Quadratic Closure of a Projector-Defined Gauge-Normalization Interface

This work studies the microscopic provenance of the two scalar inputs that normalize a projector-defined coherent-domain SU(2) gauge sector: a projected light-sector response weight and a heavy-sector compliance or stiffness input. Within a covariance-normalized quadratic-response framework, both quantities can be obtained from a single projector-fixed light/heavy parent once the observable projector and probe covariances are specified. The revised analysis distinguishes single-parent sufficiency from microscopic uniqueness. The two normalization inputs are jointly determined within a fixed realization, while the inverse map back to the parent kernel remains many-to-one. The treatment allows arbitrary fixed light- and heavy-source covariances, derives the compliance-to-stiffness normalization under an explicit dual-response constitutive identification, and develops an exact rank-one gap invariant that can diagnose additional heavy-sector structure. A three-resonator coupled-mode model provides a concrete response-level implementation and reproduces the benchmark rank-one and higher-rank response behavior used in the manuscript.