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\begin{document}

\section{Radial compression and index transport for product-metric
symbols}\label{radial-compression-and-index-transport-for-product-metric-symbols}

\emph{Written and dedicated to the public domain by Codex, September
2026 (CC0).}

The product metric measures \(x\)- and \(\xi\)-derivatives with separate
weights. A symbol can therefore belong to its class without belonging to
the isotropic class used in the scaled Weyl calculation. We build the
original radial compression exactly, prove uniform control in the
product metric and fixed-positive-scale control in the isotropic metric,
then compare Fredholm indices by trace-norm convergence of the two
ordered error powers. The cutoff inverse \(\chi\) and the radial profile
\(\psi\) serve different purposes and are kept separate. The left and
Weyl quantizations are linked by an explicit compact correction.

Read \href{weyl-metric-products.pdf}{Weyl products for a varying metric}
for \(W31\)--\(W33\) and \(W40\),
\href{gauss-transform-estimates.pdf}{Quadratic Fourier multipliers at a
moving scale} for the quadratic-phase remainder,
\href{metric-operator-bounds.pdf}{Metric operator bounds} for boundedness
and compactness, \href{weyl-trace-criterion.pdf}{Weyl kernels, operator
traces, and a finite trace-class test} for \(CT1\)--\(CT2\), and
\href{traces-and-complexes.pdf}{Traces that survive passage to
cohomology} for the powers-of-errors index identity. This lesson retains
the full original \(x,\xi\) and matrix coordinates throughout.

\subsection{1. The original product metric and coordinate
bounds}\label{the-original-product-metric-and-coordinate-bounds}

Let \(z=(x,\xi)\in\mathbb R^{2n}\), \(n\ge1\), and preserve the two
separate weights: \[
 G_z={|dx|^2\over1+|x|^2}
             +{|d\xi|^2\over1+|\xi|^2},\qquad
 g_z={|dx|^2+|d\xi|^2\over1+|x|^2+|\xi|^2},\qquad
 a\in S(1,G;\operatorname{End}\mathbb C^\nu).
 \tag{RC1}
\] The definition of \(S(1,G)\), by evaluating its derivative seminorms
on normalized coordinate directions and then using the multilinear
expansion, is exactly \[
 \|\partial_x^\alpha\partial_\xi^\beta a(x,\xi)\|
       \le C_{\alpha,\beta}
             \langle x\rangle^{-|\alpha|}
             \langle\xi\rangle^{-|\beta|},
 \quad
 \langle x\rangle=(1+|x|^2)^{1/2},
 \quad \langle\xi\rangle=(1+|\xi|^2)^{1/2}.
 \tag{RC2}
\] The corresponding isotropic \(S(1,g)\) condition requires a
derivative of total order \(k\) to be \(O((1+|x|^2+|\xi|^2)^{-k/2})\).
These are different requirements when one coordinate group is large and
the other is bounded.

Choose a positive smooth radial function \(\psi\) on \(\mathbb R^{2n}\)
such that \(\psi(z)=1\) for \(|z|\le1\) and \(\psi(z)=|z|^{-1}\) for
\(|z|\ge2\), with a positive smooth interpolation on the annulus. For
\(0\le\varepsilon\le1\), put \[
 \psi_\varepsilon(z)=\psi(\varepsilon z),\qquad
 F_\varepsilon(z)=
    (\psi_\varepsilon(z)x,\psi_\varepsilon(z)\xi),\qquad
 a_\varepsilon(z)=a(F_\varepsilon(z)).
 \tag{RC3}
\] At \(\varepsilon=0\), \(\psi_0=1\), \(F_0(z)=z\), and \(a_0=a\). For
\(\varepsilon>0\) and \(|\varepsilon z|\ge2\), the original formula is
exactly \(F_\varepsilon(z)=z/(\varepsilon|z|)\); the exterior is
compressed to a sphere of radius \(1/\varepsilon\).

\subsection{2. A uniform product-weight bound for the radial
map}\label{a-uniform-product-weight-bound-for-the-radial-map}

For each multiindex \(\gamma\), the specified radial profile satisfies
\[
 |\partial^\gamma\psi(s)|
       \le C_\gamma\psi(s)\langle s\rangle^{-|\gamma|}.
 \tag{RC4}
\] On \(|s|\le2\) this follows from smoothness and a positive lower
bound for \(\psi\); on \(|s|\ge2\) it follows by differentiating the
homogeneous function \(|s|^{-1}\). After scaling, each derivative
supplies a factor \(\varepsilon\). For any \(x,\xi\) and
\(0\le\varepsilon\le1\),
\(\varepsilon\langle x\rangle\le\langle\varepsilon z\rangle\) and
\(\varepsilon\langle\xi\rangle\le\langle\varepsilon z\rangle\). Apply
the first inequality once for each \(x\)-derivative and the second once
for each \(\xi\)-derivative in (RC4). This proves the full mixed
estimate \[
 |\partial_x^\alpha\partial_\xi^\beta
          \psi_\varepsilon(x,\xi)|
 \le C_{\alpha,\beta}\psi_\varepsilon(x,\xi)
       \langle x\rangle^{-|\alpha|}
       \langle\xi\rangle^{-|\beta|}
 \quad\text{uniformly for }0\le\varepsilon\le1.
 \tag{RC5}
\] For \(\varepsilon=0\), all positive-order derivatives vanish and the
same estimate holds.

Write \(F_{\varepsilon,x}=\psi_\varepsilon x\) and
\(F_{\varepsilon,\xi}=\psi_\varepsilon\xi\). Differentiate each product.
A term in which no derivative hits \(x\) is bounded by
\(\psi_\varepsilon|x|\) times the full product weight in (RC5). A term
in which one derivative hits \(x\) is bounded by
\(\psi_\varepsilon\langle x\rangle\) times that same weight, because the
derivative used on \(x\) removes one \(x\)-weight. Since the positive
profile is uniformly bounded, both \(\psi_\varepsilon|x|\) and
\(\psi_\varepsilon\langle x\rangle\) are bounded by
\(C\langle\psi_\varepsilon x\rangle\). The analogous statement holds for
\(\xi\). Thus, for every positive total derivative order, \[
 \begin{aligned}
 \|\partial_x^\alpha\partial_\xi^\beta
       F_{\varepsilon,x}(z)\|
 &\le C_{\alpha,\beta}
       \langle F_{\varepsilon,x}(z)\rangle
       \langle x\rangle^{-|\alpha|}
       \langle\xi\rangle^{-|\beta|},\\
 \|\partial_x^\alpha\partial_\xi^\beta
       F_{\varepsilon,\xi}(z)\|
 &\le C_{\alpha,\beta}
       \langle F_{\varepsilon,\xi}(z)\rangle
       \langle x\rangle^{-|\alpha|}
       \langle\xi\rangle^{-|\beta|}.
 \end{aligned}
 \tag{RC6}
\] No distinction between inner and outer regions was discarded: (RC4)
proves both uniformly.

\subsection{\texorpdfstring{3. Uniform \(G\)-symbol bounds for the
actual
composition}{3. Uniform G-symbol bounds for the actual composition}}\label{uniform-g-symbol-bounds-for-the-actual-composition}

Apply the multivariable chain rule to \(a\circ F_\varepsilon\). Every
term of an \((\alpha,\beta)\) derivative is an actual derivative
\(\partial_X^\mu\partial_\Xi^\nu a(F_\varepsilon(z))\) times \(|\mu|\)
derivatives of \(F_{\varepsilon,x}\) and \(|\nu|\) derivatives of
\(F_{\varepsilon,\xi}\), whose derivative multiindices sum to
\((\alpha,\beta)\). Equation (RC2) at \(F_\varepsilon(z)\) supplies
\(\langle F_{\varepsilon,x}\rangle^{-|\mu|}
\langle F_{\varepsilon,\xi}\rangle^{-|\nu|}\). Equation (RC6) supplies
exactly the opposite positive powers, so they cancel without comparing
the \(x\) and \(\xi\) lengths. The remaining product of the separate
source weights is precisely the bound in (RC2). Summing the finitely
many chain-rule terms at each order proves \[
 \sup_{0\le\varepsilon\le1}
 \sup_{x,\xi}
 \langle x\rangle^{|\alpha|}
 \langle\xi\rangle^{|\beta|}
 \|\partial_x^\alpha\partial_\xi^\beta
             a_\varepsilon(x,\xi)\|
       <\infty
       \quad\text{for every }\alpha,\beta .
 \tag{RC7}
\] Therefore \(a_\varepsilon\) is a bounded family in the
\emph{original} product-metric class \(S(1,G)\), including its endpoint
\(a_0=a\). Local smooth convergence to \(a\) as
\(\varepsilon\downarrow0\) is even exact on every fixed compact set once
\(\varepsilon\) is small enough that \(|\varepsilon z|\le1\) there; no
global symbol-topology or operator-norm convergence is claimed.

\subsection{\texorpdfstring{4. The separate
fixed-positive-\(\varepsilon\) isotropic
conclusion}{4. The separate fixed-positive-\textbackslash varepsilon isotropic conclusion}}\label{the-separate-fixed-positive-varepsilon-isotropic-conclusion}

Fix \(\varepsilon>0\). On \(|z|\ge2/\varepsilon\),
\(F_\varepsilon(z)=z/(\varepsilon|z|)\) is smooth homogeneous of degree
zero and has bounded image of radius \(1/\varepsilon\). Its \(k\)-th
derivative is \(O_\varepsilon(|z|^{-k})\) for every \(k\ge1\). All
derivatives of \(a\) are bounded on the compact image sphere. The chain
rule therefore gives \[
 \|\partial_z^\gamma a_\varepsilon(z)\|
       \le C_{\varepsilon,\gamma}|z|^{-|\gamma|}
       \quad (|z|\ge2/\varepsilon).
 \tag{RC8}
\] On the remaining compact ball, every derivative is bounded and
\((1+|z|^2)^{-|\gamma|/2}\) has a positive minimum depending on
\(\varepsilon\). Combining the regions proves \[
 a_\varepsilon\in S(1,g)
       \quad\text{for every fixed }\varepsilon>0 .
 \tag{RC9}
\] The constants in (RC8)--(RC9) need not be uniform as
\(\varepsilon\downarrow0\).

The exclusion of zero is real. In dimension one take
\(a(x,\xi)=\arctan x\), or this scalar function times an identity
matrix. Its \(x\)-derivatives satisfy (RC2), and all \(\xi\)-derivatives
vanish, so \(a\in S(1,G)\). But \(\partial_xa(0,\xi)=1\) for every
\(\xi\), contradicting the \(S(1,g)\) requirement
\(|\partial_xa(0,\xi)|\le C(1+|\xi|^2)^{-1/2}\). Thus \[
 S(1,G)\not\subset S(1,g),\qquad
 a_0=a\ \text{need not belong to }S(1,g).
 \tag{RC10}
\] This example also shows why the uniform conclusion in (RC7) cannot be
silently promoted to a uniform isotropic one.

