This work presents a self-contained derivation of the inverse fine-structure constant from the spectral invariants of circular tensor networks on the moduli space of Fano 3-folds. The construction proceeds through the following algebraic chain. 1. Geometric Lagrangian and Polyakov duality The starting point is the geometric Lagrangian for an oscillating circle of radius \ (R\): ₆₄₎ () = 4R²² + 1RR²² + 14R² - R²². \ For \ (R = \), the zero-mode reduction of the Polyakov action yields the duality: R²2 = 4R² T = 8, \ where \ (T\) is the string tension. The on-shell geometric action is: ₆₄₎ = ₀^2 L₆₄₎ () \, d = 4³ + ² +. \ 2. Circular MPS and spectral duality A circular Matrix Product State (MPS) is constructed by discretizing the circle. The bond dimension is \ (D = 45\), determined by the adjoint representation of \ (SO (10) \): \ (adj SO (10) ) = 45. \ Under spectral projection onto the top-\ (D\) subspace, the dominant eigenvalue \ (_\) of the transfer operator satisfies the duality: \^-1 = _ -. \ Numerically, \ (_ = 140. 1778962228\), yielding: \^-1 = 137. 0359991678, \ which matches the CODATA 2022 central value to within \ (10^-8\). 3. Fano 2-22 as algebraic moduli space The Fano 3-fold 2-22 (Mori-Mukai ID-69) has Minkowski period sequence whose coefficients \ (c₅, c₆, c₇\) factorize exactly as: ₅ = 24 45, ₆ = (43 45) (45 + 64), ₇ = 24 (43 45) (45 - 10). \ These identifications yield: \ (45 = (adj SO (10) ) \): the MPS bond dimension; \ (24\): the number of transverse modes of the bosonic string; \ (64 = 8² = T²\): the square of the string tension; \ (Iₕ = 4/ (3) \): the hinge invariant from the 5D Lagrangian. 4. The 5D Lagrangian and hinge invariant The 5D gauge theory has gauge group \ (SO (10) \) and Lagrangian: ₅₃ = -14g² Tr (F₌₍F^MN) + 12 Tr (DM DM) - 4 (Tr (²) - (⁴+1) ) ² + L₇₈₍₆₄, \ where the hinge term is: ₇₈₍₆₄ = ² | e^-i/6d - Iₕ _ |², Iₕ = 43. \ Integration over the projective phase yields a contact term with geometric invariants: \ S = 22 - 7, ₓ₇ = 12 (88 - 7), W = 2425 ₓ₇ S. \ 5. Difference operator and characteristic polynomial The difference operator \ (X = A - B\) with = 22\, (ᵦ I₂), B = 7\, (I₂ ᵦ) \ has characteristic polynomial: \X (x) = x⁴ - 30x² + 1. \ Its root \ (S = 22 - 7\) is the entanglement deficit, measuring the gap between the Tsirelson bound \ (22\) and the entanglement eigenvalue \ (7\) realized at the spinorial phase \ (= /6\). 6. Graph sequence and convergence to the continuum The same algebraic structure generates a sequence of weighted regular graphs whose adjacency matrices satisfy trace constraints. These graphs converge in cut norm to the graphon: (x, y) = W \, e^- d (x, y) \ on the 3-sphere \ (S³\) of radius \ (R = \), where \W = 2425 ₓ₇ S, = ⁴+1 2425 ₓ₇ S. \ The spectral gap \ (₁/₀ = 1. 997651035107\) is consistent with the isotropy of a homogeneous spatial section. 7. Numerical verification and falsifiability All results are expressed in closed algebraic form with no free parameters. A public verification suite generates 18 independent, falsifiable outputs at 50-digit precision, including: closed-form \ (^-1\), historical CODATA analysis, MPS spectral convergence, Monte Carlo simulation, sensitivity scans (\ (\), \ (\), \ (D\) ), PF–DS numerical and symbolic equivalence, statistical test for \ (D=45\), scan of 105 Fano families, cut norm convergence, spectral gap, graphon parameters, and a minimal reproducible script. 8. Main result The numerical evaluation yields: \^-1 = 137. 0359991678, \ which matches the CODATA 2022 central value \ (137. 035999177\) to within \ (10^-8\). The derivation uses no adjustable parameters: all constants are determined by the algebraic structure of the Fano 2-22 moduli space, the Polyakov string tension \ (T = 8\), and the hinge invariant \ (Iₕ = 4/ (3) \).
Massimiliano Blandino (Mon,) studied this question.