Lagrangian of the Fine-Structure Constant: Analytical Derivation of Quantum Gravity (PEPS 5D Formulation)

We present a unified Lagrangian formulation in which electromagnetism and gravity emerge from complementary projections of a single 5-dimensional Projected Entangled Pair State (PEPS). The fundamental cell is a Planck sphere of radius l_p, and its informational volume is axiomatically defined as I_s = π/6 from spinorial holography. The complete unified Lagrangian is the sum of three terms: L_unified = L_volume + L_EM + L_hinge where:- L_volume = ½∂_μΦ∂^μΦ - μ²/2 (Φ² - (π⁴+1))² (tesseract volume potential, gravitational sector)- L_EM = √(-g(t)) (ḋ² - 1/64 (∇d)²) (Polyakov string with tension T=8, electromagnetic sector from Alpha Series, concept DOI: https://doi.org/10.5281/zenodo.19802606)- L_hinge = Φ² |e-iπ/6 Ψ_d - (4/(3π)) Ψ_Φ|² (hinge term that locks the string state Ψ_d to the gravitational volume state Ψ_Φ at the minimal spinorial resolution angle χ = π/6) The hinge term L_hinge is the mathematical suture that unifies the two sectors. It does not stand alone; it couples L_volume and L_EM into a single coherent structure. Solving the combined action on-shell yields the bare gravitational coupling: ln(G_bare m_e² / ħ c) = -π/3 (π⁴ + 1) No free parameters are introduced. The fine-structure constant α (derived in the Alpha Series) and the bare gravitational coupling α_G (derived here) are two eigenvalues of the same topological operator, distinguished only by the dimensionality of their projection: 1D for electromagnetism (circular MPS, bond dimension D=45), 5D for gravity (PEPS, Planck sphere cell). The construction is closed, falsifiable, and requires no external parameters beyond π, the string tension T=8, and the overlap angle π/6. Keywords: Quantum gravity, fine-structure constant, PEPS, MPS, topological invariants, Lagrangian unification, Alpha Series Alpha Series: https://doi.org/10.5281/zenodo.19802606