Panel review: Bourgain’s Fourier–Walsh spectrum paper

Reviewed edition: Jean Bourgain, On the Fourier–Walsh spectrum of the Moebius function, arXiv 1112.1423v1, 6 December 2011, 21 pages.

Disposition

The panel found a defective local approximation, and a separately checked replacement preserves the estimates needed immediately downstream. The full headline theorem is still unresolved in this audit.

The original proof, the replacement proof and statement truth have separate statuses. This is a constructive repair of one proof component, not a refutation or a complete verification of the paper.

What was repaired

In the periodic Fourier-series interpretation used later, the conditions (4.6) impose incompatible endpoint values on a continuous periodic function. Sol 6.1 at max constructed a shifted Fejér smoothing with a larger auxiliary Fourier cutoff. Opus 5.5 at max independently accepted all five replacement lemmas: the kernel construction, uniform error on dyadic grids including the jumps, product coefficient bound, block coefficient normalizations and transferred error for g. The final local frequency bounds (4.14) survive.

Read the mathematical repair and all lemma proofs and individual checker assessments. The repair dependency overlay is separate from the original source map.

What remains open

The characterwise Case-I application of Balog–Perelli’s Theorem 4 still needs original-source inspection. Other panel concerns include normalization, sharp-to-smooth arc justification and final logarithmic parameter bookkeeping. An unverified import is not a demonstrated false citation, and these remaining objections must be adjudicated individually.

The local repair does not establish the full theorem’s quantitative cutoff or its corollary. No counterexample to the main theorem has been produced.

Coverage

Eight initial full-text attempts used Sol, Opus, Fable and Gemini in two prompt conditions at high effort. Four returned usable assessments: Sol in both conditions, Opus baseline, Gemini source-first. Opus source-first, both Fable attempts and Gemini baseline reached their output caps. These failures provide no mathematical verdict. Sol and Opus flagged the periodic approximation; Gemini’s completed response did not. The disagreements remain visible in the individual assessments.

The additional targeted construction and checker both completed at max effort. Receiving the full manuscript does not guarantee complete coverage. Identifying front matter was withheld, while references and mathematical content remained recognizable; this was not fully blind or an unseen-capability benchmark.

Accounting and evidence

Initial full-text conservative accounting: $3.571123. Successful repair-call accounting: $0.584531. An earlier request-validation rejection retains its full $0.508588 allowance; total repair accounting is $1.093119, below $2. Total campaign accounting is $4.664242, below $25. These are token estimates and retained allowances, not invoices.

The checker wrapped its JSON in Markdown. Offline normalization removed only that complete fence, retained strict schema checking and preserved the original answer; it made no additional provider call. The public records contain checkable visible arguments and usage, with credentials, hidden reasoning and private grading maps excluded.

All mathematical assessments and curator checks here have AI origin. No independent human grade or formal certificate is claimed. The source PDF, original AMS-TeX, reviewed normalization and source hashes are preserved; the paper’s original dependency map is not overwritten.