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  pdftitle={X-A-ST: Supported classical covariance of a fixed-ramp relational scalar},
  pdfauthor={},
  pdfsubject={A local Grade S transfer on a prescribed conical spacetime}
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\title{\textbf{X-A-ST: Supported classical covariance of a\\
fixed-ramp relational scalar}\\[5pt]
\large A local Grade S transfer through second coupling order\\
on a prescribed conical spacetime}
\author{Ngo Cheung}
\date{}

\begin{document}
\maketitle

\begin{abstract}
We give a supported transfer of one fixed classical
relational-coordinate prescription to a regular laboratory
of the prescribed I-B conical spacetime. The calculation
is performed on a declared Minkowski completion, with
signature \((+,-,-,-)\), and then restricted to the
laboratory; the host retains its independently prescribed
signature \((-,+,+,+)\), worldsheet defect, and future
relation. No global conical inverse is assumed.

At fixed ramp width \(\varepsilon>0\), the divergence-form
coordinate problem has unique formal coefficients through
order \(\kappa^2\). On the gauge algebra fixed before the
calculation, we prove the complete second-order coordinate
intertwiner, off-shell invariance of the label-smeared
relational scalar, and covariance of the switched scalar
solution with unchanged free input. A separate
varied-action derivation displays every switch contribution.
An explicit edge-zone parameter has the strictly negative
integrated preflight defect
\(-\int_0^\varepsilon\sigma'(s)\alpha(s)\,\dd s\);
this disproves an unrestricted extension of the preflight
identity without changing the lower-order prescription.
Pure-gauge and sourced diagnostics verify the retained
quadratic terms.

The contribution is a precise supported realization and
sign, support, and operator audit of Cheung's X-A result,
not a new claim of priority for its classical covariance.
Its potential usefulness is to provide a reproducible
classical baseline that distinguishes a successful
plateau-restricted identity from regulator-induced
breaking. In particular, smoothness is proved for
observation-smeared first-derivative kernels, not for the
unsmeared two-point wave kernel. A finite destructive-clock
instrument and reference-preserving regular transport
encode supplied evaluations without acquiring a continuum
field. The release is local Grade S only: no time-ordered
product, normalized retarded map, sharp-width limit,
quantum Ward identity, or spacetime-emergence theorem is
constructed.
\end{abstract}

\statementbox{Release and attribution}{%
The classical target is constructed through
\(\kappa^2\) at each fixed \(\varepsilon>0\), on the
explicitly frozen plateau algebra. The source result is
attributed to N.~Cheung's X-A manuscript.

The conical transfer is a supported presentation on a
declared flat completion. It is not an intrinsic
construction on the entire regular conical product.
The finite Enter interface reads installed encodings
of evaluations already defined mathematically.
It does not establish a field-to-qubit acquisition map.}

\tableofcontents

%=====================================================================
\section{Purpose, background, and scope}
\label{sec:intro}
%=====================================================================

Relational coordinates can remove a simultaneous
coordinate change of fields only when the coordinate
prescription itself transforms correctly. Initial data,
switching, and regularization are therefore part of the
statement, rather than ancillary notation. The
field-dependent-coordinate literature provides useful
context, but does not prove covariance of the particular
switched and regulated problem considered here
\cite{BFR,FrobLima}.

The starting classical prescription is Cheung's X-A:
a compact switch multiplies the complete
curved-minus-flat scalar density before variation,
whereas an independent smooth time ramp regulates the
coordinate density \cite{XA}. We retain this distinction.
In particular, neither substituting \(gh\) into a metric
nor convolving metric arguments defines the prescription
below.

The operational background is the Enter calculus, in
which destructive readout consumes the measured quantum
names and returns a classical record with any unmeasured
residual \cite{CheungEnter,I}. We use ordinary finite Born
instruments and prove the channel identities needed in
this manuscript. The operational characterization of the
Born strategy is not a premise of the classical proof.

The philosophy of Enter emphasizes distinctions between
readout and recovery, between an input--output channel
and a thermodynamic implementation, and between an
analogy and an identified physical model
\cite{EnterPhilosophy}. Here those distinctions prevent
three unwarranted conclusions:
\begin{enumerate}
\item a finite clock encoding does not define the continuum
time coordinate used by the Cauchy problem;
\item a finite record of a classical evaluation does not
constitute a measurement of that functional on a field;
\item a prescribed spacetime supporting a protocol is not
spacetime derived from the protocol.
\end{enumerate}

\begin{definition}[Local Grade S]
\label{def:grade}
For this manuscript, a local Grade S package consists of:
\begin{enumerate}[label=\textup{(S\arabic*)}]
\item a prescribed host with a supported regular laboratory;
\item classical objects with explicit domains, equations,
initial data, and transformation laws;
\item one finite destructive-clock factor and a stated
evaluation dictionary;
\item complete finite instruments, including their
untouched-reference extensions;
\item a scheduled-locality theorem and a nonempty
reference-preserving route whose physical dependency
containment is separately certified.
\end{enumerate}
Grade E, metric emergence, and a quantum field
construction are not included in this definition.
\end{definition}

No companion-sheet theorem is needed as a hidden
premise: the host choices and analytical prescription
used here are stated explicitly, and the classical
identities are proved below.

%=====================================================================
\section{The prescribed host and the flat-completion ledger}
\label{sec:host}
%=====================================================================

Fix \(0<\delta_{\mathrm c}<2\pi\),
\(\alpha_{\mathrm c}=1-\delta_{\mathrm c}/(2\pi)\),
and \(c>0\). The regular host is
\begin{equation}
 \mathcal M^\circ
   =\RR_T\times C_{\delta_{\mathrm c}}^\circ\times\RR_z,
 \qquad
 G=-c^2\dd T^2+\dd r^2+
      \alpha_{\mathrm c}^2r^2\dd\varphi^2+\dd z^2,
 \qquad r>0.
 \label{eq:host}
\end{equation}
Its stipulated completion has singular set
\[
 \mathcal W=\RR_T\times\{o\}\times\RR_z.
\]
Thus the defect is a worldsheet, not a worldline.
The time orientation is increasing \(T\).
These are I-B-type inputs \cite{IB}.

\begin{proposition}[Failure of global hyperbolicity]
\label{prop:nonGH}
The regular product \eqref{eq:host} has a noncompact
causal diamond and is not globally hyperbolic.
\end{proposition}

\begin{proof}
Fix a regular spatial point of radius \(r_*>0\), and two
events there separated by more than \(2r_*/c\).
At an intermediate time, points on the same radial ray
with radii decreasing to zero lie in the causal diamond
between the two events. They have no subsequence
converging in the regular product.
\end{proof}

\begin{definition}[Supported regular chart]
\label{def:chart}
Choose an injective product developing chart
\[
 \iota:\mathcal L\longrightarrow\mathcal M^\circ,
 \qquad
 \mathcal L=(-a,T_*+a)\times B_R(\mathbf x_*),
 \qquad a>0,
\]
whose relevant compact closure stays off \(\mathcal W\).
Set \(t=c(T-T_\Sigma)\) on its image.
In developed coordinates,
\[
 \iota^*G=-\dd t^2+\dd\mathbf x^2.
\]

Choose a relatively compact open observation region
\(\mathcal O\Subset(0,T_*)\times B_R(\mathbf x_*)\)
such that
\[
 D_*:=J^-_{\eta}(\overline{\mathcal O})
                        \cap([0,T_*]\times\RR^3)
 \Subset\mathcal L.
 \label{eq:retainedshadow}
\]
Only tests \(f\in C_c^\infty(\mathcal O;\RR)\) are
used for the supported observable.

All nested retarded calculations for such a test are
retained in \(D_*\). No independently adjustable lateral
boundary datum is introduced.
\end{definition}

The chart can be chosen with enough regular margin to
contain the compact supports of the diagnostics below.
For those diagnostics the same retained-shadow
condition is imposed before their tests are selected.

\begin{definition}[Declared flat completion]
\label{def:completion}
The analytical completion is the separate Minkowski
space
\[
 \widehat M=\RR^{1+3},
 \qquad
 \eta=\operatorname{diag}(1,-1,-1,-1),
 \qquad
 \MT=[0,T_*]\times\RR^3,
 \qquad
 \Sigma=\{t=0\}.
\]
All coordinate and scalar Cauchy problems are solved
on this completion. Smooth fields are defined on a
neighbourhood of \(\MT\), without decay assumptions.
A compact laboratory test is transported to
\(\widehat M\) and extended by zero.
The initial surface is the complete flat
\(\Sigma\), not a claimed Cauchy surface of
\(\mathcal M^\circ\).
\end{definition}

This makes explicit the completion pattern of IV-B
\cite{IVB}. It is not a global developing map.

\begin{proposition}[Signature ledger]
\label{prop:signature-ledger}
The two signature conventions are related on the chart by
\[
 \iota^*G=-\eta,
\]
but are not silently substituted for one another.
All density expansions, Euler signs, and Cauchy
inverses below use \(\eta\) and
\[
 \Box=\partial_t^2-\Delta_{\mathbf x}.
\]
The host's unit-lapse stationary proper time is \(T\);
the developed comparison variable is \(t=c(T-T_\Sigma)\).
The formal metric jet introduced below is not a command
that evolves \(G\).
\end{proposition}

\begin{proof}
The displayed chart metric gives the first identity.
The remaining statements specify which objects enter
the analytical equations and which enter physical
port placement. Reversal of the overall signature
leaves their common null directions unchanged, but
does not justify changing an Euler or Green-kernel sign.
\end{proof}

\begin{proposition}[No global transfer from the chart]
\label{prop:no-global-transfer}
The developing chart does not supply a global Minkowski
representation, an intrinsic Newton inverse on the cone,
or a global field-controlled unitary.
\end{proposition}

\begin{proof}
Parallel transport around the missing conical axis
has nontrivial rotational holonomy for
\(0<\delta_{\mathrm c}<2\pi\). A global isometry into
Minkowski space would identify this connection with
one having trivial holonomy, which is impossible.

A local isometry also does not fix nonlocal inverse
boundary conditions. For example, two flat-completion
sources agreeing in a chart but differing by a
nonnegative bump outside it can have different
Newton potentials inside it. Finally, a chart supplies
neither a field representation nor a coupling Hamiltonian
or controlled unitary on such a representation.
\end{proof}

The nonimplication, rather than a theorem prohibiting
every future intrinsic construction, is the point of
the last two assertions.

%=====================================================================
\section{Frozen classical prescription}
\label{sec:freeze}
%=====================================================================

\subsection{Formal fields and density coefficients}

Work over
\[
 \mathbb D_2=\RR[\kappa]/(\kappa^3).
\]
An equality modulo \(\kappa^3\) is coefficient equality,
not a numerical error estimate.

The configuration variables are
\[
 q_{\mu\nu}=\eta_{\mu\nu}+\kappa h_{\mu\nu}
                              +\kappa^2k_{\mu\nu},
 \qquad
 \phi=\phi_0+\kappa\phi_1+\kappa^2\phi_2,
\]
where \(h,k\) are arbitrary smooth real symmetric tensors
and \(\phi_j\) arbitrary smooth real scalars.
There is no Einstein equation, free-shell quotient,
gauge fixing, or decay condition on this off-shell
configuration space.

