Quadratic Closure of a Projector-Defined Gauge-Normalization Interface

Andrew Morton · Accepted for publication · AIP Advances · Quadratic response realization

Shows how one positive quadratic model can supply both response inputs of a specified gauge-normalization interface.

Current program context

The later Action-Level and UV Matching analyses make complete gauge-source normalization an independent requirement. Jointly realizing the two scalar inputs does not by itself establish the full physical source operator.

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.

Scope

The theorem assumes the projector, source covariances and constitutive matching relation, and establishes joint realization rather than a unique microscopic parent. Its resonator example is a calculated benchmark with a conditional, testable response relation.

Paper and supporting files

The Zenodo deposit includes a rank-diagnostic notebook for the quadratic-closure and coupled-resonator benchmark. The resonator example is calculated, not newly collected experimental data.

Read the current paper and supporting files on Zenodo

Related reading

Gauge geometry on coherent domains

What determines a gauge coupling?

Matching across coherence boundaries

Abstract checked 28 August 2026 · v6 · public release 7 August 2026.