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% Substantive corrections incorporated:
% 1. A constant quadratic form on an affine carrier is flat; nonzero
%    curvature requires additional spatially varying geometric data.
% 2. A conformal class does not determine the Einstein tensor.
% 3. Linearized matching requires an extracted perturbation or metric
%    jet, not merely an extracted cone.
% 4. The comparison uses the linearized Einstein tensor, with an
%    explicitly declared coupling, rather than an unspecified curvature.
% 5. Stress-energy interpretation requires more than assigning numerical
%    functions to process matrices or signalling strengths.
% 6. Conservation is necessary for a field equation, but is not a
%    sufficient stress-energy identification or matching theorem.
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% 8. A failed selected matching, infeasibility in a fixed ansatz, and
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\hypersetup{
  pdftitle={XXII-E: Dynamics matching on an extracted Lorentz cone},
  pdfauthor={Ngo Cheung},
  pdfsubject={Stage 4 of a limited kinematic emergence programme}
}

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\title{\textbf{XXII-E: Dynamics matching\\
on an extracted Lorentz cone}\\[4pt]
\large Stage 4 of a limited kinematic emergence programme}
\author{Ngo Cheung}
\date{}

\begin{document}
\maketitle

\begin{abstract}
This note specifies a dynamics-comparison layer for a prospective
operational Lorentz-cone extraction and its separately justified
transport realization. A single affine cone determines a conformal
class of constant quadratic forms, not a curved metric field:
its translation-invariant representative is flat. Moreover, even
a full pointwise cone field does not determine the Einstein tensor,
because position-dependent conformal rescaling preserves the cones
while changing curvature. We prove these obstructions and formulate
the additional metric-jet and source interfaces needed for a
linearized Einstein comparison. An explicit finite-dimensional
comparator, gauge identity, conservation test, matching residual,
ansatz-feasibility criterion, and error certificate are established.
Unrestricted matching at one point is shown to be algebraically
manufacturable and therefore insufficient as an emergence claim.
Process data alone also do not identify an energy observable, and
transport convergence alone does not control curvature. No
operational stress-energy rule, successful metric-jet extraction,
or countermodel satisfying a complete extraction--transport--dynamics
package is constructed. The release is a proved collection of
scoped obstructions and conditional comparison tools, not an
Einstein-dynamics or holographic reconstruction theorem.
\end{abstract}

\section{Precise claim and logical status}
\label{sec:purpose}

XXII-A specifies the operational Lorentz-cone selection target and
its faithful-generation and non-circularity requirements
\cite[\loc{target:selection}, \loc{def:representation}]
{CheungXXIIA}.
XXII-B distinguishes operational data, additional assumptions,
derived properties, and open obligations
\cite{CheungXXIIB}.
XXII-C fixes a tripartite qubit process-matrix family and its
component-order extraction
\cite{CheungXXIIC}.
XXII-D separates cone extraction from compatible quantum transport
and physical calibration
\cite{CheungXXIID}.

The present claim under construction is not
\[
 \text{extracted Lorentz cone}
 \quad\Longrightarrow\quad
 \text{Einstein equation}.
\]
Nor does successful reference-preserving transport supply that
implication.

The corrected objective is:
\begin{quote}
On a named nonempty admissible operational domain, first obtain the
required extraction and compatible transport realization. Then
construct an independently specified geometric perturbation or
metric-jet rule and an operationally justified source rule.
For a predeclared linearized Einstein comparison, either prove
their matching on that domain or provide a correctly scoped
mismatch or infeasibility witness.
\end{quote}

The existence of such an operational package is not proved here.
In particular, the alternatives ``matching or obstruction'' are
not an automatic completed deliverable: a construction can remain
unresolved.

\paragraph{A necessary correction to the proposed curvature claim.}
The expression ``curvature of the extracted conformal quadratic
form'' has two different possible meanings.
A constant quadratic form translated across its affine carrier has
zero curvature.
If instead a varying representative of a local conformal class is
intended, its derivative data and conformal factor must be supplied.
The cone does not determine those data.

Consequently, the dynamics layer has a missing geometric interface
in addition to the missing source interface.
This remains true even if XXII-A--D were successfully instantiated.

\paragraph{Source discipline.}
The Cheung manuscripts are identified by supplied filename, version
where available, and internal labels.
Documentary scope records are not mathematical proof-transfer
premises.
The calculations printed here prove their own stated claims.
They do not independently verify predecessor manuscripts.

The process-matrix and complete-channel frameworks are those of
\cite{OCB,Watrous}.
The Einstein operator used below is a declared classical comparison
operator, with conventions stated explicitly; standard treatments
include \cite{Carroll,Tong}.
Using that operator does not derive its dynamical status from
operational data.

\section{Conditional operational input}
\label{sec:input}

\subsection{The frozen class and its limitations}

The XXII-C class is
\[
 \ProcessClass=(\ProcessSet)^{\N}.
\]
Each member \(W_e\in\ProcessSet\) has three parties
\[
 \mathsf J=\{A,B,C\},
 \qquad
 H_{a_I}=H_{a_O}=\C^2.
\]
With the transposed unnormalized Choi convention, validity means
\[
 W_e\geq0,\qquad
 \Tr[W_e(M_A\otimes M_B\otimes M_C)]=1
\]
for all positive channel Choi matrices satisfying
\(\Tr_{a_O}M_a=I_{a_I}\).
The access \(\Access\) consists of the separately tested, one-use
local instruments and unconditional classical receiver records
declared in XXII-C
\cite[\loc{def:process}, \loc{spec:access}, \loc{def:class}]
{CheungXXIIC}.

The direct influence relation has no cross-experiment edges.
Its strongly connected component quotient, ordered by reachability,
is \((E_D,\preceq_D)\).
Every component has at most three elements in its reflexive future,
and every chain has at most three distinct components
\cite[\loc{thm:order}]{CheungXXIIC}
\cite[\loc{lem:finite-futures}]{CheungXXIID}.

An experiment identifier is not a spacetime coordinate.
The data contain no derivative operation with respect to those
identifiers, no Hamiltonian assignment, and no continuum metric field.
The absence of cross-experiment tests is a declaration of access,
not a geometric separation measurement.

\subsection{Conditional extraction and transport}

\begin{specification}[Conditional kinematic input]
\label{spec:kinematic}
Fix \(m\geq3\).
For a datum \(D\), suppose a successful predeclared extraction has
provided
\[
 (V_D,\iota_D,K_D,T_D),
 \qquad
 T_D:V_D\longrightarrow\R^m
\]
with \(T_D\) invertible, \(K_D\) proper, and
\begin{align}
 p\preceq_D q
 &\quad\Longleftrightarrow\quad
 \iota_D(q)-\iota_D(p)\in K_D,
 \label{eq:faithful}\\
 V_D
 &=\Span\{\iota_D(q)-\iota_D(p):p,q\in E_D\},
 \label{eq:span}\\
 K_D
 &=\overline{\Cone
 \{\iota_D(q)-\iota_D(p):
                   p\preceq_D q,\ p\ne q\}},
 \label{eq:generated}\\
 T_D(K_D)&=\Lor_m
 =
 \{(t,x):t\geq\|x\|_2\}.
 \label{eq:lorentz}
\end{align}
Suppose also that a separately justified realization
\(\Realization\) meets the compatible transport contract of XXII-D
on a declared nonempty target family.

If a revised carrier or access model is used, its maps,
faithfulness, composition validity, and compatibility with
\eqref{eq:generated} must be stated.
No nonempty joint domain satisfying these conditions is exhibited
in this note.
\end{specification}

In the coordinates of \(T_D\), choose the representative
\[
 \eta_{\mu\nu}=\operatorname{diag}(1,-1,\ldots,-1),
 \qquad 0\leq\mu,\nu\leq m-1.
\]
Equivalently, on \(V_D\), the bilinear representative is
\[
 \eta_D(u,v)
 =
 (T_Du)_0(T_Dv)_0
 -
 \sum_{i=1}^{m-1}(T_Du)_i(T_Dv)_i.
\]
The cone determines its positive constant rescalings as one
conformal class, not a preferred member, observer, or physical
system of units
\cite[\loc{rem:conformal}]{CheungXXIIA}
\cite[\loc{prop:calibration}]{CheungXXIID}.

The value \(m=4\) is not derived by this convention.
All calculations below hold for the declared dimension \(m\geq3\).

\section{What the affine cone cannot supply}
\label{sec:geometry}

\subsection{The translation-invariant representative is flat}

The vector space \(V_D\) has its ordinary smooth affine structure
as a mathematical object. Using it as an auxiliary comparison
carrier does not identify it with an operational continuum spacetime.

\begin{theorem}[Flatness of the constant affine representative]
\label{thm:flat}
Translate any constant representative \(c\eta_D\), \(c>0\), across
the affine carrier. Its Levi-Civita connection coefficients vanish
in affine coordinates, and its Riemann, Ricci, and Einstein tensors
are zero.

