Openai 2026 asymptotically minimal maxima real littlewood polynomials

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corollary_7_1: There are sign polynomials P_N of every length whose normalized modulus |P_N|/sqrt N tends to one in L^p on the unit circle for every fixed finite p > 0; deduced from Theorem 1.1, and claimed to contradict el Abdalaoui.

corollary_8_1: The maximum merit factor over binary words of length N tends to infinity through all integer lengths, with no rate; a claimed disproof of Turyn's bounded-merit-factor conjecture, deduced from Theorem 1.1 by the fourth moment.

theorem_1_1: For every eta > 0 and every sufficiently large length N there is a sign polynomial of length N with maximum modulus at most (1+eta) sqrt N on the unit circle; a claimed negative answer to Problem 1150.

OpenAI, Asymptotically minimal maxima of real Littlewood polynomials, OpenAI Math Release preprint, September 23, 2026. Released under the Apache License 2.0 at https://github.com/openai/math (revision adc7f1241), folder preprints/Asymptotically-minimal-maxima-of-real-Littlewood-polynomials-September-23-2026; the held PDF, paper.pdf in the release, is retained as openai_2026_asymptotically_minimal_maxima_real_littlewood_polynomials.pdf, and the release's TeX bundle in that folder is the TeX source cited on this card.

@misc{OAI:Asymptotically-minimal-maxima-of-real-Littlewood-polynomials-September-23-2026,
  author = {{OpenAI}},
  title = {{Asymptotically minimal maxima of real Littlewood polynomials}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Asymptotically-minimal-maxima-of-real-Littlewood-polynomials-September-23-2026/paper.pdf}{OAI:Asymptotically-minimal-maxima-of-real-Littlewood-polynomials-September-23-2026}},
  year = {2026}
}

Attestation, as the release states it: the release's root README says the repository holds manuscripts "produced by an internal OpenAI model", that the collection "includes results at different stages of verification", that not all manuscripts have Lean formalizations and, in its words, "Some of the unformalized results could have issues". The manuscript's own README in the release folder gives only the title, the author "OpenAI", the date September 23, 2026 and the citation block above; it adds no statement about human assistance. The manuscript names no author beyond "OpenAI", carries no arXiv identifier and no journal, and cites a companion release manuscript (a three-torus diffeomorphism with simple Lebesgue spectrum) for a different realization of its spectral consequence. These are the source's own provenance attestations, recorded as history and not as this corpus's review. No refereed publication, arXiv version or independent review of the manuscript is recorded here and nothing on this card is independently reviewed.

