The Hodge Quartet

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). The count "four" is convention-dependent — it reflects the programme's individuation (proton as irreducible, neutrino as one slot), not an observer-independent tally (§2.1); the claim defended here is the narrower one that under that individuation, the stable-particle count matches the S³ cohomology-group count and the massive/massless block structure is forced by duality. 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 the leading-order masslessness of the neutrino, from topology. The equality |e_p| = |e_e| follows from Poincaré duality. These consequences hold once the particle-to-form dictionary is fixed; the dictionary is the conjectured, adjustable part (§2.1, §7), so the correspondence is offered as a consistency between a rigid topology and a chosen reading, not as the topology independently predicting the spectrum. The projection from S³ to the icosahedral surface S² drops the surface mode-count to three, rendering the neutrino invisible from the surface — a conjectured topological origin for weak interaction (the specific restriction map is an open item; see §4). The Poincaré dodecahedral space S³/2I, the quotient by the binary icosahedral group, preserves all four Betti numbers (it is a homology sphere), connecting this cohomological structure to the icosahedral lattice of the 2I Universe Programme. The neutrino's small mass is treated as a leading-order-massless result (b₂ = 0 on bare S³) with a named correction from the icosahedral quotient: the companion derivation [NEUTRINO-PHASE-CARRIER] proposes mν₃ = mₑ α³/4 with generation scaling r = 2^(d/2) (d = 5). A pre-registered menu null [NEUTRINO-MENU-NULL] finds the mass-form integers (3,4) to be the unique within-1% survivor of an integer menu (0.294% match) within the αᵖ/q family, refuting a divisor-fishing reading of the mass (the form itself being motivated, not null-tested); the generation scaling is the only natural small form near the observed ratio but matches at only ~2%, disclosed as coarser. OBSERVED throughout (see §3.1).