Rank Is Not Enough: Adaptive Policy Selection at Singular Ties

Rank is not enough to predict the difficulty of selecting among tied policies. We construct finite MDPs with identical state and action spaces, transition-rank budgets, symmetry multiplicities, Fisher information, exact policy tie, and comparison-gradient norm, yet whose fixed-radius local minimax regrets differ by a factor ${\Theta }_R(\sqrt{d/r})$. The governing object is the Fisher-relative geometry between the contingent cone ${\text{cK}}_0$ of feasible transition perturbations and the polytope ${\text{fG}}_0$ of tied policy gradients. For equivariant sparse low-rank transition models, we first prove that the oracle-local root-$n$ decision problem converges exactly to the Gaussian game determined by $({\text{cK}}_0,{\text{fG}}_0)$. We then construct one unknown-center rule that attains this oracle value uniformly on quantitative stable support–rank–policy strata. At singular rank intersections, however, we prove a positive adaptation tax: no branch-agnostic procedure can simultaneously attain the neighboring oracle risks. A commitment lift realizes the rank-diffusion gap in the complete deterministic policy class. Finally, we give a certified finite algorithm for the Gaussian game and finite-sample categorical experiments that recover adaptive attainment, adaptation failure, and the predicted $\sqrt{d/r}$ scaling.