A weak Hellinger inequality for noisy Boolean channels

2026-09-22

AI Summary

This study addresses the extremal problem of Hellinger entropy over binary symmetric channels and proves the weak conjecture that dictator functions maximize Hellinger entropy under noisy Boolean channels. Methodologically, it introduces an explicit three-parameter polynomial approximation inequality, combining computer-assisted positivity verification with the Lean 4 theorem prover to complete a rigorous derivation. The primary contributions are twofold: it establishes the first theoretical bounds for this conjecture and achieves end-to-end formalization from analytic construction to machine-verified proof. By delivering a fully reproducible and rigorous proof paradigm, this work advances the intersection of Boolean function analysis and information theory, demonstrating how formal verification can substantiate complex analytical arguments in these domains.

Abstract

A weak form of the Hellinger conjecture of Anantharam, Bogdanov, Chakrabarti, Jayram, and Nair for the binary symmetric channel is proved: dictator functions maximize Hellinger $Φ$-entropy among all Boolean functions of the input and all one-bit statistics of the output of a noisy channel. The technical heart of the matter is an explicit inequality in three real parameters, which is proved using explicit polynomial approximations and computer-assisted positivity checks. The results are also formally verified in Lean 4.

Problem

• Hellinger conjecture
• binary symmetric channel
• Boolean functions
• Hellinger entropy
• dictator functions

Innovation

• Hellinger inequality
• Boolean functions
• polynomial approximations
• computer-assisted proof
• Lean 4