Learning Capacity of Totally Geodesic Maps into the Circle

We classify the squared geodesic loss class of smooth totally geodesic maps from a connected Riemannian manifold without boundary into the unit circle. Its pseudodimension is two exactly when there is no nonzero integral-period parallel one-form; otherwise its fat-shattering dimension at margin 1/24 is infinite. On any closed domain admitting a nonconstant map, a Dirichlet-energy cutoff gives matching logarithmic orders of capacity. The proof combines lattice packing and phase thresholds with an explicit base-four shattering construction. We also determine the closure of affine-circle interpolation values and give a sharp two-point obstruction to an unrestricted interpolation assertion in the 2008 report. The known infinite capacity of circle-to-circle maps is credited. Arbitrary target manifolds in OWR-2041-002 remain unclassified here. AI-assisted, self-audited preprint; no independent review, formal verification or absolute-priority certification is claimed.