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\hypersetup{
  pdftitle={X-D-ST: The sharp-width limit of the completed scalar tests},
  pdfauthor={Ngo Cheung},
  pdfsubject={Normal-flux transmission, fixed-support convergence,
  and a supported local Grade S transfer}
}

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\title{\textbf{X-D-ST: The sharp-width limit\\
of the completed scalar tests}\\[5pt]
\large Normal-flux transmission and fixed-support convergence\\
on the inherited X-A-ST/X-C-ST host}
\author{Ngo Cheung}
\date{}

\begin{document}
\maketitle

\begin{abstract}
We establish the sharp-width limit of the completed
scalar-linear observable of X-C-ST on the supported
host already fixed by X-A-ST. The calculation takes
place on the declared Minkowski completion; the conical
host, its worldsheet, signature, and future relation
remain prescribed inputs.

For fixed smooth classical backgrounds, a fixed interaction
switch, and observation and interaction supports separated
from the initial surface, both completed coefficient tests
converge at rate \(O(\varepsilon)\) in every seminorm of
one regulator-independent compact-test space. Their classes
therefore converge in the original free-scalar quotient
topology. The proof first integrates the coordinate normal
flux and then estimates the shrinking layer. It derives
zero limiting positive-order positions and the velocities
\[
 -a_1^{0\rho}[h]\big|_\Sigma,\qquad
 \bigl(a_1^{00}[h]a_1^{0\rho}[h]-A_2^{0\rho}[h,k]\bigr)
                                                    \big|_\Sigma .
\]
The quadratic velocity survives, whereas profile-dependent
position moments vanish. A conformal example shows that
the \(O(\varepsilon)\) rate cannot generally be improved
to \(o(\varepsilon)\).

The limit is the continuous-coordinate,
continuous-normal-flux transmission prescription.
Its observation-smeared linear background kernel can
jump at the initial surface but contains no hypersurface
delta. It is generally different from the unchanged
zero-velocity multiplied-equation prescription.
The completed background identity passes to the limit
on the gauge algebra frozen before the calculation.

The principal limit theorem is credited to Cheung's
X-D. The contribution here is its proved supported
transfer, an explicit topology and defect audit, and
a finite complete-channel encoding of supplied evaluations.
Its prospective usefulness is a reproducible sharp
scalar baseline that does not confuse test convergence
with quantum metric admission or characteristic
composition. No time-ordered hierarchy, loop regulator,
continuum-field acquisition, or spacetime-emergence
construction is supplied.
\end{abstract}

\statementbox{One limit, not another dynamical completion}{%
X-B-ST supplies one first-vertex scalar map.
X-C-ST supplies one second-vertex map and one iterated
first-vertex coefficient. Their sum represents the
second scalar response once.

X-D-ST takes the sharp-width limit of the resulting
completed tests. It does not add another copy of that
response, define another \(R_2\), or install a product
hierarchy. The metric coefficients remain classical.}

\tableofcontents

%=====================================================================
\section{Purpose, Enter background, and the inherited host}
\label{sec:host}
%=====================================================================

\subsection{Purpose and attribution}

The analytical source is Cheung's X-D, version 3,
\emph{Convergence of the completed ramp relational scalar
as the regulator width tends to zero} \cite{XD}.
The predecessor presentation is supplied by
X-A-ST, X-B-ST, and X-C-ST \cite{XAST,XBST,XCST}.
The present manuscript states the data needed for the
limit and proves its layer, propagation, transmission,
and topology assertions. The parent limit theorem is
not assumed as a premise.

% Online source check: X-D is identified as version 3 and distinguishes
% fixed-input scalar-test convergence from unchanged sharp admission.
% ([enter.grok.me](https://enter.grok.me/browse/X-D))

The contribution is not a new priority claim for the
parent limit. It is a supported realization of that
result with explicit completion-normal data, preserved
counting, and narrowly stated operational consequences.

The identifier X-D-ST is used here for this proposed
manuscript. No external registration, submission,
acceptance, or registry update is asserted.

\subsection{The Enter discipline used here}

Enter distinguishes destructive classical readout from
quantum return. The measured quantum names are consumed
from the returned quantum interface; the declared output
contains a classical record and the retained residual
\cite{EnterI,EnterPhilosophy}.

The philosophy relevant here is the separation of:
\begin{enumerate}
\item a mathematically defined evaluation from a measurement
that acquires unknown field data;
\item a complete channel from its record marginal;
\item a channel identity from a thermodynamic implementation;
\item a finite operational model on a supplied spacetime
from a derivation of that spacetime.
\end{enumerate}
The spacetime Enter model I-B supplies the fixed-host
clock and transport interpretation \cite{IB}.
It does not supply the sharp-width limit.

% The interpretive source explicitly separates destructive readout,
% recovery, thermal implementation, and geometric identification.
% ([enter.grok.me](https://enter.grok.me/enter))

Only ordinary finite Born instruments are used below.
Their complete channel identities are proved directly.
No abbreviated reconstruction of the Enter
characterization theorem is attempted.

\subsection{No new cone}

\begin{assumption}[Inherited supported presentation]
\label{ass:host}
Retain exactly the X-A-ST/X-C-ST host instance:
\begin{enumerate}[label=\textup{(H\arabic*)}]
\item the regular time-oriented conical product
\((\mathcal M^\circ,G)\), its stipulated worldsheet
\(\mathcal W\), and its supplied future relation;
\item the supported injective developing chart
\[
 \iota:\mathcal L\longrightarrow\mathcal M^\circ,
\]
whose retained compact reserve stays off \(\mathcal W\);
\item the separately declared analytical completion
\[
 \widehat M=\RR^{1+3},\qquad
 \eta=\operatorname{diag}(1,-1,-1,-1),\qquad
 M_{T_*}=[0,T_*]\times\RR^3,\qquad
 \Sigma=\{t=0\};
\]
\item the inherited signature and time ledger
\[
 \iota^*G=-\eta,\qquad
 t=c_{\mathrm{host}}(T-T_\Sigma).
\]
The host keeps signature \((-,+,+,+)\), while the
analytical equations use \((+,-,-,-)\);
\item a retained observation region
\(\mathcal O\Subset(0,T_*)\times\RR^3\), with
\[
 D_*:=J_\eta^-(\overline{\mathcal O})\cap M_{T_*}
                                             \Subset\mathcal L.
\]
Compact laboratory tests are transported to
\(\widehat M\) and extended by zero.
\end{enumerate}
The regular conical product is not globally hyperbolic.
All global Green operators and the free scalar field
used below belong to \(\widehat M\), not to that regular
product.
\end{assumption}

The non-global-hyperbolicity diagnosis and the chart
contract are retained from X-A-ST. No conical geometry
is reconstructed. The worldsheet is not replaced by a
worldline, and no global conical inverse is introduced.

For a laboratory test \(f\), define
\[
 K=\supp f,\qquad D_f=J_\eta^-(K)\cap M_{T_*}.
\]
The supported calculations below use only \(D_f\),
or a fixed slightly larger retained shadow when
neighbourhood seminorms are needed.

\begin{definition}[Local Grade S in this file]
\label{def:grade}
A local Grade S release consists of:
\begin{enumerate}[label=\textup{(S\arabic*)}]
\item the inherited regular supported host;
\item explicitly defined limiting scalar tests, topology,
normalization, and background identity;
\item a finite installed dictionary of supplied evaluations;
\item a complete destructive-clock instrument, including
untouched references;
\item scheduled locality and a nonempty regular,
reference-preserving route under declared finite
operational permissions.
\end{enumerate}
No metric emergence, Einstein equation, or Grade E
requirement is included.
\end{definition}

%=====================================================================
\section{Frozen data, completed tests, and the topology sheet}
\label{sec:data}
%=====================================================================

\subsection{Fixed inputs and the shrinking ramp}

Let \(m\ge0\), and fix smooth real symmetric tensors
\(h,k\) on a neighbourhood of the closed slab.
They have no imposed field equations or spatial decay.
Set
\[
 q=\eta+\kappa h+\kappa^2k,\qquad
 \kappa^3=0.
\]
The coupling is formal. The limit concerns its
coefficient tests, not a numerical perturbation series.

Fix
\[
 f\in C_c^\infty(\mathcal O;\RR),\qquad
 g\in C_c^\infty((0,T_*)\times\RR^3;[0,1]).
\]
Retain the same nonempty relatively compact open plateau
\(U\) and algebra
\[
 \overline U\Subset
 \mathcal L\cap\operatorname{int}\{g=1\},\qquad
 \gp=C_c^\infty(U;T\widehat M).
\]
The group \(\Gp\) is generated through order
\(\kappa^2\) by flows of \(\kappa\xi+\kappa^2\zeta\),
with \(\xi,\zeta\in\gp\).

Choose \(\delta>0\), once and for all, such that
\begin{equation}
 K\cup\supp g\cup\overline U
                 \subset\{2\delta<t<T_*\}.
 \label{eq:separation}
\end{equation}
Only \(0<\varepsilon<\delta\) will be considered.
Thus the same group lies above every ramp in this tail
of the family. It is not enlarged as the width shrinks.

Use the predecessor profile
\[
 \sigma_\varepsilon(t)=\chi(t/\varepsilon),\qquad
 \chi(s)=
 \begin{cases}
 0,&s\le0,\\[1mm]
 \displaystyle
 \frac{e^{-1/s}}{e^{-1/s}+e^{-1/(1-s)}},&0<s<1,\\[3mm]
 1,&s\ge1.
 \end{cases}
\]
The proof also applies to any other \emph{fixed} smooth
profile with the same constant extensions and endpoint
values. Constants may depend on that profile.
No uniform theorem for arbitrary
\(\varepsilon\)-dependent profiles is asserted.

\subsection{Density coefficients}

Indices are raised with \(\eta\). Put
\(H=\eta^{\mu\nu}h_{\mu\nu}\), and abbreviate
\begin{align}
 a^{\mu\nu}
   &=a_1^{\mu\nu}[h]
      =\tfrac12H\eta^{\mu\nu}-h^{\mu\nu},
 &
 b&=\tfrac12H,
 \nonumber\\
 b_2[h]&=\tfrac18H^2-\tfrac14h_{\alpha\beta}h^{\alpha\beta},
 \nonumber\\
 a_2^{\mu\nu}[h]
   &=h^\mu{}_\lambda h^{\lambda\nu}
      -\tfrac12Hh^{\mu\nu}+b_2[h]\eta^{\mu\nu},
 \nonumber\\
 A^{\mu\nu}
   &=A_2^{\mu\nu}[h,k]
       =a_1^{\mu\nu}[k]+a_2^{\mu\nu}[h],
 &
 B&=b_1[k]+b_2[h].
 \label{eq:density}
\end{align}
These are the coefficients of
\[
 \sqrt{-\det q}\,q^{\mu\nu}
     =\eta^{\mu\nu}+\kappa a^{\mu\nu}+\kappa^2A^{\mu\nu}.
\]
Only formal density coefficients are used; a globally
nondegenerate numerical metric at finite \(\kappa\)
is not required.

\subsection{Coordinate equations}

Let
\[
 X_\varepsilon^\rho
    =x^\rho+\kappa w_\varepsilon^\rho
                         +\kappa^2v_\varepsilon^\rho.
\]
The inherited divergence-form prescription is
\begin{equation}
 \partial_\mu\left(
  [\eta^{\mu\nu}
       +\sigma_\varepsilon(\kappa a^{\mu\nu}
                                      +\kappa^2A^{\mu\nu})]
                    \partial_\nu X_\varepsilon^\rho\right)
                                                    =0\formal,
 \label{eq:coordinate-prescription}
\end{equation}
with flat coordinate Cauchy data at \(t=0\).
Equivalently,
\begin{align}
 \Box w_\varepsilon^\rho
    &=-\partial_\mu(\sigma_\varepsilon a^{\mu\rho}),
 \nonumber\\
 \Box v_\varepsilon^\rho
    &=-\partial_\mu\left[
       \sigma_\varepsilon
          (A^{\mu\rho}+a^{\mu\nu}\partial_\nu w_\varepsilon^\rho)
                              \right],
 \label{eq:coordinates}\\
 w_\varepsilon(0)&=\partial_tw_\varepsilon(0)=0,\qquad
 v_\varepsilon(0)=\partial_tv_\varepsilon(0)=0.
 \nonumber
\end{align}
Here
\[
 \Box=\partial_t^2-\Delta_{\mathbf x}.
\]
Existence and uniqueness follow successively from the
smooth flat Cauchy problem, locally by finite propagation
when the backgrounds do not decay \cite{BGP}.
The switch \(g\) does not occur in
\eqref{eq:coordinates}.

\subsection{The already completed scalar sector}

The Klein--Gordon operator and its positive Green
operators are
\[
 P_m=\Box+m^2,\qquad \ER,\EA,
\]
with
\begin{equation}
 P_m\EA c=P_m\ER c=c,\qquad
 \EA P_ma=\ER P_ma=a,\qquad
 \ER^{\mathrm T}=\EA
 \label{eq:green}
\end{equation}
for compact smooth tests. Their supports are respectively
advanced and retarded, and their test-to-smooth maps
are continuous \cite{BGP}.

The regulator-independent Euler coefficients are
\begin{align}
 L_1u
   &=\partial_\mu(g a^{\mu\nu}\partial_\nu u)+gm^2bu
 \nonumber\\
   &=g\bigl[\partial_\mu(a^{\mu\nu}\partial_\nu u)+m^2bu\bigr]
          +(\partial_\mu g)a^{\mu\nu}\partial_\nu u,
 \nonumber\\
 L_2u
   &=\partial_\mu(g A^{\mu\nu}\partial_\nu u)+gm^2Bu
 \nonumber\\
   &=g\bigl[\partial_\mu(A^{\mu\nu}\partial_\nu u)+m^2Bu\bigr]
          +(\partial_\mu g)A^{\mu\nu}\partial_\nu u.
 \label{eq:Euler}
\end{align}
In particular, \(L_1,L_2\) do not depend on
\(\varepsilon\). Their action-Hessian signs are
\(-L_1,-L_2\), as fixed in X-B-ST/X-C-ST.

