A Geometric Framework for the Riemann zeta Function via Dirichlet Helix Dynamics

Hardy's theorem establishes that the Riemann zeta function has infinitely many zeros. The present work studies the geometric mechanism underlying these zeros.In the first part, the Dirichlet vector walk S_n(s) = sum_{k ≤ n} k^{-s} is analyzed geometrically on the critical line. Under the stabilization mechanism identified in the correlation dynamics, the walk assumes a rigid helical form S_n(t) ≈ sqrt(n) * exp(-i t log n) * c(t) which represents perfect geometric cancellation. This helical configuration produces vanishing in the analytic continuation and can occur only when Re(s) = 1/2. In the second part, the perfect helix is used as an analytic probe. Subtracting the canonical helical carrier from the analytic remainder R_n(s) = S_n(s) - n^(1-s)/(1-s) yields a second-stage residual. Using the Euler–Maclaurin expansion, we show that after subtraction of the perfect helix the residual has leading term proportional to n^{-s} and therefore cannot asymptotically vanish. Thus any nontrivial zero must arise from exact cancellation by the perfect helical mechanism itself. Since this mechanism is critical-line selective, zeros can occur only when Re(s) = 1/2.