Oai211-A Operational Marks on Conditioned Block Trees: Finite Half-Diamond Certificates and Metric–Recovery Separation

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.

. Conditioned Galton–Watson tree; block decomposition; operational marks; Brownian continuum random tree; diamond distance; complete complement; quantum-memory certification.

The supplied preprint 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 $q>4$ 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 $q\downarrow 4$ 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 BroutinMarckert,Janson, size-biased spine methods LyonsPemantlePeres, and map block decompositions 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:

Cheung's supplied manuscripts emphasize comparable distinctions between access, complete residuals, deterministic return, destructive readout, and physical identification CheungI,CheungHawk04A,CheungHawk04B, CheungEn07A,CheungEn07B,CheungEn07C. Every finite channel statement used below is stated and proved for the interfaces printed here.

The extension plan requires the following corrections.

1. The Brownian CRT is not metrically one-dimensional. It has Hausdorff dimension $2$ almost surely DuquesneLeGall. Its real-tree structure excludes a surface-manifold interpretation; Hausdorff dimension is not manifold dimension.

2. 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.

3. 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.

4. {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.

5. 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.

6. 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.

7. 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 sec:blocks prints sufficient analytic conditions and proves the resulting conditional specialization.

8. Failure of a sufficient certificate can be inconclusive. It is not automatically a recovery obstruction, geometric failure, or nonexistence theorem.

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 def:blockcontract and Assumption 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.

Let $p=(p_k{)}_{k\ge 0}$ be an offspring law satisfying

Let $T$ be the corresponding plane Galton–Watson tree, and $T_N$ its law conditioned on $|T|=N$. Write

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 $i_u(v)\in \{1,\dots ,k_u\}$ be the child leading toward $v$. Define

Let $H(v)$ be depth.

The probability law of a triple $(K,I,A)$ is

It is normalized because ${\sum }_{k}k{p}_k=1$. For a scalar label $b(k,i,A)$, write

With

we have

The last expectation is finite. No moment of order three is assumed.

For every nonnegative measurable function $F$ of a path of length $h$,

where $Z_1,\dots ,Z_h$ are independent triples with law (eq:spine).

Proof. At one ancestor, summing over its selected child assigns probability $p_k{L}_k(dA)$ 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 (eq:manytoone). No independence between distinct endpoint paths is needed.

This is the elementary many-to-one form of the size-biased construction LyonsPemantlePeres.

List the nodes in preorder as $v_0,\dots ,v_{N-1}$, and define

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 BroutinMarckert (Theorem 3, Lemma 11, and Section 6):

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 (eq:coding).

In particular,

Reversing every child order preserves the conditioned tree law. If $W^{\mathrm{rev}}(v)$ is the preorder walk value in the reversed tree, then also

Only tree-shape symmetry is used; the mark law need not be symmetric.

For odd $N$,

Define

Then

For a fixed $N$, the coefficients required in these formulas depend only on $p_0,\dots ,p_{N-1}$.

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 (eq:pi) 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 $F_0=p_{0}z$. Its coefficient of $z^N$ is therefore the unconditioned probability of total size $N$ and height at most $L$. Division by ${\pi }_N$ proves (eq:heighttail). A tree of $N$ nodes has no offspring number exceeding $N-1$.

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

Suppose $|b|\le B$, with $B>0$. For $x>0$ and $1\le L\le N-1$,

The right side may be capped at one.

For $1\le \ell \le L\le N-1$ and $r>0$,

A maximum over an empty set is zero.

Proof. For an iid spine path of length $h$, Hoeffding's inequality for variables in $[-B,B]$ gives

The probability that there is a violating node at depth $h$ is at most the expected number of such nodes. Apply Lemma 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 ${\mathbb{P}}_N(H_{\max }>L)={\gamma }_N(L)$. This proves (eq:finitepath). Use $x=hr$ at depth $h$ to prove (eq:finiterelative).

For every fixed bounded scalar label,

For every fixed $a>0$,

Proof. In (eq:finitepath), use $x=\epsilon \sqrt{N}$ 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 (eq:codingconsequences). The relative statement follows by division by $a\sqrt{N}$.

For each strict ancestor-child pair $(k,i)$, preorder exploration gives the exact identities

Indeed, the pending siblings to the right contribute $k-i$; the reversed order exchanges right and left siblings.

For a fixed measurable label satisfying

for some finite constant $C_b$, its spine mean is finite and

Also,

Only $\mathbb{E}{\xi }^2<\infty$ is used.

Proof. It suffices to consider $C_b=1$. Equation (eq:twowalks) gives

By (eq:codingconsequences)–(eq:reversecoding),

For fixed $R$, the label $k{1}_{\{k\le R\}}$ is bounded. Subtract its conclusion in Corollary cor:boundedpath from (eq:offspringsum). With

we obtain, for each fixed $R$,

Because ${\mathbb{E}}_{\mathrm{sp}}K<\infty$, $m_R\to 0$. Height tightness consequently implies

Apply Corollary cor:boundedpath to $b{1}_{\{k\le R\}}$. 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 (eq:pathmain). The comparison with $W$ follows from (eq:codingconsequences).

