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\title{\textbf{Oai365-A}\\
Conditional boundary-to-interior identification\\
of an operational Lorentz-cone representation\\
\large Explicit quantum-process pilots, gauge interfaces,\\
transport certificates, and obstruction audits}
\author{Ngo Cheung}
\date{14 October 2026}

\begin{document}
\maketitle

\begin{abstract}
We formulate and prove a conditional boundary-to-interior
identification theorem, keeping quantum-process data, elliptic
boundary data, affine event geometry, and physical dynamics distinct.
The general theorem requires a calibrated extraction of a complete
same-patch Dirichlet-to-Neumann energy form and an independently
established uniqueness theorem on a specified coefficient class.
Neither requirement is supplied by a catalogue entry.

An explicit countable family of separately tested tripartite qubit
processes provides a nonempty operational pilot. Its unconditional
signalling determines three-event chains; tomography of the source
states and operationally distinguishing channel parameters gives a
predeclared, permutation-covariant, faithful generated affine
representation. Rich pure-qubit preparations generate the
four-dimensional Lorentz cone. A restricted one-parameter dictionary
then identifies a smooth scaled-ball metric from an operational
parameter, without invoking the general inverse claims of Family 365.
This dictionary installs a comparison class, not spacetime emergence.

We also prove finite-generation, finite-future, symmetry, and
finite-mode boundary-data obstructions; same-patch diffeomorphism and
gauge covariance; calibrated finite-shot bounds; and complete-channel
transport certificates. An explicit moving-carrier model realizes
buffered null-endpoint approximations only after additional transport
resources are declared. Regularity and metric-jet counterexamples
prevent transfers to rough-conductivity uniqueness, scalar-potential
recovery, Einstein dynamics, or the inherited characteristic quantum
coefficient. The general operational inverse bridge remains
conditional, and the separate analytic admission chain remains open.
\end{abstract}

\tableofcontents

\section{Claim freeze, source status, and dependencies}
\label{sec:status}

\subsection{The three layers}

The proposed programme contains three mathematically different layers:
\begin{enumerate}
\item \textbf{Operational and representation layer.}
A declared quantum-process family, its accessible intervention
statistics, and an extracted event order are represented by a
faithful generated affine cone.
\item \textbf{Boundary identification layer.}
An independently justified observation rule identifies a complete
zero-frequency same-patch Dirichlet-to-Neumann energy form with the
operational data. An inverse theorem, on a specified coefficient
class, then identifies a Riemannian metric and possibly a unitary
connection modulo their actual invariances.
\item \textbf{Transport and dynamics layer.}
Additional resources realize quantum transport under a separately
calibrated geometric interpretation. Additional metric-jet and
source rules are needed for any dynamical comparison.
\end{enumerate}

An implication between these layers requires a printed mapping
theorem. Similar terminology is not such a theorem.

\subsection{Family 365 is documentary, not a premise}

The inspected catalogue records attribute to OpenAI:
\begin{itemize}
\item 365-01, dated 5 October 2026:
\emph{Determination of a metric and a unitary connection from one
boundary patch};
\item 365-02, dated 24 September 2026:
\emph{Smooth anisotropic uniqueness in the Calder\'on problem from
one boundary patch};
\item 365-03, dated 23 September 2026:
\emph{Nonuniqueness for bounded measurable scalar conductivities in
three dimensions}.
\end{itemize}
The first two abstracts concern \emph{Riemannian} elliptic inverse
problems, not Lorentzian causal reconstruction. The catalogue
describes the third as a bounded-measurable scalar-conductivity
counterexample. Its displayed Lean scope record names 365-03, not
365-01 or 365-02 \cite{OAI365a,OAI365b,OAI365c}.

These statements are neither independently verified here nor used as
theorem premises. The inspected pages did not provide an accessible
complete proof through their displayed source sections. The supplied
excerpts contain selected arguments, but not an executed verification
of the full inverse claims.

For comparison, established inverse-problem literature includes
smooth isotropic full-boundary uniqueness \cite{SylvesterUhlmann},
real-analytic connection-Laplacian recovery
\cite{GabdurakhmanovKokarev}, and Poisson-embedding methods
\cite{LassasLiimatainenSalo}. Their hypotheses must not be silently
replaced by the stronger catalogue descriptions.

\subsection{Cheung manuscript roles}

The operational target and its acceptance discipline are documented in
Cheung's XXII-A--F \cite{CheungXXIIA,CheungXXIIB,CheungXXIIC,
CheungXXIID,CheungXXIIE,CheungXXIIF}. We use the supplied identifiers
and statements as comparisons. The proofs below establish their own
interfaces.

In particular:
\begin{itemize}
\item XXII-C uses a \emph{countably indexed family} of separately tested
tripartite processes. It is not one finite process with infinitely
many event occurrences.
\item Instrument settings are not additional event occurrences.
Experiment identifiers are not physical times or positions.
\item Strongly connected components yield a partial order, but neither
a manifold nor a metric.
\item XXII-F proves a conditional implication. Its nonempty operational
instantiation is a separate question.
\item En10-A and En10-B provide finite-estimation and
physical-identification comparisons, not an already constructed
boundary apparatus
\cite{CheungEn10A,CheungEn10B}.
\end{itemize}

The fact that source passages for En10-A are supplied with the present
question does not retroactively make the identifier
\texttt{En10-A@1} a theorem premise of XXII-F. We re-prove the
elementary statistical statements used here.

\subsection{Exact supported claim}

The manuscript proves:
\begin{enumerate}
\item an explicit nonempty, enriched quantum-process pilot with a
faithful generated Lorentz representation;
\item an exact boundary dictionary on one predeclared
one-parameter smooth comparison class;
\item a general \emph{conditional} boundary-identification theorem;
\item conditional transport certificates and an explicitly
resource-dependent transport pilot;
\item obstructions that delimit every stronger interpretation.
\end{enumerate}

It does not prove general smooth same-patch inverse uniqueness,
construct a general physical DN acquisition apparatus, derive
Einstein dynamics, or construct the inherited characteristic quantum
coefficient.

\section{Quantum-process data and the event carrier}
\label{sec:operational}

\subsection{Finite process convention}

Each laboratory \(a\in\{A,B,C\}\) has qubit input and output spaces.
For a completely positive map \(\mathcal M\), use the unnormalized
Choi operator
\[
J(\mathcal M)=
\sum_{i,j=0}^1\kb i j\otimes\mathcal M(\kb i j).
\]
A channel satisfies
\[
J(\mathcal M)\ge0,\qquad
\Tr_{\mathrm{out}}J(\mathcal M)=I_{\mathrm{in}}.
\]

A tripartite process is a positive operator \(W\) on the six-qubit
input-output space such that
\[
P_W(a,b,c)=
\Tr\!\left[
W\bigl(J(\mathcal M_A^a)\otimes
       J(\mathcal M_B^b)\otimes
       J(\mathcal M_C^c)\bigr)
\right]
\]
is normalized for every triple of local instruments. We use this
explicit Choi convention throughout. In this convention the initial
state factor of the comb below is transposed.

This is a finite process-matrix interface of the type introduced in
\cite{OCB}. Nothing in its definition assigns spatial positions.

Let \(I\) be countable. The datum consists of the separately tested
family \((W_e)_{e\in I}\), the declared local access, its complete
probability functionals, and the derived strengths below.

\subsection{Unconditional signalling}

For distinct parties \(a,b\), and remaining party \(c\), compare two
sender channels \(S,T\), one receiver instrument \((B_z)_z\), and the
same third-party channel \(M_c\). Define
\[
\Delta_e(a,b)
=
\sup_{S,T,(B_z),M_c}
\frac12\sum_z
\left|
\Tr\!\left[
W_e\bigl((J(S)-J(T))_a\otimes J(B_z)_b\otimes J(M_c)_c\bigr)
\right]
\right|.
\]
The other parties' outcomes are summed. There is no normalized
postselection.

\begin{lemma}[Binary reduction and attainment]
\label{lem:strength}
The strength is the maximum of the absolute difference of one
receiver-event probability. It lies in \([0,1]\) and is attained.
\end{lemma}

\begin{proof}
For finite probability laws, total variation equals the maximum
absolute difference over subsets of outcomes. Aggregate such a subset
into one binary instrument element.

The Choi operators of channels and binary instrument elements form
closed bounded sets in finite-dimensional spaces. For a binary element
\(B\), the inequalities are
\[
B\ge0,\qquad \Tr_{\mathrm{out}}B\le I.
\]
Every such \(B\) can be completed to an instrument: if
\(R=I-\Tr_{\mathrm{out}}B\ge0\), add \(R\otimes\tau\), where
\(\tau\) is a fixed output state. The resulting optimization is
continuous on a compact set, proving attainment.
\end{proof}

Draw an edge \((e,a)\to(e,b)\) when \(\Delta_e(a,b)>0\).
No edge joins different trials.

\begin{proposition}[Component order and finite futures]
\label{prop:order}
Collapse strongly connected components and order the quotient by
reachability. The result is a partial order. Every component has at
most three components in its future, including itself.
\end{proposition}

\begin{proof}
Reachability is reflexive and transitive. Mutual reachability is
exactly the equivalence relation being collapsed, so the quotient is
antisymmetric. A directed path stays within one trial, whose graph has
three vertices.
\end{proof}

This re-proves the order interface of XXII-C
\cite{CheungXXIIC}; it does not identify that order with supplied
geometric causality.

