# Normal extension and the complete boundary defect

Let \(t\ge0\) be a product boundary coordinate, \(D_t=-i\partial_t\), and let \(u\) be an interior distribution which has some distribution extension across the boundary. If
\[
 P=D_t^m+\sum_{j<m}a_j(x',t,D')D_t^j,\qquad m\ge1,
\]
has smooth finite matrix coefficients and \(Pu=f\) in the interior, with \(f\) in the dual conormal class \(\mathcal A'\), then \(u\) has exactly one extension \(U\in\mathcal A'\). It satisfies \(t^m(PU-f)=0\) as an ambient supported distribution. Moreover it is the only supported distribution extension satisfying that weighted equation. The tangential differential orders of the \(a_j\) can be arbitrary finite integers. The last section proves the invariant version for differential operators of total order \(m\) with an invertible principal normal coefficient.

Sections 5.1–5.5 of AN03-U034, *Global boundary operators, compressed wave fronts, and normal extension*, provide the normal-extension argument. The proof uses transpose operators, nested compact sets in the bootstrap, a supported distribution and the full boundary defect. Original source credit: Codex, September 2026, CC0. Current source connections and clarifications: AN-04 course-writing task and OpenAI Codex, 5 October 2026, CC0.

The mathematical antecedent is the Hörmander III, 2007 eBook, ISBN 978-3-540-49938-1, Section 18.3; its use in the higher-order Cauchy problem is in Section 23.2. The proof is supplied here in full, using the exact earlier programme proofs below.

## Definitions and exact prerequisites

The conormal amplitude and complete test-space component proves the dyadic Besov norms (C2), smooth coefficient and coordinate bounds at every real index, and conormal completeness. The [dual conormal component](../20261005-conormal-test-foundations/dual-conormal-distributions-and-jets.md) proves the exact tangent generators (GA1)–(GA2), positive-support approximation (GS1)–(GS7), dual pairing and interior injectivity (GD1)–(GD4), actual boundary trace and intrinsic derivatives (GD5)–(GD12). We retain its conormal index convention \(\kappa_k=-k-n/4\) and smooth-test threshold \(m_0=-(n+2)/4\).

All distribution pairings are complex-linear, against the dual density: \(D_t^{\mathrm t}=-D_t\). Hilbert adjoints are explicitly identified when used for Sobolev norm estimates. The [full Hilbert duality proof](../20261005-mixed-halfspace-foundations/halfspace-support-and-duality.md), [Fourier Sobolev embedding](../20261005-restored-first-order-cauchy/spacetime-symbol-composition.md), finite distribution-order and coordinate proof, and [locally finite partitions PS5](../20261004-free-intrinsic-graph/prerequisites/global-principal-symbol.md) are included earlier components. Their component licences and credits remain attached to them. The proof map supplies exact source hashes and unique proof locators.

Here extendibility means existence of an ambient distribution with the stated interior restriction; no special boundary values of that temporary extension are assumed. The symbols \(f,U\) denote supported representatives when used in ambient equations. The compressed wave-front and \(\mathcal N\) conclusions require the further compressed operator calculus and are not asserted by this component.

## 1. The exact local conormal seminorms

Fix \(K=K_0\times[0,c/2]\) compactly contained in the coordinate
chart, and use a slightly larger compact \(K'\). For
\(\kappa_k=-k-n/4\), the conormal topology on supported tests is
generated by
\[
 p_{k,K,L}(\phi)
 =\sum_{|\alpha|\le L}
       \|x_n^{\alpha_n}D^\alpha\phi\|_{B^{\kappa_k}_{2,\infty}}
 \quad(L=0,1,2,\ldots).
 \tag{GE1}
\]
Smooth chart cutoffs are inserted in each norm. Formula (GA2) proves
that these weighted normal derivatives span exactly the same
filtered family as all words in \(D'\) and \(x_nD_n\). No lower term
of the triangular polynomial relation is dropped. Tangential
differential operators \(a(x,D')\), of *any* fixed finite order,
and their formal transposes map each \(\mathcal A^k_K\) continuously
into \(\mathcal A^k_{K'}\). To verify this, expand a tangent word
through \(a\): every resulting term is another finite tangent word
with smooth coefficients, including the derivatives of those
coefficients. Multiplication by a compact smooth coefficient is
bounded on each \(B^{\kappa_k}_{2,\infty}\) by (C2)–(C3).
The number of seminorms needed depends on the actual tangential
order; no relation to the normal order is assumed.