\subsection{5. Product metric and its Planck
weight}\label{product-metric-and-its-planck-weight}

Keep the two coordinate groups and the original product metric \[
 G_z={|dx|^2\over\langle x\rangle^2}
             +{|d\xi|^2\over\langle\xi\rangle^2},
 \qquad
 G_z^\sigma
   =\langle\xi\rangle^2|dx|^2
       +\langle x\rangle^2|d\xi|^2,
 \qquad
 h_G(z)={1\over\langle x\rangle\langle\xi\rangle}.
 \tag{PT1}
\] The symplectic dual follows by inverting the two diagonal blocks and
exchanging them under the standard symplectic form. The ratio
\(G/G^\sigma\) is \(h_G^2\) in either block, so \(h_G\le1\) and
\(h_G(z)\to0\) as \(|z|\to\infty\).

Here are the structural checks used below. If \(G_z(v)\le\delta^2\) for
\(\delta<1\), then \(|v_x|\le\delta\langle x\rangle\) and
\(|v_\xi|\le\delta\langle\xi\rangle\). The inequalities
\((1-\delta)\langle x\rangle\le\langle x+v_x\rangle\le(1+\delta)\langle x\rangle\)
and the identical \(\xi\) inequalities prove slow variation. For
arbitrary \(z,w\), each ratio \(\langle x\rangle/\langle y\rangle\) and
its inverse is bounded by \(1+|x-y|\), and similarly for \(\xi,\eta\).
Since \(G_w^\sigma(z-w)\ge|z-w|^2\), these ratios and the coefficients
of \(G,G^\sigma,h_G^{\pm1}\) are bounded by fixed powers of
\(1+G_w^\sigma(z-w)\). Thus \(G\) and every fixed \(h_G\) power meet the
one-metric W31--W33 and B26 hypotheses, with constants independent of
the radial parameter used later.

\subsection{6. Original cutoff inverse and a uniformly compact cutoff
term}\label{original-cutoff-inverse-and-a-uniformly-compact-cutoff-term}

Assume \(a\in S(1,G;\operatorname{End}\mathbb C^\nu)\) is invertible
with uniformly bounded inverse outside a ball. Choose a separate scalar
cutoff \(\chi\), supported in its invertibility region and equal to one
for \(|z|\ge R_0\), and define \(b=\chi a^{-1}\) there, extended by zero
where \(\chi=0\). Differentiating the inverse in its actual matrix order
and using the separate coordinate derivative bounds RC2 proves
\(b\in S(1,G)\) and \[
 ba=ab=\chi I_\nu .
 \tag{PT2}
\] Let \(\psi\) be the radial compression profile of RC3,
\(F_\varepsilon(z)=\psi(\varepsilon z)z\), and \[
 a_\varepsilon=a\circ F_\varepsilon,\qquad
 b_\varepsilon=b\circ F_\varepsilon,\qquad
 \chi_\varepsilon=\chi\circ F_\varepsilon,
 \quad 0\le\varepsilon\le1.
 \tag{PT3}
\] RC7 applies separately to \(a\), \(b\), and \(\chi\), so all three
families are uniformly bounded in \(S(1,G)\). Also
\(b_\varepsilon a_\varepsilon=a_\varepsilon b_\varepsilon=\chi_\varepsilon I_\nu\)
pointwise.

The cutoff part is even more stable than mere boundedness. Set
\(c=\min_{r\ge1}r\psi(r)>0\); this minimum is positive because
\(r\psi(r)=1\) for \(r\ge2\) and is positive on the compact annulus
\(1\le r\le2\). Choose \(\varepsilon_*>0\) with
\(\varepsilon_*R_0<\min(1,c)\). If \(0<\varepsilon\le\varepsilon_*\) and
\(|\varepsilon z|\ge1\), then
\(|F_\varepsilon(z)|=\varepsilon^{-1}(|\varepsilon z|\psi(\varepsilon z))\ge c/\varepsilon>R_0\),
hence \(\chi_\varepsilon(z)=1\). If \(|\varepsilon z|\le1\), then
\(F_\varepsilon(z)=z\); whenever \(|z|\ge R_0\) the same conclusion
holds. Therefore \[
 1-\chi_\varepsilon=1-\chi
       \quad\text{globally for }0\le\varepsilon\le\varepsilon_*,
 \tag{PT4}
\] where the right side has one fixed compact support. This exact
identity prevents an escaping cutoff defect.

\subsection{7. Both error sides and trace-class
powers}\label{both-error-sides-and-trace-class-powers}

W31--W33 in the product metric, in each written matrix order, give
uniform \(S(h_G,G)\) remainders from the two zeroth products
\(\chi_\varepsilon I_\nu\). With the actual bounded Weyl operators
\(A_\varepsilon=a_\varepsilon^w\), \(B_\varepsilon=b_\varepsilon^w\),
the same compact-approximation and kernel argument used in IP8
identifies symbol product with operator composition. Thus \[
 \begin{aligned}
 E_{1,\varepsilon}
 &=I-B_\varepsilon A_\varepsilon
       =(r_{1,\varepsilon})^w,\\
 E_{2,\varepsilon}
 &=I-A_\varepsilon B_\varepsilon
       =(r_{2,\varepsilon})^w,\\
 r_{j,\varepsilon}-(1-\chi_\varepsilon)I_\nu
 &\in S(h_G,G)\quad\text{uniformly in }\varepsilon .
 \end{aligned}
 \tag{PT5}
\] By (PT4), the compact cutoff term belongs to \(S(h_G,G)\) with one
fixed seminorm bound for \(0\le\varepsilon\le\varepsilon_*\). Hence both
full error symbols are uniformly in \(S(h_G,G)\) on this smaller
interval. W31 and the operator identity preserve the original factor
order under powers: \[
 E_{j,\varepsilon}^{\,N}
       =\bigl(r_{j,\varepsilon}^{\#N}\bigr)^w,
 \qquad
 r_{j,\varepsilon}^{\#N}
       \in S(h_G^N,G)
 \quad\text{uniformly for }0\le\varepsilon\le\varepsilon_* .
 \tag{PT6}
\]

Take the explicit integer \(N=2n+2\) for the comparison argument. If
\(q\in S(h_G^N,G)\), its derivative of \(x\)-order \(p\) and
\(\xi\)-order \(q'\) is bounded by
\(C\langle x\rangle^{-N-p}\langle\xi\rangle^{-N-q'}\). For any four
multiindices in the \emph{original} CT1 sum, with total degree at most
\(n+1\), multiplication by \(x^\alpha\xi^\beta\) gives \[
 |x^\alpha\xi^\beta
       \partial_x^{\alpha'}\partial_\xi^{\beta'}q(x,\xi)|
 \le C\langle x\rangle^{|\alpha|-N-|\alpha'|}
       \langle\xi\rangle^{|\beta|-N-|\beta'|}
 \le C\langle x\rangle^{-(n+1)}
       \langle\xi\rangle^{-(n+1)}.
 \tag{PT7}
\] The right side is in \(L^2(\mathbb R^{2n})\), separately in each
\(n\)-dimensional group. CT1--CT2 therefore make both error powers trace
class, uniformly bounded in trace norm. T28 proves Fredholmness of every
\(A_\varepsilon\) for \(0\le\varepsilon\le\varepsilon_*\) and gives \[
 \operatorname{ind}A_\varepsilon
  =\operatorname{Tr}E_{1,\varepsilon}^{\,N}
   -\operatorname{Tr}E_{2,\varepsilon}^{\,N}.
 \tag{PT8}
\] This larger power is chosen only to obtain an elementary uniform
product-weight \(L^2\) dominator; it does not replace the original
\(n+1\) threshold proved for the isotropic calculation in IP11--IP14.

\subsection{\texorpdfstring{8. Trace-norm convergence, without an
operator-norm claim for
\(A_\varepsilon\)}{8. Trace-norm convergence, without an operator-norm claim for A\_\textbackslash varepsilon}}\label{trace-norm-convergence-without-an-operator-norm-claim-for-a_varepsilon}

RC3 gives \(a_\varepsilon=a\), \(b_\varepsilon=b\), and
\(\chi_\varepsilon=\chi\) on every fixed compact phase-space set once
\(\varepsilon\) is small. The W31 product is continuous in the local
smooth topology on bounded symbol sets. Apply this successively to the
two errors and their \(N\)-fold products. Every derivative of
\(r_{j,\varepsilon}^{\#N}\) converges locally uniformly to the
corresponding derivative of \(r_{j,0}^{\#N}\). The common global
majorant in (PT7) makes dominated convergence applicable to each of the
finitely many weighted \(L^2\) norms in CT1. Hence \[
 \mathcal N_{n+1}
   \bigl(r_{j,\varepsilon}^{\#N}
         -r_{j,0}^{\#N}\bigr)\longrightarrow0,\qquad
 \|E_{j,\varepsilon}^{\,N}
       -E_{j,0}^{\,N}\|_{\mathcal S_1}\longrightarrow0 .
 \tag{PT9}
\] The second limit is CT2 with the exact Weyl symbols. Taking traces in
(PT8), the integer \(\operatorname{ind}A_\varepsilon\) tends to the
integer \(\operatorname{ind}A_0\). Therefore the two integers are equal
for every sufficiently small positive \(\varepsilon\): \[
 \operatorname{ind}(a\circ F_\varepsilon)^w
       =\operatorname{ind}a^w
       \quad(0<\varepsilon\le\varepsilon_0)
       \quad\text{for some }\varepsilon_0>0 .
 \tag{PT10}
\] For a chosen enclosing sphere \(\partial B_R\), reduce
\(\varepsilon_0\) so that \(\varepsilon_0R\le1\). Then
\(F_\varepsilon(z)=z\) on that sphere, and \[
 (a\circ F_\varepsilon)|_{\partial B_R}
       =a|_{\partial B_R},\qquad
 ((a\circ F_\varepsilon)^{-1}d(a\circ F_\varepsilon))
       |_{\partial B_R}
       =(a^{-1}da)|_{\partial B_R}.
 \tag{PT11}
\] The second equality concerns the pullback to the tangent bundle of
the sphere. The restriction of \(F_\varepsilon\) to this sphere is
exactly its identity map, so its tangent differential is exactly the
identity, including the endpoint \(\varepsilon R=1\). Under a strict
inequality it is also the identity on a neighborhood. At equality,
smoothness of \(\psi\) and its constant value for \(r\leq1\) give
\(\psi(1)=1\) and \(\psi'(1)=0\); the full first differential in (RC3)
is therefore the identity there as well. No neighborhood assertion is
needed at that endpoint. Equations (PT10)--(PT11) are the exact analytic
and boundary receiving maps needed for the product-metric extension.
Section 14 receives the completed finite ordered isotropic coefficient;
an exterior boundary evaluation is not used to prove these maps.