Set
\[
 B_\varepsilon^\mu
       =w_\varepsilon^\nu\partial_\nu w_\varepsilon^\mu
                                              -v_\varepsilon^\mu,
\]
and define the actual bare representatives
\begin{align}
 c_{1,\varepsilon}
      &=\partial_\mu(fw_\varepsilon^\mu),
 \nonumber\\
 c_{2,\varepsilon}
      &=-\partial_\mu(fB_\varepsilon^\mu)
          +\tfrac12\partial_\mu\partial_\nu
                                (fw_\varepsilon^\mu w_\varepsilon^\nu).
 \label{eq:bare}
\end{align}
They are supported in \(K\).

The completed tests inherited from X-C-ST are
\begin{align}
 d_{1,\varepsilon}
    &=c_{1,\varepsilon}-L_1\EA f,
 \nonumber\\
 d_{2,\varepsilon}
    &=c_{2,\varepsilon}-L_1\EA c_{1,\varepsilon}
                        -L_2\EA f+L_1\EA L_1\EA f.
 \label{eq:completed}
\end{align}
Write
\[
 c_\varepsilon=f+\kappa c_{1,\varepsilon}
                         +\kappa^2c_{2,\varepsilon},\qquad
 e_\varepsilon=f+\kappa d_{1,\varepsilon}
                         +\kappa^2d_{2,\varepsilon}.
\]
Then
\begin{equation}
 e_\varepsilon=\Rq c_\varepsilon,\qquad
 \Rq=(1+L[q]\EA)^{-1},\qquad
 L[q]=\kappa L_1+\kappa^2L_2
 \quad\formal.
 \label{eq:moller}
\end{equation}

This formula recalls the existing test recursion.
It does not reconstruct its response maps. In particular,
\(-L_2\EA f+L_1\EA L_1\EA f\) is the existing
second-order pair, counted once.

All intermediate insertions remain actual off-shell
tests. Replacing a supplied insertion by its free class
before applying a response rule is not allowed:
\[
 R_1(V_{1,g};F_{P_ma})=\field(-L_1a)
\]
need not vanish \cite{XBST,XCST}.

\subsection{One fixed compact-test space}

Let
\begin{equation}
 C=K\cup\bigl(\supp g\cap J_\eta^-(K)\bigr).
 \label{eq:C}
\end{equation}
This compact set is independent of the ramp width and
lies above \(t=2\delta\).

For a compact \(C\subset\widehat M\), let
\[
 \DD_C=\{d\in C_c^\infty(\widehat M;\CC):\supp d\subset C\},
 \qquad
 p_N(d)=\max_{|\alpha|\le N}\sup_{\widehat M}|\partial^\alpha d|.
\]
The derivatives are ambient smooth derivatives.
No smooth structure is assigned to the possibly nonsmooth
boundary of \(C\).

The inherited degree-one free-field sector has topology
\begin{equation}
 \Qm=\DD/P_m\DD,\qquad
 \DD=C_c^\infty(\widehat M;\CC),\qquad
 \field(d)=[F_d].
 \label{eq:quotient}
\end{equation}
It is the linear sector of the scalar free algebra
\(\Afree\). Its equation relation is
\(\field(P_ma)=0\), and its commutator is
\[
 [\field(c),\field(d)]
       =i\hbar\pair{c}{(\EA-\ER)d}\,\mathbf1.
\]
No topology on a newly enlarged interacting algebra is
introduced \cite{FR}.

\begin{lemma}[Closed equation subspace]
\label{lem:closed}
The subspace \(P_m\DD\) is closed in \(\DD\).
Consequently \(\Qm\) is Hausdorff. Convergence in
\(\DD_C\) passes through the continuous quotient map.
\end{lemma}

\begin{proof}
The continuous map
\(\EA-\ER:\DD\to C^\infty(\widehat M)\) has kernel
\(P_m\DD\). One inclusion follows from
\eqref{eq:green}. Conversely, if
\(\EA d=\ER d=:u\), then
\[
 \supp u\subset
       J_\eta^-(\supp d)\cap J_\eta^+(\supp d),
\]
a compact causal diamond. Thus \(u\in\DD\) and
\(P_mu=d\). The kernel is closed, and the remaining
assertions follow from the quotient topology.
\end{proof}

An individual representative seminorm \(p_N\) need
not descend to the quotient. The theorem below proves
the rate for representatives and then uses the
continuous quotient map.

\begin{lemma}[Reduction of the limit problem]
\label{lem:reduction}
For tests supported in \(K\), the operators
\[
 L_1\EA,\qquad L_2\EA,\qquad L_1\EA L_1\EA
\]
are continuous into \(\DD_C\), with the causal support
specified by \eqref{eq:C}.

If \(c_{j,\varepsilon}\to c_{j,0}\) in \(\DD_K\),
then the corresponding completed tests converge in
\(\DD_C\). More precisely,
\begin{align}
 d_{1,\varepsilon}-d_{1,0}
     &=c_{1,\varepsilon}-c_{1,0},
 \nonumber\\
 d_{2,\varepsilon}-d_{2,0}
     &=c_{2,\varepsilon}-c_{2,0}
                  -L_1\EA(c_{1,\varepsilon}-c_{1,0}).
 \label{eq:reduction}
\end{align}
For every \(N\), there exist \(M_N\) and \(A_N\)
such that
\[
 p_N(L_1\EA a)\le A_Np_{M_N}(a),
 \qquad a\in\DD_K.
\]
\end{lemma}

\begin{proof}
Advanced propagation sends a compact smooth test to
a smooth function supported in its causal past.
Each \(L_j\) has compact smooth coefficients supported
in \(\supp g\). For the iteration, the second causal
past remains in \(J_\eta^-(K)\).
Green continuity, multiplication, and differentiation
give continuous maps between the stated fixed-support
Fr\'echet spaces. Continuity supplies some finite
seminorm loss; no universal derivative-loss count is
assumed. Subtraction of \eqref{eq:completed} gives
\eqref{eq:reduction}.
\end{proof}

%=====================================================================
\section{The shrinking layer: integrate the normal flux first}
\label{sec:layer}
%=====================================================================

\subsection{Localizing the spatial estimates}

Choose a compact neighbourhood of \(K\) still inside
the retained observation region. Its truncated causal
past is compact. Multiply \(h,k\) by one fixed spatial
cutoff equal to one near that entire past.

Finite propagation leaves \(w_\varepsilon,v_\varepsilon\)
unchanged on the observation neighbourhood, including
their nested second-order dependence. After this
localization all sources and solutions have uniformly
bounded spatial support on the fixed slab.

We may therefore use
\[
 \|u\|_{H^r}=\|u\|_{H^r(\RR^3)},\qquad r=0,1,\ldots,
\]
and return afterwards to the original smooth,
not necessarily decaying backgrounds.
For spatial smooth control we use
\[
 \|u\|_{C^N(\RR^3)}\le C_N\|u\|_{H^{N+2}(\RR^3)}.
\]
This follows directly from Fourier inversion and
Cauchy--Schwarz, since \(N+2>N+3/2\).

\subsection{An elementary flux estimate}

Define the free spatial wave operators
\[
 \WaveC(t)=\cos(t|D|),\qquad
 \WaveS(t)=\frac{\sin(t|D|)}{|D|},\qquad |D|=\sqrt{-\Delta}.
\]
The sine quotient has its continuous value \(t\) at
zero frequency.

\begin{lemma}[Integrated normal-source estimate]
\label{lem:flux}
Suppose
\[
 \Box u=-\partial_tF-H,\qquad
 u(0)=\partial_tu(0)=0,\qquad F(0)=0.
\]
For smooth spatially localized data,
\begin{align}
 u(t)
   &=-\int_0^t\WaveC(t-s)F(s)\,\dd s
       -\int_0^t\WaveS(t-s)H(s)\,\dd s,
 \nonumber\\
 \partial_tu(t)+F(t)
   &=-\int_0^t\WaveC'(t-s)F(s)\,\dd s
       -\int_0^t\WaveC(t-s)H(s)\,\dd s.
 \label{eq:flux-form}
\end{align}
If
\[
 M_r(F)=\sup_{0\le s\le\varepsilon}\|F(s)\|_{H^r},
\]
and likewise for \(H\), then, for \(0\le t\le\varepsilon\),
\begin{align}
 \|u(t)\|_{H^r}
    &\le tM_r(F)+\tfrac12t^2M_r(H),
 \nonumber\\
 \|\partial_tu(t)+F(t)\|_{H^r}
    &\le\tfrac12t^2M_{r+2}(F)+tM_r(H).
 \label{eq:flux-bounds}
\end{align}
No time derivative of \(F\) occurs in these bounds.
\end{lemma}

\begin{proof}
The zero-data wave formula is
\[
 u(t)=-\int_0^t\WaveS(t-s)\partial_sF(s)\,\dd s
      -\int_0^t\WaveS(t-s)H(s)\,\dd s.
\]
Integrating its first integral by parts, using
\(\WaveS(0)=0\), \(F(0)=0\), and
\(\partial_s\WaveS(t-s)=-\WaveC(t-s)\), gives the first
line of \eqref{eq:flux-form}. Differentiation gives
the second line.

The Fourier multiplier bounds are
\[
 \|\WaveC(\tau)b\|_{H^r}\le\|b\|_{H^r},\qquad
 \|\WaveS(\tau)b\|_{H^r}\le\tau\|b\|_{H^r},
\]
and
\[
 \|\WaveC'(\tau)b\|_{H^r}
            \le\tau\|b\|_{H^{r+2}}.
\]
Integrating them proves \eqref{eq:flux-bounds}.
\end{proof}

This elementary estimate is the decisive ordering of
operations: integrate the normal derivative before
estimating the layer. It avoids replacing a bounded
integrated effect by a crude inverse-width upper bound.
The parent argument is attributed to X-D
\cite[\loc{lem:flux}]{XD}.

% Source check for the parent flux argument, not an imported proof premise.
% ([enter.grok.me](https://enter.grok.me/passage/X-D-lemma-lem-flux))

\subsection{The two coordinate fluxes}

For \(\rho=0,1,2,3\), define
\begin{align}
 F_{1,\varepsilon}^\rho
    &=\sigma_\varepsilon a^{0\rho},
 &
 H_{1,\varepsilon}^\rho
    &=\sigma_\varepsilon\partial_i a^{i\rho},
 \nonumber\\
 F_{2,\varepsilon}^\rho
    &=\sigma_\varepsilon
       \left(A^{0\rho}
           +a^{00}\partial_tw_\varepsilon^\rho
           +a^{0i}\partial_iw_\varepsilon^\rho\right),
 \nonumber\\
 H_{2,\varepsilon}^\rho
    &=\sigma_\varepsilon\partial_i
       \left(A^{i\rho}
           +a^{i0}\partial_tw_\varepsilon^\rho
           +a^{ij}\partial_jw_\varepsilon^\rho\right).
 \label{eq:fluxes}
\end{align}
Spatial indices run from \(1\) to \(3\).
Both equations \eqref{eq:coordinates} have the form
in Lemma~\ref{lem:flux}, and each \(F_{j,\varepsilon}(0)\)
vanishes.

\begin{proposition}[Uniform layer control]
\label{prop:layer}
For every spatial Sobolev order, uniformly on
\(0\le t\le\varepsilon\),
\begin{align}
 w_\varepsilon&=O(\varepsilon),&
 \partial_tw_\varepsilon
       &=-\sigma_\varepsilon a^{0\,\cdot}+O(\varepsilon),
 \nonumber\\
 v_\varepsilon&=O(\varepsilon),&
 \partial_tv_\varepsilon
       &=-\sigma_\varepsilon A^{0\,\cdot}
            +\sigma_\varepsilon^2a^{00}a^{0\,\cdot}
                                                +O(\varepsilon).
 \label{eq:layer-bounds}
\end{align}
The corresponding local spatial smooth estimates also hold.
\end{proposition}

\begin{proof}
The first flux pair is bounded in every spatial Sobolev
norm independently of \(\varepsilon\). Lemma~\ref{lem:flux}
therefore gives \(w_\varepsilon=O(\varepsilon)\) and
\(\partial_tw_\varepsilon
=-\sigma_\varepsilon a^{0\,\cdot}+O(\varepsilon)\).

These bounds, in sufficiently high spatial orders,
make both \(F_{2,\varepsilon}\) and \(H_{2,\varepsilon}\)
uniformly bounded in any prescribed spatial Sobolev
order. In particular, no estimate of
\(\partial_t^2w_\varepsilon\) is needed to establish
their boundedness.

Applying the lemma again gives
\(v_\varepsilon=O(\varepsilon)\) and
\(\partial_tv_\varepsilon=-F_{2,\varepsilon}
+O(\varepsilon)\). Substituting the first-coordinate
estimates in \eqref{eq:fluxes} gives the final expression.
Sobolev embedding gives the local spatial smooth version.
\end{proof}

The positive-order normal currents are
\[
 \Pi_{1,\varepsilon}^\rho
     =\partial_tw_\varepsilon^\rho+F_{1,\varepsilon}^\rho,
 \qquad
 \Pi_{2,\varepsilon}^\rho
     =\partial_tv_\varepsilon^\rho+F_{2,\varepsilon}^\rho.
\]
They obey conservation equations
\[
 \partial_t\Pi_{j,\varepsilon}^\rho
        =-\partial_iJ_{j,\varepsilon}^{i\rho},
\]
where
\begin{align*}
 J_{1,\varepsilon}^{i\rho}
       &=-\partial_iw_\varepsilon^\rho
                                      +\sigma_\varepsilon a^{i\rho},\\
 J_{2,\varepsilon}^{i\rho}
       &=-\partial_iv_\varepsilon^\rho
          +\sigma_\varepsilon
             (A^{i\rho}+a^{i\nu}\partial_\nu w_\varepsilon^\rho).
\end{align*}
Their layer integrals are \(O(\varepsilon)\).
These are completion-normal currents, not host evolution
equations.