\subsection{Faithful generated affine representations}

\begin{definition}
\label{def:representation}
For a declared event poset \((E,\preceq)\), a faithful generated affine
representation is \((V,\iota,K)\), where \(V\) is finite-dimensional,
\(\iota:E\to V\) is injective, \(K\) is closed, pointed,
full-dimensional, and convex, and
\begin{align}
p\preceq q
&\Longleftrightarrow \iota(q)-\iota(p)\in K,
\label{eq:faithful}\\
V&=\Span\{\iota(q)-\iota(p):p,q\in E\},
\label{eq:span}\\
K&=\overline{\cone}
\{\iota(q)-\iota(p):p\preceq q,\ p\ne q\}.
\label{eq:generated}
\end{align}
The closure and the event carrier are part of the specification.
\end{definition}

These are the generation and non-circularity requirements of XXII-A
in an explicit notation \cite{CheungXXIIA}.

\begin{proposition}[Finite-generation obstruction]
\label{prop:finite}
A finite event carrier cannot satisfy
\eqref{eq:generated} with \(K\) linearly isomorphic to
\[
\Lcone_m=\{(t,x)\in\R\times\R^{m-1}:t\ge\|x\|_2\},
\qquad m\ge3.
\]
\end{proposition}

\begin{proof}
A finitely generated cone is closed and polyhedral and has only
finitely many extreme rays. The Lorentz cone has the distinct
extreme rays
\[
\R_{\ge0}(1,u),\qquad u\in S^{m-2}.
\]
For example, the nonnegative functional
\((t,x)\mapsto t-u\cdot x\) vanishes on exactly that ray.
Invertible linear maps preserve extreme rays.
\end{proof}

\begin{proposition}[Repeated-block covariance obstruction]
\label{prop:symmetry}
Suppose infinitely many indistinguishable experiment blocks can be
permuted arbitrarily while preserving the datum. There is no injective
finite-dimensional representation that is covariant under all those
permutations by affine maps.
\end{proposition}

\begin{proof}
Choose finitely many represented points that affinely span the
representation's affine hull. Only finitely many blocks contain them.
Transpose two indistinguishable blocks outside this finite set.

Its implementing affine map fixes every point of the chosen affine
basis, hence is the identity on the affine hull. It must also exchange
two distinct represented events, contradicting injectivity.
\end{proof}

The obstruction concerns a datum's automorphisms. Relabelling a datum
and transporting its distinct operational parameters is a different
operation.

\section{An explicit enriched-process Lorentz pilot}
\label{sec:pilot}

\subsection{The quantum processes}

Let \(\sigma_1,\sigma_2,\sigma_3\) be a declared Pauli frame, and set
\[
\rho_r=\frac12(I+r\cdot\sigma),\qquad
\mathcal D_p(X)=pX+(1-p)\Tr(X)I/2.
\]
Here \(\|r\|_2\le1\) and \(0<p<1\).

For each trial, prepare \(\rho_r\), send the output of \(A\) through
\(\mathcal D_p\) to \(B\), send the output of \(B\) through the identity
channel to \(C\), and discard the final output. Its process operator is
\begin{equation}
W(r,p)=
\rho_r^T{}_{A_I}\otimes
J(\mathcal D_p)_{A_OB_I}\otimes
J(\id)_{B_OC_I}\otimes I_{C_O}.
\label{eq:pilot-W}
\end{equation}
The two link Choi operators are real in the declared computational
basis.

\begin{lemma}[Validity and exact operational parameters]
\label{lem:pilot-process}
The operator \eqref{eq:pilot-W} is a valid process. Its probabilities
are
\[
\Tr\!\left[
\mathcal M_C^c\,
\mathcal M_B^b\,
\mathcal D_p\,
\mathcal M_A^a(\rho_r)
\right].
\]
Its strengths are
\[
\Delta(A,B)=p,\qquad
\Delta(A,C)=p,\qquad
\Delta(B,C)=1,
\]
and all reverse strengths vanish. The source vector \(r\) is acquired
from the root laboratory's Pauli outcome expectations.
\end{lemma}

\begin{proof}
Every tensor factor in \eqref{eq:pilot-W} is positive.
Contracting the initial factor \(\rho_r^T\) with
\(J(\mathcal M_A^a)\) produces
\(\mathcal M_A^a(\rho_r)\). Contraction with the depolarizing link,
then the identity link, gives the displayed sequential probability.
Summing all outcomes gives trace one.

For normalized states,
\[
\mathcal D_p(\rho)-\mathcal D_p(\rho')
=p(\rho-\rho').
\]
Trace distance contracts under later channels, so no sender
intervention can produce receiver total variation exceeding \(p\)
across the first link. Replacement preparation of two orthogonal
states, followed by their distinguishing measurement, attains \(p\).
For \(A\to C\), let \(B\) use the identity. The \(B\to C\) link is
the identity and permits perfect distinction. Later trace-preserving
operations cannot change earlier marginal outcome laws, giving zero
reverse strengths.

Measuring the Pauli effects \((I\pm\sigma_i)/2\) at \(A\), with
all later outcomes summed, gives expectation \(r_i\).
\end{proof}

Thus the component order in each trial is exactly a three-element
chain. The initial state is not inferred from the signalling strengths
alone.

\subsection{A predeclared enriched subclass}

\begin{specification}[Rich tagged class]
\label{spec:rich}
The datum belongs to the rich tagged class if:
\begin{enumerate}[label=\textup{(P\arabic*)}]
\item every trial has the form \eqref{eq:pilot-W}, with operationally
identified parameters \(r_e,p_e\);
\item \(\|r_e\|_2=1\), and \(\{r_e:e\in I\}\) is dense in \(S^2\);
\item the channel tags satisfy
\[
|p_e^{-1}-p_f^{-1}|\ge\frac12\qquad(e\ne f);
\]
\item the largest tag \(p_*=\max_e p_e\) lies in
\([1/2,2/3]\).
\end{enumerate}
The Pauli frame and its first axis are laboratory calibration data,
not physical spatial directions.
\end{specification}

Reciprocal separation makes the maximum well defined: a separated
subset of \((1,\infty)\) has only finitely many elements below any
fixed bound and has an attained minimum.

The conditions are exact structural assumptions, not assertions
certified by finitely many noisy shots.

\begin{proposition}[Non-vacuity]
\label{prop:nonempty}
The rich tagged class is nonempty. Indeed, for every \(s\in[1,2]\),
choose a countable dense sequence \(r_e\in S^2\) and set
\[
p_e=\frac{s}{e+s},\qquad e=1,2,\ldots.
\]
Then
\[
p_*=\frac{s}{1+s},\qquad
p_e^{-1}=1+\frac e s.
\]
\end{proposition}

\begin{proof}
Rational stereographic coordinates provide a countable dense subset:
\[
r(a,b)=
\frac{(2a,2b,1-a^2-b^2)}{1+a^2+b^2},
\qquad a,b\in\Q.
\]
Enumerate it. The reciprocal gaps are at least \(1/s\ge1/2\);
the maximum tag is the first, and it belongs to the required interval.
Validity follows from Lemma~\ref{lem:pilot-process}.
\end{proof}

This is non-vacuity of the presently specified subclass, not a claim
that an unspecified package in a predecessor manuscript has been
instantiated.

\subsection{The extraction rule}

Write each trial's operationally sorted chain as \(e_0\prec e_1\prec
e_2\). Put
\[
V=\R\times\R^3,\qquad
d_e=(1,r_e),\qquad
b_e=\frac{20}{p_e}(1,0,0),
\]
and define
\begin{equation}
\iota(e_k)=(0,b_e)+k\,d_e,\qquad k=0,1,2.
\label{eq:pilot-embedding}
\end{equation}
Define \(K\) by the actual generated-cone rule
\eqref{eq:generated}; do not append a cone independently.

The reciprocal tag, not the experiment's numerical name, determines
the affine offset.

\begin{theorem}[Faithful generated Lorentz representation]
\label{thm:pilot-cone}
For every datum in Specification~\ref{spec:rich},
\eqref{eq:pilot-embedding} is injective and satisfies
\eqref{eq:faithful}--\eqref{eq:generated}, with
\[
K=\Lcone_4.
\]
The cone is homogeneous and strictly convex in the cone sense.
The construction is covariant under trial relabellings carrying their
operational parameters and under matched common changes of the
declared Pauli frame.
\end{theorem}

\begin{proof}
Within a trial, the positive differences are \(d_e,2d_e\), so its
operational chain is represented correctly.

Across two trials the offset separation is at least ten. Since
\(k,l\in\{0,1,2\}\),
\[
\|b_f+l r_f-b_e-k r_e\|_2\ge10-4=6,
\qquad |l-k|\le2.
\]
Neither directed cross-trial difference lies in \(\Lcone_4\).
The points are also distinct.

Every generating displacement belongs to \(\Lcone_4\).
Conversely, density of the \(r_e\) places every \((1,u)\),
\(u\in S^2\), in the closed generated cone. If \(\|x\|\le t\),
write \(x/t\) as a convex combination of unit vectors when \(t>0\).
For \(x\ne0\), the combination of \(u=x/\|x\|\) and \(-u\) with
weights \((1+\|x\|/t)/2\) and \((1-\|x\|/t)/2\) suffices.
The zero case uses any antipodal pair. Hence the generated cone is
exactly \(\Lcone_4\). Its generators span \(V\).
The cross-trial estimate and the within-trial calculation now prove
faithfulness for the actual \(K\).

Two nonproportional nonzero future-null vectors have a sum in the
interior by the strict case of the triangle inequality. This is cone
strict convexity.

For homogeneity, rotate an interior vector \((t,x)\) so \(x\) lies on
the first spatial axis. With \(a=\sqrt{t^2-\|x\|^2}>0\), write
\(t=a\cosh\eta\), \(\|x\|=a\sinh\eta\).
A Lorentz boost sends \((a,0)\) to \((t,x)\), and positive scaling
relates all such \(a\). These invertible maps preserve the cone.

Trial relabelling merely transports \(r_e,p_e\).
A matched common unitary change of Pauli frame rotates both Bloch
vectors and the calibrated first axis by the same orthogonal map.
The corresponding spatial rotation, with identity on the first
coordinate, intertwines the representations.
\end{proof}

\begin{remark}[Why this is not merely renaming state positivity]
Qubit positivity supplies the bound on Bloch vectors. It does not by
itself supply an event order. Theorem~\ref{thm:pilot-cone} proves the
separate order-identification and cross-trial faithfulness statements.

The embedding is nevertheless a chosen mathematical dictionary.
Its offsets, coordinate units, and laboratory axes are not measured
physical locations. It establishes a representation of operational
comparisons, not physical spacetime.
\end{remark}

\begin{proposition}[Genuinely additional content]
The selecting package is not forced by the chain scheduler.
Replacing every \(r_e\) by one fixed pure Bloch vector leaves the
same directed strengths and the same chain scheduler, but its
generated comparable-displacement cone is a ray and does not span
four dimensions.
\end{proposition}

\begin{proof}
Lemma~\ref{lem:pilot-process} makes the strengths independent of \(r\).
With constant \(r\), all positive comparable differences are positive
multiples of one vector.
\end{proof}

Thus the richness requirement has content beyond scheduler
consistency, as required by the XXII-A/B audit
\cite{CheungXXIIA,CheungXXIIB}.

