Source authenticated

Sinai's conjecture: positive metric entropy of the standard map for a positive-measure set of parameters

Theorem 1.1: there is $k_0>0$ with $h_m(f_k)>0$ for every $k\ge k_0$ ; equivalently, by Pesin's formula, the top Lyapunov exponent is positive on a set of positive area. Corollary 1.2: a positive-area ergodic component with nonzero exponents that splits into finitely many cyclic Bernoulli pieces. It does NOT claim ergodicity of area, positive exponents almost everywhere, a uniform lower bound on the entropy, or any explicit $k_0$ , and says nothing about small or moderate $k$ .

Exact FrontierDelta

Prior state unknown→proved

Scope and record

Occurred: Sep 23, 2026

Delta type: SOURCE CLAIM

Assumptions: VibeMathed verification: lean-checked. 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: Sinai's conjecture: positive metric entropy of the standard map for a positive-measure set of parameters · Sinai's standard map entropy conjecture

Confidence: Not scored

Registry verification: lean checked · 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: Sinai's conjecture: positive metric entropy of the standard map for a positive-measure set of parameters

specialist registry · passed

No independent mathematician has checked this yet. Checked here: Theorem 1.1 and Corollary 1.2 of the TeX source read against Sinai's conjecture as cited; positivity for every $k\ge k_0$ is stronger than the positive-measure parameter set the conjecture asks for. The proof was not refereed. Lean: three Comparator challenges, StandardMapEntropy (OAI.StandardMapEntropy.main_entropy), StandardMapLyapunov and StandardMapComponents, with solution modules present at the pinned commit; none is in the formalization catalogue. The StandardMapEntropy statement was read here: it defines the same map and normalized area on the torus and asserts some $k_0>0$ with positive metric entropy for all $k\ge k_0$ , which is the headline. Metric entropy is defined in the challenge itself (finite measurable partitions, block entropies), since Mathlib has none, so that definition is the prover's own. Permitted axioms are propext, Quot.sound and Classical.choice. Not rebuilt here.

Source ↗

Sinai's conjecture: positive metric entropy of the standard map for a positive-measure set of parameters

witness · pending

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

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Lean Comparator statement StandardMapEntropy (not in formalization.yaml main results)

lean artifact · pending

lean-proof link supplied by the VibeMathed registry.

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Lean Comparator statement StandardMapLyapunov

lean artifact · pending

lean-proof link supplied by the VibeMathed registry.

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Lean Comparator statement StandardMapComponents

lean artifact · pending

lean-proof link supplied by the VibeMathed registry.

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Lean scope note for family 146

witness · pending

other link supplied by the VibeMathed registry.

Source ↗

Berger and Turaev 2019, Adv. Math. 349 (Conjecture 0.4)

specialist registry · pending

problem-record link supplied by the VibeMathed registry.

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Artifacts and verifiers

Lean Comparator statement StandardMapEntropy (not in formalization.yaml main results)

lean artifact · pending

Artifact ↗

Lean Comparator statement StandardMapLyapunov

lean artifact · pending

Artifact ↗

Lean Comparator statement StandardMapComponents

lean artifact · pending

Artifact ↗

Compute record

No linked compute attempts recorded.

Lineage and corrections

VibeMathed record: Sinai's conjecture: positive metric entropy of the standard map for a positive-measure set of parameters evidence for this event

This event attributed to Unreleased internal OpenAI model

Lean Comparator statement StandardMapComponents evidence for this event

Sinai's conjecture: positive metric entropy of the standard map for a positive-measure set of parameters evidence for this event

Lean scope note for family 146 evidence for this event

Sinai's conjecture: positive metric entropy of the standard map for a positive-measure set of parameters parent of this event

Lean Comparator statement StandardMapEntropy (not in formalization.yaml main results) evidence for this event

Lean Comparator statement StandardMapLyapunov evidence for this event

Berger and Turaev 2019, Adv. Math. 349 (Conjecture 0.4) evidence for this event

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