose the ordinary density edge as an operator-linear inverse. That edge need not be operator-linear. Its sign, and the target augmentation sign, are already included in \(A_Y\); neither is discarded.

#### CP.D. A common model with two independent right actions

Write \(\mathscr K_W=\omega_W\otimes_{\mathcal O_W}\mathcal D_W\), with its regular action \(1\) and density-adjoint action \(2\). A two-factor current model \(\mathcal Q_W\) for \(\mathscr K_W[d]\) has terms
\[
\begin{gathered}
\mathcal Q_W^{p,q}=\mathcal C_W^{p,q}\\
{}\otimes_{\mathcal O_W}(\mathcal D_W\otimes_{\mathcal O_W}\mathcal D_W),\\
\deg=p+q-2d.
\end{gathered}
\tag{CP.18}
\]
Both ordinary tensors in parentheses use left coefficient multiplication. Its two right operator actions are regular right multiplication in the two separate factors. They commute, but their right function actions are not identified. The differential is the full current differential together with the diagonal left connection
\[
\begin{gathered}
\Delta_j(P_1\otimes P_2)\\
=\partial_jP_1\otimes P_2
 +P_1\otimes\partial_jP_2.
\end{gathered}
\tag{CP.19}
\]
The finite Leibniz rule proves ordinary balancing, exactly as in OT.24.

Its augmentation is zero for \(p\ne d\). For \(p=d\), it is
\[
\begin{gathered}
B_W^q(u\otimes P_1\otimes P_2)\\
=(-1)^{dq}J_W(v),\\
v=(u\otimes P_1)\cdot_2P_2.
\end{gathered}
\tag{CP.20}
\]
The action inside \(J_W\) is density side change on the entire first coefficient. The current extension of \(J_W\), with all distributional derivatives and no product of distributions, is proved in ID.B. Formula CP.20 is the augmentation OT.25 followed by that involution. Its horizontal and vertical chain identities and exactness are therefore precisely the proved ones in OT.C–OT.E. It is a quasi-isomorphism. The first regular operator factor of CP.18 becomes action \(2\) on its target; the second becomes action \(1\). These labels are part of the augmentation.

This also identifies the model with the canonical kernel itself, rather than an unspecified automorphism of it. At holomorphic current degree zero in a volume frame,
\[
\begin{gathered}
B_W^0(\nu\otimes P_1\otimes P_2)\\
=\nu\otimes P_1^{t,\nu}P_2.
\end{gathered}
\tag{CP.21}
\]
This is exactly regular action \(1\) by \(P_2\) and adjoint action \(2\) by \(P_1\) on \(\nu\otimes1\). It is the canonical cancellation when both transfers are identities. In particular the identity kernel trace has identity normalization.

Transfer the first regular factor of \(\mathcal Q_X\) by \(T_1=T_f\), and the second by \(T_2=T_f\). These are the output and input transfers respectively. The transferred current model represents
\[
\begin{gathered}
\mathscr K_X[n]\otimes^L_{\mathcal D_X,2}T_1
 \otimes^L_{\mathcal D_X,1}T_2\\
\simeq E_f[n],
\end{gathered}
\tag{CP.22}
\]
where the last identification is OT.11. To justify termwise computation, apply CP.11 to one regular factor, with the other factor in its coefficient module. After the first tensor, reorder the two left ordinary coefficient factors; the term is induced in the remaining regular operator factor. Apply CP.11 again. Both transfers are ordinary-flat by PBW. Thus all terms are acyclic for this successive derived tensor, even though the current coefficient need not be ordinary-flat. There is no unproved flatness over the operator ring or over a tensor product of two operator rings.

The resulting terms, with \(\mathcal B=f^{-1}\mathcal D_Y\) and \(\mathcal R_f=f^{-1}\mathcal O_Y\), are
\[
\begin{gathered}
\mathcal C_X^{p,q}\otimes_{\mathcal R_f}
 (\mathcal B\otimes_{\mathcal R_f}\mathcal B),\\
\deg=p+q-2n.
\end{gathered}
\tag{CP.23}
\]
Their diagonal connection is OT.24. The positive holomorphic augmentation CP.21, together with OT.11, agrees with the Spencer resolution of \(E_f[n]\) used in OT.E. Hence these are the same canonical kernel and maps.

