Manuscript OpenAI@2026-10-07 · OpenAI.tex · remark · 6a931f8f
Remark: 119-02. Hellinger contraction with arbitrary Boolean output bias
Manuscript map · lines 4785–4794
OpenAI@2026-10-07#rem:oai-p-119-02
Authors: OpenAI. Family 119.
Title: Hellinger contraction with arbitrary Boolean output bias.
Abstract: 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 ${}rho{}in[-1,1]$, the loss ${}sqrt{1-m{}2}-{}mathbb E{}sqrt{1-(T_{}rho f){}2}$ is at most $1-{}sqrt{1-{}rho{}2}$, 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.
This abstract is a catalogue record. It is not a verification and not a premise. It does not close O.1. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted.
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