Definability Computation over Σsuf

Abstract This paper continues the algebraic refinement of the Identity-Persistence Program by running the definability program over the sufficient-regime signature Σsuf. It does not reopen the forcing theorems. Instead, it asks which forced roles remain independently declarable once theory, declaration form, and admissible equivalence are fixed. The paper establishes three main results. First, the finite fragment of the regime theory admits a natural formalization in first-order logic with transitive closure, while first-order logic alone cannot express the required accumulation reachability. This gives countermodel enumeration a completeness privilege for certifying role independence within the finite fragment, while explicit-definition search lacks a Beth-style completeness guarantee. Second, four previously unaudited joints are computed by construction: the invariant basis quotients under declared verdict-inclusive equivalence, the target role is independent, the composition role splits into fused accumulation and independent exit semantics, and evaluator access is independent of authority at envelope scope. Third, the paper states an anti-tupling criterion for admissible fusion: compression is meaningful only when the fused object is forced as one structure by a prior theorem and carries surplus structure beyond the tuple of roles it contains. The resulting ledger is index-relative: at most twelve independent declaration slots under specification-level equivalence and at most eleven under declared verdict-inclusive equivalence. The three-class quotient remains a conjecture, now sharpened into the claim that exactly two further forced-surplus fusions must exist. The paper therefore turns the algebra of regime specifications from an open intuition into a bounded computation program with explicit certificates, obstructions, and non-claims.