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\begin{document}

\ReaderArmAliases{\ReaderOriginalAnchorStart{polynomial-inverse-expansion-with-ordered-matrix-coefficients}\ReaderOriginalAnchorEnd}
\section{Polynomial inverse expansion with ordered matrix
coefficients}\label{polynomial-inverse-expansion-with-ordered-matrix-coefficients}

This lesson proves the finite inverse expansion for the polynomial and
cutoff specified below. It keeps the original tangential and normal
frequency coordinates, the leading coefficient \(I_d\), and the order of
every matrix product. All position-variable estimates are uniform on the
original \(\mathbb R_x^n\). No pseudodifferential composition theorem is
used.

\href{mixed-symbol-inversion.pdf}{Inverting mixed symbols without
commuting matrix factors} supplies the ordered inverse derivative
identity and cutoff inverse used below.

\ReaderArmAliases{\ReaderOriginalAnchorStart{AN03-PIN-001}\ReaderOriginalAnchorEnd\begingroup\expandafter\def\csname @currentHref\endcsname{AN03-PIN-001}\label{AN03-PIN-001}\endgroup\ReaderOriginalAnchorStart{an03-pin-001-exact-setting-and-both-weight-presentations}\ReaderOriginalAnchorEnd\begingroup\expandafter\def\csname @currentHref\endcsname{an03-pin-001-exact-setting-and-both-weight-presentations}\label{an03-pin-001-exact-setting-and-both-weight-presentations}\endgroup\ReaderOriginalAnchorStart{1-exact-setting-and-both-weight-presentations}\ReaderOriginalAnchorEnd\begingroup\expandafter\def\csname @currentHref\endcsname{1-exact-setting-and-both-weight-presentations}\label{1-exact-setting-and-both-weight-presentations}\endgroup}
\subsection{1. Exact setting and both weight
presentations}\label{exact-setting-and-both-weight-presentations}

Fix \(n,d\geq1\) and an integer \(m\geq1\). Write \[
 x\in\mathbb R^n,\quad \xi=(\eta,\zeta)\in\mathbb R^{n-1}\times\mathbb R,\quad
 h=\langle\eta\rangle=(1+|\eta|^2)^{1/2},\quad
 Q=\langle\xi\rangle=(1+|\eta|^2+\zeta^2)^{1/2}.
 \tag{PI1}
\] Keep also the original weights \[
 R=1+|\xi|,\qquad T=1+|\eta|,\qquad
 \Xi=\zeta+ih,\qquad |\Xi|=Q,\quad 1\leq h\leq Q.
 \tag{PI2}
\] The equality \(|\Xi|=Q\) is exact. It follows by squaring the
modulus; there is no rescaling of a frequency coordinate.

For real \(u,v\), a smooth matrix function \(f\) is in the bracket
presentation of \(S^{u,v}\) when all seminorms \[
 [f]_{u,v;\beta,\alpha,r}
 =\sup_{x,\eta,\zeta}
 \frac{\|\partial_x^\beta\partial_\eta^\alpha\partial_\zeta^r f\|}
 {Q^{u-r}h^{v-|\alpha|}}
 \tag{PI3}
\] are finite, where \(\beta\in\mathbb N_0^n\),
\(\alpha\in\mathbb N_0^{n-1}\), \(r\in\mathbb N_0\). Matrices carry
their Euclidean operator norm. The original presentation replaces the
denominator by \(R^{u-r}T^{v-|\alpha|}\), and its seminorm is written
\([f]^o_{u,v;\beta,\alpha,r}\).

For each \(s\geq0\), \(1+s^2\leq(1+s)^2\leq2(1+s^2)\). Thus
\(R/Q,T/h\in[1,\sqrt2]\). If \(a=u-r\), \(b=v-|\alpha|\), and
\(a_\pm=\max(\pm a,0)\), \(b_\pm=\max(\pm b,0)\), then \[
 2^{-(a_-+b_-)/2}Q^ah^b
 \leq R^aT^b
 \leq2^{(a_++b_+)/2}Q^ah^b.
 \tag{PI4}
\] For a negative exponent this follows by taking the reciprocal of the
corresponding positive-exponent inequality; hence it covers all real
orders and every derivative shift. Dividing by these positive weights
gives \[
 [f]^o_{u,v;\beta,\alpha,r}
 \leq2^{(a_-+b_-)/2}[f]_{u,v;\beta,\alpha,r},\qquad
 [f]_{u,v;\beta,\alpha,r}
 \leq2^{(a_++b_+)/2}[f]^o_{u,v;\beta,\alpha,r}.
 \tag{PI5}
\] The identity on actual functions is therefore the exact continuous
linear bijection between the two presentations. It commutes with every
derivative and preserves multiplication in its original order. In
particular the proof below with \(Q,h\) proves the assertion with
\(R,T\), with constants (PI5), including when either differentiated
exponent is negative.

A tangential symbol \(b(x,\eta)\in S^s_{\mathrm{tan}}\), for real \(s\),
means \[
 |b|_{s;\beta,\alpha}
 =\sup_{x,\eta}
 \frac{\|\partial_x^\beta\partial_\eta^\alpha b(x,\eta)\|}
 {h^{s-|\alpha|}}<\infty.
 \tag{PI6}
\] Its \(T\)-weight presentation satisfies the one-factor version of
(PI4)--(PI5), with exponent \(s-|\alpha|\).

