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  pdftitle={Oai211-A: Operational Marks on Conditioned Block Trees},
  pdfsubject={Finite half-diamond certificates, path sums, and metric-recovery separation}
}

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\title{\textbf{Oai211-A}\\[4pt]
Operational Marks on Conditioned Block Trees:\\
Finite Half-Diamond Certificates and\\
Metric--Recovery Separation}
\author{}
\date{Version 1 --- 12 October 2026}

\begin{document}
\maketitle

\begin{abstract}
We develop an operational marking theory for critical plane
Galton--Watson trees conditioned on their total size, assuming only
finite offspring variance. Fixed scalar labels bounded by offspring
number satisfy a uniform ancestral-path law of large numbers.
For finite-dimensional channel marks, we define a genuine
path-averaged channel and prove convergence in half-diamond distance
to its size-biased spine average, uniformly over paths of
macroscopic depth. Explicit finite-size probability bounds retain
the cost of size conditioning and an exactly defined height tail.

A second construction gives an actual serial channel:
offspring-bounded nonnegative labels determine weak depolarizing
steps with strength \(N^{-1/2}\). Their compositions admit a uniform
half-diamond comparison with a depth-dependent channel, and their
preorder error profiles converge to an explicit functional of
Brownian excursion. These are designed finite operational models,
not dynamics inferred from a metric scaling limit.

We prove a three-correlation qubit certificate with separate tree,
sampling, calibration, and implementation allowances. We also give
a conditional transfer to weighted planar-map block decompositions,
including the exact corner-to-degree sampling identity and a local
proof of the endpoint criterion for finite offspring variance.

The supplied OpenAI preprint of 24 September 2026 is recorded as
the source of a claimed \(q>4\) Fortuin--Kasteleyn map limit, not
used as an unverified theorem premise. Its application to the present
framework requires the printed analytic and model-identification
conditions. Counterexamples show that metric data, selected entropy,
and path-label concentration do not determine quantum recovery,
entanglement, or a radiation history. In particular, the Brownian
CRT has Hausdorff dimension two, not one; its tree structure does
not make it a two-dimensional manifold or a gravitational surface.
No spacetime emergence, holographic reconstruction, Page curve,
or characteristic quantum coefficient is constructed.
\end{abstract}

\paragraph{Keywords.}
Conditioned Galton--Watson tree; block decomposition; operational
marks; Brownian continuum random tree; diamond distance;
complete complement; quantum-memory certification.

\section{Introduction and the corrected target}
\label{sec:intro}

The supplied preprint
\emph{Brownian continuum random tree limits of finite
Fortuin--Kasteleyn maps above four}, attributed to OpenAI and dated
24 September 2026, states a finite-volume
Gromov--Hausdorff--Prokhorov limit for every fixed \(q>4\)
\cite{OpenAI211}. Its graph metric uses all edges, its measure is
normalized degree measure, and its stated conclusion forgets the
root and FK decoration. No uniformity as \(q\downarrow4\) is asserted.

The proposed extension asks for an operational certificate built
from its block-tree machinery while refusing gravitational
interpretations not justified by that machinery.
This paper completes a mathematically specified version of that
extension. It does not independently prove the supplied FK
scaling-limit claim.

The probabilistic ingredients have established antecedents:
conditioned-tree coding limits
\cite{BroutinMarckert,Janson}, size-biased spine methods
\cite{LyonsPemantlePeres}, and map block decompositions
\cite{AddarioBerry}.
The supplied preprint also contains the two-exploration-order
truncation argument motivating our offspring-bounded path theorem.
We include the local argument with attribution rather than claim
a new general branching-process principle.

The operational contribution is the precise choice of channels,
their error normalization, explicit finite-size bounds, and the
separation of four types of probability or approximation:
\[
 \text{random-tree law},\qquad
 \text{conditional experimental sampling},\qquad
 \text{calibration},\qquad
 \text{model/implementation comparison}.
\]
Cheung's supplied manuscripts emphasize comparable distinctions
between access, complete residuals, deterministic return,
destructive readout, and physical identification
\cite{CheungI,CheungHawk04A,CheungHawk04B,
CheungEn07A,CheungEn07B,CheungEn07C}.
Every finite channel statement used below is stated and proved for
the interfaces printed here.

\subsection{Necessary repairs}

The extension plan requires the following corrections.

\begin{enumerate}
\item \textbf{The Brownian CRT is not metrically one-dimensional.}
It has Hausdorff dimension \(2\) almost surely
\cite{DuquesneLeGall}.
Its real-tree structure excludes a surface-manifold interpretation;
Hausdorff dimension is not manifold dimension.

\item \textbf{A path sum is not a channel composition.}
A sum of local half-diamond errors is generally neither the error
of a serial channel nor an exact decoder error.
We therefore define a path mixture, and separately construct a
serial family for which the composition can be calculated.

\item \textbf{An additive law does not concentrate one root mark.}
Uniform concentration of sums at scale \(\sqrt N\) does not imply
concentration of an observable attached to the single root block.

\item \textbf{Absolute \(o_P(\sqrt N)\) control is not a uniform
relative law at shallow depth.}
Path averages are controlled uniformly only after imposing a
minimum depth. The root and bounded-depth paths are excluded from
that relative conclusion.

\item \textbf{Size conditioning and experimental conditioning differ.}
Shared ancestors make path marks dependent.
Repeated experimental trials are required to be conditionally
independent after the marked tree and the tested path are fixed.

\item \textbf{A metric limit does not instantiate a quantum interface.}
A code, channel, decoder, complement, and physical correspondence
must be supplied independently. A combinatorial corner is not
automatically a boundary quantum register.

\item \textbf{The FK specialization has an analytic gate.}
The finite-variance offspring law reported in the supplied preprint
is not silently assumed as an independently verified fact.
Section~\ref{sec:blocks} prints sufficient analytic conditions and
proves the resulting conditional specialization.

\item \textbf{Failure of a sufficient certificate can be inconclusive.}
It is not automatically a recovery obstruction, geometric failure,
or nonexistence theorem.
\end{enumerate}

\subsection{Dependency policy}

Theorems on the independently defined marked trees use identified
standard mathematical inputs from published sources, especially
the conditioned-tree coding theorem of Broutin and Marckert.
The supplied OpenAI and Cheung texts are cited for attribution,
comparison, and their stated boundaries.

The application to FK maps is conditional on
Definition~\ref{def:blockcontract} and
Assumption~\ref{ass:endpoint}.
No documentary catalogue entry is substituted for those hypotheses.
This distinction permits a coherent proved theorem without claiming
verification of the supplied FK paper or its companions.

\section{The marked tree and its spine law}
\label{sec:tree}

Let \(p=(p_k)_{k\ge0}\) be an offspring law satisfying
\begin{equation}
 p_k=0\quad(k\text{ odd}),\qquad
 p_0p_2>0,\qquad
 \sum_k kp_k=1,\qquad
 0<\sigma^2:=\sum_k(k-1)^2p_k<\infty.
 \label{eq:offspring}
\end{equation}
Let \(T\) be the corresponding plane Galton--Watson tree, and
\(T_N\) its law conditioned on \(|T|=N\).
Write
\[
 \pi_N=\mathbb P(|T|=N).
\]
The attainable sizes are exactly the positive odd integers:
every tree has even total offspring count, and full binary trees
realize every odd size. Throughout, \(N\to\infty\) means convergence
through these sizes.

At each node \(u\), independently conditional on the tree,
assign a mark \(A_u\) with law \(L_{k_u}\), where \(k_u\) is the
number of children of \(u\).
A mark may contain a block, a finite channel vector, or an auxiliary
random variable. Different children at the same node may use
correlated functions of that common mark.

For a strict ancestor \(u\) of \(v\), let \(i_u(v)\in\{1,\ldots,k_u\}\)
be the child leading toward \(v\). Define
\[
 \operatorname{path}(v)
 =\bigl((k_u,i_u(v),A_u):u\text{ a strict ancestor of }v\bigr).
\]
Let \(H(v)\) be depth.

\begin{definition}[Spine distribution]
\label{def:spine}
The probability law of a triple \((K,I,A)\) is
\begin{equation}
 \Psp(K=k,I=i,A\in E)=p_kL_k(E),
 \qquad 1\le i\le k.
 \label{eq:spine}
\end{equation}
It is normalized because \(\sum_k kp_k=1\).
For a scalar label \(b(k,i,A)\), write
\[
 \mu_b=\Esp b(K,I,A),\qquad
 B_b(v)=\sum_{\operatorname{path}(v)}b.
\]
\end{definition}

With
\[
 \nu=\frac{\sigma^2}{2},
\]
we have
\begin{equation}
 \Esp(K-I)=\Esp(I-1)=\nu,\qquad
 \Esp K=\mathbb E\xi^2=\sigma^2+1.
 \label{eq:spinemoments}
\end{equation}
The last expectation is finite. No moment of order three is assumed.

\begin{lemma}[Path-counting identity]
\label{lem:spine}
For every nonnegative measurable function \(F\) of a path of
length \(h\),
\begin{equation}
 \mathbb E\!\left[\sum_{H(v)=h}F(\operatorname{path}(v))\right]
 =
 \Esp F(Z_1,\ldots,Z_h),
 \label{eq:manytoone}
\end{equation}
where \(Z_1,\ldots,Z_h\) are independent triples with
law \eqref{eq:spine}.
\end{lemma}

\begin{proof}
At one ancestor, summing over its selected child assigns
probability \(p_kL_k(dA)\) to each pair \((k,i)\).
Independence of the offspring and marks at different nodes
allows these factors to be multiplied down the ancestral path.
Summing recursively gives \eqref{eq:manytoone}.
No independence between distinct endpoint paths is needed.
\end{proof}

This is the elementary many-to-one form of the size-biased
construction \cite{LyonsPemantlePeres}.

\subsection{The standard coding input}

List the nodes in preorder as \(v_0,\ldots,v_{N-1}\), and define
\[
 W_0=0,\qquad W_{j+1}-W_j=k_{v_j}-1,\qquad W_N=-1.
\]
Set \(H_j=H(v_j)\) and append \(H_N=0\).
Interpolate linearly.

