Task-Relative Dynamic Relevance and Decision Preservation Under Stochastic Evolution

This preprint develops a task-relative framework for determining which distinctions in a finite stochastic state must be retained to preserve specified decisions through time. Building from instantaneous task equivalence, it examines when a task-relevant quotient remains closed under stochastic evolution and introduces dynamic task closure by pulling future task directions backward through the dynamics. The construction extends to changing tasks, finite horizons, and uncertain future dynamics. A three-state example illustrates how a presently task-invisible state direction can become dynamically visible before a decision-boundary crossing occurs. The manuscript further distinguishes natural decision evolution from compression-induced distortion and develops margin-based certificates for decision preservation, including temporally and directionally aligned refinements. Randomized stress tests are used as implementation checks and to examine certificate conservatism. The proposed contribution is a decision-relative synthesis connecting dynamic task relevance, future task closure, compression, and decision preservation while remaining explicitly grounded in established work on observability, Markov-chain lumpability, invariant subspaces, and model reduction.