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Turyn's conjecture that binary merit factors are bounded

Section 8 deduces from Theorem 1.1 (real-sign polynomials of every large length with maximum $(1+o(1))\sqrt N$ ) that $\max_{A\in\{-1,1\}^N}F(A)\to\infty$ as $N\to\infty$ 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 $L^2$ density. The construction is existential: no explicit sequences, no rate and no algorithm are given.

Exact FrontierDelta

Prior state unknown→disproved

Scope and record

Occurred: Sep 23, 2026

Delta type: SOURCE CLAIM

Assumptions: VibeMathed verification: unreviewed. Publication: announcement. AI contribution: ai-discovered. VibeMathed editorial classifications, scores, notes, relations, and dataset structure are CC BY 4.0. Source statements and linked content retain their own rights.

Canonical aliases: Turyn's conjecture that binary merit factors are bounded · Turyn's merit factor conjecture

Confidence: Not scored

Registry verification: unreviewed · announcement · candidate

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Attribution

VibeMathed
registry · event recorded by

Unreleased internal OpenAI model
model · ai model contributor · OpenAI

Evidence vector

SourceReplayReproducedFormal proofStatement auditExternal checkExpert reviewPeer review

Named evidence

VibeMathed record: Turyn's conjecture that binary merit factors are bounded

specialist registry · passed

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 $N$ tends to infinity through all integer lengths, deduced from flatness in every finite $L^p$ 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 $\int |\,|P_N|/\sqrt N-1|^p\to0$ for every finite $p>0$ . This is narrower than the headline: merit factors are not defined in the Lean, and the passage from $p=4$ 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.

Source ↗

Turyn's conjecture that binary merit factors are bounded

witness · pending

Primary source named by the VibeMathed record. Presence in this import is not an independent mathematical verification.

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Lean proof (OAI.AsymptoticallyMinimalLittlewood.main)

lean artifact · pending

lean-proof link supplied by the VibeMathed registry.

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Comparator statement: AsymptoticallyMinimalLittlewood.lean

lean artifact · pending

lean-proof link supplied by the VibeMathed registry.

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Lean proof of finite-exponent flatness (OAI.AsymptoticallyMinimalLittlewoodFiniteFlatness.main)

lean artifact · pending

lean-proof link supplied by the VibeMathed registry.

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Comparator statement: LittlewoodFiniteFlatness.lean

lean artifact · pending

lean-proof link supplied by the VibeMathed registry.

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Companion: Ultraflat real Littlewood polynomials (two-sided, Oct 5)

paper · pending

paper link supplied by the VibeMathed registry.

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Companion: Nearly minimal maxima and positive minima of Littlewood polynomials

paper · pending

paper link supplied by the VibeMathed registry.

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Artifacts and verifiers

Lean proof (OAI.AsymptoticallyMinimalLittlewood.main)

lean artifact · pending

Artifact ↗

Comparator statement: AsymptoticallyMinimalLittlewood.lean

lean artifact · pending

Artifact ↗

Lean proof of finite-exponent flatness (OAI.AsymptoticallyMinimalLittlewoodFiniteFlatness.main)

lean artifact · pending

Artifact ↗

Comparator statement: LittlewoodFiniteFlatness.lean

lean artifact · pending

Artifact ↗

Compute record

No linked compute attempts recorded.

Lineage and corrections

Turyn's conjecture that binary merit factors are bounded parent of this event

Comparator statement: LittlewoodFiniteFlatness.lean evidence for this event

This event attributed to Unreleased internal OpenAI model

Companion: Ultraflat real Littlewood polynomials (two-sided, Oct 5) evidence for this event

VibeMathed record: Turyn's conjecture that binary merit factors are bounded evidence for this event

Lean proof (OAI.AsymptoticallyMinimalLittlewood.main) evidence for this event

Lean proof of finite-exponent flatness (OAI.AsymptoticallyMinimalLittlewoodFiniteFlatness.main) evidence for this event

Comparator statement: AsymptoticallyMinimalLittlewood.lean evidence for this event

Companion: Nearly minimal maxima and positive minima of Littlewood polynomials evidence for this event

Turyn's conjecture that binary merit factors are bounded evidence for this event

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