The identified standard input is the following consequence of
Broutin and Marckert's joint coding theorem and their
conditioned-degree lemma \cite[Theorem 3, Lemma 11, and Section 6]
{BroutinMarckert}:
\begin{equation}
 \left(
 \frac{W_{\lfloor Nt\rfloor}}{\sigma\sqrt N},
 \frac{\sigma H_{\lfloor Nt\rfloor}}{2\sqrt N}
 \right)_{0\le t\le1}
 \Longrightarrow(e,e)
 \quad\text{in the uniform continuous-path topology}.
 \label{eq:coding}
\end{equation}
The interpolation and endpoint conventions make the floors harmless.
The excursion \(e\) is the standard normalized Brownian excursion.
Conditional on degree counts, the tree is uniform among plane trees
with those counts; this is the passage from that theorem to
\eqref{eq:coding}.

In particular,
\begin{equation}
 H_{\max}=O_P(\sqrt N),\qquad
 \max_v|W(v)-\nu H(v)|=o_P(\sqrt N).
 \label{eq:codingconsequences}
\end{equation}
Reversing every child order preserves the conditioned tree law.
If \(W^{\rm rev}(v)\) is the preorder walk value in the reversed
tree, then also
\begin{equation}
 \max_v|W^{\rm rev}(v)-\nu H(v)|=o_P(\sqrt N).
 \label{eq:reversecoding}
\end{equation}
Only tree-shape symmetry is used; the mark law need not be symmetric.

\subsection{Exact conditioning and height probabilities}

Let
\[
 P(z)=\sum_{k\ge0}p_kz^k.
\]

\begin{lemma}[Conditioning probability and height recursion]
\label{lem:conditioning}
For odd \(N\),
\begin{equation}
 \pi_N=\frac1N[z^{N-1}]P(z)^N
 \sim\frac{2}{\sigma\sqrt{2\pi}}N^{-3/2}.
 \label{eq:pi}
\end{equation}
Define
\[
 F_{-1}(z)=0,\qquad F_L(z)=zP(F_{L-1}(z)),\quad L\ge0.
\]
Then
\begin{equation}
 \gamma_N(L):=\PN(H_{\max}>L)
 =1-\frac{[z^N]F_L(z)}{\pi_N}.
 \label{eq:heighttail}
\end{equation}
For a fixed \(N\), the coefficients required in these formulas
depend only on \(p_0,\ldots,p_{N-1}\).
\end{lemma}

\begin{proof}
The offspring increments are at least \(-1\).
Among the \(N\) circular shifts of a list totaling \(-1\), exactly
one is a valid tree exploration. Summing the iid probabilities gives
the Otter--Dwass formula \eqref{eq:pi}
\cite{Dwass}. The increment law \(\xi-1\) has lattice span two,
mean zero, and variance \(\sigma^2\).
The finite-variance lattice local limit theorem at \(-1\) yields
the stated asymptotic.

A tree of height at most \(L\) has a root and a sequence of children
whose subtrees have height at most \(L-1\).
Multiplying their generating functions and averaging over root
offspring gives \(F_L=zP(F_{L-1})\).
For \(L=0\), this gives \(F_0=p_0z\).
Its coefficient of \(z^N\) is therefore the unconditioned
probability of total size \(N\) and height at most \(L\).
Division by \(\pi_N\) proves \eqref{eq:heighttail}.
A tree of \(N\) nodes has no offspring number exceeding \(N-1\).
\end{proof}

Equations \eqref{eq:pi} and \eqref{eq:heighttail} specify finite
quantities. Inclusion of a coefficient formula is not a claim that
it has been numerically evaluated or rigorously enclosed.

\section{Uniform path sums}
\label{sec:paths}

\begin{lemma}[Finite-size bounded-label certificate]
\label{lem:finitepath}
Suppose \(|b|\le B\), with \(B>0\).
For \(x>0\) and \(1\le L\le N-1\),
\begin{equation}
 \PN\!\left(
 \max_{v\in T_N}|B_b(v)-\mu_bH(v)|>x
 \right)
 \le
 \gamma_N(L)+
 \frac{2}{\pi_N}
 \sum_{h=1}^{L}
 \exp\left(-\frac{x^2}{2B^2h}\right).
 \label{eq:finitepath}
\end{equation}
The right side may be capped at one.

For \(1\le\ell\le L\le N-1\) and \(r>0\),
\begin{equation}
 \PN\!\left(
 \max_{H(v)\ge\ell}
 \left|\frac{B_b(v)}{H(v)}-\mu_b\right|>r
 \right)
 \le
 \gamma_N(L)+
 \frac{2}{\pi_N}
 \sum_{h=\ell}^{L}
 \exp\left(-\frac{hr^2}{2B^2}\right).
 \label{eq:finiterelative}
\end{equation}
A maximum over an empty set is zero.
\end{lemma}

\begin{proof}
For an iid spine path of length \(h\), Hoeffding's inequality
for variables in \([-B,B]\) gives
\[
 \Psp\left(
 \left|\sum_{j=1}^h b(Z_j)-h\mu_b\right|>x
 \right)
 \le2\exp\left(-\frac{x^2}{2B^2h}\right)
 \quad\cite{Hoeffding}.
\]
The probability that there is a violating node at depth \(h\)
is at most the expected number of such nodes.
Apply Lemma~\ref{lem:spine} in the unconditioned tree, then bound
the probability after size conditioning by division by \(\pi_N\).
Sum over \(h\le L\), and add
\(\PN(H_{\max}>L)=\gamma_N(L)\).
This proves \eqref{eq:finitepath}.
Use \(x=hr\) at depth \(h\) to prove
\eqref{eq:finiterelative}.
\end{proof}

\begin{corollary}[Bounded path law]
\label{cor:boundedpath}
For every fixed bounded scalar label,
\[
 \max_v|B_b(v)-\mu_bH(v)|=o_P(\sqrt N).
\]
For every fixed \(a>0\),
\[
 \max_{H(v)\ge a\sqrt N}
 \left|\frac{B_b(v)}{H(v)}-\mu_b\right|
 \longrightarrow0
 \quad\text{in probability}.
\]
\end{corollary}

\begin{proof}
In \eqref{eq:finitepath}, use \(x=\varepsilon\sqrt N\)
and \(L=\lfloor C\sqrt N\rfloor\).
The sum divided by \(\pi_N\) tends to zero exponentially in
\(\sqrt N\), up to a polynomial factor.
Then let \(C\) increase, using
\eqref{eq:codingconsequences}.
The relative statement follows by division by \(a\sqrt N\).
\end{proof}

\subsection{Offspring-bounded labels without a third moment}

For each strict ancestor-child pair \((k,i)\), preorder exploration
gives the exact identities
\begin{equation}
 W(v)=\sum_{\operatorname{path}(v)}(k-i),\qquad
 W^{\rm rev}(v)=\sum_{\operatorname{path}(v)}(i-1).
 \label{eq:twowalks}
\end{equation}
Indeed, the pending siblings to the right contribute \(k-i\);
the reversed order exchanges right and left siblings.

\begin{theorem}[Uniform offspring-bounded path sums]
\label{thm:path}
For a fixed measurable label satisfying
\[
 |b(k,i,A)|\le C_bk
\]
for some finite constant \(C_b\), its spine mean is finite and
\begin{equation}
 \max_{v\in T_N}|B_b(v)-\mu_bH(v)|=o_P(\sqrt N).
 \label{eq:pathmain}
\end{equation}
Also,
\[
 \max_v\left|B_b(v)-\frac{\mu_b}{\nu}W(v)\right|
 =o_P(\sqrt N).
\]
Only \(\mathbb E\xi^2<\infty\) is used.
\end{theorem}

\begin{proof}
It suffices to consider \(C_b=1\).
Equation~\eqref{eq:twowalks} gives
\[
 \sum_{\operatorname{path}(v)}k
 =W(v)+W^{\rm rev}(v)+H(v).
\]
By \eqref{eq:codingconsequences}--\eqref{eq:reversecoding},
\begin{equation}
 \max_v\left|
 \sum_{\operatorname{path}(v)}k-(\sigma^2+1)H(v)
 \right|=o_P(\sqrt N).
 \label{eq:offspringsum}
\end{equation}

For fixed \(R\), the label \(k\mathbf1_{\{k\le R\}}\) is bounded.
Subtract its conclusion in
Corollary~\ref{cor:boundedpath} from
\eqref{eq:offspringsum}. With
\[
 m_R=\Esp[K\mathbf1_{\{K>R\}}],
\]
we obtain, for each fixed \(R\),
\[
 \max_v\left|
 \sum_{\operatorname{path}(v)}k\mathbf1_{\{k>R\}}
 -m_RH(v)
 \right|=o_P(\sqrt N).
\]
Because \(\Esp K<\infty\), \(m_R\to0\).
Height tightness consequently implies
\[
 \lim_{R\to\infty}\limsup_{N\to\infty}
 \PN\left(
 \max_v\sum_{\operatorname{path}(v)}
 k\mathbf1_{\{k>R\}}>\varepsilon\sqrt N
 \right)=0.
\]

Apply Corollary~\ref{cor:boundedpath} to
\(b\mathbf1_{\{k\le R\}}\).
The discarded absolute path sum is bounded by the displayed
offspring tail, and the discarded absolute spine mean is bounded
by \(m_R\). First take \(N\to\infty\), then \(R\to\infty\).
This proves \eqref{eq:pathmain}.
The comparison with \(W\) follows from
\eqref{eq:codingconsequences}.
\end{proof}

The two-walk truncation follows the argument in the supplied
preprint \cite[Section 5]{OpenAI211}.
It uses \(\Esp K=\mathbb E\xi^2\), not
\(\Esp K^2=\mathbb E\xi^3\).
An exponential moment or a third offspring moment is not
silently introduced.

\subsection{A root mark need not concentrate}

\begin{proposition}[Root-mark counterexample]
\label{prop:root}
Path concentration does not imply concentration of one root mark.
\end{proposition}

\begin{proof}
Take \(p_0=p_2=1/2\). At every node attach an independent
Bernoulli\((1/2)\) mark \(U_u\), and define \(b(k,i,U)=U\)
for \(k>0\).
Then \(0\le b\le k\), so Theorem~\ref{thm:path} applies.
For every conditioned size \(N>1\), the root is internal and
its mark remains Bernoulli\((1/2)\), independently of the
size-conditioning event. It does not concentrate on its mean.
\end{proof}

Thus a three-setting correlator on a single root-block factor
requires its own preparation and sampling certificate.
It is not certified by an ancestral-path law.

\section{Finite-dimensional channel marks}
\label{sec:channels}

Fix a logical space \(C\cong\bbC^d\), \(d\ge2\).
All channel marks in this section have input and output \(C\).
The choice of this finite space is additional operational data,
not a consequence of the tree.

