Openai 2026 nearly minimal maxima positive minima littlewood polynomials

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theorem_1_1: The manuscript's main claim: for every eta > 0 and every large N, some polynomial with N consecutive plus-minus-one coefficients has modulus between sqrt(N)/16 and (1+eta) sqrt(N) on the whole unit circle, including z = 1 and z = -1; a release manuscript, unverified here.

OpenAI, Nearly minimal maxima and positive minima of Littlewood polynomials, OpenAI Math Release preprint, October 5, 2026. Released under the Apache License 2.0 at https://github.com/openai/math (revision adc7f1241), folder preprints/Nearly-minimal-maxima-and-positive-minima-of-Littlewood-polynomials-October-5-2026; the held PDF, littlewood-lower-envelope.pdf in the release, is retained as openai_2026_nearly_minimal_maxima_positive_minima_littlewood_polynomials.pdf, and the release's TeX bundle in that folder is the TeX source cited on this card.

@misc{OAI:Nearly-minimal-maxima-and-positive-minima-of-Littlewood-polynomials-October-5-2026,
  author = {{OpenAI}},
  title = {{Nearly minimal maxima and positive minima of Littlewood polynomials}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Nearly-minimal-maxima-and-positive-minima-of-Littlewood-polynomials-October-5-2026/littlewood-lower-envelope.pdf}{OAI:Nearly-minimal-maxima-and-positive-minima-of-Littlewood-polynomials-October-5-2026}},
  year = {2026}
}

Attestation as the release states it. The release's root README says the manuscripts were "produced by an internal OpenAI model", that the collection "includes results at different stages of verification", that not all have Lean formalizations, and, in its words, "Some of the unformalized results could have issues". The manuscript's own README adds only the title, author, date and citation block; the TeX source names "OpenAI" as author and carries no statement on human assistance. These sentences are recorded here as the source's own historical attestations, 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. The release's catalogue (lean/formalization.yaml) lists no formalization for this manuscript. The family page the release keeps for this group of manuscripts (lean/docs/076.md) covers only the companion Asymptotically minimal maxima of real Littlewood polynomials, whose comparator statements concern the nearly minimal maximum and finite-exponent flatness, not the lower bound claimed here; read statically from the release's catalogue. The corpus's verification built the companion's declaration OAI.AsymptoticallyMinimalLittlewood.main and checked its axioms (propext, Classical.choice and Quot.sound only); it concerns the upper bound alone. 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 companion's finite-flatness statement 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 claimed here.

Companions. The manuscript is one of three manuscripts in one release family. It reproduces, with attribution, the companion's arguments for the auxiliary polynomial, the interval packing, the sampling and the rounding from Asymptotically minimal maxima of real Littlewood polynomials (the upper bound $(1+\eta)\sqrt N$ alone) and adds the lower bound; the third manuscript, Ultraflat real Littlewood polynomials, claims the two-sided bound with both constants tending to one, which would supersede the constant $1/16$ here.

Read status: claims checked for Theorem 1.1 and for the statements of Propositions 2.1, 4.1 and 5.1 and Lemmas 2.2, 3.1, 4.2, 6.1 and 6.2, read clause by clause in the TeX source (main.tex, sections/introduction.tex lines 18--28, sections/spreading.tex lines 24--73, sections/packing.tex lines 19--52, sections/sampling.tex lines 17--100, sections/correction.tex lines 20--64, sections/rounding.tex lines 12--43) on 2026-10-07; the proofs were read for their structure only and no step was checked; nothing here is independently reviewed.