\subsection{Exit data and the surviving quadratic velocity}

Write
\[
 a_\Sigma=a(0,\cdot),\qquad A_\Sigma=A(0,\cdot),\qquad
 I_\ell(s)=\int_0^s\chi(r)^\ell\,\dd r,\quad \ell=1,2.
\]

\begin{lemma}[Rescaled layer and endpoint data]
\label{lem:profile}
Uniformly for \(0\le s\le1\), in every local spatial
smooth seminorm,
\begin{align}
 w_\varepsilon^\rho(\varepsilon s)
   &=-\varepsilon I_1(s)a_\Sigma^{0\rho}
                                              +O(\varepsilon^2),
 \nonumber\\
 \partial_tw_\varepsilon^\rho(\varepsilon s)
   &=-\chi(s)a_\Sigma^{0\rho}+O(\varepsilon),
 \nonumber\\
 v_\varepsilon^\rho(\varepsilon s)
   &=\varepsilon\left[
       -I_1(s)A_\Sigma^{0\rho}
       +I_2(s)a_\Sigma^{00}a_\Sigma^{0\rho}\right]
                                              +O(\varepsilon^2),
 \nonumber\\
 \partial_tv_\varepsilon^\rho(\varepsilon s)
   &=-\chi(s)A_\Sigma^{0\rho}
       +\chi(s)^2a_\Sigma^{00}a_\Sigma^{0\rho}
                                               +O(\varepsilon).
 \label{eq:rescaled}
\end{align}
The limiting positive-order initial positions are zero,
and the exit velocities converge to
\begin{equation}
 \boxed{
 \mathsf v_1^\rho=-a_\Sigma^{0\rho},\qquad
 \mathsf v_2^\rho
       =-A_\Sigma^{0\rho}+a_\Sigma^{00}a_\Sigma^{0\rho}.}
 \label{eq:velocities}
\end{equation}
These limiting data depend on the profile endpoints,
not its interior moments.
\end{lemma}

\begin{proof}
The fixed smooth coefficient difference
\(a(\varepsilon s)-a_\Sigma\), and similarly for \(A\),
is \(O(\varepsilon)\) in every localized spatial norm.
Use this in \eqref{eq:layer-bounds}.
The estimates in \eqref{eq:flux-bounds} also apply with
the current \(t\) in place of \(\varepsilon\); their
remainders are \(O(t)\). Integrating the velocities
from zero therefore gives the position expansions
with \(O(\varepsilon^2)\) remainders.

At \(s=1\), \(\chi(1)=1\).
The profile moments are multiplied by
\(\varepsilon\) in the positions, whereas the
velocities approach \eqref{eq:velocities}.
\end{proof}

The quadratic term in \(\mathsf v_2\) must not be
halved. For a direct audit, the expanded second source
contains both
\[
 -\sigma_\varepsilon'a^{00}\partial_tw_\varepsilon^\rho
 \quad\text{and}\quad
 -\sigma_\varepsilon a^{00}\partial_t^2w_\varepsilon^\rho.
\]
The first equation implies
\[
 \partial_t^2w_\varepsilon^\rho(\varepsilon s)
    =-\varepsilon^{-1}\chi'(s)a_\Sigma^{0\rho}+O(1).
\]
Consequently the singular part of the second source is
\[
 \varepsilon^{-1}\left[
  -\chi'(s)A_\Sigma^{0\rho}
       +2\chi(s)\chi'(s)a_\Sigma^{00}a_\Sigma^{0\rho}\right].
\]
Since
\[
 \int_0^1\chi'\,\dd s=1,\qquad
 \int_0^1\chi\chi'\,\dd s=\tfrac12,
\]
the two quadratic contributions sum to the full term
in \eqref{eq:velocities}. This audit is consistent with,
but not a substitute for, the integrated flux estimate.

%=====================================================================
\section{Propagation and the fixed-support convergence theorem}
\label{sec:convergence}
%=====================================================================

\subsection{The limiting Cauchy problems}

Define \(w_0,v_0\), smooth on the right-hand closed slab,
by
\begin{align}
 \Box w_0^\rho
    &=-\partial_\mu a^{\mu\rho},
 &
 w_0^\rho(0)&=0,
 &
 \partial_tw_0^\rho(0)&=\mathsf v_1^\rho,
 \nonumber\\
 \Box v_0^\rho
    &=-\partial_\mu
          (A^{\mu\rho}+a^{\mu\nu}\partial_\nu w_0^\rho),
 &
 v_0^\rho(0)&=0,
 &
 \partial_tv_0^\rho(0)&=\mathsf v_2^\rho.
 \label{eq:limit-cauchy}
\end{align}
Their data are determined by Lemma~\ref{lem:profile},
not chosen independently.

The ordinary flat Cauchy problem constructs these
solutions successively. For example, if \(s_j^\rho\)
denotes the corresponding right-hand source and
\(u_1=w_0,u_2=v_0\), then
\begin{align}
 u_j^\rho(\delta)
     &=\WaveS(\delta)\mathsf v_j^\rho
           +\int_0^\delta\WaveS(\delta-s)s_j^\rho(s)\,\dd s,
 \nonumber\\
 \partial_tu_j^\rho(\delta)
     &=\WaveC(\delta)\mathsf v_j^\rho
           +\int_0^\delta\WaveC(\delta-s)s_j^\rho(s)\,\dd s.
 \label{eq:delta-data}
\end{align}
Thus no initial-layer datum is silently lost before
propagation to the observation region.

\begin{proposition}[Propagation of the layer limit]
\label{prop:propagation}
After the fixed spatial localization, for every
integer \(r\ge0\),
\begin{align}
 \sup_{\varepsilon\le t\le T_*}
 \bigl(&\|w_\varepsilon(t)-w_0(t)\|_{H^r}
       +\|\partial_t(w_\varepsilon-w_0)(t)\|_{H^r}
 \nonumber\\
       &+\|v_\varepsilon(t)-v_0(t)\|_{H^r}
       +\|\partial_t(v_\varepsilon-v_0)(t)\|_{H^r}\bigr)
                                                   =O(\varepsilon).
 \label{eq:propagation}
\end{align}
Every spacetime derivative converges at rate
\(O(\varepsilon)\) on fixed compact subsets of
\((0,T_*]\times\RR^3\).
\end{proposition}

\begin{proof}
For \(t\ge\varepsilon\), the ramp is identically one.
Set
\[
 z_1=w_\varepsilon-w_0,\qquad z_2=v_\varepsilon-v_0.
\]
The exit expansions and smoothness of the right-hand
solutions give \(O(\varepsilon)\) position and velocity
data for both \(z_j\) at \(t=\varepsilon\), in every
spatial Sobolev order.

The first difference satisfies
\[
 \Box z_1=0.
\]
For a free wave with initial time \(s\), Fourier
multipliers give, on a fixed bounded time interval,
\begin{equation}
 \sup_{s\le t\le T_*}
 \left(\|u(t)\|_{H^{r+1}}+\|\partial_tu(t)\|_{H^r}\right)
 \le C_{r,T_*}\left[
  \|u(s)\|_{H^{r+1}}+\|\partial_tu(s)\|_{H^r}
                   +\int_s^{T_*}\|\Box u(\tau)\|_{H^r}\dd\tau
                       \right].
 \label{eq:wave-energy}
\end{equation}
Indeed,
\(\WaveC\) is bounded in each Sobolev norm,
\(\WaveS:H^r\to H^{r+1}\) is uniformly bounded on
bounded time intervals, and
\(\WaveC':H^{r+1}\to H^r\) is uniformly bounded.
The constant is independent of \(s=\varepsilon\).

This proves the estimate for \(z_1\), using higher
initial norms where needed. The second equation is
\[
 \Box z_2^\rho
    =-\partial_\mu(a^{\mu\nu}\partial_\nu z_1^\rho).
\]
Its source is \(O(\varepsilon)\) in every spatial
Sobolev order. Terms with
\(\partial_t^2z_1\) are controlled by
\(\partial_t^2z_1=\Delta z_1\); all coefficient
derivatives belong to the fixed smooth backgrounds.
Equation \eqref{eq:wave-energy} proves the estimate
for \(z_2\).

Higher time derivatives follow inductively from these
equations, using higher spatial norms and the fixed
coefficient derivatives. Sobolev embedding gives
spacetime smooth convergence away from \(t=0\).
Finally finite propagation removes the auxiliary
spatial localization on the retained region.
\end{proof}

There is generally no smooth convergence of coordinate
velocities on the whole closed slab: the finite-width
velocities vanish at \(t=0\), while the limiting
right-hand velocities need not vanish.

\subsection{Limiting bare and completed tests}

Set
\[
 B_0^\mu=w_0^\nu\partial_\nu w_0^\mu-v_0^\mu,
\]
and define
\begin{align}
 c_{1,0}&=\partial_\mu(fw_0^\mu),
 \nonumber\\
 c_{2,0}&=-\partial_\mu(fB_0^\mu)
      +\tfrac12\partial_\mu\partial_\nu(fw_0^\mu w_0^\nu),
 \nonumber\\
 d_{1,0}&=c_{1,0}-L_1\EA f,
 \nonumber\\
 d_{2,0}&=c_{2,0}-L_1\EA c_{1,0}
                          -L_2\EA f+L_1\EA L_1\EA f.
 \label{eq:limiting-tests}
\end{align}
The products with \(f\) extend smoothly by zero to
the completion, because \(f\) is supported away from
the initial surface.

\begin{theorem}[Sharp-width convergence in one fixed test space]
\label{thm:convergence}
For each fixed input satisfying
\eqref{eq:separation},
\[
 c_{j,0}\in\DD_K,\qquad d_{j,0}\in\DD_C,\qquad j=1,2.
\]
For every \(N\ge0\), there is a constant \(A_N\)
and a sufficiently small positive width bound such that
\begin{equation}
 p_N(c_{j,\varepsilon}-c_{j,0})
        +p_N(d_{j,\varepsilon}-d_{j,0})
                  \le A_N\varepsilon,\qquad j=1,2.
 \label{eq:rate}
\end{equation}
Consequently,
\[
 d_{j,\varepsilon}\longrightarrow d_{j,0}
            \quad\text{in }\DD_C,\qquad
 [d_{j,\varepsilon}]\longrightarrow[d_{j,0}]
            \quad\text{in }\Qm.
\]
The limiting tests are independent of the choice of
fixed ramp profile with the stated endpoints.
\end{theorem}

\begin{proof}
Proposition~\ref{prop:propagation} controls every
coordinate derivative on a fixed neighbourhood of \(K\).
The first bare test is a finite derivative of
\(fw_\varepsilon\).
For the second, write
\[
 B_\varepsilon-B_0
   =(w_\varepsilon-w_0)\cdot\partial w_\varepsilon
      +w_0\cdot\partial(w_\varepsilon-w_0)
      -(v_\varepsilon-v_0).
\]
Uniform boundedness of the coordinate derivatives and
their \(O(\varepsilon)\) differences gives the same
rate for every derivative of \(B_\varepsilon-B_0\).
The Leibniz rule applied to \eqref{eq:bare} proves
the bare-test estimates, with support in \(K\).

Lemma~\ref{lem:reduction} proves the completed-test
estimates and their support in \(C\).
The inclusion \(\DD_C\to\DD\) and quotient map
\(\DD\to\Qm\) are continuous.

The limiting data \eqref{eq:velocities} do not depend
on interior profile moments. Cauchy uniqueness in
\eqref{eq:limit-cauchy}, followed by
\eqref{eq:limiting-tests}, proves profile independence.
\end{proof}

This reproduces the fixed-input conclusion of X-D
\cite[\loc{thm:convergence}]{XD}; no parent convergence
theorem was used in its proof.

% Source check for the attributed parent convergence statement.
% ([enter.grok.me](https://enter.grok.me/passage/X-D-theorem-thm-convergence))

The representative tests converge before taking equation
classes. No quotient cancellation rescues a divergent
representative family. For every continuous seminorm
\(q\) on \(\Qm\), continuity of
\(\DD_C\to\Qm\) also gives
\[
 q([d_{j,\varepsilon}-d_{j,0}])=O(\varepsilon).
\]
This is not an operator-norm estimate in a Hilbert
representation or a channel estimate for a field detector.

\subsection{Supported patch agreement}

\begin{proposition}[Supported transfer of the limit]
\label{prop:patch}
The coordinate coefficients on \(K\), the limiting
tests, and their compact pairings depend only on the
backgrounds near \(D_f\), with the same initial data.
Extensions on the declared flat completion agreeing
there give the same supported result.

Transport through \(\iota\) preserves these compact
pairings and their convergence. No global conical
propagator or gluing around the worldsheet is obtained.
\end{proposition}

\begin{proof}
The coordinate equations are solved successively by
flat finite propagation. Every nested retarded
dependence ending in \(K\) remains in \(D_f\).
The limiting initial data are the traces of those
same local coefficients on the relevant initial shadow.

For the response terms, all contributing interaction
vertices lie in
\(\supp g\cap J_\eta^-(K)\).
A second advanced test action remains in that causal
past. The same local coefficients and fixed completion
Green operators therefore give the same tests.

