12 July 2026. Third companion to GENERATIONS_PROGRAM.md
(lepton masses) and WHERE_R_LIVES.md (the soliton interior).
The mass program left a loud gap: it modeled the charged leptons as Q-balls, which are spin-0 bosons — but electrons are spin-½ fermions. So the Q-ball was only ever the mass and charge skeleton. This note fills in the rest of the particle picture — spin and statistics, the neutrino, parity, and quark confinement — and draws the honest boundary where a U(1) framework stops.
The through-line: one topological invariant — the winding-line linking / helicity — carries spin, statistics and chirality.
⚠ Corrected 22 Aug 2026. This opening previously read “…carries spin, statistics, baryon number, and chirality all at once”, and cited the baryo/magneto/chirality arc as having derived it. Both parts are withdrawn. Baryon number is not carried by this invariant — it has no carrier in this framework at all (see
DERIVED_SUMMARY.md§9). And the arc’s “one invariant, three faces” claim was itself withdrawn: baryon number ΣW and magnetic helicity (2π)²ΣWᵢWⱼLkᵢⱼ are two independent invariants, not one — the counterexamples are explicit (two unlinked W=+1 windings give B=2, H=0; a linked W=+1/W=−1 pair gives B=0, H=+8π²). What survives, and is the honest core of this note: spin, statistics and chirality do come from the one linking invariant. The status table below was corrected on 18 Aug; this opening line was missed, and is corrected now. Everything below is a reading of that one fact, plus the sine-Gordon mass and the mass dial. Labels: [derived] · [computed] · [consistency] · [proposed] · [floor] (an absolute value / non-abelian group the U(1) field does not fix — open in TFT as it is elsewhere). Pre-registrations:SPIN0_,NU0_,QCD0_.
spin_statistics.py)A soliton of a bosonic field can be quantized as a fermion (Finkelstein–Rubinstein) iff a 2π rotation is a non-contractible loop in its configuration space. A spherical Q-ball’s rotation is trivial → spin-0 boson. The carrier of fermions in a U(1) phase field is a twisted, current-carrying vortex loop (a vorton / Hopfion), whose self-linking (writhe + twist) sets the rotation sign.
Computed, as parameter-free topology: the Gauss linking integral is quantized
(unlinked rings Lk = 0.000, Hopf link Lk = −1.000 → ±1); the self-linking of a
twisted loop equals its twist = the internal current winding (SL = 0,1,2,3 for
winding 0,1,2,3). Then odd self-linking → (−1) under 2π rotation → spin-½
fermion; even → boson [derived]. That self-linking is the same winding-line
helicity as baryon number and chirality — one linking invariant, four three
meanings (baryon number removed 18 Aug 2026, CHRG0 — see the status table).
Resolution [consistency]: the leptons are not plain Q-balls — the true lepton is a vorton/Hopfion with odd self-linking. The Q-ball gave the mass and charge; the fermionic nature lives in the loop’s linking. (And the electron being a near-silent winding-odd state — why it is light, from the mass dial — and being a fermion are the same topological data.)
neutrino_parity.py)The mass program (see GENERATIONS_PROGRAM.md) put the three charged leptons on
one dial, with the electron light because it is 99.85% winding-reversal-odd
(the mass lives in the even channel). Push that to its end:
This gives the weak force’s two TFT-native features — the neutrino and parity — with no SU(2). The gauge machinery (W/Z, the doublet) is a [floor].
quark_confinement.py)A quark is a winding-line endpoint. Two facts:
Two payoffs. Statistics [consistency]: quark = odd-linking fermion ⇒ meson (q q̄) = boson, baryon (qqq) = fermion — as observed, pure linking parity. Unification [derived]: σ = 8√Λ is the same 8√Λ as the derived kink rest mass — one scale √Λ sets both the hadron mass and the confinement tension (Λ_QCD and the hadron mass are the same number, as in real QCD).
Honest: “quarks are wall-endpoints” is the topological-confinement seed made concrete on the derived kink — a model of what a quark is, not a derivation of QCD. Fractional charge and color SU(3) are a [floor] (the U(1) field gives integer charge only; the 3-strand/Z₃ junction is [proposed]). (Noun corrected 18 Aug 2026, UNC1: this read “integer winding”. The integrality comes from the number/Noether operator’s spectrum on the single-valued S¹, not from a topological winding — the floor is unchanged.)
spec_selfconsistent.py)WHERE_R_LIVES.md localized the last unexplained lepton-mass number, r ≈ 0.318,
to a two-sector flux and showed it collapses together with the amplitude
A ≈ √2 into one residual. The self-consistent solve settles what that residual
is [derived]: the fully self-induced dial is flat (the induced sectors
co-rotate with the dipole — a global rotation), so a non-trivial r requires
independently-excited sectors; and A = √2 plus U(1) charge quantization are
only two conditions on three amplitudes — a one-parameter family survives, r
varying along it. So r (and A) bottom out at the generation-mode excitation
amplitude — an initial-condition-class quantity, the same floor as the baryon
asymmetry η, not a dynamically-derived number. The mechanism is fully mapped;
the final number is a floor.
| statement | label |
|---|---|
| self-linking is a quantized topological invariant (Hopf link ±1, twist = winding) | computed |
| odd self-linking → spin-½ fermion; Q-ball → spin-0 boson | derived |
| spin, statistics, |
derived, except baryon number — withdrawn 18 Aug 2026 (CHRG0): the same integer cannot be self-linking parity and baryon number and electric charge. The electron (spin-½ ⇒ odd SL ⇒ odd W; B=0) forces W to be electric charge. |
| leptons are vortons/Hopfions (odd linking), not plain Q-balls | consistency |
| neutrino = pure winding-odd, massless-chiral limit of the lepton dial | derived |
| parity violation forced by chirality = winding (100% V−A, no ν_R) | derived |
| large PMNS / small CKM = neutrino near-degeneracy vs charged-lepton hierarchy | proposed |
| linear confinement, string tension σ = 8√Λ = the kink mass | computed |
| meson = boson, baryon = fermion (linking parity) | consistency |
| one scale √Λ sets both hadron mass and confinement tension | derived |
| the last mass number r (=A) = an excitation amplitude, an initial condition | derived (a floor) |
| absolute scales (ν-mass, Λ_QCD, …); non-abelian groups (SU(2)_L, color SU(3)) | floor |
TFT’s particle sector is a U(1) phase field, and it goes exactly as far as a U(1) field can: it derives structures, mechanisms, and scale-relations — what a particle is (a soliton), why some are fermions (odd linking), why the electron is light and the neutrino massless and chiral (the mass dial’s cancellation), why parity is violated (chirality = winding), why quarks confine and why the confinement scale equals the mass scale (the sine-Gordon wall). It does not fix the absolute scales (every mass scale, Λ_QCD, the mixing couplings) or supply the non-abelian groups (SU(2)_L for the full weak interaction, color SU(3) for the full strong one). Those are the named floors — open in TFT, and mostly open everywhere. The line between the two is where this note ends.
pip install numpy
mkdir -p outputs
python3 spin_statistics.py # fermions from odd self-linking; Q-ball = boson
python3 neutrino_parity.py # neutrino = pure-chiral dial limit; parity; mixing
python3 quark_confinement.py # linear confinement, sigma = 8 sqrt(Lambda) = kink mass
python3 spec_selfconsistent.py # r/A bottom out at an excitation amplitude (a floor)
Inputs: PDG masses and the repo field only. Each script prints its pre-
registered gate and verdict; JSON lands in outputs/.