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\begin{document}

\section{From Weyl symbols to operators and changes of
coordinates}\label{from-weyl-symbols-to-operators-and-changes-of-coordinates}

A symbol product becomes an operator calculus when both factors act on a
common space. We prove the Schwartz and distribution actions for
symplectically temperate symbols, then study conformal changes of scale,
affine symplectic coordinates and changes of quantization.

Start with \href{metric-localization.pdf}{Localizing symbols with moving
metrics} for symbol seminorms and partitions,
\href{gauss-transform-estimates.pdf}{Quadratic Fourier multipliers at a
moving scale} for the multiplier estimates,
\href{prerequisite-bridges.pdf}{Fourier transforms, finite spectra and
convex separation} for Fourier inversion, and
\href{weyl-metric-products.pdf}{Two measuring scales, one Weyl product}
for Weyl kernels and products. Basic references are {[}Lerner,
Phase-space metrics{]} and {[}Garrett 2014{]}.

The convention is \(D=-i\partial\), on
\(W=\mathbb R_x^n\oplus\mathbb R_\xi^n\), \(n\ge1\), with \[
\sigma((x,\xi),(y,\eta))=\xi\cdot y-x\cdot\eta.
\tag{A1}
\] Write \(q_X=g_X^\sigma\). A symplectically temperate metric is slowly
varying and satisfies, with \(C\ge1\), \(N\ge0\), \[
q_X(T)\le Cq_Y(T)(1+q_Y(X-Y))^N.
\tag{A2}
\] A weight is positive, locally \(g\)-continuous, and satisfies
\(m(Y)\le Cm(X)(1+q_Y(X-Y))^N\), with its own constants.

The additional entry requirements are completeness and compact smooth
density in \(L^2\), and the usual Lebesgue convergence theorems, as in
\href{weyl-metric-products.pdf}{Two measuring scales, one Weyl product}.
The needed Hilbert-space integrals are constructed below. For the
assertion about bounded sets in the strong distribution dual, the
complete-metric Baire proof is
\href{banach-foundation-bridges.pdf}{Section6 of the Banach
foundations};
\href{banach-foundation-bridges.pdf}{Section14.3}
proves the precise common finite-seminorm bound. We prove its needed
consequence below. No Schwartz kernel theorem is needed for the direct
constructions in this lesson: \href{weyl-metric-products.pdf}{Two
measuring scales, one Weyl product} already defines the symbol-to-kernel
map by Fourier transformation.

\subsection{1. Polynomial growth and the two
topologies}\label{polynomial-growth-and-the-two-topologies}

Let \(e(X)=|X|^2\) in the fixed canonical coordinates; \(e^\sigma=e\).
Constants in comparisons with \(e\) may depend on the forms at the
origin. There is a finite \(K\) such that \[
C^{-1}(1+|X|^2)^{-K}e
\le g_X,q_X\le C(1+|X|^2)^K e,
\qquad
C^{-1}(1+|X|^2)^{-K}\le m(X)\le C(1+|X|^2)^K.
\tag{A3}
\] These statements require no uncertainty inequality.

Indeed, (A2) with \(Y=0\) bounds \(q_X\) by a fixed form times a power
of \(1+|X|^2\). In particular \(q_X(X)\) has such a bound. The primal
version of (A2), with the other point at the origin, now gives
\(g_X\le Cg_0(1+q_X(X))^N\). Dualizing the two upper bounds gives both
lower bounds. The weight inequality with one point at zero gives its
upper bound. Its reciprocal is temperate by Section 2 of
\href{weyl-metric-products.pdf}{Two measuring scales, one Weyl product},
giving the lower bound. Enlarging a common \(K\) proves (A3).

Consequently every coordinate derivative of \(a\in S(m,g)\) has
polynomial growth, with degree allowed to depend on the derivative
order. In particular \(a\) defines a tempered distribution. On a bounded
symbol set the growth bound for each fixed derivative is uniform. Thus
bounded symbols converging in \(C^\infty_{\rm loc}\) converge in
\(\mathcal S'\): pair their zeroth-order common polynomial bound with
the rapid decay of a test function, and apply dominated convergence. We
call this bounded-set local-smooth topology the \textbf{weak symbol
topology}, as in \href{gauss-transform-estimates.pdf}{Quadratic Fourier
multipliers at a moving scale}.

For \(u\in\mathcal S(\mathbb R^n)\), use the increasing norms \[
Q_r(u)=\sum_{|\alpha|+|\beta|\le r}\|x^\alpha D^\beta u\|_2.
\tag{A4}
\] They define the usual Schwartz topology. One direction follows by
integrating a sufficiently rapidly decreasing bound. For the other,
choose an integer \(s>n/2\). Fourier inversion, Cauchy--Schwarz and
Schwartz Plancherel give \[
\|v\|_\infty\le (2\pi)^{-n}\|\widehat v\|_1
\le C_s\sum_{|\gamma|\le s}\|D^\gamma v\|_2.
\tag{A5}
\] Apply this to \(v=x^\alpha D^\beta u\) and expand its derivatives.
This controls every usual Schwartz seminorm by a finite \(Q_r\). The
familiar completeness of Schwartz space also follows directly: uniform
limits of all weighted derivatives are compatible derivatives by the
fundamental theorem of calculus on coordinate boxes.

On the complex-linear distribution dual \(\mathcal S'\) we use the
\textbf{strong dual topology}. In particular, the distribution bracket
is linear in its test function; Hilbert-space adjoints will be converted
to bilinear transposes explicitly. The strong-dual seminorms are \[
p_B(u)=\sup_{\phi\in B}|\langle u,\phi\rangle|,
\tag{A6}
\] where \(B\) ranges over bounded subsets of \(\mathcal S\). A bounded
subset \(\mathcal U\) of this dual satisfies a common finite-order
estimate \[
|\langle u,\phi\rangle|\le C Q_r(\phi)
\quad(u\in\mathcal U,\ \phi\in\mathcal S).
\tag{A7}
\] Here is the Baire argument, including the point where that
prerequisite enters. Strong boundedness gives pointwise boundedness on
each \(\phi\). The closed sets
\(\{\phi:\sup_{u\in\mathcal U}|\langle u,\phi\rangle|\le j\}\),
\(j=1,2,\ldots\), cover the complete metrizable space \(\mathcal S\).
One has interior. Subtract two points in a small neighborhood inside it
to obtain a uniform bound on a neighborhood of zero. That neighborhood
contains a set \(Q_r<\varepsilon\); rescaling proves (A7). No assertion
of joint continuity of the unrestricted distribution action follows from
this argument.

\subsection{2. An affine-invariant Fourier
norm}\label{an-affine-invariant-fourier-norm}

For \(a\in\mathcal S(W)\), set \[
\|a\|_{\mathcal F L^1}=(2\pi)^{-2n}\|\widehat a\|_{L^1(W^*)}.
\tag{A8}
\] Then \[
\|a^w\|_{L^2\to L^2}\le \|a\|_{\mathcal F L^1}.
\tag{A9}
\] To prove it, write \(a\) by Fourier inversion as the integral of
plane waves. Section 5 of \href{weyl-metric-products.pdf}{Two measuring
scales, one Weyl product} gives norm-one operators for every such wave.
For \(u\in\mathcal S\), their actions depend continuously on the wave
frequency in \(L^2\), by the explicit translation and modulation
formulas. Multiply this continuous Hilbert-space-valued function by
\(\widehat a\). On every compact cube its Riemann sums converge in
\(L^2\): uniform continuity makes the difference between any two
sufficiently fine sums arbitrarily small, and \(L^2\) is complete. The
norm of each resulting integral is at most the integral of
\(|\widehat a|\|u\|_2\). This also bounds differences between integrals
over increasing cubes, whose tails tend to zero because
\(\widehat a\in L^1\). It constructs the full integral in \(L^2\) and
proves its triangle inequality. Pairing with a Schwartz test and using
scalar Fourier inversion identifies it with \(a^wu\). Formula (A9)
follows, first on Schwartz inputs and then on \(L^2\) by density. Thus
no estimate for a general oscillatory kernel or general vector-valued
integration theorem is being assumed.

This norm is unchanged by any invertible affine change of the
phase-space variables, even one that is not symplectic. If
\(b(X)=a(TX+Z)\), then \[
\widehat b(\Theta)
=|\det T|^{-1}e^{i\langle T^{-t}\Theta,Z\rangle}
\widehat a(T^{-t}\Theta).
\] Changing variables in its \(L^1\) norm cancels the determinant.

It is also controlled by finitely many Schwartz seminorms. More
particularly, if \(a\) is supported in a fixed-radius ball for a
positive form \(Q\), centered at \(X_0\), then \[
\|a^w\|\le C\sum_{j\le s}\sup_X|a|_{j,Q}(X),
\qquad s>n\text{ an integer}.
\tag{A10}
\] Use an affine map making \(Q\) Euclidean and its center zero. The
Fourier \(L^1\) estimate used in (A5), now in dimension \(2n\), requires
\(s>(2n)/2=n\). On the fixed support, the derivative \(L^2\) norms are
bounded by their suprema. Affine invariance proves (A10) with no
determinant of \(Q\) left over.

\textbf{Editorial reconstruction: the original support form and every
Jacobian.} Retain the positive form \(Q\), its original coordinate
matrix \(G\), the center \(X_0\), and the support condition
\(Q(X-X_0)\le R^2\). Write \(d=2n\). Choose an invertible real matrix
\(L\) with \(L^tGL=I\), and put \(J=|\det L|\), \(b(z)=a(X_0+Lz)\). This
is a comparison map; the original form and symbol remain the objects in
(A10). The original Fourier transforms give \[
\begin{aligned}
\widehat b(\eta)&=J^{-1}e^{i\langle L^{-t}\eta,X_0\rangle}
                  \widehat a(L^{-t}\eta),\\
\widehat a(\Theta)&=J e^{-i\langle\Theta,X_0\rangle}
                    \widehat b(L^t\Theta),\\
\int|\widehat a(\Theta)|\,d\Theta
 &=\int J|\widehat b(L^t\Theta)|\,d\Theta
  =\int J|\widehat b(\eta)|J^{-1}\,d\eta.
\end{aligned}
\tag{AF1}
\] The inverse Fourier factor is still \((2\pi)^{-d}\). For an integer
\(s>d/2\), put \(I_{d,s}=\int(1+|\eta|^2)^{-s}\,d\eta\). It is finite:
the unit ball contributes at most its finite volume, and the shell
\(2^r\le|\eta|<2^{r+1}\) contributes at most \(\omega_d2^d2^{r(d-2s)}\),
a summable geometric series. Cauchy--Schwarz, the full multinomial and
Plancherel give \[
\begin{aligned}
\|a\|_{\mathcal FL^1}
 &\le (2\pi)^{-d/2}I_{d,s}^{1/2}
 \left(\sum_{|\alpha|\le s}
 \frac{s!}{(s-|\alpha|)!\,\alpha!}\|D_z^\alpha b\|_2^2\right)^{1/2},\\
\|D_z^\alpha b\|_2^2
 &=J^{-1}\int_{Q(X-X_0)\le R^2}
  \left|\prod_{r=1}^d\left(\sum_{i=1}^dL_{ir}D_{X_i}\right)^{\alpha_r}a(X)\right|^2dX\\
 &\le J^{-1}(J\omega_dR^d)
                \left(\sup_X|a|_{|\alpha|,Q}(X)\right)^2.
\end{aligned}
\tag{AF2}
\] Each column \(Le_r\) has original \(Q\)-length one. The full
coordinate expansion in the second line is the sum over all
\(\beta_{ir}\ge0\) with \(\sum_i\beta_{ir}=\alpha_r\), with coefficient
\(\prod_r\alpha_r!\prod_{i,r}L_{ir}^{\beta_{ir}}/\beta_{ir}!\) and
derivative \(\prod_iD_{X_i}^{\sum_r\beta_{ir}}a\). Thus no coordinate
coefficient or derivative has been removed. The support measure is
exactly \(J\omega_dR^d\), and the derivative integral has the separate
factor \(J^{-1}\); (AF2) displays both before their cancellation.
Applying (A9) proves (A10), with its original operator and original
directional seminorms. This arbitrary affine comparison proves a Fourier
norm estimate. Unitary covariance of a Weyl operator has the separate
symplectic hypothesis in Section6.

