HopfionSeries Paper XX - The Dynamical Layer of the Quark Sector: $E_6$-Native Generations, the Isospin Mass-Scale, and Static Residuals as Strong Dynamics

Abstract

Papers V and XV–XIX establish the static, topological content of the QH=3 quark/baryon sector — the trefoil T(2,3), the binary tetrahedral group 2T, the WZW model (E6)1 ⊃ SU(3)1×3, and a saddle-less (algebraic) quark mass. Paper XX assembles the sector's dynamical content and, in doing so, sharpens what the framework is and is not: the static/topological layer is the exact leading order, and the residuals are a single dynamical correction whose size tracks the sector's coupling — largest precisely where the object, unlike the charged lepton, has no stable field saddle. It records six results and one organising principle. This is a draft paper. E6-native generations, E8-bridge anchor, and a parameter-free up-quark ratio The quark's own group is E6; E8 enters only as the charged-lepton bridge that anchors the down sector (strange = muon). Generation is a per-strand E6 ribbon twist, and the up/down tower-steepness asymmetry is the ratio of WZW phase units πQ/(k h), namely (π/4)/(π/9)=9/4, matching the measured steepness (≈2.1) to 6% with no free parameter. The first-to-second up-generation gap is an integer fixed entirely by WZW T-matrix and E6 Coxeter data, giving the zero-parameter ratio mu/mc=e−2π=1.87×10−3, which agrees with the common-scale value to 5%. Isospin mass-scale from tangent orientation, and its residual computed as the confined breathing mode The up/down geometric-mean mass-scale ratio is fixed by the trefoil crossing/midpoint tangent geometry (Paper XVIII), Mdown/Mup ≈ 0.189 at leading order. The generation-universal correction is computed rather than posited: the Faddeev energy is scale-invariant, so the trefoil's size is a flat ‘breathing’ mode that confinement's tube-tension turns into an oscillation, Airy-quantized against the shared-jam wall. The transverse restoring couples to the local curvature, so the more sharply-curved up (crossing) network is held more stiffly (force ratio 1.375, inertia ratio 1.009), multiplying the leading ratio by 0.81 to give 0.153 against the measured 0.157 (≈2%), from the same tangent geometry that fixes the leading value; the residual few percent is the higher-order jam dynamics. A direct Faddeev-energy computation shows the operative quantity is tangent orientation (in-plane at crossings vs out-of-plane at midpoints), not a static energy difference. Fractional charge from colour triality The ℤ3 center of the colour CFT SU(3)1=(E6)1 forces the quark hypercharge Y≡triality/3, giving Q={+2/3, −1/3} for the colour triplet and integer charge for the colour-singlet leptons. The fractional denominator is the colour center — the three crossings of T(2,3) — and the SU(3)1 Chern–Simons edge modes are its dynamical carrier, deriving from the colour CFT what Paper XVIII had assigned by writhe. The generation-1 electromagnetic flip, and its sign as a Chern–Simons level A charge-linear coupling δm=−QΦ flips the u s, t>b intact, since Φ is generation-independent and ∼MeV — decisive at the light end, negligible at GeV. Its magnitude is compositional: Φ=αem σ(3R0+R)/3 ≈ 2.2 MeV, 90% of the QCD neutron–proton splitting, a product of two framework-derived scales (αem−1=360/φ2 and the jam scale) with no new parameter. Its sign is the SU(3)1 Chern–Simons level, locked to the fractional charge, the topological spin, and the matter orientation — one shared vacuum-chirality bit, which a parity no-go forbids deriving from the parity-even sector: an input, but a single shared one, not a per-doublet parameter. The doublet sum rule and the derived constituent scale The confined doublets obey mup+mdown=C mℓ with C≈13, cross-predicting the confined up quarks. The constituent scale follows from the same Chern–Simons and condensate structure as the charged-lepton mass, mconst=Λcondbaryonφ6e8π ≈ 237 MeV (the 8π Chern–Simons spoke), matching the phenomenological 215.5 MeV to 10% and reducing C to the single E6-native tower level, so that deriving C and pinning the tower become one problem. Several attractive numerological readings of C (the 4π, φ6, and solid-angle interpretations) are recorded as retracted. Residuals as the dynamical layer, and why the quark alone has no static saddle A diagnostic across all sectors shows the static/topological predictions are exact at leading order for static-saddle sectors (charged leptons, gauge couplings: residuals below 0.5%) and leave αs-sized residuals (20–30%) precisely in the dynamical sectors (quarks, neutrino) — the two classes separated by a factor ~40 with no overlap, the residual size tracking the sector's coupling. The mechanism behind the split is the density feedback itself: for the spread charged-lepton torus it stays responsive and stabilises a genuine Faddeev saddle (its mass predicted to 0.013%), whereas the sharp trefoil tube drives the feedback into saturation (pinned at 1/β over ~93% of the isotropic energy), so the isolated quark has no stable saddle and its mass is intrinsically algebraic rather than a configuration energy. The organising result is that the framework's residuals are not unrelated misses but one non-perturbative correction, forced into view in the quark sector because the quark alone among the massive fermions has no static saddle; a framework-internal perturbative αs, already built, changes the predictions by only a few percent and does not close them. Of the three open problems, the absolute anchor of the up-tower is fixed algebraically as the completion (de-jamming) energy Λcondbaryone8π=(constituent)/φ6 ≈ 13 MeV — the constituent scale stripped of its φ6 tower factor, with no large-number subtraction — leaving the explicit Chern–Simons/Hall form of the generation-1 coupling and the derivability of the tangent-fraction functional from the density-feedback action as the remaining dynamical handles.