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

Abstract When does a transformed presentation remain the same for a declared task, what evidence can establish that judgment, and what can a bounded observer safely do with it? This paper separates representation, evidence, control and implementation, giving each an exact mathematical contract. For declared transformations and required observations, contextual indistinguishability determines the greatest observation-preserving congruence. Its quotient is the coarsest lawful representation preserving every demanded observation under every permitted continuation. A reference–current construction applies this result directly to persistence judgments, including breaking outcomes and required provenance. An evidence channel determines the demanded distinctions extensionally exactly when its kernel refines the contextual congruence; failure supplies an indistinguishable pair requiring different answers. For explicitly presented finite systems, the paper provides quotient constructions, separating witnesses, supplementary-information bounds and exact finite-memory criteria. Under declared transition and observation semantics, robust safety is characterized by a greatest fixed point, with checkable winning and losing certificates. Suitable observer quotients preserve the safety computation. Finite controller enumeration and declared sensor catalogues yield exact bounded-memory synthesis and joint resource frontiers; adaptive acquisition is incorporated as action under explicit contracts. The paper also establishes incremental and transport laws, strategic comparison results, and conservative model-to-system safety transfer under stated coverage hypotheses. Representation minimality, safety maximality and bounded implementation attainment remain distinct. Extensional existence does not imply effective discovery or affordability, and mathematical guarantees do not establish physical correspondence. The results characterize what can be preserved, distinguished and safely controlled within declared regimes while exposing the additional assumptions required for implementation and deployment. This public edition updates the previously deposited v19.1 release. It clarifies editorial framing, historical review status and correction lineage while preserving the mathematical content. Earlier versions remain part of the publication history.