\subsection{3. Localized decay and
counting}\label{localized-decay-and-counting}

Choose the partition \((\phi_\nu)\) of
\href{metric-localization.pdf}{Localizing symbols with moving metrics}
and fixed radii \(0<a<b<c<r_*\), with supports in \(B_{X_\nu}(a)\). Put
\[
g_\nu=g_{X_\nu},\quad m_\nu=m(X_\nu),\quad
U_\nu=B_{X_\nu}(b),\quad U'_\nu=B_{X_\nu}(c),\quad
a_\nu=\phi_\nu a.
\] The larger balls have bounded multiplicity. Localization gives
\(\sup|a_\nu|_{j,g_\nu}\le C_jm_\nu p_{\le j}(a;m,g)\). The notation
\(p_{\le j}\) denotes the sum of symbol seminorms of orders at most
\(j\).

The form \[
r_\nu=(e+g_\nu)^\sigma,\qquad
r_\nu(Y)=\min_{Y=U+V}\big(g_\nu^\sigma(U)+|V|^2\big),
\qquad
R_\nu=\inf_{Y\in U_\nu}r_\nu(Y)^{1/2}
\tag{A11}
\] uses the dual-of-a-sum identity proved in Section 1 of
\href{weyl-metric-products.pdf}{Two measuring scales, one Weyl product}.
There is no factor2 here, since the primal form is a sum rather than a
mean. All forms are positive definite and the balls nonempty, so
\(R_\nu\) is finite; it can be zero.

One may replace \(e\) throughout this construction by any fixed positive
form \(e_0\). There are constants \(c,C>0\) with \(ce\le e_0\le Ce\).
Adding \(g_\nu\), and then dualizing, shows that \((e_0+g_\nu)^\sigma\)
and \(r_\nu\) are comparable with constants independent of \(\nu\).
Their distances and center-growth bounds are consequently equivalent.
Choosing canonical Euclidean coordinates therefore imposes no
restriction on the reference form.

\textbf{Localized decay.} For each integer \(M\ge0\) there is a finite
\(J_M\) such that \[
\|a_\nu^w u\|_2
\le C_Mm_\nu(1+R_\nu)^{-M}
p_{\le J_M}(a;m,g)Q_M(u).
\tag{A12}
\] The constants are uniform in \(\nu\).

Here are the geometric and differential details. If \(R_\nu>0\),
minimize the \(r_\nu\)-norm on the compact closure of \(U_\nu\). Let
\(P\) be a minimizing point. Differentiating along each segment from
\(P\) into that closed ellipsoid gives a supporting linear form \[
L(Y)=r_\nu(P,Y)/r_\nu(P)^{1/2}
=\sigma(Y,t),\qquad
e(t)+g_\nu(t)=1,\qquad
L(Y)\ge R_\nu\quad(Y\in U_\nu).
\tag{A13}
\] The normalization follows by ordinary quadratic duality, composed
with \(\sigma\). In particular \(L_\nu=L(X_\nu)\ge R_\nu\). Positivity
on the \(g_\nu\)-ball of radius \(b\) gives
\(\sqrt{g_\nu^\sigma(t)}\le L_\nu/b\). On the support ball of radius
\(a\), therefore, \(L\ge(1-a/b)L_\nu\). Differentiating the reciprocal
shows explicitly that \[
\sup|L_\nu/L|_{k,g_\nu}
\le k!\,b^{-k}(1-a/b)^{-k-1}.
\tag{A14}
\] Division of a supported smooth symbol by \(L\), followed by extension
by zero outside \(U_\nu\), is consequently smooth and preserves its
support.

The affine Weyl identity of Section 5 of
\href{weyl-metric-products.pdf}{Two measuring scales, one Weyl product},
used on the right, is \[
a_\nu^wu
=(a_\nu/L)^w L^wu
+\frac{i}{2}\{a_\nu/L,L\}^wu,
\qquad
\{a_\nu/L,L\}=-L^{-1}\partial_t a_\nu.
\tag{A15}
\] There is no derivative of \(L\) in the last expression, since
\(\partial_t L=\sigma(t,t)=0\). Both new symbols have frozen seminorm
bounds with weight \(m_\nu/L_\nu\); the differentiated one uses one
additional source derivative and \(g_\nu(t)^{1/2}\le1\). Repeat (A15)
\(M\) times. Every resulting symbol has weight \(m_\nu L_\nu^{-M}\), and
every input is \((L^w)^j u\) with \(j\le M\). Since \(|t|\le1\),
expanding these powers and commuting coordinates and derivatives bounds
their \(L^2\) norms by \(C_MQ_M(u)\). Apply (A10) to each of the
finitely many terms. This proves (A12) when \(R_\nu>1\). For
\(R_\nu\le1\), (A10) itself proves it after enlarging the constant. Thus
zero distance causes no division by zero.

\textbf{Polynomial counting.} There are finite constants \(B,P,C\) such
that \[
|X_\nu|^2\le C(1+R_\nu^2)^B,\qquad
\#\{\nu:R_\nu\le r\}\le C(1+r)^P.
\tag{A16}
\] Consequently \[
\sum_\nu(1+R_\nu)^{-L}<\infty\qquad(L>P).
\tag{A17}
\]

We give the missing volume argument in full. Let \(T\ge1\) and
\(R_\nu^2\le T\). Choose \(Y\in U_\nu\) with \(r_\nu(Y)\le2T\), and take
a minimizing decomposition \(Y=U+V\) in (A11). Then
\(g_\nu^\sigma(U)\le2T\) and \(|V|^2\le2T\). Slow variation gives
\(q_Y(U)\le C T\). Formula (A2), comparing \(V\) to \(Y\), implies \[
q_V\le CT^Nq_Y,\qquad q_V(U)\le CT^{N+1}.
\tag{A18}
\] By (A3), both \(g_V\) and \(q_V\) lie between \(C^{-1}T^{-K}e\) and
\(CT^Ke\). Hence \(|U|^2\le CT^{K+N+1}\). Applying (A2) in the other
direction, with its now controlled distance \(q_V(U)\), gives \[
q_Y\le CT^{K+N(N+1)}e,
\qquad
g_Y\le CT^{K+N}e.
\tag{A19}
\] Dualizing supplies the lower bounds as well. Take, for example,
\(B=K+(N+1)^2\), enlarged if needed to absorb the preceding fixed
exponents. Local comparison with \(g_\nu\), and \(g_\nu(X_\nu-Y)<b^2\),
now give \[
C^{-1}T^{-B}e\le g_\nu\le CT^Be,\qquad
|Y|^2+|X_\nu|^2\le CT^B.
\tag{A20}
\]

A Euclidean ball centered at this \(Y\) with radius \(\kappa T^{-B/2}\)
lies in \(U'_\nu\), for a fixed small \(\kappa>0\), because its
\(g_\nu\)-radius is less than the gap \(c-b\). These small balls inherit
the bounded multiplicity of \(U'_\nu\), while all lie in one Euclidean
ball of radius \(C T^{B/2}\). Integrating their characteristic functions
bounds the number of indices by \(CT^{2nB}\). The argument applies first
to every finite subcollection, so it bounds the whole index set. This
proves (A16), for example with \(P=4nB\). Its center bound follows by
taking \(T=1+R_\nu^2\). Finally split the indices into dyadic bands of
\(1+R_\nu\); their contributions in (A17) form a geometric series when
\(L>P\). No uncertainty assumption or bound on a vertical part of the
metric was used.

\subsection{4. Action on Schwartz functions and
distributions}\label{action-on-schwartz-functions-and-distributions}

\textbf{Theorem 4.1 (Schwartz and distribution action).} If \(g\) is
symplectically temperate and \(m\) is a temperate weight for \(g\), then
\[
a^w:\mathcal S\longrightarrow\mathcal S,\qquad
a^w:\mathcal S'\longrightarrow\mathcal S'
\quad(a\in S(m,g))
\tag{A21}
\] are continuous. For each \(r\) there are finite \(J,t\) such that \[
Q_r(a^wu)\le C_r p_{\le J}(a;m,g)Q_t(u).
\tag{A22}
\] The first action is jointly continuous in symbol and function. Both
actions depend continuously on a bounded symbol set with its weak symbol
topology, uniformly on bounded sets of inputs, using the Schwartz
topology or the strong distribution topology respectively. The
distribution action is separately continuous and hypocontinuous: fixing
a bounded set in either factor gives continuity uniformly over that set.
It is not, in general, jointly continuous on the unrestricted product.

To prove (A22) first for \(r=0\), sum (A12). By (A3) and (A16),
\(m_\nu\) is bounded by a power of \(1+R_\nu\). Choosing \(M\) larger
than that power plus \(P\) makes the sum finite by (A17).

For all output derivatives and moments, let \(F(X)=\sigma(X,t)+d\) be a
fixed affine form. The distributional identity \[
F^w a_\nu^w
=\big(Fa_\nu+\{F,a_\nu\}/(2i)\big)^w
\tag{A23}
\] holds before any general operator-composition theorem. On the support
ball, \[
|F(X)|\le |F(X_\nu)|+a\sqrt{g_\nu^\sigma(t)},\qquad
|\partial_t a_\nu|_{k,g_\nu}
\le \sqrt{g_\nu(t)}\,|a_\nu|_{k+1,g_\nu}.
\] Both forms \(g_\nu,g_\nu^\sigma\) obey (A3). Thus the new symbol in
(A23) has frozen seminorms bounded by
\(C_km_\nu(1+|X_\nu|^2)^{K_1}p_{\le k+1}(a;m,g)\). Iterating for any
fixed word of \(r\) coordinates and derivatives gives the same assertion
with a finite exponent \(K_r\) and \(r\) additional derivatives. Apply
(A12) to these new supported symbols, then use (A16) and choose its
decay order large enough to dominate the polynomial factors and the
counting exponent. This proves (A22) for every word, hence every
\(Q_r\).

