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Turyn's conjecture that binary merit factors are bounded

For a binary word A=(a0,…,aN−1)∈{−1,1}NA=(a_0,\dots,a_{N-1})\in\{-1,1\}^N with aperiodic autocorrelations Cu(A)=∑j=0N−1−uajaj+uC_u(A)=\sum_{j=0}^{N-1-u}a_ja_{j+u}, the merit factor is F(A)=N2/(2∑u=1N−1Cu(A)2)F(A)=N^2/(2\sum_{u=1}^{N-1}C_u(A)^2). Equivalently 1/F(A)=∥PA∥44/N2−11/F(A)=\|P_A\|_4^4/N^2-1 for the polynomial PA(z)=∑ajzjP_A(z)=\sum a_jz^j on the unit circle. The best proven constructions, from modified Legendre sequences, reach limiting merit factor about 6.346.34 (Jedwab, Katz and Schmidt). Turyn's conjecture, equivalently Erdos's L4L^4-norm conjecture, asserts that F(A)F(A) is bounded over all binary words of all lengths. Is the merit factor of binary sequences bounded?

Result
Disproved(see note)
Status
Candidate (review pending)
AI contribution
AI-discovered
Method
Argument
Field
Binary sequences; aperiodic autocorrelation
Posed by
Richard J. Turyn (name as used by Downarowicz and Lacroix 1998); equivalently Erdos's L4-norm conjecture
Year posed
—
Years open
—
Solved
2026-09-23
Model
Unreleased internal OpenAI model
Vendor
OpenAI
Collaborators
—
Verification
Unreviewed
Publication
Announced
Collection
OpenAI math release (October 2026), version adc7f12
Significance
35 / 100
Disclosed cost
—
Wikipedia
No dedicated article

What was actually shown

Section 8 deduces from Theorem 1.1 (real-sign polynomials of every large length with maximum (1+o(1))N(1+o(1))\sqrt N) that max⁡A∈{−1,1}NF(A)→∞\max_{A\in\{-1,1\}^N}F(A)\to\infty as N→∞N\to\infty through all integers, disproving Turyn's conjecture in this all-length form. Via Downarowicz and Lacroix it also gives a uniquely ergodic binary Morse shift with simple spectrum and absolutely continuous zero-coordinate spectral measure with L2L^2 density. The construction is existential: no explicit sequences, no rate and no algorithm are given.

What the AI did

The release README says every result in it was produced by an unreleased internal OpenAI model following a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier model results. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author. The principal manuscript is dated September 23, 2026. Two companions dated October 5, 2026 strengthen it: one adds a lower bound of sqrt(N)/16, the other makes the polynomials two-sided ultraflat; the latter says it reuses lemmas of a 'version-2' refinement of the principal paper.

Verification

No independent mathematician has checked this yet. Checked here: Section 1.2 and Section 8 of the manuscript were read against Turyn's conjecture; the paper claims the largest merit factor at length NN tends to infinity through all integer lengths, deduced from flatness in every finite LpL^p sense. The challenge LittlewoodFiniteFlatness is not in the formalization catalogue (lean/formalization.yaml); it is linked from lean/docs/076.md and its solution module exists at the pinned commit. Its statement was read here: one family of real-sign polynomials, one for each length, with ∫∣ ∣PN∣/N−1∣p→0\int |\,|P_N|/\sqrt N-1|^p\to0 for every finite p>0p>0. This is narrower than the headline: merit factors are not defined in the Lean, and the passage from p=4p=4 flatness to unbounded merit factor is a standard identity left informal. Not rebuilt here. The paper says its theorem conflicts with nonflatness claims in preprints of el Abdalaoui (arXiv:1609.03435 and two others) and examines specific issues in those arguments in its Appendix A; no public response to this release was found or searched for here. Listed as Unreviewed rather than Lean-checked because its formal statement covers only the L^p flatness step, not the unbounded merit factor.

Sources

Changelog1 change

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