For states and channels, use
\[
 \dtr(\rho,\sigma)=\tfrac12\norm{\rho-\sigma}_1,\qquad
 \ddia(\Phi,\Psi)=\tfrac12\norm{\Phi-\Psi}_\diamond.
\]
The diamond norm includes arbitrary external references.
For any Hermiticity-preserving linear map \(\Delta\), write
\[
 \|\Delta\|_{\diamond,1/2}=\tfrac12\|\Delta\|_\diamond.
\]
This notation also applies to sums that are not channels.

Let \(\Lambda(k,i,A)\) be a fixed measurable CPTP channel-valued
function. Define its spine average
\begin{equation}
 \overline\Lambda
 =\sum_{k\ge1}p_k\sum_{i=1}^k
       \int\Lambda(k,i,A)\,L_k(dA).
 \label{eq:spinechannel}
\end{equation}
The weights sum to one. Finite-dimensional integration and
convexity make \(\overline\Lambda\) a CPTP channel.

For \(H(v)>0\), define
\begin{equation}
 \Lambda_v^{\rm av}
 =\frac1{H(v)}
   \sum_{\operatorname{path}(v)}\Lambda(k,i,A).
 \label{eq:pathchannel}
\end{equation}

\begin{definition}[The path-average operation]
\label{def:averageoperation}
Conditional on a fixed marked tree and endpoint \(v\), the
operation chooses one strict ancestor-child slot of the path
uniformly and applies its channel once to the logical input.
The choice is independent of that input.
After ignoring the choice register, the returned channel is
\(\Lambda_v^{\rm av}\).
\end{definition}

This is a finite randomized operation. It is not traversal through
all blocks, simultaneous application of all marks, or a decoder
whose existence is inferred from geometry.
If the chosen slot is retained as an output flag, the channel type
changes and must be analyzed separately.

\subsection{A coordinate estimate}

Let
\[
 \Phi_d=\frac1{\sqrt d}\sum_{i=0}^{d-1}\ket i\ket i,\qquad
 P_d=\kb{\Phi_d}{\Phi_d},\qquad
 J_\Delta=(\id\otimes\Delta)(P_d).
\]
The normalized Choi matrix of a channel is a density matrix.

\begin{lemma}[Choi-coordinate control]
\label{lem:choicoord}
Let \(E_1,\ldots,E_{d^4}\) be a Hermitian Hilbert--Schmidt
orthonormal basis of matrices on \(C\otimes C\).
Set
\[
 a_j(\Lambda)=\Tr(E_jJ_\Lambda).
\]
Then \(|a_j(\Lambda)|\le1\).
For a Hermiticity-preserving \(\Delta\),
\begin{equation}
 \|\Delta\|_{\diamond,1/2}
 \le\frac{d^4}{2}
       \max_j|\Tr(E_jJ_\Delta)|.
 \label{eq:coordinatebound}
\end{equation}
\end{lemma}

\begin{proof}
Each \(E_j\) has operator norm at most its Hilbert--Schmidt
norm one, proving the scalar bound on a state.

For completeness,
\[
 \|\Delta\|_\diamond\le d\|J_\Delta\|_1.
\]
To see this, write unit referenced vectors as
\(u=(A\otimes I)\Phi_d\) and
\(v=(B\otimes I)\Phi_d\), where
\(\Tr(A^\dagger A)=\Tr(B^\dagger B)=d\).
The output of \(\kb uv\) is
\((A\otimes I)J_\Delta(B^\dagger\otimes I)\), with trace norm
at most \(d\|J_\Delta\|_1\).
A singular-value decomposition extends the inequality to all
trace-norm-one inputs and all reference dimensions.

The matrix \(J_\Delta\) has size \(d^2\), so
\[
 \|J_\Delta\|_1
 \le d\|J_\Delta\|_{\rm HS}
 \le d^3\max_j|\Tr(E_jJ_\Delta)|.
\]
Divide by two.
\end{proof}

The normalized Choi and completely bounded norm conventions agree
with the standard channel theory \cite{Watrous}.

\begin{theorem}[Finite half-diamond path-average certificate]
\label{thm:channel}
Under \eqref{eq:offspring} and the marking contract,
\begin{equation}
 \frac1{\sqrt N}
 \max_v
 \left\|
 \sum_{\operatorname{path}(v)}
       (\Lambda-\overline\Lambda)
 \right\|_{\diamond,1/2}
 \longrightarrow0
 \quad\text{in probability}.
 \label{eq:channelsum}
\end{equation}
Consequently, for every fixed \(a>0\),
\begin{equation}
 \max_{H(v)\ge a\sqrt N}
 \ddia(\Lambda_v^{\rm av},\overline\Lambda)
 \longrightarrow0
 \quad\text{in probability}.
 \label{eq:channelaverage}
\end{equation}

For \(x>0\), \(1\le L\le N-1\), the probability that the
maximum in \eqref{eq:channelsum}, before division by \(\sqrt N\),
exceeds \(x\) is at most
\begin{equation}
 \gamma_N(L)+
 \frac{2d^4}{\pi_N}
 \sum_{h=1}^{L}
 \exp\left(-\frac{2x^2}{d^8h}\right).
 \label{eq:channelabsolute}
\end{equation}
For \(1\le\ell\le L\), the probability that
\[
 \max_{H(v)\ge\ell}
 \ddia(\Lambda_v^{\rm av},\overline\Lambda)>\varepsilon
\]
is at most
\begin{equation}
 \gamma_N(L)+
 \frac{2d^4}{\pi_N}
 \sum_{h=\ell}^{L}
 \exp\left(-\frac{2\varepsilon^2h}{d^8}\right).
 \label{eq:channelrelative}
\end{equation}
Both bounds may be capped at one.
\end{theorem}

\begin{proof}
The spine mean of \(a_j(\Lambda)\) is
\(a_j(\overline\Lambda)\).
If the half-diamond norm of a centered path sum exceeds \(x\),
Lemma~\ref{lem:choicoord} implies a coordinate deviation greater
than \(2x/d^4\).
Apply Lemma~\ref{lem:finitepath} with \(B=1\) to each coordinate,
and use a union bound over \(d^4\) coordinates.
This proves \eqref{eq:channelabsolute}.

For a path of depth \(h\), a path-average error exceeding
\(\varepsilon\) gives a centered sum error exceeding
\(h\varepsilon\). The same argument proves
\eqref{eq:channelrelative}.
The asymptotic statements follow from
Corollary~\ref{cor:boundedpath}, the finite coordinate set,
and division by the minimum depth.
\end{proof}

The bounds are not optimized in dimension.
They are finite, explicit, and valid without pretending that the
conditioned ancestral triples are iid.

\begin{corollary}[Loss labels and channel errors are distinct]
\label{cor:loss}
Let
\[
 e_{\rm loc}(k,i,A)=\ddia(\Lambda(k,i,A),\id_C),
 \qquad \mu_{\rm loc}=\Esp e_{\rm loc}.
\]
Then the path average of these scalar losses concentrates on
\(\mu_{\rm loc}\) on macroscopic paths.
Also,
\[
 \ddia(\Lambda_v^{\rm av},\id_C)
 \le\frac1{H(v)}
       \sum_{\operatorname{path}(v)}e_{\rm loc},
\]
and
\[
 \left|
 \ddia(\Lambda_v^{\rm av},\id_C)
 -\ddia(\overline\Lambda,\id_C)
 \right|
 \le\ddia(\Lambda_v^{\rm av},\overline\Lambda).
\]
Generally,
\(\ddia(\overline\Lambda,\id_C)\ne\mu_{\rm loc}\).
\end{corollary}

\begin{proof}
The scalar loss lies in \([0,1]\), so the bounded path law
applies. Convexity proves the upper bound.
The reverse triangle inequality proves the last displayed
comparison. No equality follows from these inequalities.
\end{proof}

\subsection{Composition is a different operation}

For a chosen path with channels \(\Lambda_1,\ldots,\Lambda_h\),
the serial channel is
\[
 \Lambda_h\circ\cdots\circ\Lambda_1.
\]
Telescoping and CPTP contraction give
\[
 \ddia(\Lambda_h\circ\cdots\circ\Lambda_1,\id_C)
 \le\min\left\{1,\sum_{r=1}^h
                   \ddia(\Lambda_r,\id_C)\right\}.
\]
This is an upper bound, not an identity.

For example, for a qubit and \(\Lambda_r=\Ad_Z\) at every step,
each local error is one, while the serial channel is identity for
even \(h\) and \(\Ad_Z\) for odd \(h\).
The path average is always \(\Ad_Z\).
Thus neither a local error sum nor a path mixture can be renamed
as the serial error.

\section{An actual serial half-diamond certificate}
\label{sec:serial}

A meaningful serial theorem requires a composition law.
We now supply one explicitly rather than infer it from the map.

Let \(s(k,i,A)\) be a fixed nonnegative label with
\[
 0\le s(k,i,A)\le C_sk,\qquad
 \mu=\Esp s<\infty,\qquad
 S(v)=\sum_{\operatorname{path}(v)}s.
\]
Define the completely depolarizing channel
\[
 \Tdep(X)=\Tr(X)I_C/d
\]
and the designed local channel
\begin{equation}
 D_{N,s}=e^{-s/\sqrt N}\id_C+
          (1-e^{-s/\sqrt N})\Tdep.
 \label{eq:weaksteps}
\end{equation}
This \(N^{-1/2}\) weakness is part of the model definition.
It is not deduced from a geometric scaling exponent.

\begin{lemma}[Depolarizing composition and exact error]
\label{lem:depolarizing}
For \(a,b\in[0,1]\), write \(D_a=a\id+(1-a)\Tdep\).
Then
\[
 D_aD_b=D_{ab},\qquad
 \ddia(D_a,D_b)=\kappa_d|a-b|,
 \qquad \kappa_d=1-d^{-2}.
\]
In particular, \(\ddia(D_a,\id)=\kappa_d(1-a)\).
\end{lemma}

\begin{proof}
The relation \(\Tdep^2=\Tdep\) and commutation with identity give
the composition formula.

