The Weinberg Angle from 5D Octonionic Geometry: Why Local 5D Brane-Bulk Ge- ometry Forces sin2 θW = 3/8 at the Planck Scale, and Why This is the Deeper Origin of SU(5) Grand Unication

The Weinberg angle sin2 θW is one of the last undetermined parameters of the Standard Model from the framework's perspective. We derive the tree-level value sin2 θW = 3/8 at the Planck scale from the local 5D geometry of the brane-bulk system, without postulating any grand unied group. The argument is: (1) the local geometry near the brane is 5-dimensional; (2) the symmetry group preserving the bulk direction e7 while acting on the H+ brane sector is the SU(3)C stabiliser of e7 in G2; (3) the SU(3)C Killing form forces hypercharge normalisation kY = 5/3, giving sin2 θW = 3/8 exactly. This is the same result as SU(5) grand unication, but derived from a deeper structure: SU(5) GUT is a 4D eective approximation to the 5D octonionic geometry of the brane-bulk system. The running from 3/8 at mPl to the observed 0.2312 at mZ is standard renormalisation group evolution using SM beta function coecients, which the framework expects to be xed by the Fano mode particle census of Paper LII. This running is known to give ≈0.231 in the SM at two-loop order, consistent with observation. The W and Z masses follow from the viscosity gradient anisotropy of Paper LII. We explicitly retract the formula 3/(3 + 2 √ 7) from the previous version of this paper: it was arrived at by numerical pattern-matching rather than derivation and is replaced by the physically transparent argument presented here. A cosmological time variation of sin2 θW is predicted from the Hubble breathing mode. Five predictions follow (Predictions 7478, retained from v1 with two corrections). Part of the One-Octonion Brane-Bulk Framework series. Anchor DOI: 10.5281/zenodo.19120873. Community: one-octonion-brane-bulk. Author: Bharathi Dasan Jagadeesan, M.D., University of Minnesota. ORCID: 0000-0002-1143-941X.