For later use, the dyadic Fourier estimates give, for every real
\(s\),
\[
 \|w\|_{B^s_{2,\infty}}
 \le C_s\left(\|w\|_{B^{s-1}_{2,\infty}}
       +\sum_{j=1}^{n}\|D_jw\|_{B^{s-1}_{2,\infty}}\right).
 \tag{GE2}
\]
On the low-frequency block the first norm controls the left side.
On the \(l\)-th high block,
\(2^l\asymp(\sum_j|\xi_j|^2)^{1/2}\);
multiply its \(L^2\) norm by \(2^{l(s-1)}\), use Plancherel for
each \(D_j\), and take the supremum over \(l\). This proves (GE2)
without a boundary norm convention or an omitted tangential term.

## 2. The normal primitive gains one conormal order

For \(\phi\in C_c^\infty(X^\circ)\) with support in \(K\), put
\[
 \psi(x',t)=i\int_0^t\phi(x',s)\,ds,
 \qquad D_n\psi=\phi,
 \quad \psi=0\text{ below the normal support of }\phi.
 \tag{GE3}
\]
Choose a fixed normal cutoff \(\chi(t)\) equal to one on
\([0,c/2]\) and smoothly supported in \((-c,c)\), with positive-side support in \([0,c)\). Then \(\chi\psi\) has
compact support in the interior for each such \(\phi\), although
the distance of that support from the boundary need not be uniform.

We first prove that the localized Volterra operator
\(J:\phi\mapsto\chi\psi\) is bounded on
\(B^s_{2,\infty}\) for every real \(s\), with fixed chart cutoffs.
For this ambient-space estimate, choose a smooth input cutoff equal
to one on the original \(K\), extending a short distance across
\(t=0\), and write the integral as
\(i\chi(t)\int_{-\infty}^{t}\theta(s)\phi(x',s)\,ds\).
For the original interior-supported \(\phi\) this is exactly (GE3)
after multiplication by \(\chi\). The smooth cutoff is necessary in
the negative-order dual estimate; a sharp input cutoff at zero would
create an unjustified boundary multiplier.
For \(s=0\), Cauchy–Schwarz on the finite normal interval gives
\(\|\psi\|_{L^2}\le c\|\phi\|_{L^2}\). For a nonnegative
integer \(p\), differentiate \(\chi\psi\) at most \(p\) times.
Tangential derivatives commute with the integral; each positive
normal derivative of \(\psi\) is the corresponding derivative of
\(\phi\) of one lower order; derivatives of \(\chi\) multiply
the same expressions. Thus \(J:H^p\to H^p\) is bounded.
For real cutoffs the **Hilbert adjoint** has the reversed integral
\(g(s)\mapsto-i\theta(s)\int_s^c\chi(t)g(t)\,dt\).
The complex-linear distribution transpose has \(+i\) instead, without
complex conjugation. This distinction fixes the sign convention without
changing either norm estimate. Both reversed integrals have the same
integer derivative bounds, including every derivative of \(\theta\).
Hilbert duality therefore gives \(J:H^{-p}\to H^{-p}\).
For a real \(s\), choose integers
\(p<s<q\). The dyadic operator matrix estimate between the two
Sobolev endpoints is the one proved from (C2)–(C3): after weighting
by \(2^{ls-js}\), it decays geometrically in \(|l-j|\).
Summation proves the claimed \(B^s_{2,\infty}\) bound, including
negative indices. The constants depend on the fixed cutoffs, not on
how near the input support is to \(t=0\).

Now set \(s=\kappa_k=-k-n/4\). We prove the stronger precise
primitive estimate
\[
 p_{k,K',L}(\chi\psi)
 \le C_{k,K,L}\,p_{k+1,K,L'}(\phi)
 \quad\text{for a finite }L'=L'(L),
 \tag{GE4}
\]
since \(\kappa_{k+1}=s-1\). For a tangent word with no normal
derivative, \(D'^\beta(\chi\psi)=J(D'^\beta\phi)\), so the
Volterra bound controls its \(B^{s-1}\) norm by the corresponding
\(\phi\) seminorm. Apply (GE2) to gain the missing one degree.
Its tangential derivatives are \(J(D'^{\beta+e_j}\phi)\), and its
normal derivative is
\(\chi D'^\beta\phi+(D_n\chi)D'^\beta\psi\).
The same Volterra estimate controls all these
\(B^{s-1}\) norms by a finite set of (GE1) seminorms for \(\phi\).