On a Bell input, \(\Tdep\) gives \(I/d^2\).
Testing the Bell projector gives
\(\ddia(\Tdep,\id)\ge1-d^{-2}\).
The uniform average of the \(d^2\) Weyl conjugations is
\(\Tdep\). Separating the identity conjugation and using the
triangle inequality gives the matching upper bound.
Finally,
\(D_a-D_b=(a-b)(\id-\Tdep)\).
\end{proof}

\begin{theorem}[Serial path certificate and excursion profile]
\label{thm:serial}
The serial composition of \eqref{eq:weaksteps} along the path to
\(v\) is exactly
\[
 \mathcal S_{N,v}
 =e^{-S(v)/\sqrt N}\id+
      (1-e^{-S(v)/\sqrt N})\Tdep.
\]
Define its depth predictor
\[
 \mathcal S^{\rm pred}_{N,v}
 =e^{-\mu H(v)/\sqrt N}\id+
      (1-e^{-\mu H(v)/\sqrt N})\Tdep.
\]
Then
\begin{equation}
 \max_v\ddia(\mathcal S_{N,v},
                    \mathcal S^{\rm pred}_{N,v})
 \le
 \frac{\kappa_d}{\sqrt N}
       \max_v|S(v)-\mu H(v)|
 =o_P(1).
 \label{eq:serialcomparison}
\end{equation}
The exact returned-code error is
\[
 e_{N,v}=\ddia(\mathcal S_{N,v},\id)
 =\kappa_d(1-e^{-S(v)/\sqrt N}).
\]
Its linearly interpolated preorder profile, with terminal value
zero appended, converges in distribution in the uniform topology to
\begin{equation}
 t\longmapsto
 \kappa_d\left(1-\exp\left(-\frac{2\mu}{\sigma}e(t)\right)\right).
 \label{eq:serialprofile}
\end{equation}
Furthermore,
\begin{equation}
 \frac1{N-1}\sum_{v\ne v_0}e_{N,v}
 \Longrightarrow
 \int_0^1
 \kappa_d\left(1-e^{-(2\mu/\sigma)e(t)}\right)\,dt.
 \label{eq:serialintegral}
\end{equation}
\end{theorem}

\begin{proof}
Lemma~\ref{lem:depolarizing} makes the parameters multiply, yielding
the exact composition.
The function \(x\mapsto e^{-x}\) is one-Lipschitz on
\([0,\infty)\).
Therefore Lemma~\ref{lem:depolarizing} and
Theorem~\ref{thm:path} prove \eqref{eq:serialcomparison}.

Uniform approximation by the depth predictor, together with
\eqref{eq:coding}, gives \eqref{eq:serialprofile} by continuous
mapping. Appending the last zero is harmless because the normalized
last height tends to zero and the uniform approximation controls
the corresponding label sum.
Uniform convergence in distribution to a continuous profile also
gives convergence of the Riemann sums.
Removing the root and using \(N-1\) instead of \(N\) changes a bounded
average by \(O(N^{-1})\), proving \eqref{eq:serialintegral}.
\end{proof}

The limiting profile is random.
The theorem does not assert convergence of the complete profile
to a deterministic constant.

If \(0\le s\le B\), Lemma~\ref{lem:finitepath} gives a finite-size
failure probability for any chosen bound on the right side of
\eqref{eq:serialcomparison}.
For general offspring-bounded \(s\), the finite-variance theorem
supplies the stated asymptotic conclusion; no numerical rate is
invented.

\begin{remark}[The preorder parameter is not physical time]
The variable \(t\) in \eqref{eq:serialprofile} indexes preorder
locations. It is not evaporation time, proper time, or radiation
release time. The error profile is not an entropy profile and is
not a Page curve.
\end{remark}

\section{Three Bell correlations and a complete finite experiment}
\label{sec:bell}

For this section \(d=2\).
Let \(X,Y,Z\) be the Pauli matrices. The Bell projector is
\[
 P_2=\frac{I+X\otimes X-Y\otimes Y+Z\otimes Z}{4}.
\]
For a channel \(\Lambda\), set
\[
 c_x(\Lambda)=\Tr((X\otimes X)J_\Lambda),\quad
 c_y(\Lambda)=\Tr((Y\otimes Y)J_\Lambda),\quad
 c_z(\Lambda)=\Tr((Z\otimes Z)J_\Lambda).
\]
Each correlator lies in \([-1,1]\), and
\[
 F(\Lambda)=\Tr(P_2J_\Lambda)
 =\frac{1+c_x(\Lambda)-c_y(\Lambda)+c_z(\Lambda)}4.
\]
In particular, let \(F_{\rm sp}=F(\overline\Lambda)\).

\begin{lemma}[A tree-law Bell-overlap interval]
\label{lem:treebell}
For \(1\le\ell\le L\le N-1\) and \(r>0\), set
\begin{equation}
 \delta_{\rm tree}
 =
 \min\left\{1,\,
 \gamma_N(L)+\frac6{\pi_N}
       \sum_{h=\ell}^{L}e^{-hr^2/2}
 \right\}.
 \label{eq:treefailure}
\end{equation}
With probability at least \(1-\delta_{\rm tree}\),
simultaneously for all \(v\) with \(H(v)\ge\ell\),
\[
 |F(\Lambda_v^{\rm av})-F_{\rm sp}|\le\frac{3r}{4}.
\]
\end{lemma}

\begin{proof}
Apply \eqref{eq:finiterelative} to the three correlator labels
with \(B=1\), and use a union bound.
Their path-average correlators are exactly those of
\(\Lambda_v^{\rm av}\), by linearity.
The fidelity combination has sum of absolute coefficients \(3/4\).
\end{proof}

The uniform event permits selection of the endpoint after observing
the marked tree. It does not permit selecting the channel-mark rule
after seeing the marks without additional complexity control.

\subsection{The experiment contract}

Conditional on the marked tree and a selected endpoint \(v\) with
\(H(v)\ge\ell\), let
\[
 \Lambda_0=\Lambda_v^{\rm av}
\]
be the declared pre-analyzer channel.
Let \(\Lambda_\star\) be the identified implemented target channel
on the same logical input/output spaces.

Each setting \(j\in\{x,y,z\}\) has a fixed count \(m_j\ge1\)
of independent attempted trials, conditional on the complete
selection and calibration record. Its score \(Z_{ji}\in[-1,1]\)
is specified for every attempted outcome, including loss,
readout failure, and invalid-preparation flags.

Let \(a_j=\mathbb E[Z_{ji}\mid\text{selection and calibration}]\).
On a calibration/transfer event \(G\), assume
\begin{equation}
 \left|\frac{1+a_x-a_y+a_z}{4}-F(\Lambda_0)\right|
 \le b_{\rm cal},
 \qquad
 \ddia(\Lambda_\star,\Lambda_0)\le b,
 \qquad
 \mathbb P(G^c)\le\beta.
 \label{eq:calibration}
\end{equation}
These are proved or explicitly assumed allowances.
They are not determined by the map.

A sufficient bias conversion is the following.
If the actual common pre-analyzer state \(\rho\) satisfies
\(\dtr(\rho,J_{\Lambda_0})\le\delta\), and the complete score
observable \(B_j\) differs in operator norm from the named Pauli
correlation observable by at most \(\eta_j\), then
\[
 b_{\rm cal}=\delta+\frac{\eta_x+\eta_y+\eta_z}{4}
\]
is valid. Indeed, the Bell projector is an effect, so its expectation
changes by at most \(\delta\); the three score replacements cost
at most the displayed sum.

This separates prepared Choi-state accuracy from correlation
analyzer accuracy. It is comparable to the supplied Oai265-B
protocol, whose report is not independently rerun here
\cite{Oai265B}.

\begin{lemma}[Bell overlap and half-diamond error]
\label{lem:fidelity}
For a channel \(\Lambda:C\to C\) of logical dimension \(d\), with
\(F=\Tr(P_dJ_\Lambda)\),
\begin{equation}
 1-F\le\ddia(\Lambda,\id)
 \le\min\{1,d\sqrt{1-F}\}.
 \label{eq:bellerror}
\end{equation}
\end{lemma}

\begin{proof}
The Bell-projector effect proves the lower bound.
For a state \(\rho\) and rank-one projector \(P\),
\[
 \dtr(\rho,P)\le\sqrt{1-\Tr(P\rho)}.
\]
This follows by decomposing \(\rho\) into pure states, using the
pure-state distance formula, trace-norm convexity, and
square-root concavity; it is the pure-target
Fuchs--van de Graaf bound \cite{FuchsVanDeGraaf}.
Combine it with
\(\ddia(\Lambda,\id)\le d\,\dtr(J_\Lambda,P_d)\),
proved in Lemma~\ref{lem:choicoord}.
\end{proof}

\begin{theorem}[Finite half-diamond experiment certificate]
\label{thm:experiment}
Fix \(\alpha_{\rm s}\in(0,1)\), and put
\[
 \widehat c_j=\frac1{m_j}\sum_{i=1}^{m_j}Z_{ji},
 \qquad
 r_j^{\rm s}=
 \sqrt{\frac2{m_j}\log\frac6{\alpha_{\rm s}}},
 \qquad
 R_{\rm s}=\frac{r_x^{\rm s}+r_y^{\rm s}+r_z^{\rm s}}4,
\]
\[
 \widehat F=\frac{1+\widehat c_x-\widehat c_y+\widehat c_z}{4}.
\]
Define
\begin{align*}
 F_L&=\max\left\{
 0,\ \widehat F-R_{\rm s}-b_{\rm cal},\
 F_{\rm sp}-3r/4
 \right\},\\
 F_U&=\min\left\{
 1,\ \widehat F+R_{\rm s}+b_{\rm cal},\
 F_{\rm sp}+3r/4
 \right\}.
\end{align*}
If \(F_L>F_U\), report an inconsistent interval and do not certify.

Otherwise, on an event of probability at least
\[
 1-\delta_{\rm tree}-\alpha_{\rm s}-\beta,
\]
the implemented target error obeys
\begin{equation}
 \max\{0,1-F_U-b\}
 \le\ddia(\Lambda_\star,\id_2)
 \le\min\{1,2\sqrt{1-F_L}+b\}.
 \label{eq:finitecertificate}
\end{equation}
A sufficient acceptance rule for threshold \(\varepsilon\) is
\[
 F_L\le F_U,\qquad
 \min\{1,2\sqrt{1-F_L}+b\}\le\varepsilon.
\]
The probability of accepting while the actual target error exceeds
\(\varepsilon\) is at most
\(\delta_{\rm tree}+\alpha_{\rm s}+\beta\).

