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

We define the Hilbert space H_Fano of the 105 deformation families of Fano 3-folds (Mori-Mukai classification). On this space we construct the algebraic collapse operator Π̂_coll = Σᵢ₌₁¹⁰⁵ δ(D_PF⁽ⁱ⁾, D_DS) |F_i⟩⟨F_i|, where D_PF⁽ⁱ⁾ is the Picard-Fuchs operator of the i-th Fano and D_DS is the Dyson-Schwinger operator derived from the PEPS-5D Lagrangian that unifies electromagnetism and quantum gravity (EM+QG): L_PEPS-5D = 1/4 Fμν Fμν + 1/(16πG) R + Φ^2 |e-iπ/6 Ψ_d − 4/(3π) Ψ_Φ|^2 + L_MPS(D=45). We show that for the Fano 3-fold 2-22 (ID-69) the identity D_PF⁽²⁻²²⁾ ≡ D_DS holds, while for every other Fano j ≠ 2-22 the two operators differ. Consequently Π̂_coll = |F₂₋₂₂⟩⟨F₂₋₂₂|. The measurement of the fine-structure constant α (equivalently, the observation of α⁻¹ = ln λ_max − π from the MPS circular duality) acts as the collapse process. For any coherent state |Ψ_Fano⟩ = Σ_i α_i |F_i⟩ with α₂₋₂₂ ≠ 0, the post-measurement state is |Ψ_phys⟩ = ( Π̂_coll |Ψ_Fano⟩ ) / ‖ Π̂_coll |Ψ_Fano⟩ ‖ = eiθ |F₂₋₂₂⟩, i.e. the collapse to the single Fano 2-22 occurs deterministically. No free parameters are introduced. The algebraic collapse operator does not rely on any numerical tolerance: it is defined by the exact coincidence of the two differential operators. The Hilbert space H_Fano is finite (105 dimensions) and the projection is spectral. A systematic scan of all 105 Minkowski period sequences confirms that the factorization identities in terms of the PEPS-5D structural parameters (D = 45, T = 8, π I_0 = 4/3, L_10 = 10, n_trans = 24) hold only for the Fano 2-22 (ID-69). All results are reproducible; the complete Python script is associated with this work (DOI: 10.5281/zenodo.20016894).