M26 R94 — current-agent challenge of work-01, 2026-10-25
Reviewer /root; NOT independent. Frozen candidate was read against the exact original card.

Disposition
The argument presently supplies a complete candidate for the original M26 quantifiers. No mathematical hole was found in the checks below. This is a technical adaptation of existing overlap-gap methodology, not a claim to invent tree coupling or Ramsey extraction. Independent proof review and a wider priority review remain outstanding. The original stays in the open bank pending that review. No round closes on this assessment alone.

A. Why the old cyclic-coupling obstruction does not apply
Only a tree with T branches, Q edges per branch, and a common root is needed. There is no requirement that two nonadjacent nodes use any prescribed pair coupling. For fixed full disorder, the Markov tree measure has the exact Born marginal at each node by induction. Hence a union bound for node energy failures is legitimate. In particular, the proof never treats a conditional child law as an unconditional Born law. Such a replacement would be false and would invalidate a forced-success resampling argument. The unconditional union bound avoids it completely.

B. Quantum support equality
The statement must concern the entire outside reduced state. Single-site equalities would not suffice for the supported Hamming coupling. The backwards conjugation proof does give joint equality. Within each cost layer all Pauli-Z terms commute, so a term disjoint from the pre-layer observable support can be cancelled even if another cost term subsequently expands the support. This prevents unintended within-layer cascades. The p+1 excluded radius is conservative; it covers all changed cost terms touching the resampled vertex block at all p positions. Repeated indices merely remove some Pauli support and cannot increase the cone. Mixers are identical products of one-site gates.

An exact rational two-layer eight-qubit control changes a repeated-index term, verifies distinct complete output distributions and equal joint marginal on the two outside qubits, and checks normalization without floating point. This validates an interface convention, not the all-n theorem.

C. Unbounded degrees and Poisson path counts
It would be wrong to replace the maximum degree by a fixed constant. The proof instead bounds the total number of short occurrence paths from a deterministic block. In the factorial-moment expansion, covariance terms require a shared occurrence. Distinct occurrences are independent marks of the Poisson process even when they realize identical tuple values. The cancellation of disjoint-occurrence terms is an identity of Poisson factorial integrals, not an assumption of disjoint vertex neighborhoods. Once an occurrence is shared, the union of the two incidence paths is connected, so v<=(q-1)e+1. Each coincidence pattern contributes O(n); there are finitely many for fixed p,q. Restricting marked roots to a deterministic block cannot increase the bound. The O(n) variance and O(n/Q) mean therefore imply the simultaneous o(1) failure bound for fixed Q. Floors contribute O(1), absorbed before taking n large.

The constants can be astronomical as p increases. They enter Q, T and the required onset n, but NOT q0(alpha). This is exactly the distinction needed by M26.

D. Large-deviation estimate and endpoint dependence
The common random graph prevents unconditional independence of endpoint Hamiltonians. Nevertheless the marked Poisson formula integrates that common graph exactly. At the endpoint independent branch signs make E(W_E^2)=m for every tuple; higher correlations remain in the fourth-order remainder. The proof retains them. The symmetric sign distribution cancels odd Taylor powers; |W_E|<=m bounds the fourth remainder uniformly. Choosing d>=max(1,alpha^2 m^3) makes the error at most n*m*alpha^2/192, so a true exponential bound survives the exponential union over strings.

The covariance mask is a deterministic vertex suffix. For each pair of strings and every suffix, -D<=u<=1-D. In the close-distance window a<=D<=b<1/2, both endpoints have absolute value at most 1-a. Thus odd q is covered without same-instance flip symmetry, and both parities are valid. The exact identity E[1_(E subset R) chi_s(E)chi_t(E)]=u^q holds because the tuple coordinates are independent with replacement. It would need modification for a fixed simple hypergraph, which is not the model used here.

The rational exponent arithmetic is correct: 1/2-5/32-1/192=65/192; subtracting the entropy allowance 9/192 leaves 7/24. Counts use all ordered m-tuples, allowing identical strings at the endpoint; this only strengthens the exclusion. The binomial-ball entropy bound is valid for b<1/2. No ordinary central-limit theorem is used to estimate exponential tails.

