Proof of Semantic Response Equivalence for Functor Models

This document formalizes semantic response invariance within the Functor Model Architecture (FcMA). Executable systems are modeled via an evaluation map from parameter space and input space to output space. The induced parameterization map associates each parameter state with its realized executable function. Observational equivalence is defined as equality of realized functions across all inputs. A governed set of parameter transforms is introduced and shown to form a monoid under composition. This monoid acts on the realized function class, enabling a structured interpretation of parameter evolution. Within this framework, we distinguish equivalence-preserving transforms (representation changes that leave outputs invariant) from governing transforms (structured semantic updates). The resulting orbit structure provides a precise algebraic mechanism for bounding functional evolution and identifying structural drift. Deterministic, probabilistic, and stateful evaluation semantics are treated within a unified formulation. The central result establishes that admissible reparameterizations preserve observable model behavior. Consequently, architectural refactoring, parameter restructuring, or governance-layer modifications do not alter semantic response. This document provides a formal mathematical foundation for governed parameter evolution in FcMA systems. Certain architectural components referenced in the broader FcMA framework are patent pending.LEAN verification: https://github.com/AutonomicAI/equivalence-lean.git johnharby@autonomicaillc.com