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\title{\textbf{Oai212-A}\\
Quantum-to-first-passage interfaces:\\
exact engineered reductions, controlled comparisons,\\
and the boundary of shape-regularity transfer}
\author{Ngo Cheung}
\date{14 October 2026}

\begin{document}
\maketitle

\begin{abstract}
We develop the four proposed quantum-to-first-passage interfaces
without treating an external catalogue entry as a verified theorem.
For an explicitly controlled CNOT circuit, we prove a realization-wise
identity between a normalized squared commutator and exponential
first-passage infection on the undirected square lattice. For a
Poisson random-Clifford circuit, we derive a finite-volume comparison
with the same infection process, including its dependence on volume,
time, and local dimension. An independent-pair quench realizes
independent exponential local R\'enyi-2 entanglement costs; its
declared membrane optimization is exactly first-passage percolation,
and finite-dimensional truncations have explicit preparation and
optimization errors. Local measurements produce an atom at zero
and therefore invalidate an exact exponential reduction. An
advertised-service, serial-teleportation network has additive
first-passage route costs, whereas concurrent link generation
already violates additivity on a two-link chain.

The macroscopic transfer theorems are proved from the standard
first-passage shape theorem. Strict convexity and boundary
differentiability remain conditional on independent verification of
the corresponding claims in manuscript 212-02. These constructions
do not establish a generic entanglement-growth membrane, a
measurement-induced critical theory, or optimal routing in an
unrestricted repeater network. They do not construct the separate
characteristic quantum coefficient discussed in the supplied
Cheung manuscripts. Thus the requested unconditional completion of
the entire programme is not claimed.
\end{abstract}

\tableofcontents

\section{Status, dependencies, and the exact claim being submitted}
\label{sec:status}

\subsection{External record versus mathematical premise}

The supplied record identifies Family 212, manuscript 212-02, as an
OpenAI manuscript dated 24 September 2026, entitled
\emph{Strict convexity and differentiability of the planar exponential
first-passage limit shape}. The companion states that it does not
verify the proof or its formalization
\cite{OpenAI212,Companion212}.
The separate no-bigeodesics manuscript 212-01 is not used.

This paper neither verifies 212-02 nor reproduces its missing
intermediate arguments. In particular, a contraction argument for an
auxiliary quantity cannot be used here without the definitions and
proofs establishing that contraction on its complete admissible class.

We separate the following external assertions.

\begin{definition}[External regularity assertions]
\label{def:external}
Let $\mu_F$ be the time-constant norm for undirected nearest-neighbour
first-passage percolation on $\Z^2$ with i.i.d.\ edge law $F$.
\begin{enumerate}[label=\textup{(H\arabic*)}]
\item\label{hyp:exp}
For $F=\Exp(\lambda)$, $\lambda>0$, the unit ball of $\mu_F$
is strictly convex, and $\mu_F$ is Fr\'echet differentiable on
$\R^2\setminus\{0\}$.
\item\label{hyp:gamma}
For $F=\GammaLaw(\kappa,\lambda)$, $\kappa,\lambda>0$,
$\mu_F$ is Fr\'echet differentiable on $\R^2\setminus\{0\}$.
\end{enumerate}
These are conditional interfaces to the claims of 212-02, not
theorems proved or adopted unconditionally in this paper.
\end{definition}

The existence of the norm and the standard shape theorem are
established classical inputs, distinct from
Definition~\ref{def:external}
\cite{CoxDurrett,ADH}.

\subsection{What is proved here}

The unconditional results below concern explicitly specified models:
\begin{enumerate}
\item a flag-controlled CNOT growth circuit;
\item a Poisson random-Clifford support process and its finite-volume
large-dimension comparison;
\item a local independent-pair quench and its declared entropy-cost
optimization;
\item a monitored version of that quench, for which the exact
exponential interface fails;
\item a serial teleportation timing contract and a counterexample to
its unrestricted concurrent extension;
\item finite-volume, channel-error, preparation-error, and statistical
transfer bounds.
\end{enumerate}

These are not identifications of unspecified quantum models.
Constructing an engineered example proves existence within its
specified class; it does not prove universality.

\subsection{Documentary Cheung comparisons}

The supplied Cheung manuscripts are cited for their stated scope and
for elementary interface patterns that are re-proved when used below.
No theorem concerning another model is silently imported.

In particular:
\begin{itemize}
\item En05-A, En05-B, and En06-B distinguish scheduled envelopes,
prescribed speeds, and finite transport
\cite{CheungEn05A,CheungEn05B,CheungEn06B}.
\item En08-A distinguishes scheduler consistency from cone selection;
En08-B exhibits accumulation of local errors
\cite{CheungEn08A,CheungEn08B}.
\item VII-J distinguishes scheduled ancestry, operational influence,
and an independently supplied physical causal relation
\cite{CheungVIIJ}.
\item VIII-A and En09-A distinguish records from complete residual
outputs \cite{CheungVIIIA,CheungEn09A}.
\item XIX-C distinguishes finite data from a determined continuum
evolution \cite{CheungXIXC}.
\item En07-B distinguishes channel equality from thermodynamic cost
\cite{CheungEn07B}.
\item XIV-H, XIV-I, XIV-J, and the supplied XV-B revision concern a
different characteristic-composition programme
\cite{CheungXIVH,CheungXIVI,CheungXIVJ,CheungXVB}.
\end{itemize}

The present paper does not discharge that programme. The supplied
status conventions remain:
\[
\boxed{
\begin{gathered}
\text{O.1 remains NOT\_CLOSED},\\
\text{the downstream quantum-admission exit to XV-B is not opened},\\
\text{the universal XVI-N obligations and Route B are not discharged},\\
\text{no inherited gravitational quantum coefficient is admitted}.
\end{gathered}}
\]
This does not withdraw any separately stated conditional Route A
result in XVI-N \cite{CheungXVIN}.

\section{Classical first-passage process and finite-volume control}
\label{sec:fpp}

Let
\[
E_2=\bigl\{\{u,v\}:u,v\in\Z^2,\ |u-v|_1=1\bigr\}.
\]
Attach independent weights $\tau_e\sim\Exp(\lambda)$ to $e\in E_2$.
For a finite nearest-neighbour path $\gamma$, define
\[
\tau(\gamma)=\sum_{e\in\gamma}\tau_e,\qquad
T(u,v)=\inf_{\gamma:u\to v}\tau(\gamma).
\]
Since weights are positive, loops can be erased without increasing
cost. Let
\[
I(t)=\{x:T(0,x)\le t\},\qquad
A(t)=\bigcup_{x\in I(t)}
\left(x+[-\tfrac12,\tfrac12]^2\right).
\]

\subsection{A constructive finite-volume tail bound}

Write
\[
\Lambda_L=[-L,L]^2\cap\Z^2,\qquad
N_L=(2L+1)^2,\qquad M_L=4L(2L+1).
\]
Here $M_L$ counts undirected edges with both endpoints in $\Lambda_L$.

\begin{lemma}[Escape bound and non-explosion]
\label{lem:escape}
Fix $s>2\lambda$ and put
\[
r_s=\frac{3\lambda}{\lambda+s}<1.
\]
For $L\ge1$ and $t\ge0$,
\[
\Prob\!\left(
\text{a path from $0$ of cost at most $t$ reaches }
\partial\Lambda_L
\right)
\le
\frac{4}{3(1-r_s)}e^{st}r_s^L.
\]
Consequently $I(t)$ is finite almost surely for every finite $t$.
\end{lemma}

\begin{proof}
Any path reaching $\partial\Lambda_L$ has, after loop erasure,
at least $L$ edges. There are at most $4\cdot3^{k-1}$
self-avoiding paths of length $k$ starting at $0$. For each such path,
independence and Markov's inequality give
\[
\Prob\bigl(\tau(\gamma)\le t\bigr)
\le e^{st}\E e^{-s\tau(\gamma)}
=e^{st}\left(\frac{\lambda}{\lambda+s}\right)^k.
\]
Sum over $k\ge L$. The resulting bound tends to zero as $L\to\infty$.
An infinite $I(t)$ would produce paths reaching arbitrarily large
boundaries with cost at most $t$. This event has probability zero.
Take a countable intersection over integer $t$.
\end{proof}

\begin{remark}[Gamma version]
For $\GammaLaw(\kappa,\lambda)$ weights the same proof works with
\[
r_s=3\left(\frac{\lambda}{\lambda+s}\right)^\kappa,
\]
choosing $s$ sufficiently large that $r_s<1$.
\end{remark}

Let $T_L$ be passage time restricted to $\Lambda_L$.

\begin{corollary}[Exact agreement on a confinement event]
\label{cor:box}
On the event that no path of cost at most $t$ reaches
$\partial\Lambda_L$, the finite and infinite infected sets agree up to
time $t$:
\[
\{x\in\Lambda_L:T_L(0,x)\le u\}=I(u),
\qquad 0\le u\le t.
\]
The probability of disagreement is at most the bound in
Lemma~\ref{lem:escape}.
\end{corollary}

\begin{proof}
Restricted passage times are no smaller than unrestricted ones.
If an unrestricted path of cost at most $u\le t$ leaves the box,
its initial segment reaches the boundary with cost at most $t$.
On the stated event every relevant path remains inside.
\end{proof}

For $t=nt_0$ and $L=\lceil an\rceil$, choosing
\[
a>\frac{st_0}{-\log r_s}
\]
makes the escape allowance exponentially small in $n$.
This is an explicit finite-volume approximation, not an assertion
that a finite circuit determines a continuum model.