The tree interval may be omitted. In that case its failure
allowance is omitted as well.
\end{theorem}

\begin{proof}
Conditional Hoeffding inequalities give
\[
 \mathbb P(|\widehat c_j-a_j|>r_j^{\rm s}
          \mid\text{selection and calibration})
 \le\alpha_{\rm s}/3.
\]
A union bound and integration over the conditioning record give
simultaneous statistical coverage with failure probability at most
\(\alpha_{\rm s}\).
On this event and \(G\), the experimental interval covers
\(F(\Lambda_0)\).
Lemma~\ref{lem:treebell} supplies the independent type of
tree-law interval. On the intersection of the three valid events,
their intersection is nonempty and
\[
 F_L\le F(\Lambda_0)\le F_U.
\]
Apply Lemma~\ref{lem:fidelity}, then the reverse triangle inequality
and \(\ddia(\Lambda_\star,\Lambda_0)\le b\).
The union bound proves joint coverage and the invalid-acceptance
bound.
\end{proof}

The square-root conversion concerns the tested channel.
The transfer allowance \(b\), already in half-diamond units,
is added outside the square root.

\subsection{Failure records and resource claims}

A finite operational programme supplies complete history maps
\(K_h\) satisfying
\[
 \sum_hK_h^\dagger K_h=I.
\]
The returned channel is the sum over every history, with all
failure outputs declared.
If the experiment consumes a returned quantum register to obtain
a record, that later destructive operation is not the earlier
quantum return.

These distinctions are explicitly developed in Cheung's
En02-B/C and Landmark I
\cite{CheungEn02B,CheungEn02C,CheungI}.
The present information-theoretic mixture specifies a channel.
A resource bound additionally requires its own instruction and
live-register ledger. No such laboratory implementation is inferred
from tree distance.

\begin{remark}[Small sample counts]
For equal counts \(m\) and the Hoeffding-only upper certificate with
zero allowances, the best possible value
\(\widehat F=1\) gives
\[
 F_L\le1-R_{\rm s},\qquad
 R_{\rm s}=\frac34
       \sqrt{\frac2m\log\frac6{\alpha_{\rm s}}}.
\]
Acceptance at \(\varepsilon<1\) therefore requires
\(R_{\rm s}\le\varepsilon^2/4\).
This is a restriction of this stated certificate, not a universal
sample lower bound for all quantum-memory protocols.
The supplied Oai265-B execution report and its version-specific
nonacceptance claim are documentary evidence only
\cite{Oai265B}; no new dataset or execution is claimed here.
\end{remark}

\section{Conditional transfer to weighted planar-map blocks}
\label{sec:blocks}

This section specifies the application gate.
Its conditional statements do not promote the supplied FK proof
to a verified premise.

\begin{definition}[Weighted corner-substitution contract]
\label{def:blockcontract}
A weighted rooted planar-map class supplies:
\begin{enumerate}
\item one trivial zero-edge map of weight one;
\item rooted nontrivial blocks, including the one-link and
one-loop blocks, with positive weights \(w(B)\);
\item a unique recursive corner substitution, with one child slot
per block corner and trivial insertions retained as leaves;
\item multiplicative map weight equal to the product of its block
weights;
\item a deterministic order of each rooted block's \(2m\) corners
when it has \(m\) edges.
\end{enumerate}
Write
\[
 a_n=\sum_{|E(M)|=n}w(M),\quad
 b_m=\sum_{|E(B)|=m}w(B),\quad b_0=1,
\]
\[
 T(z)=\sum_{n\ge0}a_nz^n,\qquad
 \Phi(y)=1+\sum_{m\ge1}b_my^m.
\]
Then the substitution contract gives the formal identity
\begin{equation}
 T(z)=\Phi(zT(z)^2).
 \label{eq:substitution}
\end{equation}
\end{definition}

The rooted block-tree organization is standard
\cite{AddarioBerry}.
The contract states the rooting and empty-insertion conventions
actually needed here.

\begin{assumption}[Endpoint coefficient inputs]
\label{ass:endpoint}
For some \(u>0\),
\[
 \tau:=T(u)<\infty,
\]
and for some \(c>0\), all real \(x<1\) sufficiently close to one
satisfy
\begin{equation}
 \sum_{n\ge1}na_nu^nx^n\ge c(1-x)^{-1/2}.
 \label{eq:lowerinput}
\end{equation}
Also \(b_1>0\).
\end{assumption}

\begin{lemma}[Quadratic inverse criterion]
\label{lem:inverse}
Under Definition~\ref{def:blockcontract} and
Assumption~\ref{ass:endpoint}, put \(v=u\tau^2\).
Then
\[
 \Phi(v)=\tau,\qquad
 2v\Phi'(v)=\Phi(v),\qquad
 \Phi''(v)<\infty,
\]
where derivatives at the endpoint denote nonnegative coefficient
series.
\end{lemma}

\begin{proof}
Integrate \eqref{eq:lowerinput} to obtain, near \(u\),
\[
 \tau-T(z)
 =\int_{z/u}^1\frac1x
       \sum_{n\ge1}na_nu^nx^n\,dx
 \ge c_1\sqrt{u-z}.
\]
All exchanges are justified by nonnegativity.

The function \(y(z)=zT(z)^2\) increases from zero to \(v\).
The formal substitution identity extends to interior positive
arguments and then to \(\Phi(v)=\tau\) by monotone convergence.
Its inverse is
\[
 f(y)=\frac{y}{\Phi(y)^2}.
\]
Since \(f\) is increasing,
\[
 f'(y)=\frac{\Phi(y)-2y\Phi'(y)}{\Phi(y)^3}\ge0.
\]
This bounds \(\Phi'(y)\) near \(v\), so its increasing derivative
series has a finite endpoint limit.

Moreover,
\[
 v-y(z)
 =(u-z)\tau^2+z(\tau^2-T(z)^2)
 \ge c_2(\tau-T(z))
\]
near \(u\). Hence
\[
 0\le u-f(y)\le C(v-y)^2.
\]
The finite endpoint limit of \(f'\) must be zero, proving
\(2v\Phi'(v)=\Phi(v)\).

Finally,
\[
 f''(y)=
 -\frac{4\Phi'(y)}{\Phi(y)^3}
 -\frac{2y\Phi''(y)}{\Phi(y)^3}
 +\frac{6y\Phi'(y)^2}{\Phi(y)^4}.
\]
If \(\Phi''(v)=\infty\), then \(f''(y)\to-\infty\).
For every large \(K\), sufficiently near \(v\),
\(f''(y)\le-K\).
Integrating twice, using \(f'(v-)=0\), gives
\[
 u-f(y)\ge\tfrac K2(v-y)^2,
\]
contradicting the preceding fixed quadratic upper bound.
\end{proof}

This is the endpoint argument reported in
\cite[Lemma 4.2]{OpenAI211}, included locally.
It requires no analytic continuation across \(v\).

Define
\begin{equation}
 p_{2m}=\frac{b_mv^m}{\Phi(v)},\qquad p_{2m+1}=0.
 \label{eq:blockoffspring}
\end{equation}
Then
\[
 \mathbb E\xi=1,\qquad
 \sigma^2=1+\frac{4v^2\Phi''(v)}{\Phi(v)}\in(0,\infty).
\]
At a node with \(2m>0\) children, choose block \(B\) with
probability \(w(B)/b_m\).
Additional operational marks may be chosen independently at nodes,
conditional on their selected blocks and offspring numbers.

\begin{theorem}[Exact conditional weighted representation]
\label{thm:blockrepresentation}
Under the block and endpoint contracts, the marked tree conditioned
on \(N=2n+1\) nodes produces exactly the weighted \(n\)-edge map law
proportional to \(w(M)\).
Every \(n\ge1\) is attainable.

Consequently all tree and channel theorems above apply to this
independently specified enriched map model.
\end{theorem}

\begin{proof}
A block with \(m\) edges has \(2m\) child slots.
Thus
\[
 N-1=\sum_{\text{nodes}}2m=2n.
\]
The probability factor for a prescribed nontrivial block is
\[
 \frac{b_mv^m}{\Phi(v)}\frac{w(B)}{b_m}
 =\frac{v^mw(B)}{\Phi(v)}.
\]
A leaf contributes \(1/\Phi(v)\).
Their product is
\[
 \Phi(v)^{-N}v^nw(M).
\]
The first two factors are constant at fixed \(N\), proving the law.
Because \(p_0p_2>0\), a full binary tree with \(n\) internal nodes
has positive probability for every \(n\).
\end{proof}

Exponential tilting conditional on tree size is the standard
simply generated-tree mechanism \cite{Janson}.
The endpoint hypotheses, not a catalogue label, justify its use.

\subsection{The exact degree-sampling identity}

Let \(\pi(u)\) be the pole of a block node, or the attachment
vertex for a trivial insertion.

\begin{proposition}[Corner mass and operational transport]
\label{prop:corners}
For any nontrivial map in the contract, the pushforward of uniform
mass on its \(2n\) nonroot block-tree nodes is
\[
 \mu_M(x)=\frac{\deg_M(x)}{2n}.
\]

If each nonroot node \(u\) has a finite channel \(\Gamma_u\), define
\[
 \Gamma_x=
 \frac1{\deg_M(x)}
 \sum_{\substack{u\ne u_0\\\pi(u)=x}}\Gamma_u.
\]
Then
\begin{equation}
 \sum_x\mu_M(x)\Gamma_x
 =\frac1{2n}\sum_{u\ne u_0}\Gamma_u.
 \label{eq:degreechannel}
\end{equation}
\end{proposition}

\begin{proof}
Every nonroot node occupies exactly one parent-block corner,
including empty insertions.
At each map vertex, the total number of these corners is its number
of incident darts. Every dart belongs to exactly one block.
A loop contributes two.
Thus the pole multiplicity is \(\deg_M(x)\).
Summing the channel mixtures with these multiplicities gives
\eqref{eq:degreechannel}.
\end{proof}

This identity transports a \emph{specified sampling law}.
It does not identify degree mass with quantum entropy.
Different coding nodes can have the same map pole; a node-channel
assignment does not descend to vertices unless its averaging or
compatibility rule is supplied.

For the serial model of Section~\ref{sec:serial}, the right side of
\eqref{eq:degreechannel} is itself depolarizing, with coefficient
\[
 \frac1{N-1}\sum_{u\ne u_0}e^{-S(u)/\sqrt N}.
\]
Its half-diamond error converges to the random integral in
\eqref{eq:serialintegral}. This is an exact, declared finite
degree-sampling operation, not an entropy--area prescription.

