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The Hellinger conjecture for Boolean functions under product noise

Claims Theorem 1.1: for every n≥1n\ge1, every f:{−1,1}n→{−1,1}f:\{-1,1\}^n\to\{-1,1\} with m=Efm=\mathbb E f and every ρ∈[−1,1]\rho\in[-1,1], 1−m2−E1−(Tρf)2≤1−1−ρ2\sqrt{1-m^2}-\mathbb E\sqrt{1-(T_\rho f)^2}\le1-\sqrt{1-\rho^2}, with equality for signed coordinates. By the Anantharam et al. implication it reproves the Courtade-Kumar bound (Corollary 8.1). It does NOT treat non-Boolean (soft) functions or non-uniform inputs, and its proof depends on computer-checked arithmetic certificates.

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Prior state unknown→proved

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Occurred: Sep 24, 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: The Hellinger conjecture for Boolean functions under product noise · Hellinger conjecture

Confidence: Not scored

Registry verification: unreviewed · announcement · candidate

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Unreleased internal OpenAI model
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The Hellinger conjecture for Boolean functions under product noise parent of this event

This event attributed to Unreleased internal OpenAI model

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

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

Claims Theorem 1.1: for every n≥1n\ge1, every f:{−1,1}n→{−1,1}f:\{-1,1\}^n\to\{-1,1\} with m=Efm=\mathbb E f and every ρ∈[−1,1]\rho\in[-1,1], 1−m2−E1−(Tρf)2≤1−1−ρ2\sqrt{1-m^2}-\mathbb E\sqrt{1-(T_\rho f)^2}\le1-\sqrt{1-\rho^2}, with equality for signed coordinates. By the Anantharam et al. implication it reproves the Courtade-Kumar bound (Corollary 8.1). It does NOT treat non-Boolean (soft) functions or non-uniform inputs, and its proof depends on computer-checked arithmetic certificates. parent of this event

Anantharam et al. (2017), A conjecture regarding optimality of the dictator function under Hellinger distance evidence for this event

VibeMathed record: The Hellinger conjecture for Boolean functions under product noise evidence for this event

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