\subsection{The established shape input}

The classical shape theorem supplies a deterministic norm $\mu$
such that, almost surely, for every $\varepsilon>0$ and sufficiently
large $t$,
\[
(1-\varepsilon)t\Bcal
\subseteq A(t)
\subseteq(1+\varepsilon)t\Bcal,
\qquad
\Bcal=\{x:\mu(x)\le1\}.
\]
The harmless cell convention is included in this statement
\cite{CoxDurrett,ADH}.

We shall also use its uniform passage-time consequence:
for every fixed $R<\infty$,
\begin{equation}
\sup_{\substack{x\in\Z^2\\ |x|_2\le Rn}}
\frac{|T(0,x)-\mu(x)|}{n}\longrightarrow0
\quad\text{almost surely}.
\label{eq:uniform-shape}
\end{equation}
To see this consequence, apply the shape inclusions at all sufficiently
large passage times. They compare $T(0,x)$ and $\mu(x)$
multiplicatively outside the finite set infected before a fixed
threshold. The remaining finite set contributes $o(n)$; then let
$\varepsilon\downarrow0$.

\section{The macroscopic transfer theorem and geometric corrections}
\label{sec:transfer}

\subsection{Uniform comparison, not merely a sandwich}

\begin{theorem}[Transfer of an asymptotically identical passage front]
\label{thm:transfer}
Let $S_n(x)$ be nonnegative random arrival times, coupled to $T$.
For every fixed $R$, suppose there are $b_n(R)=o(n)$ and
$\epsilon_n(R)\to0$ such that
\[
\Prob\!\left(
\sup_{|x|_2\le Rn}|S_n(x)-T(0,x)|>b_n(R)
\right)\le\epsilon_n(R).
\]
Then for $t_0>0$ and $\delta>0$, with probability tending to one,
\[
\{x:\mu(x/n)\le t_0-\delta\}
\subseteq
\{x:S_n(x)\le nt_0\}
\]
and
\[
\{x:S_n(x)\le nt_0\}
\subseteq
\{x:\mu(x/n)\le t_0+\delta\},
\]
within any fixed observation box $|x|_2\le Rn$.
If the rescaled arrival sets are also contained in one fixed compact
set with probability tending to one, their filled versions converge
in Hausdorff distance in probability to $t_0\Bcal$.

For every fixed $a\in(0,1)$, the corresponding annealed level sets
\[
F_{n,a}(t_0)=
\left\{\frac{x}{n}:
\Prob(S_n(x)\le nt_0)\ge a\right\}
\]
have the same inner and outer macroscopic inclusions, under the same
uniform compact-containment condition.
\end{theorem}

\begin{proof}
On the comparison event and the event in
\eqref{eq:uniform-shape}, the sum of the two uniform errors is
less than $\delta n$ for sufficiently large $n$.
If $\mu(x/n)\le t_0-\delta$, this implies $S_n(x)\le nt_0$.
If $\mu(x/n)\ge t_0+\delta$, it implies $S_n(x)>nt_0$.
This proves the random-set inclusions.

Compact containment allows one observation box to contain the
entire set with high probability. The two norm balls approach
$t_0\Bcal$ in Hausdorff distance as $\delta\downarrow0$,
and lattice-cell errors are $O(n^{-1})$.

The probability of the common good event tends to one.
Uniformly over the inner set the arrival probability therefore
tends to one; uniformly over the outer complement it tends to zero.
Eventually these bounds lie on the appropriate sides of $a$.
\end{proof}

\begin{remark}[Averaging is not a realization-wise identification]
The last conclusion concerns deterministic level sets of
arrival probabilities. It does not identify one random front with
its average, nor control fluctuations on a smaller-than-linear scale.
\end{remark}

\begin{corollary}[Conditional regularity transfer]
\label{cor:regularity}
Under Theorem~\ref{thm:transfer}, independent verification of
\ref{hyp:exp} implies that the common limiting front has no flat
boundary segment and has a $C^1$ boundary.
Independent verification of \ref{hyp:gamma}, for a Gamma target,
implies the $C^1$ conclusion, but not the no-flat-segment conclusion.
\end{corollary}

\begin{proof}
The theorem identifies the limiting body with a positive scalar
multiple of the target body. Positive dilation preserves strict
convexity and $C^1$ regularity. The Gamma assertion contains no
strict-convexity hypothesis.
\end{proof}

\subsection{Differentiability and strict convexity are different}

A norm, as a function on all of $\R^2$, is not strictly convex along
positive rays. The correct meaning of a \emph{strictly convex norm}
here is that its unit ball is strictly convex.

\begin{lemma}[Differentiable norm gives a $C^1$ unit boundary]
\label{lem:c1}
If a norm $\nu$ is differentiable at every nonzero point, then
$\nabla\nu$ is continuous there and $\{\nu=1\}$ is a $C^1$ curve.
This does not imply strict convexity of the unit ball.
\end{lemma}

\begin{proof}
The gradients are locally bounded because a norm is Lipschitz.
If $x_j\to x\ne0$ and a subsequence of $\nabla\nu(x_j)$ converges
to $p$, the supporting inequalities
\[
\nu(y)\ge\nu(x_j)+\nabla\nu(x_j)\cdot(y-x_j)
\]
pass to the limit. Thus $p$ is a subgradient at $x$.
Differentiability makes it unique, namely $\nabla\nu(x)$.
Every convergent subsequence has this limit, proving continuity.

Positive homogeneity gives
$\nabla\nu(x)\cdot x=\nu(x)$.
The gradient is therefore nonzero on $\{\nu=1\}$, and the
$C^1$ implicit-function theorem applies.

For the last assertion, take the Minkowski functional of a
centrally symmetric stadium: a disk plus a nontrivial horizontal
segment. Its boundary is $C^1$ but has horizontal flat segments.
The unique-supporting-line criterion gives a differentiable
Minkowski functional.
\end{proof}

\begin{proposition}[Fixed comparison errors do not transfer regularity]
\label{prop:nonstable}
A bounded multiplicative comparison with a smooth strictly convex
body does not imply that the compared body is smooth or strictly
convex. Even arbitrarily small Hausdorff error does not imply either
property for the approximating bodies.
\end{proposition}

\begin{proof}
Inscribed regular polygons converge in Hausdorff distance to the
Euclidean disk. Each polygon has flat edges and corners.
For sufficiently many sides it lies between
$(1-\varepsilon)$ and $(1+\varepsilon)$ dilates of the disk.
Thus neither conclusion follows from a fixed comparison allowance.
\end{proof}

This is why a comparison must identify the \emph{limiting} norm,
as in Theorem~\ref{thm:transfer}, rather than merely bound it between
two different norms.

\subsection{Radial speed, normal speed, and promoted cones}

For a Euclidean unit vector $u$, the radial macroscopic speed is
\[
v_{\mathrm{rad}}(u)=\frac1{\mu(u)}.
\]
It is continuous because every norm is continuous and positive away
from zero. This continuity does not require 212-02.

The supporting-line velocity in unit normal direction $n$ is instead
\[
v_{\mathrm{normal}}(n)=h_{\Bcal}(n)
=\sup_{\mu(x)\le1}n\cdot x.
\]
These generally differ. Neither is, without additional physical
structure, an observer-invariant speed.

\begin{proposition}[Correct cone-promotion boundary]
\label{prop:cone}
For any norm $\nu$ on $\R^2$,
\[
K_\nu=\{(t,x)\in\R\times\R^2:t\ge\nu(x)\}
\]
is a three-dimensional closed proper convex cone.
If its base is strictly convex, it is strictly convex in the
usual proper-face sense. The dimension requirement in the
It\^o--Louren\c{c}o classification is therefore satisfied after this
promotion. The missing hypothesis is homogeneity, not dimension.

Strict convexity and a $C^1$ base alone do not imply that $K_\nu$
is linearly isomorphic to a Lorentz cone.
\end{proposition}

\begin{proof}
Convexity follows from the triangle inequality and homogeneity of
$\nu$. The cone is closed, pointed, and has nonempty interior.
A nontrivial higher-dimensional proper face would cut the section
$t=1$ in a nontrivial boundary segment; strict convexity excludes it.

The classification of strictly convex homogeneous proper cones
requires homogeneity \cite{ItoLourenco}. No such property follows
from the preceding facts.

For a counterexample, take
$\nu(x)=(x_1^4+x_2^4)^{1/4}$.
Its unit ball is smooth and strictly convex.
The associated $4$-cone is not homogeneous and is not a Lorentz
cone up to linear isomorphism \cite{ItoLourenco}.
\end{proof}

This corrects the dimensional shortcut in the proposed plan while
preserving its substantive refusal of an automatic Lorentzian
interpretation.

\section{Bridge 1: an exact controlled operator-spreading model}
\label{sec:bridge1}

\subsection{The local timer rule}

\begin{specification}[Flag-controlled CNOT growth circuit]
\label{spec:cnot}
On a finite connected induced lattice graph $G=(V,E)$:
\begin{enumerate}
\item Place one payload qubit at each vertex.
Attach an independent positive timer $\tau_e$ to each undirected edge.
\item Initialize classical flags $f_0=1$ and $f_x=0$ for $x\ne0$.
The flags are independent of every payload input.
\item When the first endpoint of an unarmed edge becomes flagged,
start its one-shot timer.
\item If its timer expires while exactly one endpoint is flagged,
apply a CNOT from that endpoint to the unflagged endpoint, then
flag the latter. If both endpoints are already flagged, cancel the
unused operation.
\item Keep all controller predecessors in the classical dependency
description. No quantum-state-dependent flag update is permitted.
\end{enumerate}
Timers and gates use an external calibrated time parameter.
The ideal CNOT has zero duration in this timing contract.
Finite-duration hardware is a separate approximation problem.
\end{specification}

\begin{lemma}[Timer arrivals are undirected first-passage times]
\label{lem:timers}
For every positive timer realization, the flagging time of $x$
in Specification~\ref{spec:cnot} equals $T_G(0,x)$.
The successful edges form a rooted tree.
For independent continuous timers, ties occur with probability zero.
\end{lemma}

\begin{proof}
A successful edge $u\to v$ flags $v$ at time $S(u)+\tau_{\{u,v\}}$,
so the successful ancestry chain gives a path of cost $S(v)$.
Thus $T_G(0,v)\le S(v)$.

