Gaussian Convexity Principles for Sharp Moderate Deviations of Gaussian Maxima and Critical SK Free Energy Variance

What the math gives to ML

The paper gives a correlation-sensitive moderate-deviation law for the maximum of a centered Gaussian vector: when the expected maximum is already a nontrivial fraction of the independent-Gaussian scale, upward excursions have a sharper exponent than the generic Borell bound. This can transfer to attention logits and MoE router logits, where correlated scores may occasionally produce one-token or one-expert domination despite marginal normalization. The practical adaptation is an extreme-logit calibration rule that estimates the expected maximum and penalizes or rescales logits above a mathematically motivated threshold. The guarantee is exact only for approximately Gaussian logits with bounded marginal variance, so the proposed experiment should test both Gaussian synthetic logits and trained neural representations.