A Proof of the Most Informative Boolean Function Conjecture

2026-09-21

AI Summary

本文通过计算机辅助证明了Courtade-Kumar猜想，即I(g(X);Y)≤1-H₂(p)，采用微分方程方法和布尔噪声半群的差异化来解决最信息量布尔函数问题。

Abstract

Let $X$ be uniform on $\{-1,1\}^n$, let $Y$ be obtained by passing its coordinates independently through a binary symmetric channel with crossover probability $p$, and let $g:\{-1,1\}^n\to\{0,1\}$ be a Boolean function. We give a computer-assisted proof of the Courtade--Kumar conjecture $I(g(X);Y)\le1-H_2(p)$, where $H_2$ is binary entropy, with equality attained by dictator functions. The present work builds on the differential-equation method, itself a limiting form of the auxiliary-receiver approach in network information theory using a continuum of degraded receivers. The proof proceeds from a local inequality to a dimension-independent bound on entropy production. Differentiation along the Boolean noise semigroup expresses entropy production as an average of edge costs. The key estimate is therefore an unrestricted Bellman inequality with two mean constraints and two entropy constraints, allowing arbitrary couplings of the edge variables. This paper and its supplement provide the proofs and computational verification records. The document is lengthy because it is designed to be entirely self-contained, deriving all proofs from first principles and reproducing the proofs of cited results. The supplementary material supporting the computer-assisted parts of the proof are available online.

Problem

• Boolean function
• mutual information
• binary symmetric channel
• entropy

Innovation

• computer-assisted proof
• Courtade--Kumar conjecture
• entropy production bound
• Boolean noise semigroup
• unrestricted Bellman inequality