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.