\subsection{9. Left quantization keeps the same
index}\label{left-quantization-keeps-the-same-index}

The kernel identity W40 is exact for tempered symbols:
\(\operatorname{Op}_0(a)=\operatorname{Op}_{1/2}(a_{\mathrm W})\) with
\[
 a_{\mathrm W}
  =\exp\!\left(-{i\over2}\langle D_x,D_\xi\rangle\right)a .
 \tag{PT12}
\] For the product metric (PT1), one \(x\)- and one \(\xi\)-derivative
of \(a\) give the full factor \(h_G\). The quadratic-phase estimate of
the owned Gauss theorem applies because its parameter is at most
\(h_G/4\le1/4\); its first-order Taylor remainder, or the integral over
the multiplier parameter with the same estimate, gives \[
 a_{\mathrm W}-a\in S(h_G,G).
 \tag{PT13}
\] The distributional identity W40 and bounded-symbol approximation
identify both sides as bounded operators. Since \(h_G\to0\) at full
phase-space infinity, the finite-matrix compactness result B35 makes
\((a_{\mathrm W}-a)^w\) compact. Fredholm index stability under compact
perturbation then proves \[
 \operatorname{ind}\operatorname{Op}_0(a)
       =\operatorname{ind}a^w .
 \tag{PT14}
\] This retains the original left and Weyl quantizations as distinct
operators connected by a proved compact morphism; it does not equate
their symbols.

\subsubsection{The exact product-weight trace threshold and the smaller
error
power}\label{the-exact-product-weight-trace-threshold-and-the-smaller-error-power}

The preceding \(N=2n+2\) argument remains valid with every original CT1
term. A separate factorization proves the stronger criterion \[
 \begin{gathered}
 q\in S(h_G^N,G;\operatorname{End}\mathbb C^\nu),\qquad N>n
 \quad\Longrightarrow\quad q^w\text{ is trace class},\\
 \|q^w\|_1\leq C_{n,\nu,N}\,p_J(q;h_G^N,G),
 \qquad h_G=(\langle x\rangle\langle\xi\rangle)^{-1}.
 \end{gathered}                                                   \tag{PT15}
\] Here \(p_J\) is a finite collection of the original symbol seminorms;
\(N\) may be any real number exceeding \(n\). We prove both the estimate
and continuity in the bounded local smooth topology. These give both
original error powers at \(N=n+1\), without requiring the larger CT1
dominator (PT7).

Put \(s=N/2>n/2\), and define the actual configuration-space and Fourier
multipliers \(M_s f(x)=\langle x\rangle^sf(x)\) and
\(F_s f=\mathcal F^{-1}(\langle\xi\rangle^s\widehat f)\). Their maximal
domains are respectively \(\{f:\langle x\rangle^sf\in L^2\}\) and
\(\{f:\langle\xi\rangle^s\widehat f\in L^2\}\), with the fixed original
Plancherel coefficient. Multiplication and Plancherel prove they are
closed; the inverse multipliers \(M_{-s},F_{-s}\) are bounded, have norm
at most one, and map onto those domains with both inverse identities.
Every multiplier preserves Schwartz functions, since its exact weight is
smooth with polynomially bounded derivatives.

The original \(q^w\) also preserves Schwartz functions. In fact all
Euclidean derivatives of \(q\) are bounded when \(N>0\), and the exact
Weyl commutators are \[
 [x_j,q^w]=i(\partial_{\xi_j}q)^w,\qquad
 [D_j,q^w]=-i(\partial_{x_j}q)^w.
 \tag{PT16}
\] They follow by applying CT10 on the left and the same complete kernel
calculation on the right. Repeatedly move each output coordinate or
derivative to the input using these identities; every resulting symbol
is an actual derivative of \(q\), bounded in the Euclidean order-zero
class. The owned boundedness theorem bounds its action on every input
Schwartz monomial in \(L^2\). Thus all output polynomial-weighted
derivatives are in \(L^2\); local Sobolev embedding, applied at each
higher order with those weights, gives all Schwartz seminorms. This
proves preservation before applying either unbounded multiplier.

Keep the full original order and set \[
 \begin{gathered}
 b_N=\langle\xi\rangle^s\#\langle x\rangle^s\#q,
 \qquad B_N=F_sM_sq^w\text{ on }\mathcal S,\\
 b_N\in S(\langle x\rangle^s\langle\xi\rangle^s h_G^N,G)
        =S(h_G^{N-s},G)=S(h_G^s,G),\qquad B_N=b_N^w,\\
 A_N=M_{-s}F_{-s},\qquad
 K_{A_N}(x,y)=(2\pi)^{-n}\langle x\rangle^{-s}
       \int e^{i(x-y)\cdot\xi}\langle\xi\rangle^{-s}I_\nu\,d\xi.
 \end{gathered}                                                   \tag{PT17}
\] The positive weights and all their powers satisfy the original
\(G\)-temperateness bounds proved in Section 5. Their derivative bounds
are \(C_\alpha\langle x\rangle^{s-|\alpha|}\) or
\(C_\beta\langle\xi\rangle^{s-|\beta|}\); hence the one-metric product
theorem W31 applies in exactly the written order. Its operator identity
on Schwartz vectors gives the displayed \(B_N\) equality. In the kernel
of \(A_N\), the Fourier integral is understood as its \(L^2\) inverse
transform; the multiplier \(\langle\xi\rangle^{-s}\) is in \(L^2\)
because \(2s=N>n\).

Partial Plancherel for that kernel and the complete Weyl formula CI5 for
\(b_N\) give \[
 \begin{split}
 \|A_N\|_2^2
 &=\nu(2\pi)^{-n}
       \int\langle x\rangle^{-2s}\,dx
       \int\langle\xi\rangle^{-2s}\,d\xi<\infty,\\
 \|B_N\|_2^2
 &=(2\pi)^{-n}\int
       \operatorname{tr}_{\mathbb C^\nu}(b_N^*b_N)(x,\xi)
                                      \,dx\,d\xi\\
 &\leq C(2\pi)^{-n}p_J(q;h_G^N,G)^2
       \int\langle x\rangle^{-2s}\,dx
       \int\langle\xi\rangle^{-2s}\,d\xi<\infty.
 \end{split}                                                     \tag{PT18}
\] The two integrals are separate original \(n\)-dimensional integrals,
not an isotropic replacement. W31 bounds the full \(b_N\) seminorm by
finitely many original \(q\) seminorms, and its Hilbert--Schmidt fiber
norm supplies the stated finite matrix constant.

On Schwartz vectors, the actual cancellations are adjacent and ordered:
\[
 A_NB_N=M_{-s}F_{-s}F_sM_sq^w
             =M_{-s}M_sq^w=q^w.                               \tag{PT19}
\] Both \(A_N\) and the Hilbert--Schmidt extension of \(B_N\) are
bounded. The original \(q^w\) is bounded by the owned metric bound,
since \(h_G^N\leq1\). Thus equality on the dense Schwartz domain extends
to the original Hilbert space. The trace-ideal product inequality and
(PT18) prove (PT15).

There are also actual unbounded-domain maps for every input, not merely
a formal equality on Schwartz vectors. Given \(f_j\in\mathcal S\)
tending to \(f\in H_\nu\), (PT19) gives \(M_sq^wf_j=F_{-s}B_Nf_j\),
whose right side tends to \(F_{-s}B_Nf\). Closedness of \(M_s\) proves
\(q^wf\in D(M_s)\) and the same identity at \(f\). Then
\(F_sM_sq^wf_j=B_Nf_j\to B_Nf\); closedness of \(F_s\) proves
\(M_sq^wf\in D(F_s)\) and \(F_sM_sq^wf=B_Nf\). This gives both full
receiving domains for (PT17) and (PT19).

Suppose a family \(q_\varepsilon\) is bounded in this original
\(S(h_G^N,G)\) class and converges locally smoothly to \(q_0\). Applying
the bounded-set continuity of W31 with the two fixed multipliers shows
\(b_{N,\varepsilon}\to b_{N,0}\) locally smoothly, bounded in
\(S(h_G^s,G)\). The squared dominator
\(Ch_G^{2s}=C\langle x\rangle^{-N}\langle\xi\rangle^{-N}\) is integrable
in both coordinate groups. Dominated convergence and the exact CI5
factor therefore give \[
 \begin{gathered}
 \|B_{N,\varepsilon}-B_{N,0}\|_2^2
   =(2\pi)^{-n}\|b_{N,\varepsilon}-b_{N,0}\|_{L^2;\mathrm{HS}}^2
                    \longrightarrow0,\\
 \|q_\varepsilon^w-q_0^w\|_1
     \leq\|A_N\|_2\|B_{N,\varepsilon}-B_{N,0}\|_2
                    \longrightarrow0.                       \tag{PT20}
 \end{gathered}
\] Apply this to each original ordered error symbol
\(q_\varepsilon=r_{j,\varepsilon}^{\#N}\) in (PT6), now with the integer
\(N=n+1>n\). The same W31 continuity and local equality of the original
input symbols used in Section 8 prove its local smooth convergence.
Equations (PT15) and (PT20) consequently prove uniform trace class and
trace-norm convergence of both error powers at \(n+1\). The
powers-of-errors identity gives (PT8) at this smaller exponent, and
integer trace continuity gives precisely the same (PT10) and (PT11)
receiving maps. The original \(2n+2\) calculation remains an
independently proved larger-power case.

The exponent condition in (PT15) is exact for the whole symbol class.
For any real \(N\), the original positive symbol
\(q_N(x,\xi)=\langle x\rangle^{-N}\langle\xi\rangle^{-N}I_\nu
             =h_G^NI_\nu\) belongs to \(S(h_G^N,G)\) by its full
separate derivative bounds. If its Weyl operator were trace class, the
positive coherent-state argument CI9--CI11, as proved in WT11, would
imply \[
 \|q_N^w\|_1\geq(2\pi)^{-n}\nu
          \int\langle x\rangle^{-N}\,dx
          \int\langle\xi\rangle^{-N}\,d\xi.                   \tag{PT21}
\] Each \(n\)-dimensional tail converges exactly for \(N>n\), diverges
logarithmically at \(N=n\), and diverges by a power for \(N<n\). The
coherent pairings remain defined for every real \(N\), since the symbol
has polynomial growth and the Wigner function is Gaussian. Thus
\(N\leq n\) has an explicit member of the class whose Weyl operator is
not trace class. This proves the precise whole-class boundary without
changing any original product metric, symbol or Fourier factor.

\begin{figure}
\centering
\pandocbounded{\includegraphics[keepaspectratio,alt={The ordered product-weight factorization with both original coordinate integrals}]{../figures/product-weight-trace-threshold.png}}
\caption{The ordered product-weight factorization with both original
coordinate integrals}
\end{figure}

The diagram records the actual \(A_N,B_N\), their Hilbert-space types,
both domain maps, the two original coordinate integrals and the endpoint
\(N>n\). Equations (PT15)--(PT21) prove its sufficiency, bounded-set
trace-norm continuity and sharpness. This is an editorial strengthening
of the larger-power comparison; the original calculation remains
identifiable. No novelty claim is made.