\section{Cone classification and conformal information}
\label{sec:classification}

\begin{theorem}[Conditional classification]
\label{thm:classification}
Suppose an independently constructed representation
\((V,\iota,K)\) satisfies Definition~\ref{def:representation},
\(\dim V=m\ge3\), and \(K\) is homogeneous and cone-strictly-convex.
Then \(K\) is linearly isomorphic to \(\Lcone_m\).
\end{theorem}

\begin{proof}
Apply the homogeneous strictly convex cone classification of
Ito and Louren\c{c}o \cite{ItoLourenco}. Its hypotheses concern the
constructed cone. They do not construct the representation.
\end{proof}

The explicit pilot proves its selecting properties directly; it does
not need this external theorem for its conclusion.

\begin{lemma}[Quadratic-form ambiguity]
\label{lem:quadratic}
In coordinates with \(K=\Lcone_m\), a nonzero quadratic form vanishing
on the future-null boundary is a scalar multiple of
\[
q(t,x)=t^2-\|x\|_2^2.
\]
If it is positive on the cone interior, the scalar is positive.
\end{lemma}

\begin{proof}
Write the form as
\(a t^2+2t b\cdot x+x^TCx\).
Evaluation at \((1,u)\) and \((1,-u)\), \(\|u\|=1\), gives \(b=0\).
Then \(u^TCu=-a\) for every unit \(u\), so \(C=-aI\).
The sign follows from evaluation at \((1,0)\).
\end{proof}

The scale is constant on this one affine carrier. A field of cones
would require additional local identifications and could leave a
position-dependent conformal factor. No numerical physical speed is
selected by a changeable ratio of spatial and temporal coordinate
units \cite{CheungEn08A,CheungXXIIE}.

\section{The same-patch elliptic boundary object}
\label{sec:DN}

\subsection{Variational definition}

Let \(M\) be compact, connected, and smooth, with nonempty smooth
boundary and dimension \(n\ge3\). Fix a nonempty open
\(\Gamma\subset\partial M\). Let \(g\) be a smooth Riemannian metric.
On \(M\times\C^2\), let
\[
\nabla_A=d+A,\qquad A^*=-A
\]
be a smooth unitary connection. Hermitian pairings are conjugate
linear in the first argument.

Set
\[
\HP=\overline{C_c^\infty(\Gamma;\C^2)}^{\,H^{1/2}(\partial M;\C^2)}.
\]
Thus boundary inputs vanish off the measured patch in the trace
sense.

For \(f\in\HP\), let \(u_f\) be the harmonic extension:
\[
\nabla_A^*\nabla_Au_f=0,\qquad u_f|_{\partial M}=f.
\]
Define
\begin{equation}
B_{g,A}^{\Gamma}(f,h)=
\int_M\langle\nabla_Au_f,\nabla_Au_h\rangle_g\,dV_g,
\qquad Q_{g,A}^{\Gamma}(f)=B_{g,A}^{\Gamma}(f,f).
\label{eq:DNform}
\end{equation}
This is the zero-frequency same-patch DN energy form. Its observation
variable \(h\) is also supported on \(\Gamma\).

\begin{lemma}[Well-posedness and continuity]
\label{lem:DNwell}
The extension exists uniquely and depends linearly and continuously on
\(f\). The form \(B_{g,A}^{\Gamma}\) is bounded, Hermitian, and
nonnegative on \(\HP\).
\end{lemma}

\begin{proof}
On zero-trace sections, the scalar Poincar\'e inequality and
\(|d|u||\le|\nabla_Au|\) bound \(\|u\|_2\) by
\(\|\nabla_Au\|_2\). Since \(A\) is bounded, the identity
\(du=\nabla_Au-Au\) then gives an equivalent \(H^1_0\) norm.
Consequently the connection energy is coercive there.

Choose a bounded trace extension \(F\) of \(f\), and solve for
\(v\in H^1_0\) the equation
\[
\int_M\langle\nabla_Aw,\nabla_Av\rangle\,dV_g
=
-\int_M\langle\nabla_Aw,\nabla_AF\rangle\,dV_g
\quad(w\in H^1_0).
\]
The Hilbert-space variational theorem gives a unique \(v\);
\(u_f=F+v\) is independent of the chosen extension.
The coercive estimate and bounded trace extension give continuous
dependence. The stated properties of \(B\) follow from its integral
definition.
\end{proof}

The standard Sobolev trace and variational framework used here is also
the framework for the connection-Laplacian inverse problem
\cite{GabdurakhmanovKokarev}.

\begin{lemma}[Polarization]
\label{lem:polarization}
The energy determines the whole complex form:
\[
B(f,h)=\frac14\bigl[
Q(f+h)-Q(f-h)-iQ(f+ih)+iQ(f-ih)
\bigr].
\]
\end{lemma}

\begin{proof}
Expand each quadratic expression using sesquilinearity.
The first difference is \(4\operatorname{Re}B(f,h)\).
The second, with the displayed signs, contributes
\(4i\operatorname{Im}B(f,h)\).
\end{proof}

\subsection{Actual invariances}

Let \(\Phi:M\to M\) be a smooth diffeomorphism fixing \(\Gamma\)
pointwise, and let \(U:M\to U(2)\) be smooth with \(U|_\Gamma=I\).
Set
\[
g'=\Phi^*g,\qquad
A'=U^{-1}\Phi^*A\,U+U^{-1}dU.
\]

\begin{proposition}[Same-patch diffeomorphism and gauge invariance]
\label{prop:DNcov}
For every \(f,h\in\HP\),
\[
B_{g',A'}^\Gamma(f,h)=B_{g,A}^\Gamma(f,h).
\]
\end{proposition}

\begin{proof}
The transformed extension is
\[
u'_f=U^{-1}(u_f\circ\Phi).
\]
It has trace \(f\): the transformation is the identity on \(\Gamma\),
and a boundary diffeomorphism fixing \(\Gamma\) maps its complement
to itself. The connection derivative transforms as
\[
\nabla_{A'}u'_f=U^{-1}\Phi^*(\nabla_Au_f).
\]
Unitarity and change of variables prove equality of energies and
pairings. The weak harmonic equations and uniqueness in
Lemma~\ref{lem:DNwell} identify the transformed solutions.
\end{proof}

This proof establishes invariance, not the converse uniqueness
asserted in 365-01.

\begin{remark}[Different symmetry groups]
The group of patch-fixing diffeomorphisms and unitary gauges is not
the affine covariance group of an event representation.
A gauge acts on internal fibres; an affine transformation acts on
event coordinates. An identification between their actions requires
an additional intertwining rule.
\end{remark}

\subsection{Connection transport}

For a piecewise smooth curve \(\gamma:[0,1]\to M\), define
\(T_A^\gamma\) by
\[
\frac{dT}{dt}=-A_{\gamma(t)}(\dot\gamma(t))T,\qquad T(0)=I.
\]

\begin{proposition}[Gauge-covariant parallel transport]
\label{prop:parallel}
The matrix \(T_A^\gamma\) is unitary. Under
\(A'=U^{-1}AU+U^{-1}dU\),
\[
T_{A'}^\gamma
=
U(\gamma(1))^{-1}T_A^\gamma U(\gamma(0)).
\]
In particular, transport between endpoints on \(\Gamma\) is unchanged
when \(U|_\Gamma=I\).
\end{proposition}

\begin{proof}
Skew-Hermiticity of \(A(\dot\gamma)\) gives
\(\frac{d}{dt}(T^*T)=0\).
Differentiate
\(U(\gamma(t))^{-1}T_A^{\gamma|_{[0,t]}}U(\gamma(0))\).
It obeys the transformed differential equation and initial condition.
\end{proof}

Connection parallel transport is not yet a physical quantum wire
along \(\gamma\), and specifies no arrival time.

\section{A rigorous restricted boundary dictionary}
\label{sec:restricted}

The general inverse problem is unnecessary for the following pilot.

Let \(M=\overline{B(0,1)}\subset\R^3\), choose any nonempty open
\(\Gamma\subset S^2\), and predeclare the comparison class
\[
g_s=s^2\delta,\qquad A=0,\qquad 1\le s\le2.
\]
The ball, patch, trivialization, and constant-scale ansatz are part of
the comparison model.

\begin{theorem}[Exact scaled-ball identification]
\label{thm:scaled}
For data in Specification~\ref{spec:rich}, put
\[
s(D)=\frac{p_*}{1-p_*}.
\]
The rule
\[
\mathcal B(D;f,h)=s(D)\,B_{\delta,0}^\Gamma(f,h)
\]
is exactly \(B_{g_{s(D)},0}^\Gamma\).
Different \(s\)'s have different same-patch forms and cannot be
identified by a patch-fixing isometry.

For the witness family of Proposition~\ref{prop:nonempty}, this rule
recovers its parameter \(s\).
\end{theorem}

\begin{proof}
For a constant metric scaling in dimension \(n\),
\[
|du|_{s^2g}^2\,dV_{s^2g}
=s^{n-2}|du|_g^2\,dV_g.
\]
The minimizing harmonic extension is unchanged because the energy
has been multiplied by a positive constant. Thus in dimension three
the DN energy is multiplied by \(s\).

Choose a nonconstant smooth patch-supported boundary section.
Its Euclidean energy is positive: a zero-energy harmonic extension
would be constant. Therefore equality of the forms forces equality
of \(s\).

A diffeomorphism fixing the open patch has identity derivative on
its tangent spaces. A patch-fixing isometry would therefore equate
\(s_1^2\delta\) and \(s_2^2\delta\) on a nonzero tangential vector,
forcing \(s_1=s_2\).
The formula for the witness family's maximum tag gives its claimed
recovery.
\end{proof}

\begin{remark}[Exact scope]
This is an explicit map from operational data to a genuine
same-patch boundary form, with a direct identification proof on its
restricted class. It is not a derivation of that class from unknown
boundary physics. The reference Euclidean form and the
one-parameter metric ansatz are installed in the comparator
dictionary.

Consequently this theorem does not establish general metric
emergence, general smooth anisotropic uniqueness, or a physical
boundary acquisition instrument. It does not invoke 365-01 or
365-02.
\end{remark}

Pure-gauge connections
\(A=U^{-1}dU\), \(U|_\Gamma=I\), may be included without changing
the data, by Proposition~\ref{prop:DNcov}. This adds the correct gauge
ambiguity but does not recover a nontrivial connection class.

