Functor Model Distribution Equivalence and Governed Functional Evolution

This document demonstrates the distribution equivalence of functor and stochastic models. It then formalizes the structural relationship between parameter evolution and functional evolution in the Functor Model Architecture (FcMA). We show that, under governed parameter transforms, functional change is the image of parameter change under the parameterization map Φ, and that observational equivalence of model instances is exactly equality of realized functions. The governed transforms form a monoid; we identify precisely the condition under which this monoid acts on the realized function class (compatibility with observational equivalence), and use the resulting orbit structure to give an algebraic account of governed learning, drift detection, and governance constraints. The result is model-type agnostic, relying only on parametric evaluability. It is an identification of functional and parameter evolution as two views of one mechanism — not a claim that the function representation has expressive power different from, or equal to, the parameter representation.