Deriving Geometry: From One Axiom to Complex Numbers, the Prime Alphabet, Euclidean Space, and the Structure of the Riemann Zeros

This paper derives geometry from a single axiom: a fundamental entity is a chiral wave — a rotation with a fixed, unchangeable handedness. From this one sentence, with no further assumptions, the following are established. Frame-independent chirality requires the wave's symmetry group to act transitively on S², forcing SO(3)/SU(2) symmetry and all-plane rotation — proved formally as Theorem 1. The imaginary unit i is not an algebraic postulate: two perpendicular oscillating faces are forced by the constant-magnitude requirement (Theorem 2), and the operator connecting them necessarily satisfies i² = −1. Euler's formula is not an algebraic identity but the description of a helix. The dimensional structure of space emerges from a ladder of rotations: each dimension is generated by rotating the previous shape (line → circle → sphere), and the angle between adjacent dimensions determines the required algebra — 90° forces complex numbers, 120° forces the three-mode structure that produces the triangle. The standing wave modes of the sphere are identified precisely as the spherical harmonics Y_ℓ^m. The prime numbers are proved to be the irreducible letters of an angular alphabet: the angles 360°/p that no combination of other prime angles can produce (Theorem 3: log p / log q is irrational for distinct primes). The Fundamental Theorem of Arithmetic is reinterpreted as a wave-independence statement; the geometric picture illuminates why unique factorization holds, while the classical algebraic proof proceeds via Euclid's lemma. Euclidean geometry is shown to be the infinite-wave limit of spherical wave geometry — every Euclidean axiom is a convergence statement. The Riemann Hypothesis is approached geometrically. The Riemann sphere is the natural domain of ζ as a meromorphic function — not an analogy. The functional equation is the equatorial reflection symmetry of this sphere. The paper identifies the Euler product as a chirality-conservation law and uses the Davenport-Heilbronn theorem (1936) as the critical test: Dirichlet series satisfying the functional equation but lacking an Euler product have zeros off the critical line, establishing that the Euler product — not the sphere alone — is the confinement mechanism. The remaining open step is identified precisely: proving that the Euler product forces strict horizontal monotonicity of |ζ(σ+it)| for 0 < σ < 1/2 (which, by Matiyasevich-Saidak-Zvengrowski 2014, is equivalent to RH). All claims carry explicit status labels: derived (proved from the axiom), reinterpreted (existing mathematics explained by the framework), confirmed (established independently by experiment or prior mathematics), or open.