Cumulative-Exposure Thresholds in Post-Quantum Migration Risk: An Occupation-Time Model with Exact Passage Laws
A Markov model that allows a vulnerable cryptographic deployment to return to a safe state can inadvertently erase risk accumulated before remediation. We formulate a cumulative-exposure threshold model for post-quantum migration analysis. A two-state continuous-time chain alternates between protected and exposed operation, but a separate, nondecreasing occupation-time variable records total exposed time. A threshold event occurs when this variable first reaches a specified budget. We derive the exact passage-time distribution as a Poisson mixture of Erlang laws, its Laplace transform, mean, variance, and a finite-horizon transform. If compromise also arrives with constant hazard per unit of exposed time, the probability of compromise before the budget is exhausted is $1-e^{-\phi B}$, independent of the recovery rate; recovery delays the event in calendar time without eliminating recorded exposure. An extension to finite-state operational regimes follows from the Feynman--Kac formula. Numerical values in the paper are synthetic and illustrate the mathematical mechanism rather than estimate the security of any standardized algorithm. The contribution is a testable risk-model architecture and exact benchmark for assessing migration policies when exposure cannot be reset by repair.