Yang-Mills-Higgs: A Geometric Theory of Binary Labels on Non-Contractible Spaces

We reformulate binary classification on a manifold M as a Yang-Mills-Higgs variational problem. Labelled data is encoded as a functor from the fundamental groupoid of M to the one-object groupoid B(Z_2), whose monodromy class in H^1(M, Z_2) is a topological obstruction to realising the classifier by a sign function. The classifier-section and the connection jointly minimise a Yang-Mills-Higgs energy subject to hard data conditions: the matter sector carries the classification content, while the Yang-Mills sector is bounded below in each topological class by the Bogomolny inequality and selects the gauge background. This recovers the companion paper's harmonic interpolation as the contractible-base, flat-connection reduction. Two structural payoffs follow. First, the curvature 2-form of the selected connection has a precise dictionary with transformer attention: it is the antisymmetric part of the attention bilinear, and the abelian/non-abelian split of curvature corresponds to the single-head/multi-head split of attention. Second, XOR on the torus is solved in closed form by the covariantly harmonic section of the double-Mobius bundle, with minimum energy 2*pi^2 verified numerically to machine precision, whereas an MLP trained on the same data finds a structurally different boundary that ignores the toroidal identifications. Worked examples run an example ladder (circle, torus, S^2 Dirac monopole, S^4 BPST instanton); a matter-sector proximity-scaling theorem is proved in the two-point case.