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

This paper develops a finite dependency atlas for sufficient declared-regime specifications. Its governing question is whether the fifteen semantic roles required by the Identity-Persistence Program can be faithfully represented by a finite typed diagram of quotient maps and kernel inclusions, and which obligations—such as interaction, existence, priority, or global assembly—necessarily escape that local representation. The paper first establishes the structural instruments that a corpus-native computation requires: a complete lattice of semantically closed role sets, a canonical implication basis, and a quotient-dependency diagram in which arrows are kernel inclusions and absent arrows carry two-point certificates. It then develops a closed interaction theory for finite deterministic semantics at any declared finite partition. The theory is organized by a Galois connection. Equal semantic outputs generate coordinate equivalences, and product formation reconstructs the least product relation containing the semantic kernel. Noninteraction is exact reconstruction; interaction is failure of exact reconstruction. This characterization yields a polynomial-time decision procedure, independently checkable chain certificates, a canonical closure defect, and a proof that interaction admits no uniformly bounded induced-substructure certificates. The generated-equivalence domain is canonically the congruence lattice of the many-sorted empty-signature algebra on the declared components. The paper uses this structure to characterize transformer-relative completeness, categorical transport, partition-indexed closure monotonicity, merge exactness, and the interaction frontier—the antichain of minimal partitions at which the semantics becomes noninteractive. It also proves that interaction is asymptotically generic on total full grids with fixed finite output alphabet and, under a fixed finite relational encoding, is definable with transitive closure but not in first-order logic. Interaction is distinguished from global assembly. Interaction asks whether an existing semantic relation factors across a declared decomposition; assembly asks whether compatible local relations possess any global realization. The latter is governed jointly by context-hypergraph structure and the polymorphisms of the admitted relation language. The paper does not claim that the fifteen-role corpus relation has been instantiated or computed. Seven declarations remain deliberately blank. The result is therefore a draft whose abstract instruments and finite interaction theory are complete within their declared scope, while corpus computation and independent proof verification remain outstanding.