Execution Categories for Stochastic Program Generators: Quotient Semantics for Deterministic Executable Identity

Stochastic program generators produce syntactically diverse proposals for identical computations. Prior work established operational machinery for this setting: a grounding gate enforcing admissibility boundaries between AI proposals and formal execution (Paper 1), and canonical execution semantics providing deterministic replay identity via content-addressed hashing (Paper 2). This paper formalizes the semantic structure those systems imply. We define a registry-indexed execution category whose objects are typed executable artifacts and whose morphisms are admissible pipeline transformations. The operational equivalence generated by the system's rewrite rules -- alias resolution, argument-order normalization, canonical form selection -- forms a congruence: equivalent subexpressions remain equivalent under arbitrary well-typed pipeline composition. The resulting quotient category gives precise meaning to deterministic execution identity. Content-addressed hashing, applied to canonical representatives, serves as a computable operational witness of quotient membership under standard collision-resistance assumptions. A projection connects stochastic proposals to their execution classes, with fibers measuring collapse from surface diversity to canonical identity. The grounding gate restricts this projection to admissible proposals, and the empirical measurements from Papers 1-2 -- replay determinism across 50 runs, canonicalization collapse across 1,200 LLM generations, hash stability under registry evolution -- become interpretable as structural properties of the quotient and its fibers. All constructions are operational, registry-relative, and grounded in the empirical systems of Papers 1-2. The semantics do not claim universal program equivalence; they formalize exactly the invariances that a registry-indexed execution system declares and enforces.