The Kerr-Riemann Equations: A Continuous 4D Phase-Space Framework for Modeling Asymmetric Strain in the Critical Strip

Abstract

While classical analytic number theory frequently relies on static asymptotic bounds to evaluate the critical strip of the Riemann zeta function $\zeta(s)$, navigating the chaotic interference of off-axis trajectories remains computationally and analytically opaque. In this paper, we introduce the Kerr-Riemann Equations, establishing a robust suite of continuous geometric metrics anchored analytically to the Digamma function $\psi(s)$. By unfreezing the real coordinate $\sigma$, we define the Kerr Manifold—a determinate, non-conformal 4D Euclidean phase-space that embodies Riemann's complex sphere, translating lateral complex displacement directly into calculable physical strain. From this framework, we derive the Unified Sigma Isolator to algebraically model irreducible Pythagorean strain and formulate the Lock-Step Mapping to quantify conjugate amplitude skew. Furthermore, by applying Cauchy's Argument Principle to this rigidly budgeted Euclidean space, we introduce new topological constraints—namely the $4\pi$ Orbit Toll and Arc Length Starvation—which establish absolute bounds on the kinetic phase velocity required to sustain theoretical off-line zero doublets. Validating these structural metrics against Odlyzko's computational dataset of 2,001,052 empirical zeros, we demonstrate that this geometric architecture imposes severe mechanical barriers to off-axis trajectories, providing a rigorous new mathematical foundation for investigating the structural stability of the critical strip.