E. Ramsey and continuity requirements
For each pair of branches, normalised Hamming distance changes by at most 2eta at each step, by the triangle inequality. Since 2eta<b-a, a pair starting at zero and ending above b must visit [a,b]. Assign any visited time as its colour, or 0 if none. The endpoint exclusion rules out a zero-colour m-clique. The elementary pivot bound with T=(2(Q+1))^((Q+1)m+2) is more than sufficient for a monochromatic m-clique. Its nonzero colour identifies one common time for all pairs. There is no requirement to approximate a continuous curve, concentrate pair overlaps, or glue incompatible contexts. Finite T,Q factors disappear against exp(-c n), and finite QT factors multiply o_n(1) node failures. Their order of choice is essential.

F. Imported source interface and normalization
The specific imported results are Theorems1 and2 of arXiv:2204.10306, inspected at equations(3.1),(3.2),(3.12),(3.13) and the Poisson model in section2/AppendixE. The second moment is the expectation of (H/n)^2 in the quantum state, so it includes Born outcome fluctuations. It is not merely concentration of the quantum expectation. The sparse ordered-occurrence ensemble and weight 1/sqrt(d) agree with the source exactly. Both it and the original Gaussian ensemble have variance n for every string. The assumptions concern symmetry of the coupling distribution; they do not require even q. No sqrt(q), q!, or sqrt(2) factor from the separate regular-graph theorem is introduced.

The proof first shows V_d<=alpha for every fixed d>=d0, p and fixed angles. The established d->infinity identity then bounds each dense limiting energy. Taking a supremum of quantities all bounded by alpha is valid without uniform convergence. No n-dependent angles, growing p or logarithmic depth have been established.

G. Source priority and qualification
The main conceptual ingredients have strong antecedents. Mints–Anschuetz, arXiv:2609.10838v1 section4, already use tree paths and multicolour Ramsey extraction. Those ideas must be cited prominently. Our inspected statements there concern quantum hypergraph Max-Cut with even arity; they do not state M26's odd classical Gaussian conclusion. The candidate's concrete additional work is the exact marked-Poisson exponential bound for masked signed tuples, the unconditional valid-marginal tree argument, the treatment of growing maximum degree by short occurrence-path counts, and the composition with the exact ordered-model energy theorem. These are modest technical additions/application work; no broad novelty claim is justified.

The source's displayed conditional-child assertion and Gaussian-tail passage cannot be imported without additional justification: a child conditional on a parent need not have its marginal distribution, and a central-limit approximation with additive o(1) error does not supply an exp(-c n) bound. The candidate does not rely on either. This review is not a claim that the source's main theorem is false; those passages may admit repairs, including methods similar to the present argument.

Chen–Huang–Marwaha 2310.01563v1 Theorem1.5 and Corollary1.7 explicitly require even predicates; the surrounding discussion calls removal of that condition further work. The 2025 limit-swapping paper 2404.06087 retains even-order obstruction and numerical evidence for odd order. The newly inspected 2609.35131 concerns mean-field AOA rather than the quantum theorem. No located primary statement directly subsumes the full M26 claim. This search is not exhaustive priority clearance, especially for an application of a September2026 preprint.

H. Residual risks and disposition
No gap remains inside the written candidate, subject to the imported proven energy results. The most valuable next review is an independent check of the Poisson connected-pattern variance lemma and the simultaneous tree/landscape probability space, plus whether the same conclusion is an already known corollary of a more general stability theorem. Repeating small numerical QAOA tests would not settle these questions. The original may be labelled candidate complete, not independently verified or publication-ready. This mechanism should be routed to verification rather than recycled into another discovery round without new input.

Controls: check-interfaces.py/json. All 912 masked-covariance identities, 124 sign-variance cases, exact cone control, rational tail constants and 49 Ramsey-size arithmetic cases pass. They are subordinate to the analytic proof. Candidate pins remain unchanged.