Indices are raised using \(\eta\), and parentheses
include a factor \(1/2\). Write
\[
 H=\eta^{\mu\nu}h_{\mu\nu},
 \qquad h:h=h_{\mu\nu}h^{\mu\nu}.
\]
Set
\begin{align}
 b_1[h]&=\tfrac12H,
 &
 b_2[h]&=\tfrac18H^2-\tfrac14h:h,
 \nonumber\\
 a_1^{\mu\nu}[h]&=\tfrac12H\eta^{\mu\nu}-h^{\mu\nu},
 &
 a_2^{\mu\nu}[h]
   &=h^\mu{}_\lambda h^{\lambda\nu}
          -\tfrac12Hh^{\mu\nu}+b_2[h]\eta^{\mu\nu}.
 \label{eq:densitycoeff}
\end{align}
Polarization is normalized by
\(a_2[h,h]=a_2[h]\) and \(b_2[h,h]=b_2[h]\).

\begin{lemma}[Density expansion]
\label{lem:density}
For \(v_q=\sqrt{-\det q}\) and
\(A_q^{\mu\nu}=v_qq^{\mu\nu}\),
\begin{align}
 v_q
 &=1+\kappa b_1[h]
        +\kappa^2\bigl(b_1[k]+b_2[h]\bigr),\nonumber\\
 A_q^{\mu\nu}
 &=\eta^{\mu\nu}+\kappa a_1^{\mu\nu}[h]
       +\kappa^2\bigl(a_1^{\mu\nu}[k]+a_2^{\mu\nu}[h]\bigr).
 \label{eq:densityexp}
\end{align}
\end{lemma}

\begin{proof}
The inverse is
\[
 q^{\mu\nu}
 =\eta^{\mu\nu}-\kappa h^{\mu\nu}
   +\kappa^2\bigl(h^\mu{}_\lambda h^{\lambda\nu}
                                      -k^{\mu\nu}\bigr).
\]
Expanding
\(\exp\{\tfrac12\tr\log(I+\kappa\eta^{-1}h+
\kappa^2\eta^{-1}k)\}\) gives \(v_q\).
Multiplication gives \(A_q\).
\end{proof}

Define
\begin{equation}
 C^\mu[h]=\partial_\nu h^{\nu\mu}-\tfrac12\partial^\mu H
          =-\partial_\nu a_1^{\nu\mu}[h],
 \qquad
 (K\xi)_{\mu\nu}=\partial_\mu\xi_\nu+\partial_\nu\xi_\mu.
 \label{eq:CK}
\end{equation}
A direct calculation gives
\[
 C^\mu[K\xi]=\Box\xi^\mu.
 \label{eq:CKidentity}
\]

\subsection{Switch, ramp, and gauge algebra}

Fix \(m\ge0\), \(0<\varepsilon<T_*\), and
\[
 g\in C_c^\infty(\MTo;[0,1]),
 \qquad P_g=\operatorname{int}\{g=1\}\ne\varnothing.
\]
For the supported realization, the retained interaction
support is in the laboratory. Contributions outside a
given \(D_f\) will in any event not enter its evaluation.

Define
\[
 \chi(s)=
 \begin{cases}
 0,&s\le0,\\[1mm]
 \displaystyle
 \frac{e^{-1/s}}{e^{-1/s}+e^{-1/(1-s)}},&0<s<1,\\[3mm]
 1,&s\ge1,
 \end{cases}
 \qquad \sigma(t)=\chi(t/\varepsilon).
 \label{eq:ramp}
\]
It is smooth and flat at both endpoints;
\(\sigma'(t)>0\) on \(0<t<\varepsilon\), and
\(\sigma=1\) for \(t\ge\varepsilon\).

\begin{definition}[Plateau and frozen gauge group]
\label{def:plateau}
Choose a nonempty relatively compact open set \(U\) with
\[
 \overline U\Subset
   \mathcal L\cap P_g\cap\{\varepsilon<t<T_*\}.
 \label{eq:plateau}
\]
Set \(\gp=C_c^\infty(U;T\widehat M)\).
Its elements are extended by zero to the completion.

For \(\xi,\zeta\in\gp\), let
\(F_\kappa\) be the time-one flow of
\(\kappa\xi+\kappa^2\zeta\).
The formal pullbacks
\[
 F_\kappa^*=
  I+\kappa\Lie_\xi+
       \kappa^2(\Lie_\zeta+\tfrac12\Lie_\xi^2)
                    \formal
\]
generate \(\Gp\).
This is the frozen plateau algebra; it is not the
Killing algebra of the cone.
\end{definition}

In particular, \(U\) is open with compact closure.
It is not called a nonempty compact open subset of
Minkowski space.

\begin{lemma}[Preservation of the fixed backgrounds]
\label{lem:group}
Every admitted \(F_\kappa\) is the identity near \(\Sigma\)
and outside a compact subset of \(U\), preserves the slab,
and satisfies
\[
 F_\kappa^*g=g,\qquad F_\kappa^*\sigma=\sigma.
\]
The truncated pullbacks form a group.
\end{lemma}

\begin{proof}
A compactly supported smooth vector field has a complete
flow, which fixes the complement of its support.
Here the flow is supported where both \(g\) and \(\sigma\)
are one. It is the identity near the slab endpoints.
Closure of the truncated pullbacks follows by expanding
their product: the order-two bracket contribution is
\(\tfrac12[\xi,\xi']\), again supported in \(U\).
\end{proof}

For scalars put \(X_\xi=\xi^\mu\partial_\mu\); for scalar
densities put \(D_\xi r=\partial_\mu(\xi^\mu r)\).
The full coefficient transformations are
\begin{align}
 h'&=h+K\xi,\nonumber\\
 k'&=k+\Lie_\xi h+K\zeta+\tfrac12\Lie_\xi^2\eta,
 \label{eq:metriclaw}\\
 \phi_0'&=\phi_0,\qquad
 \phi_1'=\phi_1+X_\xi\phi_0,\nonumber\\
 \phi_2'&=\phi_2+X_\xi\phi_1+
                 X_\zeta\phi_0+\tfrac12X_\xi^2\phi_0.
 \label{eq:scalarlaw}
\end{align}
Deleting the quadratic terms in \eqref{eq:metriclaw}
would change the problem.

\subsection{Two different density operators}

\begin{definition}[Regulated coordinate density]
\label{def:Aeps}
The auxiliary coordinate density and its operator are
\begin{equation}
 A_\varepsilon[q]^{\mu\nu}
   =\eta^{\mu\nu}+\sigma(A_q^{\mu\nu}-\eta^{\mu\nu}),
 \qquad
 D_\varepsilon[q]\psi
   =\partial_\mu(A_\varepsilon[q]^{\mu\nu}\partial_\nu\psi).
 \label{eq:Aeps}
\end{equation}
The output is a scalar density relative to \(\dd^4x\).
No convolution occurs.
\end{definition}

\begin{definition}[Coordinate Cauchy prescription]
\label{def:coordinates}
The four components of
\[
 \Co_q^\mu=x^\mu+\kappa w^\mu+\kappa^2v^\mu
\]
are scalar functions satisfying
\begin{equation}
 D_\varepsilon[q]\Co_q^\mu=0\formal,\qquad
 \Co_q^\mu|_\Sigma=x^\mu|_\Sigma,\qquad
 \partial_t\Co_q^\mu|_\Sigma=\delta^\mu_0.
 \label{eq:coordinateproblem}
\end{equation}
The component label \(\mu\) does not prescribe a vector-field
pullback. The switch \(g\) does not enter this problem.
\end{definition}

For the scalar dynamics define instead
\begin{align}
 A_g[q]^{\mu\nu}
  &=\eta^{\mu\nu}+g(A_q^{\mu\nu}-\eta^{\mu\nu}),\nonumber\\
 r_g[q]&=1+g(v_q-1),\nonumber\\
 \mathscr L_g[q,\phi]
  &=\tfrac12\bigl(A_g[q]^{\mu\nu}\partial_\mu\phi\partial_\nu\phi
                                      -m^2r_g[q]\phi^2\bigr).
 \label{eq:switchedaction}
\end{align}
Thus
\[
 \mathscr L_g=(1-g)\mathscr L_\eta+g\mathscr L_q.
\]
The action \(S_g^{\mathrm{sc}}=\int_{\MT}\mathscr L_g\)
is understood variationally; its numerical value on
arbitrary smooth fields need not be finite.
Compactly supported variations and the finite changes
used below are well defined.

The sign convention is
\begin{equation}
 \frac{\delta S_g^{\mathrm{sc}}}{\delta\phi}
     =-E_g[q]\phi,\qquad
 E_g[q]\phi
     =\partial_\mu(A_g[q]^{\mu\nu}\partial_\nu\phi)
                                      +m^2r_g[q]\phi.
 \label{eq:Euler}
\end{equation}

%=====================================================================
\section{Flat Cauchy construction and coefficient equations}
\label{sec:cauchy}
%=====================================================================

\begin{lemma}[Smooth zero-data inverse]
\label{lem:cauchy}
For \(\mu\ge0\), smooth source \(s\) on the slab and smooth
initial data on \(\Sigma\), the equation
\[
 (\Box+\mu^2)u=s
\]
has a unique smooth solution.
Its value on a compact set depends only on the source
and data in the corresponding truncated flat causal past.
For zero data denote it by \(\mathsf E_\mu s\).

On compact sets this solution map is continuous in the
smooth topologies. No spatial decay is required.
\end{lemma}

\begin{proof}
For compactly supported smooth data and source, spatial
Fourier transformation gives the solution using
\[
 \Omega_\mu=(-\Delta+\mu^2)^{1/2},\qquad
 (\mathsf E_\mu s)(t)
    =\int_0^t
       \frac{\sin((t-s')\Omega_\mu)}{\Omega_\mu}
                          s(s')\,\dd s'.
 \label{eq:Duhamel}
\]
The quotient has value \(t-s'\) at zero frequency.
Differentiation of the Fourier formula gives smoothness.
The usual differentiated energy estimate, or the same
Fourier formula in Sobolev norms followed by Sobolev
embedding, gives finite-seminorm bounds on compact
time intervals.

For finite propagation, apply the energy identity with
energy density
\[
 \tfrac12\bigl((\partial_tu)^2+|\nabla u|^2+\mu^2u^2\bigr)
\]
to a truncated backward cone. Its lateral flux has the
required nonnegative sign. Zero source and data on that
cone imply zero solution at its vertex. Adding an
\(u^2\) term and applying Gronwall also covers the
massless low-frequency estimate.

For arbitrary smooth data, fix a compact target set,
cut off the source and data on a neighbourhood of its
compact truncated past, and use the compact-data
solution. Finite propagation makes the result
independent of the cutoff there. Such solutions agree
on overlaps, giving a global smooth solution on the slab.
The same argument proves uniqueness and local
continuity. This is the standard Cauchy mechanism
used on the declared completion, not on the non-globally
hyperbolic host \cite{BGP}.
\end{proof}

For a compact interior source \(a\), let
\(\mathsf E_{\mathrm A,\mu}a\) be the advanced solution,
with zero terminal data after the source. On the slab
it is smooth, compactly supported in the truncated past
of \(\supp a\), and vanishes near \(t=T_*\).

\begin{lemma}[Transposition]
\label{lem:transpose}
For a compact interior test \(a\) and smooth source \(s\),
\[
 \int_{\MT}a\,\mathsf E_\mu s\,\dd^4x
 =\int_{\MT}(\mathsf E_{\mathrm A,\mu}a)s\,\dd^4x.
 \label{eq:transpose}
\]
The formula also holds for smooth compact tests flat at
\(t=0\), extended by zero to negative times.
\end{lemma}

\begin{proof}
Use Green's identity with \(u=\mathsf E_\mu s\) and
\(b=\mathsf E_{\mathrm A,\mu}a\).
The terminal term vanishes because \(b\) is terminally
zero; the initial term vanishes because \(u\) has zero
Cauchy data. Spatial boundary terms vanish after a
cutoff outside the compact retained past. The
flat-at-zero case follows by the same identity or
smooth cutoff approximation.
\end{proof}

Write \(\Ezero=\mathsf E_0\), \(\Em=\mathsf E_m\), and
\(\EA=\mathsf E_{\mathrm A,0}\).

\subsection{Coordinate equations, with every ramp term}

Set \(B^{\mu\nu}=a_1^{\mu\nu}[k]+a_2^{\mu\nu}[h]\).
Then
\[
 D_\varepsilon[q]=\Box+\kappa\mathsf V_\varepsilon[h]
                          +\kappa^2\mathsf W_\varepsilon[h,k],
\]
where
\begin{align}
 \mathsf V_\varepsilon[h]\psi
 &=\sigma\partial_\alpha(a_1^{\alpha\beta}[h]\partial_\beta\psi)
       +\sigma'a_1^{0\beta}[h]\partial_\beta\psi,\nonumber\\
 \mathsf W_\varepsilon[h,k]\psi
 &=\sigma\partial_\alpha(B^{\alpha\beta}\partial_\beta\psi)
       +\sigma'B^{0\beta}\partial_\beta\psi.
 \label{eq:coordinateoperators}
\end{align}

\begin{proposition}[Unique formal coordinate coefficients]
\label{prop:coordeq}
The coordinate problem has the unique solution
\begin{align}
 w^\mu
  &=-\Ezero\partial_\alpha(\sigma a_1^{\alpha\mu}[h]),
 \nonumber\\
 v^\mu
  &=-\Ezero\bigl(
       \mathsf V_\varepsilon[h]w^\mu
           +\mathsf W_\varepsilon[h,k]x^\mu\bigr).
 \label{eq:coordinatesolutions}
\end{align}
Equivalently, with zero data at each positive order,
\begin{align}
 \Box w^\mu
  &=\sigma C^\mu[h]-\sigma'a_1^{0\mu}[h],\nonumber\\
 \Box v^\mu
  &=\sigma\left(
       C^\mu[k]-\partial_\alpha a_2^{\alpha\mu}[h]
       -\partial_\alpha(a_1^{\alpha\beta}[h]\partial_\beta w^\mu)
          \right)\nonumber\\
  &\quad-\sigma'\left(
       a_1^{0\mu}[k]+a_2^{0\mu}[h]
                  +a_1^{0\beta}[h]\partial_\beta w^\mu\right).
 \label{eq:coordinateequations}
\end{align}
Also,
\[
 v[h,k]=w[k]+v[h,0].
 \label{eq:affinek}
\]
\end{proposition}

\begin{proof}
Insert \eqref{eq:coordinateoperators} and
\(\Co=x+\kappa w+\kappa^2v\) into
\eqref{eq:coordinateproblem}. Since \(\Box x^\mu=0\),
coefficient comparison gives the displayed equations.
Lemma~\ref{lem:cauchy} proves existence and uniqueness
successively. Only the source
\(-\partial_\alpha(\sigma a_1^{\alpha\mu}[k])\)
contains \(k\), proving the final identity.
\end{proof}

The defining operator contains one derivative of
\(\sigma\), not a discarded ramp term. No explicit
\(\sigma''\) term is generated by expanding its single
divergence. Derivatives of solutions or of coefficient
identities may of course differentiate \(\sigma'\);
the statement is about the displayed defining operators.

