The balanced Hellinger conjecture compares the overlap of two output laws induced by a Boolean decision under binary symmetric noise. We prove that a single input coordinate minimizes this overlap among antipodal decisions, whose values reverse when all input signs are reversed, throughout the noise interval. The proof uses a normalized nuclear norm of the cross-class block of the Gram matrix of square-root channel densities. Multiplication by an antipodal decision exchanges the even and odd Fourier spaces preserved by the Gram matrix. A coordinate exchange attains the minimum cost of matching their spectra. A matrix contraction extends the sharp bound to arbitrary independent binary-input channels. Fixed coordinate flips and randomized decisions follow from the same argument and concavity. These extensions give bounds for information and testing, including linear Gaussian observations, and a weighted edge-boundary inequality.

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