The unaugmented map \(F_f^{(2)}\) pushes its current coefficient by \(\kappa_r f_*\), preserving the two operator factors. It is a chain map by the two pushforward identities in CP.C and the diagonal connection CP.19. The actual kernel trace is
\[
\begin{gathered}
\epsilon_f=B_YF_f^{(2)}:\\
Rf_!(E_f[n])\to\mathscr K_Y[m].
\end{gathered}
\tag{CP.24}
\]
Its input factor \(T_2\) acts by \(1\) and its output factor \(T_1\) acts by \(2\), exactly as in OT.26.

For a bounded complex \(H\) with these two commuting right source actions, denote their parallel transfer and supported image by
\[
\begin{gathered}
U_f(H)\\
=Rf_!\left(\begin{gathered}
H\otimes^L_{\mathcal D_X,2}T_1\\
{}\otimes^L_{\mathcal D_X,1}T_2
\end{gathered}\right).
\end{gathered}
\tag{CP.25}
\]
The two copies of the target actions remain separately labelled. Apply CP.3 to the two transfers, using two finite left Spencer directions. Their finite totalization proves
\[
U_gU_f(H)\xrightarrow{\sim}U_{gf}(H).
\tag{CP.26}
\]
Each projection is linear in the other untouched action. This proves the claim without identifying its two function actions. Reassociation of the two resolution directions has its usual Koszul sign; the common transferred current model CP.23 computes it with degree-zero transfers after the acyclicity just proved.

Now compute \(U_g(\epsilon_f)\) on that model. The coproduct and c-soft arguments of CP.C apply to two PBW indices. After transferring both regular target factors, its map is \(\kappa_r f_*\) on the current and the identity on the two final \(\mathcal D_Z\) factors. The composite unaugmented map is therefore exactly \(F_{gf}^{(2)}\), by CP.16. The middle map \(U_g(B_Y)\) cancels its derived inverse in the roof defining \(\epsilon_g\). This cancels the entire augmentation, including its \((-1)^{mq}\) and its \(J_Y\), with all two-action labels. We obtain
\[
\begin{gathered}
\epsilon_{gf}
=\epsilon_g\circ U_g(\epsilon_f),\\
U_{gf}(\mathscr K_X[n])\to\mathscr K_Z[k].
\end{gathered}
\tag{CP.27}
\]
This proves the full two-action kernel-trace identity. Equality of the raw constants alone would not have supplied this cancellation; the specified common model does.

#### CP.E. Compatibility of the canonical dual comparison

Assume \(M\) is bounded coherent, its cohomology support lies in a closed \(S\) proper over \(Z\) under \(gf\), and both \(f_+M\) and \((gf)_+M\) are bounded coherent. By CP.8 the latter also identifies \(g_+f_+M\). Thus all three canonical OT comparisons in the following equality are defined:
\[
\begin{gathered}
\beta_{gf}(M)\\
=\beta_g(f_+M)\circ g_+(\beta_f(M)).
\end{gathered}
\tag{CP.28}
\]
The source and target use CP.8 and its dual. We verify equality of the derived morphisms, including attaching maps.

The defining construction OT.13 and OT.27–OT.28 consists of Hom–tensor comparison, evaluation, supported cup, the two-action kernel trace, and currying. Hom–tensor is an isomorphism on a finite projective representative of a coherent source; local finite operator perfection proves its canonical derived form. Each uncurried evaluation sends a homogeneous map and transferred input to its value, tensored with the independent output and input transfers. Applying a further transfer reassociates these same factors. The comparison for the composite and the successive comparisons therefore have identical evaluation maps before the final kernel trace.

For clarity, their interchange signs can be checked before any transfer resolutions are contracted. Let the input coefficient have degree \(b\), the first output transfer degree \(a\), the next output transfer degree \(c\), and the first input transfer degree \(e\). Moving the input coefficient past the two output factors and then placing the two transfers of the first map together gives
\[
(-1)^{(a+c)b+ce}.
\tag{CP.29}
\]
The successive construction gives \((-1)^{ab}(-1)^{cb}(-1)^{ce}\), the same sign. These are total cochain degrees, including shifts; untouched factors retain their order. This is precisely the tensor–Hom sign OT.13 and the reassociation sign in CP.26. The ordinary operator factors themselves have degree zero, but the calculation also covers every resolution degree. The Leibniz identity for homogeneous evaluation then gives the same full complex differential on both paths.

