Functor Reasoning Models: Topology-Guided Structured Reasoning Beyond Chain-of-Thought

This paper introduces Functor Reasoning Models (FRMs), a structured reasoning framework that generalizes chain-of-thought inference using topology-guided reasoning states, contextual neighborhood systems, and governed probabilistic transitions. Contemporary reasoning models typically implement reasoning by generating large numbers of hidden autoregressive tokens which are repeatedly rescanned through dense attention mechanisms. While effective, such systems incur substantial computational and energy cost and provide limited intrinsic auditability. Functor Reasoning Models instead treat reasoning as navigation over structured reasoning topologies composed of explicit reasoning-state objects and governed morphisms. Local reasoning neighborhoods, quotient-level semantic equivalence, and contextualized probabilistic structures replace unrestricted dense attention rescanning. The framework provides a mathematical and architectural basis for sparse contextual reasoning, replayable reasoning trajectories, and energy-aware structured inference systems.