The Finite Dependency Atlas of a Sufficient Regime: The Instruments, the Seven Declarations, and a Closed Interaction Theory

Abstract The Finite Dependency Atlas develops the corpus-native instruments required to compute the internal dependency structure of a sufficient declared regime. It follows the Indexed Specification Algebra, but does not infer or fabricate the fifteen-role corpus relation that its computation requires. Instead, it proves the abstract instruments in advance, records seven declaration blanks, and specifies the ordered protocol that will consume them. The paper’s central contribution is a closed theory of product-kernel interaction for finite deterministic semantics at any declared finite partition. Equal semantic outputs generate coordinate equivalences; those equivalences determine the least product over-approximation of the semantic kernel; and interaction is precisely failure of exact reconstruction through that generated product closure. This yields an exact characterization of noninteraction, a polynomial-time union–find decision procedure, independently checkable chain certificates, and a proof that no uniformly bounded induced-substructure certificate characterizes interaction. The construction is organized by a Galois connection between semantic relations and tuples of coordinate equivalences. Its closed elements are the noninteractive systems, while its closure defect is the canonical obstruction. The associated quotient is the least-identifying product quotient, and the abstract domain is canonically the congruence lattice of the many-sorted empty-signature algebra on the declared components. The paper further develops transformer-relative completeness, categorical transport, partition-indexed closure monotonicity, merge exactness through realized synchronization, the interaction frontier as a canonical antichain of minimal noninteractive partitions, and a five-axis reindexing taxonomy. Additional results show that interaction is asymptotically generic on total full grids with fixed finite output alphabet, remains computationally tractable despite unbounded local certificate complexity, and is definable in first-order logic with transitive closure but not in first-order logic under the paper’s fixed finite relational encoding. Global assembly remains a distinct local-to-global problem governed jointly by context-hypergraph geometry and the algebra of the admitted relation language. The paper does not extend the program’s forcing results or claim that the corpus computation has been performed. It is a draft whose abstract instruments are complete; the computation awaits seven author-supplied declarations, and independent line-by-line proof verification has not yet been performed.