\subsection{The FK record and the missing verification step}

For the FK model in the supplied preprint, set
\[
 t=\sqrt q,\qquad
 p=\frac{t}{t+2},\qquad u=\frac{1-p}{16},
\]
and define
\[
 w(M)=
 \sum_{A\subseteq E(M)}
 t^{\,2k_M(A)+|A|-|V(M)|-1}.
\]
The decoration-summed FK weight is \(t\,w(M)\); the extra factor is
constant across maps. Joining two maps at one vertex adds their
exponents in this formula, so \(w\) is multiplicative.
The one-link and one-loop block weights are each \(1+t\), giving
\(b_1=2(1+t)>0\).

The supplied preprint reports:
\begin{enumerate}
\item the exact word coefficient identity;
\item a Catalan upper bound giving \(T(u)<\infty\);
\item strict inventory drift for \(p>1/2\);
\item ladder-cut matching giving \eqref{eq:lowerinput};
\item the exact corner-substitution representation.
\end{enumerate}
These are attributed source claims
\cite[Proposition 2.2, Proposition 2.4, Proposition 3.1,
Lemma 4.1, and Proposition 4.3]{OpenAI211}.
They are not independently verified by recording them.

\begin{corollary}[Conditional FK operational specialization]
\label{cor:FK}
If the FK weights above are independently shown to satisfy the
block contract and endpoint inputs in this same finite model, then
Theorem~\ref{thm:blockrepresentation} identifies its marked
\(n\)-edge map with \(T_{2n+1}\).
For any additional fixed operational marking kernel satisfying the
printed hypotheses, Theorems~\ref{thm:channel},
\ref{thm:serial}, and \ref{thm:experiment} apply.

Without that verification or explicit assumption, this corollary
does not claim a proved operational theorem for the original FK law.
\end{corollary}

\begin{proof}
Apply Lemma~\ref{lem:inverse},
Theorem~\ref{thm:blockrepresentation}, and then the independent
marked-tree theorems.
\end{proof}

No identification of the added channel marks with pre-existing
physical FK degrees of freedom is asserted.

\section{Metric convergence and quantum recovery are separate}
\label{sec:separation}

\subsection{The source record and continuum type}

The supplied Family 211 records name:
\begin{itemize}
\item \emph{Canonical conformal limits of subcritical FK planar maps};
\item \emph{Metric-measure limits of subcritical FK and spanning-tree
planar maps};
\item \emph{The critical Liouville quantum sphere and geometric limits
of FK maps at \(q=4\)};
\item the attached \(q>4\) Brownian-CRT preprint;
\item optional random-walk and spectral FK--Ising companions.
\end{itemize}
The companion texts, full hypotheses, and proof dependencies are
not reconstructed here \cite{Dossier}.
Calling this collection a phase-diagram record does not verify a
completed phase-diagram theorem.

The relation \(p=\sqrt q/(\sqrt q+2)\) puts \(q=4\) at \(p=1/2\).
The inventory transition is part of the established inventory
context \cite{Sheffield}. It is not a spacetime-transition theorem.

GHP convergence concerns compact metric probability spaces.
The supplied limit is a real tree. Its Hausdorff dimension is
almost surely two \cite{DuquesneLeGall}.
One must therefore replace the proposed assertion that it has no
intrinsic metric dimension greater than one by the precise
assertion:
\[
 \boxed{\text{tree structure is not surface-manifold structure}.}
\]
Its dimension does not produce a Liouville field, conformal atlas,
spectral triple, or gravitational action.

Moreover, GHP convergence to a tree does not by itself prove that
every combinatorial cycle has vanishing diameter.
For example, a ladder graph with two long rails and unit rungs,
rescaled by the inverse rail length, converges to an interval:
the rungs collapse, while its outer simple cycle spans nearly the
entire limiting interval.
For the particular block model, a separately proved bound on the
maximum block diameter can control its cycles.
Such a conclusion must be attributed to that bound, not to GHP
convergence alone.

\subsection{A metric and stationary measure do not determine a clock}

Consider a two-vertex graph with distance one and uniform measure.
For every \(c>0\), the symmetric continuous-time generator
\[
 L_c=c\begin{pmatrix}1&-1\\-1&1\end{pmatrix}
\]
has that same metric space and stationary measure, but its nonzero
eigenvalue is \(2c\).
Thus metric-measure information alone does not determine the
time normalization or spectrum of a process.
The random-walk and spectral companions require additional
analytic inputs; they are not automatic consequences of a GHP
record.

\subsection{The complete-complement theorem, locally stated}

A finite recovery interface is an isometry
\[
 V:C\longrightarrow R\otimes B,
\]
with accessible and complementary channels
\[
 \mathcal N(X)=\Tr_B(VXV^\dagger),\qquad
 \mathcal C(X)=\Tr_R(VXV^\dagger).
\]
Every preparation purifier and unreturned quantum output belongs
to the complete \(B\).

\begin{theorem}[Exact full-code recovery]
\label{thm:exact}
A deterministic CPTP decoder on \(R\) recovers every code input,
including external references, exactly if and only if
\[
 \mathcal C(X)=\Tr(X)\tau_B
\]
for one fixed state \(\tau_B\) and every \(X\in\B(C)\).
If \(r=\rank\tau_B\), exact recovery implies \(\dim R\ge dr\).
\end{theorem}

\begin{proof}
Dilate an exact decoder by \(W:R\to C\otimes G\).
The composite \((W\otimes I_B)V\) dilates identity.
On each basis input its decoded marginal is pure, so
\[
 (W\otimes I_B)V\ket i=\ket i\ket{\chi_i}.
\]
Exact recovery of off-diagonal matrix units forces
\(\langle\chi_j,\chi_i\rangle=1\); all vectors coincide.
Tracing \(CG\) gives the constant complement.

Conversely, diagonalize
\(\tau_B=\sum_{a=1}^r\lambda_a\kb aa\), omitting zero eigenvalues,
and write
\[
 V\ket i=\sum_a\sqrt{\lambda_a}\ket{u_{ia}}_R\ket a_B.
\]
Constancy on all matrix units implies
\(\langle u_{jb},u_{ia}\rangle=\delta_{ij}\delta_{ab}\).
These \(dr\) vectors are orthonormal.
On their span, the isometry \(Uu_{ia}=\ket i\ket a\), followed by
tracing the second factor, recovers the input.
Complete the decoder on the orthogonal complement by fixed-state
preparation. The result is CPTP on all accessible inputs.
Matrix-unit equality persists with every reference.
\end{proof}

This is the established exact quantum-error-correction condition
\cite{KnillLaflamme}. Comparable complete-output formulations
occur in Cheung's Hawk04-A and En07-A/C
\cite{CheungHawk04A,CheungEn07A,CheungEn07C}.
Their model-specific interfaces are not identified with the map.

\begin{definition}[Record-only completion]
\label{def:completion}
Fix a graph, metric-measure record, or marked-tree record.
An independent quantum completion appends a finite isometry
\(V:C\to R\otimes B\), or finite states on declared factors,
without a correspondence theorem linking them to that record.

A quantum quantity is determined by the record alone only if its
value is the same in every such completion with the named logical
and output dimensions fixed.
\end{definition}

\begin{proposition}[Same record, opposite recovery]
\label{prop:goodbad}
Every fixed metric-measure or marked-tree record admits independent
completions with optimal recovery errors \(0\) and \(1-d^{-2}\),
on the same logical and output dimensions.
For every pure logical input, selected accessible entropy can be
zero in both completions.
\end{proposition}

\begin{proof}
Take \(R\cong B\cong C\), and define
\[
 V_{\rm good}\psi=\psi_R\otimes\ket0_B,\qquad
 V_{\rm bad}\psi=\ket0_R\otimes\psi_B.
\]
The first accessible channel is identity and is exactly recoverable.
The second is replacement. Every decoder of a replacement is a
replacement, whose Bell overlap is \(d^{-2}\).
Thus every decoder has error at least \(1-d^{-2}\).
Preparation of \(I_C/d\) attains the bound, as in
Lemma~\ref{lem:depolarizing}.

For pure inputs, the accessible output in either completion is pure,
so its entropy is zero.
Neither isometry changes the fixed combinatorial or metric record.
\end{proof}

This is the same record-only separation architecture used in the
supplied Oai265-A comparison \cite{Oai265A}.
It is a nonidentification theorem relative to
Definition~\ref{def:completion}, not an exclusion of future bridge
theorems.

\subsection{Cuts and entanglement}

Fix a finite graph cut with a nonempty vertex set on each side,
and attach one qubit to each vertex.
The all-zero product state has zero entanglement entropy across
that cut. A Bell pair between one selected vertex on each side,
with every other qubit in \(\ket0\), has entropy one bit across
the same cut. The graph metric and degree measure are identical.

Hence an articulation cut, even in a tree, does not determine
quantum entanglement.
The exact corner mass is classical sampling mass, not a density
operator and not an entropy.

Family 265's stated gapped-lattice area-law and PEPS results remain
separate documentary records \cite{Dossier,Oai265A}.
They provide neither a state on this map nor a radiation history.
A graph cut is not a horizon, and a size index \(n\) is not an
evaporation clock.

\subsection{Selected marginals do not specify the complete channel}

For any fixed selected state \(\tau_S\), choose a purification
\(\chi_{SB_0}\).
The two isometries
\[
 \psi\longmapsto\chi_{SB_0}\otimes\psi_{R_1}\otimes\ket0_{B_1},
 \qquad
 \psi\longmapsto\chi_{SB_0}\otimes\ket0_{R_1}\otimes\psi_{B_1}
\]
have the same selected \(S\) marginal for every input, but opposite
full accessible recovery behavior.
This applies even when \(\tau_S\) is a chosen thermal density.
It does not construct a Hawking state.

Cheung's En07-C explicitly separates a thermal input from a complete
complementary-channel verdict \cite{CheungEn07C}.
The selected packet and apparatus distinctions in Em03-A--C
likewise do not supply an identified quantum channel on the present
map \cite{CheungEm03ABC}.

\subsection{Classical destructive output}

If only a classical measurement record is accessible, every decoded
quantum channel has the form
\[
 \Lambda(X)=\sum_z\Tr(E_zX)\sigma_z.
\]
On a referenced input it is separable across reference and output.
A separable state has Bell overlap at most \(1/d\), so its full-code
recovery error is at least \(1-1/d\).
This concerns that declared record-only interface, not any surviving
quantum factors
\cite{CheungI,CheungHawk04B}.