\section{General boundary extraction: a conditional theorem and obstructions}
\label{sec:generalbridge}

\subsection{A named inverse hypothesis, not a catalogue transfer}

Fix a specified smooth coefficient class \(\mathscr C\) on a declared
manifold and patch. Use the equivalence generated by the transformations
in Proposition~\ref{prop:DNcov}.

\begin{definition}[Inverse hypothesis \(\mathsf U_{\mathscr C}\)]
The map
\[
[g,A]\longmapsto B_{g,A}^\Gamma
\]
is injective on \(\mathscr C\) modulo that equivalence.
\end{definition}

This hypothesis must be proved or imported with its exact
coefficient, boundary, and regularity assumptions before use.
No Family 365 claim is adopted as its proof.

\begin{specification}[Operational boundary-form package]
\label{spec:boundary}
A boundary-form package consists of:
\begin{enumerate}[label=\textup{(E\arabic*)}]
\item a named operational process class and observation access;
\item a fixed countable dense complex-rational vector subspace
\(\mathcal T\subset\HP\);
\item for each \(f\in\mathcal T\), a declared attempted-run
experiment with an unconditional probability \(p_D(f)<1\);
\item a proved observation identity
\[
p_D(f)=\frac{Q_{g,A}^\Gamma(f)}
                 {1+Q_{g,A}^\Gamma(f)}
\]
for some \((g,A)\in\mathscr C\), with consistent patch labels,
amplitude normalization, and calibration;
\item boundedness of the resulting polarized form on \(\HP\);
\item compatibility of relabellings and gauges with the operational
observation identities.
\end{enumerate}
A different known injective response function may replace the
displayed one if its inverse and error allowance are stated.
\end{specification}

\begin{theorem}[Conditional boundary identification]
\label{thm:boundary}
Under Specification~\ref{spec:boundary}, the exact operational
probabilities determine \(B_{g,A}^\Gamma\).
If \(\mathsf U_{\mathscr C}\) is additionally established, they
determine \([g,A]\) in that coefficient class.

The implication is proved. Existence of a general physical package
satisfying Specification~\ref{spec:boundary} is not proved here.
\end{theorem}

\begin{proof}
Invert the response:
\[
Q(f)=\frac{p_D(f)}{1-p_D(f)}.
\]
Polarization gives \(B(f,h)\) for \(f,h\in\mathcal T\).
Boundedness and density extend this uniquely to \(\HP\).

If two coefficient pairs produce the same probabilities, the
extensions of their forms are equal. Hypothesis
\(\mathsf U_{\mathscr C}\) then identifies their equivalence classes.
The compatibility clause makes the conclusion independent of the
permitted data presentations.
\end{proof}

\begin{remark}[A formal encoding is not acquisition]
Given an already known form \(Q\), one can prepare the qubit state
\[
(1-p(f))\kb00+p(f)\kb11,\qquad
p(f)=Q(f)/(1+Q(f)),
\]
and insert it at the root of a valid tripartite comb. Measuring it
encodes the energy in a finite record law.

This proves formal representability of an already supplied scalar
as a probability. It does not construct a physical transducer from
an unknown elliptic medium to that law. Inverting such a preparation
would merely recover information put into the preparation.
\end{remark}

\subsection{Why a fixed finite amplitude interface is insufficient}

\begin{lemma}[Infinite rank of the boundary energy]
\label{lem:infinite-rank}
The form \(B_{g,A}^\Gamma\) has infinite rank.
Its zero-energy subspace has complex dimension at most two.
\end{lemma}

\begin{proof}
If \(Q(f)=0\), then \(\nabla_Au_f=0\).
A parallel section on a connected manifold is determined by its value
in one fibre, since it is transported along every path by a linear
ordinary differential equation. The space of parallel sections has
dimension at most two. Its boundary traces therefore have dimension
at most two.

The patch-supported trace space \(\HP\) is infinite-dimensional.
A nonnegative form on it whose kernel has dimension at most two
cannot have finite rank.
\end{proof}

\begin{theorem}[Finite linear-amplitude obstruction]
\label{thm:finite-mode}
There do not exist a finite \(d\), a linear map
\(K:\HP\to\C^d\), and a positive matrix \(E\) such that
\[
Q_{g,A}^\Gamma(f)=\langle Kf,E Kf\rangle
\quad\text{for every }f\in\HP.
\]
\end{theorem}

\begin{proof}
The right-hand form has rank at most \(d\); alternatively its kernel
contains the finite-codimensional subspace \(\ker K\).
This contradicts Lemma~\ref{lem:infinite-rank}.
\end{proof}

This rules out a specified finite linear mode encoding, not every
nonlinear finite-parameter dictionary. In particular it does not
contradict Theorem~\ref{thm:scaled}, whose entire energy dependence
outside one scalar parameter is supplied by the reference model.

A bare normalized finite-dimensional state encoding also cannot
equate a fixed probability with energy at all amplitudes:
normalization erases \(f\mapsto cf\), whereas
\(Q(cf)=|c|^2Q(f)\).
An amplitude register, variable gain, nonlinear response, or another
specified resource is required.

\subsection{Finite information is not a complete boundary operator}

A fixed tripartite qubit process is a \(64\times64\) Hermitian matrix.
All its one-use probabilities are linear functionals of finitely many
real coordinates. Infinitely many instrument settings do not create
infinitely many independent process coordinates.

This observation alone is \emph{not} a cardinality proof excluding
every imaginable map from process matrices to metric classes.
The proved obstruction is Theorem~\ref{thm:finite-mode}, with its
explicit linear-amplitude hypotheses. A general inverse
identification claim requires its own observation law and comparison
class.

\section{Riemannian recovery does not select Lorentzian causality}
\label{sec:signature}

\begin{proposition}[No natural pointwise signature change]
\label{prop:signature}
For \(n\ge2\), there is no rule assigning to each positive-definite
inner product \(g\) on an \(n\)-dimensional vector space a Lorentzian
bilinear form on that same space, equivariantly under all linear
coordinate changes, using \(g\) alone.
\end{proposition}

\begin{proof}
At \(g=I\), equivariance forces the assigned form to be invariant under
every orthogonal transformation. Reflection of individual coordinates
forces off-diagonal entries to vanish, and coordinate permutations
force all diagonal entries to agree. Thus the form is \(\lambda I\),
which is definite or degenerate, not Lorentzian.
\end{proof}

Additional choices can produce Lorentzian comparators:
\begin{itemize}
\item with a unit one-form \(\alpha\), use \(g-2\alpha\otimes\alpha\);
\item with an added time factor, use
\[
G=-dT^2+g
\quad\text{on }\R\times M.
\]
\end{itemize}
The former introduces a direction; the latter introduces a dimension,
a clock, and a product structure.

For the restricted pilot, predeclare
\[
G_s=-dT^2+s^2|dy|^2.
\]
The map
\[
(t,x)\longmapsto (T,y)=(t,x/s)
\]
carries the abstract pilot cone to the future cone of \(G_s\).
This is a verified comparator identification, not a derivation of
the added time coordinate.

\begin{proposition}[Static data do not determine the lapse]
\label{prop:lapse}
Even exact knowledge of a spatial pair \((g,A)\) and its DN form
does not choose among
\[
G_N=-N(x)^2dT^2+g,\qquad N(x)>0.
\]
At a fixed spatial point, these comparators have different coordinate
null slopes when their lapse values differ.
\end{proposition}

\begin{proof}
The spatial operator \(\nabla_A^*\nabla_A\) and
\eqref{eq:DNform} do not contain \(N\).
The null equation is
\[
N(x)^2\,dT^2=g_x(dx,dx),
\]
which depends on \(N\).
\end{proof}

Thus even a successful smooth boundary inverse theorem would not,
without additional input, select an operational causal cone or a
clock calibration.

\subsection{Prospective kinetic Hamiltonian}

The Dirichlet connection-energy form defines the nonnegative
self-adjoint Dirichlet realization \(L_{g,A,D}\). Hence
\[
H_{\mathrm{cmp}}=\frac{\hbar^2}{2m_0}L_{g,A,D},
\qquad m_0>0,
\]
defines a comparator unitary group
\(e^{-iTH_{\mathrm{cmp}}/\hbar}\).
The coefficient \(m_0\), the external parameter \(T\), and adoption
of this Schr\"odinger law are additional modelling choices.

Patch-fixing isometries and gauges act by unitary conjugation on the
corresponding Dirichlet Hilbert spaces, as follows from the energy
identity. This is a mathematical covariance statement. It does not
supply the physical Hamiltonian of the operational processes or a
relativistic matter propagation law.

\section{Finite-shot acquisition and complete-channel error}
\label{sec:statistics}

\subsection{A directly observable pilot tag}

On the witness family of Proposition~\ref{prop:nonempty}, use its
declared first trial. Let \(A\) prepare \(\ket0\), and let \(B\)
measure the computational basis. The probability of outcome zero is
\[
q=\frac{1+p_*}{2}.
\]
All other outcomes and later operations are included, not
postselected.

\begin{theorem}[Pilot parameter certificate]
\label{thm:pilot-stat}
Use \(M\) independent attempted runs and let \(\overline Z\) be the
frequency of outcome zero. Define
\[
\widehat p=
\operatorname{proj}_{[1/2,\,2/3]}(2\overline Z-1),
\qquad
\widehat s=\frac{\widehat p}{1-\widehat p}.
\]
If the actual Bernoulli mean differs from \(q\) by at most
\(b_{\mathrm{cal}}\), then, with probability at least \(1-\alpha\),
\[
|\widehat s-s|
\le18\left(
b_{\mathrm{cal}}+
\sqrt{\frac{\log(2/\alpha)}{2M}}
\right).
\]
The right side may be truncated at one.
\end{theorem}

\begin{proof}
Hoeffding's inequality bounds
\(|\overline Z-q|\) by
\(b_{\mathrm{cal}}+\sqrt{\log(2/\alpha)/(2M)}\)
with the stated coverage \cite{Hoeffding}.
Projection cannot increase the error from \(p_*\), so
\(|\widehat p-p_*|\le2|\overline Z-q|\).
The derivative of \(p/(1-p)\) is at most nine on the specified
interval.
\end{proof}

Locating a maximum in an arbitrary countable data family is not
claimed to be a finite algorithm. The theorem uses the explicitly
identified first trial of the witness family.

This is the same affine Bernoulli-inference mechanism that can be
compared with En10-A; En10-B requires a further physical acceptance
package before identifying its parameter with a physical source
\cite{CheungEn10A,CheungEn10B}.