Distribution transport through a chart is defined by
transposition on tests and densities. It rewrites the
retained pairing. On a fixed compact reserve, the chart
and its inverse have bounded derivatives of every
required finite order, so the associated fixed-support
test topologies are equivalent.
\end{proof}

%=====================================================================
\section{Identification: continuous coordinates and normal flux}
\label{sec:identification}
%=====================================================================

\subsection{The sharp divergence-form problem}

Let \(\theta(t)=\mathbf1_{\{t>0\}}\), and put
\[
 A_{\mathrm{sh}}^{\mu\nu}
      =\eta^{\mu\nu}
         +\theta(t)(\kappa a^{\mu\nu}+\kappa^2A^{\mu\nu}).
\]
Extend the limiting coordinate map by
\[
 X_{\mathrm{sh}}^\rho=
 \begin{cases}
 x^\rho,&t<0,\\
 x^\rho+\kappa w_0^\rho+\kappa^2v_0^\rho,&t>0.
 \end{cases}
\]
Its positive-order positions are continuous at
\(\Sigma\).

\begin{theorem}[Transmission characterization]
\label{thm:transmission}
The map \(X_{\mathrm{sh}}\) is the unique formal
piecewise smooth continuous map equal to \(x\) for
\(t<0\) and satisfying
\begin{equation}
 \partial_\mu\left(
 A_{\mathrm{sh}}^{\mu\nu}\partial_\nu X_{\mathrm{sh}}^\rho
                 \right)=0\formal
 \label{eq:transmission}
\end{equation}
as a distribution, where the flux is formed first as
a locally integrable piecewise smooth function.
Its completion-normal flux is continuous across
\(\Sigma\).
\end{theorem}

\begin{proof}
The derivative of a continuous piecewise smooth
coordinate function is a piecewise smooth locally
integrable function; there is no coordinate-jump
delta in that first derivative.
Thus the displayed flux is defined coefficientwise
by ordinary multiplication of piecewise smooth
functions.

Its divergence has a surface delta whose coefficient
is the jump of its normal component. On the negative
side that component is \(\eta^{0\rho}=\delta^\rho_0\).
On the positive side its initial value is
\[
 \delta^\rho_0
   +\kappa(\mathsf v_1^\rho+a_\Sigma^{0\rho})
   +\kappa^2\left(
        \mathsf v_2^\rho+A_\Sigma^{0\rho}
                           +a_\Sigma^{00}\mathsf v_1^\rho\right).
\]
Tangential positive-order derivatives vanish initially,
because the positive-order positions are identically
zero on \(\Sigma\). The jump vanishes exactly for
\eqref{eq:velocities}. The bulk equations are
\eqref{eq:limit-cauchy}, proving
\eqref{eq:transmission}.

Conversely, continuity forces the zero positive-order
initial positions. Flux matching determines the two
velocities successively. The smooth right-hand Cauchy
problems then have unique solutions.
\end{proof}

The normal here is the declared completion covector
\(n=\dd t\). These velocities are coordinate-coefficient
data, not a newly derived host lapse or a measured
observer velocity. The host time conversion remains
the inherited one.

The flux-first interpretation is important: it does
not multiply a surface delta by an unspecified
discontinuous velocity.

\subsection{The smeared linear background kernel}

Define
\[
 \mathcal C^\rho[h]=-\partial_\mu a_1^{\mu\rho}[h],
 \qquad
 (\mathcal C^*b)^{\alpha\beta}
       =-\partial^{(\alpha}b^{\beta)}
             +\tfrac12\eta^{\alpha\beta}\partial_\mu b^\mu.
\]
Parentheses include the factor \(1/2\).

\begin{proposition}[A possible jump, but no hypersurface delta]
\label{prop:kernel}
For a compact smooth covector test
\(\ell_\rho\) supported in the open slab, set
\(b_\rho=\EAzero\ell_\rho\). Then
\begin{equation}
 \int\ell_\rho w_0^\rho\,\dd^4x
     =\int h_{\alpha\beta}\,
              \theta(t)(\mathcal C^*b)^{\alpha\beta}\,\dd^4x.
 \label{eq:sharp-kernel}
\end{equation}
Thus the observation-smeared linear background kernel is
\[
 q_{\ell,0}=\theta\,\mathcal C^*(\EAzero\ell).
\]
It is compactly supported, smooth on each side of
\(\Sigma\), and locally bounded. It can jump at
\(\Sigma\), but it has no hypersurface-delta summand.

The finite-width kernels satisfy
\[
 q_{\ell,\varepsilon}
         =\sigma_\varepsilon\mathcal C^*(\EAzero\ell),
 \qquad
 \|q_{\ell,\varepsilon}-q_{\ell,0}\|_{L^1}=O(\varepsilon).
\]
\end{proposition}

\begin{proof}
For positive width, Green transposition and compact
integration by parts give
\begin{align*}
 \int\ell_\rho w_\varepsilon^\rho\,\dd^4x
   &=-\int b_\rho
          \partial_\mu(\sigma_\varepsilon a^{\mu\rho})\,\dd^4x\\
   &=\int\sigma_\varepsilon
                         (\partial_\mu b_\rho)a^{\mu\rho}\,\dd^4x\\
   &=\int h_{\alpha\beta}\,
                    \sigma_\varepsilon(\mathcal C^*b)^{\alpha\beta}
                                                           \,\dd^4x.
\end{align*}
The source is past-trivial and the ramp is flat at zero,
so there is no initial boundary term. The advanced
solution vanishes after the observation source.

After restricting to \(t\ge0\), its advanced support
has a compact truncated causal past. The kernel
difference is bounded and supported in a layer of
thickness \(\varepsilon\), with one fixed compact
spatial reserve. This proves the \(L^1\) estimate.
Theorem~\ref{thm:convergence}, applied to the coordinate
pairing, and dominated convergence then give
\eqref{eq:sharp-kernel}.

The limiting kernel is an explicitly locally integrable
function. Its formula has a step, not a surface delta.
\end{proof}

The assertion concerns an observation-smeared first
background derivative. The unsmeared two-point
coordinate kernel remains distributional.
Moreover, derivatives of \(q_{\ell,0}\) can contain
surface deltas. ``No hypersurface delta in the kernel''
does not mean that its wavefront set is empty or that
all its distributional derivatives are delta-free.

\subsection{Not the unchanged multiplied-equation prescription}

Let \(\widetilde w_0\) solve
\[
 \Box\widetilde w_0^\rho=\mathcal C^\rho[h],\qquad
 \widetilde w_0^\rho(0)=\partial_t\widetilde w_0^\rho(0)=0.
\]
Then
\begin{equation}
 w_0^\rho-\widetilde w_0^\rho
           =-\WaveS(t)a_\Sigma^{0\rho}.
 \label{eq:homogeneous-difference}
\end{equation}
Both solve the same positive-time bulk equation, but
have different velocities.

The difference is also visible after transposition.
For \(n=\dd t\), define
\[
 \mathcal B_\Sigma(b)^{\alpha\beta}
       =-n^{(\alpha}b^{\beta)}
                   +\tfrac12\eta^{\alpha\beta}n_\mu b^\mu.
\]
The multiplied-equation kernel is
\begin{equation}
 \mathcal C^*(\theta b)
     =\theta\mathcal C^*b
          +\delta_\Sigma\mathcal B_\Sigma(b|_\Sigma),
 \label{eq:multiplied-kernel}
\end{equation}
whereas \eqref{eq:sharp-kernel} lacks the last term.

These statements identify which sharp prescription is
obtained. They do not establish sharp joint
microcausal admission in quantum metric slots.
The corresponding source distinction is the one
recorded in X-D \cite[\loc{thm:identification}]{XD}.

% Source check for the attributed transmission identification.
% ([enter.grok.me](https://enter.grok.me/passage/X-D-theorem-thm-identification))

%=====================================================================
\section{Gauge passage and the defect ledger}
\label{sec:gauge}
%=====================================================================

\subsection{Formal pullbacks and the complete metric law}

For compact interior vector fields \(\xi,\zeta\), let
\[
 X_\xi=\xi^\mu\partial_\mu,\qquad
 D_\xi r=\partial_\mu(\xi^\mu r),
\]
and write
\begin{align}
 F^*&=1+\kappa X_\xi+\kappa^2Z_\xi,
 &
 Z_\xi&=X_\zeta+\tfrac12X_\xi^2,
 \nonumber\\
 F^\sharp&=1+\kappa D_\xi+\kappa^2\Delta_\xi,
 &
 \Delta_\xi&=D_\zeta+\tfrac12D_\xi^2.
 \label{eq:pullbacks}
\end{align}
Here \(F^\sharp\) is density pullback.
Compact integration by parts gives
\[
 \pair{F^\sharp a}{F^*u}=\pair{a}{u}\formal.
\]

The transformed fixed backgrounds are
\begin{align}
 h'&=h+K\xi,\qquad
 (K\xi)_{\mu\nu}=\partial_\mu\xi_\nu+\partial_\nu\xi_\mu,
 \nonumber\\
 k'&=k+\Lie_\xi h+K\zeta+\tfrac12\Lie_\xi^2\eta.
 \label{eq:metriclaw}
\end{align}
When the generators are fixed, these backgrounds are
independent of \(\varepsilon\).

\subsection{A recalled identity with an explicit representative}

The X-C-ST three-term variation can be stated entirely
at test level. Let
\[
 E[q]=P_m+L[q],\qquad s_\varepsilon[q]=\EA e_\varepsilon[q],
\]
and define
\begin{align}
 \Theta_{F,\varepsilon}[q]
     &=c_\varepsilon[q']-F^\sharp c_\varepsilon[q],
 \nonumber\\
 \Sigma_F[q]
     &=E[q']F^*-F^\sharp E[q].
 \label{eq:defects}
\end{align}
The switch defect \(\Sigma_F\) is independent of the
ramp width.

\begin{lemma}[Three-term test identity]
\label{lem:variation}
For compact interior generators,
\begin{equation}
 e_\varepsilon[q']-e_\varepsilon[q]
   =P_m\bigl((F^*-1)s_\varepsilon[q]\bigr)
      +\mathcal R_{q'}\Theta_{F,\varepsilon}
      -\mathcal R_{q'}(\Sigma_Fs_\varepsilon[q])
       \quad\formal.
 \label{eq:variation}
\end{equation}
Every positive-order coefficient of the first term
is in \(P_m\DD\).
\end{lemma}

\begin{proof}
The formal advanced solution of \(E[q]s=c_\varepsilon[q]\)
is \(s=\EA\Rq c_\varepsilon=\EA e_\varepsilon\).
Indeed,
\[
 E[q]\EA e_\varepsilon
             =(1+L[q]\EA)e_\varepsilon=c_\varepsilon.
\]
Its coefficientwise advanced uniqueness follows
successively from free advanced uniqueness and the
compact differential coefficients of \(L[q]\).

Writing \(s'=\EA e_\varepsilon[q']\), the defect
definitions give
\[
 E[q'](s'-F^*s)
          =\Theta_{F,\varepsilon}-\Sigma_Fs.
\]
The right-hand side is a compact formal source:
the coordinate defect has compact observation-test
support, and the switch defect has compact generator
coefficient support. Advanced uniqueness yields
\[
 s'-F^*s
   =\EA\mathcal R_{q'}
                     (\Theta_{F,\varepsilon}-\Sigma_Fs).
\]
Apply \(P_m\), and use \(P_ms=e_\varepsilon\).
Every coefficient of \((F^*-1)s\) is smooth and
supported in the fixed compact generator support.
\end{proof}

This is the predecessor identity, included with its
short test-level argument to make the passage auditable
\cite[\loc{prop:variation}]{XCST}. It is not another
construction or another counting of the response maps.

\subsection{Switch derivatives are not suppressed}

Write
\[
 \Sigma_F=\kappa S_1+\kappa^2S_2.
\]
Let \(Q_j\) be the corresponding unswitched Euler
coefficients, and set \(N_j=Q_j-L_j\). Thus
\begin{align}
 N_1u&=(1-g)Q_1u-(\partial_\mu g)a^{\mu\nu}\partial_\nu u,
 \nonumber\\
 N_2u&=(1-g)Q_2u-(\partial_\mu g)A^{\mu\nu}\partial_\nu u.
 \label{eq:N}
\end{align}
Unswitched scalar-density naturality gives
\begin{align}
 S_1
   &=(g-1)Q_1[K\xi]
          +(\partial_\mu g)a_1^{\mu\nu}[K\xi]\partial_\nu,
 \nonumber\\
 S_2
   &=-(N_2'-N_2)-N_1'X_\xi+D_\xi N_1.
 \label{eq:switch-defect}
\end{align}
The latter expression includes derivatives such as
\[
 -\xi^\rho(\partial_\rho\partial_\mu g)
                          a^{\mu\nu}\partial_\nu u
\]
arising from \(D_\xi N_1\). Thus second derivatives
of the switch are not discarded.

The combinations \(S_j\) have compact generator
coefficient support even though \(1-g\), and in
general \(g-\sigma_\varepsilon\), do not have compact
support. If \(g=1\) near the generators, then
\(S_1=S_2=0\).

\subsection{Passage on the originally frozen algebra}

\begin{theorem}[The limiting background identity]
\label{thm:ward}
For \(F\in\Gp\), generated by
\(\xi,\zeta\in\gp\), the limiting completed field
\[
 \widehat A_0[q]
     =\field(f)+\kappa\field(d_{1,0}[h])
                            +\kappa^2\field(d_{2,0}[h,k])
\]
is background invariant through order \(\kappa^2\),
with the free field held fixed.