Conversely, suppose a vertex of minimum $T_G$ among those not yet
flagged should have been reached earlier. On a minimizing path,
let $\{u,v\}$ be the first edge crossing from flagged to unflagged
vertices. Its timer began when $u$ was flagged.
Since $S(u)\le T_G(0,u)$ at that stage of the induction,
it expires no later than the path's proposed arrival at $v$.
If the operation was cancelled, $v$ was already flagged earlier;
otherwise it flags $v$. Both contradict the supposed delay.
Induction over the finite graph proves equality.

Each non-root vertex is targeted exactly once, by an already flagged
vertex. Arrival times strictly increase along successful edges,
so there is no directed cycle. Continuous independent weights make
distinct simple-path sums unequal almost surely: condition on an
edge present in only one of the two paths. There are only finitely
many simple paths in the finite graph.
\end{proof}

Let $U_\tau(t)$ be the chronological product of all successful CNOTs
up to time $t$, with later gates acting on the left.
Define
\[
Z_x(t)=U_\tau(t)^\dagger Z_xU_\tau(t),\qquad
C_\tau(x,t)=
\frac1{4\,2^{|V|}}
\Tr\!\left([Z_x(t),X_0]^\dagger[Z_x(t),X_0]\right).
\]

\begin{theorem}[Exact realization-wise OTOC--FPP identity]
\label{thm:exact-otoc}
For Specification~\ref{spec:cnot},
\[
C_\tau(x,t)=\ind_{\{T_G(0,x)\le t\}},
\qquad
\E C_\tau(x,t)=\Prob(T_G(0,x)\le t).
\]
The equality holds on every positive-timer realization, with a
specified tie convention if necessary.
\end{theorem}

\begin{proof}
A CNOT from $u$ to $v$ obeys
\[
\operatorname{CNOT}_{u\to v}^\dagger Z_v
\operatorname{CNOT}_{u\to v}=Z_uZ_v,
\qquad
\operatorname{CNOT}_{u\to v}^\dagger Z_u
\operatorname{CNOT}_{u\to v}=Z_u.
\]
Conjugate $Z_x$ backwards through the chronological tree circuit.
If $x$ has been flagged, the only gates that enlarge its backwards
$Z$ product are the parent edges on its unique ancestry chain.
Consequently $Z_x(t)$ is the product of $Z$ over that chain,
including $0$ and $x$.
If $x$ has not been flagged, it has neither been a successful target
nor a successful control, and $Z_x(t)=Z_x$.

For $x\ne0$, the resulting Pauli product anticommutes with $X_0$
exactly when $x$ has been flagged. For $x=0$, it always
anticommutes, matching $T_G(0,0)=0$.
A Hermitian unitary Pauli product that anticommutes with $X_0$
has commutator squared norm $4I$; a commuting one has zero.
Lemma~\ref{lem:timers} finishes the proof.
\end{proof}

\begin{remark}[What has and has not been copied]
This is copying of computational-basis influence in a CNOT circuit.
It is not broadcasting an arbitrary unknown quantum state.
The exact OTOC identity does not constitute an identity-transport
channel from the root to every infected site.
\end{remark}

\subsection{A declared complete-channel lift}

A classical process and a quantum channel have different codomains.
To state a complete-channel comparison, define the target lift
\[
\Psi_{G,t}(\rho)=
\E_\tau\!\left[
\kb{r(\tau,t)}{r(\tau,t)}
\otimes U_\tau(t)\rho U_\tau(t)^\dagger
\right],
\]
where $r$ is the finite flag-and-parent record. All payload qubits
are returned. This target is specified before any approximation.

\begin{corollary}[Complete equality on the declared interface]
\label{cor:cnotchannel}
The ideal circuit implements $\Psi_{G,t}$ exactly, including every
untouched reference. Its half-diamond error against that target is
zero.
\end{corollary}

\begin{proof}
Conditioned on the timers the controller is independent of $\rho$
and applies the stated unitary with the stated record.
Averaging gives the displayed linear channel. Tensoring with a
reference identity preserves the equality.
\end{proof}

\subsection{Finite resources and macroscopic consequence}

The finite graph uses $|V|$ payload qubits, at most $|V|-1$
successful CNOTs, $|V|$ flags, and at most $|E|$ one-shot timers.
Parent records can be stored with at most four choices per non-root
site on the square lattice. Exact real-valued timers are an ideal
resource; finite clock precision must be reported separately.

For $\Lambda_L$, these counts are $N_L$ and $M_L$.
An OTOC Hadamard test may prepare the system maximally mixed and
measure the real trace of
\[
Z_x(t)X_0Z_x(t)X_0.
\]
It requires a specified controlled-unitary implementation and
inverse circuit access. Purifying the maximally mixed input uses
$N_L$ additional qubits; a test ancilla adds one.
The test has a bounded binary score with mean $C_\tau(x,t)$.
Its implementation time is not the growth time $t$.

\begin{corollary}[Exact-model limiting front]
\label{cor:cnotshape}
For $\tau_e\sim\Exp(\lambda)$, the infinite-volume controlled
CNOT front and every fixed annealed OTOC level $a\in(0,1)$
have macroscopic body $\Bcal$.
Finite boxes with $L=\lceil an\rceil$ and the explicit choice of
$a$ following Corollary~\ref{cor:box} give the same limit at
time $nt_0$.

The no-flat-face and $C^1$ conclusions are conditional on
\ref{hyp:exp}. For Gamma timers, only the $C^1$ conclusion follows
conditionally from \ref{hyp:gamma}.
\end{corollary}

\begin{proof}
Theorem~\ref{thm:exact-otoc} identifies the front exactly with the
classical process. The escape estimate supplies finite-volume
control, the classical shape theorem supplies the limiting body,
and Corollary~\ref{cor:regularity} supplies only the stated
conditional regularity.
\end{proof}

This closes an \emph{engineered} operator-spreading interface.
It does not identify a generic chaotic circuit with this
flag-controlled construction.

\section{Bridge 1 supplement: a standard random-circuit comparison}
\label{sec:clifford}

The random-circuit literature relates averaged operator spreading to
a classical support process \cite{NahumVijayHaah}. The process is
not, at finite local dimension, monotone infection.

\begin{specification}[Poisson random-Clifford circuit]
Let $q=2^k$. Each site has $k$ qubits.
Every edge of a finite graph has an independent rate-$\lambda$
Poisson clock. At each clock event apply an independent uniformly
chosen two-site Clifford gate. Start with a nonidentity Pauli
operator at $0$ and identity elsewhere. Let $S_q(t)$ be the support
of its forward conjugate by the circuit.
\end{specification}

\begin{lemma}[Exact support update]
\label{lem:clifford-update}
At an edge event, a pair of empty sites remains empty.
If at least one endpoint is occupied, the new two-site support has
probabilities
\[
\Prob(10)=\Prob(01)=\frac1{q^2+1},
\qquad
\Prob(11)=\frac{q^2-1}{q^2+1}.
\]
These probabilities do not depend on the incoming nonzero pair.
\end{lemma}

\begin{proof}
Clifford conjugation acts transitively on nonidentity Pauli
operators. One elementary justification is to use single-qubit
Cliffords to turn nonidentity tensor factors into $X$ factors,
then CNOTs and swaps to reduce their product to one $X$ factor.
Inverting such reductions maps any nonidentity Pauli to any other.
Uniform Clifford sampling therefore gives a uniform nonidentity
two-site Pauli.

There are $q^4-1$ such Paulis. The support counts are
$q^2-1$, $q^2-1$, and $(q^2-1)^2$, respectively.
Divide by $q^4-1$.
\end{proof}

In particular, an occupied site can become empty with positive
probability. This is an explicit obstruction to an exact
finite-$q$ identification with a monotone FPP infected set.

For completeness, support is an OTOC observable after a declared
probe average. For a Pauli $P$ on $V$, let
\[
D_x(P,B)=\frac1{4q^{|V|}}
\Tr([P,B_x]^\dagger[P,B_x]),
\]
and average $B$ uniformly over nonidentity local Paulis. Then
\[
\frac{2(q^2-1)}{q^2}\E_B D_x(P,B)
=\ind_{\{x\in\supp P\}}.
\]
Indeed, exactly $q^2/2$ local Paulis anticommute with a fixed
nonidentity local Pauli; identity is not among them.

\begin{theorem}[Explicit finite-volume FPP comparison]
\label{thm:clifford-coupling}
On a graph with $M$ edges, couple $S_q$ to monotone infection
$I_{\mathrm P}$ driven by the same Poisson clocks.
At an event touching an infected site, the monotone process sets
both endpoints occupied. Then
\[
\Prob\!\left(
S_q(u)\ne I_{\mathrm P}(u)\text{ for some }0\le u\le t
\right)
\le
\min\left\{1,\frac{2\lambda Mt}{q^2+1}\right\}.
\]
The law of $I_{\mathrm P}$ is the law of the infected set for
i.i.d.\ undirected $\Exp(\lambda)$ FPP weights.
\end{theorem}

\begin{proof}
While the two supports agree, events touching no occupied endpoint
do nothing in both processes. At any other event, the Clifford
support agrees with the monotone update with probability
$(q^2-1)/(q^2+1)$. Couple these outcomes directly.
The probability of first disagreement is bounded by
$2/(q^2+1)$ times the expected number of edge events,
which is $\lambda Mt$.

