The Curvature of a Move Set: Dirichlet–Wilson Bounds and Certified Trivialization Costs for Non-Commutative Transport on Graphs

Local search improves a candidate solution by applying moves from a fixed set, but its theory usually describes the search space through the objective. Moves need not commute: the two orders of a pair of moves may end at different candidates. We ask whether the resulting loop defect can be bounded from both sides, computed cheaply and certified against true optima. We model the moves as non-commutative transport on a base graph that records which move sequences are assumed equivalent. Loop holonomy measures the mismatch, and we study the minimum cost of making every loop close. A fractional packing of cycle defects and a representation-spectrum bound certify this cost from below; a spanning-tree gauge and a cycle-balanced section bound it from above. The bracket is exact on a single cycle; with the discrete metric, the cost equals a weighted frustration index on finite groups and vanishes on the state graph of any move set. Thus curvature depends on the relations the base graph assumes. We also connect the cost to three other readings: the point-level non-commutativity defect equals plaquette holonomy, the connection Laplacian separates into cut and holonomy branches with a corrected Cheeger inequality, and an ordered cohomological certificate bounds the distance a local algorithm must travel without exceeding graph distance. On 450 rotation instances a dual certifies the optima to within $3\times 10^{-8}$, and interval arithmetic proves exactness for 490 instances. The cheap lower certificates recover on average 96 to 100 percent of the certified optimum. On the commutation lattice of two rotations, the certified cost per plaquette reaches eight times the single-plaquette value on a $32\times 32$ box. These results give a computable, certifiable curvature relative to the relations a method assumes.