Take the given polynomial \[
 p(x,\eta,\zeta)=\sum_{k=0}^{m}p_k(x,\eta)\zeta^k,\qquad
 p_m=I_d,\qquad p_k\in S^{m-k}_{\mathrm{tan}}.
 \tag{PI7}
\] Let \(\chi\in C_c^\infty(\mathbb R_\xi^n;\mathbb C)\) be the fixed
frequency cutoff, \(\theta=1-\chi\), and
\(\mathcal V=\operatorname{supp}\theta\). The meaning of the cutoff
inverse hypothesis is that \(p(x,\xi)\) is invertible at every
\((x,\xi)\) with \(\xi\in\mathcal V\), and \[
 \|p(x,\xi)^{-1}\|\leq C_0Q^{-m}\quad(\xi\in\mathcal V).
 \tag{PI8}
\] If the given invertibility statement is formulated outside a fixed
compact set, the cutoff is chosen equal to one on a neighborhood of that
set, exactly as required to define its cutoff inverse. No invertibility
is required where \(\theta\) vanishes on a neighborhood. Define the
original function \[
 \tau(x,\xi)=\theta(\xi)p(x,\xi)^{-1}\quad(\xi\in\mathcal V),
 \qquad \tau(x,\xi)=0\quad(\xi\notin\mathcal V).
 \tag{PI9}
\] The expansion below concerns this same \(\tau\), without alteration
of \(p\) or \(\chi\).

\ReaderArmAliases{\ReaderOriginalAnchorStart{AN03-PIN-002}\ReaderOriginalAnchorEnd\begingroup\expandafter\def\csname @currentHref\endcsname{AN03-PIN-002}\label{AN03-PIN-002}\endgroup\ReaderOriginalAnchorStart{an03-pin-002-all-derivatives-of-the-scalar-weights-and-complex-powers}\ReaderOriginalAnchorEnd\begingroup\expandafter\def\csname @currentHref\endcsname{an03-pin-002-all-derivatives-of-the-scalar-weights-and-complex-powers}\label{an03-pin-002-all-derivatives-of-the-scalar-weights-and-complex-powers}\endgroup\ReaderOriginalAnchorStart{2-all-derivatives-of-the-scalar-weights-and-complex-powers}\ReaderOriginalAnchorEnd\begingroup\expandafter\def\csname @currentHref\endcsname{2-all-derivatives-of-the-scalar-weights-and-complex-powers}\label{2-all-derivatives-of-the-scalar-weights-and-complex-powers}\endgroup}
\subsection{2. All derivatives of the scalar weights and complex
powers}\label{all-derivatives-of-the-scalar-weights-and-complex-powers}

We first provide constants valid even when the displayed symbol order
becomes negative. For a real number \(s\) and nonnegative integer \(j\),
let \((s)_{\underline j}=s(s-1)\cdots(s-j+1)\), with empty product one.
For a tangential multiindex \(\alpha\), differentiation of
\(h^s=(1+|\eta|^2)^{s/2}\) gives the exact finite identity \[
 \partial_\eta^\alpha h^s
 =\sum_{k+2\ell=\alpha}
 \frac{\alpha!}{k!\ell!}
 (s/2)_{\underline{|k|+|\ell|}}
 (2\eta)^k
 (1+|\eta|^2)^{s/2-|k|-|\ell|}.
 \tag{PI10}
\] Here \(k,\ell\) are tangential multiindices and their equation is
componentwise. To prove it, expand the finite Taylor polynomial of the
scalar function \(z^{s/2}\) at \(z=1+|\eta|^2>0\), substitute
\(z+2\eta\cdot y+|y|^2\), and compare the coefficient of \(y^\alpha\).
Only terms through total degree \(|\alpha|\) can contribute, so the
calculation is a finite Taylor identity for derivatives and does not
require convergence of an infinite series.

Since \(|\eta_j|\leq h\), each summand is bounded by a constant times
\(h^{s-|\alpha|}\). Explicitly put \[
 K(s,\alpha)=
 \sum_{k+2\ell=\alpha}
 \frac{\alpha!}{k!\ell!}
 |(s/2)_{\underline{|k|+|\ell|}}|\,2^{|k|}.
 \tag{PI11}
\] Then \(|\partial_\eta^\alpha h^s|\leq K(s,\alpha)h^{s-|\alpha|}\),
for every real \(s\); \(K(s,0)=1\). For \(n=1\) the tangential
multiindices are empty, \(h=1\), and the same formulas have their
empty-index meanings.

Since \(\Xi\) is in the open upper half-plane, define
\(\Xi^s=\exp(s\log\Xi)\) with its argument in \((0,\pi)\), for every
real \(s\). Its modulus is \(Q^s\). Normal differentiation yields
\(\partial_\zeta^r\Xi^s=(s)_{\underline r}\Xi^{s-r}\). For tangential
derivatives, the finite Taylor chain rule gives, when \(|\alpha|>0\), \[
 \partial_\eta^\alpha\Xi^{s-r}
 =\sum_{\ell=1}^{|\alpha|}
 \frac{(s-r)_{\underline\ell}i^\ell}{\ell!}
 \sum_{\substack{\nu_1+\cdots+\nu_\ell=\alpha\\|\nu_i|>0}}
 \frac{\alpha!}{\nu_1!\cdots\nu_\ell!}
 \Xi^{s-r-\ell}\prod_{i=1}^\ell\partial_\eta^{\nu_i}h.
 \tag{PI12}
\] The inner lists are ordered. The factor \(1/\ell!\) is the scalar
Taylor factor; the multinomial coefficient comes from multiplying the
\(\ell\) positive-degree Taylor polynomials of \(h\). Thus no
multiplicity has been suppressed.

