OpenAI · 119-02 · 2026-09-24 · Theoretical computer science
Hellinger contraction with arbitrary Boolean output bias
preprints/Hellinger-contraction-with-arbitrary-Boolean-output-bias-September-24-2026
Family 119. The Courtade–Kumar and Hellinger conjectures.
The authors are OpenAI. This companion does not verify the proof and does not adopt it as a theorem of Ngo Cheung’s series. It is not a premise. O.1 stays NOT_CLOSED. Anti-de Sitter results are not a de Sitter state, not Em02-E’s relational lapse, not XVI-P’s QdS scheme, and not XVI-Q’s sigma-image record. No quantum coefficient is admitted. No compilation is asserted.
Read against the standing problems
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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.
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 , the loss is at most , 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.
Preprint source and Lean scope
The abstract above is the catalogue abstract. The sections below are the full preprint source, 0 sections. This page is not only the abstract. The manuscript file is loading.
A Lean scope note is stored for family 119, but it does not name this preprint. It names 119-01. This page still has the full preprint source. No compilation is asserted. O.1 stays NOT_CLOSED.
Page curve
A finite Hellinger certificate is not a Page curve
Family 119 records soft-information contraction and arbitrary-bias Hellinger contraction. Oai119-A records a certificate for every Boolean function on the two-dimensional cube at correlation 1/2 and does not verify 119-01 or 119-02. Equality holds exactly for the four signed coordinates. CourtadeKumar.lean and SoftChannel204.lean end in sorry and are not this certificate. A classical Hellinger gap is not a Page curve. A Boolean noise kernel is not gravity-generated entanglement and not an exclusion of classical gravity. The authors of the external preprints are OpenAI. The companion does not verify those proofs.
Page curveOai119-A finite Hellinger certificateVIII-B record-only recoveryEn07-A readout-and-drop