The Boltzmann structure of sampling: Intrinsic $p$-value and its emergent closed-form expression

What the math gives to ML

The paper turns constrained categorical sampling into a likelihood-ratio-like statistic obtained from nested information projections onto linear moment families. Its transferable asset is the combination of exponential-family projections, KL geometry, and dimension-aware chi-square calibration for deviations in multiple learned features. In neural training, this can replace ad hoc sums of squared moment errors with a statistically normalized constraint loss whose scale accounts for feature covariance and sample size. The most direct experiment is a constraint-aware generative or conditional model in which structural moments are enforced while auxiliary moments are monitored or penalized through the projected KL statistic.