Contextual Fraction on Permutation Gain Graphs: Exact Algorithms, Query Lower Bounds, and Dynamic Maintenance

What the math gives to ML

The paper identifies a tractable structure in which a globally constrained assignment problem over a graph reduces to computing the states fixed by the holonomy subgroup generated by cycle permutations. This suggests a specialized graph-network module for permutation-labeled relational data: instead of reasoning over exponentially many global assignments, compute cycle holonomies and explicitly restrict predictions or latent states to their common fixed set. The transferable asset is not contextuality itself, but the gauge-invariant reduction from many local transport constraints to a compact fixed-point mask, together with linear-time construction and constant-time compatibility queries after preprocessing. This is most useful for graph models whose edge labels are known permutations, such as multi-view matching, symbolic relational reasoning, or finite-state world models.