The Hodge Quartet: Proton, Photon, Neutrino, and Electron as the Four Topological Necessities on S³

The de Rham cohomology of the 3-sphere S³ has exactly four groups: H⁰, H¹, H², H³, with Betti numbers (1, 0, 0, 1). This is a theorem, not a model. We conjecture that these four cohomology groups correspond to the four stable fundamental particles of nature: the proton (0-form, point mode), the photon (1-form, line mode), the neutrino (2-form, surface mode), and the electron (3-form, volume mode). Poincaré duality on S³ pairs H⁰ ↔ H³ and H¹ ↔ H², yielding two pairs: a massive matter pair (proton–electron, b = 1) and a massless mediator pair (photon–neutrino, b = 0). The vanishing Betti numbers b₁ = b₂ = 0 derive the masslessness of the photon and neutrino from topology. The equality |e_p| = |e_e| follows from…