For the second assertion, the timer construction of
Lemma~\ref{lem:timers} has, for exponential timers, independent
exponential residual clocks on its active boundary edges.
This follows inductively from memorylessness after the next minimum
clock and from independence of newly armed edges.
Its jump rate to an uninfected vertex is $\lambda$ times the number
of infected neighbours. The Poisson construction has the same
finite-state generator and the same initial state.
Thus the two laws agree.
\end{proof}

\begin{corollary}[A controlled large-dimension scaling]
\label{cor:largeq}
Take $L_n=\lceil an\rceil$, $t=nt_0$, and $q_n=2^{k_n}$ with
\[
\frac{q_n^2}{n^3}\longrightarrow\infty.
\]
Choose $a$ to make the escape allowance vanish.
Then the random-Clifford support at time $nt_0$, and its normalized
annealed OTOC levels, have limiting body $t_0\Bcal$ in probability.
Regularity remains conditional on \ref{hyp:exp}.
\end{corollary}

\begin{proof}
Here $M_{L_n}=O(n^2)$, so the coupling allowance is
$O(n^3/q_n^2)=o(1)$. Add the exponentially small escape allowance.
On the resulting common good event the complete support history
equals the confined FPP history. Apply the shape theorem and
Theorem~\ref{thm:transfer}.
\end{proof}

\begin{remark}[The order of limits is essential]
At fixed $q$, the displayed comparison does not vanish as
$t$ and volume grow. This paper therefore proves no
finite-$q$ strict-convexity or differentiability theorem for this
circuit.

Moreover, the comparison is a support-history or observable
comparison. It is not a diamond comparison between the full
Clifford payload channel and the CNOT channel of
Section~\ref{sec:bridge1}. Their quantum residuals differ.
A complete-channel ledger must record that distinction.
\end{remark}

\section{Bridge 2: an engineered entanglement-cost membrane}
\label{sec:bridge2}

The membrane literature derives directed spacetime optimizations
and, at finite local dimension, free-energy descriptions
\cite{ZhouNahum2018,ZhouNahum2020}.
Those results do not supply the independent undirected exponential
weights required here. We instead give a named engineered model
and delimit its interpretation.

\subsection{Independent geometric-pair quench}

For each edge $e=\{u,v\}$ of a finite lattice graph, reserve one mode
at $u$ and one mode at $v$, distinct from the modes for every other
edge. Let $w_e\sim\Exp(\lambda)$ independently and set
\[
z_e=\tanh(w_e/2)\in(0,1).
\]
Prepare
\[
\ket{\psi_e}
=
\sqrt{1-z_e}\sum_{r=0}^{\infty}z_e^{r/2}\ket r_u\ket r_v,
\qquad
\ket{\Psi}=\bigotimes_{e\in E}\ket{\psi_e}.
\]

This preparation is a local finite-time quench in a precise
bounded-Hamiltonian model. Put
\[
c_e=\sqrt{1-z_e},\qquad
\ket{v_e}
=\frac{\ket{00}-\ket{\psi_e}}{\sqrt{2(1-c_e)}}.
\]
For a prescribed duration $t_{\mathrm q}>0$, use
\[
H_e=\frac{\pi\hbar}{t_{\mathrm q}}\kb{v_e}{v_e}.
\]
Then
\[
e^{-it_{\mathrm q}H_e/\hbar}
=I-2\kb{v_e}{v_e},
\qquad
e^{-it_{\mathrm q}H_e/\hbar}\ket{00}=\ket{\psi_e}.
\]
The edge Hamiltonians commute because they act on disjoint mode
factors, even when their geometric edges meet at a vertex.
Each is bounded and self-adjoint; the finite product evolution is
therefore well defined.

\begin{theorem}[Exact local entropy costs and cut additivity]
\label{thm:quench}
Use natural logarithms and
$S_2(\rho)=-\log\Tr(\rho^2)$.
The one-endpoint entropy of pair $e$ is exactly $w_e$.
For every vertex bipartition $A\mid A^c$,
\[
S_2(\rho_A)=\sum_{e\in\partial A}w_e.
\]
\end{theorem}

\begin{proof}
The reduced pair state has eigenvalues
$(1-z_e)z_e^r$, $r\ge0$, so
\[
\Tr(\rho_{e,u}^2)
=\frac{(1-z_e)^2}{1-z_e^2}
=\frac{1-z_e}{1+z_e}
=e^{-w_e}.
\]
Thus its R\'enyi-2 entropy is $w_e$.

A pair wholly on one side of the bipartition is a pure tensor factor
on that side and contributes zero. A crossing pair contributes its
one-endpoint reduced state. These factors are independent tensor
factors. Their purities multiply, so their $S_2$ entropies add.
\end{proof}

\begin{definition}[Declared post-quench membrane optimization]
\label{def:membrane}
Identify the bulk planar dual of the square lattice with a translated
copy of $\Z^2$. The dual edge $e^*$ has cost equal to the
one-endpoint R\'enyi-2 entropy $w_e$ of the corresponding pair.
For dual vertices $a,b$, define
\[
\mathcal M(a,b)=
\inf_{\gamma^*:a\to b}\sum_{e^*\in\gamma^*}w_e.
\]
This is the model's declared entropy-cost membrane functional.
\end{definition}

\begin{theorem}[Exact engineered membrane--FPP reduction]
\label{thm:membrane}
The functional $\mathcal M$ of Definition~\ref{def:membrane}
is undirected square-lattice FPP with i.i.d.\ $\Exp(\lambda)$
weights. Its macroscopic norm is $\mu$.
Conditional on \ref{hyp:exp}, its limiting cost ball is strictly
convex with $C^1$ boundary.

For a finite simply connected planar domain and two dual boundary
points separating specified boundary arcs, the minimum of the
compatible vertex-cut entropies equals the corresponding restricted
dual shortest-path cost.
\end{theorem}

\begin{proof}
Every dual edge is in bijection with a primal edge.
The local entropy identity in Theorem~\ref{thm:quench} preserves
the sampled independent exponential costs exactly. The admissible
dual paths are undirected nearest-neighbour paths.
This proves equality of the optimization problems and hence of
their macroscopic norms.

In the finite domain, a compatible cut has an interface joining
the specified boundary points, possibly with extra closed interface
components. Positivity permits deletion of the latter.
Every simple compatible dual joining path separates the prescribed
arcs and determines a vertex cut. Theorem~\ref{thm:quench} identifies
its entropy with the path cost. Minimizing on both sides gives the
finite-domain statement. Boundary edges must use the specified
domain convention; no full-plane conclusion is inferred from an
unspecified boundary graph.

The last regularity statement is
Corollary~\ref{cor:regularity}.
\end{proof}

\begin{remark}[Interpretation boundary]
For arbitrary interior dual endpoints, $\mathcal M(a,b)$ is a
declared optimization of local entanglement costs. An open interior
path is not automatically the entropy of one global vertex
bipartition.

The theorem concerns a post-quench cost model. It does not derive
the time-dependent entropy of a generic many-body system, the
usual spacetime membrane, or its finite-temperature free energy.
The exponential law was engineered through independent controls;
it was not inferred from chaotic quantum dynamics.
\end{remark}

\subsection{Finite-dimensional preparation and entropy errors}

For $D\ge1$, truncate a pair to
\[
\ket{\psi_{e,D}}
=
\sqrt{\frac{1-z_e}{1-z_e^D}}
\sum_{r=0}^{D-1}z_e^{r/2}\ket r_u\ket r_v.
\]
The same Householder construction prepares it by a bounded local
gate on two $D$-dimensional systems.

\begin{proposition}[Explicit truncation ledger]
\label{prop:truncation}
For each edge,
\[
|\langle\psi_e|\psi_{e,D}\rangle|^2=1-z_e^D,
\qquad
d_{\mathrm{tr}}(\psi_e,\psi_{e,D})=z_e^{D/2}.
\]
The truncated local cost is
\[
w_{e,D}
=w_e-\log\frac{1+z_e^D}{1-z_e^D}.
\]
On $w_e\le K$, put $z_K=\tanh(K/2)$. For a graph with $M$ edges,
\[
d_{\mathrm{tr}}(\Psi,\Psi_D)\le\sqrt M\,z_K^{D/2},
\]
and
\[
0\le w_e-w_{e,D}\le
d_D(K):=\log\frac{1+z_K^D}{1-z_K^D}.
\]
On a finite dual graph with $N^*$ vertices,
\[
0\le\mathcal M(a,b)-\mathcal M_D(a,b)
\le (N^*-1)d_D(K).
\]
The probability that some sampled edge violates $w_e\le K$ is
at most $Me^{-\lambda K}$.
\end{proposition}

\begin{proof}
The overlap is the norm of the retained part of the geometric
Schmidt expansion. The pure-state trace-distance identity gives
the first two formulas.

Direct summation gives
\[
\Tr(\rho_{e,u,D}^2)
=
\frac{1-z_e}{1+z_e}
\frac{1+z_e^D}{1-z_e^D},
\]
proving the entropy formula.