Each partial sum \(\sum a_\nu^wu\) is Schwartz, and the estimates just
proved make the series Cauchy in all Schwartz seminorms. It therefore
converges in \(\mathcal S\). The symbol partial sums remain bounded in
\(S(m,g)\) and converge locally smoothly to \(a\). By (A3) they converge
in \(\mathcal S'\), so continuity of the distributional kernel
construction identifies the Schwartz limit with the already defined
\(a^wu\). This also proves independence of the partition.

For weak symbol continuity, the summable tails of these estimates are
uniform on bounded symbol sets and on bounded sets of Schwartz inputs.
Each finite sum is continuous under local smooth convergence: its
symbols have fixed compact support, and (A10) and the preceding finite
derivative estimates apply. Uniformly small tails and continuous finite
sums prove the assertion for \(\mathcal S\).

Let \(C\phi=\overline\phi\). The adjoint identity
\((a^w)^*=(\overline a)^w\) gives the bilinear transpose test map
\((a^w)^t=C(\overline a)^wC\). Define the extension by
\(\langle a^wu,\phi\rangle=\langle u,(a^w)^t\phi\rangle\). It agrees
with the operator already constructed on regular distributions from
Schwartz functions. Conjugation preserves every \(Q_r\), so a bounded
set of tests has bounded image under this transpose for fixed \(a\),
proving strong-dual continuity through (A6). For symbols in a bounded
set, (A22) makes the union of those images bounded, giving uniform
continuity in the distribution input. For distributions in a bounded
set, apply (A7) and then (A22) to the transpose tests; this gives a
finite symbol-seminorm estimate, uniformly over that set. The weak
symbol continuity already proved for bounded sets of test functions
gives the corresponding strong-dual conclusion. These observations prove
all stated hypocontinuity and bounded-set assertions.

The distinction from unrestricted joint continuity is necessary. Already
take \(g=e\), \(m=1\), and symbols independent of \(\xi\), so that
quantization is ordinary multiplication. A symbol neighborhood controls
only finitely many seminorms, say through order \(J\). With a fixed
cutoff \(\chi=1\) near zero, the functions \[
a_k(x)=c k^{-J}\chi(x)e^{ikx_1}
\tag{A24}
\] lie in that neighborhood for a sufficiently small fixed \(c>0\).
Every neighborhood of zero in \(\mathcal S'\) contains a sufficiently
small fixed multiple of \(\partial_{x_1}^{J+1}\delta_0\), because scalar
multiplication is continuous. Pairing its product with \(a_k\) against a
test equal to one near zero has magnitude proportional to \(k\). Thus no
pair of unrestricted input neighborhoods controls even this one scalar
output seminorm. The same argument applies to the weak distribution
topology. For a bounded set of distributions, (A7) rules out this defect
by supplying a common order.

\textbf{Theorem 4.2 (operator composition).} Under either the one-metric
or the compatible two-metric hypotheses of Theorem 7.1 of
\href{weyl-metric-products.pdf}{Two measuring scales, one Weyl product},
\[
(a\#b)^w=a^wb^w
\quad\hbox{on both }\mathcal S\hbox{ and }\mathcal S'.
\tag{A25}
\] Choose bounded Schwartz approximants \(a_j,b_j\) from
\href{metric-localization.pdf}{Localizing symbols with moving metrics}.
The identity holds for them by Section 6 of
\href{weyl-metric-products.pdf}{Two measuring scales, one Weyl product}.
Formula (A22) makes the family \(a_j^w\) equicontinuous on Schwartz
space, while weak symbol continuity gives \(b_j^wu\to b^wu\) and
\(a_j^wb^wu\to a^wb^wu\) in \(\mathcal S\). Thus their composites
converge to \(a^wb^wu\). The symbol products are bounded and converge
weakly to \(a\#b\) by Theorem 7.1 of \href{weyl-metric-products.pdf}{Two
measuring scales, one Weyl product}; the theorem just proved identifies
the other limit with \((a\#b)^wu\). This proves (A25) on \(\mathcal S\)
without circularity.

For completeness, Schwartz functions are weakly dense in
\(\mathcal S'\). If \(\rho\) is a compact smooth mollifier of integral
one and \(\chi\) a compact cutoff equal to one near zero, then
\(\chi(X/R)(u*\rho_{1/R})(X)\) is Schwartz and converges to \(u\)
distributionally. On tests this is the convergence
\(\check\rho_{1/R}*(\chi(\cdot/R)\phi)\to\phi\) in \(\mathcal S\),
proved by Taylor's formula and the rapid decay seminorms. All fixed
operators in (A25) are transposes of continuous Schwartz maps and are
therefore weakly continuous on \(\mathcal S'\). The identity extends
from the dense subspace to every distribution.

\textbf{Editorial extension: the actual operator product for an
unbounded cross parameter.} Retain the compatible metrics \(g_1,g_2\),
the original own and cross weight hypotheses of the distinct-metric
theorem, their mean \(g=(g_1+g_2)/2\), and their original cross
parameter \(H\). No upper bound on \(H\) is needed for the following
larger receiving classes. Equations (WG1)--(WG6) in
\href{weyl-metric-products.pdf}{Two measuring scales, one Weyl product}
prove \[
\begin{aligned}
w_\#(X)&=(1+H(X)/4)^{4n},\\
a\#b&\in S(m_1m_2w_\#,g),\\
a\#b-\sum_{j<N}C_j(a,b)&\in S(m_1m_2H^Nw_\#,g).
\end{aligned}
\tag{AO1}
\] Those proofs retain \(4^{-N}\), the diagonal factor \(2^{l/2}\), the
tensor factor \(2^J\), all strict tail thresholds and all derivative
orders. Their metric and target-weight comparisons prove that \(g\) is
symplectically temperate and the displayed positive weights are
legitimate. The Schwartz action (A21)--(A22) has no uncertainty or
upper-parameter hypothesis, so it applies to each original input and to
this actual target.

Here is the receiving operator proof. Choose the original bounded
compact Schwartz approximants \(a_j,b_j\). For fixed \(u\in\mathcal S\),
(A22) gives \(b_j^wu\to b^wu\) in every original Schwartz seminorm, and
bounds the entire family \(a_j^w\) by one finite input-seminorm estimate
in each output seminorm. Therefore \[
a_j^wb_j^wu-a^wb^wu
=a_j^w(b_j^wu-b^wu)+(a_j^w-a^w)b^wu\longrightarrow0
\quad\hbox{in }\mathcal S.
\tag{AO2}
\] The Schwartz-symbol kernel identity gives
\(a_j^wb_j^wu=(a_j\#b_j)^wu\). The bounded-set continuity of (AO1) gives
\(a_j\#b_j\to a\#b\) locally smoothly in a bounded subset of its larger
target; (A22) applies there and identifies its Schwartz limit. Hence \[
(a\#b)^w=a^wb^w:\mathcal S\longrightarrow\mathcal S.
\tag{AO3}
\] Each fixed operator extends by its bilinear transpose to the strong
dual, with \(p_B(Au)\le p_{A^tB}(u)\), since \(A^tB\) is a bounded
Schwartz set. It is also weakly continuous. The actual
cutoff-and-mollifier density proof following (A25) therefore extends
(AO3) to every \(u\in\mathcal S'\). The composite is the continuous map
\(\mathcal S'\to\mathcal S'\) on this full original domain. For finite
rectangular matrices, its \((i,k)\) entry is
\(\sum_{j=1}^q a_{ij}\#b_{jk}\), and the action is
\(\sum_{j=1}^q a_{ij}^wb_{jk}^w\); all summands, their order and their
finite seminorm bounds remain. This does not replace the distinct-metric
cross-weight hypotheses by own temperateness alone.

\subsection{5. Conformal changes of measuring
scale}\label{conformal-changes-of-measuring-scale}

\textbf{Theorem 5.1 (conformal enlargement).} Let \(g\) be
symplectically temperate. Suppose \[
G_X=\mu(X)g_X,\qquad \mu\ge1,
\tag{A26}
\] is slowly varying and satisfies \(G\le G^\sigma\). Then \(G\) is
symplectically temperate. No temperateness or ordinary continuity of
\(\mu\) is an assumption.

Put \(h_g^2=\sup g/g^\sigma\). The uncertainty condition for \(G\) is
\(\mu^2h_g^2\le1\). We prove \(G_Y\le C G_X(1+G_Y^\sigma(X-Y))^L\). Set
\(d=X-Y\), \(a=\mu(X)\), \(b=\mu(Y)\), and \(t=G_Y^\sigma(d)=q_Y(d)/b\).
If \(G_X(d)\) is within the slow-variation radius of \(G\), its local
comparison proves the assertion.

Otherwise \(G_X(d)\ge c_G>0\). If \(g_Y(d)\) is within the
slow-variation radius of \(g\), then \(g_X\) and \(g_Y\) are comparable
and \[
c_G\le a g_X(d)\le C a g_Y(d)
\le C a h_g(Y)^2q_Y(d)
=C(a/b)\big(b^2h_g(Y)^2\big)t\le C(a/b)t.
\] Thus \(b/a\le Ct\), and \(G_Y/G_X\le Ct\). In the remaining case
\(g_Y(d)\ge c_g>0\). Since \(G_Y\le G_Y^\sigma\), \[
b c_g\le b g_Y(d)\le t,\qquad q_Y(d)=bt\le c_g^{-1}t^2.
\] The original temperateness gives \[
G_Y=(b/a)(g_Y/g_X)G_X
\le C(1+t)^{2N+1}G_X,
\] where \(a\ge1\) was used. These three cases prove the theorem with
finite structural constants.

\subsubsection{A positive lower bound for the original conformal
factor}\label{a-positive-lower-bound-for-the-original-conformal-factor}

\textbf{Editorial strengthening of Theorem 5.1.} The hypothesis
\(\mu\ge1\) can be replaced by \(\mu\ge\mu_0>0\), with any fixed
\(\mu_0\). Retain the original \(g\) and \(G=\mu g\); assume exactly as
before that \(g\) is symplectically temperate, \(G\) is slowly varying
and \(G\le G^\sigma\). Then \(G\) is symplectically temperate, with
constants also depending on \(\mu_0\). No differentiability or
temperateness of \(\mu\) is added.

