The Proton as Three-Phase Icosahedral Motor: Colour Field Strengths, Mass, and Gluon Modes from First Principles

The proton's colour structure is conventionally described by three abstract labels — red, green, blue — assigned by SU(3) gauge symmetry with equal field strengths. This paper derives the colour field strengths from first principles. The sine wave is shown to be the unique real projection of self-referential rotation: the only periodic function satisfying f′ = f under rotation is e^iθ, whose amplitude modulation is sine. Physical waves are sinusoidal not by assumption but because they are real projections of complex rotation in Hamilton quaternion space. From this foundation, the six irreducible colour positions of the proton are derived as the twin prime pairs (5,7), (11,13), (17,19) — the complete set of mod 24 prime residues within the confinement boundary, selected by Eisenstein three-phase cancellation (1 + ω + ω² = 0). The golden angle modulation f(n) = sin(nπ/φ²) of these positions gives field tensions with the exact ratio F_RED/F_BLUE = 2.000, producing a u/d quark momentum ratio of 2.000 — the experimental lower bound from DIS measurements — with no free parameters. The zero-crossing structure of f(n) derives charge assignment: RED and BLUE cross zero (charged quarks), GREEN does not (neutral gluon mediator). Chirality and the u,u,d quark content are derived from the asymmetry f(1) + f(23) ≠ 0. The twin prime pair midpoints (6, 12, 18) stand in the ratio 1:2:3 — the first three harmonics of the 24-period standing wave. This identification unifies the twin prime colour domains with the eight gluon vibration modes of the quasi-crystal proton and yields a binding selection rule: the proton mass is M_p = 2E_R + E_B − 12E_G = 938.78 MeV (observed 938.27 MeV, error 0.054%), derived with no free parameters.