For the product preparation, the squared overlap is
$\prod_e(1-z_e^D)$. Therefore
\[
d_{\mathrm{tr}}(\Psi,\Psi_D)
=\sqrt{1-\prod_e(1-z_e^D)}
\le\sqrt{\sum_e z_e^D}.
\]
The cost discrepancy is increasing in $z_e^D$.

A minimizing path can be chosen simple, also when a truncated cost
vanishes. It has at most $N^*-1$ edges. Evaluating the untruncated
cost on a truncated minimizer proves the upper bound; pointwise
cost monotonicity proves the lower bound.
Finally use the exponential tail and a union bound.
\end{proof}

For the declared preparation channel, whose input is the fixed
vacuum preparation interface, the full output trace-distance bound
is also the half-diamond bound. It is not an arbitrary-input
unitary-channel approximation.

\begin{corollary}[One explicit growing-resource choice]
\label{cor:quench-resources}
Suppose $M_n,N_n^*=O(n^2)$.
For $n\ge2$, choose
\[
K_n=\frac6\lambda\log n,\qquad
D_n=
\left\lceil
\frac{8\log n}{-\log\tanh(K_n/2)}
\right\rceil.
\]
Then the tail allowance is $O(n^{-4})$, the good-event preparation
allowance is $O(n^{-3})$, and the optimization error is $o(n)$.
Thus these finite-dimensional cost optimizations have the same
macroscopic norm as the exact exponential target.
\end{corollary}

\begin{proof}
The tail bound is $M_n n^{-6}=O(n^{-4})$.
The choice of $D_n$ ensures $z_{K_n}^{D_n}\le n^{-8}$.
Hence the preparation bound is $O(n)n^{-4}=O(n^{-3})$.
For large $n$,
$d_{D_n}(K_n)\le4n^{-8}$, so the optimization error is $O(n^{-6})$.
\end{proof}

There are $2M_n$ mode registers and at most
$2M_n\lceil\log_2 D_n\rceil$ encoded qubits.
The larger oscillator, energy, control, and clock ledgers are not
determined by this Hilbert-space count.

\section{Bridge 3: measurements and the zero-cost obstruction}
\label{sec:bridge3}

The classical-percolation toy reduction for measured circuits is
not the positive exponential model
\cite{SkinnerRuhmanNahum}. The following monitored version of
Section~\ref{sec:bridge2} makes the obstruction exact.

Independently for every edge, choose $m_e\sim\operatorname{Bernoulli}(p)$.
If $m_e=1$, measure one endpoint mode in the number basis and retain
the complete outcome. If $m_e=0$, do not measure the pair.

\begin{theorem}[Exact monitored cost law]
\label{thm:measured}
For fixed sampled controls and every supported complete measurement
outcome, the conditional pure state's local entanglement costs are
\[
\widetilde w_e=(1-m_e)w_e.
\]
After summing all outcomes in the complete attempted-run law,
these costs are independent with common law
\[
F_p=p\delta_0+(1-p)\Exp(\lambda).
\]
For $p>0$, this is not an exponential or a positive-shape Gamma law.
\end{theorem}

\begin{proof}
A number outcome $r$ on one endpoint of a measured geometric pair
leaves the normalized pair state $\ket r\ket r$, which has zero
entanglement. An unmeasured pair retains cost $w_e$.
Products of the selective maps preserve the product-over-edges
structure.

The masks and controls were sampled independently.
Summing the complete quantum outcomes gives total probability one
for every mask and control realization, so it does not alter their
joint marginal law. The stated independent mixture follows.

This does not assert that selecting one prescribed outcome word
leaves the control distribution unchanged. Such postselection
would require a separate posterior calculation.
The atom at zero excludes both specified positive continuous laws.
\end{proof}

\begin{proposition}[Finite-volume mismatch cannot be hidden]
\label{prop:zero-tv}
On $M$ edges, the total-variation distance between the monitored cost
law and the product exponential cost law is at least
\[
1-(1-p)^M.
\]
At $p=1$, every passage cost vanishes, and the resulting limit is
not a norm.
\end{proposition}

\begin{proof}
Use the event that at least one cost is exactly zero.
Its probabilities are $1-(1-p)^M$ and zero, respectively.
For $p=1$, every path has zero cost.
\end{proof}

Even away from a proposed critical point, $p>0$ does not restore
the exact law required by 212-02. A coupling gives
$\widetilde w_e\le w_e$, hence $\widetilde T\le T$,
but this one-sided comparison does not transfer strict convexity
or differentiability by Proposition~\ref{prop:nonstable}.

The complete monitored instrument is well defined and retains all
outcomes. Its \emph{conditional entanglement} is not the entropy of
the unconditional mixed output. Neither a measurement threshold nor
critical exponents are proved here. Family 213 is not an input to
these assertions.

\section{Bridge 4: serial teleportation and concurrent timing}
\label{sec:bridge4}

\subsection{A timing contract that really is additive}

\begin{specification}[Advertised-service serial repeater model]
\label{spec:repeater}
Fix a finite lattice graph.
\begin{enumerate}
\item Each edge has one positive service duration $\tau_e$,
sampled independently from $\Exp(\lambda)$ and advertised to the
route controller independently of the message.
\item When activated, the edge service prepares a perfect Bell pair
after exactly $\tau_e$ time units.
\item Quantum memories are perfect. Bell measurements, Pauli
corrections, and the classical messages needed for those corrections
have zero duration in this ideal timing contract.
\item The controller chooses a route before dispatch, using the
advertised durations. Along that route, service is activated only
after the message has arrived at the preceding vertex.
\item All Bell outcomes and all surviving registers are included in
the complete protocol specification. No successful branch alone
is treated as a deterministic channel.
\end{enumerate}
This is an externally timed service model, not a derivation of
physical link-generation statistics or an optimal concurrent
repeater protocol.
\end{specification}

\begin{theorem}[Exact serial route reduction]
\label{thm:serial}
For a fixed simple route $\gamma$, the completion time in
Specification~\ref{spec:repeater} is
\[
S_\gamma=\sum_{e\in\gamma}\tau_e.
\]
The complete corrected teleportation protocol implements the
identity message channel at completion, including every untouched
reference. Minimizing over the advertised routes gives
\[
S_x=\min_{\gamma:0\to x}S_\gamma=T_G(0,x).
\]
\end{theorem}

\begin{proof}
The first edge service starts at time zero.
The next starts only after the preceding teleportation completes.
The zero-duration convention for the other steps makes completion
times additive.

For one perfect teleportation step, the four Bell outcomes have
uncorrected path maps $\frac12 P_j$ on the message, up to the fixed
Bell-basis convention. Applying the corresponding inverse Pauli
gives four corrected path maps $\frac12 I$.
Their complete sum is
\[
\sum_{j=1}^4\frac14\rho=\rho.
\]
The identity holds after tensoring with any reference.
Composition proves the route claim; records are retained with
their complete probabilities.

There are finitely many simple routes in the finite graph, and
positive costs allow loop erasure. The advertised controller can
therefore select a minimizing route with a fixed tie rule.
Its cost is precisely the finite FPP passage time.
\end{proof}

At a deadline $t$, the reduced destination task channel can be written
\[
\mathcal N_{x,t}(\rho)=
\Prob(T_G(0,x)\le t)\,
\kb{\mathrm{ok}}{\mathrm{ok}}\otimes\rho
+
\Prob(T_G(0,x)>t)\,
\kb{\mathrm{fail}}{\mathrm{fail}}\otimes\omega_*\,\Tr\rho,
\]
where $\omega_*$ is a declared failure state.
On failure the message may still survive elsewhere; that register
belongs to the complete protocol output, not to the reduced
destination task.

For a route of $\ell$ edges, the protocol uses $\ell$ Bell pairs and
$2\ell$ Bell-outcome bits. A sequential implementation can recycle
local workspace, but supplier holdings and perfect storage are
separate resources. There is no unknown-state duplication.

\begin{corollary}[Serial-model shape]
The serial reachability probabilities have the exponential FPP
limiting body and conditional regularity of
Corollary~\ref{cor:regularity}, with finite-volume error controlled
by Lemma~\ref{lem:escape}.
\end{corollary}

\begin{proof}
Apply Theorem~\ref{thm:serial} and the classical shape theorem.
\end{proof}

\subsection{Why concurrent generation changes the optimization}

\begin{proposition}[Two-link concurrent counterexample]
\label{prop:parallel}
Consider a chain of two independent rate-$\lambda$ link generators
with ideal storage and instantaneous swapping.
If both generators start at time zero, the Bell connection is ready
at
\[
S_{\parallel}=\max\{\tau_1,\tau_2\}.
\]
Serial generation takes
$S_{\mathrm{serial}}=\tau_1+\tau_2$.
For $t\ge0$,
\[
\Prob(S_{\parallel}\le t)=(1-e^{-\lambda t})^2,
\]
whereas
\[
\Prob(S_{\mathrm{serial}}\le t)
=1-e^{-\lambda t}(1+\lambda t).
\]
Their means are $3/(2\lambda)$ and $2/\lambda$.
Thus concurrent independent exponential link generation does not
imply additive FPP route cost.
\end{proposition}

\begin{proof}
With storage, the first completed pair can wait for the second.
Both are available at their maximum completion time.
Independence gives the first distribution.
Convolution of two exponential densities gives the second.
Integrating either survival function gives its stated mean.
\end{proof}

Additional swapping failures, memory decoherence, cutoffs, shared
hardware, purification, and classical signaling delays can further
alter timing and fidelity \cite{BrandCoopmansElkouss,LiCoopmansElkouss}.
They are not represented by declaring one independent exponential
weight per geometric edge.

