{"id":"openai-math/Hellinger-contraction-with-arbitrary-Boolean-output-bias","pdf":"https://github.com/openai/math/blob/main/preprints/Hellinger-contraction-with-arbitrary-Boolean-output-bias-September-24-2026/main.pdf","url":"https://github.com/openai/math/tree/main/preprints/Hellinger-contraction-with-arbitrary-Boolean-output-bias-September-24-2026","tags":["information-theory","boolean-functions","hellinger-distance","harmonic-analysis"],"score":0,"title":"Hellinger contraction with arbitrary Boolean output bias","domain":"Theoretical computer science","family":"119","origin":"ai-generated","source":"openai-math","authors":["OpenAI"],"reviews":0,"subject":"cs.CC","summary":"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.","category":"research","comments":0,"disputed":false,"division":"cs","mentions":0,"discussed":0,"published":"2026-09-24","reactions":{},"formalized":false,"family_title":"The Courtade–Kumar and Hellinger conjectures"}