Supported cups compose by their counits and the supported-image comparison already used in CP.2. Their natural chain diagrams commute before quasi-isomorphisms are inverted. Consequently, after the identical evaluations, the successive uncurried pairing uses \(\epsilon_gU_g(\epsilon_f)\), whereas the composite pairing uses \(\epsilon_{gf}\). They agree by CP.27. Currying gives CP.28. This argument is an identity of the original evaluation constructions. It does not infer equality of arbitrary derived morphisms merely from agreement on cohomology, on induced objects, or on the endpoints of triangles. Changing flat or projective representatives commutes with these same evaluation and counit diagrams, so the global identity retains all homotopies and every bounded-complex attaching morphism.

There is a useful consequence without assuming regularity of an intermediate image. Let \(M\) have regular holonomic cohomology and the support above. GG and AP, followed by ID.18, apply to \(f\) and \(gf\): their proper fibre supports give coherent ordinary generators for the original input. They prove bounded coherent images and invertibility of \(\beta_f(M)\) and \(\beta_{gf}(M)\). CP.8 makes \(g_+f_+M\) bounded coherent. OT constructs \(\beta_g(f_+M)\) using that coherence, without a generator premise for its construction. Formula CP.28 and the two known isomorphisms prove
\[
\begin{gathered}
\beta_g(f_+M)\\
\text{is an isomorphism}.
\end{gathered}
\tag{CP.30}
\]
No regularity or good filtration of \(f_+M\) was used to obtain this consequence. Proper-image regularity remains its own assertion.

#### CP.F. A nonflat intermediate coefficient and a normalization check

Take \(f:\{0\}\hookrightarrow\mathbf C\) and \(g:\mathbf C\to\{*\}\). The composite is the identity on a point. The ordinary coefficient of \(T_f\) is the residue module at zero, which is not flat over \(\mathcal O_{\mathbf C}\): tensoring multiplication by \(z\) kills its nonzero scalar class. CP.5 still applies, because the second transfer \(T_g=\mathcal O_{\mathbf C}\) is ordinary-flat. Thus this example includes the nonflat first-map coefficient.

Here \(r=-1\), \(s=1\), and
\[
\begin{gathered}
\kappa_{-1}=-2\pi i,\\
\kappa_1=-(2\pi i)^{-1},\\
\kappa_1\kappa_{-1}=1=\kappa_0.
\end{gathered}
\tag{CP.31}
\]
The point current pushes to the current of type \((1,1)\) on the line and integrates back to one. The intermediate density augmentation has sign \(-1\); its inverse carries the same sign, so they cancel in CP.17. The full two-action augmentation and its transpose cancel in CP.27. Hence the composite is the identity trace in both models, consistent with the exact residue normalization ID.20. This is an illustration of the general model calculation, not a replacement for it.

We have proved analytic supported and support-proper ordinary image composition, one- and two-action trace composition, and compatibility of the canonical duality comparison. The proofs apply to the full transfers and actual PBW coefficient unions, including nonflat coefficients and negative relative dimensions. The operator adjunction units and counits, preservation of regularity and the full analytic Riemann–Hilbert equivalence require their corresponding arguments. Analytic transfer and trace constructions also appear in [Schapira's freely accessible notes, §§2.4 and 2.6](https://perso.imj-prg.fr/pierre-schapira/wp-content/uploads/schapira-pub/lectnotes/Dmod.pdf); the exact supported projection, current augmentations, action labels and comparison identities used here are proved above.


![The common two-action current model, exact augmentation cancellation and composite trace degrees](assets/operator-duality-composition.png)

The upper rows show the same ordered output and input transfers through the intermediate current resolution. The middle augmentation is the complete map CP.20, including its density sign and intrinsic transpose; its derived inverse cancels it in CP.27. The lower panel is the nonflat point–line–point example CP.31. Drawing source, SVG and exact degree, constant and cochain-sign data specify these panels. Finite samples illustrate the general chain-map arguments.