Likewise, a one-parameter family of spatially constant representatives
has zero curvature at every parameter value. Its first metric
variation has zero linearized Einstein tensor.

Thus a matching of this unchanged constant geometry to a nonzero
source cannot be repaired by choosing another positive constant
normalization.
\end{theorem}

\begin{proof}
For constant metric components,
\[
 \Gamma^\rho_{\mu\nu}
 =
 \frac12 g^{\rho\sigma}
 \bigl(
 \partial_\mu g_{\sigma\nu}
 +\partial_\nu g_{\sigma\mu}
 -\partial_\sigma g_{\mu\nu}
 \bigr)=0.
\]
The curvature constructed from derivatives and products of these
coefficients is zero.
Constant rescaling leaves the components constant.

In a parameter family \(g_\lambda\) of constant forms,
all spacetime derivatives remain zero for every \(\lambda\).
The variation \(h=\left.\partial_\lambda g_\lambda\right|_0\)
is also spatially constant, so every second-derivative term in the
linearized Einstein operator vanishes.
A parameter derivative is not a spacetime derivative.
\end{proof}

\begin{corollary}[Scoped flat-branch obstruction]
\label{cor:flat-source}
Suppose a proposed dynamics package keeps only the constant affine
geometry of Theorem~\ref{thm:flat}, uses a fixed nonzero comparison
coupling \(\kappa_{\mathrm{cmp}}\), and produces a nonzero source
\(\tau_D\) on its comparison domain.
Then it cannot satisfy
\[
 G^{(1)}[h_D]=\kappa_{\mathrm{cmp}}\tau_D
\]
with \(h_D\) restricted to spatially constant variations.
\end{corollary}

\begin{proof}
The left side is zero by Theorem~\ref{thm:flat}, whereas the right
side is nonzero.
\end{proof}

This is an obstruction to the stated flat branch.
It is not a proof that every operationally justified metric extension
or future dynamics rule is impossible.

\subsection{Even a local cone field does not determine Einstein curvature}

A local geometric extension requires additional structure.
The following calculation takes place on a declared auxiliary open
set \(U\subset\R^m\); it is not an extracted operational spacetime.

Use the curvature convention
\[
 R^\rho{}_{\sigma\mu\nu}
 =
 \partial_\mu\Gamma^\rho_{\nu\sigma}
 -\partial_\nu\Gamma^\rho_{\mu\sigma}
 +\Gamma^\rho_{\mu\alpha}\Gamma^\alpha_{\nu\sigma}
 -\Gamma^\rho_{\nu\alpha}\Gamma^\alpha_{\mu\sigma},
 \qquad
 R_{\sigma\nu}=R^\rho{}_{\sigma\rho\nu}.
 \label{eq:curvature-convention}
\]

\begin{proposition}[Same cone field, different linearized Einstein tensor]
\label{prop:conformal}
For every smooth real \(f\), the metrics
\[
 g_\lambda=e^{2\lambda f}\eta
\]
have exactly the same pointwise future cones as \(\eta\).
Their first variation is \(h_{\mu\nu}=2f\eta_{\mu\nu}\), and
\[
 G^{(1)}_{\mu\nu}[2f\eta]
 =
 (m-2)\bigl(
 \eta_{\mu\nu}\Box f-\partial_\mu\partial_\nu f
 \bigr),
 \qquad
 \Box=\eta^{\alpha\beta}\partial_\alpha\partial_\beta.
 \label{eq:conformal-einstein}
\]
This expression need not vanish.

Consequently, even knowledge of the entire pointwise cone field
does not determine the Einstein tensor of a metric representative.
\end{proposition}

\begin{proof}
Multiplication by the positive function \(e^{2\lambda f}\)
does not change the sign or zeros of a quadratic form and preserves
the chosen future orientation.

The first-order connection is
\[
 \Gamma^{(1)\rho}_{\mu\nu}
 =
 \delta^\rho_\nu\partial_\mu f
 +\delta^\rho_\mu\partial_\nu f
 -\eta_{\mu\nu}\partial^\rho f.
\]
Using \eqref{eq:curvature-convention},
\[
 R^{(1)}_{\mu\nu}
 =
 -(m-2)\partial_\mu\partial_\nu f-\eta_{\mu\nu}\Box f,
 \qquad
 R^{(1)}=-2(m-1)\Box f.
\]
Subtracting \(\frac12\eta_{\mu\nu}R^{(1)}\) gives
\eqref{eq:conformal-einstein}.

For example, take \(f=(x^0)^2\) near the origin.
Then \(\Box f=2\), and for each spatial index \(i\),
\[
 G^{(1)}_{ii}[2f\eta]=-2(m-2)\ne0,
\]
whereas \(G[\eta]=0\).
\end{proof}

The distinction is between a positive \emph{constant} ambiguity
for one affine quadratic form and a positive \emph{function} in a
local metric extension.
Allowing the latter introduces new data; it is not merely fixing
the former's units.

The conformal calculation is independent of any field equation.
It also shows why ``Einstein curvature of a conformal class'' is
not a well-defined representative-independent source for the
proposed matching.

\section{Stress-energy is an additional extraction problem}
\label{sec:source}

\subsection{Numerical observables are not automatically stress-energy}

Process positivity concerns an operator on the process input/output
spaces. It does not identify those spaces with tangent directions
or assign energy, momentum, local density, or volume.

For example, XXII-C proves \(\Tr W=8\).
The normalized operator \(W/8\) can be used to form mathematical
expectation values. That normalization does not identify it as a
matter state on an extracted spacetime
\cite[\loc{lem:base}]{CheungXXIIC}.

\begin{proposition}[Energy interpretation is not identified by the
frozen process data]
\label{prop:energy}
The unchanged XXII-C data do not determine the energy observable of
a laboratory qubit.

More precisely, the same valid process and complete instrument
tables admit distinct appended energy assignments with different
expectation values, because energy assignments and energy-conservation
restrictions are not part of the frozen declaration.
\end{proposition}

\begin{proof}
Use \(W_0=I_{\mathcal H}/8\).
A definite-order realization prepares an independent maximally mixed
input qubit for each laboratory and discards its outgoing qubit.
For local instrument Choi matrices,
\[
 P_{W_0}(\boldsymbol x|\boldsymbol s)
 =
 \frac18\prod_{a\in\mathsf J}\Tr M^a_{x_a|s_a},
\]
which is the product of the corresponding maximally mixed input
probabilities.

Append either of the laboratory energy assignments
\[
 H^{(0)}=0,
 \qquad
 H^{(\omega)}=\omega\kb11,
 \qquad \omega>0.
\]
Their input expectations are respectively \(0\) and \(\omega/2\).
Neither assignment changes \(W_0\), an admitted instrument map,
or any frozen probability table.
The frozen declaration contains no restriction requiring the
instruments to conserve either assigned energy.

If a physical implementation is required to obey an energy balance,
its apparatus, work supply, generators, and preparation constraints
must be added and checked. They are absent from this data set.
\end{proof}

The proposition is an underdetermination result over energy-labelled
completions. It does not exclude an energy observable identified
after additional operational response data or microscopic dynamics
are supplied.

Similarly, channel equality by itself specifies no thermal reservoir,
Hamiltonian, temperature, or heat ledger.
Landauer-type statements require a thermodynamic realization with
such additional assumptions \cite{ReebWolf}.
The supplied channel-versus-cost scope is recorded in En07-A;
En07-B has a separate endpoint-recovery role
\cite{CheungEn07A,CheungEn07B}.

\begin{definition}[Source rule and interpretation ledger]
\label{def:source}
A proposed source rule \(\Source\) assigns to admitted data either:
\begin{enumerate}
\item a symmetric covariant tensor in a declared finite jet or
      comparison space; or
\item a symmetric tensor field or distribution on a justified
      geometric extension.
\end{enumerate}
Its ledger must state:
\begin{enumerate}[label=\textup{(S\arabic*)}]
\item the actual process, preparation, instrument, and residual data
      used, including every operative enlargement of \(\Access\);
\item the explicit rule producing its components and its covariance
      under the permitted matched equivalences;
\item the energy-momentum interpretation, units, localization,
      density or smearing convention, and apparatus contributions;
\item the balance or response principle supporting that interpretation;
\item the status of symmetry, conservation, renormalization, and any
      energy condition invoked;
\item whether the output is a real expectation tensor, an operator-valued
      object, or a statistical estimate, with its domain and allowances.
\end{enumerate}
Absent an interpretation theorem, the output is called a
\emph{source candidate}, not a derived physical stress-energy tensor.
\end{definition}

Standard stress-energy constructions use additional dynamical
structure, for example a matter action and metric variation
\cite[Section 4.5]{Tong}.
In locally covariant quantum field theory, metric response through
relative Cauchy evolution provides another precise framework
\cite{BFV}.
Neither framework is already contained in a three-party process
matrix.