More explicitly,
\begin{align}
 d_{1,0}'-d_{1,0}
    &=P_m(X_\xi\EA f),
 \nonumber\\
 d_{2,0}'-d_{2,0}
    &=P_m\bigl(X_\xi\EA d_{1,0}+Z_\xi\EA f\bigr).
 \label{eq:limit-ideal}
\end{align}
The functions inside \(P_m\) are compactly supported.
\end{theorem}

\begin{proof}
For every \(\varepsilon<\delta\), the generators lie
where both \(g\) and \(\sigma_\varepsilon\) equal one.
The coordinate density transforms naturally there,
and the transformation is the identity elsewhere.
The fixed coordinate Cauchy data are unchanged.
Coefficientwise Cauchy uniqueness therefore gives the
inherited coordinate intertwiner and
\(\Theta_{F,\varepsilon}=0\).
The switch defect also vanishes.

Lemma~\ref{lem:variation} becomes
\[
 e_\varepsilon[q']-e_\varepsilon[q]
       =P_m\bigl((F^*-1)\EA e_\varepsilon[q]\bigr).
\]
Theorem~\ref{thm:convergence} applies separately to
both fixed background pairs. Green continuity gives
smooth convergence of \(\EA e_\varepsilon\) on the
compact generator support. Multiplication and
differentiation by \(F^*-1\) then give convergence
in one compact-test space. Passing to the limit
proves
\[
 e_0[q']-e_0[q]=P_m\bigl((F^*-1)\EA e_0[q]\bigr).
\]
Expanding \eqref{eq:pullbacks} yields
\eqref{eq:limit-ideal}. Their classes vanish in
\(\Qm\).
\end{proof}

The group has not been shrunk or enlarged after the
calculation. The off-shell difference need not be zero:
it is the explicit equation-ideal representative.
This is not a pullback automorphism of the unchanged
free solution space, an interacting BRST identity,
or a quantum metric Ward theorem.

\subsection{Off-plateau passage and propagated gaps}

Write
\[
 \Theta_{F,\varepsilon}
       =\kappa\theta_{1,\varepsilon}
                         +\kappa^2\theta_{2,\varepsilon}.
\]
Its coefficients are
\begin{align}
 \theta_{1,\varepsilon}
    &=c_{1,\varepsilon}'-c_{1,\varepsilon}-D_\xi f
 \nonumber\\
    &=\partial_\mu\bigl(
          f(w_\varepsilon[K\xi]-\xi)^\mu\bigr),
 \nonumber\\
 \theta_{2,\varepsilon}
    &=c_{2,\varepsilon}'-c_{2,\varepsilon}
                         -D_\xi c_{1,\varepsilon}-\Delta_\xi f.
 \label{eq:theta}
\end{align}

\begin{proposition}[Limiting defect identity for fixed generators]
\label{prop:defect-limit}
For fixed compact interior generators, the coordinate
defect tests converge in a fixed compact-test space.
Let their limits be \(\theta_{j,0}\).
With \(L_1'=L_{1,g}[h']\), one has
\begin{align}
 d_{1,0}'-d_{1,0}
   ={}&P_m(X_\xi\EA f)+\theta_{1,0}-S_1\EA f,
 \label{eq:defect1}\\
 d_{2,0}'-d_{2,0}
   ={}&P_m\bigl(X_\xi\EA d_{1,0}+Z_\xi\EA f\bigr)
       +\theta_{2,0}-S_2\EA f-S_1\EA d_{1,0}
 \nonumber\\
      &-L_1'\EA\bigl(\theta_{1,0}-S_1\EA f\bigr).
 \label{eq:defect2}
\end{align}
For this fixed compact interior generator class,
\(\theta_{1,0}=\theta_{2,0}=0\).
Switch defects can nevertheless remain.
\end{proposition}

\begin{proof}
Convergence of the bare tests for both fixed background
pairs and continuity of the density-pullback
coefficients give convergence in \eqref{eq:theta}.
The compact sources
\(\Sigma_F\EA e_\varepsilon\) also converge, since
their coefficient supports are fixed.
The operator \(\mathcal R_{q'}\) is independent of
\(\varepsilon\) and continuous on each required compact
source stage. Thus \eqref{eq:variation} passes to the
limit. Extracting its two coefficients gives
\eqref{eq:defect1}--\eqref{eq:defect2}.

A fixed compact subset of the open slab has positive
distance from \(\Sigma\). For sufficiently small
\(\varepsilon\), the coordinate transformation is
therefore supported where \(\sigma_\varepsilon=1\).
Coordinate naturality, which does not require an
interaction plateau, gives
\(\Theta_{F,\varepsilon}=0\) for those widths.
Hence its limits vanish. The same argument does not
force \(g=1\) on those supports, so it does not remove
\(\Sigma_F\).
\end{proof}

\statementbox{An essential edge quantifier}{%
A fixed compact interior gauge parameter eventually lies
above the shrinking ramp. It is incorrect to conclude that
every finite-width edge defect survives as a nonzero defect
for that same fixed parameter in the sharp limit.

A parameter that stays inside the ramp must move with
\(\varepsilon\). Its transformed backgrounds are then
generally not fixed inputs of Theorem~\ref{thm:convergence}.
Neither observation is a higher-order repair of an already
nonzero lower coefficient at a fixed prescription.}

At any positive width, the exact second variation
retains
\[
 -L_1'\EA\theta_{1,\varepsilon}.
\]
In the conditional situation where the generators lie
in an existing interaction plateau, the free-class
formula is
\[
 \delta_F\field(d_{2,\varepsilon})
    =\field\bigl(\theta_{2,\varepsilon}
                         -L_1'\EA\theta_{1,\varepsilon}\bigr).
\]
That isolation is not available for a ramp-supported
parameter in the present small-width tail:
\eqref{eq:separation} makes \(g=0\) on the ramp.
The full switch contribution must then be kept.

For the inherited edge family
\[
 \xi_\varepsilon
       =\alpha(t/\varepsilon)\beta(\mathbf x)\partial_1,
 \qquad
 0\ne\alpha\ge0,\quad\alpha\in C_c^\infty((0,1)),\quad
 \int\beta\,\dd^3x=1,
\]
the coordinate gap satisfies
\begin{equation}
 \int_{\RR^3}
  (w_\varepsilon^1[K\xi_\varepsilon]
                  -\xi_\varepsilon^1)(t,\mathbf x)\,\dd^3x
       =-\int_0^1\chi'(s)\alpha(s)\,\dd s<0,
       \qquad t\ge\varepsilon.
 \label{eq:moving-edge}
\end{equation}
This is the X-A-ST edge witness with its width dependence
made explicit \cite{XAST}.
Derivatives of \(K\xi_\varepsilon\) generally grow as
inverse powers of \(\varepsilon\).
It is not a fixed smooth-background family covered by
the rate theorem.

A coordinate integral alone also does not prove that
every completed smearing has a nonzero free class.
Smearing, switch terms, and equation-zero
representatives must be checked on the actual datum.

%=====================================================================
\section{Diagnostics and sharpness of the rate}
\label{sec:diagnostics}
%=====================================================================

\subsection{Complete pure gauge}

For \(\xi\in\gp\), take the full two-jet
\[
 h=K\xi,\qquad k=\tfrac12\Lie_\xi^2\eta.
\]
It vanishes near the initial surface, so both layer
sources and both limiting velocities vanish there.
The coordinate intertwiner gives, for every small width,
\[
 w_\varepsilon=\xi,\qquad
 v_\varepsilon=\tfrac12X_\xi\xi.
\]
The completed coefficients have zero free class,
including their limiting classes by
Theorem~\ref{thm:ward}. This diagnostic checks the
complete two-jet and the zero-layer case.
It is not a replacement for the general layer proof.

The choice \(h=K\xi,k=0\) is not a second-order
pure-gauge jet.

\subsection{Conformal temporal diagnostic}

Let \(m=0\), \(h=\psi(t)\eta\), and \(k=0\) near a
causal cylinder containing the retained calculation.
Spatial cutoffs outside that cylinder give admissible
localized backgrounds without changing the result.
Then
\[
 a^{\mu\nu}=\psi\eta^{\mu\nu},\qquad A^{\mu\nu}=0.
\]
The massless second scalar vertex vanishes.

Suppose \(g=g_0(t)\) on that cylinder, and define
\begin{align*}
 \Psi_\varepsilon(t)&=\int_0^t\sigma_\varepsilon(s)\psi(s)\,\dd s,
 &
 J_\varepsilon(t)&=\int_0^t
                         (\sigma_\varepsilon(s)\psi(s))^2\,\dd s,\\
 \Gamma(t)&=\int_0^t g_0(s)\psi(s)\,\dd s,
 &
 J_g(t)&=\int_0^t(g_0(s)\psi(s))^2\,\dd s.
\end{align*}
Direct integration of the coordinate and scalar equations
gives
\[
 w_\varepsilon^0=-\Psi_\varepsilon,\qquad
 v_\varepsilon^0=J_\varepsilon,\qquad
 \rho_{1,g}[t]=-\Gamma,\qquad \rho_{2,g}[t]=J_g.
\]
All relevant spatial components vanish on this
calculation.

On the smooth massless solution \(u=t\),
\begin{align}
 \pair{d_{1,\varepsilon}}{t}
      &=\int f(\Psi_\varepsilon-\Gamma)\,\dd^4x,
 \nonumber\\
 \pair{d_{2,\varepsilon}}{t}
      &=\int f\left[
          J_g-J_\varepsilon
             +\Psi_\varepsilon\psi(\sigma_\varepsilon-g_0)
                                      \right]\dd^4x.
 \label{eq:diagnostic}
\end{align}
For instance, the bare second term is
\(\Psi_\varepsilon\sigma_\varepsilon\psi-J_\varepsilon\),
the mixed term is
\(-\Psi_\varepsilon g_0\psi\), and the iterated
scalar response is \(J_g\).
This verifies their relative signs and the counting.

Let
\[
 \Psi_0(t)=\int_0^t\psi(s)\,\dd s,\qquad
 J_0(t)=\int_0^t\psi(s)^2\,\dd s.
\]
On the observation support the limiting evaluations are
\begin{align}
 \pair{d_{1,0}}{t}
      &=\int f(\Psi_0-\Gamma)\,\dd^4x,
 \nonumber\\
 \pair{d_{2,0}}{t}
      &=\int f\left[
         J_g-J_0+\Psi_0\psi(1-g_0)\right]\dd^4x.
 \label{eq:diagnostic-limit}
\end{align}
The layer differences in \(\Psi_\varepsilon,J_\varepsilon\)
are \(O(\varepsilon)\), exactly as the theorem predicts.
The right-hand initial velocities are
\[
 \partial_tw_0^0(0)=-\psi(0),\qquad
 \partial_tv_0^0(0)=\psi(0)^2.
\]
The multiplied-equation first coordinate instead has
zero initial velocity.

\begin{proposition}[The general rate is not \(o(\varepsilon)\)]
\label{prop:optimal}
There are fixed admissible data for which
\(d_{1,\varepsilon}-d_{1,0}\) is not
\(o(\varepsilon)\) in \(\DD_C\), or after a fixed
free-solution evaluation.
\end{proposition}

\begin{proof}
Use the conformal family with \(\psi=\psi_*\ne0\)
on a fixed neighbourhood of \(t=0\), and a real
observation test with \(\int f\,\dd^4x=1\).
For all sufficiently small widths,
\[
 \Psi_\varepsilon(t)-\Psi_0(t)
       =-\varepsilon\psi_*
                       \left(1-\int_0^1\chi(s)\,\dd s\right)
\]
at every observation time. The switch contribution
does not depend on the width. Equation
\eqref{eq:diagnostic} therefore gives
\[
 \pair{d_{1,\varepsilon}-d_{1,0}}{t}
       =-\varepsilon\psi_*
                       \left(1-\int_0^1\chi(s)\,\dd s\right).
\]
For the stated ramp, the coefficient is nonzero.
Pairing with \(t\) is continuous on \(\DD_C\);
hence the representative difference cannot be
\(o(\varepsilon)\).

Equivalently, on \(K\),
\[
 c_{1,\varepsilon}-c_{1,0}
       =\varepsilon\psi_*
          \left(1-\int_0^1\chi(s)\,\dd s\right)\partial_tf.
\]
This gives the same conclusion directly.
\end{proof}

For a smooth free solution \(u\), the evaluation
\[
 \ev_u\field(d)=\pair{d}{u}
\]
is well defined on the degree-one quotient:
compact equation tests vanish by integration by parts.
It is not a state on the Wick algebra or a quantum
expectation value.

%=====================================================================
\section{Three topologies and separate comparison problems}
\label{sec:topologies}
%=====================================================================

\subsection{The actual topology ledger}

The three named topologies in this comparison are:
\begin{enumerate}
\item the fixed-support smooth topology of \(\DD_C\);
\item its image under the continuous map to the
free-scalar quotient \(\Qm\);
\item the different boundary-sensitive topology
\(\tau_\Sigma\) of the IX-A/IX-A-ST actual-functional
domain.
\end{enumerate}
Polynomial smoothing growth and characteristic
asymptotics are additional comparison estimates,
not alternative names for these topologies.

\begin{center}
\small
\renewcommand{\arraystretch}{1.12}
\begin{tabularx}{\linewidth}{@{}p{.22\linewidth}p{.32\linewidth}X@{}}
\toprule
Domain or topology & Object & Verdict and boundary\\
\midrule
\(\DD_C\) &
The fixed-background completed scalar tests &
Convergent at \(O(\varepsilon)\) in every seminorm\\
\(\Qm\) &
The same degree-one free classes &
Convergent by a continuous quotient map\\
IX-A-ST's \(\tau_\Sigma\) &
Its unchanged sharp functional and regular approximants &
Its nonconvergence result is not replaced by this limit\\
Joint quantum metric kernels &
No such limit object is constructed here &
No sharp metric-slot admission theorem\\
Characteristic cross-line composition &
A different multi-slot distribution problem &
Not an inference from scalar-test convergence\\
\bottomrule
\end{tabularx}
\end{center}

\subsection{IX-A-ST and its separator}

The IX-A-ST record preserves a specific unchanged
sharp insertion in a topology that detects its
initial boundary charge \cite{IXAST,IXA}.
Its continuous separator distinguishes that insertion
from the regular approximants whose corresponding
boundary coordinates vanish.