Consequently the engineered serial model closes its own interface,
but the original unrestricted concurrent-network obligation remains
unresolved. A restricted protocol's FPP cost is not an optimality
theorem over all quantum-network protocols.

\section{A common complete-channel, calibration, and shot ledger}
\label{sec:ledger}

Use the half-diamond convention
\[
\ddia(\Phi,\Psi)=\frac12\|\Phi-\Psi\|_\diamond.
\]
All comparisons require common input and output interfaces, including
the declared classical records and quantum residuals.

\begin{lemma}[Global composition and law errors]
\label{lem:errors}
Suppose two complete protocols have corresponding CPTP stages
$\Phi_j,\widehat\Phi_j$ with
$\ddia(\Phi_j,\widehat\Phi_j)\le\eta_j$.
Then
\[
\ddia(\Phi_m\cdots\Phi_1,
\widehat\Phi_m\cdots\widehat\Phi_1)
\le\min\{1,\sum_{j=1}^m\eta_j\}.
\]

If a protocol is sampled using input-independent laws $P,Q$ and
the same conditional complete channels $\Phi_\omega$, then
\[
\ddia\left(\int\Phi_\omega\,dP,
\int\Phi_\omega\,dQ\right)
\le\|P-Q\|_{\TV}.
\]
If $M$ independent weight laws are each calibrated within
total variation $\delta$, the product-law contribution is at most
\[
1-(1-\delta)^M\le M\delta.
\]
\end{lemma}

\begin{proof}
Insert the stages one at a time and telescope the difference.
Composition with CPTP maps contracts diamond norm.
Adaptive classical controls can be included in the stages on the
common classical--quantum interface.

For the law bound, take a maximal coupling of $P,Q$.
On the agreement event the conditional channels coincide; on the
disagreement event the half-diamond distance is at most one.
Integrate. Independent maximal couplings on the edges give the
product bound.
\end{proof}

The law assertion does not follow from marginal calibration alone
when joint independence has not been established.

\begin{proposition}[Vanishing local error can remain macroscopic]
\label{prop:accumulation}
Let
\[
\mathcal R_D(X)=\Tr(X)I_D/D,\qquad
\mathcal P_p=(1-p)\id+p\mathcal R_D.
\]
For $p_m=\alpha/m$,
\[
\mathcal P_{p_m}^{\,m}
=(1-p_m)^m\id+
\bigl(1-(1-p_m)^m\bigr)\mathcal R_D.
\]
Although each stage has half-diamond allowance at most $\alpha/m$,
\[
\liminf_{m\to\infty}
\ddia(\mathcal P_{p_m}^{\,m},\id)
\ge(1-e^{-\alpha})(1-D^{-2})>0
\]
for $D\ge2$ and $\alpha>0$.
\end{proposition}

\begin{proof}
Use $\mathcal R_D^2=\mathcal R_D$ to obtain the composition formula.
Apply the channel to half of a maximally entangled state and test
the maximally entangled projector. The replacement output has
overlap $D^{-2}$. Finally
$(1-\alpha/m)^m\to e^{-\alpha}$.
\end{proof}

This is the same obstruction pattern emphasized in En08-B
\cite{CheungEn08B}, with a proof independent of that manuscript.

\subsection{Rates and clock scales}

If calibration establishes the stronger coupling
\[
\widehat\tau_e=c\tau_e
\]
for one common $c>0$, then pathwise
\[
\widehat T=cT,\qquad
\widehat\mu=c\mu,\qquad
\widehat\Bcal=c^{-1}\Bcal.
\]
For exponential rates,
\[
\tau_e^{(\lambda)}=\lambda^{-1}\tau_e^{(1)},
\qquad
\mu_\lambda=\lambda^{-1}\mu_1,
\qquad
\Bcal_\lambda=\lambda\Bcal_1.
\]
This is a global scale calibration, not a robustness theorem for
arbitrary changes in weight law or dependence.

Clock quantization can change timer order, route records, and
deadline flags. A path-cost error alone is not automatically a
complete-channel error. An implementation must either certify
the same branches and event order, or price their disagreement
probability in the complete-channel comparison.

\subsection{Finite-shot bounds}

\begin{lemma}[Uniform bounded-record shot certificate]
\label{lem:shots}
Suppose $K$ declared scores lie in $[0,1]$.
For each score, use $N$ independent attempted runs from the same
fixed actual model. If its actual mean differs from the target mean
by at most $b$, then, simultaneously for all $K$ scores,
\[
|\widehat m_j-m_j^{\mathrm{target}}|\le h+b
\]
with probability at least $1-\alpha$, provided
\[
N\ge\frac1{2h^2}\log\frac{2K}{\alpha}.
\]
If the allowance $b$ depends on calibration valid with probability
at least $1-\beta$, the unconditional coverage is at least
$1-\alpha-\beta$.
\end{lemma}

\begin{proof}
Hoeffding's inequality bounds each two-sided deviation from the
actual mean by $2e^{-2Nh^2}$ \cite{Hoeffding}.
Use a union bound and then the mean allowance.
Add the calibration-failure probability.
\end{proof}

A finite test cannot certify an exact distributional identity or
independence of an unlimited edge family. Those remain model
assumptions unless supplied by a construction.

For the quench model, a two-copy SWAP measurement estimates local
purity $r_e=e^{-w_e}$. If $\widehat r_e$ has error at most $\eta$
and $w_e\le K$, with $\eta\le e^{-K}/2$, then
\[
|-\log\widehat r_e-w_e|\le2e^K\eta.
\]
Since the SWAP outcome lies in $\{-1,1\}$, estimating purity to
$\eta$ for all $M$ edges is ensured by
\[
N\ge\frac2{\eta^2}\log\frac{2M}{\alpha}
\]
two-copy tests per edge. This displays the increasing cost of
estimating large entropy weights.

\subsection{Required ledger}

Every finite implementation shall report at least
\[
\begin{array}{ll}
\text{Geometry:}& |V|,\ |E|,\ L,\ \text{boundary convention};\\
\text{Quantum resources:}&
\text{local dimensions, live registers, suppliers, residuals};\\
\text{Operations:}&
\text{gate count, measurement tree, routing, inverse access};\\
\text{Timing:}&
\text{clock law, precision, storage, swapping, signaling times};\\
\text{Accuracy:}&
\text{declared target lift and complete-channel allowance};\\
\text{Calibration:}&
\text{joint weight-law uncertainty and independence assumptions};\\
\text{Statistics:}&
\text{attempted runs, bounded scores, failure coverage};\\
\text{Limit:}&
\text{single realization versus annealed front, and scaling order}.
\end{array}
\]

In the exact mathematical models, ideal channel error is zero against
their explicitly declared lifts. This does not certify a physical
device. For the random-Clifford supplement, the support-history
allowance is explicit, but no full-payload diamond allowance against
an FPP lift has been proved.

\section{Layer discipline and refusal of unsupported physics}
\label{sec:layers}

The following objects stay separate:
\begin{enumerate}
\item the quantum channel or instrument and its specified observable;
\item the classical random optimization and its joint weight law;
\item the macroscopic norm and the regularity justified by the mapping.
\end{enumerate}

The engineered constructions above demonstrate that a classical
random metric can be installed in a quantum protocol. They do not
show that every such protocol selects that metric.

Scheduled ancestry bounds support but need not equal operational
influence. An identity block can have a two-site scheduling
footprint while producing no cross-site influence. Physical causal
containment requires an independently supplied embedding and
timing contract. These are the distinctions documented in VII-J,
En05-A, and XIX-C
\cite{CheungVIIJ,CheungEn05A,CheungXIXC}.

A record-only comparison also cannot replace a complete residual
comparison. For example, a one-outcome identity channel and a
one-outcome dephasing channel have the same deterministic record;
on input $\kb{+}{+}$ their residual outputs have trace distance
$1/2$ \cite{CheungVIIIA}. This is why the target lift is printed
in Section~\ref{sec:bridge1}.

Channel equality does not determine a thermodynamic implementation:
appending an unobserved energy-transfer process leaves a selected
channel unchanged while changing reservoir heat or fuel usage.
A heat or reusable-memory reset claim therefore needs Hamiltonians,
reservoirs, initial correlations, and resource-return conditions
\cite{CheungEn07B}.

\begin{proposition}[No metric dynamics from the limiting body]
\label{prop:no-gravity}
The constructions and shape-transfer statements in this paper
provide no metric field, conformal scale, Einstein tensor,
observer-frame action, or gravitational field equation.
\end{proposition}

\begin{proof}
Their mathematical outputs are norms, random arrival times,
specified channels, and classical records. None of the definitions
assigns a position-dependent Lorentzian metric or a frame action.

Even if an additional Lorentz-cone identification were supplied,
a constant affine quadratic form would have constant metric
coefficients. Its Christoffel symbols and curvature would vanish.
Conversely a pointwise cone field would fix at most conformal
causal data: multiplying a metric by a positive function preserves
its null cones while changing derivative-dependent curvature.
Thus neither operation supplies the missing metric dynamics.
\end{proof}

No Page curve, entanglement wedge, evaporation channel,
gravity-generated-entanglement test, or solution to a standing
quantum-gravity problem follows. Families 213 and 265 are not
premises of any result here.

\section{The separate characteristic-composition programme}
\label{sec:analytic}

The long supplied excerpts also describe six analytic obligations:
a full joint characteristic map, a joint section and controlled
remainder, uniform transverse estimates, contact compatibility,
an output wavefront theorem, and quantum admission.
They concern a different retained gravitational coefficient.

