From ambiguous utterances to governed reuse classes: canonicalization, quotient invariance, and conditional decidability
Semantic caching defines answer reuse on embedding similarity: two utterances share a stored answer when a similarity score clears a threshold, with no notion of authorization, versioning, or of what makes two demands the same. This note changes the object on which reuse is defined: in a governed domain, reuse should operate on a mathematically characterized quotient of resolved conversational demands, not on a similarity heuristic. Three independently defined relations on resolved utterances -- reading identity, resolution identity, and reuse identity -- form a refinement chain, strict under realized nondegeneracy conditions checkable on deployment logs; the pipeline's outputs are invariant along the chain, and reuse identity is exactly the kernel of the resolution map into the governed answer partition, so the reuse quotient is the utterance-side object that partition induces, not a relabeling of it. Reuse identity licenses the governed query key and its certified answer space; reuse of a particular answer requires resolution identity or an applicability certificate. The supporting layer is stated at exactly the strength proved: exact-denotation normal forms; join aggregation as a design operator, with closure-stable cells characterizing no-escape; total computability of the full pipeline relative to an untrusted proposal layer; policy admissibility for arbitrary proposers -- and provably not factual grounding or intent fidelity; and elicitation terminating after finitely many informative replies, sound under target consistency.