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OpenAI · 119-01 · 2026-09-24 · Theoretical computer science

Sharp binary-information contraction on the discrete cube

preprints/Sharp-binary-information-contraction-on-the-discrete-cube-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 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.

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.

This preprint is named in the Lean scope note for 119: The Courtade–Kumar and Hellinger conjectures. The authors are OpenAI. This companion does not verify the formalization. It is not a premise. The OAI library is not imported. No compilation is asserted. O.1 stays NOT_CLOSED.

CourtadeKumar.lean · CourtadeKumar.lean

Statement file

SoftChannel204.lean · SoftChannel204.lean
Statement file
  • 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