Abstract
Effective gauge-normalization interfaces can leave two scalar inputs with ambiguous microscopic provenance: a projected light-sector response weight and a heavy-sector compliance. We examine a coherent-domain SU(2) construction with a light sector defined by a rank-2 projector and a matching relation containing these inputs. For fixed observable projector and light- and heavy-source covariances satisfying stated nondegeneracy conditions, a single positive quadratic light/heavy parent determines both quantities. Low-frequency Schur reduction fixes the reduced light susceptibility, while the inverse heavy block fixes scalar compliance; normalized response contractions produce the matching inputs. This establishes single-parent sufficiency and joint determination within a fixed realization. The parent-to-scalar map remains many-to-one, so microscopic uniqueness and kernel reconstruction do not follow. Under an explicit constitutive identification of scalar heavy compliance with the component-level dual-response coefficient, we derive the generator-trace and transverse-projection conversion from compliance to stiffness. In a one-heavy-mode completion, the ratio of the heavy-mediated off-diagonal light-Hessian contribution to heavy compliance is independent of the heavy gap for fixed light-heavy couplings. In a calibrated gap scan with the direct channel independently characterized, variation of this ratio excludes the static rank-one completion with gap-independent couplings. A three-resonator coupled-mode construction provides a concrete implementation: two light resonators coupled through a detuned auxiliary resonator realize the rank-one Schur map, and a representative microwave benchmark with 80 and 60 MHz couplings at 1 GHz detuning produces a 4.8 MHz-induced coupling. The result is a conditional provenance theorem and rank-sensitive diagnostic for the normalization interface.