HopfionSeries Paper XVIII - Preimage Topology and Confinement: Preimage Topology and Confinement: The $Q_H=3$ Sector of the Density-Feedback Hopfion - DRAFT

Paper XVIII develops the dynamical content of QH=3 confinement: why the baryon sector is confined, what topological reaction rules govern its decay, and what experimental signatures distinguish it from the lepton sector. Using the Jones polynomial evaluated at the WZW root of unity q5=e2πi/5, the paper shows that rationality of the Jones invariant tracks fusion of the cable strands in a satellite construction rather than which QH sector is involved, forces pion emission as a topological necessity in baryon formation, derives the complete jet energy cascade from the trefoil's crossing geometry, and obtains a zero-free-parameter estimate of the up/down–vs–strange-type quark mass scale ratio from the trefoil's tangent geometry. This is a draft paper with several open problems noted below, two of which are addressed in the companion draft Paper XIX. Jones polynomial arithmetic: fusion, not sector, splits rational from irrational Computing the trefoil T(2,3) correctly as a (2,3)-cable satellite of one component of the QH=2 Hopf link (rather than as an abstract free-standing knot) gives |Jonessat(q5)|=1/φ. Directly computing the Hopf link's own Jones invariant gives the same modulus, |JonesT(2,2)(q5)|=1/φ: the two sectors are related by a single phase, Jonessat(q5)=q5−1·JonesT(2,2)(q5), not by an inversion of the golden ratio. A satellite twist family reveals where the rational/irrational split actually lies: varying the half-twist count n, every fused configuration (genuine torus knot, odd n) has an irrational, golden-ratio-related modulus (n=3 trefoil: 1/φ; n=5 cinquefoil: φ), while every unfused configuration (cable strands not yet merged, even n) has a rational invariant (n=4: V=1; n=6: V=−1). The split tracks fusion, not which QH sector is involved — and the integer value −1 originally associated with the trefoil itself actually occurs at n=6, an unfused, non-baryon configuration. The independent, established confinement criterion remains dimq(3/2)=1 from Paper XVII; it is numerically distinct from the Jones modulus (1≠1/φ) and no identification between the two is asserted. Whether the physical formation path passes through the rational n=4 intermediate, and whether this explains why the QH=3 gradient-flow solver finds a flat energy plateau rather than a sharp saddle, is recorded as an open conjecture, not an established result. Pion emission is topologically forced The satellite/cable construction from knot theory gives the reaction rule directly: QH=2+QH=2 → QH=3+QH=1. Threading one component of the Hopf link T(2,2) through the solid-torus neighbourhood of the other with the (2,3)-cable framing produces the trefoil T(2,3), and a QH=1 unknot is necessarily left over. Pion emission is therefore a topological necessity, not a dynamical accident: the reaction is the unique minimal charge-conserving channel forming a baryon from two QH=2 lepton-sector objects, with no free parameters. Energy cascade and jet ordering: 3-3-3-3 The ρJ4 field is minimum at the three crossing vertices (maximum curvature, pure colour, topological bottleneck) and maximum in the A-segment bodies. Crossings act as topological lenses — energy arrives via A-segments, is redirected at the crossing, and exits at B-segment departure zones — not as ejection points. The full jet energy ordering from the N=256 gradient-flow cascade is four groups of three: (1) primary B-departure exits (jets 1–3, hard, coplanar at z=+r0); (2) distal lobe secondaries (jets 4–6, ~16°–19° off the three-jet plane); (3) crossing approach-side entry disturbances (jets 7–9, softer); (4) midpoint torsion-twist residuals (jets 10–12, softest). The topological upper bound is 12 jets per baryon, a prediction with no perturbative QCD analogue. φ* prediction and the Botner framework The three crossing vertices of T2,3 sit 120° apart with tangents in the z=+r0 horizontal plane. When ρ+ρ− pairs are produced from baryon-initiated string-breaking, the decay-plane angle distribution follows (1+β3cos 3φ*)/2π with β3 ≈ 0.58 from the A/B arc-segment energy ratio — a zero-free-parameter prediction. Photon-initiated production gives (1+βcos 2χ)/2π (the cos 2χ structure measured by Botner/DELPHI 1992). The two signals differ in angular frequency (3 vs 2) and have no common parameters, making them unambiguously distinguishable. The Jeremie anomaly in four-jet BZ angle distributions at m3+m4>10 GeV is consistent with the topologically distinct 4th jet at a discrete predicted deflection angle, though a dedicated experimental analysis confirming this remains an open problem. Isospin mass-scale ratio from writhe geometry The trefoil has a ℤ2 writhe asymmetry: three crossing vertices sit at z=+r0, while midpoints and distal lobes sit at z=0. Their exit tangent z-components are exactly 0 (crossings), −0.525 (midpoints), and +0.321 (lobes). Taking the mean-squared out-of-plane fraction of the midpoint/lobe network gives ⟨ġz2⟩M,L=½(0.5252+0.3212)=0.189. This predicts the geometric-mean mass scale ratio between the down-type and up-type quark triplets as Mdown/Mup ≈ 0.189, against the PDG-measured value 0.157 — a 21% match with zero free parameters, the first numerical connection between the trefoil's exact geometry and the observed two-triplet quark mass hierarchy. Speculative correspondences connecting the 12-site fraying pattern to six-quark flavour counting (the 3+3 midpoint-breaking combinatorial) and the ℤ2 writhe asymmetry to isospin charge assignment are recorded as falsifiable conjectures. Six open problems are listed in the paper, including a first-principles derivation of β3, quantitative QH=3 production thresholds, and experimental confirmation of the Jeremie anomaly; the companion draft Paper XIX resolves the multi-component Jones-polynomial open problem and reformulates the quark-mass-from-midpoint-breaking open problem, but leaves the others open.