\subsection{10. Worked example: a product symbol outside the isotropic
class}\label{worked-example-a-product-symbol-outside-the-isotropic-class}

Set \(n=\nu=1\) and \[
 a(x,\xi)=2+\arctan x,\qquad
 2-\frac{\pi}{2}\le a(x,\xi)\le2+\frac{\pi}{2}.
 \tag{RT1}
\] Every positive \(x\)-derivative of \(\arctan x\) is bounded by
\(C_k\langle x\rangle^{-k}\), and every positive \(\xi\)-derivative
vanishes. Thus (RC2) holds and \(a\) is uniformly invertible everywhere.
Yet \(\partial_xa(0,\xi)=1\) for every \(\xi\), so \(a\notin S(1,g)\) by
the exact isotropic first-derivative requirement. For each
\(\varepsilon>0\), (RC9) puts \(a\circ F_\varepsilon\) in \(S(1,g)\);
(PT10) proves that its Weyl index equals the index of \(a^w\) for
sufficiently small positive \(\varepsilon\). Since this original symbol
depends only on \(x\), its Weyl operator is multiplication by
\(2+\arctan x\), whose inverse is multiplication by
\((2+\arctan x)^{-1}\). Both are bounded on \(L^2(\mathbb R)\). Hence \[
 \operatorname{ind}a^w
 =\operatorname{ind}(a\circ F_\varepsilon)^w
 =0\qquad(0<\varepsilon\le\varepsilon_0).
 \tag{RT2}
\] No positivity assertion about the compressed Weyl operator is needed
for its index equality.

\subsection{11. Exercise with complete solution: compare both Planck
weights}\label{exercise-with-complete-solution-compare-both-planck-weights}

\textbf{Exercise.} For \(n=1\), compare the original product-metric
Planck weight \(h_G\) in (PT1) with the isotropic weight
\(h_g=(1+x^2+\xi^2)^{-1}\) from (RC1) along the rays \((x,\xi)=(R,0)\)
and \((R,R)\), \(R\to\infty\). Decide whether they are uniformly
comparable.

\textbf{Solution.} The exact values are \[
 \begin{array}{c|cc}
 (x,\xi)& h_G & h_g\\ \hline
 (R,0)&(1+R^2)^{-1/2}&(1+R^2)^{-1}\\
 (R,R)&(1+R^2)^{-1}&(1+2R^2)^{-1}.
 \end{array}
 \tag{RT3}
\] The first-ray ratio \(h_G/h_g=\sqrt{1+R^2}\) diverges; the second-ray
ratio \((1+2R^2)/(1+R^2)\) tends to \(2\). Therefore no two constants
bound both weights above and below on the full phase space. The
comparison must keep the two coordinate groups instead of replacing
\(G\) by \(g\) at \(\varepsilon=0\).

\subsection{12. Antecedent and the received matrix
formula}\label{antecedent-and-the-received-matrix-formula}

Equations (PT10)--(PT14) prove the analytic and quantization receiving
maps. The completed \href{scaled-weyl-index-degree.pdf}{scaled Weyl
coefficient proof}, (DE11) and (OC1)--(OC31), gives the exact finite
ordered coefficient and its analytic index in every dimension. Section
14 below proves the receiving map with the separate cutoff and radial
profile. The later \href{matrix-weyl-index-theorem.pdf}{matrix Weyl
boundary lesson} concerns the exterior evaluation of that coefficient;
no theorem from that later lesson is used in the proofs here.

\subsection{13. Complete operator receivers and further
consequences}\label{complete-operator-receivers-and-further-consequences}

This supplement proves the operator composition needed for the growing
multipliers in (PT17), writes out the cutoff inverse calculation, and
strengthens the radial comparison. It retains every original object and
calculation in Sections 1--11. The earlier inputs are the proved Fourier
inversion and Plancherel formulas in
\href{prerequisite-bridges.pdf}{Fourier prerequisites}, the Gauss
extension and remainders (G19)--(G26) in
\href{gauss-transform-estimates.pdf\#section*.42}{Quadratic
Fourier multipliers}, the symbol product and its full factors
(W27)--(W34), (WG1)--(WG4) in
\href{weyl-metric-products.pdf\#section*.72}{Weyl
products}, the operator bound (B26) and finite-matrix compactness proof
(B35) in \href{metric-operator-bounds.pdf}{Metric operator bounds}, the
kernel and trace formulas (CI1)--(CI12), (CT1)--(CT2) in
\href{weyl-trace-criterion.pdf}{Weyl traces}, and the trace-ideal product
and index identity (T7), (T28)--(T30) in
\href{traces-and-complexes.pdf}{Traces and complexes}. Index constancy
and compact perturbations are proved in
\href{fredholm-stability.pdf\#section*.13}{Finite
defects}, (F6) and (F10). These are programme proofs, rather than
substitutes by external citations.

\subsubsection{13.1. The full product weights and the quantization
map}\label{the-full-product-weights-and-the-quantization-map}

For real \(p,q\), put \[
 m_{p,q}(x,\xi)=\langle x\rangle^p\langle\xi\rangle^q,\qquad
 \mathcal S_{p,q}=S(m_{p,q},G).
 \tag{RS1}
\] The symbol may have any fixed finite matrix input and output sizes.
For the square maps below, the actual Hilbert space is
\(H_\nu=L^2(\mathbb R^n;\mathbb C^\nu)\). The same coordinate expansion
as in (RC2) gives exactly \[
 \|\partial_x^\lambda\partial_\xi^\kappa u(x,\xi)\|
 \leq C_{\lambda,\kappa}\,
       \langle x\rangle^{p-|\lambda|}
       \langle\xi\rangle^{q-|\kappa|}.
 \tag{RS2}
\] The bracket is one-Lipschitz, by its gradient or by the triangle
inequality in \(\mathbb R^{n+1}\). Thus each bracket ratio and its
inverse is at most \(1+|x-y|\), or \(1+|\xi-\eta|\), respectively.
Raising these two inequalities to the absolute values of the two real
exponents proves the full temperateness of \(m_{p,q}\), with real
exponent \((|p|+|q|)/2\) after the bound
\((1+|z-w|)^2\leq2(1+G_w^\sigma(z-w))\). The corresponding constant is
\(2^{(|p|+|q|)/2}\); if an integer exponent is desired, its ceiling has
the same upper bound. The local comparisons in Section 5 prove local
continuity. This verifies the weight hypotheses for every real \(p,q\),
including the two positive weights in (PT17).

Keep the original quantization parameters \(\tau,s\) and set
\(c=\tau-s\). For \(c\ne0\), the phase \(A_c(P,Q)=cP\cdot Q\) on the
original dual variables has symmetric map \[
 B_c=\frac c2\begin{pmatrix}0&I_n\\I_n&0\end{pmatrix},
 \qquad
 G_z^{A_c}=\frac4{c^2}G_z^\sigma,\qquad
 h_{G,A_c}=\frac{|c|}{2}h_G.
 \tag{RS3}
\] Indeed \(G_z(B_c(P,Q))=(c^2/4)(|Q|^2/\langle x\rangle^2+
|P|^2/\langle\xi\rangle^2)\); taking its ordinary dual gives precisely
the displayed two blocks. The ratio to \(G_z\) in each original
direction is \(c^2h_G^2/4\). Every full dual-distance bound in Section 5
remains a phase-dual bound, with its constant multiplied by the
corresponding power of \(\max(1,c^2/4)\). Hence (G24)--(G26) apply on
the whole original phase space, with \(h_*=|c|/2\) and ambient dimension
\(2n\). Their counting factor is \((1+|c|/2)^{2n}\); their other
structural constants also retain their indicated dependence on \(c\).

Consequently the actual distributional Fourier multiplier
\(T_{A_c}=\exp(ic\langle D_x,D_\xi\rangle)\) satisfies, for every
integer \(L\geq0\) and every derivative order \(k\), \[
 \begin{split}
 T_{A_c}u&\in\mathcal S_{p,q},\\
 R_L^cu&=T_{A_c}u-
       \sum_{j<L}\frac{(ic\langle D_x,D_\xi\rangle)^j}{j!}u
       \in S(m_{p,q}h_G^L,G),\\
 p_k(R_L^cu;m_{p,q}h_G^L,G)
 &\leq C_{L,k,c}\left(\frac{|c|}{2}\right)^L
             p_{\leq J_{L,k}}(u;m_{p,q},G).
 \end{split}
 \tag{RS4}
\] The fixed phase factor remains in the bound. The Gauss theorem
supplies both finite-seminorm continuity and local smooth continuity on
bounded source sets. For \(c=0\), the multiplier is exactly the
identity; \(R_0^0u=u\) and \(R_L^0u=0\) for \(L\geq1\). These cases
require no zero positive weight.

To identify the Gauss extension with the distributional multiplier, use
the bounded compact approximants constructed below. Their common bound
is a fixed polynomial in \(x,\xi\). Every Schwartz test therefore
supplies an integrable majorant, so the approximants converge in
\(\mathcal S'\). Fourier transformation and multiplication by
\(e^{iA_c}\) are continuous on that space, because their transpose maps
preserve Schwartz space. The Gauss limit is locally smooth and has the
same polynomial bound, so it has the same distributional limit. The
kernel change (W40) is therefore the exact identity
\(\operatorname{Op}_s(T_{A_c}u)=\operatorname{Op}_\tau(u)\) for these
original product-weight symbols, with its original sign.

The full product estimate also has an explicit receiver here. For two
copies of \(G\), its original cross parameter is \(H=h_G\), its
product-space phase parameter is \(h_G/4\), its diagonal metric is
\(2G\), and the product space has dimension \(4n\). Set \[
 w_G(z)=(1+h_G(z)/4)^{4n},\qquad
 1\leq w_G(z)\leq(5/4)^{4n}.
 \tag{RS5}
\] For \(u\in\mathcal S_{p_1,q_1}\), \(v\in\mathcal S_{p_2,q_2}\), (WG4)
reads \[
 \begin{split}
 p_k\!\left(R_L(u,v);
        m_{p_1+p_2,q_1+q_2}h_G^Lw_G,G\right)
 &\leq C_{L,k}\,4^{-L}2^{k/2}2^J
       p_{\leq J}(u;m_{p_1,q_1},G)
       p_{\leq J}(v;m_{p_2,q_2},G),\\
 p_k\!\left(R_L(u,v);
        m_{p_1+p_2,q_1+q_2}h_G^L,G\right)
 &\leq(5/4)^{4n}
       p_k\!\left(R_L(u,v);
        m_{p_1+p_2,q_1+q_2}h_G^Lw_G,G\right).
 \end{split}
 \tag{RS6}
\] Here \(R_0(u,v)=u\#v\), and \(R_L\) retains the original finite
ordered coefficients. The integer \(J\) obeys the provider's (WG3) with
the actual product metric and weights. The second inequality follows by
multiplying each directional-derivative quotient by \(w_G(z)\), without
differentiating that denominator. Thus all the full factors remain, even
when the displayed bounded weight is used to receive the old class.

