Temporal Aliasing Is Arithmetic: Optimal Source-Time Design for Unseen-Time Control

A controlled finite-state continuous-time Markov process may be observed only through endpoint transitions at legal source holding times, while deployment uses an unqueryable duration. Let $\text{ClosedTimeGroup}D$ be the closed additive group generated by the source times. Equality of source skeletons propagates to $\text{ClosedTimeGroup}D$: if $\text{ClosedTimeGroup}D=\mathbb{R}$, the generator is identified, while $\text{ClosedTimeGroup}D=h\mathbb{Z}$ identifies one base skeleton and its compositional times. Under an explicit rate–time condition, every noncompositional target admits irreducible three-state models with identical adaptive source transcripts but opposite target actions. We characterize the resulting Bellman and arithmetic answer quotients. On finite observable quotients with known legal-source laws, a sparse target-excluding Track-and-Stop rule matches the answer-information lower bound, while Diophantine distance to harmful resonances controls finite-sample conditioning. Finally, on a coupled unknown-damping/logarithm-branch family with known parametric form, a profile-design confidence-sequence learner operates on raw categorical endpoints and attains the pointwise answer-information constant even though exact branch information is zero.