This section records the relevant algebraic interface and the
precise non-transfer. It does not rename an FPP construction as
one of those analytic objects.

\subsection{Reconstruction descent is a necessary condition}

\begin{proposition}[Exact descent criterion]
\label{prop:descent}
Let $\Rec:P\to K$ be a surjective linear reconstruction map.
A linear map $\widetilde M:P\to Y$ factors as
$\widetilde M=M\Rec$ if and only if
\[
\ker\Rec\subseteq\ker\widetilde M.
\]
When it factors, $M$ is unique.
\end{proposition}

\begin{proof}
A factorization annihilates every reconstruction-null presentation.
Conversely, define $M(\Rec p)=\widetilde M(p)$.
Two presentations of the same kernel differ by an element of
$\ker\Rec$, so the definition is independent of the presentation.
Surjectivity proves uniqueness.
\end{proof}

This is the elementary criterion documented in XIV-J and XV-B
\cite{CheungXIVJ,CheungXVB}, proved here without importing a
characteristic product.

\begin{lemma}[Presentation degree is not intrinsic]
\label{lem:null}
In one normal variable,
\[
q\delta'(q)+\delta(q)=0.
\]
Under a reconstruction convention
\[
\Rec(M_{\mathrm R})=\delta(q)\otimes R,\qquad
\Rec(i\theta M_{\mathrm R})=\delta'(q)\otimes R,
\]
with $R\ne0$ and physical multiplication by $q$,
\[
\Rec\bigl((qi\theta+1)M_{\mathrm R}\bigr)=0.
\]
The separate degree-zero and degree-one summands reconstruct to
nonzero opposite distributions.
\end{lemma}

\begin{proof}
For a test $\varphi$,
\[
\langle q\delta',\varphi\rangle
=-\left.\frac{d}{dq}(q\varphi(q))\right|_{q=0}
=-\varphi(0).
\]
Tensor with $R$.
The distribution $\delta\otimes R$ is nonzero by testing against
a product of tests with nonzero pairings.
\end{proof}

This qualifies a highest-degree presentation splitting.
It does not invalidate estimates on the named internal module,
nor prove a universal obstruction to every joint construction.

\subsection{A conditional affine correction is not a supplied map}

Let $H=\ker\Rec$, and let $F_0$ be a preliminary lift whose restriction
to $H$ takes values in a correction space $Y_0$.
Fix a permitted vector space
$\mathcal A\subseteq\operatorname{Hom}(P,Y_0)$.

\begin{proposition}[Relative-lifting criterion]
\label{prop:relative}
A descending member of $F_0+\mathcal A$ exists if and only if
\[
[F_0|_H]=0
\quad\text{in}\quad
\operatorname{Hom}(H,Y_0)/
\{a|_H:a\in\mathcal A\}.
\]
An individual candidate descends precisely when
\[
F_0|_H+a|_H=0.
\]
If $F_*$ is one descending candidate, all such candidates are
\[
F_*+\{a\in\mathcal A:a|_H=0\}.
\]
\end{proposition}

\begin{proof}
Apply Proposition~\ref{prop:descent} to $F_0+a$.
The required cancellation equation has a solution exactly when
$F_0|_H$ belongs to the restriction image, since that image is a
vector space. Subtract one solution to obtain the ambiguity.
\end{proof}

The criterion does not construct $F_0$, evaluate its null defect,
or prove that a permitted correction space is exhaustive for the
frozen graph. Membership means the actual image, not merely its
closure. These are the distinctions in the supplied XV-B revision
\cite{CheungXVB}.

\begin{proposition}[Off-locus agreement does not determine extension]
\label{prop:off-locus}
Suppose two distribution-valued candidate maps agree after restriction
to an open set $U$. They need not agree globally, or satisfy the same
global reconstruction-null identities. A correction supported on a
closed set $C$ cannot cancel a defect detected by a test supported
away from $C$.
\end{proposition}

\begin{proof}
A nonzero distribution supported outside $U$ has zero restriction
to $U$ and can be added without changing that restriction.
Locally, a smooth omitted submanifold admits nonzero compactly
supported delta-type distributions.

For the second statement, every distribution supported on $C$
annihilates a test supported away from $C$. Thus it cannot change
a nonzero pairing of the defect with that test.
\end{proof}

\subsection{The six gates remain distinct}

The supplied XIV-H theorem stops at its uniform internal amplitude
requirement. XIV-I reports the narrower I2 internal construction,
while leaving the joint characteristic map at I3 unresolved.
XIV-J reports an off-characteristic reference for
$K_6+K_8+K_9$, not the full composition of
\[
K_{47}+K_5+K_6+K_8+K_9
\]
with the grouped cross-lines
\cite{CheungXIVH,CheungXIVI,CheungXIVJ}.

The correct gate ledger is therefore:
\[
\begin{array}{cl}
1.&\text{Full joint map on tests meeting the characteristic locus:
not supplied here;}\\
2.&\text{Joint section and complete remainder:
not supplied here;}\\
3.&\text{Uniform transverse estimates for that same joint object:
not supplied here;}\\
4.&\text{Permitted-contact compatibility:
not supplied here;}\\
5.&\text{Complete output wavefront theorem after pushforward:
not supplied here;}\\
6.&\text{Normalized admission in the stated quantum output space:
not established here.}
\end{array}
\]

The random metric of Sections~\ref{sec:fpp}--\ref{sec:bridge4}
has neither the domain nor the singular distributions of that
coefficient. Its shape theorem provides none of the six objects.
Likewise an internal Fourier pole or an algebraic descent criterion
does not supply a generalized product of the retained cross-lines.

The existence of standalone analytic interface results in a manuscript
labelled XV-B is compatible with the downstream quantum-admission
exit remaining closed. A manuscript label is not evidence that its
entry prerequisites have been discharged.

\section{Assembly, completion test, and final verdict}
\label{sec:assembly}

\begin{theorem}[Oai212-A assembly]
\label{thm:assembly}
For the models and interfaces specified in this paper:
\begin{enumerate}[label=\textup{(\roman*)}]
\item The flag-controlled CNOT circuit has an exact
realization-wise squared-commutator identity with undirected FPP,
and an exact complete-channel identity against its declared lift.
\item The Poisson random-Clifford circuit has the support-history
allowance $2\lambda Mt/(q^2+1)$ and an FPP macroscopic limit under
the stated growing-dimension scaling. No fixed-dimension or
full-payload-channel regularity transfer is proved.
\item The independent-pair quench has exact exponential local
R\'enyi-2 costs. Its declared dual membrane optimization is FPP.
Its finite-dimensional approximations have the stated global
preparation and optimization allowances.
\item The monitored quench has an independent zero/exponential
mixture cost law. For $p>0$ the exact positive exponential and
Gamma interfaces fail.
\item The advertised-service serial teleportation model has
additive FPP route cost and reference-preserving quantum transport.
Concurrent generation does not obey this additive law even on
a two-link chain.
\item Any asymptotically identical passage-time interface satisfying
Theorem~\ref{thm:transfer} has the target macroscopic body.
Strict convexity and $C^1$ regularity are transferred only
conditionally on the corresponding external regularity assertion.
\item None of these results constructs the inherited characteristic
quantum coefficient or discharges a gravitational admission gate.
\end{enumerate}
\end{theorem}

\begin{proof}
Assertions (i)--(v) are respectively
Theorem~\ref{thm:exact-otoc} and
Corollary~\ref{cor:cnotchannel},
Theorem~\ref{thm:clifford-coupling} and
Corollary~\ref{cor:largeq},
Theorems~\ref{thm:quench}--\ref{thm:membrane} and
Proposition~\ref{prop:truncation},
Theorem~\ref{thm:measured} and
Proposition~\ref{prop:zero-tv},
and Theorem~\ref{thm:serial} with
Proposition~\ref{prop:parallel}.
Assertion (vi) is Theorem~\ref{thm:transfer} and
Corollary~\ref{cor:regularity}.
Section~\ref{sec:analytic} identifies the different domain and
the unconstructed analytic objects, proving the release boundary
in (vii).
\end{proof}

\begin{center}
\small
\begin{tabularx}{\linewidth}{@{}p{.15\linewidth}p{.26\linewidth}X@{}}
\toprule
Obligation & Proved here & Not thereby completed\\
\midrule
Operator spreading &
Exact controlled CNOT example; quantitative random-Clifford comparison &
Generic finite-dimensional stochastic circuits; hardware timing;
fixed-$q$ shape regularity\\[4pt]
Entanglement membrane &
Engineered post-quench entropy-cost optimization and finite truncation &
Generic dynamical spacetime membrane, its correlations, or its
finite-temperature free energy\\[4pt]
Measurements &
Exact mixed cost law and zero-cost obstruction &
Measurement threshold, critical exponents, or smoothness of the
mixed-law norm\\[4pt]
Communication &
Exact advertised serial timing and quantum transport &
Unrestricted concurrent repeater optimization and realistic
storage/swapping/signaling\\[4pt]
Shape regularity &
Conditional transfer from the identified limiting norm &
Independent verification of 212-02\\[4pt]
Characteristic coefficient &
Elementary descent and relative-lifting criteria &
Full joint map, section, transverse estimates, contact gluing,
output wavefront, and admission\\
\bottomrule
\end{tabularx}
\end{center}

\subsection*{Submission claim}

The submission supported by the proofs is a paper about explicit
interfaces, examples, comparison bounds, and obstructions.
It is not a proof that all four original general-purpose quantum
bridges have been completed. Replacing an unresolved mapping by
a conditional theorem would not complete it; replacing a generic
model by an engineered example must be declared, as it has been
throughout this paper.

