Panel review: Quadratic stabilization of the canonical Foulkes–Howe map

Reviewed 13 October 2026 against the complete frozen native manuscript at repository commit 1581d4cbdd69a0faf34b2089614fe2a814316a6a. Human assessment remains pending. This page contains AI mathematical assessments; the neutral reading guide and blank worksheet are separate.

Actual model coverage

Each completed response was an independent source-only review of the complete manuscript, with no other referee or human findings supplied. The requests asked for claim-level checks, precise source anchors, explicit evidence for any objection, and a distinction between demonstrated defects and incomplete verification.

Opus and Fable initially received max effort with a 16,000-token wire cap, then an explicitly recorded 32,768-token attempt where the first cap was exhausted. A final capped high-effort pass was used after repeated max-effort exhaustion. Native thinking is included in those caps and billed output. Model names do not imply that every requested pass completed; no mathematical verdict is inferred from a resource failure. Requests permit only one native inference each, and extra adapter attempts are blocked.

Mathematical source assessment

The written proof survives independent checking of the map identification, differential sign and induction degree inequalities. No explicit invalid implication, counterexample under the stated hypotheses or essential missing assumption was established. The bound is a sufficient surjectivity range for the specified map; it does not resolve every ordinary Foulkes comparison outside that range or assert optimality. This is an AI-assisted assessment, not a proof-kernel certificate.

Read the independent source assessment for the checked identities, source locators, external-input qualifications and scope limits. Its initial pass was saved before opening panel responses. Reproducible local calculations and results support the arithmetic/sign checks; they are not a Lean certificate.

Source adjudication and qualifications

The original responses are preserved. Source-anchor corrections and the treatment of concrete objections are recorded in the panel-source adjudication.

Separate from neutral human reading materials. AI source adjudication is not a human verdict. The independent source assessment was saved before any panel response was opened. All bounded campaign attempts have finished, and every completed response for this paper is accounted for below.

Sol completed response

Input: review.json, inspected 13 October 2026. It distinguishes the four proof-bearing numbered statements and final numbered scope remark, with correct native numbering and all ten numbered equations. No concrete counterexample or invalid implication is alleged.

The derivative sign, first-sequence destination degrees and second-sequence accumulated degree gap agree with independent source checking. Its sharpened second gap is p-q=b-(i-1)r-2j, for 1≤i≤r, 0≤j≤r-1, whose minimum is b-r²-r+2≥2 at the endpoint b=r(r+1). This is consistent with the manuscript's looser separate bounds; it does not require a stronger range. Its explicit endpoint pairs at (a,b)=(2,2) and (3,6) also satisfy the strict criterion.

Sol makes the representation-comparison corollary conditional on the complete-reducibility input. That reflects an uninspected cited proof rather than a demonstrated defect in the splitting argument. The main theorem precedes and does not use this input. My independent assessment likewise records that applicability is classical and clear, while the exact Macdonald locator was not audited.

Some of Sol's external-source uncertainty is narrowed by the bounded primary-source inspection: RSW §2 does give the divided-power multiplication map and normalization formulation, and its Theorem 3.9 has the stated regularity scope. Ikenmeyer's Theorem 1.3 propagates injectivity in the opposite direction. This verifies those formulations, not the complete proofs of the cited sources. Exact Brion/Landsberg/other textbook locators remain outside the completed checks.

Gemini completed response

Input: review.json. It gives eight positive checks and no objections. The two degree ledgers and binary factorization agree with the independent source pass. Its numbering is wrong: “Lemma 1” is Lemma 2 (lem:shift); “Proposition 2” is Proposition 3 (prop:vanishing); “Corollary 3” is Corollary 4 (cor:comparison). Theorem 1 and its stable equation labels are correct. Preserve its response and supply corrected anchors separately.

Its brief image-criterion explanation omits the explicit pure-power spanning step used to identify the image. The full source supplies that step; the omission limits how much independent reconstruction this summary demonstrates. Its corollary description should retain the source's exact representation class: finite-dimensional polynomial GL(V) representations over C, not arbitrary representations of any group. An empty external-obligation array does not establish that the cited textbook theorem or bibliography was audited.

Grok completed response

Input: review.json. Ten entries cover the image/annihilator criterion, differential lemma, complete induction and corollary; no concrete objection is raised. Its local algebra and endpoint inequalities agree with the independent source pass. It correctly separates the imported complete-reducibility input from the written splitting deduction.

Numeric anchors are systematically shifted: “Lemma 3” is Lemma 2, “Proposition 4” is Proposition 3, and “Corollary 5” is Corollary 4. The stable labels recover the intended statements. The unnumbered diagonal criterion does not increment the manuscript's shared theorem counter.

There is also a compressed explanation in the differential-lemma entry: from Dh=0 and p>q, the response says “both sides vanish.” What actually forces that is the sign comparison -||Eh||²=(p-q)||h||², with a nonpositive left side and a nonnegative right side; equality therefore gives h=0 and Eh=0. The native proof gives this argument correctly. This is an imprecision in the model's explanation, not an identified defect in the manuscript.

Opus 5.5 high-effort completed response

Input: review.json. Thirteen entries cover the main chain with correct theorem numbering and stable equation labels. Its detailed second-sequence degree formula agrees with the independent check. The only objection is explicitly an unverified_step about bibliographic/significance claims, with no asserted dependency impact on the internal proof. RSW and Ikenmeyer formulations have been inspected in the independent assessment; the exact Landsberg, Brion and CIM locators remain unverified here. The complete-reducibility locator likewise remains a scholarly check, separate from applicability of the classical theorem.

Outside-range sanity-check qualification: the response says Steps 2–3 go through at (a,b)=(5,5) because “the Step 3 base works for m=1.” In the written induction, however, r=4,m=1 fails the Step 3 requirement m≥r(r-1)=12. That entry is not a valid application of the manuscript's induction hypothesis. A separate polarization argument could obtain the relevant conclusion because every element of Sym¹ V is already a one-vector product, but that additional argument is not given in the entry. The safe source-grounded statement is that (5,5) is outside the theorem's range and fails the written Step 3 admissibility check as well as the Step 4 degree ledger. This qualification affects the model's extra sanity-check description, not the proof at admissible parameters.

Final panel coverage

Four completed responses were inspected: Sol, Gemini, Grok and Opus high. Initial max-effort Opus/Fable and expanded max-effort Opus/Fable attempts all ended with token-cap/resource failures. Fable's final high-effort attempt also exhausted its cap; its native stop reason is max_tokens and its application status is provider_error. It returned no complete mathematical assessment. Such failures have no mathematical verdict and are not abstentions or rejections. No score, provider majority or human/panel agreement statistic is inferred.

This accounts for nine native attempts: four returned reviews and five resource failures. The saved wire payloads confirm that Opus and Fable each had max-effort attempts at 16,000 and 32,768 output-token caps, followed by one high-effort attempt at 16,000. The successful Opus high response does not erase its earlier resource outcomes. No additional attempt is pending or proposed.

Recorded API cost

All captured native attempts for this paper, including incomplete max-effort attempts, give a usage-based estimate of $5.5515. Per-attempt usage and pricing sources distinguish known amounts from unknown charges. This is not an account invoice; Codex chat/subagent usage is excluded.

Sources

The original manuscript and hashes, independently checked primary sources listed in the source assessment, and complete model responses form the evidence record. Bibliographic similarity is not evidence of what the generating model read or how it discovered its construction. No literature-priority claim is made.