The Courtade–Kumar and Hellinger conjectures

What OpenAI claims

Proves the Courtade–Kumar conjecture: among Boolean functions of independent uniform bits, a single coordinate retains the most mutual information after independent bit-flip noise. A stronger theorem treats randomized binary summaries at fixed initial information. The Hellinger conjecture is also proved for every Boolean output bias and noise correlation. (Lean)

Papers (2)

1. Sharp binary-information contraction on the discrete cube

   24 September 2026 · Preprint, not peer reviewed · PDF

   We prove sharp contraction of the information carried by a binary channel under independent symmetric noise on a uniform discrete cube. At fixed initial information, a noisy coordinate retains the most information. The Boolean specialization resolves the Courtade–Kumar conjecture and gives an output-entropy refinement. We also establish a stronger mean-dependent entropy-production bound. The proof combines an explicit three-point optimizer for the local joining problem, two entropy capacities, and a common-output thinning inequality, followed by dimension induction and integration along the noise semigroup.

2. Hellinger contraction with arbitrary Boolean output bias

   24 September 2026 · Preprint, not peer reviewed · PDF

   We prove the Hellinger conjecture for Boolean functions on the uniform discrete cube, with arbitrary output bias. For a Boolean function of mean m and every ρ ∈ [-1,1], the loss √(1-m²)-𝔼√(1-(T_ρ f)²) is at most 1-√(1-ρ²), with equality for signed coordinates. The proof combines asymmetric dimension induction, a calibrated noise-semigroup energy estimate, and finite exact arithmetic certificates. The Hellinger inequality also yields the Courtade–Kumar information bound.

Status

Preprint. The papers have not been independently reviewed. Treat the result as OpenAI's claim until other mathematicians have checked it. If a video exists, it illustrates the statement; it is not the proof.