\subsection{Scalar equations, with every switch term}

Let \(P_m=\Box+m^2\). The expansion is
\[
 E_g[q]=P_m+\kappa\mathsf V_{1,g}[h]
                       +\kappa^2\mathsf V_{2,g}[h,k],
\]
with
\begin{align}
 \mathsf V_{1,g}[h]\psi
 &=g\left(
      \partial_\mu(a_1^{\mu\nu}[h]\partial_\nu\psi)
                            +m^2b_1[h]\psi\right)
       +(\partial_\mu g)a_1^{\mu\nu}[h]\partial_\nu\psi,
 \nonumber\\
 \mathsf V_{2,g}[h,k]\psi
 &=g\left(
      \partial_\mu(B^{\mu\nu}\partial_\nu\psi)
                      +m^2(b_1[k]+b_2[h])\psi\right)
       +(\partial_\mu g)B^{\mu\nu}\partial_\nu\psi.
 \label{eq:scalaroperators}
\end{align}

For a fixed smooth real free input \(P_m\phi_0=0\),
the switched scalar solution is
\begin{align}
 \rho_g[q;\phi_0]
  &=\phi_0+\kappa\rho_{1,g}+\kappa^2\rho_{2,g},\nonumber\\
 \rho_{1,g}
  &=-\Em\mathsf V_{1,g}[h]\phi_0,\nonumber\\
 \rho_{2,g}
  &=\Em\mathsf V_{1,g}[h]\Em\mathsf V_{1,g}[h]\phi_0
                  -\Em\mathsf V_{2,g}[h,k]\phi_0.
 \label{eq:scalarsolution}
\end{align}

\begin{proposition}[Scalar existence and support]
\label{prop:scalarconstruction}
Equation \eqref{eq:scalarsolution} is the unique formal
solution of \(E_g[q]\rho_g=0\) with the free input's
Cauchy data and zero positive-order data.
Its positive-order sources are smooth and supported
in \(\supp g\), and
\[
 \supp\rho_{j,g}\subset J_\eta^+(\supp g)\cap\MT,
 \qquad j=1,2.
\]
\end{proposition}

\begin{proof}
At orders one and two the equations are respectively
\[
 P_m\rho_{1,g}=-\mathsf V_{1,g}[h]\phi_0,\qquad
 P_m\rho_{2,g}
  =-\mathsf V_{1,g}[h]\rho_{1,g}
                  -\mathsf V_{2,g}[h,k]\phi_0.
\]
Apply Lemma~\ref{lem:cauchy}. Every coefficient of the
source operators contains \(g\) or a derivative of \(g\).
\end{proof}

No coupled Einstein solution is inferred.

%=====================================================================
\section{Preflight and the retained edge obstruction}
\label{sec:preflight}
%=====================================================================

\begin{proposition}[Exact preflight on post-ramp gauges]
\label{prop:preflight}
If \(\xi\in C_c^\infty(\MTo;T\widehat M)\) and
\(\supp\xi\subset\{t\ge\varepsilon\}\), then
\[
 \Co_1[K\xi]=w[K\xi]=\xi.
 \label{eq:preflight}
\]
In particular, this holds for every \(\xi\in\gp\).
\end{proposition}

\begin{proof}
All derivatives of \(\xi\), and hence \(a_1[K\xi]\),
are supported where \(\sigma=1\).
Therefore
\[
 \Box w^\mu[K\xi]
   =-\partial_\alpha(\sigma a_1^{\alpha\mu}[K\xi])
   =C^\mu[K\xi]=\Box\xi^\mu.
\]
Both sides have zero Cauchy data. Uniqueness proves
the assertion.
\end{proof}

This is the preflight identity of the fixed ramp
prescription \cite[{\loc{prop:preflight}}]{XA}.

\begin{proposition}[Explicit edge-zone defect]
\label{prop:edge}
Let \(0\ne\alpha\in C_c^\infty((0,\varepsilon))\)
be nonnegative, and choose
\(\beta\in C_c^\infty(\RR^3)\) with
\(\int\beta\,\dd^3x=1\). Set
\[
 \xi=\alpha(t)\beta(\mathbf x)\partial_1.
\]
For every \(t\in[\varepsilon,T_*]\),
\begin{equation}
 \int_{\RR^3}
    \bigl(w^1[K\xi]-\xi^1\bigr)(t,\mathbf x)\,\dd^3x
       =-\int_0^\varepsilon\sigma'(s)\alpha(s)\,\dd s<0.
 \label{eq:edge}
\end{equation}
This parameter is outside \(\gp\).
\end{proposition}

\begin{proof}
The source for \(w[K\xi]\) is compact, so its solution
has compact spatial support on each bounded time slice.
Put \(W^\mu(t)=\int w^\mu[K\xi](t,\mathbf x)\,\dd^3x\)
and \(\Xi^\mu(t)=\int\xi^\mu(t,\mathbf x)\,\dd^3x\).
Spatial divergences integrate to zero. From
\[
 a_1^{0\mu}[K\xi]
   =(\partial_\rho\xi^\rho)\eta^{0\mu}
                          -\partial^0\xi^\mu-\partial^\mu\xi^0
\]
one obtains
\[
 \int a_1^{0\mu}[K\xi]\,\dd^3x=-\dot\Xi^\mu.
\]
Thus
\[
 \ddot W^\mu=(\sigma\dot\Xi^\mu)^{\boldsymbol\cdot},
 \qquad
 \dot W^\mu=\sigma\dot\Xi^\mu,
 \qquad
 W^1(t)=\int_0^t\sigma(s)\alpha'(s)\,\dd s.
\]
For \(t\ge\varepsilon\), integration by parts gives
\eqref{eq:edge}. Its strict sign follows from
\(\sigma'>0\) in the ramp and \(\alpha\ge0\), \(\alpha\ne0\).
The support of \(\xi\) is disjoint from the prescribed
post-ramp \(U\).
\end{proof}

The formula retains the parent edge witness
\cite[{\loc{prop:edge}}]{XA}. It does not classify the
kernel of the preflight defect on every possible
ramp-supported parameter.

\begin{corollary}[A smeared edge failure and its order]
\label{cor:edgeorder}
There are a smooth scalar \(\phi_0\) and a supported
observation test \(f\) for which
\[
 \int f(\xi-w[K\xi])^\mu\partial_\mu\phi_0\,\dd^4x\ne0.
 \label{eq:smearededge}
\]
No correction starting at \(\kappa^2\), or at positive
\(\hbar\)-order in a formal double expansion, cancels
this order-\(\kappa\), order-\(\hbar^0\) coefficient
while the lower-order prescription is held fixed.
\end{corollary}

\begin{proof}
At a time strictly after \(\varepsilon\), \eqref{eq:edge}
implies that \(w^1[K\xi]\) is nonzero somewhere.
Choose \(\phi_0=x^1\) and a nonnegative bump \(f\) where
that component has a fixed nonzero sign.
The chart reserve may be chosen in advance to contain
the compact wave support and the resulting test's
truncated past.

At the flat metric the first relational variation under
the simultaneous first-order field pullback is precisely
\eqref{eq:smearededge}. Formal coefficients of different
powers cannot cancel it. A universal extension to all
such parameters is therefore excluded without changing
the lower-order data.
\end{proof}

This is the coefficientwise logic illustrated in VI-B,
not an identification with VI-B's different switched
Einstein--scalar defect \cite{VIB}.

%=====================================================================
\section{Relational expansion and covariance: Route I}
\label{sec:routeI}
%=====================================================================

\subsection{Formal inverse and observable}

The coordinate inverse exists algebraically:
\begin{equation}
 \Co_q^{-1}(z)
 =z-\kappa w(z)+\kappa^2\bigl(w^\nu\partial_\nu w-v\bigr)(z).
 \label{eq:inverse}
\end{equation}
No globally invertible numerical coordinate map is
required.

Define
\begin{equation}
 \Phi_{g,\varepsilon}[q,\phi](z)=\phi(\Co_q^{-1}(z)),
 \qquad
 \Obs_{f,g,\varepsilon}[q,\phi]
      =\int f(z)\Phi_{g,\varepsilon}[q,\phi](z)\,\dd^4z.
 \label{eq:observable}
\end{equation}
The label-space test \(f\) is fixed, not pulled back.
No \(q\)-volume density is inserted in this pairing.
The off-shell expression depends on \(g\) only through
the declared gauge and evaluation contract.

Taylor expansion gives
\begin{align}
 \Phi^{(0)}&=\phi_0,\nonumber\\
 \Phi^{(1)}&=\phi_1-w^\mu\partial_\mu\phi_0,\nonumber\\
 \Phi^{(2)}&=\phi_2-w^\mu\partial_\mu\phi_1
       +(w^\nu\partial_\nu w^\mu-v^\mu)\partial_\mu\phi_0
       +\tfrac12w^\mu w^\nu\partial_\mu\partial_\nu\phi_0.
 \label{eq:relationalcoeff}
\end{align}
For the dynamical evaluation, replace \(\phi_1,\phi_2\)
by \(\rho_{1,g},\rho_{2,g}\).
In particular, the forced scalar correction is not
omitted.

\subsection{Naturality and plateau covariance}

For an orientation-preserving diffeomorphism, a
contravariant tensor density of weight one transforms as
\[
 (F^*A)^{\mu\nu}(x)
  =\det DF(x)\,(DF^{-1})^\mu{}_\alpha
                   (DF^{-1})^\nu{}_\beta A^{\alpha\beta}(F(x)).
\]
Consequently,
\begin{equation}
 \partial_\mu((F^*A)^{\mu\nu}\partial_\nu F^*\psi)
   =F_{\mathrm{dens}}^*
          \bigl(\partial_\mu(A^{\mu\nu}\partial_\nu\psi)\bigr).
 \label{eq:naturality}
\end{equation}
This follows from the chain rule and change of variables,
or from integration against compact tests. Metric
density and volume satisfy
\[
 A_{F^*q}=F^*A_q,\qquad v_{F^*q}=F_{\mathrm{dens}}^*v_q.
\]
For formal jets these identities follow coefficientwise
by applying the genuine identities on each compact
neighbourhood for sufficiently small numerical
\(\kappa\), then taking Taylor coefficients.
No globally uniform numerical nondegeneracy is needed.

\begin{lemma}[Plateau covariance of the coordinate operator]
\label{lem:plateaucoord}
For \(F\in\Gp\),
\[
 D_\varepsilon[F^*q](F^*\psi)
            =F_{\mathrm{dens}}^*(D_\varepsilon[q]\psi)\formal.
 \label{eq:coordinatecov}
\]
\end{lemma}

\begin{proof}
Since \(F^*\sigma=\sigma\),
\[
 F^*A_\varepsilon[q]-A_\varepsilon[F^*q]
       =(1-\sigma)(F^*\eta^{-1}-\eta^{-1}).
\]
The right side vanishes on \(U\) because \(\sigma=1\),
and outside \(U\) because \(F=\id\) there.
Apply \eqref{eq:naturality}.
\end{proof}

\begin{lemma}[Plateau covariance of the scalar Euler operator]
\label{lem:plateaueuler}
For \(F\in\Gp\),
\begin{align}
 E_g[F^*q](F^*\psi)
      &=F_{\mathrm{dens}}^*(E_g[q]\psi),\nonumber\\
 \mathscr L_g[F^*q,F^*\psi]
      &=F_{\mathrm{dens}}^*\mathscr L_g[q,\psi]
                      \formal.
 \label{eq:scalarcov}
\end{align}
\end{lemma}

\begin{proof}
The same calculation with \(g\) gives
\[
 F^*A_g[q]-A_g[F^*q]
       =(1-g)(F^*\eta^{-1}-\eta^{-1})=0.
\]
For the volume coefficient the corresponding difference
is \((1-g)(F_{\mathrm{dens}}^*1-1)=0\).
Naturality proves both identities.
\end{proof}