The logical implication is elementary: if a continuous
linear map \(b_\Sigma\) satisfies
\[
 b_\Sigma(F_\lambda)=0,\qquad b_\Sigma(F_{\mathrm{sh}})\ne0,
\]
then \(F_\lambda\) cannot converge to
\(F_{\mathrm{sh}}\) in that topology.

The present family is not identified with that pair.
Its sharp linear background kernel is
\(\theta\mathcal C^*b\), while the unchanged
multiplied-equation kernel carries the additional
surface term in \eqref{eq:multiplied-kernel}.
Moreover, the theorem freezes the classical backgrounds
and concerns smooth scalar output tests, not convergence
of the full mixed metric--scalar functional kernels.
No map identifying the IX-A-ST topology with \(\DD_C\)
or \(\Qm\) is asserted.

\subsection{Polynomial growth, scalar diagnostics, and raw divergence}

The following supplied records are comparisons only
\cite{Em01B,Em01C,Em02D}:
\begin{itemize}
\item Em01-B, version 3, gives polynomial smoothing
growth for its retained contraction. A bound by
a negative power of a smoothing scale is not
convergence through a characteristic locus.
It is not the \(O(\varepsilon)\) estimate proved here.
\item Em01-C, version 3, separates its scalar aligned-null
expansion from the source-specific complete-sum
characteristic estimate. Its scalar diagnostic is
not the completed physical coefficient and not
the present coordinate kernel.
\item Em02-D, version 4, has a fixed-model separated
detecting test with
\[
 t_\lambda(\Phi_A)
       =c_A\lambda^{-3}+O(\lambda^{-2}),\qquad c_A\ne0.
\]
That raw family has no weak limit. A correction supported
on the excluded attachment cannot change a detecting
test supported away from it.
\end{itemize}
The last nonconvergence follows because weak convergence
would make \(t_\lambda(\Phi_A)\) bounded, contrary
to the displayed asymptotic.
An \(O(\lambda^{-2})\) remainder does not by itself
establish a subtracted finite part.
None of these distribution families is replaced or
repaired by Theorem~\ref{thm:convergence}.

\subsection{Smooth multipliers do not supply missing products}

For a fixed smooth multiplier \(a\) and a distribution
\(u\),
\[
 \WF(au)\subseteq\WF(u).
\]
This standard microlocal fact \cite{Hormander},
also recorded as a conditional tool in XVI-A
\cite{XVIA}, is not a subtraction prescription.
It supplies no uniform estimate for an arbitrary
scale-dependent multiplier family, no extension of an
undefined product, and no loop regulator.

The ramp disappears from the coordinate density in
a specified Cauchy limit. It was never a regulator
for the characteristic quantum graph or for a
Feynman-kernel square.

%=====================================================================
\section{VII-C audit and the remaining analytical boundary}
\label{sec:viic}
%=====================================================================

\subsection{The comparison assumptions}

For clarity, the logical contents of VII-C's four
normal-topology extension requirements are reproduced
here as \emph{comparison requirements}, not assumptions
of the present theorem \cite{VIIC}.

Its model includes a quantum metric block
\(H_{hh}=P\Delta_0^+\), a scalar block, and a sharp
mixed insertion \(B_f\) defined with its zero-Cauchy
coordinate prescription. Its signature convention is
separate from the analytical convention of this file.

A proposed extension in that record must have:
\begin{enumerate}[label=\textup{(E\arabic*)}]
\item a Hausdorff locally convex formal algebra containing
the old microcausal polynomial algebra with its fixed
Wick product;
\item admitted sharp insertions whose order-\(\hbar^0\)
symbols equal the specified \(B_f\) on all smooth
off-shell configurations;
\item sequentially continuous admission on the specified
mixed-kernel normal-topology stages, agreeing with
the regular inclusion and sending the prescribed
mollified sharp kernels to those insertions;
\item bounded-set hypocontinuous multiplication and a
continuous field-independent order-\(\hbar^3\)
coefficient extraction, agreeing with the old
extraction.
\end{enumerate}

For suitable separated tests, its prescribed word has
\[
 \ev_0[\hbar^3]\bigl(
     B_{f,\lambda}\star_H W_g\star_H B_{f',\lambda}\bigr)
       =\mathcal C(f,g,f')\log(\lambda^{-1})+O(1),
 \qquad \mathcal C(f,g,f')>0,
\]
where \(W_g=\int g\,h_{00}^2\,\dd^4x\).
That is its named extension obstruction, not a claim
that every smaller scalar construction is impossible.

\subsection{Why the hypotheses do not match}

\begin{center}
\small
\renewcommand{\arraystretch}{1.12}
\begin{tabularx}{\linewidth}{@{}p{.08\linewidth}p{.39\linewidth}X@{}}
\toprule
Item & VII-C requires & X-D-ST actually uses\\
\midrule
(E1) &
The specified scalar--metric multiplet and Wick matrix &
The inherited scalar algebra; \(h,k\) are classical\\
(E2) &
The exact unchanged sharp mixed functional &
Fixed-background smooth scalar tests built from
normal-flux transmission coordinates\\
(E3) &
Its mollified mixed kernels in its normal topology &
Ramp coefficient tests in one \(\DD_C\)\\
(E4) &
The mandatory triple product and continuous
fully contracted extraction &
No such word, quantum metric block, or coefficient
extension is formed\\
\bottomrule
\end{tabularx}
\end{center}

Thus VII-C does not apply and is not contradicted.
Its logarithm is neither subtracted nor reinterpreted
as an error in \eqref{eq:rate}. A signature dictionary
does not identify different field contents,
prescriptions, kernels, or topologies.

VII-I's finite-width free algebra and its transmission
law are likewise a different construction
\cite{VIIIndexI}. The present velocity formulas
cannot be assigned to that object without an
explicit prescription comparison.

The X-B-ST diagonal-product boundary also stands:
the ordinary wavefront criterion does not define
the Feynman-kernel square on the diagonal
\cite{XBST,Hormander}. No limit of the present
degree-one tests constructs that omitted product
or a \(T_2,T_3\) family.

\subsection{Equation ideals are not metric admission}

The compact representatives in \eqref{eq:limit-ideal}
can be used as equation-ideal identities by a successor
that declares the same scalar quotient.
They do not prove admission after making a metric
slot quantum. X-F's separate use of equation-ideal
representatives or fixed-width tree constructions
has its own domains and field content \cite{XF}.
It does not inherit this sharp scalar limit as a
metric-admission theorem.

No inherited characteristic quantum coefficient is
admitted here. This qualification does not deny the
smooth scalar classes and their convergence that have
actually been constructed.

%=====================================================================
\section{Finite Enter encoding of supplied limiting evaluations}
\label{sec:instrument}
%=====================================================================

\subsection{From the analytic estimate to a declared table}

A finite dictionary consists of supplied descriptions
\[
 \mathcal E_i=(h_i,k_i,f_i,g_i,u_i,j_i),
 \qquad j_i\in\{1,2\},
\]
with real smooth free solutions \(u_i\), and with each
entry satisfying its supported fixed-input contract.
Define the already mathematical values
\[
 a_{i,\varepsilon}
        =\pair{d_{j_i,\varepsilon}}{u_i},\qquad
 a_{i,0}=\pair{d_{j_i,0}}{u_i}.
\]

\begin{proposition}[Finite evaluation stability]
\label{prop:evaluations}
For any such fixed finite dictionary, there are
\(M<\infty\) and a positive width bound such that
\[
 \max_i|a_{i,\varepsilon}-a_{i,0}|
                                     \le M\varepsilon.
 \label{eq:evaluation-rate}
\]
\end{proposition}

\begin{proof}
For each fixed compact support \(C_i\), pairing with
the fixed smooth solution is a continuous functional
on \(\DD_{C_i}\). In particular, a containing compact
box gives a bound by a constant times \(p_0\).
Apply \eqref{eq:rate} to each entry and take the
maximum over finitely many constants.
\end{proof}

The installed table stores supplied exact values,
rational reports, or certified intervals.
The theorem is not a general algorithm for evaluating
arbitrary smooth inputs, and finite descriptions of
arbitrary smooth fields or arbitrary real numbers
are not assumed.

The dictionary can always be nonempty: for zero
backgrounds, both completed positive-order tests
are zero. The conformal diagnostics provide
nontrivial mathematical entries when those data
are selected in advance.

\subsection{Complete destructive-clock channel}

Use the inherited finite clock
\(C_{\mathrm{clk}}\cong\CC^{2^{n_C}}\), \(n_C\ge1\),
with orthonormal calibrated basis
\(\{\ket{\tau_k}\}\).
Let \(S\) contain every surviving finite register,
and define
\[
 D_k=\bra{\tau_k}_{C_{\mathrm{clk}}}\otimes I_S,
 \qquad P_k=\proj{\tau_k}.
\]

\begin{theorem}[Complete clock instrument]
\label{thm:instrument}
The channel
\begin{equation}
 \mathcal M(X)
    =\sum_k\proj{k}_R\otimes D_kXD_k^\dagger
    =\sum_k\proj{k}_R\otimes
       \Tr_{C_{\mathrm{clk}}}
            [(P_k\otimes I_S)X(P_k\otimes I_S)]
 \label{eq:clock}
\end{equation}
is CPTP and satisfies
\[
 \Tr_R\mathcal M(X)=\Tr_{C_{\mathrm{clk}}}X.
\]
The identities hold on all operators after adjoining
any untouched reference. The measured clock names
are consumed; no quantum clock factor is returned.
\end{theorem}

\begin{proof}
The recorded Kraus maps are \(\ket{k}_R\otimes D_k\).
Their squared moduli sum to
\[
 \sum_kP_k\otimes I_S=I_{C_{\mathrm{clk}}S}.
\]
Expand an arbitrary operator as
\[
 X=\sum_{k,\ell}\kb{\tau_k}{\tau_\ell}\otimes X_{k\ell}.
\]
The \(k\)-th output block and the indicated projected
partial trace both equal \(X_{kk}\). Summing proves
the discard identity. The same calculation includes
any reference factor.
\end{proof}

This is the I-B/En09-A pattern, proved on the present
finite interface \cite{IB,En09}.
The conditional state on a positive-probability branch
is its subnormalized block divided by its trace.
No normalized state is assigned to a zero-probability
branch. Conditional reference steering is compatible
with the unchanged unconditional marginal.

% I-B supplies this finite-interface pattern, not the scalar limit.
% ([enter.grok.me](https://enter.grok.me/browse/I-B/))

If distinct calibrated labels \(t_k\) must equal the
time of one fixed readout event whenever their outcomes
occur, at most one code has positive probability.
Otherwise the table uses programme/history calibration.
Reading it does not consume the geometric time axis.

The channel is not universally invertible: the clock
states \((\ket{\tau_0}\pm\ket{\tau_1})/\sqrt2\), with
the same residual, have identical outputs.
The record alone also cannot determine an arbitrary
residual. Finally, no finite clock matrices satisfy
\([T_C,H_C]=iI_C\), by taking traces.

\subsection{A specific complete-channel limit for the table}

Choose one fixed scale \(B_{\mathrm{tab}}>0\) large
enough that the finitely many reports
\[
 v_{k,\varepsilon}=a_{i(k),\varepsilon}/B_{\mathrm{tab}},
 \qquad
 v_{k,0}=a_{i(k),0}/B_{\mathrm{tab}}
\]
lie in \([-1,1]\) for all sufficiently small widths.
The dictionary and scale are installed data.

For \(y=\pm1\), set
\[
 p_y(v)=\frac{1+yv}{2},
\]
and define
\[
 \mathcal I_v(X)
    =\sum_{k,y}\proj{k,y}_R\otimes
                            p_y(v_k)D_kXD_k^\dagger.
\]

\begin{proposition}[Reference-stable finite-table convergence]
\label{prop:table}
The channel \(\mathcal I_v\) is CPTP, and
\begin{equation}
 \ddia(\mathcal I_v,\mathcal I_{\widetilde v})
       =\tfrac12\max_k|v_k-\widetilde v_k|,
 \qquad
 \ddia(\Lambda,\Gamma)
                  =\tfrac12\|\Lambda-\Gamma\|_\diamond.
 \label{eq:table-error}
\end{equation}
Consequently the explicitly encoded ramp and limiting
tables obey
\[
 \ddia(\mathcal I_{v_\varepsilon},\mathcal I_{v_0})
            \le \frac{M}{2B_{\mathrm{tab}}}\varepsilon.
\]
All statements include untouched references.
\end{proposition}

\begin{proof}
A complete Kraus family is
\[
 \sqrt{p_y(v_k)}\,\ket{k,y}_R\otimes D_k.
\]
For an arbitrary input operator with a reference
included, the difference channel has orthogonal
blocks
\[
 \frac{y(v_k-\widetilde v_k)}2X_{kk}.
\]
Its trace norm is at most
\[
 \max_k|v_k-\widetilde v_k|
                          \sum_k\|X_{kk}\|_1
       \le\max_k|v_k-\widetilde v_k|\,\|X\|_1.
\]
The final inequality is trace-norm contraction of
pinching, an average of unitary conjugations.
A clock-basis input at a maximizing index saturates
the estimate. Divide by two.
Equation \eqref{eq:evaluation-rate} gives the rate.
\end{proof}

This channel statement depends on the explicitly
chosen probabilistic encoding. Nearby real numbers
encoded as different deterministic orthogonal words
need not produce nearby channels.