Formalization, as the release lists it: the release's Lean catalogue (lean/formalization.yaml) names this manuscript and pairs it with one comparator entry, the configuration lean/ComparatorChallenges/AsymptoticallyMinimalLittlewood.json, declaration OAI.AsymptoticallyMinimalLittlewood.main, solution file OAI/Analysis/Littlewood/Main.lean. Its family page "Real ultraflat Littlewood polynomials and unbounded binary merit factors" (lean/docs/076.md) says the formalization gives, for every fixed $\eta>0$, real sign polynomials of every sufficiently large length with maximum modulus at most $(1+\eta)\sqrt N$ (the content of Theorem 1.1), and a second statement choosing one all-length family whose normalized modulus tends to one in every finite $L^p$ mean (the content of Corollary 7.1). The comparator statement files it names are lean/ComparatorChallenges/AsymptoticallyMinimalLittlewood.lean (namespace OAI.AsymptoticallyMinimalLittlewood, MainStatement: for every $\eta>0$ there is $N_0\ge1$ such that every $N\ge N_0$ has a sign vector $\varepsilon\colon\mathrm{Fin}\,N\to\mathbb R$ with $\lVert\sum_k\varepsilon_kz^k\rVert\le(1+\eta)\sqrt N$ on $\lVert z\rVert=1$) and lean/ComparatorChallenges/LittlewoodFiniteFlatness.lean (one sign family indexed by all $N$ with the $L^p$ integral of $\bigl||P_N|/\sqrt N-1\bigr|$ over the circle tending to zero for every real $p>0$); the second is named by the family page only, not by the catalogue file. Each comparator file carries the statement alone, its theorem closed by sorry; each comparator configuration (and, for Main.lean, the catalogue file) names a solution module under lean/OAI/Analysis/Littlewood/ (Main.lean and FiniteFlatness.lean, the second deducing its statement from the first); the configurations permit the axioms propext, Quot.sound and Classical.choice. The family page says the results are existential and give no convergence rate or signing algorithm. All of this is read statically from the release's catalogue. The corpus's verification built the declaration OAI.AsymptoticallyMinimalLittlewood.main and checked its axioms (propext, Classical.choice and Quot.sound only). For Problem 1150 that verification covers the question, answered no: for every $\eta>0$ and every length $N\ge N_0$ there are real $\pm1$ signs whose polynomial $\sum_{k<N}\varepsilon_kz^k$ (degree $N-1$) has modulus at most $(1+\eta)\sqrt N$ on all of $|z|=1$, so for every $c>0$ and every large degree $n$ some $\pm1$ polynomial of degree $n$ has circle maximum at most $(1+c)\sqrt n$, and no $c>0$ works. For Problem 230 it covers the question, answered no: for every $c>0$ and every large $n$ (in particular some $n\ge2$) there are unimodular coefficients $a_1,\dots,a_n$, in fact real $\pm1$, with $\max_{|z|=1}\bigl|\sum_{1\le k\le n}a_kz^k\bigr|$ at most $(1+c/2)\sqrt n$, which is below $(1+c)\sqrt n$. The records are kept on the claim pages of Problem 1150 and Problem 230, not on this card; the finite-flatness declaration of LittlewoodFiniteFlatness.lean is not named in that record and has no build or fidelity audit recorded here, and no declaration of the release states the lower bound Problem 228 asks for.

Companions: the release groups this manuscript in one family with two October 5, 2026 manuscripts, Nearly minimal maxima and positive minima of Littlewood polynomials and Ultraflat real Littlewood polynomials. The release's family description covers the three as a whole; by the companions' titles and the abstracts the release lists for them, the first adds a uniform lower bound to the present upper bound and the second makes the family two-sided ultraflat. Neither companion was read for this card, and the present manuscript cites neither (it predates both).

Read status: claims checked for Theorem 1.1, Proposition 2.1, Lemma 2.2, Corollary 7.1, Corollary 8.1 and Corollary 8.2, read clause by clause in the TeX source (introduction.tex lines 1--39, reduction.tex lines 1--38, flatness.tex lines 1--13, merit-morse.tex lines 1--36 and 57--86, with the labels thm:main, prop:almost-signs, lem:rounding, cor:finite-flatness, cor:merit, cor:morse) on 2026-10-07; the statements of Lemma 3.1, Theorem 3.2, Proposition 4.1, Lemma 4.2, Lemma 5.1 and Lemma 6.1 were read for what they supply, and every proof was read for its structure only with no step checked; nothing here is independently reviewed.

Contents

The manuscript has 26 PDF pages: a table of contents (p. 1), Sections 1--8 (pp. 2--23), Appendix A (pp. 23--24) and references (pp. 25--26). Results are numbered by section.

