Identity Persistence under Declared Transformations: Contextual Quotients, Evidence Attainment, and the Limits of Bounded Judgment

Abstract When does a transformed presentation still count as the same declared object, and what can a bounded observer establish or safely do on that basis? This paper separates representation, evidence, control and implementation within an explicit mathematical framework. Given a state space, permitted transformations and demanded observations, contextual indistinguishability determines the greatest observation-preserving congruence. Its quotient is the coarsest lawful representation in the declared class. Available evidence supports the required contextual distinctions exactly when its kernel refines that congruence. For finite explicitly presented systems, the paper provides quotient constructions, separating witnesses, the coarsest evidence-preserving lawful refinement, and a sharp minimum supplementary-alphabet bound. It distinguishes exact from sufficient realization through a restricted channel catalogue and establishes NP-completeness for budgeted exact realization in a specified finite class. A complementary control layer characterizes robust safety through a greatest fixed point, identifies safely warranted actions, and establishes conditions under which observer compression preserves the safety computation. Further results give exact finite-memory and bounded-controller synthesis criteria, joint sensor–controller resource frontiers, adaptive acquisition constructions, and conditional transfer of abstract safety guarantees to concrete systems. The results distinguish three objects often conflated: a terminal representation, a maximal safe action set, and the implementations attainable under a declared resource contract. Incremental scope change, transport, recurrence and strategic quotienting receive separate statements with explicit hypotheses. The framework does not derive the task’s declarations or establish their correspondence to the world. Its contribution is a connected account of which distinctions must be preserved, when evidence supports them, and when a bounded implementation can realize the resulting guarantees.