The switch and ramp derivatives in
\eqref{eq:coordinateoperators} and
\eqref{eq:scalaroperators} have not been discarded.
Their coefficients are unchanged where the transformation
does not act.

\begin{theorem}[Coordinate intertwiner]
\label{thm:coordinate}
For every smooth metric jet and every \(F\in\Gp\),
\begin{equation}
 \Co_{F^*q}=F^*\Co_q=\Co_q\circ F\formal.
 \label{eq:intertwiner}
\end{equation}
In coefficients,
\begin{align}
 w[h']&=w[h]+\xi,\nonumber\\
 v[h',k']&=v[h,k]+\xi^\nu\partial_\nu w[h]
                    +\zeta+\tfrac12\xi^\nu\partial_\nu\xi.
 \label{eq:coordinatelaws}
\end{align}
\end{theorem}

\begin{proof}
By Lemma~\ref{lem:plateaucoord}, each component of
\(F^*\Co_q\) solves the transformed coordinate problem.
It has the same initial data because \(F=\id\) near
\(\Sigma\). Proposition~\ref{prop:coordeq} gives uniqueness.
Expansion of \(\Co_q\circ F\) gives
\eqref{eq:coordinatelaws}.
\end{proof}

This is the supported form of the X-A coordinate
intertwiner \cite[{\loc{thm:coordinate}}]{XA}.

\begin{proposition}[Scalar intertwiner with unchanged free input]
\label{prop:scalar}
For \(F\in\Gp\),
\begin{equation}
 \rho_g[F^*q;\phi_0]=F^*\rho_g[q;\phi_0]\formal.
 \label{eq:scalarintertwiner}
\end{equation}
Thus
\begin{align}
 \rho_{1,g}'&=\rho_{1,g}+X_\xi\phi_0,\nonumber\\
 \rho_{2,g}'&=\rho_{2,g}+X_\xi\rho_{1,g}
                   +X_\zeta\phi_0+\tfrac12X_\xi^2\phi_0.
 \label{eq:rholaws}
\end{align}
\end{proposition}

\begin{proof}
Lemma~\ref{lem:plateaueuler} makes the pulled-back solution
solve the transformed equation.
Its Cauchy data are exactly those of the same
\(\phi_0\), because the transformation is initial-trivial.
Coefficientwise Cauchy uniqueness proves
\eqref{eq:scalarintertwiner}; expansion gives
\eqref{eq:rholaws}.
\end{proof}

\begin{theorem}[Second-order classical covariance]
\label{thm:main}
At fixed \(\varepsilon>0\), for the frozen data above:
\begin{enumerate}[label=\textup{(\roman*)}]
\item the coordinate and switched scalar coefficient
problems have their stated unique solutions;
\item preflight is exact on \(\gp\);
\item for every off-shell scalar jet and \(F\in\Gp\),
\[
 \Phi_{g,\varepsilon}[F^*q,F^*\phi]
         =\Phi_{g,\varepsilon}[q,\phi]\formal,
\]
and hence
\[
 \Obs_{f,g,\varepsilon}[F^*q,F^*\phi]
         =\Obs_{f,g,\varepsilon}[q,\phi]\formal;
 \label{eq:offsinvariance}
\]
\item with fixed free input, the dynamical evaluation
\[
 \Obs_{f,g,\varepsilon}[q,\rho_g[q;\phi_0]]
\]
is invariant under the same group.
\end{enumerate}
The metric is off shell throughout. No quantum object
is part of this theorem.
\end{theorem}

\begin{proof}
Existence and preflight have been proved.
Formal inversion of \eqref{eq:intertwiner} yields
\[
 \Co_{F^*q}^{-1}=F^{-1}\circ\Co_q^{-1}.
\]
Since \(F^*\phi=\phi\circ F\),
\[
 (F^*\phi)\circ\Co_{F^*q}^{-1}
      =\phi\circ F\circ F^{-1}\circ\Co_q^{-1}
      =\phi\circ\Co_q^{-1}.
\]
Pairing with the unchanged \(f\) proves the off-shell
statement. Substitute \eqref{eq:scalarintertwiner}
to prove the dynamical one.
\end{proof}

No larger gauge algebra is released here, even though
some individual lemmas use fewer plateau conditions.

\subsection{Second-order operator audit}

For either the coordinate or scalar operator write
\(P+\kappa V+\kappa^2W\), and let primes denote the
transformed coefficients. Put
\[
 X=X_\xi,\quad T_2=X_\zeta+\tfrac12X_\xi^2,\quad
 D=D_\xi,\quad \Delta_2=D_\zeta+\tfrac12D_\xi^2.
\]
The proven covariance expands to
\begin{align}
 V'&=V+DP-PX,\nonumber\\
 W'+V'X+PT_2&=W+DV+\Delta_2P.
 \label{eq:operatoraudit}
\end{align}
These are identities on arbitrary smooth test scalars,
not only on solutions.
For the coordinates, \(P=\Box\) and \(Px^\mu=0\);
inserting \eqref{eq:operatoraudit} verifies directly
that
\[
 v_{\mathrm{cand}}=v+Xw+T_2x
\]
solves the transformed second-order equation with zero
data. This checks the quadratic metric law without
deleting the terms proportional to \(P\).

%=====================================================================
\section{Varied-action derivation: Route II}
\label{sec:routeII}
%=====================================================================

Route II gives a variational derivation of the dynamics'
covariance and, applied to an auxiliary coordinate action,
of the coordinate operator covariance. It does not
infer observable invariance from action invariance
alone: the fixed Cauchy data, uniqueness, and relational
composition remain necessary.

Temporarily let \(\xi,\zeta\) be arbitrary compact
interior vector fields, not necessarily plateau-supported.
For a scalar density \(r\), define
\[
 Z_F^\mu[r]
   =\kappa\xi^\mu r+
       \kappa^2\left(\zeta^\mu r+
                       \tfrac12\xi^\mu D_\xi r\right).
\]
Then
\begin{equation}
 F_{\mathrm{dens}}^*r-r=\partial_\mu Z_F^\mu[r]\formal.
 \label{eq:Z}
\end{equation}
This is the Taylor expansion of density pullback.
All currents have compact support in the gauge support.

\begin{proposition}[Varied switched action]
\label{prop:varied}
The finite variational difference satisfies
\begin{equation}
\begin{split}
 S_g^{\mathrm{sc}}[F^*q,F^*\phi]-S_g^{\mathrm{sc}}[q,\phi]
 ={}&\int_{\MT}(1-g)
             \bigl(\mathscr L_\eta[F^*\phi]
                         -\mathscr L_\eta[\phi]\bigr)\,\dd^4x\\
 &-\int_{\MT}(\partial_\mu g)
                              Z_F^\mu[\mathscr L_q[\phi]]\,\dd^4x
 \quad\formal.
\end{split}
\label{eq:varied}
\end{equation}
Every non-divergence term contains \(1-g\) or
\(\partial g\).
\end{proposition}

\begin{proof}
Naturality of the curved density gives
\[
 \mathscr L_g[F^*q,F^*\phi]-\mathscr L_g[q,\phi]
  =(1-g)(\mathscr L_\eta[F^*\phi]-\mathscr L_\eta[\phi])
       +g(F_{\mathrm{dens}}^*\mathscr L_q-\mathscr L_q).
\]
Apply \eqref{eq:Z} and
\(g\partial_\mu Z^\mu
=\partial_\mu(gZ^\mu)-(\partial_\mu g)Z^\mu\).
The compact divergence integrates to zero.
The difference is finite even when the individual
unvaried actions have no finite numerical value.
\end{proof}

For completeness, define the symmetric bilinear expression
\[
 \mathcal B_m(u,v)
       =\partial^\mu u\,\partial_\mu v-m^2uv,
 \qquad T_2=X_\zeta+\tfrac12X_\xi^2.
\]
The two terms in \eqref{eq:varied} are explicitly
\begin{align}
 \mathscr L_\eta[F^*\phi]-\mathscr L_\eta[\phi]
  ={}&\kappa\mathcal B_m(\phi,X_\xi\phi)\nonumber\\
    &+\kappa^2\left(
        \mathcal B_m(\phi,T_2\phi)
                  +\tfrac12\mathcal B_m(X_\xi\phi,X_\xi\phi)\right),
 \label{eq:flatvariation}\\
 Z_F^\mu[\mathscr L_q[\phi]]
  ={}&\kappa\xi^\mu\mathscr L_\eta[\phi]\nonumber\\
    &+\kappa^2\left(
       \xi^\mu\mathscr L^{(1)}[h,\phi]
       +\zeta^\mu\mathscr L_\eta[\phi]
       +\tfrac12\xi^\mu D_\xi\mathscr L_\eta[\phi]\right),
 \label{eq:currentvariation}\\
 \mathscr L^{(1)}[h,\phi]
  ={}&\tfrac12\left(
       a_1^{\mu\nu}[h]\partial_\mu\phi\partial_\nu\phi
                                      -m^2b_1[h]\phi^2\right).
 \nonumber
\end{align}
For a \(\kappa\)-dependent \(\phi\), all products on the
right are themselves expanded and truncated. No
coefficient has been suppressed by treating that
\(\phi\) as independent of \(\kappa\).

\begin{corollary}[Classical plateau identity]
\label{cor:noether}
For \(F\in\Gp\),
\[
 S_g^{\mathrm{sc}}[F^*q,F^*\phi]
      -S_g^{\mathrm{sc}}[q,\phi]=0\formal
\]
off shell.
Varying \(\phi\) with a compact test recovers the
density-valued Euler covariance
\eqref{eq:scalarcov}.
\end{corollary}

\begin{proof}
Every term in \eqref{eq:flatvariation} is supported where
the gauge transformation acts; there \(1-g=0\).
The current is supported there as well, and
\(\partial g=0\). Thus \eqref{eq:varied} vanishes.

For a compact variation \(\chi\), vary the transformed
scalar by \(F^*\chi\). The Euler variation and change
of variables identify the two density-valued Euler
operators against every test, proving their equality.
\end{proof}

\begin{proposition}[First-order off-plateau display]
\label{prop:offplateau}
For a \(\kappa\)-independent scalar and
\[
 T^{\mu\nu}[\phi]
   =\partial^\mu\phi\partial^\nu\phi
                                  -\eta^{\mu\nu}\mathscr L_\eta[\phi],
\]
the first coefficient of \eqref{eq:varied} is
\begin{align}
 [\delta_FS_g^{\mathrm{sc}}]_{\kappa}
   &=\int(1-g)T^{\mu\nu}\partial_\mu\xi_\nu\,\dd^4x
 \nonumber\\
   &=\int(\partial_\mu g)T^{\mu\nu}\xi_\nu\,\dd^4x
      -\int(1-g)\xi^\nu(P_m\phi)\partial_\nu\phi\,\dd^4x.
 \label{eq:offplateau}
\end{align}
\end{proposition}

\begin{proof}
Insert the first terms of
\eqref{eq:flatvariation}--\eqref{eq:currentvariation}.
Integrate
\(\partial_\mu((1-g)\xi^\mu\mathscr L_\eta)\).
The result is the first line.
The identity
\[
 \partial_\mu T^{\mu\nu}=(P_m\phi)\partial^\nu\phi
\]
and another integration by parts give the second.
\end{proof}

This displays a classical breaking term, not a loop
anomaly or a quantum Ward defect
\cite[{\loc{prop:varied}, \loc{cor:noether}}]{XA}.