\subsection{General boundary-bank error}

For a finite set of boundary tests, suppose
\(p(f)\le1-\kappa\), \(0<\kappa<1\).
Project empirical probabilities into \([0,1-\kappa/2]\).
The inverse response \(F(p)=p/(1-p)\) has derivative at most
\(4/\kappa^2\) on that interval. Therefore
\[
|\widehat Q(f)-Q(f)|
\le\frac4{\kappa^2}|\widehat p(f)-p(f)|.
\]
If every energy used by polarization has error at most
\(\varepsilon_Q\), the resulting complex pairing has error at most
\(\varepsilon_Q\).

For \(K\) predeclared probability tests, independent \(M\)-shot
blocks and a union bound give simultaneous probability error
\[
b_{\mathrm{cal}}+
\sqrt{\frac{\log(2K/\alpha)}{2M}}
\]
with coverage at least \(1-\alpha\).
A calibration certificate valid only with probability \(1-\beta\)
reduces unconditional coverage to at least \(1-\alpha-\beta\).

These bounds concern finitely many form values. They do not give
metric stability, exact recovery from finitely many noisy records,
or a resource bound for a general inverse algorithm.

\subsection{Complete instruments rather than scalar records}

For compatible channels write
\[
\ddia(\Phi,\Psi)=\tfrac12\|\Phi-\Psi\|_\diamond.
\]
A complete target must declare every returned classical register and
quantum residual.

\begin{lemma}[Complete-channel telescoping]
\label{lem:telescoping}
If corresponding stages have half-diamond allowances
\(\eta_1,\ldots,\eta_L\), then
\[
\ddia(\widehat\Phi_L\cdots\widehat\Phi_1,
       \Phi_L\cdots\Phi_1)
\le\min\{1,\eta_1+\cdots+\eta_L\}.
\]
The allowance bounds all output event-probability differences and all
untouched-reference output trace distances.
\end{lemma}

\begin{proof}
Telescope the difference of compositions. CPTP pre- and
post-composition contract the diamond norm. The diamond norm includes
arbitrary untouched references, and measurement cannot increase trace
distance.
\end{proof}

Preservation of an unconditional reference marginal alone is weaker
than identity transmission. Likewise equal classical records need not
mean equal residual instruments \cite{CheungI,CheungVIIJ,
CheungEn09A}. For example, the identity channel and computational-basis
dephasing may return the same deterministic record while their
residual outputs on \(\ket+\) have trace distance \(1/2\).

\section{Transport: obstructions, certificates, and an added-resource pilot}
\label{sec:transport}

\subsection{The frozen finite-future obstruction survives}

\begin{theorem}[Anchored-null obstruction]
\label{thm:future}
For a fixed event \(p\) in the unchanged component carrier,
\[
F_p=\{\iota(q)-\iota(p):p\prec q\}
\]
has at most two elements. A convergent sequence in \(F_p\) is
eventually constant. Consequently no fixed original event
approximates the entire nonzero null boundary by its comparable
event differences.
\end{theorem}

\begin{proof}
Proposition~\ref{prop:order} bounds the strict future by two events.
A finite set is closed and its distinct elements have positive
minimum separation. The nonzero Lorentz null boundary contains
infinitely many vectors.
\end{proof}

The pilot's cone is generated using different trials. It does not
provide a large future to one fixed source. Adding boundary-form
uniqueness does not alter this combinatorial fact.

\subsection{Geometry does not supply quantum transmission}

\begin{proposition}[Classical-channel obstruction inside the pilot]
\label{prop:EB}
For the witness family, every first link with \(e\ge4\) is
entanglement breaking. Nevertheless its signalling strength is
positive.

Without an additional quantum side channel or preshared entanglement,
no local decoder after such a link approximates the unknown-qubit
identity channel with half-diamond error below \(1/2\).
\end{proposition}

\begin{proof}
For \(e\ge4\) and \(s\le2\),
\(p_e=s/(e+s)\le1/3\).
The normalized Choi state of \(\mathcal D_p\) is
\[
p\kb{\Omega}{\Omega}+(1-p)I_4/4,
\qquad
\ket\Omega=(\ket{00}+\ket{11})/\sqrt2.
\]
At \(p=1/3\) it is the separable mixture
\[
\frac16\sum_{r\in\{\pm e_1,\pm e_2,\pm e_3\}}
\rho_r\otimes\rho_r^T.
\]
For smaller \(p\) it is a convex mixture of that state and \(I_4/4\).
Equivalently, \(\mathcal D_p\) has a measure-and-prepare
implementation and is entanglement breaking.
The strength remains \(p_e>0\).

Local decoding preserves separability from an untouched reference.
Every separable two-qubit state has Bell overlap at most \(1/2\):
for pure product states this follows by Cauchy--Schwarz, and mixtures
preserve the bound. Testing the Bell projector against the identity
channel output gives half-diamond error at least \(1/2\).
\end{proof}

\subsection{Conditional endpoint certificate}

Fix coordinates in which the selected abstract cone is \(\Lcone_m\).
A target is an anchored pair
\[
\zeta=(a,v),\qquad v=(\tau,x),\quad \tau=\|x\|_2>0.
\]
Use the comparison norm
\(\|(t,x)\|_* = |t|+\|x\|_2\).

A compatible realization must declare:
\begin{itemize}
\item actual source and receiver coordinates \(a_{n,\zeta},b_{n,\zeta}\);
\item a finite complete channel
\(\Gamma_{n,\zeta}:\mathcal B(M_0)\to
 \mathcal B(M_0\otimes F_{n,\zeta})\);
\item an ideal complete target
\(\Theta_{n,\zeta}(X)=X\otimes\sigma_{n,\zeta}\);
\item matching factorizations, encoder and decoder allowances,
physical edge availability, and all resource counts.
\end{itemize}

\begin{theorem}[Conditional transport certificate]
\label{thm:transport}
Suppose uniformly over a nonempty declared target set:
\[
\|a_{n,\zeta}-a\|_*+
\|x_{n,\zeta}-x\|_2\le\delta_n,\qquad
|\tau_{n,\zeta}-\tau|\le w_n,
\]
where \(b_{n,\zeta}-a_{n,\zeta}=(\tau_{n,\zeta},x_{n,\zeta})\).
Suppose the complete channel has design, encoder, decoder, and
per-stage allowances \(\mu_n,\alpha_n,\beta_n,\eta_n\), with at most
\(H_n\) stages. Set
\[
\varepsilon_n=
\min\{1,\mu_n+\alpha_n+\beta_n+H_n\eta_n\}.
\]
Then the logical channel
\(\Lambda_{n,\zeta}=\Tr_F\Gamma_{n,\zeta}\) obeys
\[
\sup_\zeta
\left[
\frac{\|a_{n,\zeta}-a\|_*+
      \|x_{n,\zeta}-x\|_2+
      |\tau_{n,\zeta}-\tau|}{\ell}
+
\ddia(\Lambda_{n,\zeta},\id)
\right]
\le
\frac{\delta_n+w_n}{\ell}+\varepsilon_n
\]
for any declared comparison length \(\ell>0\).

The same channel allowance controls all message-reference outputs.
If the ideal complete channel has no failure or out-of-window
arrival, the actual probability of that event is at most
\(\varepsilon_n\).
\end{theorem}

\begin{proof}
Lemma~\ref{lem:telescoping} bounds the complete channel by
\(\varepsilon_n\). Partial trace contracts that bound and gives the
logical estimate. The geometric estimate is substitution.
Finally apply the complete-channel allowance to the declared failure
or arrival-record effect.
\end{proof}

This re-establishes the conditional certificate mechanism, not a
general realization theorem, in agreement with XXII-D/F
\cite{CheungXXIID,CheungXXIIF}.

\subsection{Moving arrival requires a declared topology}

\begin{proposition}[Arrival discontinuity]
A deterministic arrival time \(t_n\to t\), \(t_n\ne t\), need not
converge in total variation as a classical time record. A fixed
deadline can also have discontinuous acceptance probability.
\end{proposition}

\begin{proof}
The point masses \(\delta_{t_n}\) and \(\delta_t\) have total variation
distance one. If \(t_n=t+1/n\), the event \((-\infty,t]\) has
probabilities zero and one, respectively.
\end{proof}

A valid approximation may instead specify buffered windows,
finite calibrated bins with a margin, or bounded-Lipschitz time
tests. For a Lipschitz test \(f\),
\[
|f(t_n)-f(t)|\le\operatorname{Lip}(f)|t_n-t|.
\]
The topology cannot be inferred from a small coordinate error.

\subsection{An explicit added-resource realization}

\begin{specification}[Moving-carrier comparison library]
\label{spec:bus}
After the comparator \(G_s\) has been selected, allow:
\begin{enumerate}
\item stationary qubit memories at declared sites;
\item mobile qubit carriers whose centres follow preprogrammed
future-causal curves of \(G_s\), with identity internal propagation;
\item local SWAP gates only while a carrier and memory are
coincident;
\item an external calibrated clock and controllable gate generators.
\end{enumerate}
These are added resources, not operations obtained from the original
one-use processes.
\end{specification}

\begin{theorem}[Buffered null approximation in the declared library]
\label{thm:bus}
For every target \(v=(\tau,x)\in\partial\Lcone_4\setminus\{0\}\)
and integer \(L\ge1\), the library of
Specification~\ref{spec:bus} has a finite identity-transmission
protocol with:
\[
\text{spatial displacement }x/s,\qquad
\text{arrival delay }2/L,
\]
relative to the comparator-null travel time \(\tau\).
Its complete final channel is the input message at the receiver
tensor blank states on all other declared registers.

A realization retaining all registers uses \(2L+1\) qubits,
\(L\) carrier legs, and \(2L\) SWAP gates. With gate duration
\(\delta_L=L^{-2}\), the largest declared gate-generator norm is
\(\pi\hbar L^2\).
\end{theorem}

\begin{proof}
Place \(L+1\) memories along the straight spatial segment, with
successive displacement \(x/(sL)\). A carrier travels each segment
in time \(\tau/L\); its speed is comparator-null since
\(s\|x/(sL)\|=\tau/L\).

At each leg, SWAP the message from its current memory into a fresh
blank carrier, propagate it, and SWAP it into the next blank memory.
During both gates the carrier and memory are coincident and stationary.
There are \(2L\) dwell intervals of duration \(L^{-2}\).
Thus the total arrival time is \(\tau+2/L\), with the required
spatial endpoint.

A SWAP transfers the whole input operator, including every untouched
reference. Induction over the legs puts the unknown input at the last
memory and returns every earlier memory and every carrier blank.
This proves the complete channel identity.