Let the sum on the right below have its single value one at
\(\alpha=0,\ell=0\), and otherwise only the displayed positive
partitions: \[
 H(s;r,\alpha)=
 |(s)_{\underline r}|
 \sum_{\ell=0}^{|\alpha|}
 \frac{|(s-r)_{\underline\ell}|}{\ell!}
 \sum_{\substack{\nu_1+\cdots+\nu_\ell=\alpha\\|\nu_i|>0}}
 \frac{\alpha!}{\nu_1!\cdots\nu_\ell!}
 \prod_i K(1,\nu_i).
 \tag{PI13}
\] Combining (PI11)--(PI12) with \(h/Q\leq1\) proves \[
 |\partial_\eta^\alpha\partial_\zeta^r\Xi^s|
 \leq H(s;r,\alpha)Q^{s-r}h^{-|\alpha|}.
 \tag{PI14}
\] Indeed a term before the last comparison has weight
\(Q^{s-r-\ell}h^{\ell-|\alpha|}\), whose ratio to the displayed weight
is \((h/Q)^\ell\leq1\). This works for all real \(s\), including every
negative exponent used in the expansion.

The pointwise product rule gives, for arbitrary real orders, \[
 \begin{split}
 [fg]_{u+u',v+v';\beta,\alpha,r}
 \leq
 \sum_{\substack{\gamma\leq\beta,\ \epsilon\leq\alpha\\0\leq l\leq r}}
 \binom{\beta}{\gamma}\binom{\alpha}{\epsilon}\binom rl
 [f]_{u,v;\gamma,\epsilon,l}
 [g]_{u',v';\beta-\gamma,\alpha-\epsilon,r-l}.
 \end{split}
 \tag{PI15}
\] Each term preserves the order \(fg\). Its first weight exponents add
to \(u+u'-r\), and its second to \(v+v'-|\alpha|\), as literal
equalities. Also, for every real \(a\geq0\), \[
 S^{u-a,v+a}\subset S^{u,v},\qquad
 [f]_{u,v;\beta,\alpha,r}\leq[f]_{u-a,v+a;\beta,\alpha,r}.
 \tag{PI16}
\] This follows from the exact weight ratio \((h/Q)^a\leq1\), and
remains valid for negative differentiated exponents.

\ReaderArmAliases{\ReaderOriginalAnchorStart{AN03-PIN-003}\ReaderOriginalAnchorEnd\begingroup\expandafter\def\csname @currentHref\endcsname{AN03-PIN-003}\label{AN03-PIN-003}\endgroup\ReaderOriginalAnchorStart{an03-pin-003-the-original-polynomial-and-its-cutoff-inverse-estimates}\ReaderOriginalAnchorEnd\begingroup\expandafter\def\csname @currentHref\endcsname{an03-pin-003-the-original-polynomial-and-its-cutoff-inverse-estimates}\label{an03-pin-003-the-original-polynomial-and-its-cutoff-inverse-estimates}\endgroup\ReaderOriginalAnchorStart{3-the-original-polynomial-and-its-cutoff-inverse-estimates}\ReaderOriginalAnchorEnd\begingroup\expandafter\def\csname @currentHref\endcsname{3-the-original-polynomial-and-its-cutoff-inverse-estimates}\label{3-the-original-polynomial-and-its-cutoff-inverse-estimates}\endgroup}
\subsection{3. The original polynomial and its cutoff inverse
estimates}\label{the-original-polynomial-and-its-cutoff-inverse-estimates}

Set \(P_{k;\beta,\alpha}=|p_k|_{m-k;\beta,\alpha}\), finite by
hypothesis. Since
\(\partial_\zeta^r\zeta^k=(k)_{\underline r}\zeta^{k-r}\) for
\(r\leq k\) and vanishes otherwise, (PI7) gives \[
 \|\partial_x^\beta\partial_\eta^\alpha\partial_\zeta^r p\|
 \leq A_{\beta,\alpha,r}Q^{m-r}h^{-|\alpha|},\qquad
 A_{\beta,\alpha,r}
 =\sum_{k=r}^{m}(k)_{\underline r}P_{k;\beta,\alpha}.
 \tag{PI17}
\] For \(r>m\) the sum and derivative are zero. To verify the bound term
by term, divide \(h^{m-k-|\alpha|}|\zeta|^{k-r}\) by
\(Q^{m-r}h^{-|\alpha|}\). The quotient is
\((h/Q)^{m-k}(|\zeta|/Q)^{k-r}\leq1\). Thus the actual polynomial
belongs to \(S^{m,0}\).

For clarity all inverse constants can now be written with the bracket
weights. Use a combined derivative multiindex \(\mu=(\beta,\alpha,r)\).
Section 2 of \href{mixed-symbol-inversion.pdf}{Inverting mixed symbols
without commuting matrix factors}, formula (MSI6), proves the exact
finite inverse derivative formula \[
 \partial^\mu p^{-1}
 =\sum_{k=1}^{|\mu|}(-1)^k
 \sum_{\substack{\mu_1+\cdots+\mu_k=\mu\\|\mu_i|>0}}
 \frac{\mu!}{\mu_1!\cdots\mu_k!}
 p^{-1}(\partial^{\mu_1}p)p^{-1}\cdots
 (\partial^{\mu_k}p)p^{-1}
 \tag{PI18}
\] for \(\mu\ne0\), on the open set of invertible matrices. It is
obtained by taking the finite Taylor polynomial of \(p\), inverting its
constant part, and using the finite geometric identity in the nilpotent
positive-degree Taylor terms. That finite algebra proof does not commute
matrix factors.

