Functor Reasoning Models: A Sheaf-Theoretic Alternative to Chain-of-Thought Inference

Modern reasoning models typically implement reasoning through autoregressive token generation combined with repeated attention over an expanding scratchpad context. While effective, this approach incurs substantial computational and energy cost and provides limited intrinsic auditability. This paper introduces Functor Reasoning Models (FRMs), a generalized reasoning framework based on token topology, contextualized probability functors, and sheaf-theoretic reasoning structures. Rather than treating reasoning as repeated rescanning of serialized token streams, FRMs model reasoning as structured navigation over contextual topological spaces with governed probabilistic section selection. The framework rests on three pillars. Energy: FRMs reduce inference cost through two distinct channels — a per-pass aggregation reduction from to with , and, more significantly, the elimination of the reasoning-length rescanning multiplier inherent to serialized chain-of-thought. Assurance: coherent reasoning corresponds to the construction of compatible global sections over a reasoning topology, so reasoning failure becomes a detectable gluing obstruction rather than a statistically invisible bad token sequence; the system can abstain on obstruction. Dynamic extension: when reasoning reaches the boundary of the governed region, a deterministic code-generation and validation pipeline synthesizes candidate morphisms that are admitted only under governed validation. We present a quotient-level equivalence result demonstrating that, under locality and canonicalization assumptions, topology-based aggregation achieves semantic equivalence to a restricted class of attention mechanisms. We are explicit about the costs the architecture incurs — in particular, the cost of section-compatibility checking — and about the empirical parameters (morphism-to-token compression, gluing overhead) that determine realized savings. All energy claims are stated as plausible pending direct measurement; a validation methodology is given. The resulting architecture provides a potential foundation for energy-efficient, auditable, and governed reasoning systems suitable for high-assurance environments.