Exact Renewal Identity for Policy-Gated Injection in Sink-Stabilized Surface-Code CTMCs

Surface-code continuous-time Markov chain (CTMC) models with boundary injection and sink absorption provide a continuous-time view of defect transport and logical-parity flips, but policy-gated injection introduces a normalization ambiguity: should flips be normalized by attempted or accepted injections? Standard discrete-round analyses and naive "flips per attempt" ratios do not separate throughput control from per-accepted-attempt maliciousness. We establish an exact renewal identity: under empty-manifold gating (accept only when the syndrome is empty), accepted injections launch disjoint excursions, and the hazard-normalized estimator peff,acc = N flip /A acc is unbiased for the odd-parity excursion probability under the accepted-start distribution (Theorem 1). A representative CTMC dataset at L = 11, η = 0.9 shows that peff,acc remains stable (within statistical error) even when the acceptance fraction varies by more than a factor of two, while "per attempted injection" ratios change proportionally (values reported in companion application paper). We provide portable bridges to practice: mapping sustaining flip rates to per-cycle logical error probability, a decoder-integration pipeline from CTMC logs to detection events, and a reduced-model exact cross-check. We do not claim an optimal admission policy, nor do we address non-renewal (overlap) regimes in this work.