The Geometric Electroweak Sector: Derivation of the Weinberg Angle and Particle Spectrum in a 59-Dimensional Framework

This paper presents new results within the proposed 59-dimensional geometric framework for particle physics and the electroweak sector. The work derives the Weinberg mixing angle from purely geometric considerations, obtaining the relation: sin²(θW) = 1 − e^(−1/4) ≈ 0.2211 The derivation is based on the geometric relation: Δn = −D(0) / D(SO(8)) = −1/8 within the decomposition 59 = 29 + 1 + 29, where the neutral dimension D(0) = 1 and the SO(8) Triality dimension D(SO(8)) = 8. The paper also introduces the complete n-index computation for 24 particle states and establishes the electroweak base index n_base = −11 as a geometric consequence of coupling to all eleven M-theory dimensions. Three methodological routes are distinguished: Casimir-based derivations, geometric derivations, and circular verification procedures. The framework connects higher-dimensional geometry, electroweak symmetry, SO(8) Triality structure, and exponential mass relations in a unified formalism. The work further discusses open problems including renormalization-group running and independent derivations of lepton and Higgs masses.