\subsubsection{13.2. Actual Schwartz compositions for growing
symbols}\label{actual-schwartz-compositions-for-growing-symbols}

We prove the needed common domain for all \(\mathcal S_{p,q}\), rather
than applying the classical bounded-symbol theorem to
\(\langle x\rangle^s\). For \(f\in\mathcal S(\mathbb R^n)\), start with
the original left formula
\(\operatorname{Op}_0(u)f(x)=(2\pi)^{-n}\int e^{ix\cdot\xi}u(x,\xi)\widehat f(\xi)\,d\xi\).
The integral is absolutely convergent on compact sets of \(x\), with all
output derivatives, by (RS2) and Schwartz decay. Differentiating and
integrating by parts in \(\xi\) gives, for every pair \(\alpha,\beta\),
\[
 \begin{split}
 x^\alpha\partial_x^\beta\operatorname{Op}_0(u)f(x)
 &=(2\pi)^{-n}i^{|\alpha|}
   \sum_{\lambda\leq\beta}\binom{\beta}{\lambda}
   \sum_{\kappa+\theta+\omega=\alpha}
        \frac{\alpha!}{\kappa!\theta!\omega!}\\
 &\quad\times\int e^{ix\cdot\xi}
       P_{\beta-\lambda,\kappa}(\xi)
       (\partial_\xi^\theta\partial_x^\lambda u)(x,\xi)
       (\partial_\xi^\omega\widehat f)(\xi)\,d\xi,\\
 P_{v,\kappa}(\xi)
 &=\begin{cases}
 i^{|v|}\dfrac{v!}{(v-\kappa)!}\xi^{v-\kappa},
                  &\kappa\leq v,\\
 0,&\kappa\not\leq v.
 \end{cases}
 \end{split}
 \tag{RS7}
\] The coefficient \(i^{|\alpha|}\) is exactly
\((-1)^{|\alpha|}i^{-|\alpha|}\) from integration by parts. Each
boundary term vanishes by the full Schwartz decay in \(\xi\); (RS2)
gives polynomial growth for all differentiated symbol factors.

Let \(Q_{\theta,\lambda}(u)\) be the supremum in (RS2), with its whole
displayed weight as denominator, and let
\(U_{M,\omega}(f)=\sup_\xi\langle\xi\rangle^M
|\partial_\xi^\omega\widehat f(\xi)|\). For every nonzero summand put
\(E=|\beta-\lambda-\kappa|+q-|\theta|\), and choose an integer
\(M>n+\max E\) over this finite set. With \(p_+=\max(p,0)\), (RS7) gives
the complete bound \[
 \begin{split}
 |x^\alpha\partial^\beta\operatorname{Op}_0(u)f(x)|
 &\leq\langle x\rangle^{p_+}(2\pi)^{-n}
 \sum_{\lambda\leq\beta}\binom{\beta}{\lambda}
 \sum_{\substack{\kappa+\theta+\omega=\alpha\\
                 \kappa\leq\beta-\lambda}}
 \frac{\alpha!(\beta-\lambda)!}
      {\kappa!\theta!\omega!(\beta-\lambda-\kappa)!}\\
 &\quad\times Q_{\theta,\lambda}(u)U_{M,\omega}(f)
                   \int\langle\xi\rangle^{E-M}\,d\xi .
 \end{split}
 \tag{RS8}
\] Every integral is finite: on the dyadic annulus of radius \(2^j\),
its bound is a constant times \(2^{j(n+E-M)}\), a convergent geometric
sum. The original Fourier factor, polynomial derivatives, zero terms and
finite sums all remain.

This polynomial output bound implies rapid output decay, as follows. For
an integer \(k\geq0\), retain the exact polynomial \[
 \langle x\rangle^{2k}
   =\sum_{|\gamma|\leq k}
          \frac{k!}{(k-|\gamma|)!\gamma!}x^{2\gamma}.
 \tag{RS9}
\] Multiply \(x^\alpha\partial^\beta\operatorname{Op}_0(u)f\) by this
polynomial and apply (RS8) separately with \(\alpha+2\gamma\) for every
term. Division by the positive \(\langle x\rangle^{2k}\) yields a
finite-seminorm bound times \(\langle x\rangle^{p_+-2k}\). Given any
desired output decay order, choose \(2k\) larger than that order plus
\(p_+\). The Fourier prerequisite bounds every \(U_{M,\omega}\) by
finitely many original Schwartz input seminorms. Thus
\(\operatorname{Op}_0(u):\mathcal S\to\mathcal S\) is continuous, with
finite symbol-seminorm control, for all real \(p,q\). The quantization
map (RS4), followed by this left action, proves the same statement for
every fixed \(\operatorname{Op}_\tau(u)\), including the original Weyl
operator.

This action also respects bounded-set local smooth convergence. Suppose
\(u_j\to u\) locally smoothly in a bounded set of \(\mathcal S_{p,q}\),
and let the inputs range over a bounded Schwartz set. For the output
\(v_j=\operatorname{Op}_0(u_j-u)f\), the higher seminorms just proved
are uniformly bounded. If \(|x|>R\), some coordinate satisfies
\(|x_i|>R/\sqrt n\), so \[
 |x^\alpha\partial^\beta v_j(x)|
 \leq\frac{\sqrt n}{R}
             \max_i\sup_x|x^{\alpha+e_i}\partial^\beta v_j(x)|.
 \tag{RS10}
\] On \(|x|\leq R\), use (RS7), split the frequency integral at
\(|\xi|=S\), and choose \(M\) larger by one in (RS8). Its integrable
tail tends to zero uniformly in \(j,x,f\). On the remaining compact
phase box, each required symbol derivative tends uniformly to zero and
the input Fourier derivatives are uniformly bounded. First choose
\(R,S\), then \(j\). This proves convergence in every Schwartz output
seminorm, uniformly on the bounded input set. Equation (RS4) supplies
the same convergence for the Weyl action.

Choose a compact smooth scalar \(\zeta\) equal to one on the unit ball,
and put \(u_R(x,\xi)=\zeta(x/R)\zeta(\xi/R)u(x,\xi)\), \(R\geq1\). These
are Schwartz symbols. If \(k\geq1\) derivatives hit one cutoff, their
factor is \(R^{-k}\) on an annulus where its coordinate bracket is
comparable to \(R\). It is therefore bounded by the original bracket to
power \(-k\). The full product rule and (RS2) prove uniform bounds for
\(u_R\) in \(\mathcal S_{p,q}\), and local equality with \(u\) once
\(R\) is large. This constructs the bounded approximants used above,
without asserting convergence in the global symbol seminorms.

For \(u,v\) in two such classes with matching finite matrix fibers,
choose these approximants for both. The full product theorem gives
\(u_R\#v_R\to u\#v\) locally smoothly in a bounded set of the class in
(RS6). Since \(w_G\) is bounded, this is also bounded in
\(\mathcal S_{p_1+p_2,q_1+q_2}\). For each Schwartz input \(f\), write
\(U_R=u_R^w\), \(V_R=v_R^w\). The sequence \(V_Rf\) converges to
\(v^wf\) in Schwartz space and is bounded there. Consequently \[
 \begin{split}
 U_RV_Rf-u^wv^wf
 &=(U_R-u^w)V_Rf+u^w(V_Rf-v^wf)\longrightarrow0,\\
 (u\#v)^w&=u^wv^w:\mathcal S\longrightarrow\mathcal S .
 \end{split}
 \tag{RS11}
\] The first term uses uniform convergence on bounded Schwartz inputs;
the second uses the proved continuous action. The Schwartz-symbol
identity (W27) and convergence of \((u_R\#v_R)^wf\) prove the second
line. Entrywise finite sums retain every intermediate matrix index and
its order. Kernel injectivity from (W20) now proves associativity,
because the two parenthesizations have the same actual Schwartz
composite. When the symbols have bounded weights, (B26) and density
extend the identity to all of \(H_\nu\).

In particular, the three factors of (PT17) belong, respectively, to
\(\mathcal S_{0,s}\), \(\mathcal S_{s,0}\) and \(\mathcal S_{-N,-N}\).
Equation (RS11) proves the exact ordered action \(b_N^w=F_sM_sq^w\) on
Schwartz vectors, including its intermediate domains. Each of the two
product steps retains its factor from (RS6); the resulting seminorm
bound is a finite chain of those full bounds. Formula (PT18) extends
this actual \(B_N\) to a Hilbert--Schmidt map, and the two closed-domain
arguments after (PT19) extend both receiving domains to every \(H_\nu\)
input. Thus the factorization does not apply an unbounded multiplier to
an unspecified distribution.

The Schwartz receiver also supplies the sup-norm step mentioned after
(PT16) without a separate embedding assumption. For a function whose
weighted derivatives are all in \(L^2\), fix an integer \(k>n/2\) and
apply the Fourier transform to each \(x^\alpha\partial^\beta f\). Its
derivatives through \(k\) give an \(L^2\) bound for
\(\langle\xi\rangle^k\) times that transform, by the exact multinomial
comparison (RS9). Cauchy--Schwarz against \(\langle\xi\rangle^{-k}\),
whose square has a convergent dyadic tail, makes the transform
integrable. Inversion with coefficient \((2\pi)^{-n}\) bounds the
corresponding continuous output. Applying this to every weighted
derivative gives all Schwartz seminorms. These Fourier identities first
hold for smooth approximants and extend in \(L^2\); distributional
inversion identifies the original function.

\subsubsection{13.3. The cutoff inverse and both exact
errors}\label{the-cutoff-inverse-and-both-exact-errors}

For a combined coordinate multiindex \(\kappa\ne0\), differentiation of
the actual inverse gives the finite ordered identity \[
 \partial^\kappa a^{-1}
 =\sum_{m=1}^{|\kappa|}(-1)^m
   \sum_{\substack{\kappa_1+\cdots+\kappa_m=\kappa\\
                   |\kappa_i|\geq1}}
       \frac{\kappa!}{\kappa_1!\cdots\kappa_m!}
       a^{-1}(\partial^{\kappa_1}a)a^{-1}
            \cdots(\partial^{\kappa_m}a)a^{-1}.
 \tag{RS12}
\] To prove it, write \(a(z+h)=a(z)+\Delta(h)\). For small \(h\), the
actual inverse is the norm-convergent ordered series
\(\sum_{m\geq0}(-a(z)^{-1}\Delta(h))^m a(z)^{-1}\). For the derivative
of total order \(|\kappa|\) at \(h=0\), every term \(m>|\kappa|\)
vanishes since each \(\Delta(0)=0\). Leibniz's rule on each remaining
ordered term gives exactly the inner sum and its multinomial
coefficient. The norm-convergent inverse and its finite derivatives can
also be justified by differentiating the identity \(a^{-1}a=I_\nu\)
successively; the same ordered sums result. Uniform boundedness of the
inverse and (RC2) bound every term by the full separate \(x,\xi\)
weights. Multiplying by \(\chi\), with the finite full Leibniz sum,
proves the \(b\) membership claimed in Section 6. Its zero extension is
smooth because the support of \(\chi\) is a closed subset of the open
invertibility region, so \(\chi\) vanishes on a neighborhood of every
point outside that region.

