HopfionSeries Paper XV -The Q_H=3 Sector of the Density-Feedback Hopfion

@misc{manfredi2026hopfionseries,
  title = {{HopfionSeries Paper XV -The $Q_H=3$ Sector of the Density-Feedback Hopfion}},
  author = {Frederick Manfredi},
  howpublished = {Zenodo (CERN European Organization for Nuclear Research)},
  year = {2026},
  month = jun,
  doi = {10.5281/zenodo.20691001},
  url = {https://doi.org/10.5281/zenodo.20691001},
  note = {CC BY},
}

TY  - UNPB
TI  - HopfionSeries Paper XV -The Q_H=3 Sector of the Density-Feedback Hopfion
AU  - Frederick Manfredi
PY  - 2026
DA  - 2026/06/16
PB  - Zenodo (CERN European Organization for Nuclear Research)
DO  - 10.5281/zenodo.20691001
UR  - https://doi.org/10.5281/zenodo.20691001
AB  - Paper XV extends the framework from the QH=2 lepton sector (torus, binary icosahedral 2I, SU(2)3) to the QH=3 quark sector (trefoil knot T2,3, binary tetrahedral 2T, SU(3)1). The McKay chain 2T ↔ Ê6 ↔ (E6)1 ⊃ SU(3)1×3 identifies the sector’s WZW model via a conformal embedding with exact central-charge match c = 6 = 3×2, mathematically stronger than the pentagon identification used for QH=2. Quark charges from conformal weights The trinification embedding E6 ⊃ SU(3)C×SU(3)L×SU(3)R at level 1 gives the charge generator Q = T3L + T8L/√3 uniquely from SU(3)L group theory (tracelessness plus ℤ3 centre compatibility with no Standard Model hypercharge input). Because the charge-generator direction in Cartan space is μ = 2ω1, both quark charges follow from the single number (ω1, ω1)A2 = 2/3: |Qd| = h(1,0) = 1/3 and |Qu| = 2h(1,0) = h27 = 2/3. The conformal weight h(1,0)=1/3 is the sector’s topological spin e2πi/3 (the colour ℤ3 braiding phase), so fractional charge is an anyonic Wilson-loop holonomy — the quark-sector generalisation of the Aharonov–Bohm phase. This identity holds only at N = 3, the unique N where hfund(SU(N)1) = 1/N, tying the charge derivation to the same N forced by the McKay correspondence. Trefoil geometry and colour confinement The trefoil T2,3 at R0=3, r0=0.874 has L3 = 41.26, ⟨κ⟩ = 0.335, ⟨κ2⟩ = 0.120 ≈ 1/R02. All three crossings have identical curvature κmax = 0.399 (proved by the t ↦ t+π, z ↦ −z isometry), so the energy-concentration count is exactly 3, one per quark colour. Torsion cancels at leading order for the symmetric profile (a small O(κ2τ2) correction, ≈4% of ⟨κ2⟩, enters once the curvature-sourced profile correction is included). The three crossing midpoints form an equilateral triangle; the origin is the Fermat–Steiner Y-junction of the baryon string network (3 arms of length R0, total string length 3R0 = N32 = 9). Colour confinement follows: separating one tube requires simultaneously crossing all three energy barriers, the universe creates a quark–antiquark pair instead. λ3 = φ6 proved: the Bogomolny parameter The sector assignment QH=3 ↔ j=3/2 (the simple current of SU(2)3) is proved by three independent arguments. (i) Triple-fusion multiplicity: the fusion j1/2⊗3 = j1/2(×2) ⊕ j3/2(×1) at k=3 has a unique multiplicity-1 channel; the topological non-degeneracy π3(S2) = ℤ forbids the multiplicity-2 channel j=1/2. (ii) Verlinde uniqueness: |Sj, 1/2/Sj, 0| = φ only at j=0 and j=3/2; since j=0 is QH=2, QH=3 must be j=3/2. (iii) ℤ2 charge parity: QH odd forces j ∈ {1/2, 3/2}, then multiplicity selects j=3/2. The simple-current self-fusion 3/2 ⊗ 3/2 = 0 gives Nfus=1, the Linking Scale formula gives sopt2k=1, and λ3 = φ6 follows in three lines. All conditional results now unconditional λ3 = φ6 simultaneously closes four previously conditional conditions: r3 = 2/φ5 (the geometric saddle ratio); φ9√(J2a(3)J4(3)) = 3 (the QH=3 normalisation identity, proved equivalent to λ3 = φ6 by four-line algebra); V3 = 9/(5φ17) (the condensate volume); and C3* = 2.497 (the trefoil saddle concentration). WRT corroboration: three exact results From Witten–Reshetikhin–Turaev theory at k=3 and q = e2πi/5: (1) the writhe-3 framing phase θ3/23 = i (a primitive 4th root of unity — smallest non-trivial order at writhe 3; proved: h3/2 = 3/4, so 3 × 3/4 = 9/4 ≡ 1/4 mod 1); (2) the quantum dimension dimq(3/2; k=3) = 1 (string-tension minimisation for confined states, since sin(4π/5) = sin(π/5)); (3) the WRT contribution S0,3/2 θ3/23/S0,0 = i has unit modulus — the j=3/2 sector undergoes no dimensional renormalisation under trefoil surgery. A topology-preserving numerical confirmation of C3* = 2.5062 from the QH=3 profile was once the remaining step, but since λ3 = φ6 and C3* are proved analytically it is a verification, not a precondition. The companion Paper XXIII resolves the deeper point: because the feedback energy is a product the QH=3 object is a saddle, not a minimum, so gradient flow — which seeks minima — cannot converge to it, and the trefoil is transient, with no isolated static minimum to relax to. Its stationary configuration is instead located as the index-1 transition state of the pion-emission reaction T(2,2)→T(2,3); what remains open is only that transition state's energy in physical units — a formation-energy evaluation gated by tube resolution, not a static saddle-finding algorithm.
ER  -