Contents

• Section 1, Introduction (pp. 1--3). Defines a Littlewood polynomial of length $N$ as $P(z)=\sum_{k=0}^{N-1}\varepsilon_kz^k$ with $\varepsilon_k\in\{-1,1\}$, notes Parseval's $\|P\|_\infty\ge\sqrt N$, and states Theorem 1.1: for every $\eta>0$ there is $N_0(\eta)$ such that every $N\ge N_0$ admits signs with $\sqrt N/16\le|P(z)|\le(1+\eta)\sqrt N$ on $|z|=1$, including $z=\pm1$. The context subsection places the two-sided flatness question in Erdős's 1957 list (Problem 26) and Littlewood's 1966 paper, cites the Hayman--Lingham collection (Problems 4.13 and 4.31) for the real-sign questions and the Parseval bound, recalls the Rudin--Shapiro bound $\sqrt{2N}$ at dyadic lengths, Kahane's complex ultraflat polynomials and Bombieri--Bourgain, and the Balister, Bollobás, Morris, Sahasrabudhe and Tiba theorem that flat Littlewood polynomials exist with two absolute constants; it says the companion manuscript gives the upper constant $1+o(1)$ and that the two properties must come from one choice of signs. The strategy subsection fixes the notation $U_X(t)=N^{-1/2}\sum_kX_k\mathrm e(kt)$ and the defect $\mu(X)=\tfrac12\sum_k(1-|X_k|)$, previews the rounding cost $C/\sqrt N+C\sqrt{(\mu/N)\log(80N/\mu)}$, and lays out the plan: one accuracy parameter $\delta$ is chosen first and determines every auxiliary object, and $N$ grows only afterwards.
• Section 2, Spreading Fourier coefficients (pp. 4--9). Proposition 2.1: for $0<\delta<1/20$ there are a dimension $m$, pairwise nonparallel $a_1,\dots,a_r\in\mathbb Z^m$, a real trigonometric polynomial $F$ on $\mathbb T^m$ with $\|F\|_\infty\le1$, $\|F\|_2\ge1-3\delta$ and Fourier support inside $\{\pm a_i\}$, and a vector $v$ whose widths $\lambda_a=a\cdot v$ are nonzero, satisfy $\sum_a|\lambda_a|<1$ and $|\widehat F(a)|\le K_\delta\sqrt{|\lambda_a|}$ with $K_\delta=\sqrt{(1+\delta)^3/(1-\delta)}$. Lemma 2.2 is a uniform stationary-phase evaluation of $\int g(x)\mathrm e(T\beta x^2/2-ux)\,dx$. The proof builds a bounded polynomial of mean square near one by a recursion in new torus variables, spreads each coefficient over about $B^D\Delta_s$ frequencies by quadratic oscillation, and truncates. The section says it reproduces Section 4 of the companion.
• Section 3, Packing signed intervals (pp. 9--13). Lemma 3.1: pairwise nonparallel integer forms $a_1,\dots,a_r$ and widths with $2\sum_iw_i<1$ admit $H\ge1$ and $\theta_1,\dots,\theta_H\in\mathbb T^m$ such that the $2rH$ closed intervals centered at $\pm a_i\cdot\theta_h$ of length $w_i/H$ are pairwise disjoint. Theorem 3.2 is the Pippenger--Spencer hypergraph edge-coloring theorem in the form of Alon--Yuster, Lemma 2.1, the section's external input. The proof works over $\mathbb F_q$ for large primes $q$, builds a random $r$-uniform hypergraph of slots with near-regular degrees and small codegrees, deletes the exceptional vertices and takes a large matching. The data are fixed before $N$ varies, so the finite field imposes no arithmetic condition on $N$. Reproduced from the companion.
• Section 4, Sampling and the geometry of the small values (pp. 13--17). Proposition 4.1: for $0<\delta<1/100$ there are a smooth even $B:\mathbb T\to[0,K_\delta]$ and, for every $N$, coefficients $Y_{N,k}\in[-1,1]$ and smooth conjugate-symmetric $B_N$ with $|B_N|=B$, $\|U_{Y_N}-B_N\|_\infty=O_\delta(N^{-1})$, $\int B^2\ge1-7\delta$ and $N^{-1}\sum_kY_{N,k}^2\ge1-8\delta$ for large $N$; $B$ vanishes near $0$ and $1/2$, its superlevel set $\{B\ge1/8\}$ in $[0,1/2]$ is a finite union of closed intervals, the complementary gaps have total length $L\le10\delta$, and at each gap endpoint in $(0,1/2)$ the wave $B_N$ has the local form $B\,\mathrm e(\alpha_z+N\psi_z)$ with a quadratic $\psi_z$ and $\psi_z'(z)\in(0,1)$. Lemma 4.2 converts Lemma 2.2 by Poisson summation into a uniform formula for quadratic exponential sums. The proof samples $F$ along quadratic paths in $H$ blocks with smooth cutoffs; the packed arcs keep the leading waves disjoint, so $|B_N|$ is independent of $N$. The sampling argument is attributed to Section 5 of the companion; the gap description is the addition needed here.
• Section 5, A correction with small Fourier coefficients (pp. 17--23). The manuscript's own step. Proposition 5.1: under the hypotheses that Proposition 4.1 supplies, for every large $N$ there is a continuous conjugate-symmetric $R_N$ supported on the gaps and their reflections with $1/8-o(1)\le|B_N+R_N|\le K_\delta$ on $\mathbb T$, $\max_{0\le k<N}\sqrt N|\widehat R_N(k)|\le D\sqrt\delta$ for an absolute $D$, and $\sum_{k\notin\{0,\dots,N-1\}}|\widehat R_N(k)|=O_\delta(N^{-1/4})$, all coefficients real. The proof assigns each gap a derivative slot $D_j\subset(1/4,3/4)$ of length $l_j/(4L)$, builds a piecewise quadratic leading phase whose derivative takes the prescribed values at the gap's ends and runs across the slot on the middle piece, adds an affine phase adjustment with slope at most $1/l_j$ to match phases modulo integers at the transitions, turns the amplitude $1/2$ on over strips of width $N^{-3/4}$, and bounds the coefficients by a quadratic stationary-phase estimate on the one slot near $k/N$ plus first-derivative bounds elsewhere; the exterior tail follows from two integrations by parts, the boundary terms of the first canceling around the whole circle. Figure 1 (p. 20) is a schematic of the slots and tapers.
• Section 6, Rounding with a small defect mass (pp. 23--26). Lemma 6.1: an absolute $C$ such that every $X\in[-1,1]^N$ admits signs with $\max_t|\sum_k(\varepsilon_k-X_k)\mathrm e(kt)|$ at most $C(1+\sqrt{\mu\log(80N/\mu)})$, where $\mu=\mu(X)$ and the root term is zero at $\mu=0$. Lemma 6.2: a real matrix in $[-1,1]^{R\times s}$, $1\le s\le R$, has a sign vector with discrepancy at most $C_0\sqrt{s\log(2R/s)}$, derived from the existential form of the Lovett--Meka partial-coloring theorem (Theorem 4 of arXiv:1203.5747v2), with Spencer's method named as the origin. The proof rounds the defects on dyadic grids against a $40N$-row matrix of cosine and sine evaluations at $20N$ grid points, keeping the defect mass from growing by a sign reversal at each scale, and passes from the grid to the circle by the maximum principle and a Cauchy estimate. Reproduced in full from Section 6 of the companion.
• Section 7, Proof of the main theorem (p. 27). Sets $X_{N,k}=(Y_{N,k}+\sqrt N\,\widehat R_N(k))/(1+D\sqrt\delta)$, checks $X_N\in[-1,1]^N$, $U_{X_N}=(B_N+R_N)/(1+D\sqrt\delta)+o(1)$ uniformly and $\mu(X_N)/N\le4\delta+2D\sqrt\delta$, applies Lemma 6.1, and chooses $\delta$ then $N_0$ so that the lower margin exceeds $1/16$ and the upper margin stays below $1+\eta$.
• References (p. 28, twelve entries). The text cites Erdős 1957, Littlewood 1966, Hayman--Lingham 2018, Rudin 1959, Balister et al. 2020, Kahane 1980, Bombieri--Bourgain 2009, Pippenger--Spencer 1989, Alon--Yuster 2005, Spencer 1985, Lovett--Meka 2015 and the companion manuscript. The bibliography file in the TeX bundle also carries entries for Balister 2019, Erdélyi 2026, two el Abdalaoui preprints and Bonami--Révész that no sentence of the text cites and that the PDF does not print.