If an action, generators, or response model are additionally assumed,
that status must remain visible.
If an operator-valued stress tensor is claimed, its field domain,
state, composite-field prescription, and comparison with operational
records require separate proofs. An equality of real expectation
tensors is not an operator Einstein equation.

\subsection{Conservation alone does not identify the source}

\begin{proposition}[Conserved improvement ambiguity]
\label{prop:improvement}
On the flat comparison carrier, for every smooth \(f\),
\[
 I_{\mu\nu}(f)
 =
 \eta_{\mu\nu}\Box f-\partial_\mu\partial_\nu f
\]
is symmetric and satisfies
\[
 \partial^\mu I_{\mu\nu}(f)=0.
\]
There are nonzero such tensors. Therefore symmetry and conservation
alone do not uniquely identify a stress-energy candidate.
\end{proposition}

\begin{proof}
Symmetry follows from commutation of derivatives.
Moreover,
\[
 \partial^\mu I_{\mu\nu}
 =
 \partial_\nu\Box f-\Box\partial_\nu f=0.
\]
The example \(f=(x^0)^2\) gives \(I_{ii}=-2\).
Adding such a tensor to a conserved symmetric candidate preserves
those two properties while changing the candidate.
\end{proof}

A restrictive matter model, normalization, and coverage theorem can
remove particular ambiguities.
Cheung's Oai282-B supplies such a scoped free-scalar stress-class
comparison on its independently specified background
\cite[\loc{thm:main}, \loc{def:wardmeaning}]{CheungOai282B}.
It is not a stress extraction for \(\ProcessClass\), and its Ward
conclusions are not Einstein dynamics.

\section{A well-defined linearized comparison}
\label{sec:comparator}

\subsection{Metric variation and source order}

A flat-background field comparison would require an independently
constructed family
\[
 g_\lambda=\eta+\lambda h+O(\lambda^2),
 \qquad
 T_\lambda=\lambda\tau+O(\lambda^2),
 \label{eq:field-expansion}
\]
on one identified differentiable carrier.
The remainder topology must control the derivatives used in the
equation.

The zero-order source is zero in this particular flat vacuum
comparison.
A nonzero background source, cosmological term, or curved background
requires a different declared linearization.
This note does not derive the physical absence of those terms.

The parameter \(\lambda\) is a perturbation parameter, not an
experiment identifier, physical clock, or Planck constant.
A family of operational data \(D_\lambda\), if used to construct
\eqref{eq:field-expansion}, needs its own comparison maps and
differentiability theorem.

\subsection{Finite-dimensional metric jets}

Curvature can also be compared algebraically without asserting
that the full field family has been constructed.

\begin{definition}[Second metric jet]
\label{def:jet}
At a declared formal marked point of the affine comparison carrier,
set
\[
 \Jet^2
 =
 \Sym^2(V_D^*)\otimes
 \bigl(\R\oplus V_D^*\oplus\Sym^2(V_D^*)\bigr).
\]
A jet \(j\in\Jet^2\) has components
\[
 (h_{\mu\nu},h_{\mu\nu|\alpha},h_{\mu\nu|\alpha\beta}),
\]
symmetric in \(\mu,\nu\) and in \(\alpha,\beta\).
The vertical bar denotes a formal derivative slot, not a derivative
already measured in the process data.
\end{definition}

Every such jet is represented by the polynomial
\[
 h_{\mu\nu}(x)
 =
 h_{\mu\nu}
 +h_{\mu\nu|\alpha}x^\alpha
 +\frac12h_{\mu\nu|\alpha\beta}x^\alpha x^\beta
\]
at the origin. This gives a mathematical realization of the jet,
not an operational extraction of it.

\begin{definition}[Linearized Einstein comparator]
\label{def:einstein}
Use \(\eta\) to raise and contract indices, and define
\begin{align}
 R^{(1)}_{\mu\nu}(j)
 &=
 \frac12\eta^{\rho\sigma}
 \bigl(
 h_{\sigma\nu|\rho\mu}
 +h_{\sigma\mu|\rho\nu}
 -h_{\mu\nu|\rho\sigma}
 -h_{\rho\sigma|\mu\nu}
 \bigr),
 \label{eq:ricci-jet}\\
 R^{(1)}(j)&=\eta^{\mu\nu}R^{(1)}_{\mu\nu}(j),
 \label{eq:scalar-jet}\\
 \Ein_\eta(j)_{\mu\nu}
 &=
 R^{(1)}_{\mu\nu}(j)
 -\frac12\eta_{\mu\nu}R^{(1)}(j).
 \label{eq:einstein-jet}
\end{align}
Thus
\[
 \Ein_\eta:\Jet^2\longrightarrow\Sym^2(V_D^*)
\]
is a fixed linear map.
For a smooth field \(h\), write
\(\Ein_\eta[h](x)=\Ein_\eta(j_x^2h)\).
\end{definition}

Equations~\eqref{eq:ricci-jet}--\eqref{eq:einstein-jet} are the
first variation of the curvature convention
\eqref{eq:curvature-convention}.
They follow by substituting
\[
 \Gamma^{(1)\rho}_{\mu\nu}
 =
 \frac12\eta^{\rho\sigma}
 (\partial_\mu h_{\sigma\nu}
 +\partial_\nu h_{\sigma\mu}
 -\partial_\sigma h_{\mu\nu})
\]
and retaining the terms linear in \(h\).
The comparator agrees with the standard flat-background
linearized Einstein calculation
\cite[Section 6]{Carroll}
\cite[Section 5]{Tong}.

A proposed rule must now supply
\[
 j_D=\MetricRule(D),\qquad
 \tau_D=\Source(D),\qquad
 \Residual(D)=\Ein_\eta(j_D)-\kappa_{\mathrm{cmp}}\tau_D.
 \label{eq:residual}
\]
Here \(\kappa_{\mathrm{cmp}}\ne0\) is a predeclared comparison coupling.
Its normalization, physical dimensions, and possible relationship
to Newton's constant are not determined by the cone.

In dimension \(m\geq3\), matching the Einstein tensor is equivalent
to the trace-adjusted Ricci relation
\[
 R^{(1)}_{\mu\nu}
 =
 \kappa_{\mathrm{cmp}}
 \left(
 \tau_{\mu\nu}
 -\frac{\eta_{\mu\nu}}{m-2}\eta^{\alpha\beta}\tau_{\alpha\beta}
 \right).
 \label{eq:ricci-matching}
\]
It is not an equality between the full four-index Riemann tensor
and a two-index source.

\subsection{Gauge and conservation tests}

\begin{proposition}[Linearized gauge identity]
\label{prop:gauge}
For a sufficiently differentiable covector field \(\xi\), define
\[
 (\Gauge\xi)_{\mu\nu}
 =
 \partial_\mu\xi_\nu+\partial_\nu\xi_\mu.
\]
Then
\[
 \Ein_\eta[\Gauge\xi]=0.
 \label{eq:gauge}
\]
At jet level the same identity holds for the induced second metric
jet obtained from a third jet of \(\xi\).
\end{proposition}

\begin{proof}
Substituting \(h=\Gauge\xi\) into the first-order connection gives
\[
 \Gamma^{(1)\rho}_{\mu\nu}=\partial_\mu\partial_\nu\xi^\rho.
\]
The first-order Riemann tensor is zero by commutation of third
derivatives. Its Ricci and Einstein contractions are therefore zero.
The calculation depends only on the relevant third derivative
slots of \(\xi\), so it also proves the jet assertion.
\end{proof}

This mathematical gauge identity does not identify an instrument
relabelling or scheduler interleaving with a geometric gauge
transformation. Such an identification would need its own
intertwining theorem.

\begin{proposition}[Linearized Bianchi necessity]
\label{prop:bianchi}
For every \(C^3\) symmetric field \(h\),
\[
 \partial^\mu\Ein_\eta[h]_{\mu\nu}=0.
 \label{eq:bianchi}
\]
The identity also holds distributionally.

