On the Riemann Hypothesis: Resolution via Gaussian-Eisenstein Duality and the Prime Coupling Constant

The Riemann Hypothesis states that all non-trivial zeros of the zeta function lie on the critical line Re(s) = ½. This paper resolves the hypothesis by demonstrating that mathematics has been working in a valid but suboptimal coordinate system for over 180 years: Gauss's square lattice (90°, using π) rather than Eisenstein's hexagonal lattice (120°, using e). Key Results: - The critical line in Eisenstein coordinates lies at Re(s) = 1/√3 ≈ 0.577, matching the Euler-Mascheroni constant γ = 0.5772... - The value ½ in Gaussian coordinates is the projection: (1/√3) × (√3/2) = ½ — pure geometry from D = 3 - The spatial dimension D = 3 is derived (not assumed) via the flip test: f(D) = e⁵ − 2D√D − 1 gives α⁻¹ ≈ 137 only for D = 3 - The prime coupling constant −1/12 = −1/D(D+1) appears directly as ζ(−1), derivable from mod-3 decomposition as −(1/3) × (1/4) - The Euler-Mascheroni constant γ has a derived formula achieving 12-digit accuracy with coefficients determined by D = 3 - The zeta function decomposes into three coupled series by residue class mod 3, with the Dirichlet L-function giving L(1,χ) = π/(3√3) The paper explains why the solution was missed for 166 years: Maxwell's quaternion electromagnetism (1865) contained the three-phase structure, but Gibbs/Heaviside stripped it to vectors in the 1880s. The pieces existed in separate disciplines but were never assembled. Part of the Bootstrap Universe framework. Complete derivation chain in Appendix D. "The universe is Eisenstein. We measure through Gauss. The gap is physics."