\section{The assembly theorem and the unchanged open gates}
\label{sec:assembly}

\begin{theorem}[Oai211-A operational assembly]
\label{thm:assembly}
For the independently defined marked conditioned tree satisfying
\eqref{eq:offspring}, the following claims are proved:
\begin{enumerate}
\item fixed offspring-bounded scalar labels satisfy the uniform
path law \eqref{eq:pathmain};
\item fixed finite-dimensional channel marks define a CPTP spine
average and a path-average operation with the finite-size and
asymptotic half-diamond bounds of
Theorem~\ref{thm:channel};
\item the designed weak depolarizing serial family has the exact
compositions, uniform approximation, and random excursion profile
of Theorem~\ref{thm:serial};
\item a declared selected-path qubit experiment satisfying
\eqref{eq:calibration} has the complete confidence guarantee
\eqref{eq:finitecertificate};
\item under the weighted block and endpoint contracts, these
statements transfer to the exact \(N=2n+1\) enriched map model,
with degree-sampling transport \eqref{eq:degreechannel}.
\end{enumerate}

These statements certify only their named operations.
They do not determine a bulk--boundary map, a complete complement
for an unspecified physical code, an entanglement area law,
a Page curve, or a gravitational action.

Acceptance, exclusion of the specified channel at a threshold,
and an inconclusive outcome are separate decisions.
The supplied FK application remains conditional as stated in
Corollary~\ref{cor:FK}.
\end{theorem}

\begin{proof}
Items 1--5 are, respectively,
Theorem~\ref{thm:path},
Theorem~\ref{thm:channel},
Theorem~\ref{thm:serial},
Theorem~\ref{thm:experiment},
and Theorem~\ref{thm:blockrepresentation} together with
Proposition~\ref{prop:corners}.
The nonidentification claims follow from
Definition~\ref{def:completion} and
Proposition~\ref{prop:goodbad}, and from the distinct types of
objects required for those physical interpretations.

If the upper bound in \eqref{eq:finitecertificate} is at most
\(\varepsilon\), the specified implemented channel is certified.
If its lower bound is greater than \(\varepsilon\), that channel is
excluded at the threshold; another decoder has not thereby been
excluded. If neither inequality separates the threshold, the
certificate is inconclusive.
\end{proof}

\subsection{The four endpoint obligations are not constructed}

The detailed supplied En07-B text names:
\begin{enumerate}
\item complete preparation and a justified finite code;
\item complete microscopic evolution;
\item outgoing access and an access-preserving endpoint comparison;
\item a complete endpoint complement that is constant, or within
a declared replacement allowance.
\end{enumerate}
These are documentary clauses of
\cite[\sourceid{def:four}]{CheungEn07B}.
They are not established by a tree limit or by the designed serial
channels above.

Oai290-A's independently defined operational endpoint inputs
additionally distinguish model complementary control, physical
transfer, and executable deterministic return
\cite{Oai290A}.
They should not be misattributed to
\sourceid{def:endpoint} of En07-A.
The differing manuscript labels and contracts are not interchangeable.

\subsection{The modular bounded-witness record is a different theorem}

The supplied Family 290-02 record concerns a von Neumann algebra,
a faithful normal state with scalar modular centralizer, a fixed
Hilbert-space operator, and an invariant modular average on
shrinking spectral bands \cite{OpenAI290}.
Its stated conclusion supplies bounded algebra witnesses with
a controlled spectral widening.

Our labels live on a finite branching tree.
No standard-form representation, modular generator, invariant mean,
or correspondence between its \(T\) and our channels is supplied.
The path theorem therefore neither proves nor instantiates the
modular theorem.
The reported Lean-note identifier \sourceid{290-02} is provenance
information from the supplied dossier, not proof-assistant
verification \cite{Dossier}.

\subsection{Characteristic-map and quantum-admission boundaries}

The supplied analytic records retain:
\begin{itemize}
\item the historical XIV-H internal-amplitude stop at H2;
\item the XIV-I internal construction at I2, followed by a full
joint-map stop at I3;
\item the XIV-J supported-sector reference away from its
characteristic locus, with the full map unresolved at J2;
\item XV-B version 2's local lifting and extension interfaces,
with its inherited full graph still unresolved at B1.
\end{itemize}
These are version-qualified documentary statuses
\cite{CheungXIVHIJ,CheungXVB}.
The existence of XV-B as a technical manuscript is not denied.
The closed downstream admission exit is a different assertion.

The remaining chain still requires, on the same complete target:
\[
 \text{joint characteristic map}
 \longrightarrow
 \text{joint section and remainder}
 \longrightarrow
 \text{transverse estimates}
 \longrightarrow
 \text{contact compatibility}
 \longrightarrow
 \text{output wavefront theorem}
 \longrightarrow
 \text{quantum admission}.
\]
No probability estimate on tree labels defines a missing product
of distributions.

The inherited status ledger is unchanged:
\[
 \boxed{
 \begin{gathered}
 \sourceid{O.1}:\ \sourceid{NOT_CLOSED},\\
 \text{XV-B downstream admission exit: not opened},\\
 \text{XVI-N remaining obligations: not discharged},\\
 Q_HR_1^{(1)}\in\mathscr E_H:\ \text{not asserted}.
 \end{gathered}}
\]
The separate conditional XVI-N obstruction is not removed or enlarged
by the present finite marking model \cite{CheungXVIN}.

\section{Source ledger and verification declarations}
\label{sec:sources}

{\small
\renewcommand{\arraystretch}{1.12}
\begin{longtable}{@{}p{0.23\linewidth}p{0.34\linewidth}
                       p{0.37\linewidth}@{}}
\toprule
Source & Supplied role & Role in this paper\\
\midrule
\endfirsthead
\toprule
Source & Supplied role & Role in this paper\\
\midrule
\endhead
\bottomrule
\endfoot

OpenAI FK preprint, 24 September 2026 &
Finite \(q>4\) map CRT claim, endpoint criterion, block tree,
and path-sum argument. &
Attributed source and conditional application target.
No independent verification of the full proof.\\[4pt]

Family 211 companions &
Reported conformal, metric, critical, walk, and spectral limits. &
Documentary phase-diagram context; not inputs.\\[4pt]

Broutin--Marckert &
Published finite-variance joint conditioned-tree coding. &
Explicit standard mathematical input
\eqref{eq:coding}.\\[4pt]

Cheung Hawk04-A/B; En07-A/C &
Complete complements, readout versus return, thermal-input scope. &
Comparisons; the needed finite recovery theorem is proved locally.\\[4pt]

Cheung En07-B &
Four same-model endpoint obligations. &
Detailed supplied clauses recorded; not instantiated.\\[4pt]

Oai265-A/B &
Area-law separation and finite qubit certification. &
Source-qualified comparisons.
No report execution or physical calibration imported.\\[4pt]

Oai290-A; Family 290-02 &
Typed channel assembly and modular bounded witnesses. &
Different contracts; no modular/tree identification.\\[4pt]

Cheung En02-B/C; En01-B &
Complete programmes, resource ledgers, bounded statistics. &
Operational discipline and attribution, not an FK physical model.\\[4pt]

Cheung VIII-A/B/C; XVII-A &
Specified geometric dictionaries and a finite effect benchmark. &
No conical or AdS objects identified with the CRT.\\[4pt]

Cheung XIV-H/I/J; XV-B v2 &
Internal and partial distributional constructions; open full map. &
Open analytic gates retained.\\[4pt]

Cheung XVI-N &
Conditional prepared-insertion obstruction and remaining tasks. &
Unchanged external analytic boundary.\\
\end{longtable}
}

Cheung citations identify supplied manuscript identifiers, versions,
and excerpts. No missing publication metadata, registry coverage,
or proof dependency is reconstructed.

The additional catalogue claims about Family 265 and its
moment-canonicalization Lean attachment remain reported provenance
only. A moment-canonicalization file is not a formalization of the
present channel certificate or a proof of a lattice area law.
Likewise, the supplied 140-dataset Oai265-B report is not a new
execution, laboratory dataset, or calibration result here
\cite{Oai265B,Dossier}.

The mathematical release consists of the printed conditional
implications, their local proofs, and the named standard coding
input. No successful \sourceid{pdflatex} build, script execution,
proof-assistant checking, laboratory realization, independent
verification of supplied manuscripts, referee approval, or
journal acceptance is asserted.

\appendix

\section{Why reconstruction descent is not supplied by a path sum}
\label{app:descent}

For completeness, one elementary compatibility statement from the
analytic comparison can be restated and proved without constructing
its absent graph.

Let
\[
 \Rec:\mathscr P\longrightarrow\mathcal D'(W),
 \qquad \mathscr K=\im\Rec,
\]
be a linear reconstruction of presentations.
A candidate later map
\(\widetilde M:\mathscr P\to\mathcal D'(W)\)
acts intrinsically on reconstructed kernels only if it respects
every reconstruction identity.

\begin{proposition}[Descent criterion]
\label{prop:descent}
There is a unique map \(M:\mathscr K\to\mathcal D'(W)\) with
\(\widetilde M=M\Rec\) if and only if
\[
 \ker\Rec\subseteq\ker\widetilde M.
\]
Annihilation after restriction to an open set proves only
restricted descent.
\end{proposition}

\begin{proof}
A factorization annihilates every reconstruction-null presentation.
Conversely, equal reconstructions have difference in
\(\ker\Rec\), so the condition makes their later images equal.
This defines the unique factorization.

Restriction to an open set does not detect distributions supported
in its complement. Restricted annihilation need not be global
annihilation.
\end{proof}

For example, \(q\delta'(q)+\delta(q)=0\).
Thus a reconstruction with
\[
 \Rec(M_R)=\delta(q)\otimes R,\qquad
 \Rec(i\theta M_R)=\delta'(q)\otimes R
\]
has
\[
 \Rec((qi\theta+1)M_R)=0,
\]
while the separate summands can be nonzero.
Highest polynomial degree of a chosen presentation is therefore
not automatically an intrinsic symbol of its reconstructed kernel
\cite{CheungXIVHIJ,CheungXVB}.

Theorem~\ref{thm:path} concerns numerical additive functions of
offspring and marks. It supplies neither \(\widetilde M\) nor the
annihilation condition above. It supplies no characteristic
cross-line product, extension normalization, or output wavefront
bound. The characteristic-map chain remains open rather than being
closed by shared terminology.

\section*{Conclusion}

The completed operational statement is:
\[
 \boxed{
 \begin{gathered}
 \text{critical finite-variance marked tree}\\
 +\ \text{fixed finite-dimensional channel rule}\\
 \Longrightarrow\\
 \text{uniform macroscopic path-average}\\
 \text{half-diamond comparison with its spine channel}.
 \end{gathered}}
\]
A separately designed weak depolarizing serial family has an actual
composition theorem and an excursion-indexed error limit.
A calibrated repeated experiment gives a finite-confidence
certificate for its declared target channel.

