A Geometric Derivation of the Weinberg Angle from Discrete Octonionic Operators

The Weinberg angle (weak mixing angle) θW is a fundamental parameter of the Standard Model that describes the mixing between the electromagnetic and weak forces after electroweak symmetry breaking. In the conventional framework, sin2 θW is a free parameter requiring experimental determination. We present a derivation of the Weinberg angle from first principles within the Kosmoplex Theory framework, which derives 4D physical constants as projections of geometric structures in 8-dimensional octonionic space, obtaining sin2 θW (mZ ) = sin2(1)/√3π ≈ 0.23064 with zero adjustable parameters at tree level. This value arises from the geometric structure of three fundamental “glyphs”, discrete octonionic operators in the 8D substrate that project to observable numerical values in 4D spacetime through the Octonionic Binomial-Modular Transform (OBMT). We demonstrate agreement with experimental data across energy scales from 1 GeV to the Z-pole (mZ ≈ 91 GeV), with minimal post-OBMT running characterized by a single parameter. The theory predicts sin2 θW (10 GeV) ≈ 0.2343, suggesting a testable discrepancy with current experimental extractions that warrants further investigation.