\subsection{Variational recovery of coordinate covariance}

For four scalar functions \(Z^\mu\), use the auxiliary
action
\[
 S_\varepsilon^{\mathrm{aux}}[q,Z]
  =\frac12\sum_{\mu=0}^3
       \int A_\varepsilon[q]^{\alpha\beta}
                  \partial_\alpha Z^\mu\partial_\beta Z^\mu\,\dd^4x.
\]
The sum is over four scalar labels; it is not a
spacetime-vector contraction.

\begin{proposition}[Action route for the coordinate operator]
\label{prop:auxroute}
For plateau transformations the finite variation of
\(S_\varepsilon^{\mathrm{aux}}\) vanishes modulo
\(\kappa^3\). Variation in \(Z^\mu\) then proves
Lemma~\ref{lem:plateaucoord} without assuming it.
Consequently Route II, together with Cauchy uniqueness,
also yields Theorem~\ref{thm:main}.
\end{proposition}

\begin{proof}
For each \(Z^\mu\) repeat \eqref{eq:varied} with
\(m=0\) and the multiplier \(g\) replaced by \(\sigma\).
The terms contain \(1-\sigma\) or \(\partial\sigma\)
on the compact gauge support, where both vanish.
Differentiate this finite action identity with respect
to compact variations of \(Z^\mu\), transforming the
variation by \(F^*\). The Euler derivative is
\(-D_\varepsilon[q]Z^\mu\), so the density-valued
operator covariance follows by change of variables.

The coordinate and scalar equations obtained this way
have the same fixed initial data as in Route I.
Their unique solutions therefore have the same
intertwiners. Relational inversion and composition give
the observable identity.
\end{proof}

The coordinate action is auxiliary; it is not obtained
by varying the switched scalar action
\(S_g^{\mathrm{sc}}\) with respect to \(q\).

%=====================================================================
\section{Mandatory diagnostics}
\label{sec:diagnostics}
%=====================================================================

\begin{proposition}[Pure-gauge two-jet]
\label{prop:puregauge}
Take \(\xi\in\gp\), \(\zeta=0\), and
\[
 h=K\xi,\qquad
 k=\tfrac12\Lie_\xi^2\eta,\qquad
 \phi=F_\kappa^*\phi_0.
\]
Then
\[
 w=\xi,\qquad v^\mu=\tfrac12\xi^\nu\partial_\nu\xi^\mu,
 \qquad
 \Phi^{(1)}=\Phi^{(2)}=0.
 \label{eq:puregauge}
\]
The coordinate coefficients are independent of any ramp
width for which the same gauge support remains strictly
post-ramp.
\end{proposition}

\begin{proof}
The flat coordinate solution is \(x\), so
Theorem~\ref{thm:coordinate} gives
\(\Co_{F^*\eta}=F^*x\).
This gives \(w,v\). In \eqref{eq:relationalcoeff},
\[
 \phi_1=X_\xi\phi_0,\qquad
 \phi_2=\tfrac12X_\xi^2\phi_0.
\]
The first cancellation is immediate.
At second order use
\[
 X_\xi^2\phi_0
   =(\xi^\nu\partial_\nu\xi^\mu)\partial_\mu\phi_0
        +\xi^\mu\xi^\nu\partial_\mu\partial_\nu\phi_0.
\]
The terms cancel exactly. The result depends only on
the post-ramp support condition, not on the width inside
the initial edge zone.
\end{proof}

Using \(h=K\xi,k=0\) instead would not be a
second-order pure-gauge test.

\begin{proposition}[Sourced conformal diagnostic]
\label{prop:sourced}
Let \(m=0\), \(\phi_0=t\),
\(h_{\mu\nu}=\psi\eta_{\mu\nu}\), \(k=0\), for a real
compact interior bump \(\psi\). Then
\begin{align}
 a_1[h]&=\psi\eta^{-1},\qquad b_1[h]=2\psi,
 \qquad a_2[h]=0,\nonumber\\
 \mathsf V_{1,g}[h]\phi_0&=\partial_t(g\psi),\qquad
 \rho_{1,g}=-\Ezero\partial_t(g\psi),\nonumber\\
 w^0&=-\Ezero\partial_t(\sigma\psi),\qquad
 w^i=\Ezero(\sigma\partial_i\psi),\nonumber\\
 A_1&:=\rho_{1,g}-w^\mu\partial_\mu\phi_0
              =-\Ezero\partial_t((g-\sigma)\psi).
 \label{eq:sourced}
\end{align}
There are supported choices for which \(A_1\ne0\)
and its smearing is nonzero.

If instead \(\supp\psi\subset P_g\cap\{t>\varepsilon\}\),
then \(A_1=0\), while \(\rho_{1,g}\) and \(w^0\)
are nonzero when \(\partial_t\psi\not\equiv0\).
\end{proposition}

\begin{proof}
In four dimensions \(\tr h=4\psi\), \(h:h=4\psi^2\).
Substitution in \eqref{eq:densitycoeff} gives the first
line. In fact the conformal identity
\(A_{(1+\kappa\psi)\eta}=(1+\kappa\psi)\eta^{-1}\)
checks \(a_2[h]=0\) directly.

The remaining formulae follow from the coefficient
equations and \(\partial_\nu t=\delta^0_\nu\).
Choose \(\psi\) at positive post-ramp time outside
\(\supp g\), with \(\partial_t\psi\ne0\).
Then \(A_1=\Ezero\partial_t\psi\ne0\), since applying
\(\Box\) gives the nonzero source.
A bump in an open set where \(A_1\) has fixed sign
gives a nonzero observation. All supports and their
pasts can be chosen within the retained laboratory.

In the plateau alternative \(g\psi=\sigma\psi=\psi\),
so \(A_1=0\). Applying \(\Box\) to either displayed
nontrivial response proves that it is not identically
zero.
\end{proof}

A common observation plateau therefore does not force
equality of arbitrary responses produced by different
switch histories. For example, changing \(g\) by a
small bump in a compact causal-past region, while keeping
both switches equal to one near a later observation,
changes \(\partial_t(g\psi)\) in
\eqref{eq:sourced}. A nonzero resulting response can be
detected by a later test. Causal-past agreement, not only
agreement near the observation, is sufficient for
equality; this distinction is also retained in VII-A-ST
\cite{VIIAST}.

\begin{proposition}[Zero-switch check]
\label{prop:zeroswitch}
In the separate diagnostic specialization \(g=0\),
\[
 E_g=P_m,\qquad \rho_g=\phi_0.
\]
The coordinate coefficients remain those of the ramp
prescription and need not vanish. In particular, in
the massless conformal example,
\[
 A_1=\Ezero\partial_t(\sigma\psi)
\]
can be nonzero.
There is no nonempty plateau algebra of the form
\eqref{eq:plateau} when \(g=0\).
\end{proposition}

\begin{proof}
The scalar operators in \eqref{eq:scalaroperators}
vanish, while the coordinate equations contain no \(g\).
The last assertion follows from \(P_g=\varnothing\).
\end{proof}

Thus the zero-switch diagnostic is not an enlargement
of the main theorem to a nonexistent nonempty plateau.

%=====================================================================
\section{Support, continuity, and the smooth-kernel statement}
\label{sec:regularity}
%=====================================================================

For the fixed observation test define
\[
 D_f=J_\eta^-(\supp f)\cap\MT.
\]
It is compact and retained in the laboratory.

\begin{proposition}[Dependence and fixed-width continuity]
\label{prop:continuity}
The coefficients
\(\Obs^{(j)}_{f,g,\varepsilon}\), \(j=0,1,2\), depend
on the configuration only through its restriction to
every neighbourhood of \(D_f\). The same holds for the
dynamical evaluation, with its fixed free input.

At fixed \(\varepsilon>0\), \(h\mapsto w[h]\) is
continuous linear and \((h,k)\mapsto v[h,k]\) continuous
polynomial in the smooth compact-open topologies.
The smeared observable coefficients are continuous
polynomial functionals, and depend continuously
linearly on \(f\) at a fixed configuration.
\end{proposition}

\begin{proof}
The local coefficient operators do not enlarge source
dependence. Every retarded inverse at a point \(x\)
uses only \(J_\eta^-(x)\cap\MT\). If an intermediate
point lies in that past, its own truncated past lies
there as well. Applying this successively to
\eqref{eq:coordinatesolutions} and
\eqref{eq:scalarsolution}, then to
\eqref{eq:relationalcoeff}, proves dependence on \(D_f\).

The Cauchy estimates in Lemma~\ref{lem:cauchy},
continuity of multiplication and differentiation in
\(C^\infty\), and compact smearing give all stated
continuity assertions. The constants may depend on
\(\varepsilon\); derivatives of the ramp scale as
\(\varepsilon^{-r}\).
\end{proof}

\begin{lemma}[Exact patch agreement]
\label{lem:patch}
Two smooth flat completions agreeing on a neighbourhood
of \(D_f\), with the same ramp, switch there, and the
same relevant initial data, give identical observable
coefficients. Their restrictions transported by
\(\iota\) therefore give a well-defined supported
presentation of those coefficients on the host.
\end{lemma}

\begin{proof}
At each coefficient order subtract the two flat Cauchy
equations. Finite propagation makes their difference
zero on the required retained past, successively for
coordinates and scalar corrections. The relational
formula and the same test then agree.
Transport through the chart simply rewrites that
same compact pairing.
\end{proof}

This is patch agreement, not a gluing theorem around
the worldsheet. Changing initial data or allowing
independent incoming lateral data is not covered.

\subsection{The observation-smeared linear coordinate kernel}

Let \(Lh=w[h]\). For a compact smooth covector test \(a\),
put
\[
 J_\varepsilon(a)[h]=\int a_\mu w^\mu[h]\,\dd^4x.
\]
Define
\[
 (C^*b)^{\rho\sigma}
  =-\partial^{(\rho}b^{\sigma)}
                  +\tfrac12\eta^{\rho\sigma}\partial_\mu b^\mu.
\]

\begin{lemma}[Smooth smeared kernel]
\label{lem:kernel}
With \(b=\EA a\),
\begin{equation}
 J_\varepsilon(a)[h]
      =\int h_{\rho\sigma}q_{a,\varepsilon}^{\rho\sigma}\,\dd^4x,
 \qquad
 q_{a,\varepsilon}=\sigma C^*(\EA a).
 \label{eq:smearedkernel}
\end{equation}
This tensor is smooth, compactly supported on the
retained slab, and extends smoothly by zero across
\(t=0\). Its support is contained in
\(J_\eta^-(\supp a)\cap\{t\ge0\}\).
\end{lemma}

\begin{proof}
Transposition gives
\[
 \int a_\mu w^\mu
   =-\int b_\mu\partial_\alpha(\sigma a_1^{\alpha\mu}[h])
   =\int\sigma(\partial_\alpha b_\mu)a_1^{\alpha\mu}[h].
\]
There is no initial boundary contribution because
\(\sigma\) is flat and zero at \(t=0\); the terminal
one vanishes because \(b\) is terminally zero.
Insertion of \(a_1\) gives \eqref{eq:smearedkernel}.
Advanced smoothness and the flat ramp give the asserted
regularity and support.
\end{proof}

This is the precise smoothness statement in X-A
\cite[{\loc{lem:kernel}}]{XA}.
It does not assert that the unsmeared operator kernel
\(L(x,y)\) is smooth. That kernel contains derivatives
of the wave fundamental solution.

The distinction from the multiplied-equation
prescription is explicit:
\[
 \widetilde w=\Ezero(\sigma Ch)
 \quad\Longrightarrow\quad
 \widetilde L^*a=C^*(\sigma b)
      =\sigma C^*b+\sigma'\mathcal B_\Sigma(b),
 \label{eq:multiplieddifference}
\]
where
\[
 \mathcal B_\Sigma(b)^{\rho\sigma}
  =-n^{(\rho}b^{\sigma)}
          +\tfrac12\eta^{\rho\sigma}n_\mu b^\mu,
 \qquad n_\mu=\partial_\mu t.
\]
The divergence-form prescription has
\(\sigma C^*b\), not the extra term in
\eqref{eq:multiplieddifference}.
The sharp prescription's surface delta is absent here.
It would be inaccurate to insert a \(\sigma'\mathcal
B_\Sigma\) term into \eqref{eq:smearedkernel} merely
because it occurs for the different multiplied equation.
No \(\varepsilon\downarrow0\) limit is taken.

