The Hellinger conjecture of Anantharam et al. for the binary symmetric channel would imply the Courtade–Kumar conjecture that dictator functions maximize mutual information on noisy inputs. Both remain open beyond the high-noise regime.

A weak form of the conjecture is proved, in which the output Y is replaced by a one-bit statistic g(Y). Via a data processing inequality for the reverse hypercontractivity parameter, the problem reduces to an explicit inequality F(s,c,d) ≥ 0 in three real parameters. That inequality is proved using an integral representation of a normalized KL divergence, explicit polynomial approximations and computer-assisted positivity checks. The results are formally verified in Lean 4.

Dictator functions maximize Hellinger Φ-entropy among all Boolean functions of the input and all one-bit statistics of the output. Equality in the three-point inequality holds exactly when s ∈ {0,1}, or c+d=1, or s=1/2 and c=d.