Define \[
 V_0=C_0,\qquad
 V_\mu=
 \sum_{k=1}^{|\mu|}
 \sum_{\substack{\mu_1+\cdots+\mu_k=\mu\\|\mu_i|>0}}
 \frac{\mu!}{\mu_1!\cdots\mu_k!}
 C_0^{k+1}\prod_i A_{\mu_i}
 \quad(\mu\ne0).
 \tag{PI19}
\] Equations (PI8), (PI17), and (PI18) prove \[
 \|\partial_x^\beta\partial_\eta^\alpha\partial_\zeta^r p^{-1}\|
 \leq V_{\beta,\alpha,r}Q^{-m-r}h^{-|\alpha|}
 \quad(\xi\in\mathcal V).
 \tag{PI20}
\] The \(k+1\) inverse factors and \(k\) polynomial derivative factors
give exponent \(-(k+1)m+km-r=-m-r\); the tangential exponents sum to
\(-|\alpha|\). All orders and constants are therefore retained.

Choose \(M\geq1\) such that
\(\operatorname{supp}\chi\subset\{|\xi|\leq M\}\), and put
\(H_M=(1+M^2)^{1/2}\). For a frequency multiindex
\(\delta=(\epsilon,l)\), write \[
 t_\delta=\sup_\xi|\partial_\eta^\epsilon\partial_\zeta^l\theta(\xi)|,\qquad
 L_\delta=\begin{cases}1,&\delta=0,\\H_M^{|\epsilon|+l},&\delta\ne0.\end{cases}
\] The derivative of \(\theta\) is supported in this ball whenever
\(\delta\ne0\). Leibniz' rule and (PI20) give \[
 [\tau]_{-m,0;\beta,\alpha,r}
 \leq U_{\beta,\alpha,r}
 :=\sum_{\epsilon\leq\alpha,\ 0\leq l\leq r}
 \binom\alpha\epsilon\binom rl
 t_{(\epsilon,l)}L_{(\epsilon,l)}
 V_{\beta,\alpha-\epsilon,r-l}.
 \tag{PI21}
\] The weight correction on a nonzero cutoff derivative is exactly
\(Q^lh^{|\epsilon|}\leq H_M^{l+|\epsilon|}\).

The function in (PI9) is smooth globally. Outside \(\mathcal V\),
\(\theta\) is locally zero. At a point of \(\mathcal V\), the invertible
matrix \(p\) has a smooth local inverse by the determinant formula; its
product with \(\theta\) agrees with (PI9) on that neighborhood,
including points outside \(\mathcal V\). This proves smoothness at its
boundary and makes (PI21) a global estimate. It also proves the global
exact identities \[
 p\tau=\tau p=\theta I_d.
 \tag{PI22}
\]

\ReaderArmAliases{\ReaderOriginalAnchorStart{AN03-PIN-004}\ReaderOriginalAnchorEnd\begingroup\expandafter\def\csname @currentHref\endcsname{AN03-PIN-004}\label{AN03-PIN-004}\endgroup\ReaderOriginalAnchorStart{an03-pin-004-re-expression-in-the-unchanged-complex-normal-coordinate}\ReaderOriginalAnchorEnd\begingroup\expandafter\def\csname @currentHref\endcsname{an03-pin-004-re-expression-in-the-unchanged-complex-normal-coordinate}\label{an03-pin-004-re-expression-in-the-unchanged-complex-normal-coordinate}\endgroup\ReaderOriginalAnchorStart{4-re-expression-in-the-unchanged-complex-normal-coordinate}\ReaderOriginalAnchorEnd\begingroup\expandafter\def\csname @currentHref\endcsname{4-re-expression-in-the-unchanged-complex-normal-coordinate}\label{4-re-expression-in-the-unchanged-complex-normal-coordinate}\endgroup}
\subsection{4. Re-expression in the unchanged complex normal
coordinate}\label{re-expression-in-the-unchanged-complex-normal-coordinate}

Substitute the literal identity \(\zeta=\Xi-ih\) into the polynomial: \[
 \begin{split}
 p&=\sum_{j=0}^{m}P_j(x,\eta)\Xi^j,\\
 P_j&=\sum_{k=j}^{m}\binom kj\,p_k(x,\eta)(-ih)^{k-j},\qquad
 P_m=I_d.
 \end{split}
 \tag{PI23}
\] The scalar \(h\) commutes with matrices, so this binomial expansion
does not move one matrix past another. Each \(P_j\) has tangential order
\(m-j\). In fact \[
 |P_j|_{m-j;\beta,\alpha}
 \leq
 \sum_{k=j}^{m}\binom kj
 \sum_{\epsilon\leq\alpha}\binom\alpha\epsilon
 P_{k;\beta,\epsilon}\,K(k-j,\alpha-\epsilon).
 \tag{PI24}
\] The weight exponent in each term is
\((m-k-|\epsilon|)+(k-j-|\alpha-\epsilon|)=m-j-|\alpha|\), including all
its possibly negative values.

Write \[
 A_\ell=P_{m-\ell}\in S^\ell_{\mathrm{tan}},\quad1\leq\ell\leq m,\qquad
 p=\Xi^m\left(I_d+\sum_{\ell=1}^{m}A_\ell\Xi^{-\ell}\right).
 \tag{PI25}
\] Let \(a_{\ell;\beta,\alpha}\) denote the explicit right side of
(PI24) with \(j=m-\ell\). It is a finite bound for every tangential
derivative seminorm of \(A_\ell\).