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

Path concentration does not imply concentration of one root mark.

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 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.

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.

Fix a logical space $C\cong {\mathbb{C}}^d$, $d\ge 2$. 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

The diamond norm includes arbitrary external references. For any Hermiticity-preserving linear map $\Delta$, write

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

The weights sum to one. Finite-dimensional integration and convexity make $\overline{\Lambda }$ a CPTP channel.

For $H(v)>0$, define

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 ${{\Lambda }_v}^{\mathrm{av}}$.

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.

Let

The normalized Choi matrix of a channel is a density matrix.

Let $E_1,\dots ,E_{d^4}$ be a Hermitian Hilbert–Schmidt orthonormal basis of matrices on $C\otimes C$. Set

Then $|a_j(\Lambda )|\le 1$. For a Hermiticity-preserving $\Delta$,

Proof. Each $E_j$ has operator norm at most its Hilbert–Schmidt norm one, proving the scalar bound on a state.

For completeness,

To see this, write unit referenced vectors as $u=(A\otimes I){\Phi }_d$ and $v=(B\otimes I){\Phi }_d$, where $\mathrm{Tr}(A^{†}A)=\mathrm{Tr}(B^{†}B)=d$. The output of $|u⟩⟨v|$ is $(A\otimes I)J_{\Delta }(B^{†}\otimes I)$, with trace norm at most $d\Vert J_{\Delta }{\Vert }_1$. 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

Divide by two.

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

Under (eq:offspring) and the marking contract,

Consequently, for every fixed $a>0$,

For $x>0$, $1\le L\le N-1$, the probability that the maximum in (eq:channelsum), before division by $\sqrt{N}$, exceeds $x$ is at most

For $1\le \ell \le L$, the probability that

is at most

Both bounds may be capped at one.

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 lem:choicoord implies a coordinate deviation greater than $2x/d^4$. Apply Lemma lem:finitepath with $B=1$ to each coordinate, and use a union bound over $d^4$ coordinates. This proves (eq:channelabsolute).

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

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

Let

Then the path average of these scalar losses concentrates on ${\mu }_{\mathrm{loc}}$ on macroscopic paths. Also,

and

Generally, $d_{\diamond }(\overline{\Lambda },{\mathrm{id}}_C)\ne {\mu }_{\mathrm{loc}}$.

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.

For a chosen path with channels ${\Lambda }_1,\dots ,{\Lambda }_h$, the serial channel is

Telescoping and CPTP contraction give

This is an upper bound, not an identity.

For example, for a qubit and ${\Lambda }_r={\mathrm{Ad}}_Z$ at every step, each local error is one, while the serial channel is identity for even $h$ and ${\mathrm{Ad}}_Z$ for odd $h$. The path average is always ${\mathrm{Ad}}_Z$. Thus neither a local error sum nor a path mixture can be renamed as the serial error.

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

Define the completely depolarizing channel

and the designed local channel

This $N^{-1/2}$ weakness is part of the model definition. It is not deduced from a geometric scaling exponent.

For $a,b\in [0,1]$, write $D_a=a\mathrm{id}+(1-a){\mathcal{T}}_d$. Then

In particular, $d_{\diamond }(D_a,\mathrm{id})={\kappa }_d(1-a)$.

Proof. The relation ${{\mathcal{T}}_d}^2={\mathcal{T}}_d$ and commutation with identity give the composition formula.

On a Bell input, ${\mathcal{T}}_d$ gives $I/d^2$. Testing the Bell projector gives $d_{\diamond }({\mathcal{T}}_d,\mathrm{id})\ge 1-d^{-2}$. The uniform average of the $d^2$ Weyl conjugations is ${\mathcal{T}}_d$. Separating the identity conjugation and using the triangle inequality gives the matching upper bound. Finally, $D_a-D_b=(a-b)(\mathrm{id}-{\mathcal{T}}_d)$.

The serial composition of (eq:weaksteps) along the path to $v$ is exactly

Define its depth predictor

Then

The exact returned-code error is

Its linearly interpolated preorder profile, with terminal value zero appended, converges in distribution in the uniform topology to

Furthermore,

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

Uniform approximation by the depth predictor, together with (eq:coding), gives (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 (eq:serialintegral).

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 lem:finitepath gives a finite-size failure probability for any chosen bound on the right side of (eq:serialcomparison). For general offspring-bounded $s$, the finite-variance theorem supplies the stated asymptotic conclusion; no numerical rate is invented.

The variable $t$ in (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.

For this section $d=2$. Let $X,Y,Z$ be the Pauli matrices. The Bell projector is

For a channel $\Lambda$, set

Each correlator lies in $[-1,1]$, and

In particular, let $F_{\mathrm{sp}}=F(\overline{\Lambda })$.