Consequently, if a field equation
\[
 \Ein_\eta[h]=\kappa_{\mathrm{cmp}}\tau
\]
holds with fixed nonzero \(\kappa_{\mathrm{cmp}}\), then
\[
 \partial^\mu\tau_{\mu\nu}=0.
 \label{eq:source-conservation}
\]
A source violating \eqref{eq:source-conservation} cannot be matched
by any \(h\) on that field comparison domain.
\end{proposition}

\begin{proof}
Put
\[
 h=\eta^{\alpha\beta}h_{\alpha\beta},
 \qquad
 A_\nu=\partial^\mu h_{\mu\nu}.
\]
The Ricci formula gives
\[
 R^{(1)}_{\mu\nu}
 =
 \frac12(
 \partial_\mu A_\nu+\partial_\nu A_\mu
 -\Box h_{\mu\nu}-\partial_\mu\partial_\nu h).
\]
Taking a divergence,
\[
 \partial^\mu R^{(1)}_{\mu\nu}
 =
 \frac12\partial_\nu
 (\partial^\mu A_\mu-\Box h)
 =
 \frac12\partial_\nu R^{(1)}.
\]
Subtracting the divergence of the trace term proves
\eqref{eq:bianchi}.
Constant-coefficient derivatives commute on distributions as well.
Taking the divergence of the matching equation proves
\eqref{eq:source-conservation}.
\end{proof}

\begin{example}[A mathematical source rejected by conservation]
\label{ex:nonconserved}
Let \(f\in C_c^\infty(U)\) satisfy \(\partial_0f\not\equiv0\).
Set
\[
 \tau_{00}=f,\qquad \tau_{\mu\nu}=0
 \quad\text{for }(\mu,\nu)\ne(0,0).
\]
Then \(\partial^\mu\tau_{\mu0}=\partial_0f\ne0\), so no field
solution exists under the comparison of
Proposition~\ref{prop:bianchi}.

This is a comparator diagnostic, not a source extracted from an
admissible operational datum.
\end{example}

A single second jet does not itself contain the divergence data
in \eqref{eq:bianchi}. A pointwise conservation test requires the
appropriate third metric jet and first source jet, or a field
extension with compatible derivatives.

\subsection{Unrestricted single-point matching is too weak}

\begin{theorem}[Every symmetric point source has a formal matching jet]
\label{thm:surjective}
For \(m\geq3\), the map \(\Ein_\eta:\Jet^2\to\Sym^2(V_D^*)\)
is surjective.
For every symmetric tensor \(b_{\mu\nu}\), there is a polynomial
conformal perturbation with
\[
 \Ein_\eta(j_0^2h)=\kappa_{\mathrm{cmp}}b.
\]

Thus existence of an unrestricted matching jet at one point,
without an independently extracted jet rule or compatibility
conditions, is not a substantive dynamics-matching theorem.
\end{theorem}

\begin{proof}
Let \(b=\eta^{\mu\nu}b_{\mu\nu}\), and define the symmetric tensor
\[
 F_{\mu\nu}
 =
 \frac{\kappa_{\mathrm{cmp}}}{m-2}
 \left(
 \frac{b}{m-1}\eta_{\mu\nu}-b_{\mu\nu}
 \right).
\]
Its trace is
\[
 \eta^{\mu\nu}F_{\mu\nu}
 =
 \frac{\kappa_{\mathrm{cmp}}b}{(m-2)(m-1)}.
\]
Set
\[
 f(x)=\frac12F_{\alpha\beta}x^\alpha x^\beta,
 \qquad h_{\mu\nu}(x)=2f(x)\eta_{\mu\nu}.
\]
Equation~\eqref{eq:conformal-einstein} gives
\[
 \Ein_\eta(j_0^2h)_{\mu\nu}
 =
 (m-2)(\eta_{\mu\nu}\eta^{\alpha\beta}F_{\alpha\beta}
       -F_{\mu\nu})
 =
 \kappa_{\mathrm{cmp}}b_{\mu\nu}.
\]
Since the coupling is nonzero and \(b\) was arbitrary,
the map is surjective.
\end{proof}

The construction solves a chosen algebraic equation.
It does not extract \(h\) from independent operational geometry.
Likewise, defining
\(\tau_D=\kappa_{\mathrm{cmp}}^{-1}\Ein_\eta(j_D)\)
would make the residual vanish by definition, but would not prove
that this tensor is independently identified matter stress-energy.

A prescribed Einstein inverse can provide a valid conditional
classical response construction. It does not, by itself, derive
Einstein dynamics from process data.

\section{Scale, covariance, and matching verdicts}
\label{sec:verdicts}

\subsection{Constant scale is not an arbitrary fitting parameter}

\begin{proposition}[Constant rescaling of the lower-index comparator]
\label{prop:scale}
For \(c>0\),
\[
 \Ein_{c\eta}(c j)=\Ein_\eta(j),
 \qquad
 \Ein_{c\eta}(j)=c^{-1}\Ein_\eta(j).
 \label{eq:scale}
\]
The first equation corresponds to consistently rescaling both the
background and its metric variation.

Under an invertible affine coordinate change, the comparator
transforms as a covariant two-tensor, provided its background and
metric jets are transformed together.
\end{proposition}

\begin{proof}
Every term in the linearized Ricci formula contains one inverse
background factor and one metric-variation derivative.
Rescaling both cancels those factors, leaving
\(R^{(1)}_{\mu\nu}\) unchanged.
The scalar contraction gains \(c^{-1}\), while multiplication
by the background in the Einstein trace term gains \(c\).
This proves the first identity.

With the variation held fixed, all Ricci terms instead gain
\(c^{-1}\), and the trace term has the same overall factor.
This proves the second identity.

For an affine coordinate change, its Jacobian is constant.
Derivative slots transform covariantly, inverse-background factors
contravariantly, and all contracted Jacobian factors cancel.
The two uncontracted tensor indices give the stated covariance.
\end{proof}

A physical source has its own units and scaling convention.
Those must be transformed consistently before a physical comparison
is made.
Positive constant normalization cannot create curvature in a flat
representative, remove a nonconservation defect, or determine the
comparison coupling.

Position-dependent conformal rescaling is not covered by
\eqref{eq:scale}: it has derivative terms, as
Proposition~\ref{prop:conformal} demonstrates.

\subsection{A selected matching and an ansatz-feasibility problem}

For several formal marked points one may take finite products of
the jet and source spaces.
Alternatively, one may use a finite-dimensional field ansatz on a
declared auxiliary carrier and predeclared evaluation or smearing
maps.
Those choices must be fixed before fitting.

\begin{definition}[Finite comparison package]
\label{def:finite}
A finite comparison package supplies:
\begin{enumerate}
\item finite-dimensional real normed spaces \(X,Y\);
\item a fixed linear comparator \(L:X\to Y\), explicitly identified
      with the relevant restriction or projection of
      \(\Ein_\eta\);
\item predeclared maps \(a(D)\in X\) and \(s(D)\in Y\) obtained from
      the geometric and source rules;
\item a fixed nonzero coupling \(\kappa_{\mathrm{cmp}}\);
\item the residual
      \[
       r(D)=L a(D)-\kappa_{\mathrm{cmp}}s(D).
       \label{eq:finite-residual}
      \]
\end{enumerate}
The selected matching is \(r(D)=0\).
The separate ansatz-feasibility question asks whether there exists
some \(a\in X\) with \(La=\kappa_{\mathrm{cmp}}s(D)\).
\end{definition}

\begin{theorem}[Finite matching and infeasibility certificates]
\label{thm:feasibility}
For the fixed package of Definition~\ref{def:finite}:
\begin{enumerate}[label=\textup{(\roman*)}]
\item If \(r(D)\ne0\), the selected pair fails exact matching.
      This does not imply that every other \(a\in X\) fails.
\item A matching element exists in \(X\) if and only if
      \[
       \kappa_{\mathrm{cmp}}s(D)\in\Range L.
      \]
\item Infeasibility is equivalent to existence of
      \(\ell\in Y^*\) such that
      \[
       \ell\circ L=0,\qquad
       \ell(\kappa_{\mathrm{cmp}}s(D))\ne0.
       \label{eq:left-witness}
      \]
\item With an auxiliary Euclidean inner product on \(Y\),
      \[
       \inf_{a\in X}
       \|La-\kappa_{\mathrm{cmp}}s(D)\|
       =
       \|\Pi_{(\Range L)^\perp}
               (\kappa_{\mathrm{cmp}}s(D))\|.
       \label{eq:distance}
      \]
      The infimum is attained.
\end{enumerate}
\end{theorem}

\begin{proof}
The first assertion is the definition of the selected residual.
The second is the definition of the image of \(L\).

If the source vector is outside the finite-dimensional subspace
\(\Range L\), project it onto the orthogonal complement of that
subspace and pair with this nonzero projection. This gives
\eqref{eq:left-witness}.
Conversely, such a functional excludes membership in the image.