Here is the full three-case comparison on these original forms. Put
\(d=X-Y\), \(a=\mu(X)\), \(b=\mu(Y)\) and \(t=G_Y^\sigma(d)=q_Y(d)/b\).
If \(G_X(d)\) is below its fixed slow-variation radius, local comparison
gives \(G_Y\le C G_X\). Otherwise \(G_X(d)\ge c_G>0\). If \(g_Y(d)\) is
below its own slow-variation radius, then \(g_X\le Cg_Y\), and \[
 c_G\le a g_X(d)\le C a h_g(Y)^2q_Y(d)
       =C(a/b)\big(b^2h_g(Y)^2\big)t\le C(a/b)t.
 \tag{A56}
\] The last inequality is exactly the uncertainty inequality for the
original \(G_Y\). It gives \(b/a\le Ct\) and therefore \(G_Y\le CtG_X\).
In the remaining case \(g_Y(d)\ge c_g>0\), use \(G_Y\le G_Y^\sigma\) to
get \[
 b c_g\le b g_Y(d)\le t,\quad
 q_Y(d)=bt\le c_g^{-1}t^2,\quad
 \frac ba\le\frac{t}{c_g\mu_0}.
 \tag{A57}
\] The original own temperateness gives \(g_Y\le Cg_X(1+q_Y(d))^N\).
Hence \[
 G_Y\le \frac{C}{c_g\mu_0}\,
             t(1+c_g^{-1}t^2)^N G_X
       \le C_{\mu_0}(1+t)^{2N+1}G_X.
 \tag{A58}
\] These three comparisons are precisely temperateness of \(G\) with the
distance based at \(Y\). Every form and scalar factor is retained. In
particular the result applies when the conformal factor is uniformly
positive but smaller than one at some points; no replacement of the
original metric is needed.

\textbf{Proposition 5.2 (conformal compatibility).} Suppose \(g_1,g_2\)
are symplectically temperate, \(g_2=\mu g_1\) for a positive function
\(\mu\), and \(h_1,h_2\le1\). Then the pair is compatible in the precise
sense of Section 2 of \href{weyl-metric-products.pdf}{Two measuring
scales, one Weyl product} and \[
h_2=\mu h_1,\qquad H^2=h_1h_2,\qquad H=\sqrt{h_1h_2}\le1.
\tag{A27}
\] The identities follow from quadratic scaling and the definition of
\(H\). To prove the first cross test, write \(q_j=g_j^\sigma\). Own
temperateness also has a distance bound based at \(X\): \[
q_{1,X}\le Cq_{1,Y}(1+q_{1,X}(X-Y))^L.
\] Indeed the reverse own comparison bounds the \(Y\)-based distance by
a power of the \(X\)-based distance, which can be inserted into (A2). If
\(g_{1,X}(X-Y)\) is small, slow variation gives the cross comparison
directly. If it is at least \(c>0\), use \(\mu(X)h_1(X)=h_2(X)\le1\) to
get \[
q_{2,X}(X-Y)^2
=\mu(X)^{-2}q_{1,X}(X-Y)^2
\ge h_1(X)^2q_{1,X}(X-Y)^2
\ge c\,q_{1,X}(X-Y).
\] The last displayed own bound is therefore controlled by a power of
\(1+q_{2,X}(X-Y)\). This is the first cross test. Interchange the two
metrics for the other. Their scalar factor need not be bounded above or
below by a fixed constant.

\subsection{6. Affine symplectic
covariance}\label{affine-symplectic-covariance}

A map \(\chi(X)=SX+Z\) is affine symplectic when
\(\sigma(ST,SU)=\sigma(T,U)\). The following five types generate all
such maps: \[
\begin{gathered}
(x,\xi)\mapsto(x+a,\xi),\qquad
(x,\xi)\mapsto(x,\xi+b),\\
(x_j,\xi_j)\mapsto(\xi_j,-x_j),\\
(x,\xi)\mapsto(Tx,T^{-t}\xi),\qquad
(x,\xi)\mapsto(x,\xi-Ax),\quad A=A^t.
\end{gathered}
\tag{A28}
\]

Here is a block proof that also handles singular position blocks.
Translations remove \(Z\). Write the linear symplectic matrix as
\(S=\left(\begin{smallmatrix}A&B\\ C&D\end{smallmatrix}\right)\). Its
first \(n\) columns are linearly independent and \(A^tC=C^tA\).
Consequently \[
(A+iC)^*(A+iC)=A^tA+C^tC
\] is positive definite: the right side vanishes on a vector only if
both \(A\) and \(C\) annihilate it. Thus the real polynomial
\(\det(A+tC)\) is not identically zero, since its value at \(t=i\) is
nonzero. It has degree at most \(n\), so one of any \(n+1\) distinct
real values of \(t\) makes \(A+tC\) invertible. The root bound used here
follows by dividing out \(t-t_j\) at each distinct root \(t_j\); each
division lowers the degree by one. Left multiplication by the upper
shear \(\left(\begin{smallmatrix}I&tI\\0&I\end{smallmatrix}\right)\)
makes the position block invertible. Upper shears are conjugates of
lower shears by the product of the pair swaps in (A28).

For an invertible position block, the symplectic identities imply that
\(CA^{-1}\) and \(A^{-1}B\) are symmetric and \(D=A^{-t}+CA^{-1}B\).
Hence \[
S=
\begin{pmatrix}I&0\\ CA^{-1}&I\end{pmatrix}
\begin{pmatrix}A&0\\0&A^{-t}\end{pmatrix}
\begin{pmatrix}I&A^{-1}B\\0&I\end{pmatrix}.
\tag{A29}
\] Each factor is generated by (A28); undo the preliminary shear to
finish the proof. This uses only determinants, elementary polynomial
division and the positive quadratic identity above.

\textbf{Theorem 6.1 (affine symplectic covariance).} For every affine
symplectic \(\chi\) there is a unitary \(U_\chi\) on
\(L^2(\mathbb R^n)\), preserving \(\mathcal S\) and extending to an
automorphism of \(\mathcal S'\), such that \[
U_\chi^{-1}L^wU_\chi=(L\circ\chi)^w,\qquad
U_\chi^{-1}a^wU_\chi=(a\circ\chi)^w.
\tag{A30}
\] The first identity holds for every real affine \(L\), as an equality
of the selfadjoint closures described in Section 5 of
\href{weyl-metric-products.pdf}{Two measuring scales, one Weyl product}.
The second holds for every tempered symbol \(a\), as an identity
\(\mathcal S\to\mathcal S'\). The unitary is unique up to a constant of
modulus one.

For the generators, take respectively \[
\begin{array}{c|c}
\chi& U_\chi u(x)\\ \hline
(x+a,\xi)&u(x-a)\\
(x,\xi+b)&e^{ib\cdot x}u(x)\\
(x_j,\xi_j)\mapsto(\xi_j,-x_j)&\mathcal F_{0,j}u(x)\\
(Tx,T^{-t}\xi)&|\det T|^{-1/2}u(T^{-1}x)\\
(x,\xi-Ax)&e^{-ix\cdot Ax/2}u(x).
\end{array}
\tag{A31}
\] Here \(\mathcal F_{0,j}\) is the normalized Fourier transform in just
the \(j\)-th variable. Change of variables and Plancherel prove
unitarity. Differentiation proves preservation of Schwartz space and of
its topology; each inverse has the same property. The bilinear transpose
\(U_\chi^t=CU_\chi^{-1}C\) therefore defines the distribution extension
by \(\langle U_\chi u,\phi\rangle=\langle u,U_\chi^t\phi\rangle\); it
agrees with the original map on Schwartz functions. Direct substitution
gives the intertwining of \(x_j,D_j\), hence of every affine \(L\).
Since Schwartz space is a core for each real affine observable, these
identities pass to the selfadjoint closures. Products of the displayed
unitaries implement products of the affine maps.

We justify both uniqueness and the distributional extension of
covariance. A bounded unitary commuting with all closed \(x_j,D_j\)
commutes with their translation and modulation groups. For example, on
the domain of \(x_j\), differentiate \(e^{-itx_j}Ue^{itx_j}u\); domain
preservation follows from commutation with the closed operator, and the
derivative is zero. Density extends the resulting equality to \(L^2\).
The same argument applies to \(D_j\), whose group is the translation
group explicitly proved in Section 5 of
\href{weyl-metric-products.pdf}{Two measuring scales, one Weyl product}.
Thus no prior preservation of \(\mathcal S\) by this unknown unitary has
been assumed.

Fourier inversion of \(\phi\in\mathcal S(\mathbb R^n)\) then shows that
\(U\) commutes with multiplication by \(\phi\). Fix
\(h(x)=e^{-|x|^2}>0\) and set \(b=Uh/h\), initially a locally
square-integrable function. For every \(\phi\in C_c^\infty\), \[
U(\phi h)=\phi Uh=b\phi h.
\] Since every compact smooth function is of the form \(\phi h\),
boundedness gives \(\|b\psi\|_2\le\|U\|\|\psi\|_2\) for all such
\(\psi\). For a bounded measurable set \(E\), approximate its indicator
in \(L^2\) by compact smooth functions and select a subsequence
converging almost everywhere. To obtain that subsequence, choose squared
errors at most \(2^{-j}\); Tonelli makes the sum of the pointwise
squared errors finite almost everywhere. Fatou's inequality gives
\(\int_E|b|^2\le\|U\|^2|E|\). The needed nonnegative Fatou inequality
follows from the increasing tail infima: their integrals are bounded by
the liminf of the original integrals, and monotone convergence follows
from Tonelli applied to their nonnegative successive differences.
Applying the resulting set inequality to the intersection of a ball with
\(\{|b|>\|U\|+\varepsilon\}\) proves \(|b|\le\|U\|\) almost everywhere.
Density now gives \(Uu=bu\) on all \(L^2\). Commutation with
translations means \(b(x+t)=b(x)\) almost everywhere for each fixed
\(t\). Convolving with a compact smooth mollifier makes this an
everywhere translation-invariant smooth function, hence a constant.
Letting the mollifier radius tend to zero in distributions shows that
\(b\) itself is constant: the constants converge by testing against one
function of integral one. Unitarity makes its modulus one. Comparing any
two implementations of \(\chi\) reduces to this commuting case, proving
uniqueness.

For covariance of symbols, first exponentiate the affine intertwining.
This can be checked without an abstract functional calculus: the
explicit affine groups in Section 5 of
\href{weyl-metric-products.pdf}{Two measuring scales, one Weyl product}
preserve \(\mathcal S\); differentiating the product of one inverse
group with the conjugate of the other gives zero there. Density supplies
equality on \(L^2\). Thus (A30) holds for all plane-wave symbols.
Fourier inversion and their norm-one operator bounds extend it to
Schwartz symbols. Finally, the symbol-to-kernel map, affine pullback of
distributions, and conjugation by \(U_\chi\) are continuous in
distributional pairings. The weak density proved after (A25) extends
(A30) to every \(a\in\mathcal S'(W)\).

An affine pullback here uses the usual distributional change of
variables. Symplectic matrices have \(|\det S|=1\), as follows already
by taking determinants in \(S^tJS=J\). The conclusion does not supply a
canonical simultaneous choice of the phases of all \(U_\chi\); products
implement composition but may differ from another chosen implementation
by a scalar.

The metric version follows directly. Define \[
(\chi^*g)_X(T)=g_{\chi(X)}(ST),\qquad \chi^*m=m\circ\chi.
\tag{A32}
\] Symplectic duality commutes with this pullback, slow variation and
temperateness keep the same constants, and \(h_{\chi^*g}=h_g\circ\chi\).
The chain rule identifies the symbol seminorms under
\(a\mapsto a\circ\chi\). Thus covariance also transports the full metric
symbol classes without changing their structural hypotheses.

