Persistent-Transient Policy Evaluation for Markov Chains via Minimal Peripheral Quotients

We study fixed-policy evaluation for finite Markov chains that may be reducible and periodic. Classical evaluation methods with gain and bias decomposition are not always diagnostic: the gain records only invariant Ces\`aro averages, while persistent phase-dependent behavior is absorbed into the bias together with genuinely transient effects. We identify the real peripheral invariant subspace $\mathcal{K}(P)$ of the transition matrix $P$ as the source of this ambiguity. Quotienting by $\mathcal{K}(P)$ is the minimal exact quotient that removes all non-decaying modes and makes the remaining dynamics strictly stable. After choosing a gauge projection $\Pi$ with kernel $\mathcal{K}(P)$, the reward admits a unique decomposition $r=g_{\Pi }^{\star }+(I-P)v_{\Pi }^{\star }$, where $g_{\Pi }^{\star }$ is a persistent regime profile and $v_{\Pi }^{\star }$ is a gauge-fixed transient component. An exact comparison with classical normalized gain and bias shows that the new pair reallocates the same information so that all persistent modes are represented in $g_{\Pi }^{\star }$ and $v_{\Pi }^{\star }$ is transient. This decomposition reconstructs finite-horizon returns, recovers statewise average reward, admits a transient-cost interpretation, and yields a stable estimator under a generative model.