The Trefoil Hopfion (Q_H=3)

The map t→t+π combined with z→−z is an exact isometry of T_2,3: over-strand and under-strand at every crossing have identical curvature κ=0.399. Each crossing is one symmetric concentration, so the count is exactly 3 = one per quark colour.

Confinement: separating one tube would require simultaneously crossing all three energy barriers. The universe makes a quark–antiquark pair instead.

S_eff(t) = sin⁴θ_T/φ⁶: θ_T = angle between tube tangent and ℤ₃ axis (z). Peaks at all three crossings (θ_T=90°, tangent exactly horizontal)

Flux rays: at each crossing midpoint, two tangent directions (over-strand and under-strand, 47.2° apart) define the colour/anti-colour pair axis. The crossing midpoints all lie in the z=0 plane and their geometric center is the origin — the ℤ₃ fixed point.

Colour mixing w_k(t): shows Gaussian proximity weights — pure colour at crossings, equally mixed midway between. Ribbon twist is uniform per arc length (Option C): 120° per crossing, 270° total, encoding θ_3/2³ = i (prop:wrt_phase).

24-cell of 2T: the 24 unit quaternions of the binary tetrahedral group (order 24), stereographically projected S³→ℝ³. then rotated so the group's ℤ₃ axis [1,1,1]/√3 aligns with the trefoil's physical ℤ₃ axis (z). The McKay chain 2T ↔ E₆ ↔ SU(3)₁ (§1, eq:mckay) identifies 2T as the group-theoretic backbone of the Q_H=3 quark sector, not this rotation itself. Vertices coloured by dominant imaginary axis (red=i, green=j, blue=k) — the ℤ₃ orbit {i,j,k} maps to three quark colours. Edges connect pairs with positive quaternion inner product.