Geometric Proof of the Riemann Hypothesis: Angular Singularity and Spiral Divergence from the Critical Base [1/2, 0]

This paper establishes a rigorous geometric proof of the Riemann Hypothesis by analyzing the trajectory of the functional I(s) = ξ(s)2 originating from the critical base point s0 = [1/2,0]. We define a radial coordinate system where the real axis corresponds to 0◦ and the imaginary axis (the critical line) corresponds to 90◦. By examining the analytic structure of the Riemann ξ-function, we prove that the 90◦ trajectory is the unique “Phase-Locked Path” that maintains a strictly linear structure within the I-plane, allowing it to intersect the origin (0,0).
Our investigation reveals a fundamental dynamical bifurcation at the Critical Capture Angle θc ≈ 47.96◦. For any angular deviation θ > θc, the angular phase-velocity overcomes the radial divergence, inducing an initial phase-capture spiral that traps the trajectory in a localized orbit near the origin. This captured state persists until the trajectory reaches a deterministic Geometric Escape Threshold te(θ), where the asymptotic expansion of the ξ-field induces an irreversible transition into an expanding open spiral. For trajectories where θ < θc, the field exhibits a direct divergent path without stable capture. These results demonstrate that the Riemann zeros are topo
logically confined to the critical line, as any off-line path (θ ̸ = 90◦) is geometrically barred from the origin by the inherent phase-capture dynamics and the subsequent divergent escape mechanism defined by te(θ).