Categorical Belief Propagation: Sheaf-Theoretic Inference via Descent and Holonomy

We develop a categorical foundation for belief propagation on factor graphs. We construct the free hypergraph category $\Syn_\Sigma$ on a typed signature and prove its universal property, yielding compositional semantics via a unique functor to the matrix category $\cat{Mat}_R$. Message-passing is formulated using a Grothendieck fibration $\int\Msg \to \cat{FG}_\Sigma$ over polarized factor graphs, with schedule-indexed endomorphisms defining BP updates. We characterize exact inference as effective descent: local beliefs form a descent datum when compatibility conditions hold on overlaps. This framework unifies tree exactness, junction tree algorithms, and loopy BP failures under sheaf-theoretic obstructions. We introduce HATCC (Holonomy-Aware Tree Compilation), an algorithm that detects descent obstructions via holonomy computation on the factor nerve, compiles non-trivial holonomy into mode variables, and reduces to tree BP on an augmented graph. Complexity is $O(n^2 d_{\max} + c \cdot k_{\max} \cdot \delta_{\max}^3 + n \cdot \delta_{\max}^2)$ for $n$ factors and $c$ fundamental cycles. Experimental results demonstrate exact inference with significant speedup over junction trees on grid MRFs and random graphs, along with UNSAT detection on satisfiability instances.