The Courtade-Kumar conjecture: dictators are the most informative Boolean functions

Claims Theorem 1.1: for every $n$, every soft channel $u:\{\pm1\}^n\to[-1,1]$ and $\rho\in[-1,1]$, $I(T_\rho u)\le\psi(|\rho|\psi^{-1}(I(u)))$, with soft coordinates attaining equality; Corollary 1.2 gives $I(f(X);Y)\le1-h_2(\varepsilon)$ bits for Boolean $f$, plus a strictly better bound when $f$ is biased; Theorem 1.3 is a mean-dependent entropy-production inequality. It does NOT treat non-uniform inputs, asymmetric noise, or multi-bit summaries. Independent proofs of the Boolean inequality were announced in September 2026 by Chen-Gohari-Javanmard-Lin-Mirrokni-Nair-Woodruff, Ky-Tran and Mahdavifar-Beirami.

Claims Theorem 1.1: for every $n$, every soft channel $u:\{\pm1\}^n\to[-1,1]$ and $\rho\in[-1,1]$, $I(T_\rho u)\le\psi(|\rho|\psi^{-1}(I(u)))$, with soft coordinates attaining equality; Corollary 1.2 gives $I(f(X);Y)\le1-h_2(\varepsilon)$ bits for Boolean $f$, plus a strictly better bound when $f$ is biased; Theorem 1.3 is a mean-dependent entropy-production inequality. It does NOT treat non-uniform inputs, asymmetric noise, or multi-bit summaries. Independent proofs of the Boolean inequality were announced in September 2026 by Chen-Gohari-Javanmard-Lin-Mirrokni-Nair-Woodruff, Ky-Tran and Mahdavifar-Beirami.