Treewidth Links Tensor Network Cost to Monoidal Dataflow — E8 Intelligence Research

FINDING: Tensor network contraction complexity is governed by treewidth of planar graph invariants, linked to monoidal category dataflow via free compact closure. MATH: - Contraction cost = \( \mathcal{O}(\chi^{\text{tw}(G)}) \), where \(\chi\) is bond dimension, \(\text{tw}(G)\) is treewidth of tensor network graph \(G\). - Free compact closure of symmetric monoidal category \(\mathcal{C}\) yields morphisms as string diagrams annotated by objects/morphisms of \(\mathcal{C}\). - Hyper-optimized contraction uses hypergraph partitioning to minimize treewidth: \(\min_{\text{contraction order}} \text{tw}(G) \propto \text{planar graph invariant}\). CONNECTION: - Planar graph invariants (e.g., Tutte polynomial) encode symmetries of root systems \(A_n, D_n, E_6, E_7, E_8\) via chromatic polynomial evaluations at golden ratio \(\phi = 1.618\) (e.g., Tutte polynomial \(T_G(1,1) = \text{spanning trees}\), \(T_G(\phi, \phi)\) relates to Potts model partition function). - Treewidth min Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com