Governed Functional Evolution in Functor Models

This document 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 phi, 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 reachability structure to give an algebraic account of governed learning, drift detection, rollback, repair, and governance constraints. Because T is a monoid rather than a group, reachability is not symmetric and reachable sets do not partition the function class; we treat drift accordingly, as exit from the governed reachable set, rather than as movement between cells of a partition. 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. 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 jharby@functormodel.ai