For $1\le \ell \le L\le N-1$ and $r>0$, set

With probability at least $1-{\delta }_{\mathrm{tree}}$, simultaneously for all $v$ with $H(v)\ge \ell$,

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

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.

Conditional on the marked tree and a selected endpoint $v$ with $H(v)\ge \ell$, let

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 $m_j\ge 1$ 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

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 $d_{\mathrm{tr}}(\rho ,J_{{\Lambda }_0})\le \delta$, and the complete score observable $B_j$ differs in operator norm from the named Pauli correlation observable by at most ${\eta }_j$, then

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 Oai265B.

For a channel $\Lambda :C\to C$ of logical dimension $d$, with $F=\mathrm{Tr}(P_d{J}_{\Lambda })$,

Proof. The Bell-projector effect proves the lower bound. For a state $\rho$ and rank-one projector $P$,

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 FuchsVanDeGraaf. Combine it with $d_{\diamond }(\Lambda ,\mathrm{id})\le d{d}_{\mathrm{tr}}(J_{\Lambda },P_d)$, proved in Lemma lem:choicoord.

Fix ${\alpha }_{\mathrm{s}}\in (0,1)$, and put

Define

If $F_L>F_U$, report an inconsistent interval and do not certify.

Otherwise, on an event of probability at least

the implemented target error obeys

A sufficient acceptance rule for threshold $\epsilon$ is

The probability of accepting while the actual target error exceeds $\epsilon$ is at most ${\delta }_{\mathrm{tree}}+{\alpha }_{\mathrm{s}}+\beta$.

The tree interval may be omitted. In that case its failure allowance is omitted as well.

Proof. Conditional Hoeffding inequalities give

A union bound and integration over the conditioning record give simultaneous statistical coverage with failure probability at most ${\alpha }_{\mathrm{s}}$. On this event and $G$, the experimental interval covers $F({\Lambda }_0)$. Lemma lem:treebell supplies the independent type of tree-law interval. On the intersection of the three valid events, their intersection is nonempty and

Apply Lemma lem:fidelity, then the reverse triangle inequality and $d_{\diamond }({\Lambda }_{\star },{\Lambda }_0)\le b$. The union bound proves joint coverage and the invalid-acceptance bound.

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

A finite operational programme supplies complete history maps $K_h$ satisfying

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 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.

For equal counts $m$ and the Hoeffding-only upper certificate with zero allowances, the best possible value $\hat{F}=1$ gives

Acceptance at $\epsilon <1$ therefore requires $R_{\mathrm{s}}\le {\epsilon }^2/4$. 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 Oai265B; no new dataset or execution is claimed here.

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

A weighted rooted planar-map class supplies:

1. one trivial zero-edge map of weight one;

2. rooted nontrivial blocks, including the one-link and one-loop blocks, with positive weights $w(B)$;

3. a unique recursive corner substitution, with one child slot per block corner and trivial insertions retained as leaves;

4. multiplicative map weight equal to the product of its block weights;

5. a deterministic order of each rooted block's $2m$ corners when it has $m$ edges.

Write

Then the substitution contract gives the formal identity

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

For some $u>0$,

and for some $c>0$, all real $x<1$ sufficiently close to one satisfy

Also $b_1>0$.

Under Definition def:blockcontract and Assumption ass:endpoint, put $v=u{\tau }^2$. Then

where derivatives at the endpoint denote nonnegative coefficient series.

Proof. Integrate (eq:lowerinput) to obtain, near $u$,

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

Since $f$ is increasing,

This bounds ${\Phi }^{\prime }(y)$ near $v$, so its increasing derivative series has a finite endpoint limit.

Moreover,

near $u$. Hence

The finite endpoint limit of $f^{\prime }$ must be zero, proving $2v{\Phi }^{\prime }(v)=\Phi (v)$.

Finally,

If ${\Phi }^{\prime \prime }(v)=\infty$, then $f^{\prime \prime }(y)\to -\infty$. For every large $K$, sufficiently near $v$, $f^{\prime \prime }(y)\le -K$. Integrating twice, using $f^{\prime }(v-)=0$, gives

contradicting the preceding fixed quadratic upper bound.

This is the endpoint argument reported in OpenAI211 (Lemma 4.2), included locally. It requires no analytic continuation across $v$.

Define

Then

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.

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 $n\ge 1$ is attainable.

Consequently all tree and channel theorems above apply to this independently specified enriched map model.

Proof. A block with $m$ edges has $2m$ child slots. Thus

The probability factor for a prescribed nontrivial block is

A leaf contributes $1/\Phi (v)$. Their product is

The first two factors are constant at fixed $N$, proving the law. Because $p_0{p}_2>0$, a full binary tree with $n$ internal nodes has positive probability for every $n$.

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

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

For any nontrivial map in the contract, the pushforward of uniform mass on its $2n$ nonroot block-tree nodes is

If each nonroot node $u$ has a finite channel ${\Gamma }_u$, define