Define matrices \(\tau_j(x,\eta)\) recursively by \[
 \tau_0=I_d,\qquad
 \tau_j=-\sum_{\ell=1}^{\min(m,j)}A_\ell\tau_{j-\ell}
 \quad(j\geq1).
 \tag{PI26}
\] This specifies the multiplication order. A closed finite expression
for the same coefficient is \[
 \tau_j=
 \sum_{k=1}^{j}(-1)^k
 \sum_{\substack{\ell_1+\cdots+\ell_k=j\\1\leq\ell_i\leq m}}
 A_{\ell_1}\cdots A_{\ell_k}\quad(j\geq1).
 \tag{PI27}
\] To prove it, group each ordered list by its first entry in the right
side; the resulting sum is exactly (PI26). Grouping by its last entry
also proves the different, compatible recursion \[
 \tau_j=-\sum_{\ell=1}^{\min(m,j)}\tau_{j-\ell}A_\ell.
 \tag{PI28}
\] This equality is a sum identity obtained from the same ordered lists;
it does not assert pairwise commutation of its factors.

Each \(\tau_j\) is in \(S^j_{\mathrm{tan}}\). Explicit constants that
prove every derivative bound are obtained as follows. Put
\(b_{0;0,0}=1\) and \(b_{0;\beta,\alpha}=0\) for
\((\beta,\alpha)\ne(0,0)\). For \(j\geq1\), set \[
 b_{j;\beta,\alpha}=
 \sum_{\ell=1}^{\min(m,j)}
 \sum_{\substack{\gamma\leq\beta\\\epsilon\leq\alpha}}
 \binom\beta\gamma\binom\alpha\epsilon
 a_{\ell;\gamma,\epsilon}
 b_{j-\ell;\beta-\gamma,\alpha-\epsilon}.
 \tag{PI29}
\] Induction using the tangential Leibniz identity shows \[
 \|\partial_x^\beta\partial_\eta^\alpha\tau_j\|
 \leq b_{j;\beta,\alpha}h^{j-|\alpha|}.
 \tag{PI30}
\] The exponent is
\(\ell-|\epsilon|+j-\ell-|\alpha-\epsilon|=j-|\alpha|\), so the
induction is valid at every derivative order without assuming a
nonnegative exponent.

\ReaderArmAliases{\ReaderOriginalAnchorStart{AN03-PIN-005}\ReaderOriginalAnchorEnd\begingroup\expandafter\def\csname @currentHref\endcsname{AN03-PIN-005}\label{AN03-PIN-005}\endgroup\ReaderOriginalAnchorStart{an03-pin-005-exact-residual-coefficients-and-the-next-cancellation}\ReaderOriginalAnchorEnd\begingroup\expandafter\def\csname @currentHref\endcsname{an03-pin-005-exact-residual-coefficients-and-the-next-cancellation}\label{an03-pin-005-exact-residual-coefficients-and-the-next-cancellation}\endgroup\ReaderOriginalAnchorStart{5-exact-residual-coefficients-and-the-next-cancellation}\ReaderOriginalAnchorEnd\begingroup\expandafter\def\csname @currentHref\endcsname{5-exact-residual-coefficients-and-the-next-cancellation}\label{5-exact-residual-coefficients-and-the-next-cancellation}\endgroup}
\subsection{5. Exact residual coefficients and the next
cancellation}\label{exact-residual-coefficients-and-the-next-cancellation}

For an integer \(N\geq1\), let \[
 q_N=\sum_{j=0}^{N-1}\tau_j\Xi^{-m-j}.
 \tag{PI31}
\] Multiplying (PI25) by this sum on the right gives constant
coefficient \(I_d\), and all coefficients of \(\Xi^{-k}\) for
\(1\leq k<N\) vanish by (PI26). The complete remaining finite polynomial
in \(\Xi^{-1}\) is \[
 \begin{split}
 p q_N&=I_d+R_N,\\
 R_N&=\sum_{s=0}^{m-1}\rho_{N,s}\Xi^{-N-s},\\
 \rho_{N,s}
 &=\sum_{\substack{1\leq\ell\leq m,\ 0\leq j<N\\\ell+j=N+s}}
 A_\ell\tau_j.
 \end{split}
 \tag{PI32}
\] An empty sum is zero. The largest possible total index is \(m+N-1\),
which explains the precise last value \(s=m-1\). Each coefficient has
the exact asserted order \[
 \rho_{N,s}\in S^{N+s}_{\mathrm{tan}},\qquad
 \tau_N=-\rho_{N,0}.
 \tag{PI33}
\] For the second assertion, the constraint \(\ell+j=N\), \(j<N\), is
exactly \(j=N-\ell\) and \(1\leq\ell\leq\min(m,N)\), which is the
recursion for \(-\tau_N\).

The explicit all-derivative constants for these residual coefficients
are \[
 c_{N,s;\beta,\alpha}=
 \sum_{\substack{1\leq\ell\leq m,\ 0\leq j<N\\\ell+j=N+s}}
 \ \sum_{\substack{\gamma\leq\beta\\\epsilon\leq\alpha}}
 \binom\beta\gamma\binom\alpha\epsilon
 a_{\ell;\gamma,\epsilon}
 b_{j;\beta-\gamma,\alpha-\epsilon}.
 \tag{PI34}
\] Then \[
 \|\partial_x^\beta\partial_\eta^\alpha\rho_{N,s}\|
 \leq c_{N,s;\beta,\alpha}h^{N+s-|\alpha|}.
 \tag{PI35}
\] In particular this proves the required coefficient order \(S^{N+s}\),
not merely a bound after multiplication by a power of \(\Xi\).