Orthogonal decomposition of the source vector shows that its
component perpendicular to \(\Range L\) is unavoidable in every
residual. Its parallel component belongs to the image and has a
preimage in \(X\), proving \eqref{eq:distance} and attainment.
\end{proof}

The auxiliary inner product is a mathematical comparison device,
not an extracted physical metric.

The theorem is useful only after \(X,L,s\) have been fixed.
Theorem~\ref{thm:surjective} shows why allowing every independent
point jet makes the pointwise feasibility question automatic.
Operationally derived restrictions or field compatibility conditions
must carry the substantive content.

A witness for a fixed ansatz is not a no-go theorem for every
possible carrier extension, source interpretation, or geometric rule.

\subsection{What a class-internal countermodel must certify}

\begin{definition}[Admissible mismatch witness]
\label{def:countermodel}
For a claimed operational dynamics package, a class-internal
mismatch witness consists of:
\begin{enumerate}
\item a specified datum satisfying every claimed class restriction
      and additional hypothesis;
\item the package's actual successful extraction and compatible
      transport realization for that datum;
\item its independently specified geometric and source outputs;
\item a nonzero residual, or an infeasibility certificate for its
      exact claimed comparison domain;
\item a proof that the witness is invariant under the permitted
      identifications or remains nonzero in the declared gauge
      quotient.
\end{enumerate}
\end{definition}

Appending an arbitrary conformal factor, source label, or energy
observable to a predecessor example does not satisfy this definition.
Such completions can prove underdetermination of a smaller interface,
but not failure of an unspecified future package.

No witness satisfying Definition~\ref{def:countermodel} for a
complete XXII-A--D admissible domain is supplied here.
The flatness, conformal, energy, and conservation results are
proved with their narrower scopes.

\section{Error control, coverage, and derivative limits}
\label{sec:evidence}

\subsection{A finite residual certificate}

\begin{theorem}[Propagation of comparison allowances]
\label{thm:stability}
For a fixed finite package, suppose implemented or reconstructed
values obey
\[
 \|\widehat a-a(D)\|_X\leq\alpha,
 \qquad
 \|\widehat s-s(D)\|_Y\leq\beta.
\]
Set
\[
 \widehat r=L\widehat a-\kappa_{\mathrm{cmp}}\widehat s,
 \qquad
 \delta=\|L\|\alpha+|\kappa_{\mathrm{cmp}}|\beta.
\]
Then
\[
 \|r(D)-\widehat r\|_Y\leq\delta,
 \label{eq:residual-error}
\]
and
\[
 \max\{0,\|\widehat r\|_Y-\delta\}
 \leq\|r(D)\|_Y
 \leq\|\widehat r\|_Y+\delta.
 \label{eq:residual-interval}
\]
For any \(\ell\in Y^*\), a strict inequality
\[
 |\ell(\widehat r)|>\|\ell\|\delta
\]
certifies failure of exact selected matching.

If \(\ell\circ L=0\), a strict inequality
\[
 |\ell(\kappa_{\mathrm{cmp}}\widehat s)|
 >
 |\kappa_{\mathrm{cmp}}|\|\ell\|\beta
\]
instead certifies infeasibility for every \(a\in X\).
\end{theorem}

\begin{proof}
Subtract the two residuals:
\[
 r(D)-\widehat r
 =
 L(a(D)-\widehat a)
 -\kappa_{\mathrm{cmp}}(s(D)-\widehat s).
\]
The triangle and operator-norm inequalities give
\eqref{eq:residual-error}.
The triangle and reverse triangle inequalities give
\eqref{eq:residual-interval}.
Duality transfers the error bound to \(\ell\).
For the final assertion, the source pairing remains nonzero and
\(\ell\) annihilates the entire comparator image; apply
Theorem~\ref{thm:feasibility}.
\end{proof}

The bounds require the same \(L\), frame, units, and comparison
interfaces. Uncertainty in \(L\), the coupling, or the reconstruction
maps requires additional terms, not silent absorption into transport
error.

\begin{proposition}[Finite-probe coverage]
\label{prop:coverage}
Let \(Z\subseteq Y\) be a proved finite-dimensional residual subspace,
and let \(P:Y\to\R^k\) be a declared linear probe map.
Exact vanishing of all probes forces \(r=0\) for every \(r\in Z\)
if and only if \(P|_Z\) is injective.

When it is injective, there is a finite constant \(C_Z\) such that
\[
 \|r\|_Y\leq C_Z\|Pr\|_\infty
 \quad(r\in Z).
 \label{eq:coverage}
\]
Thus, if \(r(D)\in Z\), \(\|P\widehat r\|_\infty\leq u\), and
\eqref{eq:residual-error} holds, then
\[
 \|r(D)\|_Y
 \leq C_Z\bigl(u+\|P\|\delta\bigr).
 \label{eq:covered-error}
\]
\end{proposition}

\begin{proof}
Injectivity is precisely the assertion that the only element of
\(Z\) in the probe kernel is zero.
If \(Z\ne\{0\}\), the continuous function
\(r\mapsto\|Pr\|_\infty\) has a positive minimum on the compact unit
sphere of \(Z\). Its reciprocal gives \(C_Z\).
For \(Z=\{0\}\), the assertion is immediate.
Finally,
\[
 \|Pr(D)\|_\infty
 \leq\|P\widehat r\|_\infty+\|P\|\delta.
\]
\end{proof}

The hypothesis that all actual residuals lie in \(Z\) is a coverage
theorem, not a consequence of having measured finitely many probes.
The analogous distinction is retained in Cheung's finite response
and Ward audits
\cite{CheungXIXD,CheungOai282B}.

\subsection{What complete-channel error can control}

\begin{proposition}[Bounded observable transfer]
\label{prop:bounded}
Let \(\Phi,\Psi\) be channels on common complete interfaces with
\[
 \ddiamond(\Phi,\Psi)\leq\epsilon.
\]
For a fixed input state and a Hermitian output observable \(O\),
\[
 \left|
 \Tr\bigl(O\Phi(\rho)\bigr)
 -
 \Tr\bigl(O\Psi(\rho)\bigr)
 \right|
 \leq
 \bigl(\lambda_{\max}(O)-\lambda_{\min}(O)\bigr)\epsilon.
 \label{eq:bounded-transfer}
\]
The same bound holds when the input includes an untouched reference.
\end{proposition}

\begin{proof}
Let \(\Delta\) be the output difference.
Then \(\Tr\Delta=0\) and \(\|\Delta\|_1\leq2\epsilon\).
Subtract the scalar midpoint
\(c=(\lambda_{\max}(O)+\lambda_{\min}(O))/2\):
\[
 |\Tr(O\Delta)|
 =
 |\Tr((O-cI)\Delta)|
 \leq\|O-cI\|_\infty\|\Delta\|_1.
\]
The spectral radius about this midpoint is half the displayed
spectral diameter.
The diamond norm includes referenced inputs.
\end{proof}

A source obtained by bounded scores or observables can therefore
inherit channel allowances after its actual readout and reconstruction
maps are specified.
Unbounded energy or stress observables need moment, tail, domain,
and implementation control. Equation~\eqref{eq:bounded-transfer}
does not apply to them without that work.

Cheung's Oai265-B and En01-C provide documentary examples of keeping
complete records, statistical coverage, and unbounded-record
allowances separate from the target quantity
\cite{CheungOai265B,CheungEn01C}.
Their certificates are not stress-energy or curvature certificates
for this class.

\subsection{Transport convergence does not control second derivatives}

\begin{proposition}[A derivative-amplification witness]
\label{prop:derivative}
On a fixed bounded comparison region, let
\[
 f_n(x)=n^{-2}\sin(nx^1),
 \qquad
 h^{(n)}_{\mu\nu}=2f_n\eta_{\mu\nu}.
\]
Then \(h^{(n)}\to0\) uniformly together with its first derivatives,
but
\[
 \Ein_\eta[h^{(n)}]_{00}
 =
 (m-2)\sin(nx^1)
 \label{eq:derivative-witness}
\]
does not converge uniformly to zero on any region whose \(x^1\)
projection contains an interval.

Moreover \(e^{2f_n}\eta\) has exactly the same pointwise cones as
\(\eta\) for every \(n\).
\end{proposition}

\begin{proof}
The zeroth and first derivative bounds are respectively
\(O(n^{-2})\) and \(O(n^{-1})\).
Since
\[
 \partial_1^2f_n=-\sin(nx^1),
 \qquad
 \Box f_n=\sin(nx^1),
 \qquad
 \partial_0^2f_n=0,
\]
equation~\eqref{eq:conformal-einstein} gives
\eqref{eq:derivative-witness}.
The supremum of its absolute value reaches \(m-2\) on such a
region for all sufficiently large \(n\).
Positive conformal multiplication preserves the cones.
\end{proof}

The proposition concerns uniform curvature comparison.
It does not deny weaker distributional convergence of this example.