## 6. What monodromy misses in the irregular world

Consider $\partial_t e=t^{-2}e$. A horizontal coefficient is $e^{1/t}$; its ordinary monodromy is trivial. The trivial connection also has trivial monodromy, but (1.5) proves these meromorphic connections are not isomorphic. For $t=r e^{i\theta}$,
\[
|e^{1/t}|=\exp(\cos\theta/r).                                  \tag{6.1}
\]
It decays in one half-plane and grows faster than every power in the other. The rays $\theta=\pm\pi/2$ separate those growth regimes. This rank-one example has no nontrivial Stokes gluing matrix: it displays the exponential rate that ordinary monodromy forgets.

For higher rank, formal exponential decompositions can be realized by different analytic gauges on different sectors. Their transition maps carry additional Stokes data. The irregular Riemann–Hilbert problem reconstructs the meromorphic connection from enhanced topological data that retain those asymptotic distinctions. This paragraph is a description of the problem, not a proof of a classification theorem.

Deligne, in letters to Malgrange of the late 1970s, organized boundary directions by the relative exponential growth rates and proposed the associated filtered local system. Sabbah's [*Introduction to Stokes structures*](https://arxiv.org/abs/0912.2762), Lectures 1–2, develops these Stokes-filtered local systems in dimension one.

Scholze's [*Wild Betti sheaves*, §1](https://arxiv.org/abs/2505.24599), describes a universal enlargement of ordinary Betti sheaves with an exponential object and a Fourier equivalence. It recovers a construction of Tamarkin, also expressed through enhanced sheaves, and relates it to the irregular Riemann–Hilbert correspondence. The enlargement changes the coefficient category so that exponential information survives. We cite this construction as a modern framework; we do not identify ordinary finite-rank local systems with arbitrary irregular connections, or claim that this note alone proves the whole irregular correspondence.

## 7. Exercises with complete solutions

**Exercise 13.1 (easy).** Decide regularity at zero and infinity for $x\partial_x-\lambda$, $x^2\partial_x-1$, and $x^3\partial_x-x$.

**Solution.** On the punctured line their coefficients are respectively $\lambda/x$, $x^{-2}$ and $x^{-2}$. At zero the first has at most a simple pole, and the other two have double poles. They are respectively regular, irregular, irregular. Under (4.2) the coefficients are $-\lambda/y$, $-1$, $-1$, so all three are regular at infinity, with the last two nonsingular there. The last two define the same punctured connection; the statement does not identify their coherent extensions at zero.

**Exercise 13.2 (easy).** Show that $\mathcal O.e^x$ is irregular at infinity.

**Solution.** The formal symbol has $\partial_xe=e$. In the coordinate $y=x^{-1}$, $\partial_ye=-y^{-2}e$. Any rank-one lattice is $y^m\mathbb C[[y]]e$, with logarithmic coefficient $m-y^{-1}$. None is stable, so Theorem 1.1 proves irregularity. Analytically the horizontal coefficient $e^{-1/y}$ has exponential growth on sectors about the negative real $y$ axis, so Theorem 3.1 gives the same conclusion.

**Exercise 13.3 (medium).** Prove the second-order Fuchs criterion.

**Solution.** If $ta_1$ and $t^2a_2$ are holomorphic, (3.8) makes $\mathbb C\{t\}v+\mathbb C\{t\}\Theta v$ a stable lattice in the cyclic rank-two module. Conversely regularity makes the dual scalar solutions moderate. Their nonzero Wronskian $W$ is moderate and has scalar monodromy, so $W=t^\rho h$ with $h$ meromorphic. Its logarithmic derivative has at most a simple pole. Abel's identity $W'=-a_1W$ proves the first bound. A monodromy eigen-solution $w=t^\sigma g$, $g$ meromorphic and nonzero, has $w'/w$ of pole order at most one and $w''/w$ of order at most two. Dividing $Pw=0$ by $w$ gives (3.7), proving the second bound. Shrink the disc so the meromorphic unit factors have no zeros. For formal coefficients, use the continuation eigen-solutions and Laurent valuations constructed after (3.8); the same identities prove both bounds.

