# Panel review: An explicit noncoordinate polynomial with affine three-space zero fibre

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.

| Reviewer | Completed assessment | Effort of selected response | Attempts |
|---|---|---|---:|
| GPT-6.1 Sol | [no_demonstrated_error_found](../panel/sol/full_manuscript/review.json) | high | 1 |
| Opus 5.5 | [no_demonstrated_error_found](../panel/claude-opus-5.5/full_manuscript/review.json) | high | 3 |
| Fable 5.1 | No completed review; budget_exhausted | — | 3 |
| Gemini 3.1 Pro | [no_demonstrated_error_found](../panel/gemini/full_manuscript/review.json) | high | 1 |
| Grok 4.7 | [no_demonstrated_error_found](../panel/grok-4.7/full_manuscript/review.json) | high | 1 |

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 displayed construction and proof survive this independent source check. The argument gives polynomial inverse maps on the whole zero fibre and a critical point in the ambient space. No concrete error, hidden localization or essential missing hypothesis has been established. This is an AI-assisted mathematical assessment with the limits below, not a formal certificate or a priority judgment.

[Read the independent source assessment](independent-source-assessment.md) for the checked identities, source locators, external-input qualifications and scope limits. Its initial pass was saved before opening panel responses. [Reproducible local calculations](independent_calculations.py) and [results](independent_calculations.json) 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](panel-source-adjudication.md).

This file is 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.

## Gemini first completed response

Input: `review.json`, inspected 13 October 2026. The response gives twelve positive local checks and no objections. Its algebraic calculations agree with the independent universal identity checks. That agreement does not replace the separate proof of the quotient presentations or verify the bibliography.

**Source-anchor correction:** its entry labelled “Equation (3.7)” describes `X²+Y³=s(1+F)`, native `eq:ambient`, lines 306–322. That is **Equation (3.8)**. Equation (3.7), native `eq:fiber`, is the quotient-ring isomorphism at lines 301–304. Preserve the original response and record this correction separately; the described identity itself is present in the source.

**Coverage qualification:** the response's statement that all sections are fully audited is stronger than its supplied evidence. Its listed entries do not separately reconstruct every elimination/ideal equality, the adjoining-variable corollary or the full coordinate-inverse appendix. Those matters are covered explicitly in the independent source assessment. Its empty `unverified_obligations` array is not evidence of an external-citation audit or a formal certificate.

No new source defect is established by this completed response. A token-budget failure without a returned assessment is a resource outcome and has no mathematical verdict.

## Sol completed response

Input: `review.json`. It covers all three numbered results, the numbered equations and the inverse-map appendix, with no objections. Its source equation anchors correctly distinguish `eq:fiber` (3.7) and `eq:ambient` (3.8). The detailed presentation-ideal identity and nilpotent-ring example were checked independently: if `Q=hW-y-U²`, the original first relation differs from `hG-yU-x` by `-UQ`, and the transformed third relation differs by `Q(-2UG+2yW+WQ)`. This confirms that no division is hidden in that step.

Its additional `C[e]/(e²)` specialization has `h=e,x=1,y=-1,s=0`, `U=1+eT`, `G=-T`, `W=2T+eT²`, `V=-3T-3eT²`; reducing by `e²=0` verifies the three defining equations. This is a useful local boundary test, not a substitute for the general unit-ideal argument. The reported component gradient at `P` agrees with the independent exact script. Its separate `h=0` example with `a^5=-1/4` also satisfies the stated values and inverse relations.

Sol explicitly separates contextual citation obligations from the internal construction. My first pass addresses the KVZ formulation in part; the exact Maubach theorem and the other cited historical statements are still not fully verified. Their presence in an unverified list does not become a counterexample to the local proof.

## Grok completed response

Input: `review.json`. Its thirteen checks use stable TeX labels and cover the three numbered results and appendix. The cusp/translation algebra, elimination ideal, critical-point component calculation and appendix identity agree with independent source inspection. Its appendix reformulation `beta*y+(1+2s²x)(1+p)=1-4s⁴(x²+y³)` is the same free-polynomial certificate after rearrangement. There is no concrete mathematical objection to adjudicate.

Its empty external-obligation list should not be read as a bibliography audit: the response itself treats the contextual citations as outside the dependency chain, and does not present evidence of fetching or checking their native formulations.

## Opus 5.5 high-effort completed response

Input: `review.json`. This is a source-only high-effort response after recorded max-effort budget failures, not another independent provider vote. Its fourteen entries cover the complete local proof chain and appendix. The stable/numeric anchors correspond to the native source, and the critical-point component derivatives agree with the independent exact script. Its lifting-lemma elimination does not add a domain or regularity hypothesis; the ideal equality explicitly handles zero divisors. No concrete objection is raised.

Its contextual citation obligation remains separate from correctness of the displayed maps. The optional request to check fully substituted inverse composites is a redundant calculation proposed in addition to the symbolic lemma proof. It is not an unclosed indispensable step, and it would not establish novelty or priority. The independent assessment verifies the universal certificate and the two algebraic composites without expanding their large fully substituted ambient expressions.

## Final polynomial panel coverage

Four completed responses were inspected: Sol, Gemini, Grok and Opus high, each given the complete frozen manuscript. Gemini's numeric correction above is retained separately from its mathematical descriptions. Both initial max-effort Anthropic attempts and both expanded max-effort attempts exhausted their token caps; Fable's final high-effort attempt also exhausted its cap. Those five polynomial failures have no mathematical verdict. Opus's successful high-effort response does not erase the costs or resource outcomes of its earlier attempts. No provider vote, confidence average or human/panel agreement statistic determines the source assessment.

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**. Fable returned no complete mathematical assessment in any of its three attempts. Its missing review is not an abstention or rejection. 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 **$5.4079**. [Per-attempt usage and pricing sources](api-costs.json) distinguish known amounts from unknown charges. This is not an account invoice; Codex chat/subagent usage is excluded.

## Sources

The [original manuscript and hashes](../source_manifest.json), 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.