Information Provenance and Conservative Enrichment in Prospective Health

Description Prospective Health evaluates present organizational realization together with the adequacy of lawful, constitutively viable continuations under declared conditions. This fourth paper in the Health, Formally Defined series establishes a machine-checked finite benchmark for examining how information enters, survives, and is transformed throughout that measurement architecture. A five-state forest system provides a complete formal setting in which contextual admissibility, constitutive continuation, response observation, capacity representation, and Health judgments can be examined exhaustively. Two equivalent history encodings establish invariance under relocating contextual restrictions between dependent history constructors and an explicit lawfulness predicate. The resulting candidate–lawful–viable response pipeline identifies the precise contribution of each semantic filter. The benchmark establishes an exact information-refinement ladder with five detailed-response classes, four original-capacity classes, and three classes distinguished by the complete declared Health language. Machine-checked classification results identify every distinction lost through representational compression and every additional distinction retained by capacity but unused by the established Health queries. A general conservative-capacity-refinement lemma gives sufficient conditions for preserving existing Health judgments under representational enrichment. A strict forest instance demonstrates how additional response information can support new queries while preserving established adequacy through exact projection. The results establish a formally verified reference for information provenance, observation sufficiency, and conservative evolution of Health representations. Their scope is the declared finite dynamics and set-valued capacity semantics. Extensions to probability-weighted capacity require compatible stochastic laws and preservation of query-relevant events. The companion empirical-synthetic measurement study investigates these additional obligations under incomplete ecological observations, uncertain dynamics, and numerical inference.