Neutrinos as SU(2) Skyrmions: Mass, Mixing, and the Division Algebra Ladder

We propose that neutrinos are B = 1 Skyrmions in S³, motivated by the division algebra ladder: ascending from the complex Hopf fibration S¹ → S³ → S² (electron) to the quaternionic S³ → S⁷ → S⁴ (neutrino), the electron's total space becomes the neutrino's target. Since π₃(S⁴) = 0, solitons are forced into the fiber S³, yielding proven stability (π₃(S³) = ℤ), spin-1/2 (π₄(S³) = ℤ₂), and electric neutrality. The standard Skyrmion gives a j = 1/2 doublet identified with the solar pair (ν₁, ν₂), while the third neutrino ν₃ arises as a singlet excitation on the Grassmannian Gr(2,5). Key original results include: Division algebra ladder: ℝ → ℂ → ℍ → O with Hopf fibrations at each step; the electron lives at the complex level, the neutrino at the quaternionic level — a structural rather than ad hoc assignment (§3) 2+1 mass structure: j = 1/2 doublet = solar pair (ν₁, ν₂); ν₃ = separate Grassmannian singlet; the experimental hierarchy Δm²₂₁/Δm²₃₂ = 0.031 (nearly degenerate doublet) is structural, not fine-tuned (§4) θ₂₃ = 49.0° derived from Wigner-Eckart theorem (45° at leading order) + level repulsion (+0.85°) + Berry-phase cross-metric ξ_geo = 0.325 (derived, not fit); experimental: 49.0 ± 1.3° (§11.1a) θ₁₂ = 33.10° with zero free parameters: e² = 2√2 derived from the Berger sphere condition (same geometric condition as sin²θ_W = 3/13 in Paper VII); experimental: 33.44 ± 0.77° (−0.44σ) (§11.1b) θ₁₃ = 8.5° as a structural prediction of the 2+1 framework; experimental: 8.6 ± 0.1° (§4) Seesaw mechanism: c = η² (fiber energy fraction squared) from gauged S⁴ dynamics; KK overlap integral with exact gauged profiles; p = 1 excluded, p ≥ 2 consistent (§6.2–6.3) Σmν = 62 ± 5 meV: robust across (e², μ²) parameter space; three independent computations converge; at the CMB-S4 detection threshold (Planck bound 120 meV) (§11.6) Normal mass ordering from scalar potential: μ² ≥ 1 restores normal ordering across all tested parameter points; O(1) value, no fine-tuning required (§6.3) Grassmannian ν₃ identification: first Gr(2,5) excitation = adjoint singlet (C₂ = 6) from SO(5) branching, naturally interpreted as the atmospheric neutrino (§D2) Magnetic moment bound: transition moment μ_eff ≤ 4×10⁻¹⁹ μ_B (below GEMMA sensitivity) (§D5) Supernova burst discrimination: oscillation probability P(2+1) = 0.577 vs triplet 0.681; measurable in neutronization burst at ~5 events/year (§D9) Twelve independent derivations are presented with six proven quantitative predictions. Dirac nature is suggested by ℤ topology (no Majorana mass term). The paper derives all three PMNS mixing angles with 0–1 free parameters.