Geometric-Spectral Unification of the Riemann Zeta Function and Cosmic Information Theory: From the Zeta Metric to the σ-Flip Mechanism

We present a rigorous synthesis between two independent research programs: the geometricspectral series on the Riemann Hypothesis and the Cosmic Information Theory (CIT/TIC) framework. We prove that the information field Σζ (s) = − log |ζ (s) |, defined on the critical strip, induces a pseudo-Riemannian metric gij = ∂i∂jΣζ of forced Lorentzian signature. The associated d’Alembertian satisfies the field equation gΣζ = τζ, where τζ = P n δ (s − ρn) is the Gaussian curvature concentrated at the non-trivial zeros. Via Hadamard–Weierstrass decomposition, the eigenvalue problem gf = λf reduces to a generalized Lamé equation with quasi-periodic potential. Through a conformal embedding s 7→ x µ and thermal calibration T (t) = TQCD (t ∗/t), we identify the null point t ∗ ≈ 5. 5612 as the geometric trigger of the U (1) σ phase transition at TQCD ≈ 150 MeV, recovering the σ-flip mechanism, the Temporium Lagrangian, and the dark matter hierarchy mχ ∼ 1 GeV, mΣ ∼ 3. 4 meV. We state a Unified Theorem linking spectral identification to the Riemann Hypothesis and provide an explicit epistemic status matrix. The framework is conditionally complete, with the sole remaining gap being the rigorous triviality of Fuchsian monodromy for the quasi-lattice tn, a well-posed problem in modern spectral theory.