• Section 1, Introduction (pp. 2--4). Defines a Littlewood polynomial of length $N$ as $P(z)=\sum_{k=0}^{N-1}\varepsilon_kz^k$ with $\varepsilon_k\in\{-1,1\}$, and $m_N$ as the minimum over all sign choices of $N^{-1/2}\lVert P\rVert_\infty$; Parseval gives $m_N\ge1$. The real-sign question, whether $m_N\ge1+c$ eventually for an absolute $c>0$, is attributed to Erdős's 1957 problem list (Problem 22) and to Hayman--Lingham (Problems 4.13 and 4.31), with Kahane's refutation of the complex unimodular version noted. States Theorem 1.1: $\lim_{N\to\infty}m_N=1$, through all integer lengths, with no uniform lower bound on $|P|$, no rate and no algorithm asserted. Section 1.1 places the result against Shapiro--Rudin ($\sqrt{2N}$ at dyadic lengths), Balister's all-length bound $\sqrt{6N-2}-1$, the two-sided flat polynomials of Balister, Bollobás, Morris, Sahasrabudhe and Tiba, the unimodular constructions of Littlewood, Kahane and Bombieri--Bourgain (the last named's use of smoothed quadratic phases with Poisson summation being credited as a forerunner of the sampling argument), the AlphaEvolve search of Georgiev, Gómez-Serrano, Tao and Wagner at degrees up to 100, and Erdélyi's lower bound $\lVert P\rVert_\infty^2\ge N+(N-1)^{1/3}/38$, which it calls compatible since the extra term is $o(N)$. It announces that the theorem conflicts with nonflatness claims in three preprints of el Abdalaoui. Section 1.2 defines the aperiodic autocorrelations $C_u(A)$ and the merit factor $F(A)=N^2/(2\sum_{u=1}^{N-1}C_u(A)^2)$ in the Downarowicz--Lacroix normalization, recalls Turyn's conjecture (bounded merit factors) and the Jedwab--Katz--Schmidt limiting value $6.342061\ldots$, and announces the all-length disproof. Section 1.3 is the proof overview: relax to real coefficients in $[-1,1]$ with defect $\mu(X)=\tfrac12\sum_k(1-|X_k|)$, build them by sampling an auxiliary torus polynomial along quadratic phases so that no angle receives more than one stationary contribution, and round to signs by partial coloring with an error controlled by the defect.
• Section 2, Reduction (pp. 4--5). States Proposition 2.1 (almost-sign approximation): for every $0<\delta<1/20$ there are $X_N\in[-1,1]^N$ for every $N\ge1$ with $\limsup N^{-1/2}\lVert Q_{X_N}\rVert_\infty\le K_\delta$, $K_\delta=\sqrt{(1+\delta)^3/(1-\delta)}$, and $\liminf N^{-1}\sum_kX_{N,k}^2\ge1-7\delta$; and Lemma 2.2 (rounding with a small defect): an absolute $C$ such that every $X\in[-1,1]^N$ has signs with $\max_t|\sum_k(\varepsilon_k-X_k)e(kt)|$ at most $C(1+\sqrt{\mu(X)\log(80N/\mu(X))})$. Deduces Theorem 1.1 from the two: $\mu(X_N)\le4\delta N$ for large $N$, so $\limsup m_N\le K_\delta+2C\sqrt{\delta\log(20/\delta)}$, then $\delta\downarrow0$.
• Section 3, Packing signed intervals (pp. 5--9). Lemma 3.1 (signed interval packing): for integers $m\ge1$, $r\ge2$, pairwise nonparallel vectors $a_1,\ldots,a_r\in\mathbb Z^m\setminus\{0\}$ and widths $w_i>0$ with $2\sum_iw_i<1$, there are $H\ge1$ and points $\theta_1,\ldots,\theta_H\in\mathbb T^m$ such that the $2rH$ closed circle intervals centered at $\pm a_i\cdot\theta_h$ of length $w_i/H$ are pairwise disjoint. Its proof builds a random $r$-uniform hypergraph over $\mathbb F_q$ for a large prime $q$ (slots in a half-circle as vertices, compatible center tuples as edges), estimates degrees and codegrees by first and second moments, deletes atypical vertices, and takes a large matching from the Pippenger--Spencer edge-coloring theorem, quoted as Theorem 3.2 in the Alon--Yuster form.
