Coh Category: Verifier-Certified Trace Composition and Projection-Induced Semantic Cost

This work introduces the Coh category, a categorical framework in which morphisms are not assumed a priori, but are generated by verifier-certified admissible traces. A Coh system consists of an observable state space, a hidden realization layer, a proposal mechanism, an exact rational valuation, and a deterministic verifier. Transitions are proposed, verified, and only then admitted as morphisms. This enforces a verify-before-execute model in which invalid transitions produce no side effects and cannot accumulate into system state. The main structural result is that admissible traces form a category under concatenation, provided the verifier satisfies identity acceptance and compositional closure. The construction uses exact rational bookkeeping, ensuring deterministic equality and avoiding the ambiguities associated with floating-point representations. In addition to syntactic bookkeeping cost, defined additively over traces, the framework introduces a semantic cost derived from hidden realizations via projection. This semantic cost is defined as a maximum over verifier-realizable hidden trajectories and is shown to be subadditive under composition. A worked example demonstrates that semantic cost can be strictly smaller than syntactic cost, reflecting the collapse of hidden structure under projection. This establishes a principled separation between observable accounting and underlying execution cost. The contribution is a foundational categorical structure for verifier-governed execution, with explicit separation between proposal, verification, and realization layers. The framework is domain-agnostic at the structural level, while allowing domain-specific invariants to be instantiated within the verifier. This work is released for academic and research use under CC BY-NC 4.0. Commercial use is restricted. Licensing is available through NoeticanLabs.