The stationary memories number \(L+1\), and the carriers number
\(L\). The gate count is \(2L\).
For the SWAP operator \(S\),
\[
H_{\mathrm{swap}}
=\frac{\pi\hbar}{2\delta_L}(I-S)
\]
satisfies \(e^{-i\delta_LH_{\mathrm{swap}}/\hbar}=S\) and has norm
\(\pi\hbar/\delta_L\). The claimed generator bound follows.
\end{proof}

\begin{remark}[What was paid for]
The protocol uses unboundedly fast gate control as \(L\to\infty\),
declared directionally available moving carriers, an external clock,
and a geometric comparison model. With a fixed nonzero gate duration,
the displayed delay would not vanish.

It is not a realization within the original finite-future carrier,
a local relativistic field construction, or a derivation of physical
propagation from the DN form. Its resource ledger is part of its
statement, not a universal complexity bound.
\end{remark}

\subsection{Local errors may accumulate}

For
\[
\mathcal R_d(X)=\Tr(X)I_d/d,\qquad
\mathcal P_p=(1-p)\id+p\mathcal R_d,
\]
one has
\[
\mathcal P_p^L
=(1-p)^L\id+\bigl(1-(1-p)^L\bigr)\mathcal R_d.
\]
With \(p=\lambda/L\), each local error tends to zero, but testing a
maximally entangled input gives
\[
\liminf_{L\to\infty}
\ddia(\mathcal P_{\lambda/L}^L,\id)
\ge
(1-e^{-\lambda})(1-d^{-2})>0.
\]
This is the global-accumulation mechanism emphasized in En08-B
\cite{CheungEn08B}. Merely requiring \(\eta_n\to0\) is insufficient;
the sufficient certificate above requires \(H_n\eta_n\to0\).

\section{Regularity, metric jets, and dynamics separation}
\label{sec:dynamics}

\subsection{Conductivity is not automatically a smooth quantum potential}

For smooth positive scalar conductivity \(\sigma\) on a Euclidean
domain,
\[
g=\sigma^{2/(n-2)}\delta
\quad\Longrightarrow\quad
\sqrt{|g|}\,g^{ij}=\sigma\delta^{ij}.
\]
Thus the scalar conductivity energy is the corresponding Riemannian
Dirichlet energy. Smoothness and uniform positivity make this a
smooth metric comparison.

If only \(\sigma\in L^\infty\) is assumed, that formula gives only
rough coefficients. It does not produce an admissible smooth metric
in the earlier class.

Similarly, putting \(v=\sqrt{\sigma}\,u\) gives, in the smooth case,
\[
-\nabla\cdot\bigl(\sigma\nabla(v/\sqrt{\sigma})\bigr)
=
\sqrt{\sigma}\left(
-\Delta v+\frac{\Delta\sqrt{\sigma}}{\sqrt{\sigma}}v
\right).
\]
At bounded-measurable regularity,
\(\Delta\sqrt{\sigma}/\sqrt{\sigma}\) need not be an ordinary
function or even a canonically defined product of distributions.

Consequently, even if the counterexample asserted in 365-03 is
verified, it does not automatically provide two conventional smooth
quantum potentials with equal measurements. Nor does it refute a
smooth inverse theorem on a genuinely narrower class.
The distinction between smooth scalar uniqueness and lower-regularity
questions is already essential in the classical inverse literature
\cite{SylvesterUhlmann,OAI365c}.

\subsection{One affine quadratic form is flat}

\begin{proposition}[Flat affine representative]
\label{prop:flat}
Translate a constant nondegenerate quadratic form across its affine
carrier. Its Levi-Civita connection and curvature vanish in affine
coordinates. In particular, its Einstein tensor is zero.
\end{proposition}

\begin{proof}
All coefficient derivatives vanish. The Christoffel formula therefore
gives zero, and so does the curvature formula.
\end{proof}

The recovered comparator \(G_s=-dT^2+s^2|dy|^2\) is of this form.
It cannot satisfy a predeclared equation
\(G_{\mu\nu}=\kappa_{\mathrm{cmp}}T_{\mu\nu}\) with
\(\kappa_{\mathrm{cmp}}\ne0\) and a nonzero prescribed source.

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

In four-dimensional signature \((-+++)\), let
\[
\widetilde G=e^{2\varphi}\eta.
\]
It has exactly the same future cones as \(\eta\), with the same time
orientation.

\begin{proposition}[Same cones, different Einstein tensor]
\label{prop:conformal}
At a point where \(\varphi=0\) and \(d\varphi=0\),
\[
G_{\mu\nu}(\widetilde G)
=
2\bigl(\eta_{\mu\nu}\Box\varphi
             -\partial_\mu\partial_\nu\varphi\bigr).
\]
There are smooth compactly supported choices of \(\varphi\) for which
this is nonzero, although all cones agree with the flat ones.
\end{proposition}

\begin{proof}
Substitute \(e^{2\varphi}\eta\) into the Christoffel formula.
At the selected point the quadratic first-derivative terms vanish,
and contraction gives
\[
R_{\mu\nu}=-2\partial_\mu\partial_\nu\varphi
                         -\eta_{\mu\nu}\Box\varphi,
\qquad R=-6\Box\varphi.
\]
Subtracting half the scalar curvature times the metric proves the
formula.

Choose \(\varphi=a(y^1)^2/2\) near the point, multiplied by a smooth
cutoff equal to one there. Then
\(\Box\varphi=a\), and \(G_{00}=-2a\ne0\) if \(a\ne0\).
\end{proof}

This compares cones, not two equal DN forms.
It does not assert that a full smooth boundary measurement fails to
detect the conformal change.

\subsection{A Ward identity is not a source rule}

For any smooth scalar \(\chi\), the tensor
\[
I_{\mu\nu}=
\partial_\mu\partial_\nu\chi-\eta_{\mu\nu}\Box\chi
\]
is identically conserved:
\[
\partial^\mu I_{\mu\nu}=0.
\]
This follows by commuting flat derivatives.
Conservation therefore does not select a unique stress tensor.

Also \(H\) and \(H+cI\) induce identical state evolution,
since their propagators differ by a global phase. An operational
process table does not thereby determine an absolute energy
observable or a physical source normalization.

A dynamics comparison must separately specify:
\[
\mathcal J:\Data\to\{\text{metric two-jets}\},
\qquad
\mathcal S:\Data\to\{\text{source tensors}\},
\]
and then prove, or disprove on a stated domain,
\[
G^{(1)}(\mathcal J(D))
=\kappa_{\mathrm{cmp}}\mathcal S(D).
\]
Defining the source to be the left side is not an independent
operational derivation.

These are the geometric and source gaps isolated in XXII-E
\cite{CheungXXIIE}. A connection-Laplacian kinetic comparator does
not close them.

\section{Hypothesis and connection ledgers}
\label{sec:ledger}

\subsection{Acceptance ledger}

\begin{longtable}{@{}p{.07\linewidth}p{.41\linewidth}p{.45\linewidth}@{}}
\toprule
Item & Acceptance condition & Status in this paper\\
\midrule
\endfirsthead
\toprule
Item & Acceptance condition & Status in this paper\\
\midrule
\endhead
\bottomrule
\endfoot
A1 &
One named nonempty operational class, intervention access, and
one declared represented relation. &
The rich tagged comb class is explicit and nonempty.
Its represented relation is the SCC component order, not supplied
physical causality.\\[4pt]
A2 &
Matched covariance and a non-circular extraction rule. &
Proved for trial relabellings and the declared common Pauli frame.
Repeated indistinguishable blocks remain subject to
Proposition~\ref{prop:symmetry}.\\[4pt]
A3 &
Faithfulness, spanning, generation, and selecting properties. &
Proved on the pilot's countable carrier.
The wider classification theorem remains conditional on its
representation hypotheses.\\[4pt]
A4 &
Compatible transport, complete outputs, arrivals, and resource
bounds. &
Proved certificates under a supplied realization.
The moving-carrier pilot is an explicit enlargement, not an
operation of the original carrier.\\[4pt]
A5 &
Smooth elliptic coefficients, a valid boundary observation law,
and an appropriate inverse theorem. &
The scaled-ball comparison is proved directly.
General boundary acquisition and
\(\mathsf U_{\mathscr C}\) remain separately required.\\[4pt]
A6 &
Metric jets, source interpretation, and separation from physical
emergence and Einstein dynamics. &
Separation and mismatch witnesses are proved.
No general dynamics-matching package is constructed.\\
\end{longtable}

\subsection{Connection ledger}

\begin{longtable}{@{}p{.23\linewidth}p{.35\linewidth}p{.35\linewidth}@{}}
\toprule
Connection & Established mathematical interface & Non-transfer boundary\\
\midrule
\endfirsthead
\toprule
Connection & Established mathematical interface & Non-transfer boundary\\
\midrule
\endhead
\bottomrule
\endfoot
XXII-A/B/C/F &
Event-order specification, faithful generation, covariance audit;
an explicit enriched pilot is supplied here. &
The pilot is not a general quantum-to-spacetime theorem.
The original finite futures remain finite.\\[4pt]
365-01/02 &
Same-patch DN energy is the named inverse target.
Gauge and diffeomorphism invariance are proved here. &
The catalogue supplies neither the observation law nor a verified
inverse theorem for use here. The target metric is Riemannian.\\[4pt]
En08-A and Ito--Louren\c{c}o &
Conditional classification after a proper homogeneous strictly
convex cone is constructed. &
Classification does not create event displacements, access,
or physical units.\\[4pt]
XXII-D and En08-B &
Complete-channel and endpoint approximation certificates. &
Geometry does not provide quantum wires, arrival topology,
directional availability, or bounded-cost control.\\[4pt]
En10-A/B &
Calibrated source-weight inference and the need for a physical
acceptance package. &
A source weight is not a boundary energy form.
The physical identification bridge is not automatic.\\[4pt]
En09-A &
Complete clock-conditioned extraction with separate covariance
requirements. &
A clock record is not a derived metric or autonomous time law.\\[4pt]
VII-J and En05-A &
Scheduled ancestry bounds actual influence for declared channel
networks; geometric embedding is additional. &
Scheduled reachability, influence, and supplied causal relation
are not identified.\\[4pt]
365-03 &
A documentary negative control for coefficient regularity. &
No transfer from rough scalar conductivity to smooth Lorentz
geometry or conventional quantum potentials is made.\\[4pt]
XXII-E &
Metric-jet and independent source interfaces. &
Conformal cones, static inverse uniqueness, and conservation
do not imply Einstein dynamics.\\
\end{longtable}

The additional catalogue families numbered
095, 211, 212, 215, 243, 265, 280, 282, 283, and 290 remain
documentary comparison pointers from the supplied plan.
No theorem from any of them is used in this manuscript.
No result is inferred from the filenames
\texttt{Conductivity.lean} or \texttt{271-04}, and no formal execution
is asserted.

