Hellinger contraction with arbitrary Boolean output bias
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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