Optimal Depth of Neural Networks: Resource-Correct Stopping with Bounded Logit Increments

When can a network stop without executing the suffix that justifies the stop? For a fixed inspection schedule, finite observed record, confidence-resource utility, declared charges, and coupled bounded logit increments, we derive an information-optimal stopping certificate. One aligned completion attains every future confidence support. A separate sharp guard preserves the full-recall returned label, not unknown-label correctness. We compile both tests into checked integer kernels and prove their equivalence to arbitrary-precision comparisons. Five-seed experiments select schedules separately for scalar and return-conjoined policies, publicly freezing 870 choices before new evaluation. Return protection changes two of 300 main-grid schedule choices and eliminates all returned-label disagreements; precision and temperature are evaluated with newly constructed scores and supports. Actual image-to-return execution on Xeon and Neoverse CPUs makes the compiled multi-exit path faster than full execution for the registered 1,000-image workload, while retaining adverse small-batch results. A finite-family work bound and a separated design procedure distinguish pointwise certification, task risk, and statistical selection. All prior proofs, saturated-training diagnostics, and unfavorable results are preserved behind an active-claim and artifact index.