For a word with positive normal order \(a\ge1\), preserve its entire
normal weight:
\[
 x_n^aD_n^aD'^\beta\psi
   =x_n^aD_n^{a-1}D'^\beta\phi.
 \tag{GE5}
\]
The right side has a factor \(x_n\) times a tangent word in \(\phi\),
so its \(B^{s-1}\) norm is controlled. Its normal derivative is
\[
 D_n(x_n^aD_n^aD'^\beta\psi)
  =x_n^aD_n^aD'^\beta\phi
      -ia\,x_n^{a-1}D_n^{a-1}D'^\beta\phi.
 \tag{GE6}
\]
This is the full product rule, including its \(-ia\) term; when
\(a=0\) that term is absent and the earlier calculation applies.
The tangential derivative is
\(x_n^aD_n^{a-1}D'^{\beta+e_j}\phi\), again a smooth
multiple of a tangent word. A cutoff derivative contributes
\((D_n\chi)x_n^aD_n^{a-1}D'^\beta\phi\), which has the same
bound. Applying (GE2) proves the \(B^s\) estimate for each word,
and summing finitely many words proves (GE4).

The gain just proved also holds with any fixed smooth normal output cutoff
\(\eta\) in place of \(\chi\); its being one on the input support was
needed only for the later equation, not for this bound. Indeed expand
\[
 t^aD_t^a(\eta\psi)
 =\sum_{b=0}^a\binom ab\,t^a(D_t^{a-b}\eta)D_t^b\psi .
\]
For \(b\ge1\), use (GE5)–(GE6) at order \(b\), with the smooth
additional coefficient \(t^{a-b}D_t^{a-b}\eta\). For \(b=0\),
use the zero-normal-order Volterra estimate with output cutoff
\(t^aD_t^a\eta\). Smooth coefficients preserve the same Besov index.
This treats every derivative on the cutoff, including the case
\(\eta=D_t\chi\) needed in (GE9).

## 3. First-order normal equations at every test order

Let \(A_0(x,D')\) be a finite matrix of tangential differential
operators with smooth coefficients of arbitrary finite orders, and
let the extendible interior vector distribution \(u\) solve
\[
 (D_n+A_0(x,D'))u=f\quad\text{in }X^\circ,
 \qquad f\in\mathcal A'(X).
 \tag{GE7}
\]
We prove that \(u\) has a unique extension in \(\mathcal A'\).
For interior compact smooth tests \(\phi\) supported in \(K\), we
first establish the bound
\[
 |u(\phi)|\le C_{k,K}p_{k,K,L}(\phi)
 \tag{GE8}
\]
for every real \(k\), with a finite \(L\) depending on \(k,K\).
Since \(u\) is extendible, choose one ambient distribution extension
temporarily. On a fixed compact set it has finite order \(M\), so
its action on interior \(\phi\) is bounded by finitely many
\(C^M\) seminorms. Choose \(s>M+n/2\). Fourier inversion and
Cauchy–Schwarz bound those seminorms by \(H^s\), and a slightly
larger \(B^{s+\epsilon}_{2,\infty}\) norm bounds \(H^s\) by a
geometric dyadic sum. Choose \(k_0\) sufficiently negative that
\(-k_0-n/4>s+\epsilon\). The \(\alpha=0\) term of (GE1)
then proves (GE8) for \(k_0\). The extension's boundary values do
not enter this interior-test estimate.

There is no uniform-support assumption hidden in the induction.
For a requested target order \(k\), choose a fixed outer compact product
collar \(K_*\) containing the desired compact \(K\) in its relative
interior. The preceding finite-order argument on \(K_*\) fixes \(k_0\).
Choose a finite integer \(N\ge\max(0,k-k_0)\). Fit \(N+1\) nested
compact product collars between \(K\) and \(K_*\), with a positive
margin at each step. The tangential compact may be enlarged once;
normal primitives preserve its support. Start with the bound at \(k_0\)
on the largest collar, then prove the bound at \(k_0+1\) on the next
smaller collar, and continue inward. Every cutoff and its derivative
has support in the next larger collar. All constants may depend on
this finite chain. When \(N=0\), only the initial estimate and inclusion
are needed.