The conditional block representation can transport these results
to a verified weighted map model and transports corner sampling to
degree sampling exactly. It does not transform sampling mass into
entropy or a graph cut into a horizon.

The supplied \(q>4\) FK limit remains a documentary record until
its application hypotheses are independently justified.
Neither that record nor the present operational extension
constructs spacetime emergence, holographic reconstruction,
a Page curve, or an admitted characteristic quantum coefficient.

\begin{thebibliography}{99}

\bibitem{BroutinMarckert}
N.~Broutin and J.-F.~Marckert.
Asymptotics of trees with a prescribed degree sequence and
applications.
\emph{Random Structures \& Algorithms}
\textbf{44} (2014), 290--316.
doi:10.1002/rsa.20463.
arXiv:1110.5203.
Theorem 3, Lemma 11, and Section 6 are the identified coding input.

\bibitem{Janson}
S.~Janson.
Simply generated trees, conditioned Galton--Watson trees,
random allocations and condensation.
\emph{Probability Surveys}
\textbf{9} (2012), 103--252.
doi:10.1214/11-PS188.
arXiv:1112.0510.

\bibitem{LyonsPemantlePeres}
R.~Lyons, R.~Pemantle, and Y.~Peres.
Conceptual proofs of \(L\log L\) criteria for mean behavior of
branching processes.
\emph{The Annals of Probability}
\textbf{23} (1995), 1125--1138.
doi:10.1214/aop/1176988176.
arXiv:math/0404083.
Section 2 supplies the size-biased-tree context.

\bibitem{Dwass}
M.~Dwass.
The total progeny in a branching process and a related random walk.
\emph{Journal of Applied Probability}
\textbf{6} (1969), 682--686.
doi:10.2307/3212112.

\bibitem{Hoeffding}
W.~Hoeffding.
Probability inequalities for sums of bounded random variables.
\emph{Journal of the American Statistical Association}
\textbf{58} (1963), 13--30.
doi:10.1080/01621459.1963.10500830.

\bibitem{AddarioBerry}
L.~Addario-Berry.
A probabilistic approach to block sizes in random maps.
\emph{ALEA, Latin American Journal of Probability and Mathematical
Statistics} \textbf{16} (2019).
arXiv:1503.08159.
Cited for the rooted corner-indexed block-tree organization.

\bibitem{DuquesneLeGall}
T.~Duquesne and J.-F.~Le Gall.
The Hausdorff measure of stable trees.
arXiv:math/0509690.
Cited for the Brownian-CRT Hausdorff-measure and dimension facts,
not for a gravitational interpretation.

\bibitem{Sheffield}
S.~Sheffield.
Quantum gravity and inventory accumulation.
\emph{The Annals of Probability}
\textbf{44} (2016), 3804--3848.
arXiv:1108.2241.
Inventory encoding and the \(p=1/2,\ q=4\) transition context.

\bibitem{Watrous}
J.~Watrous.
\emph{The Theory of Quantum Information}.
Cambridge University Press, 2018.
Finite-dimensional channels, Choi matrices, and completely bounded
trace norm conventions.

\bibitem{FuchsVanDeGraaf}
C.~A.~Fuchs and J.~van de Graaf.
Cryptographic distinguishability measures for quantum-mechanical
states.
\emph{IEEE Transactions on Information Theory}
\textbf{45} (1999), 1216--1227.
doi:10.1109/18.761271.
arXiv:quant-ph/9712042.

\bibitem{KnillLaflamme}
E.~Knill and R.~Laflamme.
Theory of quantum error-correcting codes.
\emph{Physical Review A}
\textbf{55} (1997), 900--911.
doi:10.1103/PhysRevA.55.900.
arXiv:quant-ph/9604034.

\bibitem{OpenAI211}
OpenAI, as attributed in the supplied document.
\emph{Brownian continuum random tree limits of finite
Fortuin--Kasteleyn maps above four}.
24 September 2026.
User-supplied \sourceid{main.pdf}.
The supplied dossier places it in Family 211.
No independent verification of the complete proof or catalogue
provenance is asserted.

\bibitem{OpenAI290}
OpenAI, as attributed in the supplied document.
\emph{Bounded recovery for modular spectral averages}.
23 September 2026.
User-supplied manuscript, identified in the dossier as
Family 290, preprint \sourceid{290-02}.
Its modular statement is documentary context, not a tree-theorem
premise.

\bibitem{Dossier}
\emph{User-supplied Oai211-A development plan and comparative
source dossier}.
Includes Family 211 companion titles, Families 265 and 290
records, supplied manuscript excerpts, and retrieval label
\sourceid{6f2e9611}.
Retrieval, catalogue identifiers, and reported formalization
metadata are not mathematical verification.

\bibitem{Oai265A}
\emph{Oai265-A}, version 1.
Supplied companion manuscript excerpts, especially
\sourceid{def:determination}, \sourceid{prop:entropy}, and
\sourceid{thm:separation}.
A record-only area-law/recovery comparison.
No independent verification or physical identification imported.

\bibitem{Oai265B}
\emph{Oai265-B: Reference-Preserving Quantum-Memory Certification
from Measured Correlations}, version 2.
Supplied \sourceid{Oai265-B-v2.tex} excerpts and reported
version-1 execution summary.
The report is not independently rerun; no laboratory calibration
is supplied by its documentary use.

\bibitem{Oai290A}
\emph{Oai290-A: Typed Boundary Recovery, Complete Complements,
and Modular-Band Witnesses}, version 1.
Supplied companion draft, especially
\sourceid{thm:assembly} and \sourceid{def:endpoint}.
Its operational endpoint inputs are independently defined there
and are not En07-B's verbatim four clauses.

\bibitem{CheungI}
N.~Cheung.
\emph{Landmark I}, version 2.
Undated supplied manuscript excerpts.
Finite instruments, destructive readout, and quantum-return
boundaries.
Enter Calculus Companion.
Cited at the supplied-source level.

\bibitem{CheungHawk04A}
N.~Cheung.
\emph{Hawk04-A}, version 1.
Undated supplied \sourceid{Hawk04-A.tex} excerpts.
Especially \sourceid{thm:exact}:
complete-complement recovery and its explicit normal form.
Enter Calculus Companion.

\bibitem{CheungHawk04B}
N.~Cheung.
\emph{Hawk04-B}.
Undated supplied manuscript summaries and excerpts.
Classical readout, deterministic quantum recovery, and
reference-sensitive certification.
Enter Calculus Companion.
No physical model transferred.

\bibitem{CheungEn07A}
N.~Cheung.
\emph{En07-A}, version 1.
Undated supplied \sourceid{En07-A.tex} excerpts.
Especially \sourceid{thm:exact} and the complete-channel,
restoration, and thermodynamic distinctions.
Enter Calculus Companion.

\bibitem{CheungEn07B}
N.~Cheung.
\emph{En07-B}, version 1.
Undated supplied \sourceid{En07-B.tex} excerpts.
Especially \sourceid{def:four} and \sourceid{thm:endpoint}.
Complete preparation, evolution, access-preserving endpoint,
and complete complementary-channel obligations.
Enter Calculus Companion.

\bibitem{CheungEn07C}
N.~Cheung.
\emph{En07-C}, version 1.
Undated supplied \sourceid{En07-C.tex} excerpts.
Thermal-spectrum input and the exact complete-complement criterion;
a residual dimension count is not an area law.
Enter Calculus Companion.

\bibitem{CheungEn02B}
N.~Cheung.
\emph{En02-B}, version 1.
Supplied \sourceid{En02-B-v1.tex} excerpts.
Especially \sourceid{thm:B1} and
\sourceid{prop:observable}.
Complete programme soundness, resource ledgers, and the distinction
between logical readout and subsystem recovery.
Enter Calculus Companion.

\bibitem{CheungEn02C}
N.~Cheung.
\emph{En02-C}, version 1.
Supplied \sourceid{En02-C-v1.tex} excerpts.
Especially \sourceid{thm:C1} and the stated restricted
streaming interfaces.
No streaming-error number is transferred to a different channel.
Enter Calculus Companion.

\bibitem{CheungEn01B}
N.~Cheung.
\emph{En01-B}, version 2.
Supplied \sourceid{En01-B-v2.tex} excerpts.
Especially \sourceid{thm:bounded}.
Independent attempted runs, complete records, and robust bounded
statistics.
Enter Calculus Companion.

\bibitem{CheungEm03ABC}
N.~Cheung.
\emph{Em03-A}, version 1;
\emph{Em03-B}, version 2;
\emph{Em03-C}, version 1.
Supplied excerpts.
Fixed-background source, finite packet reductions, apparatus-model
access, and physical-correspondence boundaries.
Their field states and receiver models are not inputs here.
Enter Calculus Companion.

\bibitem{CheungVIII}
N.~Cheung.
\emph{VIII-A}, \emph{VIII-B}, and \emph{VIII-C}, version 2.
Supplied excerpts on matching contracts, the prescribed conical
dictionary, and certified fixed-packet kinematics.
These conical data are not the Brownian CRT.
Enter Calculus Companion.

\bibitem{CheungXVII}
N.~Cheung.
\emph{XVII-A}, version 1.
Supplied manuscript description and excerpts.
One finite-matrix sine effect from one declared boundary region.
Not a gravitational identification of the present channel marks.
Enter Calculus Companion.

\bibitem{CheungXIVHIJ}
N.~Cheung.
\emph{Landmark XIV-H}, \emph{Landmark XIV-I},
and \emph{Landmark XIV-J}, version 1.
Supplied excerpts.
Historical H2 stop; internal amplitude at I2; full-map stops
at I3/J2; reconstruction descent and partial supported-sector
reference.
No inherited quantum admission is imported.
Enter Calculus Companion.

\bibitem{CheungXVB}
N.~Cheung.
\emph{XV-B}, version 2.
Supplied \sourceid{XV-B.tex} excerpts.
Relative lifting, complete normal-jet null tests, and conditional
extension interfaces.
The inherited full six-branch map remains unconstructed at B1
in the supplied version.
Enter Calculus Companion.

\bibitem{CheungXVIN}
N.~Cheung.
\emph{XVI-N}, version r12.
Supplied excerpts.
Conditional massless prepared-insertion obstruction and the
separate unresolved compact-data/universal obligations.
No discharge or enlargement of its obstruction class is claimed.
Enter Calculus Companion.

\end{thebibliography}

\end{document}