• Section 4, Spreading Fourier coefficients (pp. 9--14). Proposition 4.1 (an auxiliary polynomial with controlled widths): for every $0<\delta<1/20$ a real trigonometric polynomial $F$ on some $\mathbb T^m$ with $\lVert F\rVert_\infty\le1$, $\lVert F\rVert_2\ge1-3\delta$, a containing Fourier support of pairwise nonparallel signed pairs, and a velocity $v\in\mathbb R^m$ with $\lambda_a=a\cdot v\ne0$, $\sum_a|\lambda_a|<1$ and $|\widehat F(a)|/\sqrt{|\lambda_a|}\le K_\delta$. Lemma 4.2 is a uniform stationary-phase evaluation of $\int g(x)e(T\beta x^2/2-ux)\,dx$ as $(T|\beta|)^{-1/2}$ times a unimodular phase times $g(u/(T\beta))+O(T^{-1})$, uniformly in $u$, proved by completing the square, Gaussian damping and Fourier inversion. The proof of Proposition 4.1 iterates the recursion $p_j=p_{j-1}+\tfrac12(1-p_{j-1}^2)\cos(2\pi y_j)$ until the mean square exceeds $1-\delta$, spreads each Fourier coefficient over a box of frequencies by quadratic oscillations in $D$ extra variables, bounds the mass outside the boxes by integration by parts, truncates, and tunes the curvatures so that widths and coefficient sizes match.
• Section 5, Sampling (pp. 14--18). Lemma 5.1: a Poisson-summation evaluation of smoothed quadratic exponential sums $N^{-1/2}\sum_k\chi(k/N)e(bk+\tfrac{N\lambda}2(k/N-x_*)^2)$ with error $O(N^{-1})$ uniformly for $b$ in a compact set, attributed in method to Bombieri--Bourgain. The proof of Proposition 2.1 packs the widths $|\lambda_{a_i}|$ by Lemma 3.1, fixes $H$ blocks with smooth cutoffs $\chi_h$, defines $X_{N,k}=\sum_h\chi_h(k/N)F(k\theta_h+\tfrac N2v(k/N-x_h^*)^2)$, shows by disjointness that at most one stationary term is nonzero at any angle (hence the maximum bound $K_\delta+o(1)$), and recovers the mean-square mass by Riemann sums, Lemma 5.1 for distinct curvatures and geometric summation for equal curvatures with distinct centers. Every integer $N$ is allowed; no divisibility condition enters.
• Section 6, Rounding (pp. 18--21). Lemma 6.1: an absolute $C_0$ such that every real $B\in[-1,1]^{R\times s}$ with $1\le s\le R$ has a sign vector $\xi$ with $\lVert B\xi\rVert_\infty\le C_0\sqrt{s\log(2R/s)}$, derived by iterating the Lovett--Meka partial-coloring theorem (cited in the preprint version's Theorem 4), halving the free coordinates each round. The proof of Lemma 2.2 writes $X_k=\sigma_k(1-2p_k)$, rounds the $p_k$ dyadically from scale $2^{-J}$ up, applying Lemma 6.1 at each scale to the odd coordinates on a grid of $40N$ real and imaginary Fourier rows and reversing all signs when needed so the defect mass never increases, then passes from the grid to the whole circle by a maximum-principle and Cauchy-estimate argument.
• Section 7, Flatness for every finite exponent (p. 21). Corollary 7.1: one Littlewood polynomial $P_N$ can be chosen at each length so that $\int_{\mathbb T}\bigl||P_N(e(t))|/\sqrt N-1\bigr|^p\,dt\to0$ for every fixed exponent $0<p<\infty$; proved from Theorem 1.1 and Parseval through the fourth moment.
• Section 8, Binary merit factors and Morse shifts (pp. 21--23). Records $\lVert P_A\rVert_4^4=N^2+2\sum_uC_u(A)^2$. Corollary 8.1: $\mathcal F_N$, the maximum of $F(A)$ over $A\in\{-1,1\}^N$, tends to infinity as $N\to\infty$ through every integer, with no rate. Corollary 8.2: there is a uniquely ergodic binary Morse shift whose Koopman operator has simple spectrum and whose zero-coordinate spectral measure has an $L^2$ density $h\ge0$ with $\int h=1$; proved by feeding Corollary 8.1 into Downarowicz--Lacroix's Theorem 2 and their Facts 1--2. The manuscript notes that simplicity concerns the full Koopman operator while absolute continuity is asserted only for the cyclic subspace of the zero coordinate.
• Appendix A, Comparison with contrary flatness claims (pp. 23--24). Independent of the construction. A.1 argues that the 2025 el Abdalaoui nonflatness proof (Theorem 1 of arXiv:2504.21499) uses the Bonami--Révész concentration level with the wrong quantifier order: that level is an infimum over symmetric open sets, so an upper bound on it does not bound concentration on a prescribed set, and sets containing a neighborhood of zero have concentration one (shown with Dirichlet kernels). A.2 claims a counterexample ($a_j=2^{j^2}$, all multipliers one) to the weighted criterion of Theorem 1 of arXiv:2509.04212 and a counterexample ($P=1+z$) to a signed-modulus identity displayed in its proof of Corollary 3; both examples are elementary and neither was checked here. The 2017 preprint's endpoint-sign convention is handled by changing at most two signs.

