The sharp diagonal spectral correlation inequality on the discrete cube

What the math gives to ML

The paper supplies a sharp, computable certificate that monotone Boolean predictors cannot have small covariance while simultaneously colliding in the same nonconstant Fourier modes. The transferable asset is the degree-weighted spectral collision term, which measures whether two outputs rely on the same high-order input interactions, together with the optimal constant and equality cases. A practical neural-network adaptation is a Fourier-domain regularizer or constraint for pairs of monotone heads on binary or discretized inputs, useful for controlling shared interaction structure and detecting violations of intended monotonicity. The guarantee is exact only for increasing Boolean functions, so continuous neural outputs should use it as a surrogate or apply it after Bernoulli sampling or thresholding.