Theta Closure in Modal Triplet Theory I: Gauge Couplings from Internal Geometry

This paper presents a conditional Theta-closure realization relating Standard Model gauge couplings to internal geometric overlap integrals. Measured source data are transported with SMDR v1.3 into the full-Standard-Model MS-bar scheme at Q = Mt. In the normalization used here, the resulting profile targets are I2/I1 = 0.5110273 +/- 0.0001231 and I3/I1 = 0.158335 +/- 0.001098. They are calibrated profile coordinates, not first-principles predictions of the gauge couplings. The paper formulates the admissibility and spectral-gap conditions under which an internal realization may reproduce these targets. A high-coherence twistor corner uses declared period and gauge-kinetic normalizations; its inverse-gap overlap estimate remains conditional on explicit projector and representative perturbation bounds. The scale Q = Mt is a renormalization and matching convention, not a physical coherence, gap, or quantum-gravity scale. Independent geometric selection remains open.