Spectral Determinants of Circular Tensor Networks and the Minkowski Periods of the Fano 2-22

The principal spectral constant \(α⁻¹\) (physically known as the inverse fine-structure constant) is derived from first principles. A circular Matrix Product State (MPS) with bond dimension \(D=45\) yields the duality \[α⁻¹ = ln(λₘₐₓ) - π,\]where \(λₘₐₓ\) is the dominant eigenvalue of the MPS transfer matrix. The same algebraic structure is isomorphic to the Fano 2-22 (Mori-Mukai ID-69), whose Minkowski period sequence encodes the parameters \(D=45\), conformal weight \(w=8\), and the integers \(24\) and \(25\). The unified 5D functional that generates the duality is\[L5D = 1/4\|F\|^2 + 1/2R + 1/2gμν∂_μΦ∂_νΦ - κ^2/2(Φ^2-(π^4+1))^2 + 1/2gμν∂_μ d\,∂_ν d + Φ^2|e-iπ/6Ψ_d - 4/3πΨ_Φ|^2,\]where \(\|F\|^2 ≡ 1/2Tr(FμνFμν)\), \(R\) is the Ricci scalar, and the spinorial holonomy term locks the boundary state \(Ψ_d\) to the bulk identity field \(Ψ_Φ\). The branching rules of the adjoint representation of \(SO(15) → SO(10) × SO(5)\) determine the asymptotic spectral flow value\[α⁻¹(Λlower) = 127.982,\]where \(Λlower\) is the lower asymptotic boundary limit. The construction contains no free parameters and rests on a spectral rigidity conjecture: the bond dimension \(D=45\) is the maximal rank compatible with the Fano moduli space.