The other product has its own ordered residual: \[
 q_Np=I_d+\widetilde R_N,\qquad
 \widetilde R_N=\sum_{s=0}^{m-1}\widetilde\rho_{N,s}\Xi^{-N-s},
 \qquad
 \widetilde\rho_{N,s}
 =\sum_{\substack{1\leq\ell\leq m,\ 0\leq j<N\\j+\ell=N+s}}
 \tau_jA_\ell.
 \tag{PI36}
\] It follows by (PI28), not by commuting any coefficient in (PI32). The
same derivative argument gives the same orders, with constants obtained
by interchanging the written factors and their derivative allocations in
(PI34).

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\subsection{6. The global expansion and every remainder
derivative}\label{the-global-expansion-and-every-remainder-derivative}

The exact global algebra identity follows from (PI22) and (PI32): \[
 \tau(pq_N)=\theta q_N=\tau+\tau R_N.
\] Subtract \(q_N\) and use \(\theta-1=-\chi\). The result is \[
 \boxed{\quad
 \tau=q_N+\rho_N,\qquad
 \rho_N=-\tau R_N-\chi q_N.
 \quad}
 \tag{PI37}
\] Every factor here is a globally smooth function already constructed.
This avoids use of \(p^{-1}\) in the discarded region. On the invertible
region, the first term is precisely
\(-(1-\chi)p^{-1}\sum_s\rho_{N,s}\Xi^{-N-s}\). The ordered alternative
is \[
 \rho_N=-\widetilde R_N\tau-\chi q_N,
 \tag{PI38}
\] obtained from (PI36). Both formulas equal the same actual difference
\(\tau-q_N\), although their matrix factors occupy different sides.

For full quantitative estimates, (PI14) and (PI30) imply \[
 [q_N]_{-m,0;\beta,\alpha,r}\leq
 Q_{N;\beta,\alpha,r}
 :=\sum_{j=0}^{N-1}\sum_{\epsilon\leq\alpha}\binom\alpha\epsilon
 b_{j;\beta,\epsilon}H(-m-j;r,\alpha-\epsilon).
 \tag{PI39}
\] Before applying (PI16), the \(j\)-th differentiated term has weight
\(Q^{-m-j-r}h^{j-|\alpha|}\). Its ratio to the displayed target weight
is \((h/Q)^j\), exactly.

Likewise, \[
 [R_N]_{-N,N;\beta,\alpha,r}\leq
 D_{N;\beta,\alpha,r}
 :=\sum_{s=0}^{m-1}\sum_{\epsilon\leq\alpha}\binom\alpha\epsilon
 c_{N,s;\beta,\epsilon}H(-N-s;r,\alpha-\epsilon).
 \tag{PI40}
\] The pre-comparison weight is \(Q^{-N-s-r}h^{N+s-|\alpha|}\); its
ratio to \(Q^{-N-r}h^{N-|\alpha|}\) is \((h/Q)^s\leq1\). Thus
\(R_N\in S^{-N,N}\), with every derivative and every negative exponent
controlled.

For the inverse term in (PI37), the product bound (PI15) gives \[
 [\tau R_N]_{-m-N,N;\beta,\alpha,r}\leq
 F_{N;\beta,\alpha,r},
\] where the explicit finite constant is \[
 F_{N;\beta,\alpha,r}
 =\sum_{\substack{\gamma\leq\beta,\ \epsilon\leq\alpha\\0\leq l\leq r}}
 \binom\beta\gamma\binom\alpha\epsilon\binom rl
 U_{\gamma,\epsilon,l}
 D_{N;\beta-\gamma,\alpha-\epsilon,r-l}.
 \tag{PI41}
\] In particular the inverse contribution has precisely the required
mixed order, with the rightmost coefficient factors never interchanged
with \(\tau\).

The cutoff contribution is better than required in every mixed order.
Put
\(\chi_{\epsilon,l}=\sup_\xi|\partial_\eta^\epsilon\partial_\zeta^l\chi|\).
For arbitrary real \(u,v\), direct Leibniz differentiation of
\(\chi q_N\), using (PI39), yields \[
 \begin{split}
 [\chi q_N]_{u,v;\beta,\alpha,r}
 \leq K_{N;u,v;\beta,\alpha,r}
 :=\sum_{\epsilon\leq\alpha,\ 0\leq l\leq r}
 \binom\alpha\epsilon\binom rl
 \chi_{\epsilon,l}Q_{N;\beta,\alpha-\epsilon,r-l}\\
 {}\times H_M^{(-m-u+l)_+ +(-v+|\epsilon|)_+}.
 \end{split}
 \tag{PI42}
\] To verify this exponent exactly, divide the differentiated \(q_N\)
weight \(Q^{-m-r+l}h^{-|\alpha|+|\epsilon|}\) by the requested
\(Q^{u-r}h^{v-|\alpha|}\). The quotient is
\(Q^{-m-u+l}h^{-v+|\epsilon|}\). On the support of each cutoff
derivative, \(1\leq h\leq Q\leq H_M\), so any negative power is at most
one and any positive power is bounded by its corresponding power of
\(H_M\). This proves (PI42) for real \(u,v\) without discarding negative
derivative shifts.

