# Panel review: An explicit power saving for the exact discrete Fourier transform

Review dated 13 October 2026 of the complete 31-page September 25 manuscript, frozen at OpenAI/math commit `1581d4cbdd69a0faf34b2089614fe2a814316a6a`. Human assessment remains pending. The neutral reading map and blank worksheet are separate.

## Actual independent coverage

Every completed response received the same complete expanded native source, without human findings, other referees’ responses or the Recht essay. The prompt requested precise source anchors, derivations of objections, computational-model checks and explicit unverified obligations. A returned review is an AI assessment, not a formal certificate.

| Reviewer | Returned assessment | Selected effort | Native 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 | 2 |
| Fable 5.1 | [no_demonstrated_error_found](../panel/fable/full_manuscript/review.json) | high | 2 |
| 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 began at max effort with a 65,536-token native cap. Opus exhausted that cap without returned review text; Fable’s primary request ended in provider refusal. Each received at most one high-effort fallback, charged separately. Sol, Gemini and Grok used high effort. Each job allowed one native inference request; extra adapter retries were blocked before transmission. No verdict is inferred from a refused, truncated or timed-out request. [Per-attempt usage and outcomes](api-costs.json) preserve actual coverage.

Fable reports only a spot check of Appendix A, and Opus leaves its amplification induction incompletely rederived. Complete-manuscript input does not imply exhaustive verification of every claim or external reference.

## Source assessment

The main argument supplies a coherent chain from a finite network saving to an all-length exact arithmetic algorithm. The source explicitly charges preparation and index work and uses one supplied, specified root of polynomially bounded order. Its claim does not include finite-precision stability, bounded coefficients, useful crossover lengths, or a bit-complexity improvement. These distinctions matter when comparing the theorem with earlier FFT lower bounds.

This source pass has not established a counterexample or invalid inference in the quantitative main proof. It also does not certify that proof: the complete binary-label schedule and the implementation-level operation ledger still need a dedicated check. Appendix A is a separate symbolic uniformization route and is not needed for the decimal exponent in Corollary 1.2.

[Independent source assessment](independent-source-assessment.md) gives the checked identities and remaining obligations. Its initial pass was saved before opening completed panel responses. [Exact calculation code](independent_calculations.py) and [results](independent_calculations.json) cover finite identity and integer checks; they do not certify the complete algorithm.

## Adjudication

[Source adjudication of panel claims](panel-source-adjudication.md) distinguishes a demonstrated invalid inference from a missing explanation, scope mismatch or unverified step. Original referee responses are retained unchanged.

All five completed responses report no demonstrated error. Opus and Fable identify exposition/accounting details whose proposed constructions preserve the claimed costs. Source adjudication does not establish a surviving counterexample or invalid inference; the full-network and formalization obligations remain open.

## The question and its antecedents

[Question and method provenance](context-and-provenance.md) compares the actual model with earlier FFT barriers and records the manuscript’s method lineage. The unrestricted complexity question predates this manuscript. The supplied root, unrestricted coefficients and exact arithmetic belong to its theorem contract. No floating-point, numerical-stability or useful-crossover claim is made. Discovery-process provenance is unavailable for family 130 in the checked release.

## Formalization and companion scope

No target Lean theorem build, target axiom inspection or Lean-to-manuscript comparison was performed. This campaign reviews the main manuscript, including Appendix A; the finite-tensor and integer-multiplication companions did not receive full independent reviews. The quantitative main route reproduces its network and local compiler rather than relying on the appendix’s symbolic printer.

## API usage

Captured native attempts give a usage-based estimate of **$7.5748**, including failed attempts, within the authorized $25 budget. Conservative reservations total $20.4789. Unknown-usage requests: 0. [Usage and rates](api-costs.json) are estimates, not an account invoice; Codex conversation usage is excluded.

The [source manifest](../source_manifest.json) records the source hashes, include expansion and reviewed upstream paths.