The Four-Layer Arithmetic Structure of the Fine-Structure Constant

We propose a conjecture: the fine-structure constant satisfies a four-layer sum rule α (Λφ + τ ζ ′ K (0)) + τ = 1 , where Λφ ≡ 360/φ 2 − ln φ = 137.027 is a constant built from the golden ratio φ = (1 + √ 5)/2, τ ≡ 4 kB TCMB/ER = 6.905×10 −5 is the ratio of the cosmic microwave background thermal energy to the Rydberg energy, and ζ ′ K (0) = − ln φ/2 is the derivative of the Dedekind zeta function of Q(√ 5) at its trivial zero. The four layers are: (i) 360/φ 2 = 137.508, the golden-angle geometric ground state; (ii) − ln φ, the Dirichlet regulator; (iii) τ , the cosmic thermodynamic parameter; and (iv) τ ζ ′ K (0), a cross-term coupling arithmetic and thermodynamic structures. Each successive layer reduces the residual by an order of magnitude: 3400 ppm → 69 ppm → 0.11 ppm → 0.009 ppm. The fourth layer yields α −1 = 137.035 998, matching CODATA 2022 to 0.009 ppm—a 12-fold improvement over the three-layer formula. A refined variant employing the exact Bernoulli sum 1 − φ ln φ achieves α −1 = 137.035 999 16 (δ = +0.6 ppb), within the TCMB measurement uncertainty. Two independent mathematical derivations—the Bernoulli generating function evaluated at the Dirichlet regulator RK = ln φ and the Seeley–DeWitt heat kernel on the hyperbolic symmetric space H 2 of Q(√ 5)— converge to the same first-order expression for the fourth layer, providing structural evidence beyond numerical fitting. Crucially, |ζ ′ K (0)| = ln φ/2 introduces no new parameter: it is the class number formula invariant hK RK /wK , fully determined by φ. We show that 360 = (dK + 1)/ζK (−3), where ζK (−3) = 1/60 = 1/|A5| is a special value of the Dedekind zeta function of Q(√ 5)—establishing that the geometric ground state is a purely number-theoretic quantity, independent of the degree convention. The Fermi–Dirac self-referential fixed point 1/φ shares the same discriminant ∆ = 5, so both quantum statistics select Q(√ 5). We discuss structural correspondences with the Bost– Connes system, the McKay correspondence 2I ↔ E8, a quantitative look-elsewhere analysis, five independent consistency checks, and falsifiable predictions.