Geometric Design (GD-313): Electroweak Embedding and an Induced Coupling Ratio for sin² θ_W

We present the electroweak sector of Geometric Design (codename GD-313) in a form suitablefor referee audit. The framework selects the 3–13 vacuum Gr(3, 16) via a bounded integer corridordefined by two publicly tabulated anchors. On the dynamical side, we specify an explicit embeddingof SU(2)L °ø U(1)Y into the U(16) structure compatible with the (3, 13) split and computethe induced one-loop coupling ratio from the Grassmannian coset sector. Under clearly statedassumptions (background-field gauge and cancellation of the universal prefactor in the ratio), theindex computation yields sin2 θW = 3/13 at the declared matching convention. We provide anauditable appendices package: embedding generators, index computation, and a reproducibility checklist.