Set \(u=-m-N\), \(v=N\) in (PI42). Equations (PI37), (PI41), and (PI42)
prove \[
 \begin{split}
 \|\partial_x^\beta\partial_\eta^\alpha\partial_\zeta^r\rho_N\|
 &\leq
 \left(F_{N;\beta,\alpha,r}
 +K_{N;-m-N,N;\beta,\alpha,r}\right)
 Q^{-m-N-r}h^{N-|\alpha|},\\
 \rho_N&\in S^{-m-N,N}.
 \end{split}
 \tag{PI43}
\] Together with (PI30), (PI31), and (PI37), this is the required
expansion for every \(N\geq1\): \[
 \tau=\sum_{j=0}^{N-1}\tau_j(x,\xi')\Xi^{-m-j}+\rho_N,\qquad
 \tau_j\in S^j_{\mathrm{tan}},\quad
 \rho_N\in S^{-m-N,N}.
 \tag{PI44}
\] The constants use only finitely many indicated seminorms of the
actual \(p_k\), the given \(C_0\), the actual cutoff derivatives and
radius, and the fixed derivative orders, dimension, \(m,N\). The same
coefficient sequence works for every truncation. These finite expansions
need not be a convergent infinite series.

For \(N=0\), the sum is empty and \(\rho_0=\tau\in S^{-m,0}\) by (PI21).
Thus this value is covered when the phrase ``every \(N\)'' includes
zero. If the polynomial degree is \(m=0\), then \(p=I_d\),
\(\tau=(1-\chi)I_d\), and one takes \(\tau_0=I_d\), \(\tau_j=0\) for
\(j\geq1\). For \(N\geq1\), \(q_N=I_d\), \(\rho_N=-\chi I_d\); the same
compact-support proof as (PI42) puts it in every real mixed order. This
handles the degenerate polynomial case as well.

If a later application prescribes a real number \(s\) or \(\nu\), an
integer \(N\geq\max(0,s)\), or \(N\geq\max(0,\nu)\), can be chosen
because (PI44) holds for every nonnegative integer. For example the
choices \(N=\max(0,\lceil s\rceil)\) and \(N=\max(0,\lceil\nu\rceil)\)
are exact. The mapping estimates themselves require their stated
one-sided operator arguments; the present result supplies their full
expansion at every such chosen truncation.

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\subsection{7. A complete noncommuting polynomial
example}\label{a-complete-noncommuting-polynomial-example}

Take \(d=2\), any \(n\geq1\), and \[
 U=\begin{pmatrix}0&1\\0&0\end{pmatrix},\qquad
 V=\begin{pmatrix}0&0\\1&0\end{pmatrix},\qquad
 W=U+V,\quad
 UV=\begin{pmatrix}1&0\\0&0\end{pmatrix},\quad
 VU=\begin{pmatrix}0&0\\0&1\end{pmatrix}.
 \tag{PI45}
\] Here \(U^2=V^2=0\), \(W^2=I_2\), and \(UV\ne VU\). Each of
\(U,V,UV,VU\) has operator norm one, as follows by applying it to
\((z_1,z_2)\) and computing the Euclidean norm. The same is true of
\(W\), which exchanges the two coordinates.

Keep \(h=\langle\eta\rangle\) and \(\Xi=\zeta+ih\). Define the actual
degree-two polynomial, independent of \(x\), by \[
 \begin{split}
 p&=(\Xi I_2+ihU)(\Xi I_2+ihV)\\
  &=\zeta^2I_2+ih(2I_2+W)\zeta-h^2(I_2+W+UV).
 \end{split}
 \tag{PI46}
\] Thus \(p_2=I_2\), \(p_1=ih(2I_2+W)\in S^1_{\mathrm{tan}}\), and
\(p_0=-h^2(I_2+W+UV)\in S^2_{\mathrm{tan}}\). These membership
assertions include every derivative: all positive-order \(x\)
derivatives vanish, and (PI11) gives \[
 \|\partial_\eta^\alpha p_1\|
 \leq3K(1,\alpha)h^{1-|\alpha|},\qquad
 \|\partial_\eta^\alpha p_0\|
 \leq3K(2,\alpha)h^{2-|\alpha|}.
 \tag{PI47}
\] The coefficient matrices themselves do not commute: \[
 [p_1,p_0]
 =-ih^3[W,UV]
 =-ih^3(V-U)\ne0.
 \tag{PI48}
\] Indeed \(WUV=V\), \(UVW=U\), by the displayed matrices.

Nilpotence gives exact inverses of the two factors: \[
 (\Xi I_2+ihU)^{-1}
 =\Xi^{-1}I_2-ih\Xi^{-2}U,\qquad
 (\Xi I_2+ihV)^{-1}
 =\Xi^{-1}I_2-ih\Xi^{-2}V.
 \tag{PI49}
\] For instance multiplying the first expression by \(\Xi I_2+ihU\) on
either side cancels the two linear \(U\)-terms and leaves a multiple of
\(U^2=0\). Since \(|\Xi|=Q\geq h\), each inverse has norm at most
\(2Q^{-1}\). The original \(p\) is consequently invertible everywhere
and obeys \[
 \|p^{-1}\|\leq4Q^{-2}.
 \tag{PI50}
\] The ordered product of those inverses is \[
 \begin{split}
 p^{-1}
 &=(\Xi I_2+ihV)^{-1}(\Xi I_2+ihU)^{-1}\\
 &=\Xi^{-2}I_2-ihW\Xi^{-3}-h^2VU\Xi^{-4}.
 \end{split}
 \tag{PI51}
\] Every coefficient and sign follows from multiplication in that
written order; the final coefficient is \(VU\), not \(UV\).

