Metric Resonance and the Riemann Hypothesis: A Phase–Cone Interpretation in C4 and C6 Algebras

Abstract We propose a geometric–algebraic framework in which the Riemann Hypothesis (RH) emerges as a null–energy condition within a Lorentz–type phase metric defined by the quartic complex algebra C4 (k⁴ = −1). The conjugation rule k* = −k³ induces a hyperbolic normNC4(t + xk + yk² + zk³) = −t² + x² + y² + z²,which mirrors the analytic symmetry of the Riemann zeta function under s ↦ 1 − s. In this setting, the critical line ℜ(s) = 1/2 corresponds to the null axis of the phase cone, and the nontrivial zeros of ζ(s) represent light-like states in the C4 phase manifold. We further introduce the Phase–Cone Inequality, a positive-energy condition ensuring that the analytic continuation of ζ(s) remains bounded within the causal domain of the phase metric. The extension to the C6 hybrid layer (π + ζ(3)) provides a perturbative modulation that reproduces the oscillatory amplitude of the completed zeta function ξ(s), while preserving the null condition. This approach geometrically reinterprets RH as a statement of metric equilibrium: the alignment between analytic continuation symmetry and Lorentz-type invariance. The resulting metric resonance offers a unified phase–geometric explanation for the localization of zeros on the critical line and opens the possibility of a proof via positivity and null-locking principles analogous to the positive-energy theorem in general relativity.