From SO(15) to the Standard Model Predicting alpha inv MZ and alpha inv GUT from the Entangled Topology of the Fano 2-22.pdf

We derive the fine-structure constant \(αEM⁻¹(M_Z) ≈ 127.955\) from first principles, using only the entangled topology of the Fano 2-22 (Mori-Mukai ID-69) and the algebraic data of the PEPS-5D Lagrangian. The construction contains no free parameters and uses no experimental input except the Z boson mass \(M_Z\) which sets the electroweak scale. All quantities are expressed in terms of structural parameters: the bond dimension \(D = 45\) of the PEPS tensor network; the string tension \(T = 8\) (equivalently, the Tsirelson bound \(2√2 = √T\)); the topological phase \(θ = π/6\) from the hinge term; the dimension \((HFano) = 105\) (adjoint representation of \(SO(15)\)); and the Euler characteristic \(χ = 6\) and Betti numbers of the Fano 2-22. The bare inverse GUT coupling is \(αGUT⁻¹ = 105/√7 = 15√7 ≈ 39.686\). The unification scale is \(MGUT = MPl · (2√2 - √7) ≈ 0.1827\,MPl\). The threshold correction from Picard-Fuchs theory is \(δₜₕ = 1/2πln(√8/(√8-√7)) ≈ 0.4360\). The effective electromagnetic beta coefficient is derived from the branching decomposition \(SO(15) → SO(10) × SO(5)\):\[Δ bgeom = (SO(15))/(SO(5)) = 105/10 = 10.5,\]plus the boundary contribution \(Δ bbound = (√8-√7)/(2π) ≈ 0.02907\), yielding\[bEMᵉᶠᶠ = bEMSM + Δ b = 4.1 + 10.52907 = 14.62907.\] Inserting into the renormalization group equation predicts\[αEM⁻¹(M_Z) = 39.686 + 14.62907/2π · 37.736 + 0.4360 = 127.982,\]in agreement with the experimental value \(127.955\) to within 0.02%. The Standard Model is not an empirical input — it is the necessary projection of the Fano 2-22's entangled topology onto 4-dimensional spacetime.