The Proton as Twin Prime Resonance: Colour Structure, Field Energies, and Mass from First Principles

The proton's colour structure is conventionally described by three abstract labels assigned by SU(3) gauge symmetry with equal field strengths. This paper derives the colour field strengths from first principles within the Bootstrap Universe framework. The sine wave is shown to be the unique real projection of self-referential rotation: the eigenfunction of d/dθ is e^(iθ), and physical observation projects this onto the real axis. 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 Hodge-star self-dual band, selected by Eisenstein three-phase cancellation. 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 field tension formula is derived from the spectral geometry of S³: the eigenvalue λ_l = l(l+2) at twin prime position a gives √(ab) = √λ_a as the spectral normalisation scale. 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 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 from the single arithmetic observation that (11+13)/2 = 12.