Attention as Lattice Gauge Theory: Relations Interact Only Through Wilson Loops

Abstract

Relations compose through the entity they share - Alice is Bob's mother and Bob is Carol's father, so Alice is Carol's grandmother - and the order matters, since a father's mother is not a mother's father; verifying a composed relation means closing the loop. We show that if each entity's coordinate frame may be rotated independently (per-entity frame freedom), three things follow. In the pure-gauge sector, relations that share no cycle cannot interact at all. The only gauge-invariant score of a triangle is a class function of its holonomy $H = T_{ij} T_{jk} T_{ki}$ - a Wilson loop. And the only equivariant update of an edge is $U = \Psi(H)\, T_{ij}$ with $\Psi$ conjugation-equivariant. Edge attention by Wilson loops is therefore not one design among many but the unique form the symmetry permits. Three communities hold pieces of this form: the commuting-operator embeddings of knowledge graphs are its abelian (commuting) sector; AlphaFold's triangle update and the Edge Transformer have its shape - an edge updated through the third vertex - without its constraints; and lattice QCD builds it equivariantly on lattices. We instantiate the equivariant sub-family on general relational graphs as holonomy attention. With no training and composition alone, it infers unobserved $S_4$ relations on sparse graphs at 0.88 with a fifth of the edges corrupted, at or slightly below specialist synchronization algorithms, where every commutative architecture is capped at 0.083; its holonomy score flags corrupted triangles at AUC 0.99 where commuting scores are capped at 0.76 by a second selection rule, and it separates clusters without labels.