Hellinger contraction inequality for Boolean functions
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For a Boolean function f on the uniform discrete cube and product-noise correlation ρ∈[−1,1], does H(Ef)−E H(Tρf)≤1−H(ρ) hold, where H(t)=√(1−t²)? The question includes biased outputs; signed coordinates are candidate equality cases.
Why it matters and what counts as progress
Why it matters: The sharp Hellinger inequality would imply an information bound while handling biases excluded by earlier restricted results.
What counts as progress: Prove the sharp inequality without a bias restriction, characterize equality or construct a violating Boolean function.
How the problem developed
- 12 Feb 2017 · Conjecture [1]
Venkat Anantharam, Andrej Bogdanov, Amit Chakrabarti, T. S. Jayram and Chandra Nair Formulated the Hellinger-distance dictator conjecture: a coordinate function maximizes the associated Hellinger Φ-entropy among Boolean functions on the uniform cube.
- 14 Feb 2025 · Partial method [2]
Zijie Chen, Amin Gohari and Chandra Nair Developed a differential-equation approach and a related Hellinger inequality, but left explicit scalar inequalities unresolved.
- 24 Sept 2026 · Proof claim [3]
OpenAI A manuscript claims the Hellinger inequality for every Boolean function on the uniform cube and every noise parameter, including arbitrary output bias, with equality at signed coordinates.
AI contribution
Hellinger contraction with arbitrary Boolean output bias
For every Boolean function on the uniform discrete cube, including biased outputs, and every product-noise correlation ρ∈[−1,1], the manuscript claims H(Ef)−E H(Tρf)≤1−H(ρ), where H(t)=sqrt(1−t²). Signed coordinates attain equality.
Sources
- Anantharam, Bogdanov, Chakrabarti, Jayram and Nair, A conjecture regarding optimality of the dictator function under Hellinger distance
- Chen, Gohari and Nair, A Differential Equation Approach to the Most-Informative Boolean Function Conjecture
- Hellinger contraction manuscript
- Hellinger contraction with arbitrary Boolean output bias
- Certificate-run instructions
- Original Hellinger conjecture
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