Shutdown Routing in AI Systems: Two Counterexamples and a Finite-Horizon Bound

Which observations suffice to assess a learned shutdown route? We give two constructive counterexamples and a finite-horizon bound. First, two executions agree on command receipt, inequality of command scores, maintenance count, and learner-call count, yet terminate differently. The consumed scores together with the known routing rule distinguish them. Second, two full-batch logistic-learning problems have identical multisets of work-to-STOP inner products but opposite asymptotic STOP signs. Exact separator calculations and six fixed-budget numerical trajectories establish the analytic and computational claims separately. For repeated single-example learning, a discrete bound quantifies the exponential dependence of crossing time on the normalized control threshold. These results identify limitations of specific audit summaries: score distinguishability is not routing correctness, overlap summaries do not determine learning direction, and eventual reversal need not occur within budget. They concern specified linear models and a virtual execution fixture, not observed shutdown resistance in a deployed AI system. Version 0.3.0 is a narrowed revision of the maintenance-focused preprint. It includes six fixed-budget batch trajectories (600,000 updates), 98 scalar verification cases, and a coarse-telemetry counterexample. Numerical updates and tests are not independent AI observations. All 88 historical model calls and the negative STOP result (0/24 noncompliance) are retained; no new model calls were collected for this revision. Funding: This research received no external funding. The author is an independent researcher. The source ZIP contains editable Markdown and LaTeX, Python software and tests, result JSON, supporting notes, and complete retained prior evidence. Personal GitHub repository: https://github.com/sukjoonhong/maintenance-coupled-shutdown . Software and implementation documentation are MIT licensed; manuscript and research records retain copyright without an additional reuse grant. See README_LICENSE_SCOPE.md and the package README for exact scope.