We take the chosen \(\varepsilon_*\) at most one, as required by the
original parameter range (PT3); shrinking it preserves its strict
inequality in Section 6. The core part of (PT4) is also explicit: if
\(|z|<R_0\), then \(\varepsilon|z|<1\) for every
\(0\leq\varepsilon\leq\varepsilon_*\), so \(F_\varepsilon(z)=z\) there.
Outside that ball, the two cases in Section 6 give both cutoffs equal to
one. This proves the global equality, including the region where the
original cutoff varies. On the one fixed compact support of \(1-\chi\),
\(h_G\) has a positive minimum and all coordinate brackets have a finite
maximum. Every original derivative quotient of \((1-\chi)I_\nu\) by
\(h_G\) is therefore bounded. Thus its full \(S(h_G,G)\) bound is
independent of \(\varepsilon\).

Equations (RS6) and (RS11) now give the actual errors in (PT5), their
ordered powers in (PT6), and their bounded-set local continuity used in
(PT9) and (PT20). No matrix factors are interchanged. In the latter
factorization the fixed positive multipliers are allowed by
(RS1)--(RS11). The dominated integrals are exactly those displayed in
(PT7) and (PT18), with their original separate coordinate dimensions.
This closes both original trace-norm comparison arguments.

\subsubsection{13.4. The whole admissible radial interval and all
quantizations}\label{the-whole-admissible-radial-interval-and-all-quantizations}

Fix \(0<\delta\leq M\leq\varepsilon_*\), and let \(J=[\delta,M]\). On
\(|z|\geq2/\delta\), the original radial map is
\(F_\varepsilon(z)=z/(\varepsilon|z|)\) for every \(\varepsilon\in J\).
The derivatives in \(z\) of this map and of
\(\partial_\varepsilon F_\varepsilon(z)=-z/(\varepsilon^2|z|)\) have the
complete bounds \(C_{J,\gamma}|z|^{-|\gamma|}\). The images all lie in
the fixed ball of radius \(1/\delta\); all derivatives of \(a\) on that
ball are bounded. Chain rule, including every derivative hitting the
\(\varepsilon\) factor, proves \[
 \|\partial_\varepsilon\partial_z^\gamma
                   (a\circ F_\varepsilon)(z)\|
       \leq C_{J,\gamma}\langle z\rangle^{-|\gamma|}.
 \tag{RS13}
\] On the remaining compact ball the map is smooth in
\((\varepsilon,z)\), so it has the same bound with a larger finite
constant. The fundamental theorem of calculus therefore gives a
Lipschitz bound in each original isotropic symbol seminorm. It also
gives the \(G\) seminorm bound, since
\(\langle x\rangle^{|\alpha|}\langle\xi\rangle^{|\beta|}
\leq\langle z\rangle^{|\alpha|+|\beta|}\). The operator bound (B26)
makes \(\varepsilon\mapsto A_\varepsilon\) norm continuous on \(J\).

Every \(A_\varepsilon\) on \(0\leq\varepsilon\leq\varepsilon_*\) is
Fredholm by (PT8). Its integer index is locally constant on the positive
interval by the complete finite-defect perturbation proof (F6). A
locally constant function on an interval is constant: the set where it
has its value at one point and its complement are both relatively open.
To prove that these cannot both be nonempty, take \(a\) in one set and
\(b\) in the other, ordering them so that \(a<b\), and let \(t\) be the
supremum of the first set's intersection with \([a,b]\). Openness at
\(a,b\) gives \(a<t<b\). If \(t\) is in the first set, openness produces
a larger point of that set before \(b\), contradicting the supremum. If
\(t\) is in the second set, openness excludes points of the first set
just below \(t\), again contradicting that supremum. The small-parameter
equality (PT10), already proved by trace-norm convergence, identifies
the constant. Hence the stronger full-interval result is \[
 \operatorname{ind}(a\circ F_\varepsilon)^w
     =\operatorname{ind}a^w
          \quad(0\leq\varepsilon\leq\varepsilon_*).
 \tag{RS14}
\] This retains the original \(\varepsilon_0\) assertion as a
consequence. It makes no invertibility or Fredholm assertion beyond the
stated admissible interval.

For every fixed real \(\tau\), put \(c=\tau-1/2\). The exact map
(RS3)--(RS4), with \(L=1\), gives {\small\let\split\aligned\let\endsplit\endaligned
\[
 \begin{split}
 u_\tau&=\exp\!\left(i(\tau-1/2)\langle D_x,D_\xi\rangle\right)a,
 &\operatorname{Op}_\tau(a)&=u_\tau^w,\\
 u_\tau-a&\in S(h_G,G),
 &\operatorname{ind}\operatorname{Op}_\tau(a)
     &=\operatorname{ind}a^w .
 \end{split}
 \tag{RS15}
\]
} The operator difference is compact by the finite-matrix (B35) proof,
because \(h_G\to0\). Both operators are bounded by (B26), and (F10)
proves the last equality and Fredholmness. At \(\tau=1/2\) the
difference is exactly zero. At \(\tau=0\) this is precisely
(PT12)--(PT14), including the coefficient \(-i/2\). Applying the same
reasoning to each \(a_\varepsilon\) proves all these index equalities
throughout (RS14); constants may depend on the fixed quantization
parameter. On any enclosing sphere with \(\varepsilon R\leq1\), the
exact boundary maps (PT11) remain unchanged, including the endpoint
differential proved there.

There is an actual strong operator map at the zero endpoint. Local
equality in (RC7), uniform \(G\) bounds and the Schwartz convergence
proof above give \(A_\varepsilon f\to A_0f\) for Schwartz \(f\).
Equation (B26) gives one uniform Hilbert-space norm bound. Approximate
any \(f\in H_\nu\) by a Schwartz vector and use that bound on the
difference; this proves strong convergence on all of \(H_\nu\). The
adjoint has the exact Weyl symbol \(a_\varepsilon^*\), by transposing
the original kernel. These symbols have the same uniform bounds and
local convergence, so the adjoints converge strongly too.

Global norm convergence would be a false strengthening, even for the
original example (RT1). Keep \(n=\nu=1\), \(a(x,\xi)=2+\arctan x\), and
the original coherent vectors \[
 g_{R,R^2}(y)=e^{iR^2(y-R/2)}
                \pi^{-1/4}e^{-(y-R)^2/2},\qquad R>0.
 \tag{RS16}
\] They have norm one. For fixed \(0<\varepsilon\leq1\), set
\((x,\xi)=(R+u,R^2+v)\) in the exact Gaussian Wigner pairing. For each
fixed \(u,v\), its phase length eventually lies in the exterior region,
and \[
 F_{\varepsilon,x}(R+u,R^2+v)
     =\frac{R+u}{\varepsilon
          \sqrt{(R+u)^2+(R^2+v)^2}}\longrightarrow0.
 \tag{RS17}
\] The symbol \(a_\varepsilon\) is bounded by \(2+\pi/2\) in absolute
value, independently of \(R,u,v\). The Wigner formula (CI11), with all
its factors, therefore gives by dominated convergence \[
 \begin{split}
 \langle A_\varepsilon g_{R,R^2},g_{R,R^2}\rangle
  &=\frac1{2\pi}\int_{\mathbb R^2}
       a_\varepsilon(R+u,R^2+v)\,2e^{-u^2-v^2}\,du\,dv
       \longrightarrow2,\\
 \langle A_0g_{R,R^2},g_{R,R^2}\rangle
  &=\pi^{-1/2}\int_{\mathbb R}
       (2+\arctan(R+u))e^{-u^2}\,du
       \longrightarrow2+\pi/2,\\
 \|A_\varepsilon-A_0\|&\geq\pi/2
                         \qquad(0<\varepsilon\leq1).
 \end{split}
 \tag{RS18}
\] The first integral is an exact distributional Weyl pairing: bounded
symbols paired with the Schwartz Wigner function are obtained by compact
approximation, so (CI11) applies without an integrability assumption on
the symbol. The original operator \(A_0\) is multiplication by \(a(x)\);
the modulation cancels in its displayed second pairing. The last
inequality follows from the norm bound on the pairing with each unit
vector and then the two limits. Thus the endpoint has strong convergence
of both operators and adjoints, a trace-norm receiving map for both
error powers, and a concrete uniform obstruction to operator-norm
convergence. Their types and mechanisms are all proved.

\subsubsection{13.5. Separate decay exponents, exact trace and a solved
example}\label{separate-decay-exponents-exact-trace-and-a-solved-example}

The two original coordinate groups allow a further sharp statement. For
real \(P,Q\), let \(u\in\mathcal S_{-P,-Q}\) with coefficients in
\(\operatorname{End}\mathbb C^\nu\). If \(P>n\) and \(Q>n\), put
\(s_x=P/2\), \(s_\xi=Q/2\) and retain the ordered factorization on
\(H_\nu\) {\small\let\split\aligned\let\endsplit\endaligned
\[
 \begin{split}
 A_{P,Q}&=M_{-s_x}F_{-s_\xi},&
 B_{P,Q}&=F_{s_\xi}M_{s_x}u^w=b_{P,Q}^w,\\
 b_{P,Q}&=\langle\xi\rangle^{s_\xi}
                 \#\langle x\rangle^{s_x}\#u
        \in\mathcal S_{-P/2,-Q/2},&
 A_{P,Q}B_{P,Q}&=u^w .
 \end{split}
 \tag{RS19}
\]
} Equations (RS6) and (RS11) prove its symbol and operator maps.
Partial Plancherel of the actual left kernel of \(A_{P,Q}\) and the Weyl
kernel isometry (CI5) give \[
 \begin{split}
 \|A_{P,Q}\|_2^2
   &=\nu(2\pi)^{-n}
          \int\langle x\rangle^{-P}\,dx
          \int\langle\xi\rangle^{-Q}\,d\xi,\\
 \|B_{P,Q}\|_2^2
   &=(2\pi)^{-n}\int
             \operatorname{tr}_{\mathbb C^\nu}(b_{P,Q}^*b_{P,Q})\,dx\,d\xi\\
   &\leq C(2\pi)^{-n}p_J(u;m_{-P,-Q},G)^2
          \int\langle x\rangle^{-P}\,dx
          \int\langle\xi\rangle^{-Q}\,d\xi .
 \end{split}
 \tag{RS20}
\] Both factors are Hilbert--Schmidt because each original
\(n\)-dimensional integral is finite. The exact adjacent inverse
cancellations on Schwartz vectors give (RS19), and density extends it to
\(H_\nu\). Closedness of \(M_{s_x}\) and \(F_{s_\xi}\), by the same two
limit arguments after (PT19), proves both full receiving domains for
every input. The trace-ideal product bound gives a finite-seminorm
trace-norm bound for \(u^w\). For a bounded family converging locally
smoothly, (RS6) gives local smooth convergence of \(b_{P,Q}\); its
squared norm is dominated by
\(C\langle x\rangle^{-P}\langle\xi\rangle^{-Q}\). Dominated convergence
in (CI5), followed by
\(\|A_{P,Q}(B_j-B_0)\|_1\leq\|A_{P,Q}\|_2\|B_j-B_0\|_2\), proves
trace-norm convergence. Thus (PT15) and (PT20) are the diagonal case
\(P=Q=N\) of a proved statement retaining both original weights.

