We introduce a geometric framework for the study of the Riemann zeta function based on a canonical helical structure arising in the Dirichlet partial sums on the critical scale sqrt (n). The principal asymptotic term n^ (1/2 - it) = sqrt (n) * exp (-i t log n) naturally defines a geometric helix Hₙ (t) = (sqrt (n) cos (t log n), sqrt (n) sin (t log n), log n). This helix serves as the canonical geometric carrier of the Dirichlet walk. In the first stage we show that stabilization of the Dirichlet walk along this perfect helical geometry forces the analytic remainder of the Dirichlet sum to vanish. Moreover, such a non-degenerate helical regime can occur only on the critical line Re (s) = 1/2, yielding a one-directional implication linking the perfect helix geometry to zeros of the zeta function. In the second stage we use the canonical helix to construct a projection-based second analysis of the remainder. Examining the remainder in the co-rotating helix frame leads to a geometric dichotomy between a line state (perfect helix) and a wave state. We show that the wave component coincides with the second-stage residual and that this residual cannot vanish due to a persistent oscillatory component. Consequently no additional zeros arise between the helical zeros detected in the first stage, implying that the zeros of the zeta function occur only on the critical line.
Aviad Shetrit (Fri,) studied this question.