The Hopfion Torus (QH=2)
A topological soliton of the quantum condensate · R0=3, C*=3.4318
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Hopf energy density — inner-wall crescent
Inner axis thread — tube-core energy concentration ρ<0.4
Transition ring (ρ=1, f₀=π/2 isosurface)
f₂ · Inner core ρ=0.3, f₂=−0.0045
f₂ · Inner core 2 ρ=0.5, f₂=−0.032
f₂ · Approaching transition ρ=0.7, f₂=−0.107
f₂ · Outer transition ρ=1.5, f₂=−0.093
f₂ · Outer region ρ=2.0, f₂=−0.027
f₂ · Wing extension ρ≥2.8, f₂=+0.006
Cross-section (φ=0) — Paper I colormap
600-cell (2I) 120 quaternions · 720 edges · 8 shells
φ = (1+√5)/2 — encoded in the 600-cell geometry
2I ⊂ SU(2): 120 unit quaternions = vertices of the 600-cell on S³.
Stereographic projection S³→ℝ³ reveals 8 concentric shells.
Outer icosahedron (w=−φ/2) sits at r=R₀=3.
Identity (1,0,0,0) → origin = intersection of 3 golden
rectangles = center of the Hopfion = fixed point of all 120
symmetries of 2I.
At ρ=1 transition ring:
inner wall (r=2): dens = 27.7 | outer (r=4): 7.7
Ratio 3.6×, matching Paper I Panel B
Inner axis thread (ρ<0.4): residual Hopf energy inside the tube
cross-section — the concentration that emerges at the torus centre
when the condensate is energetically perturbed toward the poles.
Visible in dynamical integrator runs as a bright central node.
The knot in the condensate whose toroidal geometry
encodes the WZW lepton sector.
Live · deformation=1.0
Δ(ρ=1) = 0.311 · rcenter = 2.689
Inner/outer = 3.6× · Shafranov = 10.4% R₀