Geometric First-Exit Structure of Persistence in Irreversible Dynamical Systems

Persistence in irreversible dynamical systems is formulated as a geometric first-exit problem from an operational identity region under CP-divisible open-system dynamics. The paper proves that a scalar cumulative entropy-production coordinate determines persistence time if and only if the operational identity region is radial with respect to the chosen divergence. In the non-radial case, persistence depends on the contingent cone structure of the admissible region, and scalar entropy-production coordinates become structurally incomplete descriptors of first-exit behavior. An explicit two-state quantum dynamical semigroup demonstrates the non-radial case by construction. The work connects entropy production, viability theory, first-exit geometry, and operational persistence in constrained dissipative systems. Empirical battery-degradation trajectories are discussed as a macroscopic illustration of the same geometric structure.