Certifying Branch and Event Identity in Finite-Budget Nonconvex Tracking

Finite-budget nonconvex tracking pipelines often accept candidates because they have low residuals or emerge repeatedly from local solves. That evidence is not the same as certifying branch identity or same-branch event identity: an unobserved branch, or a same-branch opposite-event solution, can remain compatible with the public transcript. Adaptive branch-aware certification (ABAC) makes branch and event identity explicit requirements of finite-budget delivery. Its evidence interface lets a selector allocate budget to branch/support probes and event-complement checks, or abstain when the evidence does not support delivery. Under a public-transcript ambiguity model with an explicit dominance-completability condition, we prove that worst-case zero-false delivery over transcript-compatible instances requires both witness types, and a constructive public-query hard family gives a tight m+1 lower bound for per-location evidence acquisition with m alternative branches and one opposite-event location. We instantiate ABAC with a frozen event-margin fallback and abstention. Exploratory nominal-budget results favor coverage, but omit failed-probe costs and permit seed-derived access to the observation-error direction. Our public-input evaluation charges every probe without retuning the threshold. On 640 synthetic transitions from eight held-out seeds, both the fallback and public GCT make 17 reference-matched deliveries; the fallback also makes four false deliveries, versus zero for GCT. Mean accumulated solver cost is 85.74 versus 110.64 iterations, with every transition capped at 120. The paired seed-bootstrap 95% interval for the safe-coverage difference is [-1.09,1.25] percentage points. These results support a cost–risk tradeoff, not coverage or safety dominance. Correctness uses a best-of-192 SLSQP reference, not a global certificate; all 160 prospective states across eight seed trajectories retain their labels after 320 additional starts each. The theoretical contract remains distinct from the empirical gates and their coarse labels.