The Helix Wave: Spin, Pair Production, and Sphere Covering from One Geometric Principle

Starting from one equation — v² (rot) + v² (fwd) = c² — and one geometry — the sphere volume (4/3) πR³ — this paper derives 25 results across 10 chapters with zero fitted parameters: (1) The photon has a finite helix radius R = (2/3) λ, where 2/3 is the virial ratio in D = 3 dimensions. (2) The mass–size rule m×R = (4/3) π×ℏ/c holds for all particles, giving mp/me = Re/Rp = 1836 exactly. (3) Photon spin ℏ follows from one helix turn (2π radians) ; electron spin ℏ/2 follows from the Möbius topology (4π radians). (4) Pair production threshold: the photon's helix radius equals the electron's helix radius at E = mec², exactly. (5) The helix correction λ̃ = W^ (−1/3) λ passes 7 of 10 tests, with fine structure reproduced as W^ (−1/3) −1 = α². (6) The Clausius-Mossotti dispersion relation matches BK7 glass to <0. 5% with one calibration point. (7) BEC light freeze (v = 17 m/s, Hau et al. 1999) follows from v (fwd) = cW^ (1/3) → 0 as OE → 1. (8) The shell filling rule 2n² follows from sphere coverage optimisation: with 2n² electrons, collective coverage is uniform across all shells. The framework does not replace quantum mechanics — it provides a geometric layer underneath it. The central open problem remains: deriving α = 1/137. 036 from geometry alone. Paper 2026y in the Speed Gap / Clausius-Mossotti Framework series. 10 chapters, 66 pages. All verification scripts at GitHub.