The Square Root
of Mass

RIGOR Q ≡ (m_e+m_μ+m_τ) / (√m_e+√m_μ+√m_τ)² = 2/3 holds at Q = 0.6666645 ± 0.0000051 (relative accuracy 3×10⁻⁶). In χ = √m variables this is exact equipartition of |χ⃗|² between the singlet and doublet irreps of the family group S₃; the unique S₃-invariant potential of degree ≤ 4 enforcing it is V_eq = λ(n₁ − n₂)². The relation effectively predicted the τ mass in 1981 — 1776.97 MeV against the then-measured 1784.2 ± 3.2 — and later measurements landed on the prediction.

RIGOR The canonical warp and single-scale SQUS assumption derive the near-tip law f ∝ ξ² [A/D]. A constant-ξ orbit requires h ≡ d ln f/d ln ξ = 1, and linear stability requires h′ < 0. The pure quadratic horn has h = 2 everywhere, so it supplies a barrier and turning point, not a stable circular orbit; absolute scale needs a radial completion or separate V_rad. In the stated separable winding model, concavity bounds the untwisted ratios by 4 and 9. A U(1) holonomy then gives ε_ℓ ∝ (ℓ − a)² with a = 0.1726 ± 0.0068, matching 4.878 ± 0.044 and 11.737 ± 0.182 [A/B/D].

STATUS [A/C] A two-component normalized spinor belongs to S³ ≃ SU(2); fixing component amplitudes leaves a toroidal two-phase slice. Antiperiodicity Ψ(θ + 2π) = −Ψ and Ψ(θ + 4π) = Ψ gives the fermionic 4π cycle, while a bilinear B_Γ = Ψ̄ΓΨ removes the sign and is 2π-periodic [A]. The two circles visualize separate projections of the torus's major phase θ₁ and minor cross-sectional phase θ₂, with θ₂ = 2θ₁. At Ω = 2π the small cycle has closed but the large cycle is displaced by π; at Ω = 4π both return to their starting configuration. A physical bosonic channel may instead retain a combination φ_p,q = pθ₁ + qθ₂. Interpreting this torus as SQUS fermion geometry and the circle as an emergent bosonic correlation channel remains a hypothesis [C], pending a Dirac operator, spin connection, holonomy, and gauge-field derivation.

RIGOR [A/C] A normalized two-component spinor lives on S³; at fixed amplitudes the two remaining phases sweep a Clifford torus T² ⊂ S³ — the fibered tori of the Hopf map h(Ψ) = Ψ†σΨ [A]. The spinor bilinear is the Hopf map itself: it is invariant under the fiber phase Ψ → e^iγΨ, so the 2π sign lives entirely in the fiber that the projection removes. By Sorkin's theorem the exchange loop is homotopic to the 2π-rotation loop, so any Hopf-projected channel carries trivial exchange phase, obeys symmetric statistics, and may condense — because the projection removed the topological obstruction, not because of dimensional counting [A]. A bare axisymmetric torus admits no fermionic quantization; phase decoration (n, k) restores it exactly when nk is odd, making (1, 1) the minimal fermion [A]. Identifying this internal torus with the spatial SQUS loop of the winding tower is the framework's central structural hypothesis [C/D].

RIGOR The measured fermion spectrum is log-uniform — the unique measure invariant under the reparametrization m ↔ χ [C]. Three independent measurements exhibit one and the same scale-invariant fixed-point measure along the mass axis: the log-uniformity of the spectrum, the renormalization-group freezing of the Koide ratio, and the electron's spectral dressing profile, whose plateau equals the QED anomalous dimension exactly. Neutrino values are the B₃ equipartition-cone predictions (m₁, m₂, m₃) = (0.364, 8.662, 50.13) meV, normal ordering, Σm_ν = 59.16 ± 0.24 meV.

RIGOR Table IV of the revised main paper. Direct tests are separated from conditional signatures; formal winding levels are not experimental predictions until their mixing and population rules are derived. Claims are labelled by rigor throughout: [A] theorems and exact computations, [B] computations confronted with data, [C] empirical regularities with uncertainties but no mechanism, [D] hypotheses and outlook.