**Exercise 13.4 (medium).** Exhibit the isomorphism $E_\lambda\simeq E_{\lambda+1}$.

**Solution.** Define $e_\lambda\mapsto x^{-1}e_{\lambda+1}$. Indeed
\[
\partial_x(x^{-1}e_{\lambda+1})
=-x^{-2}e_{\lambda+1}+(\lambda+1)x^{-2}e_{\lambda+1}
=\lambda x^{-2}e_{\lambda+1}.
\]
Its inverse sends $e_{\lambda+1}$ to $x e_\lambda$. Both coefficients are units on $\mathbb G_m$. They need not be units on $\mathbb A^1$, so the formula alone does not identify extensions across zero.

**Exercise 13.5 (hard).** Classify regular singular connections on the punctured formal disc by pairs $(V,T)$ with $T\in\operatorname{GL}(V)$, up to conjugacy. Specify the field.

**Solution.** Work over $\mathbb C((t))$. A stable lattice exists by Theorem 1.1. Shift residue generalized eigenspaces by powers of $t$ until their eigenvalues lie in $\Sigma$; Lemma 2.1 proves stability and termination. Solve recurrence (2.2), whose operators are invertible, to remove all positive powers of $t$ from the matrix. Assign to the resulting constant model $(V,A)$ the automorphism $e^{-2\pi iA}$.

For any prescribed invertible $T$, its generalized eigenblocks and the finite nilpotent logarithm (2.6) construct a normalized $A$. Formula (2.5) proves that all morphisms between normalized constant models are constant intertwiners. Polynomial logarithms on the $T$-blocks prove that these are exactly the $T$-intertwiners. The constructions are therefore inverse equivalences, not merely a list of objects. Changing a basis conjugates $T$, and a conjugacy gives an isomorphism of connections. A formal connection over a general characteristic-zero field does not acquire a complex topological monodromy without additional choices or field extension; the stated classification uses the complex field.

## What this lesson does not prove

Theorem 5.27 constructs an actual coherent reflexive strict coefficient lattice for a finite meromorphic connection on a reduced analytic divisor, under the analytic transverse-disc hypotheses on every local divisor component. It proves normalization at all smooth points, strict membership, saturation and logarithmic stability at all singular points, and the support statement for coherent submodules and actual classes. Arbitrary subquotients are excluded by the explicit counterexample. Section5.89, CK.A–CK.K proves the full characteristic cutoff and exact half-order correspondence. Section5.90, GG.A–GG.G proves initial coherent analytic generators near every compact set. The arbitrary analytic proper-image theorem requires its further argument.

The theorems and propositions in §§5.19–5.26 prove the actual factorial-growth symbol construction, bounded division and units, finite stalk Noetherianity, exact faithful finite-module completion at the dummy zero section, fixed neighbourhood coherent kernels, the general bounded Rees ambient comparison, and actual descent for a supplied coherent generating lattice satisfying \(tL\subset hL\). Section5.89, CK.A–CK.K proves the canonical cutoffs across singular strata and their half-order criterion. Section5.90, GG.A–GG.G supplies initial coherent ordinary generators near every compact set. Full arbitrary analytic proper/support-proper regularity requires its further argument.

Theorem 3.3 proves Fuchs's criterion in every order for formal and convergent coefficients. Theorems 5.0 and 5.1 prove canonical logarithmic extensions, the full global algebraic connection/local-system correspondence, the algebraic curve and divisorial tests, and compactification independence. The proper compactification and GAGA inputs have exact earlier programme proofs; the coherent-extension, transverse-family, meromorphic-comparison and logarithmic-jet steps are proved here.