External inputs the proofs rest on, all taken at statement level and none checked here: the Pippenger--Spencer theorem (Alon--Yuster, Lemma 2.1), the Lovett--Meka partial-coloring theorem, and Downarowicz--Lacroix's Theorem 2 with Facts 1--2 (for Corollary 8.2 only). The manuscript flags nothing as numerical, computer-assisted or conditional; its results are existential with no rate. The release folder holds no verification/ folder for this manuscript. The bibliography file in the TeX bundle carries an entry for the erdosproblems.com page of Problem 1150, but the text never cites it and the printed references omit it; the manuscript names no Erdős problem by catalogue number.

Bears on

• Problem 1150: Theorem 1.1 is a claimed negative answer to the exact question. The problem asks for $c>0$ with $\max_{|z|=1}|P(z)|>(1+c)\sqrt n$ for all large degrees $n$ and all sign polynomials; the manuscript's $m_N\to1$ (with $N=n+1$ coefficients, so $\sqrt N/\sqrt n\to1$) says no such $c$ exists. The claim is unverified here and the page's status rests on acceptance evidence.
• Problem 228: comparison with a problem already proved. The problem asks for two-sided bounds $\sqrt n\ll|P(z)|\ll\sqrt n$; Theorem 1.1 is claimed to make the upper constant $1+o(1)$ but, as the manuscript states, gives no uniform lower bound, so it does not by itself give a stronger form of the two-sided statement. The claim is unverified here and the page's status rests on acceptance evidence.
• Problem 230: comparison with a problem already disproved. The problem's class is complex unimodular coefficients and Kahane's ultraflat polynomials disprove the $(1+c)\sqrt n$ bound there; real signs are unimodular, so Theorem 1.1 is a claimed counterexample family inside the real-coefficient subclass, which the manuscript identifies as the distinct question. The claim is unverified here and the page's status rests on acceptance evidence.
• Idempotent concentration audit: Appendix A.1 makes the same quantifier objection to the 2025 preprint's concluding inference that the audit records, and supports it the same way, with a Dirichlet-kernel computation of full concentration near zero; it adds a remark that supports of density one half in $\{0,\ldots,q-1\}$ do not give the required bound either, and cites Bonami--Révész (Theorem 7 and Proposition 9) for the distinction between full concentration at zero and the uniform level.
• Source proof audit: Appendix A.2 claims to refute the weighted criterion of arXiv:2509.04212 with a different counterexample ($a_j=2^{j^2}$ in place of factorials) and claims a counterexample ($P=1+z$) to an identity displayed in its proof of Corollary 3; the examples are elementary but were not checked here.
• el Abdalaoui 2025 card: Corollary 7.1 contradicts that preprint's claimed non-$L^{2p}$-flatness of real sign polynomials for integers $p>1$, and Appendix A.1 names the step the manuscript holds responsible; the contradiction is a claim of this manuscript and is unverified here.
• Downarowicz--Lacroix 1998 card: Corollary 8.1 claims the hypothesis of that paper's Theorem 2 (binary words of unbounded merit factor), and Corollary 8.2 consumes Theorem 2 and Facts 1--2 at statement level; the manuscript adopts that paper's merit-factor normalization.
• Erdélyi 2026 card: the additive lower bound $\lVert P\rVert_\infty^2\ge N+(N-1)^{1/3}/38$ is cited as compatible with Theorem 1.1, since it is $o(N)$ above Parseval; the manuscript asserts no rate of its own, so the gap between the two is not closed.