\subsection{First derivatives through second order}

Write
\[
 B_w^\mu=w^\nu\partial_\nu w^\mu-v^\mu,\qquad
 a_\mu=f\partial_\mu\phi_0.
\]
At a fixed smooth configuration, some derivatives are
immediately
\begin{align}
 (\Obs^{(1)})'_h&=-L^*a,
 &
 (\Obs^{(2)})'_k&=-L^*a,\nonumber\\
 (\Obs^{(1)})'_{\phi_0}
   &=\partial_\mu(fw^\mu),
 &
 (\Obs^{(1)})'_{\phi_1}&=f,\nonumber\\
 (\Obs^{(2)})'_{\phi_0}
   &=-\partial_\mu(fB_w^\mu)
            +\tfrac12\partial_\mu\partial_\nu(fw^\mu w^\nu),
 \label{eq:firstderivatives}\\
 (\Obs^{(2)})'_{\phi_1}
   &=\partial_\mu(fw^\mu),
 &
 (\Obs^{(2)})'_{\phi_2}&=f.
 \nonumber
\end{align}

For completeness, if \(H\) is a symmetric metric
direction,
\begin{equation}
\begin{split}
 \Ddir_hv[H]^\mu
   =-\Ezero\partial_\alpha\bigl\{\sigma[
       a_1^{\alpha\beta}[H]\partial_\beta w^\mu
      +a_1^{\alpha\beta}[h]\partial_\beta(LH)^\mu
      +2a_2^{\alpha\mu}[h,H]]\bigr\}.
\end{split}
\label{eq:dv}
\end{equation}
Define the compact covector
\[
 r_\lambda=
  -f\partial_\lambda\phi_1
  +f(\partial_\lambda w^\mu)\partial_\mu\phi_0
  -\partial_\nu(fw^\nu\partial_\lambda\phi_0)
  +fw^\nu\partial_\lambda\partial_\nu\phi_0.
\]
Then the remaining derivative is exactly
\begin{equation}
 \Ddir_h\Obs^{(2)}[H]
       =\pair{r}{LH}-\pair{a}{\Ddir_hv[H]}.
 \label{eq:O2h}
\end{equation}

\begin{proposition}[Smooth first-derivative kernels]
\label{prop:kernels}
Every first derivative of each off-shell smeared
coefficient, at a fixed smooth configuration, has a
smooth compactly supported kernel with support in
\(D_f\). In particular, no initial-surface delta occurs
in its metric derivatives.
\end{proposition}

\begin{proof}
Equations \eqref{eq:firstderivatives} and
Lemma~\ref{lem:kernel} handle all but
\eqref{eq:O2h}. Put \(b=\EA a\).
Transposing \eqref{eq:dv} gives
\begin{align*}
 \pair{a}{\Ddir_hv[H]}
 ={}&\int\sigma\partial_\alpha b_\mu
       \left(a_1^{\alpha\beta}[H]\partial_\beta w^\mu
                      +2a_2^{\alpha\mu}[h,H]\right)\,\dd^4x\\
 &+\pair{j}{LH},\\
 j_\mu={}&-\partial_\beta
        \left(\sigma a_1^{\alpha\beta}[h]
                                     \partial_\alpha b_\mu\right).
\end{align*}
The first line is multiplication of \(H\) by an explicit
smooth tensor coefficient. The source \(j\), extended
by zero to negative times, is smooth and flat at
\(t=0\), and is compactly supported below the terminal
surface. The extended transposition statement and
Lemma~\ref{lem:kernel} apply again to \(\pair{j}{LH}\).
Thus \eqref{eq:O2h} is represented by a smooth compact
tensor.

All constructions use nested advanced pasts of the
original test, so their supports lie in \(D_f\).
Flatness of \(\sigma\) eliminates initial boundary
distributions.
\end{proof}

For the dynamical evaluation, the explicit formulas
\eqref{eq:scalarsolution} define a polynomial extension
to arbitrary smooth \(\phi_0\). Its first derivatives
also have smooth compact representatives: differentiate
the displayed finite compositions, transpose each
\(\Em\), and integrate the local differential
operators by parts. Every dynamical vertex carries
the compact interior factor \(g\); coordinate
transpositions are those already proved. Restriction
to the free solution subspace is therefore legitimate.
This does not assert a unique kernel for variations
parametrized instead by initial Cauchy data.

Smooth first derivatives at fixed configuration do
not prove a joint wavefront condition for every
higher derivative, a quantum admission theorem,
or existence of any singular contraction.

%=====================================================================
\section{Host stability and non-identifications}
\label{sec:fixedhost}
%=====================================================================

\begin{proposition}[Fixed host]
\label{prop:fixedhost}
Replacing the ramp width, a regulator description, an
evaluation-table entry, or a proposed subtraction-scale
label does not evolve \(G\), \(\mathcal W\), or its
geometric time axis.
Such a replacement may change the auxiliary classical
prescription and its values.
\end{proposition}

\begin{proof}
The host data \eqref{eq:host} are fixed before the
analytical prescription. No rule constructed here
returns a new host metric. The changed analytical
coefficient is a different entry in the prescribed
calculation or description register.
\end{proof}

The auxiliary \(q\), its formal coordinate functions,
and any lapse derived from them must not be identified
with the host clock or with a reconstructed spacetime
metric. The original-rod and cosmological lapse
programmes have their own backgrounds and prescriptions
\cite{LapseRecords}.

\begin{proposition}[Stress identities do not impose Einstein dynamics]
\label{prop:noeinstein}
Symmetry, conservation, and zero trace of a tensor on
the prescribed host do not imply Einstein's equation.
\end{proposition}

\begin{proof}
In a regular flat chart choose a nonzero constant null
covector \(\ell\) and \(t_{\mu\nu}=\ell_\mu\ell_\nu\).
It is symmetric, conserved, nonzero, and traceless,
while the local Einstein tensor vanishes.
For nonzero gravitational coupling the proposed
Einstein equation fails.
\end{proof}

The ramp is not the convolution prescription of
X-A-conv, and this theorem does not repair the latter's
gauge defect. X-Enter's conditional relational readout
of a convolution coefficient is not a readout of
\eqref{eq:observable} \cite{ConvolutionRecords}.

The operators and observables here are not VII-B's
sharp insertion, IX-A's adapted carrier element, or
IX-E-2's locked kernel \(K\).
The IX free partial product and its causal-interface
analysis are negative controls only, not constructed
objects of this paper \cite{SharpRecords}.
The VII-C obstruction retains its own original
hypotheses and is not removed.

%=====================================================================
\section{A finite Enter interface for supplied evaluations}
\label{sec:instrument}
%=====================================================================

\subsection{What is encoded}

Fix a finite catalogue of smooth configurations, tests,
and prescription labels. For each catalogue entry
\(i\) and coefficient order \(j=0,1,2\), the classical
number
\[
 a_{ij}=\Obs^{(j)}_{f_i,g_i,\varepsilon_i}
                              [h_i,k_i,\phi_i]
\]
is mathematically defined by the preceding construction.
An installed table stores finitely many rational
approximations, intervals, or exact values that have
been supplied independently.

Neither existence of these numbers nor a finite
lookup circuit is an algorithm for acquiring arbitrary
smooth fields. No universal finite encoding of real
numbers or smooth configurations is assumed.

The catalogue is nonempty without any numerical
execution: take \(m=0\), \(h=k=0\),
\(\phi_0=0\) or \(1\), \(\phi_1=\phi_2=0\), and a
fixed test with integral one. The exact coefficient
triples are respectively \((0,0,0)\) and \((1,0,0)\).
These are evaluations of the classical functional,
not measurements on a continuum quantum field.

\subsection{Complete destructive-clock instrument}

Let one live clock be \(C\cong\CC^{2^q}\), \(q\ge1\),
with an orthonormal calibration basis
\(\{\ket{\tau_k}\}\). Let \(S\) contain all surviving
finite registers, including any finite payload, and set
\[
 D_k=\bra{\tau_k}_C\otimes I_S,\qquad P_k=\proj{\tau_k}.
\]

\begin{theorem}[Clock instrument]
\label{thm:instrument}
The map
\begin{equation}
 \mathcal M_C(X)
  =\sum_k\proj k_R\otimes D_kXD_k^\dagger
  =\sum_k\proj k_R\otimes
        \Tr_C[(P_k\otimes I)X(P_k\otimes I)]
 \label{eq:clock}
\end{equation}
is completely positive and trace preserving. It obeys
\[
 \Tr_R\mathcal M_C(X)=\Tr_CX
\]
on all operators, including every untouched-reference
extension. Its quantum output contains \(S\), not \(C\).
\end{theorem}

\begin{proof}
The recorded Kraus operators are \(\ket k_R\otimes D_k\);
their squared moduli sum to \(I_{CS}\).
Writing \(X=\sum_{k,\ell}\kb{\tau_k}{\tau_\ell}
\otimes X_{k\ell}\) identifies each output block with
\(X_{kk}\). Summation proves the discard identity.
The same calculation includes any reference factor.
\end{proof}

This reproduces the relevant I-B/En09-A interface with
a complete residual, rather than just an outcome law
\cite{IB,En09}.

Choose distinct history labels \(t_k\) and a finite
dictionary \(k\mapsto(i(k),t_k,\text{table report})\).
At a fixed readout event these are programme/history
labels, unless an additional event-time promise holds.

\begin{proposition}[Clock limitations]
\label{prop:clocklimits}
\begin{enumerate}[label=\textup{(\roman*)}]
\item If every occurring \(t_k\) must equal the time of
one fixed event, at most one code has positive
probability.
\item The clock channel has no left inverse on all clock
states.
\item Its record marginal does not determine its residual.
\item No finite matrices satisfy
\([T_C,H_C]=iI_C\).
\end{enumerate}
\end{proposition}

\begin{proof}
Distinctness of the labels proves (i).
The two clock inputs
\((\ket{\tau_0}\pm\ket{\tau_1})/\sqrt2\), with the same
surviving state, have identical outputs, proving (ii).
Inputs \(P_0\otimes\rho\) and \(P_0\otimes\widetilde\rho\)
have the same deterministic record but arbitrary
different residuals, proving (iii).
Taking the trace of the proposed commutator proves (iv).
\end{proof}

Conditioning can change a reference's conditional state;
the discard identity concerns its unconditional marginal.
No thermodynamic reset cost follows merely from this
channel identity.

\subsection{Fixed-table approximation}

For an explicitly supplied finite table
\(v_{ki}\in[-1,1]\), let \(X_{ki}\) be the input blocks
after clock readout and catalogue-basis pinching.
Define
\[
 \mathcal I_v(X)
  =\sum_{k,i,y=\pm1}
       \proj{k,i,y}_R\otimes
                    \frac{1+yv_{ki}}2X_{ki}.
\]

\begin{proposition}[Fixed-interface error bound]
\label{prop:table}
The map \(\mathcal I_v\) is CPTP. On the same complete
interfaces,
\[
 \ddia(\mathcal I_v,\mathcal I_{\widetilde v})
    =\tfrac12\max_{ki}|v_{ki}-\widetilde v_{ki}|,
 \qquad
 \ddia(\Lambda,\Gamma)=\tfrac12\|\Lambda-\Gamma\|_\diamond.
 \label{eq:tableerror}
\]
\end{proposition}

\begin{proof}
The branches have Kraus operators
\[
 \sqrt{\frac{1+yv_{ki}}2}\,
  \ket{k,i,y}(\bra{\tau_k}\bra i)\otimes I_S.
\]
They are complete.
Orthogonality of the output blocks bounds the trace norm
of the difference, including a reference, by
\[
 \max_{ki}|v_{ki}-\widetilde v_{ki}|
                              \sum_{ki}\|X_{ki}\|_1.
\]
Pinching is an average of unitary conjugations and
contracts trace norm. A basis input at an index attaining
the maximum saturates the bound.
\end{proof}

The table can represent scaled approximations of supplied
classical evaluations. Its correctness does not
establish a preparation channel from unknown field data.

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

\subsection{Locality of the complete finite network}

For a finite site set with local matrix algebras,
let each tick be a product of channels on a prescribed
partition into blocks. Record registers are retained
as part of the complete output.
For a final set \(Y\), define its backwards ancestor
sets recursively:
\[
 N_{t,t}(Y)=Y,\qquad
 N_{s,t}(Y)
   =\bigcup\{B:\ B\text{ is a block at tick }s+1,\
                          B\cap N_{s+1,t}(Y)\ne\varnothing\}.
\]

\begin{theorem}[Scheduled locality]
\label{thm:locality}
The complete network satisfies
\[
 (\Phi_{s+1}^*\cdots\Phi_t^*)(\AA(Y))
                       \subseteq\AA(N_{s,t}(Y)).
 \label{eq:scheduledlocality}
\]
An intervention outside the complete backwards
dependency set cannot change the corresponding joint
output law. Untouched references are included by
retaining their joint input reduction.

If every dependency port has a declared host location
and each dependency edge has a certified future-causal
regular route, scheduled dependencies are contained
in the prescribed host causal relation.
\end{theorem}

\begin{proof}
A channel adjoint is unital. On a block disjoint from
the current support, the pulled-back observable is the
identity; on a meeting block support enlarges only
inside that block. Reverse induction proves
\eqref{eq:scheduledlocality}. Testing every joint output
observable proves the intervention statement.
Reference factors are carried unchanged through the
same argument.

