A Sharp Fisher-Information Inequality for Boolean Pairs: Proof of the Amari--Kobayashi Conjecture

Consider two terminals that observe correlated uniform binary vectors and separately compress their observations to one bit. We prove that, for every blocklength and every correlation parameter in $(-1,1)$, the Fisher information retained by the resulting pair of bits is maximized by matching parity functions of an appropriate degree, thereby resolving a conjecture of Amari and Kobayashi. The proof combines a Fourier-based bound on the derivative of the normalized covariance with a scalar inequality for signed polynomials subject to a global range constraint.