\subsection*{Debugging conclusions}

The following corrections are essential:
\begin{enumerate}
\item Strict convexity refers to the norm's unit ball, not strict
convexity of the homogeneous norm function along rays.
\item Gamma differentiability excludes corners, but does not by itself
exclude flat faces.
\item Radial speed and normal supporting-line velocity are different.
\item Promoting a planar norm ball to an epigraph cone produces
dimension three. Homogeneity remains missing.
\item Small finite comparison errors do not transfer smoothness or
strict convexity to the approximants.
\item Finite-$q$ random-circuit support can shrink and is not monotone
FPP infection.
\item Conditional trajectory entropy is not unconditional output
entropy.
\item Concurrent link generation gives maxima and protocol-dependent
waiting times, not automatically sums.
\item An observable or record comparison is not a complete-channel
comparison.
\item A reconstruction identity, internal amplitude estimate, or
off-locus reference is not the full characteristic quantum map.
\end{enumerate}

\section*{Declarations}

\paragraph{Authorship and identifier.}
Oai212-A is the requested manuscript identifier. No authorship,
approval, or endorsement by N.~Cheung or by the authors of the
external papers is asserted.

\paragraph{Mathematical dependency.}
The standard FPP shape theorem and the cited cone classification are
external established mathematical inputs. The regularity claims
of 212-02 are separately marked conditional. Cheung manuscript
comparisons do not identify distinct models.

\paragraph{Model changes.}
The controlled CNOT circuit, independent-pair quench, monitored
quench, and advertised serial repeater are explicit engineered
models. Their exact identities do not substitute for missing
reductions of generic models.

\paragraph{Physical scope.}
No invariant physical speed, Lorentzian metric field, Einstein
dynamics, thermodynamic optimum, Page curve, holographic dictionary,
or standing quantum-gravity solution is claimed.

\paragraph{Analytic scope.}
No full characteristic cross-line product, interacting Ward datum,
or inherited gravitational quantum coefficient is admitted.
O.1 remains NOT\_CLOSED; the downstream admission exit remains
closed.

\paragraph{Execution and verification.}
No successful \texttt{pdflatex} execution, proof-assistant run,
independent numerical execution, referee approval, or journal
acceptance is asserted. The source is provided as one self-contained
\texttt{pdflatex} file. The claimed new results are the explicit
proofs above, with their stated hypotheses.

\begin{thebibliography}{99}

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OpenAI,
\emph{Strict convexity and differentiability of the planar exponential
first-passage limit shape},
manuscript 212-02, 24 September 2026.
External preprint record; its regularity assertions are conditional
inputs in the present paper.
\url{https://github.com/openai/math}

\bibitem{Companion212}
\emph{Enter Calculus Companion},
Family 212, manuscript 212-02.
Catalogue and scope record, not independent proof verification.
Accessed 14 October 2026.
\url{https://enter.grok.me/openai/212-02}

\bibitem{CoxDurrett}
J.~T.~Cox and R.~Durrett,
``Some limit theorems for percolation processes with necessary and
sufficient conditions,''
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\bibitem{ADH}
A.~Auffinger, M.~Damron, and J.~Hanson,
\emph{50 Years of First-Passage Percolation},
University Lecture Series \textbf{68},
American Mathematical Society, 2017.
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\bibitem{NahumVijayHaah}
A.~Nahum, S.~Vijay, and J.~Haah,
``Operator spreading in random unitary circuits,''
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\bibitem{ZhouNahum2018}
T.~Zhou and A.~Nahum,
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T.~Zhou and A.~Nahum,
``The entanglement membrane in chaotic many-body systems,''
\emph{Physical Review X} \textbf{10} (2020), 031066.
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\bibitem{SkinnerRuhmanNahum}
B.~Skinner, J.~Ruhman, and A.~Nahum,
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\bibitem{BrandCoopmansElkouss}
S.~Brand, T.~Coopmans, and D.~Elkouss,
\emph{Efficient computation of the waiting time and fidelity in
quantum repeater chains},
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\url{https://arxiv.org/abs/1912.07688}

\bibitem{LiCoopmansElkouss}
B.~Li, T.~Coopmans, and D.~Elkouss,
\emph{Efficient optimization of cut-offs in quantum repeater chains},
2020 preprint.
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\bibitem{ItoLourenco}
M.~It\^o and B.~F.~Louren\c{c}o,
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\bibitem{CheungEn05A}
N.~Cheung,
\emph{Manuscript En05-A}, version 1.
Supplied excerpts: \texttt{thm:cone},
\texttt{thm:invariantspeed}.
Scheduled locality and metric-envelope versus invariant-speed
distinctions. Documentary comparison, not an imported model.
Enter Calculus Companion.
\url{https://enter.grok.me}

\bibitem{CheungEn05B}
N.~Cheung,
\emph{Manuscript En05-B}, version 1.
Supplied excerpt: \texttt{thm:main},
isotropic matched operational existence for prescribed $c$.
Documentary comparison.
Enter Calculus Companion.
\url{https://enter.grok.me}

\bibitem{CheungEn06B}
N.~Cheung,
\emph{Manuscript En06-B}, version 1.
Supplied excerpt: \texttt{thm:T},
finite transport contract on the specified increasing-stencil family.
Documentary comparison.
Enter Calculus Companion.
\url{https://enter.grok.me}

\bibitem{CheungEn08A}
N.~Cheung,
\emph{Manuscript En08-A}, version 1.
Supplied excerpts: \texttt{thm:main} and conditional
cone-representation statements.
Scheduler consistency is not Lorentz-cone selection.
Documentary comparison.
Enter Calculus Companion.
\url{https://enter.grok.me}

\bibitem{CheungEn08B}
N.~Cheung,
\emph{Manuscript En08-B}, version 1.
Supplied excerpts: \texttt{thm:main},
\texttt{prop:accumulation}.
Prescribed-boundary SWAP transport and accumulation of local error.
The relevant accumulation pattern is re-proved here.
Enter Calculus Companion.
\url{https://enter.grok.me}

\bibitem{CheungVIIJ}
N.~Cheung,
\emph{Manuscript VII-J}, version 1.
Supplied excerpt: \texttt{thm:certificate}, together with its
scheduled, operational, and physical-causality distinctions.
Documentary comparison.
Enter Calculus Companion.
\url{https://enter.grok.me}

\bibitem{CheungVIIIA}
N.~Cheung,
\emph{Manuscript VIII-A}, version 3.
Supplied excerpt: \texttt{A:thm:residual}.
Record agreement does not imply residual-channel agreement.
Enter Calculus Companion.
\url{https://enter.grok.me}

\bibitem{CheungEn09A}
N.~Cheung,
\emph{Manuscript En09-A}, version 1.
Supplied excerpt: \texttt{thm:main}, relational destructive-Enter
instrument and its consumption and covariance hypotheses.
Documentary comparison.
Enter Calculus Companion.
\url{https://enter.grok.me}

\bibitem{CheungXIXC}
N.~Cheung,
\emph{Manuscript XIX-C}, version 1.
Supplied excerpt: \texttt{thm:main},
finite incidence does not determine continuum evolution.
Documentary comparison.
Enter Calculus Companion.
\url{https://enter.grok.me}

\bibitem{CheungEn07B}
N.~Cheung,
\emph{Manuscript En07-B}, version 1.
Supplied excerpt: \texttt{prop:no-landauer-from-channel}.
Channel equality does not determine a thermal implementation cost.
Enter Calculus Companion.
\url{https://enter.grok.me}

\bibitem{CheungXIVH}
N.~Cheung,
\emph{Landmark XIV-H: Internal kernel relations on the operator route},
version 1, supplied manuscript excerpts.
Labels: \texttt{prop:K5exact}, \texttt{thm:H2},
\texttt{thm:release}.
Documentary predecessor scope; no full characteristic map is
imported.
Enter Calculus Companion.
\url{https://enter.grok.me}

\bibitem{CheungXIVI}
N.~Cheung,
\emph{Manuscript XIV-I}, version 1, supplied excerpts.
Labels: \texttt{prop:pole}, \texttt{thm:I2},
\texttt{thm:I3}, \texttt{thm:release}.
Constructed internal amplitude module and unresolved joint
characteristic map, on their stated domains.
Enter Calculus Companion.
\url{https://enter.grok.me}

\bibitem{CheungXIVJ}
N.~Cheung,
\emph{Manuscript XIV-J}, version 1, supplied excerpts.
Labels: \texttt{prop:descent}, \texttt{lem:null},
\texttt{thm:reference}, \texttt{thm:conditionalextension},
\texttt{thm:J2}, \texttt{cor:noadmission}.
Descent and supported-sector reference are not the full joint map.
Enter Calculus Companion.
\url{https://enter.grok.me}

\bibitem{CheungXVB}
N.~Cheung,
\emph{Manuscript XV-B}, version 2, supplied excerpts.
Labels: \texttt{thm:lifting}, \texttt{thm:jetnull},
\texttt{thm:pointcorrection}, \texttt{thm:release}.
Standalone relative-lifting and extension interfaces;
the graph-specific characteristic construction remains unresolved.
Enter Calculus Companion.
\url{https://enter.grok.me}

\bibitem{CheungXVIN}
N.~Cheung,
\emph{Manuscript XVI-N}, revision r12, supplied excerpts.
Label: \texttt{thm:massless-obstruction}, with its explicit Route A
hypotheses and undischarged Route B discussion.
A conditional instance is not universal completion.
Enter Calculus Companion.
\url{https://enter.grok.me}

\end{thebibliography}

\end{document}