For the last assertion, concatenate the supplied causal
routes for the dependency edges.
\end{proof}

Scheduled ancestry is an upper bound, not an
identification with a whole physical causal past.
The separation is consistent with VII-J-ST and En08-A,
but no schedule from those manuscripts is imported
\cite{VIIJST,En08A}.

\subsection{Target-adapted finite route}

Fix a straight developed spatial segment of length
\(\ell>0\) in a regular convex tube. All processing
and transport events below are chosen inside the
retained regular laboratory.

After clock extraction and local lookup, allow a positive
buffer before departure at developed time \(t_{\mathrm d}\).
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 registers \(A_j\cong\CC^{2^p}\),
\(p\ge1\). Put
\[
 a_{\mathrm{hop}}=\ell/n,\qquad
 \Delta t=a_{\mathrm{hop}}.
\]
Initialize \(A_1,\ldots,A_n\) in a fixed blank
\(\omega\), independently of \(A_0\) and all references.
At tick \(j\), exchange \(A_{j-1}\) and \(A_j\).

This route adopts an explicit ideal permission:
a finite two-register SWAP is available on the declared
tick-boundary ports when their dependencies have these
regular causal paths. Each cross-port dependency is
null, and every same-station dependency is timelike.
The permission is an operational input, not a derived
continuous-time Hamiltonian implementation.

The finite classical report is forwarded on separately
declared causal classical wires. Copying applies only
to its diagonal classical algebra.

\begin{theorem}[Reference-preserving target-adapted transport]
\label{thm:transport}
For every initial payload--reference state
\(\rho_{A_0E}\), after tick \(j\) the state is
\[
 \rho_{A_jE}\otimes
          \bigotimes_{\substack{0\le i\le n\\i\ne j}}\omega_{A_i}
\]
under the fixed register identifications.
Thus the complete logical payload channel is exactly
the identity, including its arbitrary reference.
The report is forwarded without changing its joint law.
Arrival is at developed time \(t_{\mathrm d}+\ell\),
or host time \(T_{\mathrm d}+\ell/c\).
All dependency routes stay in the regular part.
\end{theorem}

\begin{proof}
Initially the assertion is the preparation.
At hop \(j\), a SWAP moves the entire current payload
factor, including its correlations with \(E\), into the
next independent blank and leaves the previous factor
blank. Induction proves the formula.
The final statement follows at \(j=n\).

A classical copy preserves diagonal records and their
joint correlations; it is not cloning an arbitrary
unknown quantum state. The chosen spacings and tick
durations make every cross dependency null in the
developed chart and therefore causal in the host.
\end{proof}

The reference \(E\) may include the complete retained
clock record and inaccessible spectators.
The construction is a transport of finite registers,
not a propagation theorem for \(\rho_g\) or \(\Co_q\).
It is the target-adapted finite interface of I-B,
not a claim of exact arbitrary-null routing on one
fixed lattice \cite{IB,En08B}.

\subsection{Resources and implementation allowances}

For a \(q\)-qubit clock, \(r\) catalogue bits,
\(b\) report bits, \(w_{\mathrm{lookup}}\) work qubits,
and \(p\)-qubit payload registers, a conservative
preallocated reserve is
\[
 q+r+(n+1)p+w_{\mathrm{lookup}}+(n+1)b+1.
 \label{eq:resource}
\]
The last qubit can implement the optional binary table
probe. There are \(n\) payload-register exchanges,
\(n\) report transmissions, and the declared finite
local lookup/readout operations.
The external reference and any external table storage
are separately counted. Processing, readout, and the
buffer occur before departure and are included in the
experiment's finite duration.

For compatible complete channels, factorwise replacement
and diamond-norm contraction give
\begin{equation}
 \ddia(\widetilde\Gamma,\Gamma)
 \le
 \min\left\{1,\epsilon_{\mathrm{enc}}+
     \epsilon_{\mathrm{dec}}+\epsilon_{\mathrm{table}}
                         +\sum_j\epsilon_j\right\}.
 \label{eq:telescoping}
\end{equation}
Indeed, subtract the two compositions, replace one
factor at a time, and use the triangle inequality.
An event-probability bound follows only for an event
in the compared complete output, including its arrival
or failure flag if such a flag is claimed.
This is finite channel accounting \cite{Watrous},
not a field approximation or a hardware synthesis
certificate.

%=====================================================================
\section{Release and successor handoff}
\label{sec:release}
%=====================================================================

\begin{theorem}[X-A-ST local Grade S release]
\label{thm:release}
For the prescribed conical host, supported completion
presentation, fixed switch and ramp, fixed Cauchy data,
and plateau algebra specified before the calculation:
\begin{enumerate}[label=\textup{(\roman*)}]
\item the coordinate and scalar coefficient problems are
uniquely constructed through \(\kappa^2\);
\item preflight is exact on the frozen algebra, while the
excluded edge class has the explicit defect
\eqref{eq:edge} and the coefficientwise obstruction
of Corollary~\ref{cor:edgeorder};
\item the full coordinate intertwiner and off-shell
relational covariance hold through \(\kappa^2\);
the switched scalar solution intertwines with unchanged
free input;
\item the varied-action identity displays all terms
proportional to \(1-g\) or \(\partial g\), and its
auxiliary coordinate counterpart displays the
corresponding ramp terms;
\item the mandatory pure-gauge, sourced, and zero-switch
diagnostics have the stated exact values and scopes;
\item the coefficients satisfy the retained-past support
bound and fixed-\(\varepsilon\) continuity; their first
derivative kernels at a smooth configuration have the
specified smooth compact representatives;
\item the finite encoding has a complete destructive-clock
instrument, scheduled locality, a target-adapted
reference-preserving regular route, and a finite
resource/error ledger.
\end{enumerate}
These statements form local Grade S, not Grade E.

No intrinsic conical Green operator, time-ordered
product \(\mathsf T_2\), normalized retarded product
\(R_1\), quantum Ward identity, boundary algebra, or
sharp-regulator limit is constructed.
\end{theorem}

\begin{proof}
The host and completion are fixed in
Section~\ref{sec:host}.
Propositions~\ref{prop:coordeq} and
\ref{prop:scalarconstruction} prove (i).
Section~\ref{sec:preflight} proves (ii).
Theorem~\ref{thm:main} proves (iii).
Section~\ref{sec:routeII} proves (iv).
Section~\ref{sec:diagnostics} proves (v).
Proposition~\ref{prop:continuity},
Lemma~\ref{lem:patch}, Lemma~\ref{lem:kernel},
and Proposition~\ref{prop:kernels} prove (vi).
The finite channel and transport sections prove (vii),
including every clause of Definition~\ref{def:grade}.

All field-theoretic operations used are classical
smooth coefficient constructions on the completion.
The excluded quantum objects have not been defined
by any of those operations.
\end{proof}

\begin{center}
\small
\renewcommand{\arraystretch}{1.12}
\begin{tabularx}{\linewidth}{@{}p{.37\linewidth}XX@{}}
\toprule
Object & Verdict & Boundary\\
\midrule
Fixed-width classical covariance &
Constructed &
Plateau group fixed in advance\\
Edge-zone extension of preflight to all compact gauges &
Disproved &
Explicit witness, not every edge parameter\\
Supported host presentation &
Constructed &
Declared flat completion; no global conical inverse\\
Smooth first-derivative kernels &
Constructed &
Smeared and at fixed configuration\\
Finite Enter encoding and route &
Constructed under declared permissions &
No field acquisition or microscopic gate derivation\\
Sharp-width limit &
Not taken &
No uniform removal theorem\\
\(\mathsf T_2\), \(R_1\), quantum Ward identity &
Not constructed here &
Separate analytical programme\\
\bottomrule
\end{tabularx}
\end{center}

\subsection{Handoff to a possible X-B-ST}

A successor receives:
\begin{enumerate}
\item the exact off-shell and dynamically evaluated
coefficients \eqref{eq:relationalcoeff};
\item the smooth configuration space, fixed free-input
space, supported tests, and completion contract;
\item the support and fixed-width continuity statements;
\item the full classical gauge identity, together with
the retained edge and off-plateau formulae;
\item the single first-order switched scalar vertex
\begin{equation}
 \mathcal V_{1,g}(h;\phi)
   =\int g\,\mathscr L^{(1)}[h,\phi]\,\dd^4x
   =-\frac12\int g\,h_{\mu\nu}T_0^{\mu\nu}[\phi]\,\dd^4x,
 \label{eq:vertex}
\end{equation}
whose scalar Euler derivative is
\[
 \frac{\delta\mathcal V_{1,g}}{\delta\phi}
                         =-\mathsf V_{1,g}[h]\phi.
\]
\end{enumerate}
It does not receive a constructed \(R_1\), an admitted
quantum metric insertion, or a selected singular product.

The existing X-B source record concerns a separately
specified scalar-only fixed-regulator construction
\cite{XB}. Its results are neither denied nor imported
as achievements of X-A-ST.

\subsection{Standing analytical boundaries}

No quantum coefficient of the inherited characteristic
graph is admitted by this release. XV-B is not opened
as a downstream quantum-admission exit, XVI-N is not
discharged, and O.1 remains \loc{NOT_CLOSED} in the
inherited scope.

This paper passes neither the historical H2 requirement
nor the full I3/J2 characteristic-map gates.
That sentence does not retract XIV-I's narrower
constructed internal-amplitude result.
The version-specific records remain separate
\cite{BoundaryRecords}.

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

%=====================================================================
\appendix
\section{Direct quadratic cancellation}
\label{app:quadratic}
%=====================================================================

This calculation makes the quadratic cancellation
independent of any notation for an inverse map.

Let \(B_w=w\cdot\partial w-v\). Equations
\eqref{eq:coordinatelaws} imply
\[
 B_w'
   =B_w+w\cdot\partial\xi+
                    \tfrac12\xi\cdot\partial\xi-\zeta.
\]
The order-one difference of
\(\phi_1-w\cdot\partial\phi_0\) vanishes.

At order two, insert
\eqref{eq:scalarlaw} and \(w'=w+\xi\) into
\eqref{eq:relationalcoeff}.
The coefficient of \(\partial_\mu\phi_0\) left after
subtracting the unprimed expression is
\[
 \zeta^\mu+\tfrac12(\xi\cdot\partial\xi)^\mu
 -(w\cdot\partial\xi)^\mu-(\xi\cdot\partial\xi)^\mu
 +(w\cdot\partial\xi)^\mu
 +\tfrac12(\xi\cdot\partial\xi)^\mu-\zeta^\mu=0.
\]
The terms with derivatives of \(\phi_1\) are
\(X_\xi\phi_1-\xi\cdot\partial\phi_1=0\).
The second-derivative terms in \(\phi_0\) are
\[
 \tfrac12\xi^\mu\xi^\nu
 -w^\mu\xi^\nu-\xi^\mu\xi^\nu
 +w^\mu\xi^\nu+\tfrac12\xi^\mu\xi^\nu,
\]
contracted with the symmetric Hessian; they vanish.
This proves the coefficient cancellation using the
complete quadratic laws.

