Theta Closure in Modal Triplet Theory V: Redundant Determination of the Weak Mixing Angle

We test the Θ-closure program of Modal Triplet Theory (MTT) by imposing a redundant constraint entirely within the Standard Model gauge sector. In earlier work, a single parameter Θ was fixed using the nonabelian gauge coupling ratios at a common matching scale. Here we ask whether the same Θ, without retuning or additional free parameters, also reproduces the observed value of the weak mixing angle sin⁡2θW\sin^2\theta_Wsin2θW. Using one-loop renormalization group evolution with canonical hypercharge normalization, we derive a predicted sin⁡2θW(MZ)\sin^2\theta_W(M_Z)sin2θW(MZ) from the Θ-fixed gauge couplings. We carefully distinguish between a round-trip consistency check and a genuinely non-circular redundancy test, fixing only one unavoidable normalization and using Θ-fixed overlap ratios to determine the remaining couplings. The resulting prediction agrees with experiment within the expected theoretical uncertainty of the one-loop approximation. This demonstrates that the Standard Model gauge sector is overconstrained within the Θ-closure framework and provides a sharp internal consistency test with clear falsification criteria.