Here is the single induction step. Assume (GE8) at order \(k\) on the
larger collar, and take the input \(\phi\) in the smaller one.
Let \(\psi\) be (GE3),
\(\chi\) the fixed cutoff, and use the bilinear distribution pairing.
The formal tangential transpose \(A_0^{\mathrm t}\) retains every
coefficient derivative and reverses the matrix maps. Since
\(D_n(\chi\psi)=\phi+(D_n\chi)\psi\) everywhere (because
\(\chi\phi=\phi\)), the exact equation (GE7) gives
\[
 u(\phi)
  =-f(\chi\psi)
   +u\bigl(A_0^{\mathrm t}(\chi\psi)
              -(D_n\chi)\psi\bigr).
 \tag{GE9}
\]
All arguments of \(u\) in this formula are smooth and supported
in the interior. The functional \(f\) is continuous on compact
\(\mathcal A^k\) tests for every \(k\): for \(k\ge m_0\)
this is its defining property, and for \(k<m_0\) the inclusion
\(\mathcal A^k\hookrightarrow\mathcal A^{m_0}\) is continuous.
The full \(A_0^{\mathrm t}\) is continuous on \(\mathcal A^k\)
by Section 1, regardless of its tangential order. The primitive bound
with output cutoffs \(\chi\) and \(D_n\chi\) controls the two
arguments in (GE9). Apply the induction bound on the larger collar,
and the bound for \(f\) there, followed by (GE4). This gives (GE8)
at \(k+1\) on the smaller collar, with a finite, possibly increased \(L\).
For the chosen finite chain this proves (GE8) at \(k_0+N\) on \(K\).
The continuous inclusion \(\mathcal A^k\hookrightarrow
\mathcal A^{k_0+N}\) gives the bound at the requested real order.
Since the requested order and inner compact were arbitrary, this proves
(GE8) on every compact at every real order.

Construct the supported extension rather than identify it with the
temporary ambient one. For a compactly supported smooth boundary test
\(\phi\), the notation \(\mathcal Q_\varepsilon\phi\) below means
\(\mathcal Q_\varepsilon(H\phi)\), where \(H\phi\) is zero extension.
It is smooth and interior-supported. All small \(\varepsilon\) have
one fixed compact output enlargement; their distance from the boundary
may tend to zero. By (GD2) and (GS7) they converge to \(H\phi\)
in \(\mathcal A^{k_1}\) for every \(k_1>m_0\). The bound (GE8)
on that output compact makes their pairings Cauchy, so define
\[
 U(\phi):=\lim_{\varepsilon\downarrow0}
                   u(\mathcal Q_\varepsilon\phi)
 \tag{GE10}
\]
The same estimate on a common enlargement makes the limit independent
of the smoothing family: subtract any two interior-supported smooth
approximants converging to \(H\phi\) in the same \(\mathcal A^{k_1}\).

More generally for any compactly supported \(v\in\mathcal A^k\),
use convergence in \(\mathcal A^{k+\delta}\), \(\delta>0\),
and the bound (GE8) at \(k+\delta\) to define the same limit
\(L(v)=\lim_\varepsilon u(\mathcal Q_\varepsilon v)\).
For continuity at the original order \(k\), apply (GE8) at \(k\)
to these approximants. The full commutator formula (GS2), bounded
Fourier multipliers and coordinate cutoffs bound every output
\(\mathcal A^k\) seminorm by finitely many input seminorms at that
same order, uniformly in \(\varepsilon\). Taking the limit therefore
gives an actual finite \(\mathcal A^k\) bound for \(L\). On intersections
of test orders these definitions agree by using a common larger order.
This constructs one continuous pairing with the full filtered test
space, including every endpoint, without assuming endpoint convergence.

To construct its ambient supported representative directly, for an
ambient compact smooth test \(\Phi\) set
\(U(\Phi)=L(H(\Phi|_{t\ge0}))\).
The smooth-to-conormal bound (GD2) controls this pairing by finitely
many ordinary smooth seminorms on a fixed compact, so it is an ambient
distribution. It is supported in \(t\ge0\), and its pairing with a
smooth boundary test depends only on the restriction of \(\Phi\).
Thus it has precisely the required dual conormal bounds (GD3).
On an interior test, positive mollification converges in every ordinary
smooth seminorm with compact support still away from the boundary.
The action of the original interior \(u\) consequently converges to
its original value, proving the claimed interior restriction.
Interior injectivity (GD4) proves uniqueness in \(\mathcal A'\).
No additional supported-distribution duality theorem is invoked.

