View source for 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 ([https://github.com/openai/math 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 √(2''N'') 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 ''L''<sup>4</sup>-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''<sup>''p''</sup> 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 <code>1581d4c</code>: [https://github.com/openai/math/tree/1581d4cbdd69a0faf34b2089614fe2a814316a6a/preprints/Asymptotically-minimal-maxima-of-real-Littlewood-polynomials-September-23-2026 Asymptotically minimal maxima of real Littlewood polynomials], [https://github.com/openai/math/tree/1581d4cbdd69a0faf34b2089614fe2a814316a6a/preprints/Nearly-minimal-maxima-and-positive-minima-of-Littlewood-polynomials-October-5-2026 Nearly minimal maxima and positive minima of Littlewood polynomials] and [https://github.com/openai/math/tree/1581d4cbdd69a0faf34b2089614fe2a814316a6a/preprints/Ultraflat-real-Littlewood-polynomials-October-5-2026 Ultraflat real Littlewood polynomials].
* '''Formalization.''' The repository's [https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/076.md 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''<sup>''p''</sup> mean; it states that the results are existential and give no effective rate or signing algorithm. Its Comparator statements are [https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/ComparatorChallenges/AsymptoticallyMinimalLittlewood.lean AsymptoticallyMinimalLittlewood.lean] and [https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/ComparatorChallenges/LittlewoodFiniteFlatness.lean LittlewoodFiniteFlatness.lean].
* '''Catalogue status.''' The Justin Sun Prize record on GitHub lists the real-sign question as entry [https://github.com/TheJustinSunPrize/awards/blob/fd8fa66abb4e21df828fdebc25298e9ad75521bd/problems/catalog-0901-1000.md#JSP-000955 JSP-000955], with status "Solved" (independent mathematical review pending) and a Lean proof "reported complete; independent verification pending", checked 19 October 2026 at commit <code>fd8fa66</code>.

== Public follow-up work ==
{| class="wikitable"
! Date !! Work !! Relation !! Public claim !! Public status (checked 19 October 2026)
|-
| 2026-10-07 || [https://github.com/eliottcassidy2000/math/blob/9cbd9ad8c253e24dbc737e98e4624f7bd170a9fb/01-canon/theorems/THM-4561-the-sierpinski-skew-hadamard-tower-in-the-littlewood-problem.md THM-4561] in the research-notes repository [https://github.com/eliottcassidy2000/math eliottcassidy2000/math] (commit <code>9cbd9ad</code>) || 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)√(2''N''). || GitHub repository (notes; not a paper).
|-
| 2026-10-07 || J. G. George, ''Conditional corollaries from the October 2026 OpenAI mathematics release'' ([https://doi.org/10.5281/zenodo.23219812 Zenodo 10.5281/zenodo.23219812], version 1.0.0; [https://github.com/econgreg-hub/Math-Theory/tree/7c9c9b92695bf0e965b91c8707c1101c84e450bc econgreg-hub/Math-Theory], commit <code>7c9c9b9</code>) || Use (conditional corollary) || Assuming the October 5 ultraflatness theorem, the note states that circulant {−1, 1} matrices whose first rows are the ultraflat coefficient sequences have condition number tending to one and satisfy ‖''CC''*/''N'' − ''I''‖ → 0 in every sufficiently large order. It states that the flat-polynomial method for circulants is already explicit in work of Steinerberger, and that the collection's circulant Hadamard theorem would rule out exact orthogonality for ''N'' > 4. || Zenodo record and GitHub repository. The note states that it is pinned to commit <code>adc7f12</code>, that ChatGPT developed the displayed deductions, that it has not undergone external peer review, and that it makes no priority claim.
|-
| 2026-10-08 || [https://github.com/mathvet/mathvet MathVet] (GitHub; review of openai/math at commit <code>adc7f12</code>, tables at repository commit <code>c76f1e4</code>, 14 October 2026) || Review (fidelity of the formal statements) || Rates this family "weaker-statement" and records sandboxed Comparator passes for <code>AsymptoticallyMinimalLittlewood</code> and <code>LittlewoodFiniteFlatness</code>. Its note states that only the maximum-modulus bound and finite-''L''<sup>''p''</sup> 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'' ([https://doi.org/10.5281/zenodo.23286062 Zenodo 10.5281/zenodo.23286062], version 0.1.0; [https://github.com/jeff-kline/ultraflat-littlewood-rate 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{10<sup>12</sup> log<sup>2</sup>(2/ε)}}} works, so that the error can be taken as 2 exp{−10<sup>−6</sup>(log log log ''N'')<sup>1/2</sup>} 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 [https://github.com/JJardine919/openai-math-audit/blob/a06a8c6e77974d4c6021338a55a8e91b634f67f7/reviews/rank_508.json rank_508] and [https://github.com/JJardine919/openai-math-audit/blob/a06a8c6e77974d4c6021338a55a8e91b634f67f7/reviews/rank_517.json rank_517] in [https://github.com/JJardine919/openai-math-audit JJardine919/openai-math-audit] (archived at [https://doi.org/10.5281/zenodo.23422995 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.

[[Category:OpenAI results]]