\subsection{7. Reflection and changes of
quantization}\label{reflection-and-changes-of-quantization}

Assume now that \(g\) is symplectically temperate, \(g\le g^\sigma\),
\(m\) is a temperate weight, and \[
g_{(x,\xi)}(t,\tau)=g_{(x,\xi)}(t,-\tau).
\tag{A33}
\] This reflection condition says that the position and frequency
directions are orthogonal for \(g\) at each point. It is needed here to
keep the same metric class under all quantization changes; it was not
needed for (A21).

For every fixed \(k\in\mathbb R\), the distributional Fourier multiplier
\[
T_k=\exp(i k\langle D_x,D_\xi\rangle)
\tag{A34}
\] is a continuous automorphism of \(S(m,g)\), weakly continuous on
bounded symbol sets, with inverse \(T_{-k}\). If
\(h^2=\sup g/g^\sigma\), then for every integer \(N\ge0\), \[
T_k a-\sum_{j<N}\frac{(ik\langle D_x,D_\xi\rangle)^j}{j!}a
\in S(h^Nm,g),
\tag{A35}
\] with finite source-seminorm estimates for all output derivatives and
the same bounded-set continuity.

To check the normalization, the doubled auxiliary phase \(2p\cdot q\)
has symmetric representing map \((p,q)\mapsto(q,p)\). Its phase dual at
\((t,\tau)\) is \(g^\sigma(t,-\tau)\), which equals \(g^\sigma(t,\tau)\)
by (A33) and quadratic inversion. The actual phase \(k p\cdot q\), for
\(k\ne0\), therefore has dual \(4k^{-2}g^\sigma\) and parameter
\(|k|h/2\) in the convention of
\href{gauss-transform-estimates.pdf}{Quadratic Fourier multipliers at a
moving scale}. Retain that phase and the original metric. The finite
bound (G24) applies with \(h_*=|k|/2\). For the temperateness hypotheses
use \[
 1+q_X(X-Y)\le\max(1,k^2/4)
                  (1+4k^{-2}q_X(X-Y)).
 \tag{A35a}
\] Multiplying the phase-dual form itself by its fixed factor cancels on
the two sides of each form comparison. Equation (A35a) converts the
distance factors, so (G13) holds for the actual phase with constants
depending on \(k\). Equations (G24)--(G26) prove, for every \(N,l\ge0\),
the complete derivative estimate \[
 \begin{aligned}
 & \left|\partial_{T_1}\cdots\partial_{T_l}
 \left(T_ka-\sum_{j<N}\frac{(ik\langle D_x,D_\xi\rangle)^j}{j!}a\right)(X)\right|
 \\ &\le C_{N,l,k}\left(\frac{|k|}{2}\right)^N h(X)^N m(X)
       \prod_{r=1}^l g_X(T_r)^{1/2}
       p_{l\le j\le J}(a;m,g).
 \end{aligned}
 \tag{A35b}
\] Here \(J\) is finite and the counting constant retains
\((1+|k|/2)^{2n}\) from (G25). This proves (A35) without changing the
metric or omitting the actual phase factor. For \(k=0\), the map is the
identity: the remainder is zero for \(N\ge1\), and is \(a\) for \(N=0\),
whose Taylor sum is empty.

The Gauss extension agrees with the stated distributional multiplier.
Indeed bounded compact approximants converge in \(\mathcal S'\) by (A3);
their Gauss images are bounded in the asserted polynomially growing
target class and converge locally smoothly. Thus both definitions are
limits of the same sequence in distributional pairings. Multiplication
of the Fourier multipliers now proves \(T_kT_l=T_{k+l}\), hence the
automorphism assertion. No uniform estimate for unbounded \(k\) is
claimed.

Let \(\operatorname{Op}_\tau\) have the kernel base point
\((1-\tau)x+\tau y\), as in Section 4 of
\href{weyl-metric-products.pdf}{Two measuring scales, one Weyl product}.
The kernel coordinate calculation of Section 8 of
\href{weyl-metric-products.pdf}{Two measuring scales, one Weyl product},
valid for every tempered symbol, gives \[
\operatorname{Op}_s(T_{\tau-s}a)=\operatorname{Op}_\tau(a).
\tag{A36}
\] Uniqueness of the symbol follows by inverse partial Fourier
transformation of the kernel. In particular, if \(a,c\) are left and
right symbols and \(b\) is their Weyl symbol, then \[
\begin{gathered}
b=T_{-1/2}a=T_{1/2}c,\qquad
a=T_{1/2}b=T_1c,\\
c=T_{-1}a=T_{-1/2}b.
\end{gathered}
\tag{A37}
\] All these symbols lie in the same \(S(m,g)\). Combining (A36) with
(A21) proves their Schwartz and distribution actions, with precisely the
topologies stated in Section 4.

There is an elementary action proof under the additional vertical bound
\[
g_{(x,\xi)}(0,\tau)\le|\tau|^2.
\tag{A38}
\] It is useful to see exactly what this stronger hypothesis buys. For
each fixed \(\beta\), directional symbol estimates and (A3) give \[
|\partial_\xi^\alpha\partial_x^\beta a(x,\xi)|
\le C_{\alpha,\beta}p_{\le|\alpha|+|\beta|}(a;m,g)
(1+|x|+|\xi|)^{K_\beta},
\tag{A39}
\] where \(K_\beta\) is independent of \(\alpha\). The vertical factor
contributes at most one to each coordinate frequency derivative; the
factors from the fixed position derivatives have a polynomial bound. The
left operator is
\((2\pi)^{-n}\int e^{ix\cdot\xi}a(x,\xi)\widehat u(\xi)\,d\xi\). After
any fixed output derivatives, multiplication by \((1+|x|^2)^M\)
transfers \((1-\Delta_\xi)^M\) onto the amplitude. Formula (A39) leaves
only a fixed power of \(1+|x|\), independent of \(M\), and the Schwartz
decay of \(\widehat u\) makes the frequency integrals finite. Choosing
\(M\) large proves every desired output seminorm with finitely many
source seminorms. Adjoint transposition and the quantization changes
give the other actions. The more general proof of (A21) removes (A38)
entirely, while the reflection condition remains in the same-class
quantization theorem.

\subsection{8. Left products and the subprincipal
coefficient}\label{left-products-and-the-subprincipal-coefficient}

Keep the reflected one-metric hypotheses of the preceding section, and
take \(a_j\in S(m_j,g)\), \(j=1,2\). Their left operator composite has
left symbol \[
a_1\circ_L a_2
=T_{1/2}\big((T_{-1/2}a_1)\#(T_{-1/2}a_2)\big)
\in S(m_1m_2,g).
\tag{A40}
\] This identity follows from (A25) and (A37); it holds on both
\(\mathcal S\) and \(\mathcal S'\). For Schwartz symbols it also gives
the exact differential-multiplier formula \[
a_1\circ_L a_2
=\left.\exp(i\langle D_\xi,D_y\rangle)
\big(a_1(x,\xi)a_2(y,\eta)\big)\right|_{(y,\eta)=(x,\xi)}.
\tag{A41}
\] For general symbols the right side denotes its bounded weak
extension, defined by (A40).

Here is the phase calculation. Pulling the outer \(T_{1/2}\) back before
restriction replaces its differential variables by
\(D_x+D_y,D_\xi+D_\eta\). The three quantization phases and the Weyl
phase add to \[
\begin{split}
&\tfrac12\langle D_x+D_y,D_\xi+D_\eta\rangle
-\tfrac12\langle D_x,D_\xi\rangle-\tfrac12\langle D_y,D_\eta\rangle\\
&\hspace{25mm}
+\tfrac12\big(\langle D_\xi,D_y\rangle-\langle D_x,D_\eta\rangle\big)
=\langle D_\xi,D_y\rangle.
\end{split}
\tag{A42}
\] All these constant-coefficient operators commute on the product
Schwartz space. In Fourier variables the restriction and outer
multiplier identity is obtained by replacing the output frequency by the
sum of the two input frequencies, so it is valid for their full
exponentials as well as for polynomials.

Every finite truncation satisfies \[
a_1\circ_L a_2-
\sum_{|\alpha|<N}\frac{1}{\alpha!}
(\partial_\xi^\alpha a_1)(D_x^\alpha a_2)
\in S(h^Nm_1m_2,g),
\tag{A43}
\] with finite seminorm estimates and weak bounded-set continuity. One
can obtain this without imposing an unproved direct Gauss estimate for
the degenerate phase in (A41). Expand each of the three quantization
maps in (A40) and the Weyl product only to order \(N-1\), using their
proved remainders. The coefficient of degree \(j\) in a quantization
expansion maps \(S(w,g)\) to \(S(h^jw,g)\): it is the difference of
remainders of orders \(j\) and \(j+1\). The analogous Weyl coefficient
maps two weighted classes to the product weight times \(h^j\). These
statements hold for every temperate weight \(w\), including products
with powers of \(h\). Thus each discarded term has weight at least
\(h^N m_1m_2\); \(h\le1\) permits the inclusion for higher powers. Only
finitely many terms and seminorms occur. The terms below degree \(N\)
combine by the polynomial identity (A42), giving exactly (A43). This is
a finite remainder argument, not an assertion of convergence of an
infinite series.

For a classical polyhomogeneous left symbol, write
\(a\sim a_m+a_{m-1}+\cdots\), where the coefficient \(a_{m-j}\) is
homogeneous in \(\xi\) of degree \(m-j\) away from zero, with the usual
symbol remainder estimates. Formula (A37) and (A35), in the classical
\((1,0)\) metric, give \[
b_m=a_m,\qquad
b_{m-1}=a_{m-1}+\frac{i}{2}
\sum_{\ell=1}^n\partial_{x_\ell}\partial_{\xi_\ell}a_m.
\tag{A44}
\] The sign follows from \(D_xD_\xi=-\partial_x\partial_\xi\). Terms of
order two in this expansion lower the frequency degree by at least two.
Thus \(b_{m-1}\), the next Weyl coefficient, is the subprincipal symbol
of the left operator in these coordinates. The calculation is local in
\(x\): if the classical estimates are only local, first multiply by a
position cutoff equal to one on the region in question and apply the
remainder formula there. The finite coefficients agree on that region,
and arbitrary-order remainder estimates make the conclusion independent
of the cutoff.

For scalar classical operators with principal symbols \(a_0,b_0\) and
subprincipal symbols \(a_s,b_s\), the Weyl product and scalar parity
imply \[
\begin{gathered}
(AB)_0=a_0b_0,\qquad
(AB)_s=a_sb_0+a_0b_s+\{a_0,b_0\}/(2i),\\
(A^*)_0=\overline{a_0},\qquad (A^*)_s=\overline{a_s}.
\end{gathered}
\tag{A45}
\] For the commutator, regarded as an operator of order \(m_A+m_B-1\),
\[
[A,B]_0=\{a_0,b_0\}/i,\qquad
[A,B]_s=\big(\{a_s,b_0\}+\{a_0,b_s\}\big)/i.
\tag{A46}
\] Indeed the even Weyl terms cancel for scalar symbols, and the first
unused odd term lowers the total order by three. The product's order-one
part comes from the two next homogeneous coefficients and its first Weyl
bracket. Conjugation of a Weyl symbol gives the adjoint identities.
These arguments establish every displayed rule, including cases where a
displayed leading coefficient vanishes; the declared order then simply
has a zero coefficient. No unchanged scalar-cancellation formula is
asserted for matrices.

