The Topological and Categorical Stack of Functor Reasoning Models

Abstract

This note organizes the current mathematical architecture of Functor Reasoning Models (FRMs). The construction begins with a metric topology on tokens, passes through finite-dimensional local representations whose linear maps compose by matrix multiplication, and places an intrinsically defined reasoning topology above that computational layer. Two parameter topologies separate representational change from semantic change. The resulting picture can be understood as compiler discipline for learning: computation occurs in a comparatively fine representational layer, while behavior is classified, versioned, and governed in a coarser semantic quotient. The goal in this work is primarily seeking out different approaches that may consume less energy.