An endpoint channel limit, or even uniform coordinate and
first-derivative control, therefore cannot be renamed a curvature
limit. A metric reconstruction theorem must prove the derivative
topology required by its claim.

\subsection{Finite samples do not fix a field equation}

\begin{proposition}[Finite-point invisibility]
\label{prop:finite-samples}
Let \(F\) be a finite set in an open comparison region \(U\subset\R^m\).
There are smooth conformal metric families that agree with \(\eta\)
on neighborhoods of every point of \(F\), have the same pointwise
cones everywhere, and have nonzero first-order Einstein curvature
elsewhere.
\end{proposition}

\begin{proof}
Choose \(z\in U\setminus F\) and a ball about \(z\) whose closure
is contained in \(U\) and misses \(F\).
Choose a smooth compactly supported bump \(\chi\) in that ball,
equal to one near \(z\).
Set
\[
 f(x)=(x^0-z^0)^2\chi(x),
 \qquad
 g_\lambda=e^{2\lambda f}\eta.
\]
The metrics equal \(\eta\) near \(F\), and their cones are unchanged.
Near \(z\), Proposition~\ref{prop:conformal} gives the nonzero
spatial components \(-2(m-2)\) of their first variation's Einstein
tensor.
\end{proof}

Agreement at finitely many points, even including all derivative
jets there, is not a full field matching theorem.
A finite-dimensional ansatz and an injective coverage map can
support a narrower claim; both must be proved.

\subsection{First-order matching is not nonlinear dynamics}

\begin{proposition}[An invisible second-order curvature change]
\label{prop:nonlinear}
Near the origin compare
\[
 g^{(0)}_\lambda=\eta,
 \qquad
 g^{(1)}_\lambda=e^{2\lambda^2(x^0)^2}\eta.
\]
Both have zero first metric variation at \(\lambda=0\), the same
pointwise cones, and the same first-order vacuum matching.
Nevertheless,
\[
 G_{ii}[g^{(1)}_\lambda](0)
 =
 -2(m-2)\lambda^2
 \quad(1\leq i\leq m-1),
 \label{eq:second-order}
\]
whereas \(G[g^{(0)}_\lambda]=0\).
\end{proposition}

\begin{proof}
The first parameter derivative of both metrics at zero is zero.
At the coordinate origin,
\[
 g^{(1)}_\lambda(0)=\eta,\qquad
 \partial_\alpha g^{(1)}_\lambda(0)=0.
\]
Thus the connection is zero there and the curvature contains no
connection-product terms.
Its dependence on the second metric derivatives at that point
is exactly the flat first-variation expression.
Those derivatives equal the second derivatives of
\(2\lambda^2(x^0)^2\eta\).
Equation~\eqref{eq:conformal-einstein} gives
\eqref{eq:second-order}.
\end{proof}

No first-order residual certificate supplies the second-order
metric, source, gauge, constraint, or remainder obligations of a
nonlinear Einstein equation.

\section{The dynamics contract and obligations D1--D6}
\label{sec:obligations}

\begin{specification}[Additional dynamics interface]
\label{spec:dynamics}
A proposed operational dynamics package must fix:
\begin{enumerate}[label=\textup{(J\arabic*)}]
\item a named nonempty admissible data domain, with all extraction,
      transport, and dynamics premises explicitly classified as
      declared, assumed, or derived;
\item the unchanged carrier or a justified revision, and any local
      differentiable extension needed for derivative comparisons;
\item an independent geometric rule \(\MetricRule\), producing
      compatible metric jets or a perturbative metric family;
\item an independent source rule \(\Source\) with the interpretation
      ledger of Definition~\ref{def:source};
\item one predeclared comparison operator, coupling, gauge convention,
      perturbative order, source order, and background;
\item the precise meaning of matching: full tensor identity,
      field or distributional identity, finite-ansatz identity,
      finite probe equality, or a tolerance bound;
\item covariance, operational equivalence, calibration, boundary,
      derivative, and implementation conditions actually used.
\end{enumerate}
No successful instance of this complete interface is constructed here.
\end{specification}

A manifold, local cone field, or metric extension introduced as a
mathematical comparison domain must not be reported as an operationally
derived spacetime unless its connecting construction is supplied.
The use of the Einstein comparator is likewise not a derivation
of the governing action or dynamics.

\begin{specification}[Dynamics obligations D1--D6]
\label{spec:obligations}
A later theorem must discharge the following.
\begin{enumerate}[label=\textup{(D\arabic*)}]
\item \textbf{Independent geometry and stress-energy rules.}
      Supply the geometric derivative data missing from the affine
      cone, and the source data missing from the process interface.
      Prove their interpretation and covariance.
      A numerical function of \(W_e\), \(\Delta_e\), or a residual
      state is not stress-energy merely by its name.
      Do not define the source as the comparator output solely to
      force matching.

\item \textbf{Matching or a correctly scoped obstruction.}
      Prove the actual residual identity on the declared nonempty
      domain, or supply a witness satisfying
      Definition~\ref{def:countermodel}.
      Distinguish failure of a selected rule from infeasibility of
      every element of a fixed ansatz.
      Neither is automatically a universal obstruction to future
      additional structure.

\item \textbf{Scale, coupling, frame, and units.}
      Retain the conformal ambiguity and the separate coupling.
      Fix any calibration before applying the comparison, with
      its assumed or derived status.
      Do not use a per-datum fitted normalization to conceal a
      tensor or conservation defect.
      No numerical invariant physical speed follows from this step.

\item \textbf{Carrier, derivative, and interface compatibility.}
      Prove every claimed extension, locality, reconstruction,
      derivative limit, field integrability, and conservation
      property.
      Finite-dimensional jets are not a continuum field equation.
      Complete-channel allowances become source or geometric
      allowances only through actual bounded readout and reconstruction
      theorems or appropriately controlled unbounded interfaces.

\item \textbf{Failure tests and scope retention.}
      Retain the transport tests of XXII-D and the additional
      flatness, conformal, energy, conservation, coverage,
      derivative, and perturbative-order tests proved here.
      Repair a violated test under stated extra premises or restrict
      the claim accordingly.

\item \textbf{Classical and documentary comparisons only.}
      Keep prescribed Einstein--scalar jets, metric-valued inverse
      responses, conical packets, and classical lapse records
      separate from operational derivation.
      No theorem transfers solely because the constructions share
      notation, a continuation label, or a documentary relation.
\end{enumerate}
\end{specification}

\subsection{The retained transport tests}

The dynamics layer does not supersede XXII-D's tests
\cite[Failure tests F1--F6]{CheungXXIID}:
\begin{enumerate}
\item generated boundaries do not imply attained null protocols;
\item a fixed finite stencil need not approximate every null direction;
\item moving arrivals can prevent uniform comparison through an event;
\item small step errors can accumulate in a specified noise model;
\item tolerance is not exact causal containment;
\item endpoint limits do not establish a unique continuum evolution.
\end{enumerate}

The unchanged component carrier also retains the fixed-source
finite-future obstruction
\cite[\loc{thm:anchored-obstruction}]{CheungXXIID}.
A geometric extension introduced for curvature must explain how
its events and comparisons relate to that carrier.

Scheduled ancestry, unconditional operational influence, and an
independently supplied geometric causal relation remain distinct
\cite{CheungVIIJ,CheungXXIIA}.
A curvature comparison does not certify their equality.

\subsection{Non-circularity and non-vacuity}

The following do not discharge the dynamics contract:
\begin{itemize}
\item restricting admissibility by the desired equality without proving
      a nonempty independently described operational domain;
\item choosing arbitrary coordinates and conformal factors to fit a
      preselected stress tensor;
\item defining the stress tensor from the Einstein residual;
\item applying a prescribed Einstein inverse and reporting its defining
      equation as a newly derived law;
\item checking finitely many probes without a residual-coverage theorem.
\end{itemize}

Some of these operations can be legitimate in an explicitly
conditional classical model. Their interpretation must remain that
of the conditional model, not operational emergence.

The relevant obstruction does not have to await a completed transport
programme when it concerns an algebraic implication already formulated
precisely. Theorems~\ref{thm:flat} and
\ref{thm:surjective}, for example, establish their scopes directly.
A countermodel to a complete operational package, however, must
actually satisfy that package's premises.

\section{Documentary comparisons and component ledger}
\label{sec:ledger}

The outline describes III and VI as classical quotient or
Einstein--scalar jet constructions, IV Model 3 as a metric-valued
prescribed-inverse comparison, and Em02-E as a classical relational-lapse
comparison on an installed background.
Those are documentary descriptions of distinct models.
No unsupplied theorem or identification is imported
\cite{CheungClassicalComparisons,CheungXIXE}.