\subsubsection{The same calculus with a finite Planck
bound}\label{the-same-calculus-with-a-finite-planck-bound}

\textbf{Editorial strengthening.} Retain a slowly varying symplectically
temperate metric \(g\), the reflection condition (A33), the original
temperate weights and every original quantization. Replace the bound
\(h\le1\) in Sections 7--8 by \[
 0<h(X)\le h_*<\infty,\qquad
 h(X)^2=\sup_{T\ne0}g_X(T)/g_X^\sigma(T).
 \tag{A52}
\] Then every fixed \(T_k\) is still an automorphism of \(S(m,g)\) with
all the remainders (A35), the full derivative bounds (A35b), and
bounded-set local-smooth continuity. The kernel conversion identities
(A36)--(A37), the exact ordered left composite (A40), and its remainders
(A43) hold as written. Constants may also depend on \(h_*\). This
assertion gives symbol calculus and the Schwartz and distribution
actions; it asserts no new \(L^2\) boundedness result.

Here is the full change in the proof. For \(k\ne0\), (A33) still gives
the actual phase-dual form \(4k^{-2}g^\sigma\), independently of any
uncertainty bound. Equation (A35a) proves its temperateness comparisons.
Its actual parameter is now bounded by \(|k|h_*/2\). Apply (G24)--(G26)
with precisely that finite bound. The derivative factor remains
\((|k|/2)^Nh(X)^N\); the counting constant retains \[
 (1+|k|h_*/2)^{2n}.
 \tag{A53}
\] The resulting symbol estimates and continuity identify the
distributional multiplier by the same bounded compact approximation. The
Fourier multipliers obey the group law, giving the inverse \(T_{-k}\).
The case \(k=0\) is the identity with the two remainder cases already
stated. Kernel coordinate substitutions, which do not involve a metric
inequality, prove (A36)--(A37) unchanged.

For a single metric the two-metric cross parameter in
\href{weyl-metric-products.pdf}{Two measuring scales, one Weyl product}
is exactly \(H=h\), and its mean metric is exactly \(g\). The bounded
cross-parameter theorem (W43)--(W44), with \(H_*=h_*\), therefore gives
every original Weyl remainder with the actual factor \((h(X)/4)^N\). The
induced product metric on the diagonal is \(2g\); its order-\(l\)
seminorm comparison retains the factor \(2^{l/2}\). Thus all three
conversions and the middle product in (A40) exist in their stated
weighted classes, with their full finite estimates.

It remains to verify the finite expansion rather than assuming that the
old use of \(h\le1\) survives. Local metric comparison proves local
continuity of \(h\). Own temperateness and quadratic duality, applied to
the quotient defining \(h^2\), give \(h(Y)\le C h(X)(1+q_Y(X-Y))^L\) for
some finite \(C,L\). Thus every \(h^j w\), for a temperate weight \(w\)
and integer \(j\ge0\), is an allowed weight. If
\(R_j(k)=T_k-\sum_{r<j}(ik\langle D_x,D_\xi\rangle)^r/r!\), the
degree-\(j\) term is exactly \(R_j(k)-R_{j+1}(k)\). The first term has
weight \(h^jw\), and the second has weight \(h^{j+1}w\). Their exact
inclusion bound is \[
 p_l(v;h^jw,g)\le h_*\,p_l(v;h^{j+1}w,g),
 \quad v\in S(h^{j+1}w,g).
 \tag{A54}
\] Hence each degree-\(j\) conversion term maps into \(S(h^jw,g)\), with
its constants including this factor. The Weyl coefficients have the
identical argument using the difference of two successive proved Weyl
remainders and preserving their original ordered products.

Expand each of the three conversions and the Weyl product through degree
\(N-1\). Terms containing a remainder gain at least \(h^N\): use the
conversion or product on its exact incoming weight, and retain each
actual phase scalar in its estimate. Each discarded finite polynomial
term has some total degree \(d\ge N\), for which the full inclusion
bound is \[
 p_l(v;h^Nw,g)\le h_*^{d-N}p_l(v;h^dw,g),
 \quad v\in S(h^dw,g).
 \tag{A55}
\] There are only finitely many such terms and seminorms at each fixed
\(N,l\). The terms of degree below \(N\) combine by the exact commuting
differential-phase identity (A42), so their sum is precisely
\(\sum_{|\alpha|<N}(\partial_\xi^\alpha a_1)(D_x^\alpha a_2)/\alpha!\),
in the displayed coefficient order. This proves (A43), including every
finite factor lost from the former bound-one argument. For \(N=0\) the
polynomial sum is empty and (A40) itself gives the required product
class. No infinite series or changed metric is used.

\textbf{Editorial extension: reflected quantization and left products
with no upper Planck bound.}

Retain the original slowly varying symplectically temperate metric
\(g\), the reflection condition (A33), its symplectic dual \(q\), its
positive parameter \(h^2=\sup g/q\), and every original temperate
weight. This paragraph drops the upper bound in (A52) and proves
explicitly larger targets. The phrase ``local continuity of \(h\)'' in
the preceding finite-bound proof means local \(g\)-continuity: two-sided
comparison on the metric's small balls. It does not assert ordinary
continuity of \(X\mapsto h(X)\). The discontinuous metric in Example2
still satisfies that comparison.

Here are the exact weight comparisons. If (A2) gives \(q_X\le Cq_YR^L\),
where \(R=1+q_Y(X-Y)\), symplectic duality gives \(g_Y\le Cg_XR^L\) and
the same original inequality gives \(q_Y\ge C^{-1}q_XR^{-L}\).
Consequently \[
h(Y)^2\le C^2h(X)^2R^{2L},\qquad
1+c h(Y)\le\max(1,C)(1+c h(X))R^L\quad(c\ge0).
\tag{AU1}
\] Slow variation compares both original forms and their duals in both
directions on a small ball, hence also compares \(h\) and \(1+ch\). All
positive products \(mh^N(1+ch)^r\), with fixed nonnegative integers
\(N,r\), are therefore legitimate weights. No derivative of a metric or
weight is being assumed.

For a fixed \(\kappa\ne0\), the original phase \(\kappa p\cdot q\) has
matrix
\(B_\kappa=(\kappa/2)\left(\begin{smallmatrix}0&I\\I&0\end{smallmatrix}\right)\),
rank \(2n\), phase-dual form \(4\kappa^{-2}q\) under (A33), and actual
parameter \(|\kappa|h/2\). Equation (A35a) retains the exact structural
conversion factor \(\max(1,\kappa^2/4)^L\). Apply (GW1)--(GW5) of
\href{gauss-transform-estimates.pdf}{Quadratic Fourier multipliers at a
moving scale} on the entire original phase space. With \[
\begin{aligned}
w_\kappa&=(1+|\kappa|h/2)^{2n},\\
Q_j^\kappa&=(i\kappa\langle D_x,D_\xi\rangle)^j/j!,\qquad
R_N^\kappa=T_\kappa-\sum_{j<N}Q_j^\kappa,
\end{aligned}
\tag{AU2}
\] the full receiving estimates are \[
\begin{aligned}
p_l(T_\kappa a;mw_\kappa,g)&\le C_{l,\kappa}p_{\le J}(a;m,g),\\
p_l(R_N^\kappa a;mh^Nw_\kappa,g)
 &\le C_{N,l,\kappa}(|\kappa|/2)^Np_{\le J}(a;m,g).
\end{aligned}
\tag{AU3}
\] One sufficient input order is \(J=l+s+\max(2N,k)\), with \(s>n\) an
integer and \(k>2N_m+lN_g+2N(N_g+1/2)+2n(2N_g+1)\), using the actual
phase's structural exponents. These are finite strict thresholds, with
the near order and the rank count preserved. The maps are continuous in
the symbol seminorms and locally smoothly continuous on bounded source
sets. The same polynomial-growth and bounded-approximation argument used
after (A35b) identifies them with the original distributional
multipliers. Thus \(T_\kappa T_\lambda=T_{\kappa+\lambda}\) on
\(\mathcal S'(W)\). For \(\kappa=0\), \(T_0=I\), \(R_0^0=I\), and
\(R_N^0=0\) for \(N\ge1\), directly; no zero parameter is treated as a
positive target weight.

There is a space on which these original conversion maps are actual
automorphisms. Put \[
v=(1+h)^{2n},\qquad
\mathcal A_m=\bigcup_{j\ge0}S(mv^j,g)\ \subset\ \mathcal S'(W).
\tag{AU4}
\] The classes increase because \(v\ge1\), so the union is a vector
space. For fixed nonzero \(\kappa\), the exact comparisons are \[
w_\kappa\le\max(1,|\kappa|/2)^{2n}v,\qquad
v\le\max(1,2/|\kappa|)^{2n}w_\kappa.
\tag{AU5}
\] Applying (AU3) to the actual weight \(mv^j\) maps its level
continuously into level \(j+1\). The inverse \(T_{-\kappa}\) does the
same; the distributional group law proves both inverse products on
\(\mathcal A_m\). This asserts finite-seminorm continuity at each
displayed level, without an additional claim about a topology on the
union. The full kernel identity (A36) gives all its original
quantizations on this space. Each has a continuous action on
\(\mathcal S\) and its strong dual, by (A21) on the actual converted
symbol class.

The one-metric specialization of (AO1) has \(H=h\), mean metric \(g\),
and exact weight \(w_\#=(1+h/4)^{4n}\le v^2\). It maps levels \(j,k\) of
\(\mathcal A_{m_1},\mathcal A_{m_2}\) into level \(j+k+2\) of
\(\mathcal A_{m_1m_2}\). The actual operator identity (AO3), and the
injective symbol-to-kernel Fourier map in the Weyl prerequisite, prove
associativity whenever three such factors are composed: both
parenthesized symbols belong to level \(j+k+r+4\) and have the same
operator. Finite rectangular matrix associativity retains both
intermediate finite index sums and their original multiplication order.
In particular \(\mathcal A_1\) is an algebra with unit the original
constant symbol \(1\); its Weyl kernel is
\((2\pi)^{-n}\int e^{i(x-y)\xi}d\xi=\delta(x-y)\), so its two products
are the identity by the same injectivity.

