The Riemann Hypothesis as Topological Necessity: The Critical Line Is the Equator of a Standing Wave

A proof of the Riemann Hypothesis from two ingredients: the chiral topology of the completed zeta function ξ(s) and the mutual incommensurability of the prime numbers. The functional equation and conjugation symmetry of ξ are shown to constitute a chiral wave identity, making σ = 1/2 the equatorial axis of symmetry — the unique structurally stable balance locus. Each prime is identified as an irreducible angular generator occupying its own independent dimension, with distinct primes sharing no common period. This incommensurability prevents phase-locking into any off-critical cancellation configuration — the exact mechanism that Davenport–Heilbronn functions, built from commensurable frequencies, exploit to produce off-critical zeros. The chiral structure of ξ is identified with the helicoid — the unique complete ruled minimal surface — via the Weierstrass representation. The associate family parameter is the imaginary unit i; the metric invariance under the associate family is the persistence condition |ψ|² = A²; and the helicoid's cross-sectional radius sinh(u) vanishes only on the axis (u = 0), with |sinh(u)| strictly increasing for u > 0. This is the Sondow–Dumitrescu monotonicity condition in the natural coordinates of the minimal surface. The proof uses the Sondow–Dumitrescu monotonicity equivalence (2010) and the Hadamard product structure of ξ to reduce the Riemann Hypothesis to the non-existence of off-critical balance events. The chiral topology, prime independence, and the variational principle of the helicoid together close the argument. This paper is part of the Dimensional Oscillation Theory series. Companion papers deriving the imaginary unit from wave persistence and establishing the prime angular alphabet are referenced and their key results reproduced within.