A Topological Realization of the Riemann Zeta Function via Non-Resonant Geometries of Self-Inverting Toroidal Vortices.

The Riemann Hypothesis asserts that all non-trivial zeroes of the zeta function, zeta(s), lie on the critical line Re(s) = 1/2. While traditional approaches rely heavily on arithmetic and algebraic number theory, this paper introduces a complementary topological and geometric framework. We model the distribution of numbers as an unclosed, open-ended three-dimensional helical spiral wrapping around a static, unbending control axis governed by a primary residue node (3 mod 9). By evaluating the binary doubling orbit under the map T: Z_9 -> Z_9, T(x) = 2x (mod 9), we isolate a deterministic, self-cancelling material manifold M = {1, 2, 4, 8, 7, 5} whose unit-vector sum vanishes identically on the complex unit circle: Sum[e^(i2pi*k/9)] = 0 for k in M (the Net-Zero Theorem). We propose that the critical line Re(s) = 1/2 represents the absolute geometric mirror axis of a self-inverting toroidal vortex, and that non-trivial zeroes manifest as nodes of destructive wave interference while prime numbers emerge as discrete non-resonant vertices sustaining infinite helical expansion without structural drift. We state this formally as a conjecture and delineate the open operator-theoretic steps required to elevate the framework to a complete proof.