Operator Distinguishability Collapse and Quadratic Fisher Scaling in Parametrized Markov Kernels

We consider finite-state Markov kernels whose parameter dependence enters through a scalar gate applied to a fixed kernel perturbation. In this setting, policy distinguishability admits an exact reduction to the gate difference. For logistic gates, we derive a closed-form Fisher information along the parameter axis and show that, in the high-friction regime, Fisher information scales quadratically with operator distinguishability. Thus operator collapse and Fisher degeneracy are structurally linked. We further prove that the Fisher–Rao distance to the infinite-friction limit is finite. The results rely only on finite-state Markov structure and exponential-family behavior.