The Perfect Helix Regime in the Dirichlet Walk and the Zeros of the Riemann Zeta Function

We study the Riemann zeta function through a geometric viewpoint based on the Dirichlet partial sums

S_n(s) = sum_{k<=n} k^{-s},

interpreted as a vector walk in the complex plane. In this framework the behavior of the walk is naturally described in a co-rotating frame, where the dominant structure of the system appears as a rigid logarithmic helix.

The first part of the analysis shows that stabilization of the Dirichlet walk forces a perfect helical configuration

S_n(t) = sqrt(n) * exp(-i t log n) * (c(t) + o(1)),

which corresponds to exact geometric cancellation of the walk.

In the second part, this perfect helix is used as an analytic probe. After subtracting the canonical helical carrier arising from the analytic continuation of the series, we analyze the remaining term using the Euler-Maclaurin expansion. The resulting second-stage residual has leading term of order n^{-s} and therefore cannot vanish asymptotically.

Consequently, vanishing can occur only through exact cancellation by the perfect helical mechanism itself. Since the helical regime is critical-line selective, this geometric mechanism confines the nontrivial zeros of the zeta function to the critical line.