Topology, guards, and a finite stability allowance

Choose a metric $d_Q$ on the compared geometric descriptions within one fixed admissible branch or stratum. Give state functionals the dual norm

Specify the controller topology separately. In the following certificate, its finite command choices and interfaces are aligned. This alignment requires an exact agreement argument, a robust guard margin, or a separate certified branch-mismatch law. It is not inferred from a small state norm.

An equality guard such as $s=0$ is generally discontinuous under arbitrary clock perturbations. The event example preserves its exact clock; noisy event detection is not included.

Suppose two aligned executions have original accessible state distance $e_k$, geometric distance $d_k$, and the same current interfaces. Suppose certified bounds at each step give

with nonnegative constants. Let

Then

A fixed final readout has record discrepancy at most $e_T$.