## 4. Full normal order by the actual companion system

Let \(m\ge1\) and retain the original normal-monic equation
\[
 P=D_n^m+\sum_{j=0}^{m-1}a_j(x,D')D_n^j,
 \qquad Pu=f\text{ in }X^\circ,
 \quad f\in\mathcal A'(X).
 \tag{GE11}
\]
Each \(a_j\) is an arbitrary finite-order tangential differential
operator with smooth matrix coefficients; the orders of different
\(a_j\) are not required to be at most \(m-j\). Set
\(u_j=D_n^ju\), \(0\le j<m\). Each \(u_j\) is extendible,
because an ordinary derivative of an ambient extension remains an
ambient extension of the interior derivative. The exact first-order
companion system is
\[
 D_n\begin{pmatrix}u_0\\u_1\\\vdots\\u_{m-1}\end{pmatrix}
 +\begin{pmatrix}
 0&-I&0&\cdots&0\\
 0&0&-I&\cdots&0\\
 \vdots&\vdots&\vdots&\ddots&\vdots\\
 a_0&a_1&a_2&\cdots&a_{m-1}
 \end{pmatrix}
 \begin{pmatrix}u_0\\u_1\\\vdots\\u_{m-1}\end{pmatrix}
 =\begin{pmatrix}0\\0\\\vdots\\f\end{pmatrix}.
 \tag{GE12}
\]
For \(m=1\) this is just (GE11), with its single matrix entry
\(a_0\); the displayed larger companion matrix is read for
\(m\ge2\). Section 3 applies componentwise to its actual
tangential matrix, giving a unique vector extension
\(U_j\in\mathcal A'\) for every \(j\).

The corrected derivative (GD9) has the same interior restriction as
\(U_{j+1}\) for \(j<m-1\). Both belong to \(\mathcal A'\), so
interior injectivity (GD4) gives the exact equality
\[
 \nabla_n^{\mathrm{int}}U_j=U_{j+1},
 \qquad D_nU_j-U_{j+1}
          =-i\,(U_j|_{x_n=0})\otimes\delta(x_n),
 \quad x_n(D_nU_j-U_{j+1})=0.
 \tag{GE13}
\]
Similarly the final companion equation gives
\[
 \nabla_n^{\mathrm{int}}U_{m-1}
        +\sum_{j=0}^{m-1}a_j(x,D')U_j=f,
 \quad x_n\left(D_nU_{m-1}
        +\sum_{j=0}^{m-1}a_j(x,D')U_j-f\right)=0.
 \tag{GE14}
\]
The tangential operators act on \(\mathcal A'\) by (GD9), with
no normal delta correction of their own.

We keep the powers of \(x_n\) through the elimination. The full
commutator is
\[
 D_n(x_n^{j+1}W)
   =x_n^{j+1}D_nW-i(j+1)x_n^jW.
 \tag{GE15}
\]
Start with \(U_0=D_n^0U_0\). If
\(x_n^j(U_j-D_n^jU_0)=0\), then (GE13) implies
\(x_n^{j+1}(U_{j+1}-D_nU_j)=0\), while (GE15)
implies \(x_n^{j+1}D_n(U_j-D_n^jU_0)=0\). Hence induction
gives
\[
 x_n^jU_j=x_n^jD_n^jU_0\qquad(0\le j<m).
 \tag{GE16}
\]
Apply (GE15) once more at \(j=m-1\) to replace the first term
of (GE14) after multiplication by \(x_n^m\). For each lower
term \(j<m\), \(x_n\) commutes with \(a_j(x,D')\), and
\(x_n^m a_j(U_j-D_n^jU_0)
=x_n^{m-j}a_jx_n^j(U_j-D_n^jU_0)=0\).
Therefore the complete weighted equation is
\[
 x_n^m(PU_0-f)=0.
 \tag{GE17}
\]
No original tangential coefficient or lower normal term was removed.

