Geometric Mass Generation and Topological Interaction Diagrams in the Standard Model from M-Theory Vacuum Dynamics on G₂ Manifolds

A complete derivation of fifteen Standard Model parameters from M-theory compactified on the G₂-holonomy manifold T⁷/(ℤ₃ ⋉ I*) with Betti numbers (b₂, b₃) = (27, 451), selected as the unique physical vacuum with probability exceeding 1 − 10⁻²⁰. The fine-structure constant α⁻¹ = 137.039 (0.002%) has been derived from the gauge kinetic function, the hierarchy C_LIG/C_QG = √α from zero-mode localisation at ADE singularities, and six quark Yukawa couplings including the Cabibbo angle sin θ_c = (α/(2√2))1/4 = 0.2254 (0.4%). Three lepton masses, top quark mass (0.23σ), Higgs mass (0.16σ), and muon g−2 (0.39σ) obtained with zero adjustable parameters. Bayes factor B ≈ 10³⁹. Version 2 (April 7, 2026): Three key derivations added. (1) The factor 1/√2 in the electron mass formula derived from SUGRA constraint C_LIG + C_QG = 1/√2, ℤ₃ orbifold topology forcing an intermediate scattering vertex for w = 2 windings, and coherent re-emission amplitude at the vertex (Eq. 38-39). (2) Rule 1 (stable/unstable vertex assignment) derived from Born-Oppenheimer modulus dynamics: stationary states (dρ/dt = 0) yield C_LIG^tree, decaying states drive KK excitations yielding C_LIG^phys. (3) Two α⁻¹ formulas connected via algebraic identity (Eq. 7). Limitations updated: items (iv)-(vii) reflect derived status. Eight appendices (A-H). 31 pages, 6 figures, 201 references.