Response Quotients for Exact Semantic Representation

Abstract

Given a state space, admitted interface space, outcome space, and response map, we identify states that agree under every admitted query. The resulting response quotient carries a uniquely descended Full Admitted Response. We define response spaces and response-equivalence-preserving maps, show that quotient formation is functorial, and prove that, for a fixed response map, the quotient is terminal among surjective response factorizations with the stated arrow orientation. We distinguish this full-response quotient from the realized semantic carrier image and characterize when an independently specified continuation map factors through that image. Response-representation invariance and a carrier-presentation theorem identify when concrete logical, algorithmic, or physical carriers present the same quotient. Response-compatible dynamics descend uniquely, whereas construction, preparation, access, evolution, and decoding costs remain additional resource data. The contribution is a typed synthesis of standard kernel, quotient, image, coequalizer, and factorization constructions for exact semantic representation, not a new quotient universal property. These results yield the Exact Representation Principle: semantic representation and realization dynamics are distinct but compatibility-constrained structures.