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The Hellinger conjecture for Boolean functions under product noise

Let XX be uniform on {−1,1}n\{-1,1\}^n, let TρT_\rho be the noise operator with correlation ρ∈[−1,1]\rho\in[-1,1], and write H(t)=1−t2H(t)=\sqrt{1-t^2}. Anantharam, Bogdanov, Chakrabarti, Jayram and Nair (2017) conjectured that for every Boolean ff with mean mm, H(m)−E H(Tρf)≤1−1−ρ2H(m)-\mathbb E\,H(T_\rho f)\le1-\sqrt{1-\rho^2}, with equality for dictators: a single coordinate maximizes the loss of Hellinger affinity. They showed it implies the Courtade-Kumar conjecture. Known results included a Gaussian-isoperimetric lower bound (Chen-Nair) and the sharp balanced low-noise consequence (Durcik-Ivanisvili-Roos-Xie). Does the Hellinger inequality hold for every Boolean function, of any bias, and every ρ\rho?

Result
Proved(see note)
Status
Candidate (review pending)
AI contribution
AI-discovered
Method
Argument
Field
Boolean functions; information theory
Posed by
Venkat Anantharam, Andrej Bogdanov, Amit Chakrabarti, T. S. Jayram and Chandra Nair (ITA Workshop manuscript, February 2017)
Year posed
2017
Years open
9y
Solved
2026-09-24
Model
Unreleased internal OpenAI model
Vendor
OpenAI
Collaborators
—
Verification
Unreviewed
Publication
Announced
Collection
OpenAI math release (October 2026), version adc7f12
Significance
25 / 100
Disclosed cost
—
Wikipedia
No dedicated article

What was actually shown

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.

What the AI did

The release README says the manuscripts were produced by an unreleased internal OpenAI model, the vast majority by one fixed procedure using on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier results produced by the models. Its named exceptions to that procedure (the zeta zero-free region work, whose Re(s) > 11/12 write-up was human edited, and the Hodge conjecture for CM abelian varieties) do not concern this family. The manuscript is credited to OpenAI with no human author named. Its README and source-lineage file say the proof combines analytic sections with six exact-arithmetic programs and a runner, all supplied with the manuscript, and that it has no dependency on other release articles.

Verification

No independent mathematician has checked this yet. Theorem 1.1 was read against the conjecture of Anantharam et al.: for every n≥1n\ge1, Boolean ff with mean mm and ρ∈[−1,1]\rho\in[-1,1], H(m)−EH(Tρf)≤1−H(ρ)H(m)-\mathbb E H(T_\rho f)\le1-H(\rho), with signed coordinates attaining equality; it is the full unbalanced form. This manuscript has no Lean formalisation (the family's Lean covers the companion's Courtade-Kumar theorem). The proof relies on finite exact-arithmetic certificates; the README gives a verification runner (Python with SymPy plus C++), which was not run here. The release README warns that some unformalised results could have issues.

Sources

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