\section{Conditional assembly and completion verdict}
\label{sec:assembly}

\begin{theorem}[Boundary-to-interior conditional assembly]
\label{thm:assembly}
Fix one operational domain and one predeclared extraction package.
Assume:
\begin{enumerate}[label=\textup{(\roman*)}]
\item its event representation is faithful, spanning, and generated;
\item its constructed cone is proper, homogeneous, strictly convex,
and has dimension at least three;
\item its probabilities satisfy
Specification~\ref{spec:boundary} on a specified smooth
coefficient class \(\mathscr C\);
\item \(\mathsf U_{\mathscr C}\) is independently established;
\item any identification between the affine event cone and a
Lorentzian interpretation of the recovered Riemannian geometry
is separately specified and proved;
\item any transport realization satisfies the complete-channel,
endpoint, calibration, and resource hypotheses of
Theorem~\ref{thm:transport}.
\end{enumerate}
Then:
\begin{enumerate}
\item the represented cone is Lorentzian and determines a constant
quadratic form up to positive scale on its affine carrier;
\item the operational boundary bank identifies \([g,A]\) in
\(\mathscr C\);
\item the supplied geometric interpretation has exactly its proved
covariance and no stronger one;
\item the supplied finite protocols satisfy the complete-channel
and endpoint certificate of Theorem~\ref{thm:transport}.
\end{enumerate}
No Einstein dynamics, physical metric emergence, or inherited
quantum-coefficient admission follows from these hypotheses alone.
\end{theorem}

\begin{proof}
Theorem~\ref{thm:classification} and
Lemma~\ref{lem:quadratic} give the first conclusion.
Theorem~\ref{thm:boundary} gives the second.
The third follows only through the separately supplied identification;
Propositions~\ref{prop:signature} and \ref{prop:lapse} explain why it
cannot be omitted. Theorem~\ref{thm:transport} proves the fourth.

The hypotheses include no independent source law or metric-jet
dynamics. Propositions~\ref{prop:flat} and \ref{prop:conformal} and
the conserved-improvement example show why cone information alone
cannot replace them. No hypothesis defines the separate
characteristic quantum product.
\end{proof}

\begin{corollary}[Explicit restricted realization and its boundary]
Theorems~\ref{thm:pilot-cone} and \ref{thm:scaled} give a nonempty
explicit operational representation and restricted boundary
identification example without invoking any Family 365 claim.
After adding Specification~\ref{spec:bus}, the transport pilot gives
buffered null-endpoint approximation under its stated resource growth.

These facts do not instantiate the general smooth inverse bridge
from unknown physical boundary data, and do not eliminate the original
finite-future obstruction.
\end{corollary}

\begin{proof}
Use Proposition~\ref{prop:nonempty} for non-vacuity and the cited
theorems for the constructions.
Theorem~\ref{thm:scaled} expressly uses a supplied comparison class;
Theorem~\ref{thm:bus} expressly adds new transport resources.
Theorem~\ref{thm:future} continues to apply to the unchanged carrier.
\end{proof}

\subsection{Disposition of the proposed workflow}

\begin{enumerate}
\item \textbf{Freeze the claim.}
Sections~\ref{sec:status} and \ref{sec:assembly} distinguish the
conditional theorem, explicit restricted pilots, and absent general
physical instantiation.
\item \textbf{Hypothesis ledger.}
Section~\ref{sec:ledger} gives A1--A6 with separate acceptance
conditions and actual statuses.
\item \textbf{Finite process examples and audits.}
Section~\ref{sec:pilot} constructs finite processes in a countable
family. Sections~\ref{sec:operational} and
\ref{sec:generalbridge} prove the relevant finite and symmetry
obstructions.
\item \textbf{Classification and transport.}
The selecting properties are proved on the pilot and classified
conditionally in general. The transport construction and certificate
retain all additional access and resource assumptions.
\item \textbf{Connection ledger.}
Section~\ref{sec:ledger} records named-object connections without
automatic theorem transfer.
\item \textbf{Companion emergence paragraph.}
The following is a proposed replacement/addition, not an assertion
that the external companion has been edited.
\end{enumerate}

\subsection*{Proposed companion paragraph}

\begin{quote}
A sharpened intermediate objective is boundary-to-interior
identification of an operationally represented Lorentz cone:
one must connect a declared quantum intervention package to a complete
same-patch zero-frequency DN energy form, apply a separately justified
inverse theorem on its actual regularity class, and supply an
independent identification of the recovered Riemannian data with the
event representation and its calibrated transport model.
Oai365-A supplies an explicit enriched-process Lorentz pilot, a
restricted scaled-ball boundary dictionary, and conditional general
and transport certificates; it does not supply general physical
boundary acquisition or derive Einstein dynamics.
The OpenAI 365 entries remain unverified documentary resources, not
premises. This does not solve standing problem 6.
\end{quote}

\section{Debugging summary}
\label{sec:debug}

The following corrections are necessary for a defensible submission:
\begin{enumerate}
\item The 365 inverse target is Riemannian. Changing signature requires
new data or a declared comparison choice.
\item Rank two is an internal bundle rank, not the dimension of the
boundary test space or of spacetime.
\item A countable family of finite processes is not one finite
generated event cone.
\item Infinite instrument settings are not infinitely many causal
event occurrences.
\item State positivity is not an event cone until faithfulness and
generation are proved.
\item The explicit pilot's trial offsets use observed channel tags,
not unobserved geometric positions or bare experiment numbering.
\item Exact richness, reciprocal separation, and closure are structural
hypotheses, not finite empirical certificates.
\item A restricted boundary dictionary can be exact while still
installing its comparison class.
\item DN uniqueness identifies coefficient classes; it does not select
a quantum channel among available futures.
\item An internal connection's parallel transport is not automatically
physical message transport.
\item Arrival coordinates require a time-record topology and
calibration margin.
\item Vanishing per-step error is not global channel convergence.
\item Smooth uniqueness and rough-conductivity nonuniqueness concern
different coefficient classes.
\item A conformal cone field does not specify the metric two-jet,
source law, or Einstein tensor.
\item A complete conditional proof is not a proof that every premise
has an operational realization.
\end{enumerate}

\appendix

\section{The inherited characteristic quantum programme is separate}
\label{app:analytic}

The supplied XIV-H/I/J and XV-B passages concern a different
distributional construction
\cite{CheungXIVH,CheungXIVI,CheungXIVJ,CheungXVB}.
None of the finite process matrices, Sobolev DN forms, or comparator
transport channels constructed above is its complete characteristic
graph.

For completeness, the elementary algebraic interface is:
a linear candidate \(\widetilde M:P\to Y\) factors through a surjective
reconstruction \(\Rec:P\to K\) if and only if
\[
\ker\Rec\subseteq\ker\widetilde M.
\]
Necessity follows by applying the factorization to a null
presentation. For sufficiency, set
\(M(\Rec p)=\widetilde M(p)\); the kernel condition gives
well-definedness.

The identity
\[
q\delta'(q)+\delta(q)=0
\]
shows why highest polynomial presentation degree alone need not descend.
It follows directly by testing against \(\varphi\):
\[
\langle q\delta',\varphi\rangle
=-(q\varphi)'(0)=-\varphi(0).
\]
This condition does not construct a missing distributional product.

\begin{center}
\small
\begin{tabularx}{\linewidth}{@{}p{.06\linewidth}p{.47\linewidth}X@{}}
\toprule
Gate & Required object & Disposition here\\
\midrule
1 & Full joint characteristic map on tests meeting the locus &
Not constructed; no universal nonexistence theorem asserted.\\[4pt]
2 & Joint section and complete controlled remainder &
Not supplied by the internal amplitude or supported-sector reference.\\[4pt]
3 & Uniform transverse estimates for that joint object &
Not inferred from internal frequency homogeneity.\\[4pt]
4 & Compatibility with permitted contacts and singular spectators &
Not checked for an absent full characteristic construction.\\[4pt]
5 & Complete output wavefront theorem after the actual pushforward &
Not supplied by an internal or off-locus wavefront bound.\\[4pt]
6 & Normalized admission \(Q_HR_1^{(1)}\in\mathscr E_H\) &
Not established.\\
\bottomrule
\end{tabularx}
\end{center}

Thus the supplied I2 internal-amplitude advance is not withdrawn,
but the full I3/J2 map and its later admission chain are not completed.
The technical XV-B manuscript may exist while its downstream
quantum-admission exit remains unopened.

The conditional Route A obstruction in XVI-N is not changed.
Its universal completion obligations and Route B are not discharged
\cite{CheungXVIN}. No conclusion here updates separately scoped
bulk-sector statements in other manuscripts.

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

\paragraph{Supported submission.}
The supported paper is a collection of explicit constructions,
conditional assembly theorems, and scoped obstructions.
It is not an unconditional completion of the general operational
inverse problem.

\paragraph{Attribution.}
Oai365-A is the requested manuscript identifier.
No approval or endorsement by N.~Cheung, OpenAI's catalogue authors,
or the authors of the cited literature is asserted.
Cheung manuscripts are cited by supplied identifier, version where
available, and relevant internal labels.

\paragraph{Source limitations.}
The three requested catalogue pages and the registered XXII
interfaces were inspected. The catalogue claims are not thereby
verified. No unseen proof, source file, or formal execution is
represented as inspected or executed.

\paragraph{Verification and execution.}
No successful \texttt{pdflatex} execution, proof-assistant run,
numerical optimization, physical implementation, independent
referee approval, or journal acceptance is asserted.
The source is intended as one self-contained \texttt{pdflatex} file.