The abstract clock instrument and binary report can
be implemented under the inherited finite-unitary
and destructive-readout permissions.
Precision, table storage, probe preparation, and
gate implementation allowances are separately
charged. No field-to-qubit preparation or unknown
field acquisition has been supplied.

%=====================================================================
\section{Scheduled locality, regular transport, and resources}
\label{sec:transport}
%=====================================================================

\subsection{The complete network}

At each tick, let the complete finite channel factor
over a prescribed partition of finite sites into
blocks. Include all retained classical records and
all control wires in the dependency description.

For a final site set \(Y\), define backwards sets by
\[
 N_{t,t}(Y)=Y,\qquad
 N_{s,t}(Y)=
 \bigcup\{B:\ B\text{ is a block at tick }s+1,\
                         B\cap N_{s+1,t}(Y)\ne\varnothing\}.
\]

\begin{theorem}[Scheduled locality]
\label{thm:locality}
The complete network obeys
\[
 (\Phi_{s+1}^*\circ\cdots\circ\Phi_t^*)
                 (\mathcal A(Y))
                 \subseteq\mathcal A(N_{s,t}(Y)).
\]
Equal input reductions on that backwards region give
equal corresponding output reductions.
A prescribed intervention outside it cannot change
the corresponding joint output law.

If every dependency edge has a separately certified
regular future-causal host route, ancestry is contained
in the supplied host causal relation.
\end{theorem}

\begin{proof}
A channel adjoint is unital. A block disjoint from
the current observable support therefore acts on an
identity factor. A meeting block can enlarge support
only within itself. Backward induction proves the
inclusion. Testing all output observables proves
the reduction and intervention claims. References
are retained unchanged in the same calculation.

Concatenation of the separately certified routes
proves the final causal containment.
\end{proof}

An ancestor lies in the metric past of its output,
not in its future. Scheduled ancestry is an upper
bound on operational dependence, not an identification
with an entire physical causal past.
Scheduler consistency alone does not derive a cone
\cite{En08A}.

\subsection{The inherited regular route}

Choose a straight developed spatial segment of length
\(\ell>0\) inside the inherited regular convex tube.
After readout, local lookup, and a positive processing
buffer, depart at developed time \(t_{\mathrm d}\).
All selected events remain in the available laboratory
time reserve.

Install \(n+1\) stations
\[
 \mathbf x_j=\mathbf x_0+
          \frac jn(\mathbf x_n-\mathbf x_0),\qquad j=0,\ldots,n,
\]
with payload factors \(A_j\cong\CC^{2^p}\).
Initialize \(A_1,\ldots,A_n\) in independent blank
states \(\omega\). At hop \(j\), exchange
\(A_{j-1}\) and \(A_j\), using developed tick duration
\(\ell/n\).

A two-register SWAP on these declared tick-boundary
ports is an explicit inherited permission.
Cross-port dependencies are null and same-station
dependencies are timelike. A microscopic interaction
Hamiltonian is not inferred.
Classical report wires have their own causal certificates.

\begin{theorem}[Regular reference-preserving transport]
\label{thm:transport}
For any payload--reference state \(\rho_{A_0E}\),
after hop \(j\) the state is
\[
 \rho_{A_jE}\otimes
    \bigotimes_{\substack{0\le i\le n\\i\ne j}}\omega_{A_i},
\]
under the declared register identifications.
The logical payload channel is the identity, including
all reference correlations.

The classical report can be forwarded with its full
joint law. Arrival is at
\[
 t_{\mathrm d}+\ell,\qquad
 T_{\mathrm d}+\ell/c_{\mathrm{host}},
\]
and every dependency route stays in the regular part.
\end{theorem}

\begin{proof}
A SWAP moves the entire payload factor, including its
correlations with \(E\), into the next independent
blank and leaves the previous factor blank.
Induction proves the formula.

Classical forwarding preserves diagonal records and
their joint correlations; it is not cloning an unknown
quantum state. The inherited chart and selected tick
durations certify the null cross-port routes.
\end{proof}

This is the I-B/En08-B transport pattern
\cite{IB,En08B}. It preserves correlations that were
already present. It is not a theorem that gravity
generates entanglement.

\subsection{Resource and error ledger}

For an \(n_C\)-qubit clock, \(p\)-qubit station
payloads, \(b\)-bit report reserves, and lookup
workspace \(w_{\mathrm{lookup}}\), a conservative
preallocated reserve is
\[
 n_C+(n+1)p+(n+1)b+w_{\mathrm{lookup}}+1.
\]
Additional description registers, external references,
classical tables, and any supplier storage are
separately declared. The optional last qubit is a
binary report probe.

The route uses \(n\) register SWAPs and \(n\)
classical report transmissions. Readout, basis
change, lookup, buffers, and their durations are
charged. Consumed clock names do not reappear as
returned quantum factors.

For compatible complete implemented channels,
\begin{equation}
 \ddia(\widetilde\Gamma_\varepsilon,\Gamma_0)
 \le\min\left\{1,\
   \epsilon_{\mathrm{enc}}+\epsilon_{\mathrm{dec}}
   +\epsilon_{\mathrm{installed}}
   +\frac{M}{2B_{\mathrm{tab}}}\varepsilon
   +\sum_j\epsilon_j\right\}.
 \label{eq:channel-ledger}
\end{equation}
The proof is telescoping replacement of one channel
factor at a time, together with diamond-norm contraction
under channels and the triangle inequality
\cite{Watrous}.
An arrival or failure probability is controlled only
if its record belongs to the compared complete output.

No uniform table-generation algorithm, hardware
synthesis rate, or detector implementation is hidden
in \eqref{eq:channel-ledger}. A channel-error bound
also does not by itself prove exact causal containment
of a faulty implementation.

%=====================================================================
\section{Fixed host, release, and successor boundary}
\label{sec:release}
%=====================================================================

\begin{proposition}[Host stability]
\label{prop:fixedhost}
Shrinking the coordinate ramp, changing an installed
table, or introducing a description of a subtraction
scale does not evolve \(G,\mathcal W,T\), or the
prescribed future relation.
\end{proposition}

\begin{proof}
The host data were fixed before the analytical
construction. The maps constructed here return
scalar tests, their free classes, or finite
operational outputs. None returns a new host metric.
\end{proof}

\begin{theorem}[X-D-ST local Grade S release]
\label{thm:release}
For the inherited supported host and the fixed-input
family satisfying \eqref{eq:separation}:
\begin{enumerate}[label=\textup{(\roman*)}]
\item the shrinking coordinate layer has the endpoint
data \eqref{eq:velocities}; the finite quadratic
normal-flux contribution is retained;
\item the limiting coordinates solve the uniquely
identified continuous-coordinate,
continuous-normal-flux transmission problem;
\item the completed scalar tests converge in one fixed
\(\DD_C\) at rate \(O(\varepsilon)\) in every
seminorm, and hence in \(\Qm\);
\item the limiting tests are independent of the fixed
profile interior, although finite-width position
moments and error constants can depend on it;
\item the background identity passes through order
\(\kappa^2\) on the unchanged frozen algebra, with
the explicit compact equation representatives
\eqref{eq:limit-ideal};
\item the edge and switch distinctions retain the
three-term variation and its propagated first
gap, with the fixed-versus-moving parameter
quantifier stated explicitly;
\item the finite installed encoding has a complete
destructive-clock instrument, a specified
reference-stable table limit, scheduled locality,
regular reference-preserving transport, and
the declared resource/error ledger.
\end{enumerate}
These statements form local Grade S only.
\end{theorem}

\begin{proof}
Lemma~\ref{lem:flux},
Proposition~\ref{prop:layer}, and
Lemma~\ref{lem:profile} prove (i).
Theorem~\ref{thm:transmission} and
Proposition~\ref{prop:kernel} prove (ii).
Proposition~\ref{prop:propagation},
Theorem~\ref{thm:convergence}, and
Lemma~\ref{lem:closed} prove (iii)--(iv).
Theorem~\ref{thm:ward} proves (v).
Lemma~\ref{lem:variation},
Proposition~\ref{prop:defect-limit}, and
\eqref{eq:moving-edge} prove (vi).
Sections~\ref{sec:instrument}--\ref{sec:transport}
prove (vii), supplying the remaining clauses of
Definition~\ref{def:grade}.
\end{proof}

\begin{center}
\small
\renewcommand{\arraystretch}{1.12}
\begin{tabularx}{\linewidth}{@{}p{.34\linewidth}p{.20\linewidth}X@{}}
\toprule
Object & Status & Exact boundary\\
\midrule
Fixed-input completed scalar tests &
Convergent &
One \(\DD_C\), then \(\Qm\)\\
Normal-flux velocities &
Derived &
Completion-normal data, not a host update\\
Fixed-profile limit &
Independent of profile interior &
No arbitrary moving-profile theorem\\
Limiting background identity &
Proved &
Original frozen algebra, through \(\kappa^2\)\\
Unchanged multiplied-equation insertion &
Not identified with the limit &
Different initial velocities and boundary kernel\\
Quantum metric kernels &
Not admitted by this theorem &
Fixed-background scalar admission only\\
Time-ordered hierarchy and loop regulator &
Not constructed &
No extension or subtraction is selected\\
Finite Enter interface &
Constructed under permissions &
Supplied evaluations, not continuum acquisition\\
Spacetime emergence &
Not constructed &
The host remains prescribed\\
\bottomrule
\end{tabularx}
\end{center}

\subsection{Significance without priority inflation}

The useful feature is simultaneous control of the
limiting prescription, its coefficient topology, and
the identity that survives in the unchanged scalar
quotient. The proof shows why an inverse-width
derivative estimate need not describe the actual
solution limit, while retaining the genuine finite
quadratic layer contribution.

The parent result is Cheung's X-D \cite{XD}.
The supported transfer, explicit topology comparison,
edge-quantifier correction, and finite encoding
stability make that result usable without identifying
it with a sharper quantum-product programme.

\subsection{Successor package}

A successor receives:
\begin{enumerate}
\item the limiting tests \eqref{eq:limiting-tests}
and their fixed-support \(O(\varepsilon)\) estimates;
\item the normal-flux velocities and transmission
identification;
\item the delta-free, possibly jumping observation-smeared
linear background kernel;
\item the unchanged-algebra gauge passage and its
explicit equation-ideal representatives;
\item the fixed-generator defect formula and the separate
moving-edge warning;
\item the preserved counting of the first map and
second-order pair;
\item the finite complete-channel encoding contract.
\end{enumerate}

It receives no metric admission, quantum metric,
\(T_2\) family, nonlinear retarded-product hierarchy,
loop triangle, or characteristic complete-sum limit.
Any X-F-type use of the representatives must retain
that successor's separately specified domains.

\subsection{Standing non-claims}

No quantum coefficient of the inherited characteristic
graph is admitted. The scalar free-field classes and
their proved convergence are not denied by that sentence.

XV-B is not opened as the downstream quantum-admission
exit. This does not deny the existence of a technical
manuscript with that identifier. XVI-N is not discharged,
and O.1 remains \loc{NOT_CLOSED} in the stated inherited
scope.

The historical H2 obligation and the full I3/J2
characteristic-map obligations are not discharged here.
XIV-I's narrower internal-amplitude construction is
not withdrawn or relabelled as the missing full map
\cite{BoundaryRecords}.

No Einstein equation, ultraviolet completion, Page
curve, gravity-generated-entanglement test, or
spacetime emergence is derived.

%=====================================================================
\appendix
\section{Debugging and quantifier audit}
\label{app:audit}
%=====================================================================

\begin{longtable}{@{}p{.35\linewidth}p{.60\linewidth}@{}}
\toprule
Potential error & Correction\\
\midrule
\endfirsthead
\toprule
Potential error & Correction\\
\midrule
\endhead

Constructing another cone &
The host and chart are the inherited X-A-ST/I-B
instance, unchanged.\\[4pt]

Using a Cauchy theorem on the regular cone &
All analytical Cauchy and Green problems are on
the declared Minkowski completion.\\[4pt]

Changing signatures in the Euler equations &
The host keeps \((-,+,+,+)\); the analysis keeps
\((+,-,-,-)\).\\[4pt]

Calling the added defect a worldline &
The inherited product defect is a worldsheet.\\[4pt]

Importing the X-D limit as a premise &
The layer estimate, velocities, propagation, and
test convergence are proved in this file.\\[4pt]

Estimating \(\sigma_\varepsilon'\) before integrating &
The flux estimate removes that derivative from
the controlling bounds.\\[4pt]

Discarding half the quadratic layer term &
Both normal-source contributions are retained;
their sum gives \(a_\Sigma^{00}a_\Sigma^{0\rho}\).\\[4pt]

Choosing the limiting velocities independently &
They are derived exit data and characterize the
transmission problem.\\[4pt]

Confusing profile independence with no profile corrections &
Interior moments occur at order \(\varepsilon\)
in positions and can affect error constants.\\[4pt]

Claiming smooth convergence across \(t=0\) &
Coordinate velocities need not converge uniformly
there. Scalar test supports stay separated from it.\\[4pt]

Replacing \(\DD_C\) by smooth functions on \(C\) &
The tests are ambient smooth functions with
support in the fixed compact set.\\[4pt]

Inventing a universal Green derivative-loss count &
Continuity supplies some finite seminorm estimate
on each fixed test stage.\\[4pt]

Using a quotient seminorm to rescue divergent tests &
The representatives converge first, in \(\DD_C\).\\[4pt]

Adding \(\rho_{2,g}\) again &
Its one copy is already represented by the
X-C-ST second-order pair in \eqref{eq:completed}.\\[4pt]

Calling the limit the unchanged zero-velocity insertion &
Its normal-flux velocities and its kernel differ
from that prescription in general.\\[4pt]