In particular, a classical Einstein--scalar jet theorem can already
contain the gravitational equations among its premises.
A prescribed inverse on Minkowski space already fixes its background
and comparison operator.
Neither is an operational derivation from \(\ProcessClass\).

The supplied source and response records in XVI-B and XVI-D likewise
do not supply the present source or metric extraction maps.
Their smooth source-response domains are not, by notation alone,
the event carrier or comparison domain of this programme
\cite{CheungXVIBD}.

\begingroup
\small
\renewcommand{\arraystretch}{1.13}
\begin{longtable}{@{}p{.25\linewidth}p{.67\linewidth}@{}}
\toprule
Component & Role and boundary\\
\midrule
\endfirsthead
\toprule
Component & Role and boundary\\
\midrule
\endhead
\bottomrule
\endfoot

XXII-A &
Faithful generation, selection target, and the constant conformal
ambiguity. No field dynamics is inherited
\cite{CheungXXIIA}.\\[4pt]

XXII-B &
Ledger syntax, source-boundary warnings, reference stability,
tolerance, and refinement tests
\cite{CheungXXIIB}.\\[4pt]

XXII-C &
Frozen process class and component order. The identifiers do not
supply derivatives, Hamiltonians, or a local metric field
\cite{CheungXXIIC}.\\[4pt]

XXII-D &
Compatible transport contract, calibration separation, quantum
transmission obstruction, and tests F1--F6
\cite{CheungXXIID}.\\[4pt]

En08-A; En08-B &
Scheduler obstruction and prescribed-boundary approximation.
Neither supplies the metric-jet or source rule needed here
\cite{CheungEn08A,CheungEn08B}.\\[4pt]

VII-J &
Three-relation discipline. A supplied geometric embedding is
a separate certificate
\cite{CheungVIIJ}.\\[4pt]

XIX-C; XIX-D &
Refinement, interpolation, finite-coverage, and proof-transfer
discipline. No native dynamics comparison is inherited
\cite{CheungXIXC,CheungXIXD}.\\[4pt]

Oai265-A--B &
Static records do not instantiate recovery or a gravitational
dictionary; channel certificates require their own access and
statistical assumptions
\cite{CheungOai265A,CheungOai265B}.\\[4pt]

Oai282-B &
A separately specified free-scalar stress-class and Ward benchmark,
not stress-energy extraction for process data or Einstein dynamics
\cite{CheungOai282B}.\\[4pt]

En07-A; En07-B &
Channel/cost separation and conditional endpoint-recovery scope,
respectively. Neither provides energy-momentum accounting here
\cite{CheungEn07A,CheungEn07B}.\\[4pt]

III, IV, VI;
VIII-A/B/C;
Em02-E &
Documentary classical, geometric, and lapse comparisons only.
No operational correspondence theorem is supplied
\cite{CheungClassicalComparisons,CheungXXIIA}.\\[4pt]

OpenAI catalogue &
The supplied catalogue record is a documentary negative control,
not a theorem premise or stress-energy construction
\cite{CheungXXIIA,CheungXXIIB}.\\
\end{longtable}
\endgroup

A conjunction of unrelated response, channel, or Ward certificates
is not a correspondence theorem.
No holographic dictionary, entanglement wedge, area-to-curvature law,
or gravitational mechanism is supplied by this ledger.

\section{Completed release and remaining gap}
\label{sec:release}

\begingroup
\small
\renewcommand{\arraystretch}{1.12}
\begin{longtable}{@{}p{.37\linewidth}p{.16\linewidth}p{.38\linewidth}@{}}
\toprule
Item & Status & Exact scope\\
\midrule
\endfirsthead
\toprule
Item & Status & Exact scope\\
\midrule
\endhead
\bottomrule
\endfoot

Constant affine flatness &
Proved &
Theorem~\ref{thm:flat}; excludes a nonzero source on the unchanged
constant branch\\[4pt]

Conformal curvature ambiguity &
Proved &
Proposition~\ref{prop:conformal}; even pointwise cones do not fix
the Einstein tensor\\[4pt]

Energy underdetermination &
Proved &
Proposition~\ref{prop:energy}; bare process data admit different
appended energy assignments\\[4pt]

Linearized comparator and identities &
Defined and proved &
Formal metric jets, gauge identity, and field Bianchi necessity\\[4pt]

Single-point feasibility &
Proved &
Theorem~\ref{thm:surjective}; unrestricted algebraic feasibility
is not independent extraction\\[4pt]

Finite matching and error certificates &
Proved &
Fixed comparator, ansatz, norms, and covered residual space\\[4pt]

Operational metric-jet rule &
Open &
No independently derived \(\MetricRule\)\\[4pt]

Operational stress-energy rule &
Open &
No physically interpreted \(\Source\) on a joint admissible domain\\[4pt]

Complete class-internal mismatch &
Open &
No witness meeting Definition~\ref{def:countermodel}\\[4pt]

Continuum field matching &
Not constructed &
Carrier, derivative, integrability, and source interfaces absent\\[4pt]

Nonlinear Einstein dynamics &
Not established &
First-order certificates do not supply higher orders\\
\end{longtable}
\endgroup

The completed Stage 4 release is the corrected dynamics interface,
the obligations D1--D6, and the scoped calculations and certificates
printed above.
It is therefore inaccurate to say that no obstruction of any kind
has been constructed.
It is equally inaccurate to report these scoped obstructions as a
universal no-go theorem for the complete future operational target.

\begin{target}[Operational dynamics comparison]
\label{target:dynamics}
Construct one predeclared additional package with:
\begin{enumerate}[label=\textup{(E\arabic*)}]
\item a nonempty joint operational domain meeting the applicable
      extraction and transport targets;
\item a justified geometric derivative or metric-jet rule and source
      rule, with all independent inputs and assumed status disclosed;
\item a fixed comparison coupling, operator, gauge convention, and
      comparison domain;
\item either a matching theorem on that domain or a correctly scoped
      admissible mismatch or infeasibility certificate;
\item the invariance, coverage, derivative-limit, calibration, and
      evidence statements required by the claim.
\end{enumerate}
For a nontrivial curvature claim, the unchanged constant affine
geometry must be supplemented by an explicitly justified varying
geometric construction.

This target has no operational existence proof in XXII-E.
\end{target}

\paragraph{Unchanged boundaries.}
XXII-E does not construct the successful Lorentz-cone extraction or
compatible transport realization.
It does not extract a physical stress-energy tensor from quantum
data, construct an operationally identified metric field, or derive
a full nonlinear Einstein equation.

No holographic reconstruction, Page curve, gravity-generated
entanglement, invariant physical speed, or quantum evolution without
external time is established.
An observation boundary is not identified with a physical light cone.

No result changes O.1, opens XV-B, discharges XVI-N or Route B, or
alters H2, I3, or J2.
The symbol \(\kappa_{\mathrm{cmp}}\) is a declared comparison
coupling, not an admitted quantum coefficient of the separate
characteristic programme.
No successful compilation, programme execution, numerical experiment,
laboratory implementation, or machine verification is asserted
\cite{CheungXIXE}.

\paragraph{The precise remaining gap.}
Successful cone extraction and transport would still leave a
dynamics gap: the operational construction of compatible geometric
derivative data and independently interpreted matter-source data.
The constant affine quadratic form itself is flat, and its conformal
class is insufficient to determine a local Einstein tensor.

Only after those missing constructions are supplied can a
linearized matching be evaluated as an operational result rather
than as a formal fit or prescribed-inverse comparison.
A nonzero residual must then be interpreted on its exact admissible
domain; a zero residual must retain its perturbative, coverage,
and calibration limitations.

This freezes the dynamics layer without retroactively installing
the field equation whose operational status remains to be proved
or obstructed.

\begin{thebibliography}{99}

\bibitem{CheungXXIIA}
N.~Cheung.
\emph{XXII-A: Freeze the precise claim and the gap.
Stage 0 of a limited kinematic emergence programme}.
Supplied manuscript \loc{XXII-A.tex}, n.d.
Relevant labels:
\loc{target:selection}, \loc{def:representation},
\loc{thm:classification}, and \loc{rem:conformal};
conditions (T1)--(T5), (A1)--(A6).

\bibitem{CheungXXIIB}
N.~Cheung.
\emph{XXII-B: Catalogue the additional quantum data and the
hypothesis ledger.
Stage 1 of a limited kinematic emergence programme}.
Supplied manuscript \loc{XXII-B.tex}, n.d.
Relevant items: ledger syntax, Candidate D,
and failure tests F3--F5.