The whole-class thresholds are exact. The original positive symbol \[
 u_{P,Q}(x,\xi)=\langle x\rangle^{-P}
                    \langle\xi\rangle^{-Q}I_\nu
                     \in\mathcal S_{-P,-Q}
 \tag{RS21}
\] has polynomial growth for all real \(P,Q\). Here are its full
separate derivative bounds. For every real \(v\), a derivative of
\(\langle x\rangle^v\) is a finite sum of terms
\(C x^\rho(1+|x|^2)^{v/2-j}\) with \(2j-|\rho|\) equal to the derivative
order. The starting term has \(j=0,\rho=0,C=1\). Differentiating one
such term in \(x_i\) gives the two exact terms
\(C\rho_i x^{\rho-e_i}(1+|x|^2)^{v/2-j}\) and
\(C(v-2j)x^{\rho+e_i}(1+|x|^2)^{v/2-j-1}\); the first is zero when
\(\rho_i=0\). Both retain the relation at the next order. Each term is
bounded by its coefficient times \(\langle x\rangle^{v-|\alpha|}\).
Apply this finite induction separately with \(v=-P\) and \(v=-Q\) in the
two original coordinate groups. It proves every bound in (RS2) for
(RS21), without removing any derivative term. If its Weyl operator were
trace class, the absolute coherent-state bound and positive Gaussian
Wigner function in (CI9)--(CI11) give, by nonnegative Tonelli, \[
 \|u_{P,Q}^w\|_1
   \geq\nu(2\pi)^{-n}
          \int\langle x\rangle^{-P}\,dx
          \int\langle\xi\rangle^{-Q}\,d\xi .
 \tag{RS22}
\] Each coherent pairing is finite because a Gaussian dominates every
polynomial. This argument uses positivity of the symbol and of its
Gaussian pairing; it does not assume positivity of its Weyl operator. If
either coordinate integral diverges, restricting the other coordinate to
its unit ball gives a strictly positive finite lower factor and proves
divergence without an ambiguous product of infinite quantities.

For any real exponent \(t\), on \(2^j\leq|x|<2^{j+1}\) the bracket to
power \(-t\) is bounded above and below by positive constants depending
on \(t\) times \(2^{-jt}\). The annulus volume is exactly
\(|B_1|(2^{(j+1)n}-2^{jn})\), by the affine measure formula. Hence its
tail integral is comparable to \(\sum_{j\geq0}2^{j(n-t)}\): it converges
for \(t>n\), has a logarithmically growing truncated tail at \(t=n\),
and grows by a power for \(t<n\). This proves (RS22) fails to be finite
if \(P\leq n\) or \(Q\leq n\), including negative exponents. The exact
whole-class trace guarantee is therefore \(P>n\) and \(Q>n\). Equation
(PT21) is its original equal-exponent case.

For \(t>n\), the Gamma identity proved in Section 13.2 of the trace
lesson and the \(n\) original Gaussian integrals give \[
 \begin{split}
 I_n(t):=\int_{\mathbb R^n}\langle x\rangle^{-t}\,dx
 &=\frac1{\Gamma(t/2)}
       \int_0^\infty r^{t/2-1}e^{-r}
           \left(\int_{\mathbb R^n}e^{-r|x|^2}\,dx\right)dr\\
 &=\frac{\pi^{n/2}}{\Gamma(t/2)}
       \int_0^\infty r^{(t-n)/2-1}e^{-r}\,dr
  =\pi^{n/2}\frac{\Gamma((t-n)/2)}{\Gamma(t/2)} .
 \end{split}
 \tag{RS23}
\] All interchanges are nonnegative Tonelli; the two endpoints converge
precisely under \(t>n\). With \(P,Q>n\), (RS19)--(RS20) prove trace
class, (RS23) proves symbol integrability, and (CI12) now gives the
exact original trace \[
 \operatorname{Tr}u_{P,Q}^w
  =\nu(2\pi)^{-n}I_n(P)I_n(Q)
  =\nu(2\pi)^{-n}\pi^n
      \frac{\Gamma((P-n)/2)\Gamma((Q-n)/2)}
           {\Gamma(P/2)\Gamma(Q/2)} .
 \tag{RS24}
\] This is a trace, with the original matrix-rank and Fourier factors;
it is not asserted to equal the trace norm.

\textbf{Solved exercise.} Keep \(n=\nu=1\), \(P=2\), \(Q=1\), and the
exact symbol \(u_{2,1}(x,\xi)=(1+x^2)^{-1}(1+\xi^2)^{-1/2}\). It has all
the original \(G\) derivative bounds. Its square is integrable, and
(CI5) gives \[
 \|u_{2,1}^w\|_2^2
   =(2\pi)^{-1}I_1(4)I_1(2)
   =(2\pi)^{-1}\frac{\pi}{2}\,\pi
   =\frac{\pi}{4}.
 \tag{RS25}
\] Here \(I_1(2)=\pi\) follows from the antiderivative \(\arctan x\),
and the substitution \(x=\tan\theta\) gives
\(I_1(4)=\int_{-\pi/2}^{\pi/2}\cos^2\theta\,d\theta=\pi/2\), with both
endpoint limits retained. Alternatively these are the same Gamma values
in (RS23). Its weight tends to zero at full phase-space infinity, so
(B35) also proves compactness. But \(Q=n=1\) makes the original \(\xi\)
integral in (RS22) diverge logarithmically. Its Weyl operator is
consequently Hilbert--Schmidt and compact, and is not trace class. This
example keeps the two actual coordinates and proves the distinction
between those operator ideals.

\begin{figure}
\centering
\pandocbounded{\includegraphics[keepaspectratio,alt={The exact continuity maps and separate product-weight thresholds in dimension one}]{../figures/radial-product-proof-maps.png}}
\caption{The exact continuity maps and separate product-weight
thresholds in dimension one}
\end{figure}

The left panel records the proved endpoint maps (PT20), (RS14) and
(RS18). The right panel uses the original exponents \(P,Q\) with
\(n=1\); its regions concern guarantees for the entire symbol class, and
the marked symbol is the exact solved exercise (RS25). The original
equal-exponent line \(P=Q=N\) is retained. The Hilbert--Schmidt region
follows from (CI5): its weight squared is integrable precisely when
\(2P>1\) and \(2Q>1\), and (RS21) witnesses failure at either
complementary exponent. The figure is a proof diagram and exponent plot,
not a numerical claim of operator positivity. Sections 13.1--13.5 prove
all of its maps, domains, bounds and boundaries. These are editorial
consequences of the retained calculation; no novelty claim is made.

\subsection{14. The exact receiver for the completed finite
coefficient}\label{the-exact-receiver-for-the-completed-finite-coefficient}

The completed
\href{scaled-weyl-index-degree.pdf\#section*.82}{scaled
Weyl coefficient proof} proves the full finite coefficient and its
analytic index identity in all dimensions. Its (DE11) is the identity
with that coefficient; it does not identify the coefficient with an
exterior boundary integral. We now give its receiving map on the
original product-metric symbol. Its cutoff called \(\psi\) is received
here by the separate \(\chi\), and its isotropic dilation parameter is
independent of the radial parameter \(\varepsilon\). The original radial
profile in (RC3) remains \(\psi\).

Fix \(0<\varepsilon\leq\varepsilon_*\). By (RC9), both
\(a_\varepsilon,b_\varepsilon\) are in the original isotropic \(S(1,g)\)
class. By (PT4), their pointwise products are exactly \(\chi I_\nu\),
with the same compact cutoff. At every point where \(\chi(z)\ne0\), the
equality \(\chi(F_\varepsilon(z))=\chi(z)\) puts \(F_\varepsilon(z)\) in
the original invertibility region. Therefore \[
 b_\varepsilon(z)=\chi(z)a_\varepsilon(z)^{-1},
 \qquad
 \sup_{|z|\geq R_0}\|a_\varepsilon(z)^{-1}\|
    \leq\sup_{|w|\geq R_0}\|a(w)^{-1}\| .
 \tag{RS26}
\] The supremum on the right is finite: the original hypothesis bounds
the inverse outside its given ball, and the remaining closed bounded
region with \(|w|\geq R_0\) has a finite bound by continuity of the
inverse. All of that region is invertible since \(\chi=1\) there.
Section 6 proves \(|F_\varepsilon(z)|\geq R_0\) in this exterior region.
The support condition is smooth at its boundary for the same reason as
in (RS12). These are precisely the symbol, matrix, inverse and cutoff
hypotheses of the earlier isotropic proof, at a fixed positive radial
parameter. Its isotropic seminorms may depend on this parameter; none is
asserted uniform at the radial zero endpoint.

Keep \(N=n+1\), the original two orders, and the exact binary
differential coefficient {\footnotesize\let\split\aligned\let\endsplit\endaligned
\[
 \begin{split}
 (f_{1,\varepsilon},g_{1,\varepsilon})
     &=(b_\varepsilon,a_\varepsilon),&
 (f_{2,\varepsilon},g_{2,\varepsilon})
     &=(a_\varepsilon,b_\varepsilon),\\
 C_\ell(u,v)
 &=\sum_{|\alpha|+|\beta|=\ell}
       \left(\frac i2\right)^\ell
       \frac{(-1)^{|\beta|}}{\alpha!\beta!}
       (\partial_x^\alpha\partial_\xi^\beta u)
       (\partial_x^\beta\partial_\xi^\alpha v),\\
 c_{j,0,\varepsilon}&=(1-\chi)I_\nu,&
 c_{j,k,\varepsilon}&=-C_k(f_{j,\varepsilon},g_{j,\varepsilon})
                                      \quad(k\geq1).
 \end{split}
 \tag{RS27}
\]
} This is the original (W26) coefficient with \(D=-i\partial\) and
\(\sigma((x,\xi),(y,\eta))=\xi\cdot y-x\cdot\eta\); the two signs and
each factorial remain. There is no interchange of the matrix arguments.

