Ultraflat real Littlewood polynomials

Ultraflat real Littlewood polynomials are polynomials with coefficients in {−1, 1} whose modulus on the unit circle is (1 + o(1))√N uniformly, where N is the number of coefficients. Three manuscripts in OpenAI's public mathematics collection (github.com/openai/math), dated 23 September and 5 October 2026, claim that the minimal maximum modulus is asymptotically √N through all lengths, that two-sided ultraflat real Littlewood polynomials exist at every sufficiently large length, and that binary merit factors are unbounded. The collection includes a Lean formalization of the maximum-modulus statement. This page summarizes the published claims and lists publicly available follow-up work. Statements are the respective authors' claims; the publication of a paper, code or a formal proof is not presented here as independent verification.

Background

• As recounted in the introductions of the manuscripts:
  • Orthogonality gives a maximum modulus of at least √N. Erdős asked whether the maximum must exceed √N by a fixed proportion; for complex unimodular coefficients this was refuted by Kahane's ultraflat polynomials, with quantitative constructions by Bombieri and Bourgain. The question for real signs remained open (see also Problems 4.13 and 4.31 in the list of Hayman and Lingham).
  • The Shapiro–Rudin polynomials have maximum O(√N), with the bound √(2N) at dyadic lengths; Balister proved an all-length bound for their initial segments. Balister, Bollobás, Morris, Sahasrabudhe and Tiba constructed real Littlewood polynomials whose modulus lies between two fixed positive multiples of √N.
  • Turyn's conjecture, equivalently Erdős's L⁴-norm conjecture, asserts that binary merit factors are bounded; the best proven limiting merit factor for an explicit family was 6.342061… (Jedwab, Katz and Schmidt).
  • Preprints of el Abdalaoui asserted the nonexistence of flat real Littlewood sequences; the September manuscript states that its results conflict with these claims and examines the arguments in an appendix.

The original results

• Asymptotically minimal maxima. Theorem 1.1 of Asymptotically minimal maxima of real Littlewood polynomials (23 September 2026) states that for every η > 0 and every sufficiently large integer N there are signs with maximum modulus at most (1 + η)√N on the unit circle. Its corollaries state flatness in every finite L^p sense for one family of all lengths, that the largest binary merit factor at length N tends to infinity through all integer lengths (disproving Turyn's conjecture), and the existence of a binary Morse shift with simple spectrum whose zero-coordinate spectral measure is absolutely continuous. The manuscript states that the choices are existential and give no useful rate or signing algorithm.
• Positive minima. Nearly minimal maxima and positive minima of Littlewood polynomials (5 October 2026) states that for every η > 0 and every sufficiently large N there is a Littlewood polynomial whose modulus lies between √N/16 and (1 + η)√N everywhere on the unit circle.
• Ultraflatness. Ultraflat real Littlewood polynomials (5 October 2026) states that for every ε in (0, 1) and every sufficiently large N there is a Littlewood polynomial of length N whose modulus lies between (1 − ε)√N and (1 + ε)√N everywhere on the unit circle; the signs may be chosen separately at each length.
• Method, as described: a reduction to approximating a bounded auxiliary polynomial and rounding it to signs, using spreading of Fourier coefficients, a packing of signed intervals (Pippenger–Spencer), sampling of quadratic phases by Poisson summation and stationary phase, and rounding with a real-matrix discrepancy bound (Lovett–Meka).
• Sources. At commit 1581d4c: Asymptotically minimal maxima of real Littlewood polynomials, Nearly minimal maxima and positive minima of Littlewood polynomials and Ultraflat real Littlewood polynomials.
• Formalization. The repository's scope note covers the September manuscript. It states that the formalization gives real Littlewood polynomials of every sufficiently large length with maximum modulus at most (1 + η)√N, and one family of all lengths that is flat in every finite L^p mean; it states that the results are existential and give no effective rate or signing algorithm. Its Comparator statements are AsymptoticallyMinimalLittlewood.lean and LittlewoodFiniteFlatness.lean.
• Catalogue status. The Justin Sun Prize record on GitHub lists the real-sign question as entry JSP-000955, with status "Solved" (independent mathematical review pending) and a Lean proof "reported complete; independent verification pending", checked 19 October 2026 at commit fd8fa66.

Public follow-up work

Date	Work	Relation	Public claim	Public status (checked 19 October 2026)
2026-10-07	THM-4561 in the research-notes repository eliottcassidy2000/math (commit 9cbd9ad)	Related note	The note, whose metadata states that it was found by a session reading this family, studies the rows of a tower of skew-Hadamard matrices as Littlewood polynomials. It states that every row has merit factor below 2, and that after a Rudin–Shapiro sign switching every row has maximum modulus at most (3 + 2√2)√(2N).	GitHub repository (notes; not a paper).
2026-10-08	MathVet (GitHub; review of openai/math at commit adc7f12, tables at repository commit c76f1e4, 14 October 2026)	Review (fidelity of the formal statements)	Rates this family "weaker-statement" and records sandboxed Comparator passes for AsymptoticallyMinimalLittlewood and LittlewoodFiniteFlatness. Its note states that only the maximum-modulus bound and finite-Lp flatness are stated in Lean, not two-sided uniform ultraflatness, unbounded merit factors or the disproof of Turyn's conjecture, and that two of the three manuscripts are not linked in the scope notes.	The review states that it does not say the mathematics is correct beyond what Lean checked; it describes itself as a model-assisted reading without a referee yet.
2026-10-10	J. Kline, An explicit rate for ultraflat real Littlewood polynomials (Zenodo 10.5281/zenodo.23286062, version 0.1.0; jeff-kline/ultraflat-littlewood-rate)	Extension (effective version)	The README states an explicit length beyond which the October 5 ultraflatness statement holds: every N ≥ exp{exp{exp{1012 log2(2/ε)}}} works, so that the error can be taken as 2 exp{−10−6(log log log N)1/2} for large N. It states that the proof follows the original construction, with two new arguments: a hypergraph matching lemma with constants explicit in the edge size, replacing the Pippenger–Spencer theorem, and an exponential decay estimate for the auxiliary polynomials. It states that the proofs were checked by separate AI reviews and that no human expert has reviewed them.	Zenodo record and GitHub repository; the README states that admission under the project's public research standard is a release decision, not peer review.
2026-10-16	J. Jardine, Lattice24, referee reviews rank_508 and rank_517 in JJardine919/openai-math-audit (archived at Zenodo 10.5281/zenodo.23422995)	Review (AI-assisted audit)	Reports no gap in the two October 5 manuscripts, which have no Lean formalization. For the positive-minima manuscript, the reviewer states that it read all seven sections and verified the new correction step in full. For the ultraflat manuscript, it records one minor note: the stationary-phase lemma needs a smooth extension of an auxiliary phase with nonvanishing second derivative, which the text leaves implicit though it is easily arranged. The reviewer notes that the ultraflat manuscript relies on the companion for the signed-interval packing and the real-matrix discrepancy lemma.	Part of an audit of the collection's manuscripts without Lean that the audit describes as complete. The reviews are published under CC BY 4.0.

Notes

• The list covers follow-ups found in public sources up to the date above; it is not a priority or completeness claim.