The Refinement Mode of the Finite Structured-State Transformation Principle

Companion paper to the Finite Structured-State Transformation Principle theorem set. Version 4.0. March 2026. This paper derives the refinement mode of the Finite Structured-State Transformation Principle: the conditions under which a finite system can increase interior structured state toward predictive alignment with a reference. The central result is a five-term biconditional over a writable predictive increment that unifies residual structure, accessible signal, and physical execution budgets. The paper identifies the exact failure modes under which refinement halts and situates refinement as the interior-increase branch of the broader FSSTP framework. This record presents the cleaned theorem form of the refinement branch. Earlier versions in this lineage preserve derivational development and intermediate refinement stages; the present version preserves that lineage while presenting the final theorem form. Earlier derivational development of this branch, including adversarial stress-testing across multiple language models, is preserved in prior archived materials.