For each edge \(r<s\) among the \(N\) slots, choose multiindices
\(\alpha_{rs},\beta_{rs}\), and keep {\footnotesize\let\split\aligned\let\endsplit\endaligned
\[
 \begin{split}
 m&=\sum_{r<s}(|\alpha_{rs}|+|\beta_{rs}|),\\
 A_s&=\sum_{t>s}\alpha_{st}+\sum_{r<s}\beta_{rs},&
 B_s&=\sum_{t>s}\beta_{st}+\sum_{r<s}\alpha_{rs},\\
 F_{j,k,\varepsilon}^{(N)}(z)
 &=\sum_{\substack{i_1,\ldots,i_N\geq0,\
                        \alpha_{rs},\beta_{rs}\in\mathbb N^n\\
                    \sum_s i_s+m=k}}
       \left(\frac i2\right)^m
       \frac{(-1)^{\sum_{r<s}|\beta_{rs}|}}
            {\prod_{r<s}\alpha_{rs}!\beta_{rs}!}
       \prod_{s=1}^{N}
          (\partial_x^{A_s}\partial_\xi^{B_s}
                         c_{j,i_s,\varepsilon})(z).
 \end{split}
 \tag{RS28}
\]
} The last product is ordered by increasing slot number. This is
exactly (OC17), with the receiving symbols of (RS27). The complete
finite induction (OC14)--(OC17) proves this formula from the binary
coefficient; no formal series is used. Since the total intrinsic degree
is at most \(k\), at least \(N-k\) of the slots have intrinsic degree
zero when \(k<N\). A derivative of \(c_{j,0,\varepsilon}\) has support
in the original fixed \(\operatorname{supp}(1-\chi)\). Thus every
coefficient with \(k<N\) is compactly supported there. This also
verifies directly the support receiving hypothesis of (DE9), and keeps
every original cutoff contribution.

The earlier (DE5)--(DE11) and complete (OC1)--(OC31) proofs now apply to
this fixed isotropic pair, giving \(\operatorname{ind}a_\varepsilon^w\)
as its exact degree-\(n\) coefficient integral. Composing with the
proved radial and quantization maps (RS14)--(RS15) yields the complete
finite formula on the original operator: \[
 \begin{split}
 \operatorname{ind}\operatorname{Op}_\tau(a)
 &=\operatorname{ind}a^w
  =\operatorname{ind}(a\circ F_\varepsilon)^w\\
 &=(2\pi)^{-n}\int_{\mathbb R^{2n}}
      \operatorname{tr}_{\mathbb C^\nu}
        \bigl(F_{1,n,\varepsilon}^{(n+1)}
              -F_{2,n,\varepsilon}^{(n+1)}\bigr)(x,\xi)
                                             \,dx\,d\xi,\\
 &\hspace{20mm}
       \tau\in\mathbb R,\qquad0<\varepsilon\leq\varepsilon_* .
 \end{split}
 \tag{RS29}
\] The coefficient integral is absolutely defined by the just-proved
compact support; its two matrix orders, rank, signs, Fourier factor and
original coordinates remain. For every \(k<n\), the corresponding
difference of the two coefficient integrals is zero by the same proved
(DE11), while its separate summands need not vanish. At degree zero they
are pointwise equal to \((1-\chi)^{n+1}I_\nu\), as the full zero-slot
specialization of (RS28) also shows.

This is the exact finite ordered matrix formula received from the
completed earlier proof. Together with (PT11), it preserves both the
analytic index and the actual boundary restriction of the original
symbol. No exterior coefficient identity is imported from a later lesson
or an external theorem to prove any result here.

\subsection{15. Exact class comparison and the retained
examples}\label{exact-class-comparison-and-the-retained-examples}

The two-ray exercise has a sharp global counterpart. Write \(X=|x|^2\),
\(Y=|\xi|^2\) while retaining both original weights. Since
\(0\leq XY\leq(X+Y)^2/4\), with the second inequality proved by
\((X-Y)^2\geq0\), the exact product gives \[
 \begin{split}
 h_G&=((1+X)(1+Y))^{-1/2}
       =(1+X+Y+XY)^{-1/2},\\
 \frac{2}{2+X+Y}&\leq h_G\leq\frac1{\sqrt{1+X+Y}},\\
 \frac{2h_g}{1+h_g}&\leq h_G\leq\sqrt{h_g},
          \qquad h_g=(1+X+Y)^{-1}.
 \end{split}
 \tag{RS30}
\] For each fixed \(X+Y\), the lower endpoint occurs at \(X=Y\) and the
upper endpoint at \(XY=0\). In every \(n\geq1\), the original coordinate
groups realize both choices by using their first coordinate axes. Thus
these are sharp bounds at every fixed phase length, with the full
factors \(2\) and \(1\) retained. They reproduce both values and ratios
in (RT3), and do not replace either metric.

For every original direction \(T\), direct coefficient comparison gives
\(g_z(T)\leq G_z(T)\). Applying the original directional-derivative
definition consequently proves the exact continuous identity inclusion
\[
 \iota:S(1,g)\longrightarrow S(1,G),\qquad
       p_k(\iota u;1,G)\leq p_k(u;1,g).
 \tag{RS31}
\] No values, factors or derivatives of \(u\) are changed. The example
(RC10) proves this inclusion is strict. Its precise algebraic defect is
the vector space \(\mathcal Q=S(1,G)/\iota S(1,g)\), with projection
\(\pi u=[u]\). The sequence
\(0\longrightarrow S(1,g)\xrightarrow{\iota}S(1,G)
\xrightarrow{\pi}\mathcal Q\longrightarrow0\) is exact: \(\iota\) is the
identity injection, \(\ker\pi\) is its image by the equivalence defining
the quotient, and every class has a representative. The arctangent
example has a nonzero class. This is an algebraic quotient; no
closedness of that subspace in the original symbol topology is needed or
asserted.

Here is also the full derivative check for both retained arctangent
examples. For \(k\geq1\),
\(\partial_x^k\arctan x=\partial_x^{k-1}(1+x^2)^{-1}\). The exact
bracket derivative induction after (RS21), with exponent \(-2\), gives
\[
 |\partial_x^k\arctan x|
     \leq C_k\langle x\rangle^{-(k+1)}
     \leq C_k\langle x\rangle^{-k};
 \qquad \partial_\xi^\beta\arctan x=0\quad(|\beta|\geq1).
 \tag{RS32}
\] The stronger bound proves all of (RC2); the nondecaying first
derivative at \((0,\xi)\) proves (RC10) exactly as written. The original
strict positive bound for \(2+\arctan x\) follows also from
\(\pi/2=\int_0^\infty(1+t^2)^{-1}dt
=2\int_0^1(1+t^2)^{-1}dt<2\): in the second half of the first integral
use \(t=1/u\), retaining the derivative \(dt=-u^{-2}du\) and reversing
both endpoints. Thus \(2-\pi/2>0\), and the original reciprocal is
bounded. Integration in \(\xi\) in the original Weyl kernel gives the
distribution \(\delta(x-y)\), with coefficient \((2\pi)^{-1}\) exactly
cancelled by Fourier inversion. Multiplying it by \(2+\arctan((x+y)/2)\)
gives multiplication by \(2+\arctan x\). This proves both original
inverse maps on \(L^2\) and the zero index (RT2), without a positivity
claim for the compressed operator.

The quotient defect and the operator-ideal thresholds have a useful
joint example. Choose a nonnegative compact smooth \(\zeta(x)\) equal to
one near the origin, and put \[
 \phi(x)=\zeta(x)e^{x_1},\qquad
 v_\delta(x,\xi)=\phi(x)\langle\xi\rangle^{-\delta}I_\nu,
                    \qquad\delta>0 .
 \tag{RS33}
\] Every derivative of \(\phi\) has compact support, so for every real
\(P\), \(v_\delta\in\mathcal S_{-P,-\delta}\) by (RS2) and the bracket
induction. The exact weight with any \(P>0\) tends to zero at
phase-space infinity. Hence (B35) makes every \(v_\delta^w\) compact.
Formula (CI5) and its inverse kernel map prove it is Hilbert--Schmidt
exactly when \(2\delta>n\), because its squared symbol integral is
\(\nu\int|\phi(x)|^2dx\int\langle\xi\rangle^{-2\delta}d\xi\); the first
factor is finite and strictly positive. For \(\delta>n\), choose \(P>n\)
and apply (RS19)--(RS20) to prove trace class. If \(\delta\leq n\),
positivity of \(\phi\), the exact Gaussian coherent pairing and (RS22)
give an infinite lower bound with the strictly positive factor
\(\int\phi(x)dx\). Therefore \[
 \begin{split}
 v_\delta^w&\text{ is compact for every }\delta>0,\\
 v_\delta^w&\text{ is Hilbert--Schmidt }\Longleftrightarrow\delta>n/2,\\
 v_\delta^w&\text{ is trace class }\Longleftrightarrow\delta>n .
 \end{split}
 \tag{RS34}
\] For the necessity in the Hilbert--Schmidt statement, an assumed
Hilbert--Schmidt extension has its unique \(L^2\) Weyl symbol by the
inverse map after (CI5); distributional kernel injectivity forces that
symbol to be this original \(v_\delta\), so the same integral must be
finite.

Nevertheless every one of these symbols has a nonzero class in
\(\mathcal Q\). Indeed
\(\partial_{x_1}^kv_\delta(0,\xi)=\langle\xi\rangle^{-\delta}I_\nu\) for
every \(k\), since \(\phi=e^{x_1}\) near zero. Choose an integer
\(k>\delta\); the isotropic requirement would bound this by a constant
times \(\langle\xi\rangle^{-k}\), which fails at infinity. More
precisely, any finite set of distinct positive exponents gives linearly
independent quotient classes. If a nonzero linear combination were
isotropic, choose \(k\) larger than all its exponents and apply this
derivative at zero. Along \(\xi=te_1\), multiply by
\(\langle te_1\rangle\) to the smallest exponent having a nonzero
coefficient. All other terms tend to zero and that coefficient remains;
the isotropic bound tends to zero because \(k\) is larger. This
contradiction proves independence. Taking any infinite sequence of
distinct positive exponents makes \(\mathcal Q\) infinite dimensional.
This supplies the exact morphism and defect space relating the two
classes, and shows that the isotropic class obstruction persists at each
of the three proved operator-ideal regimes. These are consequences of
the full original formulas, with no novelty assertion.

\begin{figure}
\centering
\pandocbounded{\includegraphics[keepaspectratio,alt={The exact inclusion, its algebraic defect, and the same positive family through all operator-ideal thresholds}]{../figures/radial-symbol-class-defect.png}}
\caption{The exact inclusion, its algebraic defect, and the same
positive family through all operator-ideal thresholds}
\end{figure}

The diagram keeps the identity inclusion and quotient projection as
distinct exact maps. Its exponent chart is the explicit case \(n=1\) of
(RS34), with strict boundaries; all three arrows concern the same
original \(v_\delta\). The nonzero quotient classes persist throughout
every arrow, and the complete proofs above establish their finite linear
independence. No topology or algebra product is assigned to the quotient
beyond its defined vector-space structure.

\end{document}