HopfionSeries Paper XVII - The $Q_H=1$ Sector of the Density-Feedback Hopfion: Topology, Group Structure, Colour Exclusion

Paper XVII extends the framework's sector hierarchy to QH=1, proving by four independent group-theoretic arguments that this sector is the neutrino sector: it carries a single unknotted preimage circle, belongs to the A-type cyclic series (ℤ2 ↔ Â1 ↔ SU(2)1) rather than the E-type exceptional groups of the lepton and baryon sectors, is colour-free by a coprimality argument (gcd(2,3)=1), and has zero string tension. The Wess–Zumino descent forces the sector's own WZW level to k=1 via the McKay relation Qgroup=2(k+2)=6, changing the virial and profile ratio from the ambient k=3 values. The principal quantitative output is a parameter-free mass prediction mν1 ≈ 42 meV from TCMB alone, within the target sensitivity of Project 8. Neutrino sector identification Four independent arguments converge on QH=1 as the neutrino sector. (i) Preimage topology: the single unknotted preimage circle has zero writhe, no crossings, and no Y-junction, placing it at the base of the confinement hierarchy QH=1 ⊊ QH=2 ⊊ QH=3. (ii) McKay chain: the QH=1 group is ℤ2={±1}, with McKay chain ℤ2 ↔ Â1 ↔ SU(2)1, the A-type (abelian, cyclic) series, in contrast to the E-type exceptional groups organising QH=2 (2I, Ê8) and QH=3 (2T, Ê6). (iii) Colour exclusion: gcd(|ℤ2|,|ℤ3|)=gcd(2,3)=1 means colour ℤ3 has no non-trivial action on ℤ2-representations — a coprimality argument stronger than Paper XVI's structural proof. (iv) Quantum dimension: dimq(1/2; k=3)=φ≠1, confirming the sector is unconfined; both QH=1 and QH=2 share quantum dimension φ, while only QH=3 (the simple current) satisfies dimq=1. WZ descent forces k=1 and the Majorana prediction The Wess–Zumino descent for the QH=1 condensate forces its own WZW level to k=1 (not the ambient k=3): the McKay relation Qgroup=2(k+2)=6 gives k=1 uniquely, changing the sector-specific virial to V*=dimq(1/2; k=1)=1 and the profile ratio to J4/J2a=(1+√3/φ) ⋅ 21/3/φ6 ≈ 0.145. The j=1/2 primary of SU(2)1 is self-conjugate (j*=k−j=1/2=j) and generates a ℤ2 fusion group (1/2 × 1/2=0): it is a Majorana simple current, predicting Majorana neutrinos from the WZW structure alone, independently of any mass mechanism. Parameter-free neutrino mass mν1 ≈ 42 meV The c=1 thermodynamic matching gives the irrational tower level nν=42.39 and the mass mν1=TCMB(π2/90)1/4 ⋅ φ12 ⋅ e−5/144 ≈ 42 meV, derived with zero free parameters. The φ12 prefactor comes from Qgroup=6, and e−5/144 from the SU(2)1 T-matrix correction T1/2(k=1)/Qgroup=5/144. The prediction satisfies the exact experimental bound nν≥42.02 with margin 0.37 and lies within the final target sensitivity of Project 8 (mβ≥40 meV/c2 at 90% CL). The former "one-unit gap" nν≥43 vs nν=42 arose from unwarranted integer rounding resolved by Paper VI's irrationality theorem for tower levels. Atmospheric mass ratio and Qeff uniqueness The inter-sector coset SU(2)3/SU(2)1 carries four primary fields with T-matrix phases {−4,−1,11,14}/30 and gap structure (1/10, 2/5, 1/10). The atmospheric pair separated by outer gap ΔT=1/10 gives mν3/mν2=exp(4π2/10√5) ≈ 5.845, matching the experimental value 5.795 to 0.9%. The effective Q-group Qeff=2π/√5=2π/√(k+2) is uniquely forced by bosonic monodromy (irrationality, since the experimental mass ratio is not a root of unity) and S-matrix unitarity: it is the unique irrational in SU(2)3 modular data consistent with experiment. Every parameter of the coset — level assignments, Q-groups, gap structure, Qeff, and framing independence — is determined exactly by the trefoil's winding numbers (p,q)=(2,3). PMNS matrix as exact permutation; Brieskorn identity The primary-level branching SU(2)3 → SU(2)1 ⊗ M(5,6) gives the PMNS matrix exactly as a permutation: μ → ν3, τ → ν2, e → ν1, with θ23=45°, θ13=0°, θ12=0°. This is the complete coset prediction, not a leading-order approximation. Experimental deviations (of order 1/(k+2) to 1/(k+2)2) arise from physics beyond the coset branching. The identity Qcoset=h(E8)=30 is structural: both equal |π1(Σ(2,3,5))|/4, where the Brieskorn sphere Σ(2,3,5) encodes the trefoil and 2I as projections of the same object, directly connecting the strange quark's T-phase coincidence of Paper XVI to the coset geometry. The remaining open problem is the full q-series resummation converting the coset branching functions into physical mixing angles incorporating RG and condensate-background corrections beyond the permutation prediction.