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 \ (₄₌^-1 (MZ) 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 \ (MZ\) 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 \ (22 = T\) ) ; the topological phase \ (= /6\) from the hinge term; the dimension \ ( (H₅₀₍₎) = 105\) (adjoint representation of \ (SO (15) \) ) ; and the Euler characteristic \ (= 6\) and Betti numbers of the Fano 2-22. The bare inverse GUT coupling is \ (₆ₔₓ^-1 = 105/7 = 157 39. 686\). The unification scale is \ (M₆ₔₓ = M₏₋ (22 - 7) 0. 1827\, M₏₋\). The threshold correction from Picard-Fuchs theory is \ (ₓ₇ = 12 (8/ (8-7) ) 0. 4360\). The effective electromagnetic beta coefficient is derived from the branching decomposition \ (SO (15) SO (10) SO (5) \): \ b₆₄₎₌ = (SO (15) ) (SO (5) ) = 10510 = 10. 5, the boundary contribution \ (b₁₎ₔ₍₃ = (8-7) / (2) 0. 02907\), yielding₄₌^eff = b₄₌^SM + b = 4. 1 + 10. 52907 = 14. 62907. \ Inserting into the renormalization group equation predicts\₄₌^-1 (MZ) = 39. 686 + 14. 629072 37. 736 + 0. 4360 = 127. 982, 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.