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Hellinger contraction with arbitrary Boolean output bias

OpenAI · September 24, 2026

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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.

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Collected automatically by AI tools (Claude Opus 5.5 and GPT-6 Astra, 2026-10-21) and not checked by a person. May be wrong or incomplete. Report an error.

Follow-up works

  • V. K. Ky, The Hellinger Conjecture for Antipodal Boolean Functions (arXiv:2610.10618, 7 Oct 2026). Gives a spectral proof of the balanced Hellinger inequality for antipodal (sign-reversing) Boolean functions and extends it to arbitrary independent binary-input channels; Section 1.2 cites OpenAI's family-119 entry as an announcement of the Hellinger inequality for arbitrary output bias, without verifying it.

Formalization

Digests and expositions

Elsewhere

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Cite

@misc{OAI:Hellinger-contraction-with-arbitrary-Boolean-output-bias-September-24-2026,
  author = {{OpenAI}},
  title = {{Hellinger contraction with arbitrary Boolean output bias}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Hellinger-contraction-with-arbitrary-Boolean-output-bias-September-24-2026/main.pdf}{OAI:Hellinger-contraction-with-arbitrary-Boolean-output-bias-September-24-2026}},
  year = {2026}
}

From OpenAI's README for this paper:

Verification

The exact arithmetic checks require Python 3.9 or later with SymPy 1.14.0 and a C++ compiler supporting signed __int128, such as GCC or Clang. Run with assertions enabled; do not use Python's -O option or set PYTHONOPTIMIZE. From this paper directory, run:

verification_output="$(mktemp -d)"
python3 verification/verify_all.py --output-dir "$verification_output"

The output directory must be new or empty. Success is exit code zero, a final PASS, and "status": "pass" in verification.json under that output directory. The runner records source hashes, commands, and separate logs. It does not install dependencies or use network access. Use --cxx clang++ to select Clang when needed.

Lean formalization

OpenAI provides Lean files for this family: lean/docs/119.md. The Lean statements have not been checked by people against the paper's claims.

Family 119: The Courtade–Kumar and Hellinger conjectures

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.

Other papers in this family: