Joint Holonomy Matching: Complete Supervision of Finite Representation Connections

What information must a student preserve to reproduce a teacher’s representation connection up to local coordinate changes? We introduce Joint Holonomy Matching (JHM), which aligns a fundamental loop family under one shared root gauge. Using the standard spanning-tree classification of graph connections, we establish that zero JHM is equivalent to gauge equivalence of the complete observed connections. Among supervision schemes retaining a subset of a fixed fundamental generator family, the full family is deletion-minimal for a uniform completeness guarantee: omitting any generator can leave inequivalent connections indistinguishable. Independent per-loop alignment is also incomplete in general, as matching individual conjugacy classes can miss incompatible joint compositions. We prove that every finite orthogonal connection on a simple graph admits an exact realization by local orthonormal frames and their Procrustes transports. A non-Abelian $SO(3)$ realization exhibits $L_{\mathrm{separate}}=0$, $L_{\mathrm{joint}}=5-2\sqrt{2}$, and a group-commutator discrepancy of $\sqrt{6}$. A quantitative bound controls errors on new compositions of observed edges. In a protocol-frozen three-seed neural pilot, Joint reduced composition error relative to Separate by 1.16 percent on the common evaluable subset of the designated test set, with the same direction across all seeds. In this implementation, Separate required 3.35 times as much loss-and-alignment time and 1.36 times as much full-update time as Joint. The structural effect was not independently established on validation, and unequal effective geometric regularization prevents attribution to the shared-gauge constraint alone. These results establish a complete connection-level supervision target, identify compositional information lost by independent alignment, and provide preliminary neural evidence for its optimization.