Theorem 5.2 proves the connection/regular-holonomic identification, including presentations on a dense open and the Serre closure of curve-regular connections. Theorem 5.3 proves analytic regularity of every normalized SNC meromorphic bundle with its exact good filtration and characteristic cycle. Theorem 5.4 proves the natural logarithmic and meromorphic de Rham comparisons by convergent Laurent homotopies, fine resolutions and exact cubical periods. Theorem 5.5 proves the full global algebraic de Rham comparison for every curve-regular algebraic connection, including nonproper and non-quasi-projective varieties, by the finite pole-layer and complex-linear GAGA arguments. Theorem 5.6 proves full generic Gauss–Manin regularity on every smooth algebraic base, with arbitrary curve base change, all coefficients and all fibre multiplicities in the curve lattice. Theorem 5.7 proves duality preservation for every regular holonomic module and bounded complex, with every support allowed. Theorem 5.8 proves relative analytic twist generation in every projective dimension for arbitrary coherent sheaves near a projective fibre; Lemma 5.8.2 also proves coherent higher images for projective-line projections. Theorem 5.9 and Lemma 5.9.1 prove locally projective coherent analytic images, with the initial coherent-generator hypothesis explicit for operator modules. Theorem 5.10 proves coherence, boundedness and analytic transported support of the cotangent candidate. Theorems 5.11–5.12 prove the singular-inclusive isotropic dimension bound and the actual output-filtration comparison, hence analytic projective holonomicity with the initial coherent-generator or global good-filtration hypothesis. Section5.90, GG.A–GG.G proves that initial generator near every compact set. Theorem 5.13 proves the algebraic regular-holonomic SNC extension on every proper SNC pair and every open restriction, all boundary simple factors, and all algebraic curve pullbacks of these meromorphic modules. Theorems 5.14–5.17 prove full algebraic four-map regularity and the bounded-complex criterion using all smooth curve maps or only smooth locally closed curve immersions, including every singular support and every smooth separated finite-type ambient variety. Full analytic proper direct-image regularity remains required. Theorem 5.18 proves external-product and both tensor/dual tensor operations using the now-proved four-map and curve criterion. Section5.90, GG.A–GG.G proves the initial analytic generators and proper-support neighbourhood statement. Full arbitrary analytic proper regularity requires its further argument.

The Stokes description, the irregular Riemann–Hilbert problem and the wild Betti construction are cited perspectives. We prove no formal exponential decomposition, sectorial summation or irregular correspondence theorem. The earlier analytic flat-frame and Cauchy proofs are linked above. The product-growth estimate, removable-pole arguments and category gluing used here have explicit proofs in the lesson.

## References

- Schapira, [*An Introduction to Sheaves on Grothendieck Topologies*](https://webusers.imj-prg.fr/~pierre.schapira/LectNotes/SHV.pdf), freely accessible author notes, Sections 4.2–4.4 and 5.1; the support and dimension reading for Theorem 5.9.
- Norguet, [*Images de faisceaux analytiques cohérents*](https://www.numdam.org/item/SL_1957-1958__1__A7_0.pdf), freely accessible Séminaire Lelong exposition, Section II.1–II.2, printed pp. 11-10–11-14; the projective-line reading for Theorem 5.8.
- Bernstein, [*Algebraic theory of D-modules*](https://www.math.columbia.edu/~khovanov/resources/Bernstein-dmod.pdf), lecture notes, the lecture “Holonomic D-modules with regular singularities”: the definition of regular holonomic modules and Main Theorem B, stability under the four functors and duality and the curve criterion.
- Kashiwara, [*The Riemann–Hilbert problem for holonomic systems*](https://www.kurims.kyoto-u.ac.jp/~kenkyubu/kashiwara/RiemannHilbert.pdf), Publications of the RIMS 20 (1984), 319–365, §8: regularity of proper direct images of analytic regular holonomic modules.
- Deligne, [*Équations différentielles à points singuliers réguliers*](https://publications.ias.edu/sites/default/files/Number9.pdf), freely accessible IAS-hosted full text, 1970, Chapter II: regularity in dimension one and in higher dimension, the curve and compactification criteria, and the global connection correspondence.
- Schnell, [*Algebraic D-modules*](https://www.math.stonybrook.edu/~cschnell/pdf/notes/d-modules.pdf), Lectures 20–21: Fuchs's theorem and regular first-order systems, and the regularity-at-infinity examples and definitions.
- Frenkel, [*Lectures on the Langlands program and conformal field theory*](https://arxiv.org/abs/hep-th/0512172), §3.5, for the Euler D-module and its scalar solutions.
- Sabbah, [*Introduction to Stokes structures*](https://arxiv.org/abs/0912.2762), Lectures 1–2: Stokes-filtered local systems in dimension one, following Deligne and Malgrange.
- Scholze, [*Wild Betti sheaves*](https://arxiv.org/abs/2505.24599), arXiv:2505.24599, §1, for the exponential and Fourier enlargement and its enhanced-sheaf context.