Equivalent Optima, Different Policies: Certified Distribution-Preserving Reformulations for AI Decision Systems

A representation change can leave an optimization problem’s feasible decisions and costs intact while altering how a stochastic decision layer uses it. We develop separate tests for the decoded probability law and the complete-policy relative-entropy objective of finite log-linear policies with deterministic decoding and compatible positive reference laws. Equality of the weighted residual-feature spectra across decoder fibres characterizes preservation of the decoded law for every natural parameter. Given that equality, a common spectrum concentrated at one point also preserves the relative-entropy objective; a spectrum with several atoms adds a nonnegative conditional-divergence penalty. The penalties add across compatible reformulation stages, and an exact correction recovers the physical objective. For compact convex policy classes, preserving optimal values for every cost is stronger than preserving optimal physical policies. An exact rational checker and a bounded-integer compiler implement the certificates. In a designed six-period scheduling study with 20 cost vectors, four tied auxiliary bits per period give a median physical-policy total variation of 0.245 at unit regularization (95% percentile bootstrap interval, 0.224–0.254). The corrected and untied controls recover the physical implementation to floating-point accuracy. Checks with two optimizers and three initializations expose four unresolved gradient-tolerance failures. Analytically solved controls separate distortion caused by the representation from numerical optimization error.