The Indexed Specification Algebra: One Object, Seven Theorems, and a Factorization Engine for Regimes as Dependency Systems

Abstract The Indexed Specification Algebra develops the internal algebra of sufficient declared-regime specifications after the forcing, capacity, and identification-boundary results of the Identity-Persistence Program have been fixed. Rather than treating the program’s fifteen semantic roles as independent primitives, the paper represents a regime at a declared index by a single finite quotient-level dependency system, D_I: its admissible role assignments, declared role quotients, and replay semantics at a specified grain. This inversion unifies several previously separate constructions. Quotient-level recoverability becomes a semantic closure operator; replay sufficiency becomes functional determination; minimal records become minimal semantic keys together with mandatory nonsemantic fields; and refinement becomes a dependency-preserving morphism. A consolidation theorem proves that factorization-based recoverability and quotient-level semantic closure coincide. The paper also separates minimal sufficiency from interaction. The minimal-key hypergraph records which information determines an outcome, whereas interaction records failure of the semantic kernel to decompose across coordinate blocks. Separate fiberwise and section-level results establish that these structures are not identical and locate where masking can occur. The realizability analysis distinguishes arbitrary finite dependency systems, systems representable in the declared presentation language, floor-gated systems, and systems induced by corpus-sufficient regimes. The presented class receives a sound and complete representation theorem. Relative to a fixed pre-verdict signature, the paper proves a strict gate separation and constructs the universal gate quotient—the least-informative quotient sufficient to determine gate status. More generally, every declared factorization obligation of the form f = f̄ ∘ q is equivalent to the kernel inclusion ker(q) ⊆ ker(f) and has a complete labeled two-point failure certificate. For finite systems, these obligations compile into definite Horn clauses, connecting obstruction certificates with implication-theoretic reasoning. Countermodels show that mixed raw/quotient determination need not form a closure operator, that the quotient-level closure is not generally matroidal, and that minimal generating sets can have unequal cardinalities. Machine enumeration corroborates the finite results; the proofs and countermodels are stated in the paper. The paper closes the abstract presented-class layer, not the corpus-scope realizability problem. The remaining central question is whether regime-induced dependency systems are exactly those satisfying the complete relational translation of the specification floor: Reg_I ?= FloorDep_I Its corpus-native successor must instantiate the fifteen-role relation and compute its quotient-dependency diagram, closure lattice, canonical implication basis, minimal semantic keys, interaction structure, and floor-realizability certificates.