The Fano 3-fold 2-22 as the Underlying Structure of the Unified PEPS-5D Lagrangian (EM+QG)

We establish an explicit isomorphism between the PEPS-5D Lagrangian,previously derived as a unification of electromagnetism and gravity,and the Fano 3-fold 2-22 (Minkowski period ID-69) from the classificationof Coates, Corti, Galkin, and Kasprzyk (2013). The Picard-Fuchs operator of 2-22 generates the Minkowski period sequence:{G}_X(t) = 1 + 6t^2 + 24t^3 + 138t^4 + 1080t^5 + 6540t^6 + 50400t^7 + 362250t^8 + 2713200t^9 + ... The coefficients factorize exactly into the structural parameters of thePEPS-5D Lagrangian:c_5 = 1080 = 24 * 45,c_6 = 6540 = (4/3 * 45) * (45 + 8^2),c_7 = 50400 = 24 * (4/3 * 45) * (45 - 10). These factors correspond to:24 (transverse modes of the Polyakov string),D = 45 (bond dimension of the MPS for alpha⁻¹),T = 8 (string tension),pi I_0 = 4/3 (hinge factor with I_0 = 4/(3pi)),and L_10 = dim(so(4,1)) = 10 (dimension of the 5D projective Lorentz group). No free parameters are introduced. The Dyson-Schwinger equation of thePEPS-5D system coincides with the Picard-Fuchs equation of Fano 2-22,demonstrating that the Fano 3-fold is the algebraic moduli space of theunified physical system. All results are reproducible; a Python script with arbitrary precision(mpmath) and graphical output is provided as supplementary material.