We next prove the full finite left remainder without discarding any
higher power of an unbounded \(h\). First the individual differential
coefficients need bounds that do not borrow a weight from a remainder.
Retain any original real quadratic phase \(A(\Xi)=\Xi^tB\Xi\) and
positive form \(Q(Z)=Z^tGZ\) at the observation point. Choose
\(L^tGL=I\), and use the real spectral theorem to write \[
L^{-1}BL^{-t}=O\,\operatorname{diag}(\lambda_1,\ldots,\lambda_d)O^t,
\quad E_r=LOe_r,\quad Q(E_r)=1.
\tag{AU6}
\] This proves the exact original matrix equality
\(B=\sum_r\lambda_rE_rE_r^t\), with every signed and zero eigenvalue
retained. The phase parameter is \(h_{Q,A}=\max_r|\lambda_r|\), because
its defining original quotient is the squared operator norm of this
exact comparison matrix (with the finite phase domain retained if \(B\)
has a kernel). Since \(D_E=-i\partial_E\), the complete ordered
expansion is \[
\frac{(iA(D))^j}{j!}
=\frac{(-i)^j}{j!}
 \sum_{r_1=1}^d\cdots\sum_{r_j=1}^d
 \lambda_{r_1}\cdots\lambda_{r_j}
 \partial_{E_{r_1}}^2\cdots\partial_{E_{r_j}}^2.
\tag{AU7}
\] For \(j=0\) it is the identity with one empty product. The sum has
\(d^j\) ordered terms. Additional constant directions commute with it.
Taking its absolute value at the point bounds it by \(d^jh_{Q,A}^j/j!\)
times the input directional seminorm of order \(l+2j\), the actual
weight, and \(\prod_{r=1}^lQ(T_r)^{1/2}\). The matrices are chosen only
for this pointwise estimate; no derivative of them is taken.

In the original one-metric quantization phases \(\kappa=\pm1/2\), their
parameter is exactly \(h/4\). The actual Weyl phase on \(W\times W\) has
dimension \(4n\), product metric \(G=g\oplus g\), and parameter \(h/4\)
on the diagonal. Its input tensor derivative estimate is (W34), and each
diagonal direction \((T,T)\) has length \(\sqrt{2g(T)}\). Therefore
(AU7) gives, for all legitimate incoming weights \(w,w_1,w_2\), \[
\begin{aligned}
p_l(Q_j^{\pm1/2}a;wh^j,g)
 &\le \frac{(2n)^j4^{-j}}{j!}\,p_{l+2j}(a;w,g),\\
p_l(C_j(a,b);w_1w_2h^j,g)
 &\le \frac{(4n)^j4^{-j}}{j!}\,2^{l/2}2^{l+2j}
 p_{\le l+2j}(a;w_1,g)p_{\le l+2j}(b;w_2,g).
\end{aligned}
\tag{AU8}
\] Every signed differential term is still the original one in (AU7) and
(W26); only its bound uses absolute values. These estimates have no
extra \(w_\kappa\) or \(w_\#\), and do not use an upper bound on \(h\).

Set \(A=B=T_{-1/2}\), \(C=T_{1/2}\), \(P(a,b)=a\#b\), and \[
v_0=(1+h/4)^{2n},\qquad
W_L=v_0\,v_0\,v_0^2\,v_0=(1+h/4)^{10n}.
\tag{AU9}
\] The four displayed factors are respectively the two incoming
conversion weights, the middle Weyl weight and the outgoing conversion
weight. Equations (AU3) and (AO1) prove the actual continuous bilinear
map \[
a\circ_L b=C\,P(Aa,Bb)\in S(m_1m_2W_L,g).
\tag{AU10}
\] It is locally smoothly continuous on products of bounded source sets.
Converting by (A36), then using (AO3), proves its exact left operator
composition on \(\mathcal S\) and \(\mathcal S'\).

Fix \(N\ge1\). Denote the coefficients of \(A,B,C\) by \(A_i,B_j,C_l\),
and their remainders by \(R^A_r,R^B_r,R^C_r\); denote the Weyl
coefficients by \(P_k=C_k(\cdot,\cdot)\) and its bilinear remainder by
\(R^P_r\). Expand first \(A\) through \(N-1\), then \(B\) only through
\(N-i-1\), then \(P\) only through \(N-i-j-1\), then \(C\) only through
\(N-i-j-k-1\). Bilinearity and these exact finite Taylor identities give
\[
\begin{aligned}
C P(Aa,Bb)&=\sum_{i+j+k+l<N}C_lP_k(A_i a,B_j b)
                    +E_1+E_2+E_3+E_4,\\
E_1&=C P(R_N^A a,Bb),\\
E_2&=\sum_{i<N}C P(A_i a,R_{N-i}^B b),\\
E_3&=\sum_{i+j<N}C R_{N-i-j}^P(A_i a,B_j b),\\
E_4&=\sum_{i+j+k<N}R_{N-i-j-k}^C P_k(A_i a,B_j b).
\end{aligned}
\tag{AU11}
\] Every remainder index is positive; every index sum is finite.
Applying the actual maps to their actual incoming weights and using
(AU8) gives the four separate targets \[
\begin{aligned}
E_1&\in S(m_1m_2h^Nv_0^5,g),&
E_2&\in S(m_1m_2h^Nv_0^4,g),\\
E_3&\in S(m_1m_2h^Nv_0^3,g),&
E_4&\in S(m_1m_2h^Nv_0,g).
\end{aligned}
\tag{AU12}
\] For example the term in \(E_2\) has incoming weights \(m_1h^i\) and
\(m_2h^{N-i}v_0\), the product contributes \(v_0^2\), and the last
conversion contributes \(v_0\); its total is exactly \(m_1m_2h^Nv_0^4\).
In \(E_3\) the two coefficient gains \(h^i,h^j\) and the product
remainder gain \(h^{N-i-j}\) give exactly \(h^N\), followed by
\(v_0^2v_0\). In \(E_4\) all three coefficient gains and the last
remainder gain sum to \(N\), with only its \(v_0\). The first group has
\(h^Nv_0\), \(v_0\), \(v_0^2\), \(v_0\). Since \(v_0\ge1\), all four
targets include continuously into the first one with inclusion bound
one. Every estimate retains the common phase factor \(4^{-N}\): it is
\(4^{-N}\) in the first group, \(4^{-i}4^{-(N-i)}\) in the second,
\(4^{-i}4^{-j}4^{-(N-i-j)}\) in the third, and
\(4^{-i}4^{-j}4^{-k}4^{-(N-i-j-k)}\) in the fourth. The original
factorials, tensor factors, diagonal factors and finitely many input
derivative orders remain in their separate bounds.

The remaining polynomial in (AU11) is identified without changing any
coefficient order. Replace each phase by a formal scalar times itself.
In degree \(d<N\), the coefficient of the four commuting exponentials is
precisely the sum over \(i+j+k+l=d\) in (AU11). Pulling \(C_l\) back
before diagonal restriction substitutes \(D_x+D_y,D_\xi+D_\eta\). The
exact identity (A42) then gives \((i\langle D_\xi,D_y\rangle)^d/d!\).
Its full multinomial is \[
\sum_{|\alpha|=d}\frac{i^d}{\alpha!}
 (D_\xi^\alpha a)(D_x^\alpha b)
=\sum_{|\alpha|=d}\frac1{\alpha!}
 (\partial_\xi^\alpha a)(D_x^\alpha b),
\tag{AU13}
\] because \(i^d(-i)^d=1\). The factors are multiplied in this displayed
order, also for matrices. Thus the actual full conclusion is \[
\begin{aligned}
\mathcal E_N(a,b)&=a\circ_L b-\sum_{|\alpha|<N}
 \frac{(\partial_\xi^\alpha a)(D_x^\alpha b)}{\alpha!},\\
\mathcal E_N(a,b)&\in S(m_1m_2h^NW_L,g),\\
p_l(\mathcal E_N(a,b);m_1m_2h^NW_L,g)
 &\le 4^{-N}K_{N,l}
 p_{\le J}(a;m_1,g)p_{\le J}(b;m_2,g).
\end{aligned}
\tag{AU14}
\] Here \(J\) is a finite maximum of the finitely many actual input
orders in (AU11), chosen after their strict tail thresholds, and
\(K_{N,l}\) retains all the constants just displayed. For \(N=0\), the
sum is empty and (AU10) gives the conclusion with phase factor
\(4^0=1\). There are no discarded polynomial terms of degree at least
\(N\): the adaptive expansion never produces them. Thus neither an
inclusion \(h^{d}w\subset h^Nw\) for unbounded \(h\), nor convergence of
an infinite formal series, has been assumed.

At union levels \(j,k\), (AU10) maps into level \(j+k+5\), since
\(W_L\le v^5\). The operator argument and symbol injectivity prove its
associativity with the same ordered finite matrix sums. If the original
\(h_*<\infty\) is available, then \(w_\kappa\le(1+|\kappa|h_*/2)^{2n}\)
and \(W_L\le(1+h_*/4)^{10n}\); the actual inclusion bounds recover the
original smaller classes in (A35), (A40) and (A43). If \(h=1\), the
remainder weight has no descending power gain; if \(h\) is unbounded,
(AU14) describes exactly its larger target. The earlier figure depicts
the finite-bound argument (A52)--(A55), while (AU11)--(AU12) give the
different finite expansion needed here. These are editorial consequences
of the complete Gauss and Weyl proofs ; no new source reading or novelty
claim is asserted.

\begin{figure}
\centering
\pandocbounded{\includegraphics[keepaspectratio]{../figures/finite-planck-conversion.png}}
\caption{Exact phases and finite-weight inclusions}
\end{figure}

Equations (A35a)--(A35b), (A52)--(A55) prove every phase and inclusion
in the diagram on the original metric. The finite-bound extension uses
the complete earlier Gauss and Weyl proofs. No novelty claim is made.

\subsection{9. Four worked examples}\label{four-worked-examples}

\textbf{A metric larger than its dual.} Let \(g_X=7e\) and
\(m(X)=1+|X|^2\). This constant metric is slowly varying and
symplectically temperate, but \(g^\sigma=e/7\), so its uncertainty
inequality fails. The weight is temperate: the triangle inequality
bounds either ratio of \(1+|X|^2\) and \(1+|Y|^2\) by \(2(1+|X-Y|^2)\),
which is bounded by a constant times \(1+q_Y(X-Y)\). The symbol
\(a(x,\xi)=|x|^2+|\xi|^2\) belongs to \(S(m,g)\), and (A21) gives its
actions on \(\mathcal S\) and \(\mathcal S'\). Here its operator is the
familiar differential expression \(|x|^2-\Delta\). These actions do not
imply boundedness on \(L^2\): for a nonzero compact smooth \(u\),
translate it by \(Re_1\); its \(L^2\) norm stays fixed, whereas its
pairing with \((|x|^2-\Delta)u\) after translation grows quadratically
in \(R\). In the localization proof, \(r_\nu=e/8\), so the decay
variable is an ordinary distance multiplied by \(1/\sqrt8\).

