Bidirectional Path Kernels for Local Split and Merge Detection in Policy Rollouts

Offline policy rollouts can pass through the same local context and then follow several outgoing routes, or arrive there along several incoming routes, and neither endpoint dispersion nor a forward-only conditional law identifies this directed, path-valued object. We introduce the bidirectional local path-transition kernel (BLPTK): forward and reverse conditional laws of relative path segments, localized around a state–time context. At a declared path scale, minimum route mass and conductance tolerance, a spectral routine proposes partitions and an explicit mass-and-conductance test validates them, so every accepted partition certifies an empirical lower bound K+ or K- on outgoing or incoming path multiplicity — failure to accept one has no converse interpretation. No global route labels or annotated event times enter the detector. We want these junctions located because they are where reinforcement fine-tuning quietly deletes a route, and a local partition of them is something a training objective can be constrained against. On controlled directed graphs, path features classify all 361/361 probed local graph types, versus 255/361 for endpoint features on identical anchors. Across the round-two valency sweep, orthogonally embedded routes are counted exactly at full valency (1.000 from valency 2 to 8 and at 12, 0.994 at 10), with a false-positive rate of 0.091 traced to a grader–estimator threshold mismatch; crowded embeddings fail in three separable ways, including miscounted anchors of intermediate truth, a term exactness and recall are blind to. On D3IL-Avoid diffusion-policy rollouts the forward detector recovers the branchable contexts against a withheld label it never sees (AUROC 0.874 over 300 anchors, base rate 0.387); every trajectory there branches at the same moment, so a clock is optimal by construction and we report the measurement it cannot explain — within a fixed time slice, AUROC 0.879 over 220 anchors. On a simulated 7-DoF arm whose obstacle, and therefore whose junction time, moves between episodes, a clock is no longer optimal (max_t p(t)=0.410) and the detector leads the per-slice oracle clock by +0.260 agreement over 204 anchors. Freezing those outgoing regions as a deliberately naive branch-mass constraint during reinforcement fine-tuning keeps both demonstrated routes in all five seeds and passes a predeclared gate in four, where unconstrained fine-tuning deletes one in all five and, at matched dose, fixed-time regions and fork-window KL pass in zero — a probe of whether the regions are useful, not a proposed algorithm. Time reversal matches the forward count on all 204 matched balls, validating the reverse code path but giving no independent evidence for natural merges. Throughout, "certificate" means empirical partition feasibility at a declared resolution — not a confidence bound, and not recovery of a population route graph.