\paragraph{Unchanged boundaries.}
For the inherited programme discussed here,
\[
\boxed{
\begin{gathered}
\text{O.1 remains \texttt{NOT\_CLOSED}},\\
\text{XV-B's downstream admission exit is not opened},\\
\text{XVI-N's universal obligations and Route B are not discharged},\\
\text{no inherited characteristic quantum coefficient is admitted}.
\end{gathered}}
\]
No compilation is asserted. No holographic reconstruction, autonomous
quantum time, gravitational dynamics, or solution of a standing
quantum-gravity problem is claimed.

\begin{thebibliography}{99}

\bibitem{OAI365a}
OpenAI,
\emph{Determination of a metric and a unitary connection from one
boundary patch},
catalogue manuscript 365-01, 5 October 2026.
External claim; not verified or used as a premise.
Accessed 14 October 2026.
\url{https://enter.grok.me/openai/365-01}

\bibitem{OAI365b}
OpenAI,
\emph{Smooth anisotropic uniqueness in the Calder\'on problem from
one boundary patch},
catalogue manuscript 365-02, 24 September 2026.
External claim; not verified or used as a premise.
Accessed 14 October 2026.
\url{https://enter.grok.me/openai/365-02}

\bibitem{OAI365c}
OpenAI,
\emph{Nonuniqueness for bounded measurable scalar conductivities in
three dimensions},
catalogue manuscript 365-03, 23 September 2026.
External counterexample claim and formalization-scope record;
not verified or used as a premise.
Accessed 14 October 2026.
\url{https://enter.grok.me/openai/365-03}

\bibitem{ItoLourenco}
M.~Ito and B.~F.~Louren\c{c}o,
``The \(p\)-cones in dimension \(n\ge3\) are not homogeneous when
\(p\ne2\),''
\emph{Linear Algebra and its Applications}
\textbf{533} (2017), 326--335.
doi:10.1016/j.laa.2017.07.029.
\url{https://arxiv.org/abs/1701.05801}

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

\bibitem{SylvesterUhlmann}
J.~Sylvester and G.~Uhlmann,
``A global uniqueness theorem for an inverse boundary value problem,''
\emph{Annals of Mathematics} \textbf{125} (1987), 153--169.
doi:10.2307/1971291.
\url{https://annals.math.princeton.edu/1987/125-1/p04}

\bibitem{GabdurakhmanovKokarev}
R.~Gabdurakhmanov and G.~Kokarev,
\emph{On Calder\'on's problem for the connection Laplacian},
preprint, 2021; subsequently published in
\emph{Proceedings of the Royal Society of Edinburgh Section A:
Mathematics}.
Real-analytic geometric hypotheses; not the general smooth
catalogue claim.
\url{https://arxiv.org/abs/2112.13419}

\bibitem{LassasLiimatainenSalo}
M.~Lassas, T.~Liimatainen, and M.~Salo,
``The Poisson embedding approach to the Calder\'on problem,''
\emph{Mathematische Annalen} \textbf{373} (2019), 151--185.
doi:10.1007/s00208-019-01818-3.
Geometric inverse-problem comparison only.

\bibitem{Hoeffding}
W.~Hoeffding,
``Probability inequalities for sums of bounded random variables,''
\emph{Journal of the American Statistical Association}
\textbf{58} (1963), 13--30.

\bibitem{CheungXXIIA}
N.~Cheung,
\emph{XXII-A: Freeze the precise claim and the gap},
version 1, manuscript \texttt{XXII-A.tex}.
Relevant identifiers:
\texttt{def:representation}, \texttt{target:selection},
\texttt{thm:classification}, and the acceptance conditions.
Enter Calculus Companion.
\url{https://enter.grok.me/browse/XXII-A}

\bibitem{CheungXXIIB}
N.~Cheung,
\emph{XXII-B: Catalogue the additional quantum data and the hypothesis
ledger},
version 1, manuscript \texttt{XXII-B.tex}.
Relevant identifiers:
\texttt{prop:finite-generation}, \texttt{prop:tolerance},
and the Stage 2 handoff.
Enter Calculus Companion.
\url{https://enter.grok.me/browse/XXII-B}

\bibitem{CheungXXIIICDummy}
N.~Cheung,
\emph{XXII series source-identification convention}.
This entry is not a theorem citation; all operative references use
the individual manuscript identifiers below.

\bibitem{CheungXXIIC}
N.~Cheung,
\emph{XXII-C: Freeze one quantum process class and the extraction
obligations},
version 1, manuscript \texttt{XXII-C.tex}.
Relevant identifiers:
\texttt{def:class}, \texttt{def:data}, \texttt{prop:binary},
\texttt{thm:order}, \texttt{target:selection}.
Enter Calculus Companion.
\url{https://enter.grok.me/browse/XXII-C}

\bibitem{CheungXXIID}
N.~Cheung,
\emph{XXII-D: Operational approximation of an extracted Lorentz cone},
version 1, manuscript \texttt{XXII-D.tex}.
Relevant identifiers:
\texttt{thm:anchored-obstruction},
\texttt{thm:geometry-not-transport},
\texttt{thm:certificate}, and the moving-arrival and resource audits.
Enter Calculus Companion.
\url{https://enter.grok.me/browse/XXII-D}

\bibitem{CheungXXIIE}
N.~Cheung,
\emph{XXII-E: Dynamics matching on an extracted Lorentz cone},
version 1, manuscript \texttt{XXII-E.tex}.
Relevant identifiers:
\texttt{thm:flat}, \texttt{prop:conformal},
\texttt{prop:energy}, and \texttt{target:dynamics}.
Enter Calculus Companion.
\url{https://enter.grok.me/browse/XXII-E}

\bibitem{CheungXXIIF}
N.~Cheung,
\emph{XXII-F},
version 1, manuscript \texttt{XXII-F.tex}.
Relevant identifiers:
\texttt{thm:assembly}, \texttt{cor:approximation},
\texttt{prop:symmetry}, and \texttt{target:existence}.
Conditional assembly is distinct from nonempty operational
instantiation.
Enter Calculus Companion.
\url{https://enter.grok.me/browse/XXII-F}

\bibitem{CheungEn08A}
N.~Cheung,
\emph{En08-A},
version 1, manuscript \texttt{En08-A.tex}.
Relevant identifiers:
\texttt{thm:main}, \texttt{thm:weaker},
and the conclusion's affine-representation and classification
boundaries.
Enter Calculus Companion.
\url{https://enter.grok.me/browse/En08-A}

\bibitem{CheungEn08B}
N.~Cheung,
\emph{En08-B},
version 1, manuscript \texttt{En08-B.tex}.
Relevant identifiers:
\texttt{thm:main}, \texttt{cor:operational-limit},
and \texttt{prop:accumulation}.
Prescribed-boundary approximation and global error accounting.
Enter Calculus Companion.
\url{https://enter.grok.me/browse/En08-B}

\bibitem{CheungEn10A}
N.~Cheung,
\emph{En10-A},
version 1, manuscript \texttt{En10-A.tex}.
Supplied theorem \texttt{thm:main} and fixed-shot statistical
assumptions.
Finite source-weight inference, not a DN boundary apparatus.
Not a theorem premise of XXII-F.
Enter Calculus Companion.
\url{https://enter.grok.me/browse/En10-A}

\bibitem{CheungEn10B}
N.~Cheung,
\emph{En10-B: Physical Acquisition and Identification for a Declared
Positron-Source Observable},
version 1, manuscript \texttt{En10-B.tex}.
Relevant identifiers:
\texttt{thm:derived}, \texttt{thm:calibrated},
\texttt{thm:main}, \texttt{thm:release}.
The physical acceptance package remains a separate obligation.
Enter Calculus Companion.
\url{https://enter.grok.me/browse/En10-B}

\bibitem{CheungEn09A}
N.~Cheung,
\emph{En09-A},
version 1, manuscript \texttt{En09-A.tex}.
Relevant identifier: \texttt{thm:main}.
Clock-conditioned extraction, destructive consumption, and the
additional covariance hypothesis.
Enter Calculus Companion.
\url{https://enter.grok.me/browse/En09-A}

\bibitem{CheungVIIJ}
N.~Cheung,
\emph{Landmark VII-J: Scheduled dependencies and certified physical
causality in localized quantum simulation},
version 1, manuscript \texttt{VII-J.tex}.
Relevant identifiers:
\texttt{thm:schedule}, \texttt{thm:nosignal},
\texttt{thm:physical}, and \texttt{thm:release}.
Enter Calculus Companion.
\url{https://enter.grok.me/browse/VII-J}

\bibitem{CheungI}
N.~Cheung,
\emph{Landmark I},
version 2, manuscript \texttt{I.tex}.
Relevant identifiers:
\texttt{thm:main}, \texttt{prop:pullback},
\texttt{ex:equal-records}.
Record compatibility is distinct from complete residual-instrument
equality.
Enter Calculus Companion.
\url{https://enter.grok.me/browse/I}

\bibitem{CheungXIVH}
N.~Cheung,
\emph{Landmark XIV-H: Internal kernel relations on the operator route},
version 1, manuscript \texttt{XIV-H.tex}.
Supplied identifiers:
\texttt{prop:K5exact}, \texttt{thm:H2}.
Documentary analytic boundary, not a premise of the finite pilot.
Enter Calculus Companion.
\url{https://enter.grok.me}

\bibitem{CheungXIVI}
N.~Cheung,
\emph{XIV-I},
version 1, manuscript \texttt{XIV-I.tex}.
Supplied identifiers:
\texttt{thm:I2}, \texttt{thm:I3}, \texttt{thm:release}.
Internal amplitude construction is distinct from the full joint map.
Enter Calculus Companion.
\url{https://enter.grok.me}

\bibitem{CheungXIVJ}
N.~Cheung,
\emph{XIV-J},
version 1, manuscript \texttt{XIV-J.tex}.
Supplied identifiers:
\texttt{prop:descent}, \texttt{lem:null},
\texttt{thm:reference}, \texttt{thm:J2},
\texttt{cor:noadmission}.
Enter Calculus Companion.
\url{https://enter.grok.me}

\bibitem{CheungXVB}
N.~Cheung,
\emph{XV-B},
version 2, manuscript \texttt{XV-B.tex}.
Supplied identifiers:
\texttt{thm:lifting}, \texttt{thm:jetnull},
\texttt{thm:release}.
Standalone lifting interfaces do not admit the inherited
characteristic coefficient.
Enter Calculus Companion.
\url{https://enter.grok.me}

\bibitem{CheungXVIN}
N.~Cheung,
\emph{XVI-N},
revision r12, manuscript \texttt{XVI-N.tex}.
Supplied identifier:
\texttt{thm:massless-obstruction}, with its Route A hypotheses and
undischarged Route B discussion.
Enter Calculus Companion.
\url{https://enter.grok.me}

\end{thebibliography}

\end{document}