Standard formulations of supervenience treat what is fixed by the base as a point-valued higher-level property. This presupposition becomes strained in scientific contexts in which laws constrain outcomes by fixing stable chance profiles—conditional probability measures—rather than single results. I propose stochastic supervenience as a conservative extension of deterministic supervenience: relative to a background set of laws \(\:L\) , base states determine a law-like kernel from \(\:B\) to \(\:\varDelta\:\left(A\right)\) . To distinguish this framework from mere probabilistic correlation, I impose constraints ensuring that the kernel is invariant across \(\:L\) -worlds, non-degenerate, constant on \(\:L\) -indiscernible base states, and subject to anti-trivialization conditions. Crucially, these conditions block “mere relabeling” reconstructions that render the dependence symmetric, and exclude treating non-Dirac structure as ignorance over \(\:L\) -silent refinements. Deterministic supervenience is recovered as the Dirac boundary case, and I show that the dependence structure is preserved under coarse-graining when non-degeneracy is maintained. To connect the metaphysical proposal to scientific practice without reducing it to model-fitting, I introduce a compact diagnostic toolkit—normalized mutual information, a divergence spectrum with permutation baselines, body-tail comparisons for graded distributional multiple realization, and effective information under uniform interventions—to characterize the strength, structure, and intervention-salience of distributional dependence. The resulting framework preserves physicalist priority while making room for the law-governed uncertainty ubiquitous in the special sciences.