External inputs the proofs rest on: the Pippenger--Spencer theorem (Theorem 3.2, through Alon--Yuster) and the Lovett--Meka theorem (inside Lemma 6.2); everything else is standard Fourier analysis (Poisson summation, stationary phase, the maximum principle) proved or sketched in the text. The manuscript flags nothing as numerical, computer-assisted or conditional. The result is existential: $N_0(\eta)$ comes from a non-effective choice of $\delta$ and then of $N$, and no rate in $\eta$ is claimed. The release folder holds the PDF, the README and a build/ directory; it has no verification/ folder.

Bears on

• Problem 1150: the upper half of Theorem 1.1, $\max_{|z|=1}|P(z)|\le(1+\eta)\sqrt N$ for every $\eta>0$ and every large $N$, is a claimed negative answer to the exact question (no $c>0$ can work for all large $n$; the problem's degree $n$ is the length $N-1$ here). That half is re-proved here along the lines of the companion manuscript, which the text credits with it; the lower bound $\sqrt N/16$ is this manuscript's addition and is not asked by the problem. The claim is unverified here; the page's status rests on acceptance evidence, which this card does not supply.
• Problem 228: Theorem 1.1 is a claimed stronger form of the proved statement, with the explicit constants $1/16$ below and $1+\eta$ above in place of the two absolute constants of Balister, Bollobás, Morris, Sahasrabudhe and Tiba; its range of degrees $n\ge N_0(\eta)-1$ (lengths $N\ge N_0(\eta)$) matches the problem's "all large $n$", whereas the cited proof covers every degree $n\ge2$. Unverified here; the page's status rests on its cited acceptance evidence, not on this manuscript.
• Problem 230: comparison. The problem is disproved by Kahane's complex unimodular polynomials; Theorem 1.1 would give counterexamples with real coefficients $\pm1$, a subclass of the problem's unimodular coefficients, for every sufficiently large $n$. It says nothing about the problem's small $n$ and does not use the problem's normalization (indices $1$ to $n$). Unverified here; the page's status does not depend on it.