Sampling Decisions: Exact Path-Space Control for Physics-Informed Generative Sampling

Scientific generative models must turn tractable local decisions into globally correlated samples that respect physical constraints. We introduce Sampling Decisions, a finite-horizon framework in which a structured object is assembled on a growing state graph and corrected globally by an exact path-space control law. For a prescribed Gibbs target, the corrected law is the unique relative-entropy projection of a sequential prior and is realized by a Doob h-transform with a linear backward desirability recursion. The same object admits equivalent interpretations as a KL-optimal controller, a one-sided Schrodinger transport, and an ideal value or flow function for autoregressive and GFlowNet-type generation. A route-resolved formulation yields a finite-particle algorithm, whose transition kernels and terminal law converge as the path budget grows. For binary graphical models, we prove a structural cancellation theorem: every fixed singleton-product prior disappears from the population correction, so improved one-point marginals do not alter the exact generative dynamics. The relevant information is conditional and prefix dependent. Statistical physics supplies this structure through a Local-Boltzmann prior that absorbs interactions as spins are revealed, while optimal path-space control supplies the missing look-ahead field from the unrevealed subgraph. On exactly enumerable Ising grids, this physics-informed proposal increases effective sample size by factors of about ten to nearly one thousand relative to product proposals and reaches the exact-target reference band at the tested budgets. On a larger 10*10 grid, beyond exact enumeration, the same hierarchy persists: Local-Boltzmann guidance avoids the severe weight collapse of product proposals and approaches a long-run MCMC baseline on the reported diagnostics.