%=====================================================================
\section{Debugging and scope audit}
\label{app:debug}
%=====================================================================

\begin{longtable}{@{}p{.34\linewidth}p{.61\linewidth}@{}}
\toprule
Potential error & Correction in this manuscript\\
\midrule
\endfirsthead
\toprule
Potential error & Correction in this manuscript\\
\midrule
\endhead

Treating the regular cone as globally hyperbolic &
A noncompact diamond is exhibited. All Cauchy equations
are on the separately declared flat completion.\\[4pt]

Calling the defect a worldline &
The added \(z\)-direction makes the defect a worldsheet.\\[4pt]

Replacing the parent signature silently &
The host has signature \((-,+,+,+)\);
the analytical completion retains \((+,-,-,-)\)
and the positive Duhamel convention.\\[4pt]

Calling a plateau a compact open set &
The plateau is open and relatively compact, with its
closure inside the supported post-ramp switch plateau.\\[4pt]

Replacing the switched action by a multiplied equation &
The derivative-of-\(g\) terms are explicitly retained.\\[4pt]

Replacing the coordinate ramp by convolution &
The ramp multiplies the density deviation inside a
divergence. No convolution operator is used.\\[4pt]

Dropping the ramp derivative &
The term \(-\sigma'a_1^{0\mu}[h]\) is present in the
first coordinate equation.\\[4pt]

Using only \(h'=h+K\xi\) in a quadratic proof &
The transformation of \(k\), including
\(\tfrac12\Lie_\xi^2\eta\), is retained.\\[4pt]

Reading preflight as unrestricted covariance &
The edge witness disproves that extension. The frozen
plateau group was selected before the calculation.\\[4pt]

Claiming every ramp-supported gauge fails &
Only a specified nonzero witness and the failure of
a universal extension are proved.\\[4pt]

Identifying action symmetry with observable invariance &
The auxiliary coordinate identity, Cauchy uniqueness,
formal inverse, and fixed label test are also required.\\[4pt]

Omitting the scalar response &
The dynamical coefficient contains
\(\rho_{1,g}\) and \(\rho_{2,g}\), not dressing alone.\\[4pt]

Calling the unsmeared coordinate kernel smooth &
Smoothness is proved after observation smearing.
The wave operator's unsmeared kernel is distributional.\\[4pt]

Adding \(\sigma'\mathcal B_\Sigma\) to the divergence-form
smeared kernel &
That term belongs to the different multiplied-equation
prescription. The actual kernel is
\(\sigma C^*(\EA a)\).\\[4pt]

Inferring a quantum product from smooth first derivatives &
No joint quantum admission, contraction, or extension
theorem is proved.\\[4pt]

Keeping a nonempty original plateau when \(g=0\) &
The zero-switch case is a diagnostic specialization;
its plateau algebra is trivial.\\[4pt]

Removing a classical first-order defect by a later term &
Formal powers of \(\kappa\) and \(\hbar\) keep the
nonzero lower-order coefficient unchanged.\\[4pt]

Treating a completion change outside a chart as harmless
for every inverse &
Only agreement near the full retained causal past,
with the same data, gives the proved patch agreement.\\[4pt]

Treating a table as a field detector &
It stores supplied evaluations; no continuum-field
acquisition map is constructed.\\[4pt]

Identifying all history codes with one actual event time &
Exact event-time matching at one event is deterministic.
Multi-outcome readout uses a separate history dictionary.\\[4pt]

Equating scheduled reachability and a physical causal past &
Physical containment is proved only from separately
certified dependency routes.\\[4pt]

Treating SWAP endpoint algebra as microscopic dynamics &
The two-register tick permission is explicit.
No continuous-time implementation Hamiltonian is derived.\\[4pt]

Declaring every quantum construction impossible &
The release states what is not constructed here.
It does not prove universal quantum nonexistence.\\

\bottomrule
\end{longtable}

%=====================================================================
\section{Documentary comparisons and routing}
\label{app:comparisons}
%=====================================================================

The following comparisons organize scope; none replaces
a proof above.

\begin{itemize}
\item III-B's free-shell tangency and coupled-jet quotient,
VI-B's switched Einstein--scalar defect, VII-A-ST's
unswitched harmonic-coordinate intertwiner, and the
present ramp prescription concern different equation
spaces \cite{IIIB,VIB,VIIAST}.
No coupled Ward pair or metric equation is imported.
\item VIII-ST's fixed-host interpretation and VII-J-ST's
finite causal interfaces are comparisons for the
description-register and network distinctions, not
analytical premises \cite{VIIIandVIIJ}.
\item Em02-E and the XX records fix their own relational
lapse constructions. Neither defines this ramp or
makes its time argument a derived clock
\cite{LapseRecords}.
\item IX-A-ST through IX-D-ST do not supply a quantum
product for the present classical prescription.
The source-locked sharp kernels and numerical
residues are not identified with
\eqref{eq:smearedkernel} \cite{SharpRecords}.
\item En10-A's source-weight estimation is a different
finite statistical access contract. It does not
acquire \(\Obs_{f,g,\varepsilon}\)
\cite{En10}.
\end{itemize}

\begin{remark}[OpenAI comparisons are not premises]
\label{rem:not-premise}
No OpenAI catalogue proof is a premise, and no catalogue
family is independently verified here.

Oai283-A is only a comparison for a complete-channel
error ledger. Oai272-A, Oai276-A, Oai119-A, and Oai279-A
do not identify a field preparation, prove an all-orders
completion, establish gravity-generated entanglement,
or exclude classical gravity.
Oai212-A and Oai365-A do not derive this host metric.
Families 265, 271, 362, and 376 supply no premise for
this manuscript and are not invoked as Page curves,
lapse theorems, or Einstein dynamics.
\end{remark}

A proposed routing record, not an assertion of registry
installation, is:
\begin{verbatim}
identifier: X-A-ST
version: 1
authorField: empty
localGrade: S
host: prescribed I-B conical product
presentation: supported chart with declared flat completion
timeLandmarks: 1
emergenceSource: false
nearUv: false
classicalOrder: kappa^2
regulator: fixed positive time-ramp width
classicalCovariance: Constructed on frozen plateau algebra
edgeExtension: Disproved by explicit excluded witness
quantumProduct: Not constructed
quantumCertificate: RG-open
compilationAsserted: false
\end{verbatim}

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

\paragraph{Authorship.}
The author field is intentionally empty. Bibliographic
attribution to N.~Cheung does not assign authorship of
this adaptation. Authorized submission metadata must
be supplied for a signed submission.

\paragraph{Novelty and impact.}
The parent covariance, preflight, and edge findings are
credited to X-A. The present contribution is their
supported completion-based transfer, explicit signature
and source-domain bookkeeping, and a complete finite
interface with narrowly stated interpretation.
Its prospective usefulness is a controlled classical
baseline, not an established quantum-gravity prediction.

\paragraph{Bibliographic status.}
Internal manuscripts are cited by identifier and version
without invented publication dates or DOIs. Documentary
comparisons do not turn companion claims into premises.
The journal Enter article is listed separately from
the companion manuscripts.

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

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

\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, manuscript record.
\url{https://enter.grok.me/browse/X-A}.
Especially \loc{def:Aeps}, \loc{def:coordinates},
\loc{prop:preflight}, \loc{prop:edge},
\loc{thm:coordinate}, \loc{prop:scalar},
\loc{prop:varied}, \loc{cor:noether},
\loc{prop:puregauge}, \loc{prop:sourced},
\loc{prop:zeroswitch}, \loc{prop:continuity},
\loc{prop:kernels}, and \loc{lem:kernel}.

\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, manuscript record.
\url{https://enter.grok.me/browse/I-B}.
Host, signature, destructive instrument, calibration,
locality, and target-adapted transport.

\bibitem{IVB}
N.~Cheung,
\emph{IV-B}.
\loc{IV-B-v1.tex}, version 1.
Supplied manuscript description and excerpts,
Enter Calculus Companion.
\url{https://enter.grok.me/browse/IV-B}.
The declared completion and no-global-transfer pattern
are restated and proved in the present presentation;
no inaccessible theorem is assumed.

\bibitem{CheungEnter}
N.~Cheung,
``The Enter Calculus: A Linear Quantum Lambda Calculus
with an Operational Characterization of Measurement,''
\emph{Frontiers in Physics} \textbf{14} (2026).
DOI: \loc{10.3389/fphy.2026.1954943}.
Accepted-article bibliographic record.
\url{https://doi.org/10.3389/fphy.2026.1954943}.
Operational background only.

\bibitem{I}
N.~Cheung,
\emph{Manuscript I}.
\loc{I.tex}, version 2.
Enter Calculus Companion.
\url{https://enter.grok.me/browse/I}.
Operational characterization in its explicitly stated
strategy class; not a premise of the classical proof.

\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, not a theorem premise.
\url{https://enter.grok.me/enter}.

\bibitem{BFR}
R.~Brunetti, K.~Fredenhagen, and K.~Rejzner,
``Quantum Gravity from the Point of View of Locally
Covariant Quantum Field Theory,''
\emph{Communications in Mathematical Physics}
\textbf{345} (2016), 741--779.
DOI: \loc{10.1007/s00220-016-2676-x}.
Relational-observable background; not a covariance or
quantization theorem for this switched ramp prescription.

\bibitem{FrobLima}
M.~B.~Fr\"ob and W.~C.~C.~Lima,
\emph{Propagators for gauge-invariant observables in cosmology}.
\url{https://arxiv.org/abs/1711.08470}.
Comparison for field-dependent causal coordinates;
no propagator construction is transferred here.

\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}.
Cauchy and Green-operator theory under its stated
geometric hypotheses.

\bibitem{VIB}
N.~Cheung,
\emph{VI-B}.
\loc{VI-B-v1.tex}, version 1.
Enter Calculus Companion.
\url{https://enter.grok.me/browse/VI-B}.
Different switched Einstein--scalar defect and
coefficientwise order obstruction; comparison only.

\bibitem{VIIAST}
N.~Cheung,
\emph{VII-A-ST}.
\loc{VII-A-ST.tex}, version 1.
Enter Calculus Companion.
\url{https://enter.grok.me/browse/VII-A-ST}.
Plateau, switch-derivative, support, and comparison-map
ledger on its own classical equation space.

\bibitem{IIIB}
N.~Cheung,
\emph{III-B}.
\loc{III-B-v1.tex}, version 1.
Enter Calculus Companion.
\url{https://enter.grok.me/browse/III-B}.
Free-shell tangency and coupled-jet descent remain
separate from the present prescription.

\bibitem{VIIIandVIIJ}
N.~Cheung,
\emph{VIII-ST} and \emph{VII-J-ST}.
Version 1 manuscript records,
Enter Calculus Companion.
\url{https://enter.grok.me/browse/VIII-ST};
\url{https://enter.grok.me/browse/VII-J-ST}.
Fixed-host and finite-interface comparisons only.

\bibitem{VIIJST}
N.~Cheung,
\emph{VII-J-ST}.
Version 1, Enter Calculus Companion.
\url{https://enter.grok.me/browse/VII-J-ST}.
Complete scheduled ancestry and separately certified
physical containment.

\bibitem{En08A}
N.~Cheung,
\emph{En08-A}.
Version 1, Enter Calculus Companion.
\url{https://enter.grok.me/browse/En08-A}.
Scheduler consistency does not by itself determine
a Lorentz cone.

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

\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{En10}
N.~Cheung,
\emph{En10-A}.
Version 1, Enter Calculus Companion.
\url{https://enter.grok.me/browse/En10-A}.
Source-weight acquisition for a separately declared
finite statistical model.

\bibitem{LapseRecords}
N.~Cheung,
\emph{Em02-E}, version 1, and the supplied
\emph{XX-A--XX-F} manuscript records.
Enter Calculus Companion.
\url{https://enter.grok.me/browse/Em02-E};
\url{https://enter.grok.me/browse/XX-A}.
Background and lapse prescriptions are independent
of the present ramp-coordinate problem.

\bibitem{ConvolutionRecords}
N.~Cheung,
\emph{X-A-conv} and \emph{X-Enter}.
Version 1, Enter Calculus Companion.
\url{https://enter.grok.me/browse/X-A-conv};
\url{https://enter.grok.me/browse/X-Enter}.
Different convolution prescription and conditional
relational-readout contract.

\bibitem{SharpRecords}
N.~Cheung,
\emph{VII-B}, \emph{VII-C},
\emph{IX-A-ST}, \emph{IX-B-ST},
\emph{IX-C-ST}, \emph{IX-D-ST}, and \emph{IX-E-2}.
Version-qualified supplied manuscript records,
Enter Calculus Companion.
\url{https://enter.grok.me/browse}.
Sharp admission, numerical calibration, free products,
and causal-interface boundaries are documentary
comparisons, not constructions of this paper.

\bibitem{XB}
N.~Cheung,
\emph{X-B}.
\loc{X-B-v3.tex}, version 3.
Enter Calculus Companion.
\url{https://enter.grok.me/browse/X-B}.
Separately specified scalar-only fixed-regulator
quantum response; not imported into X-A-ST.

\bibitem{BoundaryRecords}
N.~Cheung,
\emph{XIV-H}, \emph{XIV-I}, and \emph{XIV-J},
version 1; technical \emph{XV-B}, version 2;
\emph{XVI-N}, revision r12; and
\emph{XVI-O}, revision o70-sync.
Supplied manuscript records,
Enter Calculus Companion.
\url{https://enter.grok.me/browse}.
Version-specific analytical boundaries and downstream
admission limits only.

\bibitem{Watrous}
J.~Watrous,
\emph{The Theory of Quantum Information}.
Cambridge University Press, 2018.
Author's manuscript and errata:
\url{https://cs.uwaterloo.ca/~watrous/TQI/}.
Finite channels, instruments, and completely bounded
trace-norm comparisons.

\end{thebibliography}

\end{document}