## 5. Uniqueness among all supported distribution extensions

Let \(V\) be the difference of two supported distribution extensions
of the same interior \(u\), each satisfying (GE17). Then
\(\operatorname{supp}V\subset\{x_n=0\}\). We derive its finite
normal structure here. Work in an open product subchart with a compact enlargement, and let
\(M\) bound the distribution order of \(V\) on that enlargement.
The equation is used for \(V\) on the open subchart; we do not assume
that an arbitrary localized distribution satisfies the same equation. Taylor-expand a test
\(h(x',t)\) through degree \(M\) at \(t=0\), with a fixed normal
cutoff \(\eta(t)=1\) near zero:
\[
 h(x',t)=\eta(t)\sum_{j=0}^{M}
       \frac{t^j}{j!}\partial_t^jh(x',0)+t^{M+1}r(x',t)
 \quad\text{near }\operatorname{supp}V.
 \tag{GE18a}
\]
The remainder has every normal derivative through degree \(M\)
zero on \(t=0\); multiplication by a cutoff supported in a shrinking
normal neighborhood and the order-\(M\) bound show that \(V\)
annihilates it: a derivative of total order at most \(M\), including
derivatives on the shrinking cutoff, is bounded by \(C\varepsilon\)
because the remainder has the factor \(t^{M+1}\). Tangential
derivatives leave that factor intact. Define tangential distributions
\(v_j(g)=(-1)^jV(\eta(t)t^jg(x')/j!)\). Then applying \(V\) to
(GE18a) gives, with no omitted coefficient,
\[
 V=\sum_{j=0}^{\mu}v_j(x')\otimes\delta^{(j)}(x_n),
 \qquad v_\mu\ne0\text{ unless }V=0.
 \tag{GE18}
\]
Here \(\mu\le M\) is the highest nonzero coefficient. The definition
of \(v_j\) is independent of the cutoff because \(V\) is supported
at \(t=0\). The representation is an identity of distributions on the smaller
open subchart. A cutoff may be used to prove it, but the following
equation is applied to that local identity for \(V\) itself.
The coefficients are tangential distributions, possibly vector
valued. The normal distribution formulas, with every factor, are
\[
 D_n^k\delta^{(j)}=(-i)^k\delta^{(j+k)},\qquad
 x_n^m\delta^{(j+k)}
 =\begin{cases}
 (-1)^m\dfrac{(j+k)!}{(j+k-m)!}\delta^{(j+k-m)},&j+k\ge m,\\
 0,&j+k<m.
 \end{cases}
 \tag{GE19}
\]
They follow by applying the distributions to a test function and
differentiating \(x_n^m\) exactly \(m\) times at zero; no
coefficient is normalized away. In \(x_n^mPV\), the original
leading term \(x_n^mD_n^m(v_\mu\delta^{(\mu)})\) has the
top normal coefficient
\[
 i^m\frac{(\mu+m)!}{\mu!}
       v_\mu(x')\otimes\delta^{(\mu)}(x_n),
 \tag{GE20}
\]
which is nonzero when \(v_\mu\ne0\). Every lower normal term
\(a_j(x,D')D_n^j\), \(j<m\), has normal order at most
\(\mu+j-m\le\mu-1\) after multiplication by \(x_n^m\).
Taylor coefficients of \(a_j\) at the boundary can lower that
order further, never raise it. Contributions from \(v_l\) with
\(l<\mu\) also have order below \(\mu\). The coefficient of
\(\delta^{(\mu)}\) in \(x_n^mPV=0\) therefore forces
\(v_\mu=0\), a contradiction. Repeating downward gives \(V=0\).
This proves uniqueness among *all* supported distribution extensions
satisfying (GE17), stronger than uniqueness only in \(\mathcal A'\).

The boundary delta derivatives are linearly independent over tangential
distributions: test against \(\eta(t)t^r g(x')/r!\) and isolate
the \(r\)-th coefficient. Thus the top-coefficient comparison is an
equality of actual distributions. This local uniqueness on each chart
proves uniqueness on the whole collar.

## 6. The full boundary defect with every coefficient in order

Write \(a_m=I\), and let \(U_r=(\nabla_t^{\mathrm{int}})^rU\)
and \(\gamma_r=U_r|_{t=0}\). All these vectors belong to
\(\mathcal A'\) by the intrinsic derivative proof. Their interior
restrictions are the ordinary derivatives of \(u\). The original
equation and injectivity therefore give
\(\sum_{j=0}^m a_j(x',t,D')U_j=f\).

