A Geometric–Dynamical Framework for the Completed Riemann Zeta Function

We study the completed Riemann zeta function ξ(s) = ½s(s−1)π⁻s/2Γ(s/2)ζ(s), which is entire, order-one, and satisfies the functional equation ξ(s) = ξ(1−s) 1,2. Defining the energy functional E(s) = |ξ(s)|² on the complex plane regarded as ℝ², we analyse the associated gradient flow. We prove, using the functional equation, that the critical line ℜ(s) = 1/2 is an invariant set of the flow. We introduce a transverse curvature H(t) measuring the local geometry of the energy landscape near the critical line. Invoking classical Morse–Smale theory 3 and the argument principle 2, we derive an index-summation formula relating the count of zeros to the count of saddle points in a compact region of the critical strip. These results do not constitute a proof of the Riemann Hypothesis; they provide a geometric framework within which the distribution of zeros is constrained by topological and dynamical considerations.