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

Title: Neutrinos as SU(2) Skyrmions: Mass, Mixing, and the Division Algebra Ladder Author: Alexander Novickis ([email protected]) 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 results: 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, not ad hoc, assignment (§3). A 2+1 mass structure: the j = 1/2 doublet is the solar pair (ν₁, ν₂); ν₃ is a Grassmannian Gr(2,5) singlet; the hierarchy Δm²₂₁/Δm²₃₂ ≈ 0.031 is structural, not fine-tuned (§4). The composed 2+1 mixing matrix: the measured θ₁₂ and θ₁₃ jointly fix (θ_in, V₀/Σ) ≈ (59.8°, 0.111). The doublet–singlet coupling is derived — flux-interaction screening of the shared Berger fiber gives V₀/Σ = (1−2x)/(8(1−x)) with x = 14α (quark activations plus the probe's own winding), within 0.16% of the required value. The solar-angle mechanism (Skyrmion ODE at the Berger point λ = √2, the same condition as sin²θ_W = 3/13) gives 33.10°; its identification with θ_in survives at the half-degree level (JUNO-era precision discriminates). θ₂₃ is not determined by the 2+1 block; the derived slot asymmetry 49.0° awaits identification (§11.1). The SU(2) Wess–Zumino–Witten term vanishes identically (d^abc = 0), so — conditional on the CP-conserving vacuum θ ∈ {0, π} — the Dirac phase is confined to δ_CP ∈ {0°, 180°}, with 180° the live branch: inside the NuFIT 6.0 region and falsifiable by DUNE/Hyper-K (§11.4a). A topological seesaw c = η²: the dim-5 Weinberg operator gives m_ν ~ η² · m_e²/M_W, hence Σm_ν = 60.5 ± 5 meV — at the CMB-S4 detection threshold (§6, §11.6.3). The neutrino is Majorana. The S³ zero-mode census gives a single left-handed Weyl mode with no independent right-handed neutrino, so a massive neutral left-handed fermion is necessarily Majorana; and the derived gauged B−L (geometric Pati-Salam, U(1) (B−L) ⊂ SU(4)) cannot remain unbroken and breaks with ΔL = 2, giving a type-I seesaw. The winding π₃(S³) = ℤ protects soliton stability, not lepton number (§5, §9, §11.6). Neutrinoless double beta decay: the effective Majorana mass is m_ββ ≈ 5 meV (normal-ordering floor ≈ 3.5 meV), below the reach of all next-generation experiments (KamLAND-Zen, LEGEND-1000, nEXO, CUPID), so a null result does not exclude the framework; a definitive Dirac determination would falsify it (§11.6). A sterile neutrino: the j = 3/2 rotational excitation, with E(3/2)/E(1/2) = 5 exactly, gives m_sterile ≈ 309 meV — reachable by direct-mass experiments (KATRIN, Project 8) but not by short-baseline oscillation searches (§D8). The Callias zero-mode mass parameter is derived: m = ∜2 (2 to the power 1/4) in fiber units at the physical Berger point, from the Kaluza-Klein anchoring chain (§5.3.2). A transition magnetic-moment bound μ_eff ≤ 4×10⁻¹⁹ μ_B (§D5) and a supernova-burst discriminator P(2+1) = 0.577 vs P(3f) = 0.681 (§D9). Twelve independent derivations are presented. In the mixing sector the doublet–singlet coupling and the solar-angle mechanism are derived, and δ_CP is structurally confined to {0°, 180°}; the flavor–doublet orientation θ_in and the physical 2-3 rotation remain open. Σm_ν = 60.5 meV is a CMB-S4-threshold test; m_ββ ≈ 5 meV and the 309 meV sterile mass are the framework's distinctive experimental targets; a definitive Dirac (0νββ = 0) determination would falsify the Majorana prediction. Keywords: neutrino, Skyrmion, Majorana neutrino, PMNS matrix, neutrino oscillation, seesaw mechanism, division algebra ladder, quaternionic Hopf fibration, S3 soliton, Berger sphere geometry, neutrinoless double beta decay, sterile neutrino, normal mass ordering, gauged B-L, Pati-Salam, topological soliton, Faddeev-Niemi model, Callias index theorem, Weinberg operator, CMB-S4