Discoveries / probability-statistics

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The Courtade-Kumar conjecture: dictators are the most informative Boolean functions

Claims Theorem 1.1: for every $n$ , every soft channel $u:\{\pm1\}^n\to[-1,1]$ and $\rho\in[-1,1]$ , $I(T_\rho u)\le\psi(|\rho|\psi^{-1}(I(u)))$ , with soft coordinates attaining equality; Corollary 1.2 gives $I(f(X);Y)\le1-h_2(\varepsilon)$ bits for Boolean $f$ , plus a strictly better bound when $f$ is biased; Theorem 1.3 is a mean-dependent entropy-production inequality. It does NOT treat non-uniform inputs, asymmetric noise, or multi-bit summaries. Independent proofs of the Boolean inequality were announced in September 2026 by Chen-Gohari-Javanmard-Lin-Mirrokni-Nair-Woodruff, Ky-Tran and Mahdavifar-Beirami.

Exact FrontierDelta

Prior state unknown→proved

Scope and record

Occurred: Sep 24, 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: The Courtade-Kumar conjecture: dictators are the most informative Boolean functions · Courtade-Kumar conjecture

Confidence: Not scored

Registry verification: lean checked · announcement · candidate

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Unreleased internal OpenAI model
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Named evidence

VibeMathed record: The Courtade-Kumar conjecture: dictators are the most informative Boolean functions

specialist registry · passed

No independent mathematician has checked this yet. Theorem 1.1 and Corollary 1.2 of the principal manuscript were read against the conjecture: the Boolean specialization gives $I_2(f(X);Y)\le1-h_2(\varepsilon)$ , with dictators attaining it. formalization.yaml lists ComparatorChallenges/CourtadeKumar.json, declaration OAI.LeanBlast.CourtadeKumar.courtadeKumarAndAttainment, file OAI/InformationTheory/BooleanNoise/Main.lean. The statement was read here: for $n\ge1$ , $0\le\varepsilon\le1/2$ and every $f$ on the cube, the mutual information (defined from the bit-flip kernel, in bits) is at most $1-h(\varepsilon)$ , and dictators and anti-dictators attain it. That is exactly the conjecture. A second challenge, SoftChannel204.json (OAI.SoftChannel204.fullMain, solution module present, not in formalization.yaml, found via lean/docs/119.md), states the soft-channel Theorem 1.1, the bias-refined bound and the entropy-production inequality; read, not rebuilt. Permitted axioms are propext, Quot.sound and Classical.choice. Not rebuilt here. The manuscript itself cites four other recent preprints announcing proofs, one dated 21 September 2026, before this one.

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The Courtade-Kumar conjecture: dictators are the most informative Boolean functions

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

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Companion: Hellinger contraction with arbitrary Boolean output bias

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paper link supplied by the VibeMathed registry.

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Lean: Courtade-Kumar inequality and attainment (formalization.yaml file)

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lean-proof link supplied by the VibeMathed registry.

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Lean comparator statement: Courtade-Kumar inequality and attainment

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lean-proof link supplied by the VibeMathed registry.

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Lean comparator statement: soft-channel contraction and entropy production

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lean-proof link supplied by the VibeMathed registry.

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Courtade and Kumar (2014), Which Boolean functions maximize mutual information on noisy inputs?

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problem-record link supplied by the VibeMathed registry.

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Artifacts and verifiers

Lean: Courtade-Kumar inequality and attainment (formalization.yaml file)

lean artifact · pending

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Lean comparator statement: Courtade-Kumar inequality and attainment

lean artifact · pending

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Lean comparator statement: soft-channel contraction and entropy production

lean artifact · pending

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Compute record

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Lineage and corrections

Courtade and Kumar (2014), Which Boolean functions maximize mutual information on noisy inputs? evidence for this event

This event attributed to Unreleased internal OpenAI model

Companion: Hellinger contraction with arbitrary Boolean output bias evidence for this event

Lean comparator statement: soft-channel contraction and entropy production evidence for this event

The Courtade-Kumar conjecture: dictators are the most informative Boolean functions parent of this event

The Courtade-Kumar conjecture: dictators are the most informative Boolean functions evidence for this event

Lean: Courtade-Kumar inequality and attainment (formalization.yaml file) evidence for this event

Lean comparator statement: Courtade-Kumar inequality and attainment evidence for this event

VibeMathed record: The Courtade-Kumar conjecture: dictators are the most informative Boolean functions evidence for this event

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