Microscopic Obstruction and Two-Scale Resolution in a Substrate-Based Derivation of the Fine-Structure Constant

We analyze a substrate-based route to the fine-structure constant in which the relevant infrared multiplicity is associated with a compact lattice gauge sector. The central issue is whether the naive multiplicity count survives once Gauss constraints, Bianchi closure, compactness, and finite defect-core structure are treated consistently. We show first that the exact physical plaquette projector has rank two at each nonzero momentum, so any fractional reduction of the naive count cannot arise from a noninteger projector trace. The correct object is instead a Brillouin-zone spectral average on the exact constrained subspace. We derive the corresponding finite-core multiplicity formula, first for a smooth isotropic core and then from the classical screened defect profile, yielding an exact one-dimensional representation for the effective multiplicity. We then show that a standard one-field compact sine-Gordon completion encounters a genuine microscopic obstruction: matching the observed deficit would require an implausibly small core if it is tied directly to the monopole- fugacity-generated screening mass on the same lattice. A minimal two-field ultraviolet completion resolves this tension by separating the compact phase sector from a heavy amplitude mode that sets the defect thickness. In this framework the observed value of the effective multiplicity reduces the problem to a single remaining ultraviolet geometric parameter, namely the radial curvature scale of the substrate vacuum manifold. The result is not yet a parameter-free derivation of α, but it establishes a consistent structure, identifies the precise point of microscopic tension, and isolates the final missing ingredient.