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…