The identity \(D_tU_r=U_{r+1}-i\gamma_r\otimes\delta(t)\)
implies, by induction on \(j\),
\[
 D_t^jU
 =U_j-i\sum_{r=0}^{j-1}
                 \gamma_r\otimes D_t^{\,j-1-r}\delta(t).
 \tag{NE1}
\]
For \(j=1\) it is the intrinsic derivative identity. Applying \(D_t\)
to the formula for \(j\) differentiates each displayed delta and
adds the term with \(\gamma_j\) from \(D_tU_j\), proving the next
formula with all signs intact. Consequently
\[
 PU-f
 =-i\sum_{j=1}^m a_j(x',t,D')
       \sum_{r=0}^{j-1}\gamma_r\otimes D_t^{\,j-1-r}\delta(t).
 \tag{NE2}
\]
Every \(a_j\) remains in its original position on the left. In
particular a \(t\)-dependent coefficient acts on the delta derivatives;
it has not been replaced by its boundary value. Its full action is
determined by the product rule
\[
 b(x',t)\bigl(v(x')\otimes\delta^{(\ell)}(t)\bigr)
 =\sum_{q=0}^{\ell}(-1)^q\binom{\ell}{q}
      \bigl((\partial_t^q b)(x',0)v\bigr)
                              \otimes\delta^{(\ell-q)}(t).
 \tag{NE3}
\]
To prove this, pair against a test, expand its \(\ell\)-th normal
derivative after multiplication by \(b\), and use
\(\delta^{(s)}(h)=(-1)^s h^{(s)}(0)\); the sign ratio is
\((-1)^\ell/(-1)^{\ell-q}=(-1)^q\). For matrix coefficients this
is the same entrywise identity with the indicated matrix order.
Tangential derivatives act on \(v\) in their original operator order.
Equations (NE2)–(NE3) exhibit every lower boundary term and show again
that \(t^m(PU-f)=0\). The original companion elimination gives the
same conclusion independently.

## 7. Leading coefficients, charts and the precise global scope

For the invariant conclusion let \(P\) have total differential order
\(m\) between bundles of the same finite rank over a manifold with
boundary, and assume its principal coefficient on the nonzero normal
covectors is invertible. In a boundary chart with defining coordinate
\(t\), collect its ordered derivatives as
\(P=a_m(x',t)D_t^m+\sum_{j<m}a_j(x',t,D')D_t^j\).
Its leading matrix remains invertible in a sufficiently small chart:
its determinant is continuous and nonzero at the boundary point.
The inverse is smooth there, since the cofactor formula divides
polynomials in smooth entries by that nonvanishing determinant;
ordinary differentiation proves every required local derivative bound.
Left multiplication by \(a_m^{-1}\) gives the proved monic equation
with source \(a_m^{-1}f\in\mathcal A'\). No derivative is moved
across \(a_m^{-1}\). Its supported solution satisfies
\(t^m(PU-f)=0\), by multiplying the normalized equation on the left
by \(a_m\), which commutes with \(t^m\).

The local supported distributions agree on overlaps after the actual
bundle and density coordinate maps. Indeed these maps preserve the
conormal test spaces and their full dual pairings by the earlier
coordinate proof, and the two local extensions have the same interior
restriction. Their difference vanishes by interior injectivity.
Choose the proved locally finite partition subordinate to the charts
and the interior, and define the global functional by the sum of local
functionals on the partitioned test. Only finitely many summands meet a
compact test; the overlap equality makes the result independent of
the partition and equal to each local functional. Finite local
conormal bounds give its full dual conormal continuity on each compact.
In the interior use the original \(u\), so this constructs the required
global supported extension, unique by the same local injectivity.

If \(\rho\) is another positive defining function in a chart, then
\(\rho=a(x',t)t\), where
\(a(x',t)=\int_0^1\partial_t\rho(x',st)\,ds\) is smooth and positive
near the boundary. Therefore
\(\rho^m(PU-f)=a^m t^m(PU-f)=0\), and conversely by multiplication
by \(a^{-m}\). This proves that the weighted equation is independent
of defining function. It holds for any smooth choices of ambient
coefficient extensions: their differences vanish to every order at
the boundary and on the interior side, so multiplication with a
supported finite-order distribution vanishes by the earlier flat-test
argument. The same argument applies after the finitely many derivatives
in each differential term. Existence and uniqueness among all supported
extensions are thus well defined independently of these choices.

The arbitrary tangential-order statement in the opening theorem is a
statement on a fixed product collar. The coordinate-invariant assertion
here uses the stated total-order and noncharacteristic hypotheses.
The remaining compressed wave-front statement is a separate theorem;
the proof above establishes the extension, all intrinsic jets and the
exact distributional boundary defect needed for it.