In the notation of (PI25), \[
 A_1=ihW,\qquad A_2=-h^2UV.
 \tag{PI52}
\] The recursion gives \[
 \begin{split}
 \tau_0&=I_2,\qquad \tau_1=-ihW,\\
 \tau_2&=A_1^2-A_2=-h^2I_2+h^2UV=-h^2VU,\\
 \tau_3&=-A_1\tau_2-A_2\tau_1
 =ih^3WVU-ih^3UVW=ih^3U-ih^3U=0.
 \end{split}
 \tag{PI53}
\] Next \(\tau_4=-A_1\tau_3-A_2\tau_2=0\), since \(UVVU=U(V^2)U=0\). The
two-step recursion then gives \(\tau_j=0\) for every \(j\geq3\). This
recovers (PI51) by the actual recursive coefficients rather than
assuming it from the factorization.

Take first the admissible cutoff \(\chi=0\), so \(\tau=p^{-1}\). The
remainders are exactly \[
 \rho_1=-ihW\Xi^{-3}-h^2VU\Xi^{-4}\in S^{-3,1},\qquad
 \rho_2=-h^2VU\Xi^{-4}\in S^{-4,2},\qquad
 \rho_N=0\quad(N\geq3).
 \tag{PI54}
\] All derivative estimates follow explicitly from (PI11) and (PI14),
followed, for the second term of \(\rho_1\), by the factor \(h/Q\leq1\).
This matches the target orders \(S^{-m-N,N}\) with \(m=2\). The finite
expansion terminates here because of the particular nilpotent factors;
no termination claim is made in the general theorem.

For any other allowed compact frequency cutoff \(\chi\), the
coefficients \(\tau_j\) stay exactly those in (PI53), because they
depend on \(p\) rather than the cutoff. For \(N\geq3\) the exact
remainder is then \[
 \rho_N=-\chi p^{-1}.
 \tag{PI55}
\] It belongs to every mixed order by (PI42). Thus even a terminating
inverse series does not make the actual cutoff remainder vanish unless
the cutoff term vanishes.

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\subsection{8. Solved exercise detecting an invalid matrix
reversal}\label{solved-exercise-detecting-an-invalid-matrix-reversal}

\textbf{Exercise.} For (PI46), compute the polynomial inverse and the
first three inverse-series coefficients at \(\eta=0,\zeta=0\). Compare
the true inverse with the expression obtained by reversing the two
inverse factors in (PI51). Verify the residual coefficient identity
\(\tau_2=-\rho_{2,0}\) before substituting this frequency.

\textbf{Solution.} At \(\eta=0\), \(h=1\); at \(\zeta=0\), \(\Xi=i\).
These are the original frequency values. Equation (PI46) gives \[
 p=-(I_2+U)(I_2+V)
 =\begin{pmatrix}-2&-1\\-1&-1\end{pmatrix},\qquad
 p^{-1}=\begin{pmatrix}-1&1\\1&-2\end{pmatrix}.
 \tag{PI56}
\] Their product in either order is \(I_2\), by direct entry
multiplication. The coefficients at this frequency are \[
 \tau_0=I_2,\qquad
 \tau_1=-i\begin{pmatrix}0&1\\1&0\end{pmatrix},\qquad
 \tau_2=-\begin{pmatrix}0&0\\0&1\end{pmatrix}.
 \tag{PI57}
\] Since \(i^{-2}=-1\), \(i^{-3}=i\), and \(i^{-4}=1\), their terms add
to \(-I_2+W-VU\), which is the inverse matrix in (PI56).

Reversing the inverse factors instead gives the different symbol \[
 q_{\mathrm{wrong}}
 =\Xi^{-2}I_2-ihW\Xi^{-3}-h^2UV\Xi^{-4}.
 \tag{PI58}
\] At the selected frequency its value and left product with \(p\) are
\[
 q_{\mathrm{wrong}}=
 \begin{pmatrix}-2&1\\1&-1\end{pmatrix},\qquad
 p q_{\mathrm{wrong}}=
 \begin{pmatrix}3&-1\\1&0\end{pmatrix}\ne I_2.
 \tag{PI59}
\] The error is exactly the replacement of the ordered matrix \(VU\) by
\(UV\); scalar powers and frequency coordinates were unchanged.

For \(N=2\), the residual coefficient constraints in (PI32) give \[
 \begin{split}
 \rho_{2,0}
 &=A_1\tau_1+A_2\tau_0
 =h^2W^2-h^2UV=h^2VU,\\
 \rho_{2,1}
 &=A_2\tau_1=ih^3UVW=ih^3U.
 \end{split}
 \tag{PI60}
\] Thus \(\tau_2=-\rho_{2,0}\), with tangential orders \(2\) and \(3\)
for the two residual coefficients. The other product has
\(\widetilde\rho_{2,0}=h^2VU\) and
\(\widetilde\rho_{2,1}=\tau_1A_2=ih^3V\). The highest residual
coefficients differ, confirming why (PI32) and (PI36) have to retain
their separate multiplication orders. Each still yields the same actual
inverse remainder through its corresponding formula (PI37) or (PI38).

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\subsection{References}\label{references}

The finite ordered inverse derivative identity is
\href{mixed-symbol-inversion.pdf}{Inverting mixed symbols without
commuting matrix factors}, (MSI6), restated in (PI18); its cutoff
estimates are supplied above in the exact notation needed here. The
separate one-sided operator mapping estimates use this construction in
subsequent units.

\end{document}