Interpreting no surface delta as smoothness &
The kernel can jump; its derivatives and wavefront
still require attention.\\[4pt]

Treating a smeared kernel as an unsmeared smooth propagator &
Only the stated observation-smeared linear kernel
has the displayed bounded representative.\\[4pt]

Inferring joint metric admission from fixed-background tests &
The theorem constructs no joint quantum metric
kernel or sharp contraction domain.\\[4pt]

Claiming every edge defect persists for fixed generators &
A fixed compact interior generator eventually
lies above the shrinking ramp.\\[4pt]

Using a moving edge family against the fixed-input rate &
Its backgrounds generally have growing derivatives
and leave the fixed-input hypothesis.\\[4pt]

Isolating an interaction-plateau edge in the present tail &
Here \(g=0\) near the shrinking layer, so the
switch defect cannot be suppressed that way.\\[4pt]

Equating polynomial smoothing growth with a topology &
A growth estimate is not one of
\(\DD_C,\Qm,\tau_\Sigma\), nor is it convergence.\\[4pt]

Applying VII-C by shared terminology &
Its quantum multiplet, exact mixed insertion,
normal topology, and mandatory word are absent.\\[4pt]

Calling a smooth multiplier a loop regulator &
Wavefront non-enlargement defines no missing
product, subtraction, or uniform scale limit.\\[4pt]

Using an installed table as a field detector &
Only supplied evaluations or enclosures are encoded.\\[4pt]

Reading all history labels as one actual event time &
Exact equality to the time of one event forces a
deterministic distinct-label outcome.\\[4pt]

Equating ancestry with a derived physical causal past &
Containment uses separately certified regular routes.\\[4pt]

Treating finite SWAP permission as derived microscopic dynamics &
The endpoint-port permission is explicit and
does not derive a Hamiltonian.\\[4pt]

Claiming new entanglement generated by gravity &
The route preserves arbitrary input reference
correlations on a fixed host.\\[4pt]

Treating an identifier as registration evidence &
This is a proposed manuscript source, not a
registry-installation certificate.\\

\bottomrule
\end{longtable}

%=====================================================================
\section{Documentary OpenAI comparisons}
\label{app:catalogue}
%=====================================================================

The supplied plan mentions OpenAI catalogue families
362, 376, and 282. Their topics are retained only as
documentary comparisons:
\begin{center}
\small
\begin{tabularx}{\linewidth}{@{}p{.17\linewidth}p{.37\linewidth}X@{}}
\toprule
Family & Topic named in the supplied catalogue description &
No inference for X-D-ST\\
\midrule
362 &
Relativistic Vlasov--Maxwell smoothness &
Not this scalar transmission theorem or a regulator\\
376 &
Forced Navier--Stokes analysis &
Not the coordinate layer estimate or an ultraviolet
completion\\
282 &
Scale-to-conformal questions and free-scalar comparisons &
Not quantum metric admission or spacetime emergence\\
\bottomrule
\end{tabularx}
\end{center}

The catalogue authorship is attributed to OpenAI
\cite{OpenAICatalogue}. These descriptions are not
independent verification of those proofs.
No catalogue theorem is a premise of the present
limit, channel, or transport results.

%=====================================================================
\section*{Declarations}
\addcontentsline{toc}{section}{Declarations}

\paragraph{Authorship.}
The author is Ngo Cheung. The title metadata and
declarations are consistent. No empty-author boilerplate
is retained.

\paragraph{Attribution and novelty.}
The fixed-input sharp-width theorem is attributed to
X-D, version 3. The predecessor scalar maps and
classical handoff are attributed separately.
The claimed contribution is the supported proof,
topology and defect audit, and finite encoding
interface, not a new general singular-product theorem.

\paragraph{Source status.}
X-D and the Enter background are cited as source
records. Comparisons available here as supplied
excerpts are identified as such; no catalogue-wide
proof verification is asserted. Internal publication
dates and DOIs are not invented.

\paragraph{Analytical scope.}
The metric coefficients are fixed smooth classical
inputs. The limit is in a named scalar compact-test
topology, then its unchanged free quotient.
No joint sharp metric admission or characteristic
quantum composition follows.

\paragraph{Operational scope.}
The clock consumes a finite encoding, not geometric
time. The finite dictionary does not acquire a
continuum field. Transport uses declared regular
finite-operation permissions.

\paragraph{Verification.}
No successful \loc{pdflatex} execution, proof-assistant
verification, symbolic-program execution, numerical
evaluation, hardware synthesis, experiment,
independent referee review, journal acceptance, or
external registration is asserted.
The mathematical claims are the explicit definitions
and written proofs in this source.

%=====================================================================
\begin{thebibliography}{99}
\addcontentsline{toc}{section}{References}

\bibitem{XD}
N.~Cheung,
\emph{Landmark X-D: Convergence of the completed ramp
relational scalar as the regulator width tends to zero}.
\loc{X-D-v3.tex}, version 3.
Enter Calculus Companion.
\url{https://enter.grok.me/browse/X-D}.
Especially \loc{lem:flux}, \loc{lem:profile},
\loc{prop:coordinate-convergence},
\loc{thm:convergence}, \loc{thm:identification},
\loc{cor:gauge}, and \loc{thm:release}.

\bibitem{XAST}
N.~Cheung,
\emph{X-A-ST: Supported classical covariance of a
fixed-ramp relational scalar}.
Supplied manuscript \loc{X-A-ST.tex}, version 1.
Host, supported chart, declared flat completion,
signature ledger, coordinate prescription, and edge
witness.

\bibitem{XBST}
N.~Cheung,
\emph{X-B-ST: One scalar retarded map on the X-A-ST host}.
Supplied manuscript \loc{X-B-ST.tex}.
Scalar admission, first-map normalization, off-shell
representatives, and the one-contraction boundary.

\bibitem{XCST}
N.~Cheung,
\emph{X-C-ST: Second-order scalar completion on the
inherited X-A-ST/X-B-ST host}.
Supplied manuscript \loc{X-C-ST.tex}, version 1.
Completed tests, scalar-map counting, three-term
variation, and fixed-algebra identity.

\bibitem{XA}
N.~Cheung,
\emph{Landmark X-A: Classical covariance of one
fixed-regulator switched relational scalar through
order \(\kappa^2\)}.
\loc{X-A.tex}, version 1.
Enter Calculus Companion.
\url{https://enter.grok.me/browse/X-A}.

\bibitem{XB}
N.~Cheung,
\emph{Landmark X-B: Fixed-regulator admission and one
retarded correction for the ramp relational scalar,
with the metric kept classical}.
\loc{X-B-v3.tex}, version 3.
Enter Calculus Companion.
\url{https://enter.grok.me/browse/X-B}.

\bibitem{XC}
N.~Cheung,
\emph{Landmark X-C: The missing second-order retarded
maps for the ramp relational scalar, with the metric
kept classical}.
\loc{X-C-v3.tex}, version 3.
Enter Calculus Companion.
\url{https://enter.grok.me/browse/X-C}.

\bibitem{IB}
N.~Cheung,
\emph{I-B: A Finite Enter Spacetime:
Calibrated Destructive Time Readout on a Prescribed
Conical Product}.
\loc{I-B-v1.tex}, version 1.
Enter Calculus Companion.
\url{https://enter.grok.me/browse/I-B}.
Prescribed geometry, worldsheet, finite clock,
calibration, locality, and target-adapted transport;
not a sharp-limit theorem.

\bibitem{EnterI}
N.~Cheung,
\emph{Manuscript I}.
\loc{I.tex}, version 2.
Enter Calculus Companion.
\url{https://enter.grok.me/browse/I}.
Operational characterization within its declared
strategy class; contextual only.

\bibitem{EnterPhilosophy}
N.~Cheung,
\emph{The Philosophy of Enter: Destructive Readout,
Quantum Recovery, and the Refusal of Identification}.
\loc{EntPhil-v2.tex}, version 2.
Introductory essay, Enter Calculus Companion.
\url{https://enter.grok.me/enter}.
Interpretive background, not a premise of the
convergence proof.

\bibitem{BGP}
C.~B\"ar, N.~Ginoux, and F.~Pf\"affle,
\emph{Wave Equations on Lorentzian Manifolds and
Quantization}.
ESI Lectures in Mathematics and Physics,
European Mathematical Society, 2007.
\url{https://arxiv.org/abs/0806.1036}.
Smooth Cauchy problems, finite propagation, Green
operators, and the compact-source exact sequence;
used on the Minkowski completion.

\bibitem{FR}
K.~Fredenhagen and K.~Rejzner,
\emph{Perturbative algebraic quantum field theory}.
arXiv:1208.1428, version 2, 2013.
\url{https://arxiv.org/abs/1208.1428}.
Free functional algebra and equation-ideal framework;
no higher interacting hierarchy is imported.

\bibitem{Hormander}
L.~H\"ormander,
\emph{The Analysis of Linear Partial Differential
Operators I: Distribution Theory and Fourier Analysis},
second edition, Springer, 1990;
Classics in Mathematics reprint, 2003.
\url{https://doi.org/10.1007/978-3-642-61497-2}.
Distributional derivatives and wavefront product,
pullback, and multiplier calculus.

\bibitem{IXAST}
N.~Cheung,
\emph{IX-A-ST}.
Version 1 supplied manuscript record,
Enter Calculus Companion.
\url{https://enter.grok.me/browse/IX-A-ST}.
Supported actual-functional domain and
boundary-sensitive topology; comparison only.

\bibitem{IXA}
N.~Cheung,
\emph{IX-A}.
\loc{IX-A.tex}, version 1.
Enter Calculus Companion.
\url{https://enter.grok.me/browse/IX-A}.
Boundary-sensitive admission and nonconvergence of
unchanged regular approximants.

\bibitem{Em01B}
N.~Cheung,
\emph{Em01-B}.
\loc{Em01-B-v3.tex}, version 3.
Supplied manuscript excerpts,
Enter Calculus Companion.
\url{https://enter.grok.me/browse/Em01-B}.
Polynomial smoothing-growth comparison; not a
characteristic convergence theorem.

\bibitem{Em01C}
N.~Cheung,
\emph{Em01-C}.
\loc{Em01-C-v3.tex}, version 3.
Supplied manuscript excerpts,
Enter Calculus Companion.
\url{https://enter.grok.me/browse/Em01-C}.
Scalar aligned-null diagnostics and the separate
unproved source-specific characteristic estimate.

\bibitem{Em02D}
N.~Cheung,
\emph{Em02-D}.
\loc{Em02-D-v4.tex}, version 4.
Supplied manuscript excerpts,
Enter Calculus Companion.
\url{https://enter.grok.me/browse/Em02-D}.
Fixed-model leading smoothing distribution and a
separated raw-family nonconvergence result.

\bibitem{VIIC}
N.~Cheung,
\emph{VII-C}.
\loc{VII-C.tex}, version 1.
Enter Calculus Companion.
\url{https://enter.grok.me/browse/VII-C}.
The specified normal-topology Wick-extension
obstruction under its own hypotheses.

\bibitem{VIIIndexI}
N.~Cheung,
\emph{VII-I}.
\loc{VII-I.tex}, version 1.
Enter Calculus Companion.
\url{https://enter.grok.me/browse/VII-I}.
Different finite-width free-algebra construction,
transmission law, and sharp-product boundaries.

\bibitem{XVIA}
N.~Cheung,
\emph{XVI-A}.
\loc{XVI-A.tex}, version 1.
Supplied manuscript record,
Enter Calculus Companion.
\url{https://enter.grok.me/browse/XVI-A}.
Conditional retarded-slot wavefront interface;
not a subtraction or a constructed coefficient.

\bibitem{XF}
N.~Cheung,
\emph{X-F}.
\loc{X-F.tex}, version 1.
Supplied manuscript excerpts,
Enter Calculus Companion.
\url{https://enter.grok.me/browse/X-F}.
Separate fixed-width tree and equation-ideal
comparisons; no transfer of sharp metric admission.

\bibitem{En08A}
N.~Cheung,
\emph{En08-A}.
Version 1, Enter Calculus Companion.
\url{https://enter.grok.me/browse/En08-A}.
Scheduler consistency and its distinction from
geometric reconstruction.

\bibitem{En08B}
N.~Cheung,
\emph{En08-B}.
Version 1, Enter Calculus Companion.
\url{https://enter.grok.me/browse/En08-B}.
Finite resource-controlled transport relative to a
prescribed null boundary.

\bibitem{En09}
N.~Cheung,
\emph{En09-A: Relational Enter}.
Version 1, Enter Calculus Companion.
\url{https://enter.grok.me/browse/En09-A}.
Complete destructive-clock instrument and conditioning
scope.

\bibitem{Watrous}
J.~Watrous,
\emph{The Theory of Quantum Information}.
Cambridge University Press, 2018.
Author's manuscript and errata:
\url{https://jhwatrous.github.io/books-and-courses.html}.
Finite instruments and diamond-norm channel
comparisons.

\bibitem{OpenAICatalogue}
OpenAI,
\emph{Mathematics catalogue}.
Supplied documentary descriptions of families
362, 376, and 282, as reproduced in the Enter
Calculus Companion.
\url{https://enter.grok.me/browse}.
Comparison-only attribution; the catalogue proofs
are not verified or used here.

\bibitem{BoundaryRecords}
N.~Cheung,
\emph{XIV-H}, \emph{XIV-I}, and \emph{XIV-J},
version 1; technical \emph{XV-B}, version 2;
and \emph{XVI-N}, revision r12.
Version-specific supplied manuscript records,
Enter Calculus Companion.
\url{https://enter.grok.me/browse}.
Documentary analytical boundaries only.
The internal-amplitude result, the missing full
characteristic maps, and downstream admission
remain distinct.

\end{thebibliography}

\end{document}