S³ geometry · 2026

Mass as the Self-Energy of the Confined Photon on S³

R. G. Measey

Abstract

In the confined-photon picture a particle is a single photon bound to a torus knot in the 3-sphere S³. We show that such a photon acquires a mass for a purely geometric reason: as a conformally coupled field it feels the S³ curvature through the conformal coupling ξ=⅙, and ξR = 1/a² leaves a clean mass gap ℏc/a. The same ξ completes the square of the S³ spectrum, k(k+2)+1=(k+1)², producing an evenly spaced tower ω_n = n·ℏc/a whose ground state (n=1) is the lightest particle and whose second rung (n=2) reproduces the electron–positron pair-production threshold 2m_e. We identify the lepton mass Generations as emerging from the conformal Willmore bending energy penalty E_bend ∝ Σk⁴, reproducing muon to −0.10% and tau to +0.52% with one coefficient and no per-particle tuning. The sequence terminates at n=2 (three generations) due to hadronic deconfinement.

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