Stochastic Domination of Gaussian Maxima: A Resolution of the Weak Simplex Conjecture

What the math gives to ML

The paper proves an exact extremal theorem for Gaussian score maxima: under a centered covariance condition, independent Gaussian coordinates stochastically maximize the maximum coordinate, with strictness away from independence. Combined with the Gaussian maximum-likelihood decoding identity, this establishes finite-SNR optimality of regular-simplex class geometry rather than only asymptotic or pairwise-covariance optimality. A transferable neural-network construction is a simplex-constrained classification head trained and evaluated under Gaussian representation noise, with the theorem predicting improved robustness at fixed class-signal energy. The result is most useful as a principled alternative to unconstrained classifier weights when deployment noise, quantization, or stochastic feature perturbations matter.