\bibitem{CheungXXIIC}
N.~Cheung.
\emph{XXII-C: Freeze one quantum process class and the extraction
obligations.
Stage 2 of a limited kinematic emergence programme}.
Supplied manuscript \loc{XXII-C.tex}, n.d.
Relevant labels:
\loc{def:process}, \loc{spec:access}, \loc{def:class},
\loc{lem:base}, \loc{thm:order}, and \loc{target:selection}.

\bibitem{CheungXXIID}
N.~Cheung.
\emph{XXII-D: Operational approximation of an extracted Lorentz cone.
Stage 3 of a limited kinematic emergence programme}.
Supplied manuscript \loc{XXII-D.tex}, n.d.
Relevant labels:
\loc{spec:extraction}, \loc{prop:calibration},
\loc{lem:finite-futures}, \loc{thm:anchored-obstruction},
\loc{thm:geometry-not-transport}, \loc{spec:realization},
and \loc{thm:certificate}.
Failure tests F1--F6.
The supplied duplicate XXII-C bibliography alias is not copied here.

\bibitem{OCB}
O.~Oreshkov, F.~Costa, and \v{C}.~Brukner.
Quantum correlations with no causal order.
\emph{Nature Communications}
\textbf{3} (2012), 1092.
\doi{10.1038/ncomms2076}.
Preprint: \url{https://arxiv.org/abs/1105.4464}.

\bibitem{Watrous}
J.~Watrous.
\emph{The Theory of Quantum Information}.
Cambridge University Press, 2018.
Chapters 2--3.
Author's text:
\url{https://cs.uwaterloo.ca/~watrous/TQI/}.

\bibitem{Carroll}
S.~M.~Carroll.
\emph{Lecture Notes on General Relativity}.
1997.
\url{https://arxiv.org/abs/gr-qc/9712019}.
Section 6 discusses flat-background linearized gravity.
The signature and curvature conventions of the present note
are fixed by its displayed formulas.

\bibitem{Tong}
D.~Tong.
\emph{Lectures on General Relativity}.
University of Cambridge, lecture notes, 2019.
\url{https://www.damtp.cam.ac.uk/user/tong/gr.html}.
Sections 4--5 discuss the Einstein equation, matter stress-energy,
and linearized gravity.
These are comparison frameworks, not operational emergence premises.

\bibitem{BFV}
R.~Brunetti, K.~Fredenhagen, and R.~Verch.
The generally covariant locality principle---a new paradigm for local
quantum field theory.
\emph{Communications in Mathematical Physics}
\textbf{237} (2003), 31--68.
\doi{10.1007/s00220-003-0815-7}.
Preprint:
\url{https://arxiv.org/abs/math-ph/0112041}.
Metric response and relative Cauchy evolution require their
locally covariant field-theoretic interface.

\bibitem{ReebWolf}
D.~Reeb and M.~M.~Wolf.
An improved Landauer principle with finite-size corrections.
\emph{New Journal of Physics}
\textbf{16} (2014), 103011.
\doi{10.1088/1367-2630/16/10/103011}.
Preprint: \url{https://arxiv.org/abs/1306.4352}.
Its thermodynamic realization assumptions are not supplied by
channel equality alone.

\bibitem{CheungEn08A}
N.~Cheung.
\emph{En08-A}, version 1, manuscript \loc{En08-A.tex}.
Enter Calculus Companion, n.d.
Relevant supplied labels:
\loc{thm:main}, \loc{thm:mainpositive},
\loc{thm:representation}, \loc{thm:weaker},
and \loc{prop:quadraticunique}.
Scheduler obstruction, conditional representation, and scope warnings.
Registry: \url{https://enter.grok.me/}.

\bibitem{CheungEn08B}
N.~Cheung.
\emph{En08-B}, version 1, manuscript \loc{En08-B.tex}.
Enter Calculus Companion, n.d.
Relevant supplied labels:
\loc{lem:identity}, \loc{lem:telescoping},
\loc{thm:main}, \loc{def:comparison},
and \loc{cor:operational-limit}.
Prescribed-boundary endpoint approximation, not cone selection
or dynamics matching.

\bibitem{CheungVIIJ}
N.~Cheung.
\emph{VII-J}, version 1, manuscript \loc{VII-J.tex}.
Enter Calculus Companion, n.d.
Relevant supplied labels:
\loc{thm:schedule}, \loc{thm:nosignal},
\loc{thm:physical}, and \loc{thm:release}.
Scheduled ancestry, operational influence, and supplied geometric
causality are separate interfaces.

\bibitem{CheungXIXC}
N.~Cheung.
\emph{XIX-C}, version 1, manuscript \loc{XIX-C.tex}.
Enter Calculus Companion, n.d.
Relevant supplied labels:
\loc{def:limit}, \loc{prop:two-limits},
\loc{prop:interpolation}, and \loc{thm:main}.
Finite-interface and refinement non-determination.

\bibitem{CheungXIXD}
N.~Cheung.
\emph{XIX-D}, version 1, manuscript \loc{XIX-D.tex}.
Enter Calculus Companion, n.d.
Relevant supplied label: \loc{thm:main}.
Finite-probe, representation, state, and proof-transfer separations.
No native dynamics construction is imported.

\bibitem{CheungEn07A}
N.~Cheung.
\emph{En07-A}, manuscript \loc{En07-A.tex}.
Enter Calculus Companion, n.d.
Relevant supplied scope record: \loc{thm:release}.
Complete restoration and readout-and-drop channel statements;
channel equality is not a thermodynamic cost certificate.

\bibitem{CheungEn07B}
N.~Cheung.
\emph{En07-B}, manuscript \loc{En07-B.tex}.
Enter Calculus Companion, n.d.
Relevant supplied scope record: \loc{thm:endpoint}.
Outgoing recovery under separately supplied same-model obligations.
Not a stress-energy or thermal-cost extraction theorem.

\bibitem{CheungOai265A}
N.~Cheung.
\emph{Oai265-A}, manuscript \loc{Oai265-A.tex}.
Enter Calculus Companion, n.d.
Relevant supplied labels:
\loc{def:determination} and \loc{thm:separation}.
Static area-law records do not instantiate recovery histories
or gravitational dictionaries.

\bibitem{CheungOai265B}
N.~Cheung.
\emph{Oai265-B}, version 2, manuscript \loc{Oai265-B-v2.tex}.
Enter Calculus Companion, n.d.
Relevant supplied labels:
\loc{ass:statistics}, \loc{thm:main},
\loc{eq:coverage}, and \loc{eq:falseaccept}.
Complete-record finite-confidence channel certificate;
documentary evidence discipline only.

\bibitem{CheungEn01C}
N.~Cheung.
\emph{En01-C}, version 1, manuscript \loc{En01-C.tex}.
Enter Calculus Companion, n.d.
Relevant supplied labels:
\loc{thm:certificate} and \loc{eq:clipping}.
Bounded complete-record certification with explicit moment
and clipping allowances; not a stress-energy extraction.

\bibitem{CheungOai282B}
N.~Cheung.
\emph{Oai282-B}, version 1, manuscript \loc{Oai282-B.tex}.
Enter Calculus Companion, n.d.
Relevant supplied labels:
\loc{thm:main}, \loc{thm:coverage},
and \loc{def:wardmeaning}.
A fixed-background free-scalar stress-class and local Ward benchmark;
not an operational Einstein equation for the present class.

\bibitem{CheungClassicalComparisons}
N.~Cheung.
\emph{III}, \emph{IV}, \emph{VI}, and \emph{Em02-E}.
Enter Calculus Companion manuscripts, n.d.
Only documentary identifiers and scope descriptions are used:
classical quotient or jet comparisons, a prescribed-inverse
metric-valued response, and a classical relational lapse.
No unsupplied theorem, proof, or operational identification
is imported.

\bibitem{CheungXVIBD}
N.~Cheung.
\emph{XVI-B}, version 1, manuscript \loc{XVI-B.tex};
and \emph{XVI-D}, version 1, manuscript \loc{XVI-D.tex}.
Enter Calculus Companion, n.d.
Relevant supplied items:
XVI-B \loc{prop:trace}, \loc{prop:pair}, \loc{cor:tf};
XVI-D \loc{eq:Slambda}, \loc{thm:response-package}.
The supplied classical source and smooth witness-response domains
are documentary comparisons, not extraction rules for this note.

\bibitem{CheungXIXE}
N.~Cheung.
\emph{XIX-E}, version 1, manuscript \loc{XIX-E.tex}.
Enter Calculus Companion, n.d.
Relevant supplied passage:
``What remains outside the release.''
Documentary programme boundaries and verification limits.

\end{thebibliography}

\end{document}