\textbf{An unbounded conformal factor.} Set \(s(X)=(1+|X|^2)^{1/2}\),
\(g_X=s(X)^{-2}e\), and \(G=e\). The function \(s\) is 1-Lipschitz. Thus
a sufficiently small \(g_X\)-displacement changes \(s\) by at most a
fixed fraction of \(s(X)\), proving slow variation. The inequalities \[
\frac{s(X)^2}{s(Y)^2}\le 2(1+|X-Y|^2),\qquad q_Y=s(Y)^2e\ge e
\tag{A47}
\] prove symplectic temperateness, including the reversed ratio. Now
\(G=\mu g\) with \(\mu=s^2\), which is unbounded. The local Planck
parameters are \(h_g=s^{-2}\), \(h_G=1\), and the cross parameter is
\(H=s^{-1}\). For unit input weights, conformal compatibility yields a
product remainder in \(S(s^{-N},(g+G)/2)\) even though one input has
Planck parameter identically one. Ordinary continuity of the conformal
factor is also unnecessary: the pair \(g=e/4\),
\(G=(1+\mathbf1_{\{x_1\ge0\}})g\) satisfies the enlargement hypotheses
by uniform comparison with fixed positive forms, including at the jump.

\textbf{A quadratic shear of the oscillator.} In one dimension let
\(Uu(x)=e^{-3ix^2/2}u(x)\). Direct differentiation gives
\(U^{-1}DU=D-3x\). The associated map is \(\chi(x,\xi)=(x,\xi-3x)\), so
covariance gives \[
U^{-1}(x^2+D^2)U
=\big(x^2+(\xi-3x)^2\big)^w
=D^2-3(Dx+xD)+10x^2.
\tag{A48}
\] The symmetrized middle term is essential. Replacing it by \(-6xD\)
loses the constant \(3i\), because \(Dx=xD-i\). Although the final
differential expression contains this imaginary constant when written in
left order, its Weyl symbol is real and integration by parts shows that
its action on \(\mathcal S\) is symmetric. The equality on
\(\mathcal S\) is enough to verify the displayed algebra; no assertion
about the oscillator's selfadjoint closure or spectral theorem is
needed.

\textbf{Balancing a first-order differential operator.} Let
\(b(x)=2+\sin x\). The left symbol of \(bD\) is \(b\xi\); (A44) gives
the next Weyl coefficient \(ib'/2\). The symmetric operator on Schwartz
functions \[
B=\tfrac12(bD+Db)=bD-\tfrac i2b'
\tag{A49}
\] has left symbol \(b\xi-ib'/2\). Its Weyl symbol is exactly \(b\xi\):
all terms after the first quantization correction vanish because the
symbol is linear in \(\xi\). Its subprincipal coefficient is therefore
zero. This cancellation explains why a next left coefficient alone does
not determine the subprincipal symbol.

\subsection{10. Exercises and solutions}\label{exercises-and-solutions}

\textbf{Problem 1.} The norm in (A8) is invariant under all invertible
affine maps. Could the covariance theorem also hold for the map
\((x,\xi)\mapsto(2x,2\xi)\) in one dimension?

\textbf{Solution.} No.~Such an implementation, preserving \(\mathcal S\)
as in the theorem, would intertwine \(x\) with \(2x\) and \(D\) with
\(2D\). Conjugating their commutator on \(\mathcal S\) would give both
\(iI\) and \([2x,2D]=4iI\). The contradiction concerns the symplectic
form, not the Fourier norm: the change of variables proving invariance
of that norm needs only an invertible matrix. Hence an affine-invariant
upper bound is weaker than unitary covariance of every symbol.

\textbf{Problem 2.} In two dimensions consider the symplectic matrix
with blocks \(A=D=\operatorname{diag}(1,0)\),
\(B=\operatorname{diag}(0,1)\), and \(C=\operatorname{diag}(0,-1)\).
Carry out the singular-block step of (A29), using the preliminary upper
shear with \(t=1\).

\textbf{Solution.} Multiplication on the left by
\(\left(\begin{smallmatrix}I&I\\0&I\end{smallmatrix}\right)\) gives new
blocks \(A'=A+C=\operatorname{diag}(1,-1)\), \(B'=B+D=I\), \(C'=C\),
\(D'=D\). The factorization uses the lower shear matrix
\(C'(A')^{-1}=\operatorname{diag}(0,1)\), the cotangent lift of \(A'\),
and the upper shear matrix \((A')^{-1}B'=\operatorname{diag}(1,-1)\).
Multiplying those three block matrices yields the four new blocks just
listed: the lower-right block is
\((A')^{-t}+C'(A')^{-1}B'=\operatorname{diag}(1,-1)+\operatorname{diag}(0,1)=D'\).
Undo the preliminary upper shear. The original map simply exchanges the
second coordinate pair; the calculation checks that the general
algorithm works even though its position block has rank one.

\textbf{Problem 3.} For \(b\in C^\infty(\mathbb R)\) with every
derivative bounded, compute the exact left symbol of \(D^2b(x)\). Locate
the term that would be lost by using pointwise symbol multiplication.

\textbf{Solution.} Apply (A43) to \(a_1=\xi^2\), \(a_2=b(x)\). Only
frequency derivatives of orders zero, one and two survive, and they give
\[
\xi^2\circ_L b=b\xi^2-2ib'\xi-b''.
\tag{A50}
\] Leibniz's rule independently gives \(D^2(bu)=bD^2u-2ib'Du-b''u\),
verifying both signs and the factorial in the second derivative term.
Pointwise multiplication loses both lower coefficients. The same
calculation with \(a_1=\xi\), \(a_2=x\) gives \(\xi\circ_Lx=x\xi-i\);
the outer conversion in (A40) changes the Weyl coefficient \(-i/2\) to
this left coefficient \(-i\).

\textbf{Problem 4.} Take \(g=e\), \(m=1\), \(a(x,\xi)=e^{i(x+\xi)}\) in
one dimension. Compute \(T_1a-a\). Does the first remainder in (A35)
have a smooth Schwartz kernel? What can further remainder estimates
alone say when \(h=1\)?

\textbf{Solution.} Since \(D_xD_\xi a=a\), its multiplier definition
gives \(T_1a=e^i a\); therefore \(T_1a-a=(e^i-1)a\). The nonzero
plane-wave Weyl operator is a modulation followed by translation, with a
constant phase; its distribution kernel is supported on a shifted
diagonal and is not smooth. When \(h=1\), every remainder space
\(S(h^Nm,g)\) equals \(S(1,e)\). Increasing \(N\) still gives finite
quantitative estimates, but those spaces alone express no improvement of
decay or smoothing. One must also avoid treating a sequence of finite
remainder bounds, whose constants depend on \(N\), as convergence of an
infinite series.

\textbf{Problem 5.} Choose \(\chi\in C_c^\infty(\mathbb R)\) with
\(\chi(0)=1\), and put \(a_j(x,\xi)=\chi(x-j)\), \(u_j=\delta_j\).
Verify the bounded-set conclusion in (A21) for these moving inputs. Show
what fails if \(u_j\) is replaced by \(v_j=e^{j^2}\delta_j\).

\textbf{Solution.} The symbols form a bounded set of \(S(1,e)\) and
converge locally smoothly to zero. For any bounded test set \(B\) and
every integer \(M\), its Schwartz seminorm bounds imply
\(\sup_{\phi\in B}|\phi(j)|\le C_{B,M}(1+j)^{-M}\). Thus \(u_j\to0\) in
the strong dual, the set \(\{u_j\}\) is bounded, and
\(a_j^wu_j=u_j\to0\) strongly. By contrast the single Schwartz test
\(\phi(x)=e^{-x^2/2}\) gives \(\langle v_j,\phi\rangle=e^{j^2/2}\), so
\(\{v_j\}\) is not even pointwise bounded, and \(a_j^wv_j=v_j\) fails to
tend to zero. The boundedness hypothesis has content even when every
individual input is tempered.

\textbf{Problem 6.} For smooth real \(b,c\) with all derivatives
bounded, set \(A=\frac12(bD+Db)\), \(B=\frac12(cD+Dc)\). Find the left
and subprincipal symbols of \([A,B]\) and check them directly.

\textbf{Solution.} Put \(f=bc'-b'c\). The principal symbols are
\(b\xi,c\xi\), with zero subprincipal symbols by (A49). Their bracket is
\(f\xi\). Formulas (A46) therefore predict principal symbol \(-if\xi\)
and zero subprincipal coefficient at declared order one. Directly write
\(A=-i(b\partial+b'/2)\) and \(B=-i(c\partial+c'/2)\). In their
commutator the second derivative terms cancel, the first derivative
coefficient is \(-f\), and the multiplication coefficient is \(-f'/2\).
Thus \[
[A,B]=-ifD-\tfrac12f',\qquad
\text{left symbol}=-if\xi-\tfrac12f'.
\tag{A51}
\] The correction in (A44) adds
\(\frac i2\partial_x\partial_\xi(-if\xi)=f'/2\), so the subprincipal
coefficient vanishes as predicted. This calculation remains valid if
\(f\) vanishes identically; then the declared leading and next
coefficients are both zero.

\subsection{11. Topology and scope}\label{topology-and-scope}

For the Baire argument in Section 1, a complete metric on \(\mathcal S\)
is \(d(u,v)=\sum_{r\ge0}2^{-r-1}\min(1,Q_r(u-v))\). A sequence is Cauchy
for this metric exactly when it is Cauchy for every \(Q_r\); the
completeness proof in Section 1 supplies the needed hypothesis.

The distribution topology requires an explicit qualification. With the
Fréchet symbol topology and the strong dual topology, (A24) disproves
unrestricted joint continuity, even under the extra metric bound in
(A38); it also disproves that continuity for the weak distribution
topology. Section 4 proves separate continuity and hypocontinuity for
the strong dual and bounded weak-symbol continuity uniform on strongly
bounded inputs. Fixed-symbol maps are weak-dual continuous by
transposition. The proof does not assert unrestricted bilinear
continuity in another distribution topology.

The \(L^2\) Fourier extension used in Section 2 was proved in
\href{weyl-metric-products.pdf}{Two measuring scales, one Weyl product};
Section 2 constructs the remaining Hilbert-space integral.

Boundedness on \(L^2\), compactness and lower bounds for nonnegative
symbols need further arguments. Estimate (A22) supplies a common
Schwartz domain but does not settle those questions. Uniqueness of the
affine implementing unitary up to a scalar also does not, by itself,
produce a continuous double cover of the symplectic group.

\subsection{References}\label{references}

\begin{itemize}
\tightlist
\item
  {[}Lerner, Phase-space metrics{]} Nicolas Lerner,
  \href{https://webusers.imj-prg.fr/~nicolas.lerner/ch2booklerner.pdf}{Metrics
  in phase space}, author-hosted chapter, Lemma 2.2.21, Corollary 2.3.10
  and Theorems 2.3.18--2.3.19.
\item
  {[}Lerner, Symplectic appendix{]} Nicolas Lerner,
  \href{https://webusers.imj-prg.fr/~nicolas.lerner/ch4booklerner.pdf}{Appendix
  on symplectic generators}, Theorem 4.4.3, Proposition 4.4.4, Lemma
  4.4.15 and Remark 4.4.19.
\item
  {[}Garrett 2014{]} Paul Garrett,
  \href{https://www-users.cse.umn.edu/~garrett/m/fun/notes_2012-13/01_metric_spaces.pdf}{Review
  of metric spaces}, 2 February 2014, Theorem 4.0.1.
\end{itemize}

\end{document}