OC01 — Furey Fermion Ladder
| Field | Value |
|---|---|
| Domain | Algebraic quantum field theory / Standard Model |
| System | Complexified octonions ℂ⊗𝕆, minimal left ideal of Cℓ(6) |
| Group | G₂ (automorphisms of 𝕆) ⊃ SU(3) × U(1) |
| H^k tier | H² (octonionic Hopf fibre) |
| ISA | Origami (Valence extension) |
| Status | Validated (one generation) |
| Opcodes | ORBIT · TWIST · MERGE · FLIP |
| Papers | Furey 2018 (EPJ C 78, 375); Furey 2016 (PhD thesis) |
Physical system
Cohl Furey constructs one generation of Standard Model fermions from the minimal left ideal S of the Clifford algebra Cℓ(6) ≅ ℂ⊗𝕆, the complexification of the octonions. The key structure is a pair of nilpotent ladder operators
\[\omega = e_1 e_3 e_5, \qquad \omega^\dagger = e_5 e_3 e_1,\]built from octonion basis elements e₁, e₃, e₅, satisfying
\[\omega^3 = 0, \qquad \omega^\dagger{}^3 = 0, \qquad \{\omega,\,\omega^\dagger\} = 1.\]The nilpotency ω³ = 0 produces exactly three grades of the minimal left ideal, which Furey identifies with the three colour states of a quark (grades 0, 1, 2) and the colourless lepton (grade 3, annihilated by ω).
The SU(3) colour symmetry acts on the fibre via the G₂ automorphism group of 𝕆: the Lie algebra of G₂ contains SU(3) as the subgroup that fixes the preferred octonion unit e₇. The U(1) hypercharge Y is the diagonal generator that distinguishes quark grades from the lepton.
One full generation (8 Weyl fermions: 3 quark colours + 1 lepton, each in two chiralities) = one traversal of the minimal left ideal by the ω/ω† ladder.
Why this lives in H², not H⁰ or H¹
The octonion algebra 𝕆 underlies the third Hopf fibration S¹⁵ → S⁸, whose structure group is Spin(7) ⊃ G₂. This is the full H² tier of the ISA (see §9 of the Hopf ISA paper). Furey’s ω and ω† navigate the internal structure of this fibre — they are intra-tier operators, moving between grades within H².
Compare:
- SNAP↑: the 2-cell that enters H² from H¹ (e.g. crossing the magic threshold in quantum computation, or passing through an exceptional point in a PT-symmetric laser). SNAP changes which tier the computation lives in.
- TWIST (Furey ω): a 1-cell that moves within H² by a discrete crossing change — changing the grade of the minimal ideal without leaving the octonionic sector.
Furey never uses SNAP. Her entire programme is intra-H².
Target category
Mod(Cℓ(6)) — the category of left Cℓ(6)-modules, with the minimal left ideal S as the distinguished object. Morphisms are Cℓ(6)-module maps. The G₂ automorphism group acts by outer automorphisms.
Interpretation functor
F: Origami ISA → Mod(Cℓ(6)) defined by:
| Opcode | F(opcode) |
|---|---|
| ORBIT | Set up the minimal left ideal S in its grade-0 state (vacuum = no quarks). The eigenvalue is the G₂ representation label. |
| TWIST | Apply ω: grade k → grade k+1 within S. One TWIST = one colour charge unit. Reverse TWIST applies ω†. |
| MERGE | Project two grades to a singlet via ω ∧ ω†: the Frobenius multiplication μ collapsing two ideal grades into one colourless state. |
| FLIP | Read out the hypercharge Y eigenvalue: the FLIP counit extracts the U(1) quantum number of the current grade. |
Note: BIND (entanglement between two distinct minimal ideals) would describe two-generation mixing — a natural extension but not Furey’s focus.
ISA programme
-- One generation of Standard Model fermions
SETUP: ORBIT[S | grade=0, G₂-rep=octonion] -- vacuum: empty minimal ideal
QUARK1: TWIST[ω | grade 0→1] -- first colour (e.g. red quark)
QUARK2: TWIST[ω | grade 1→2] -- second colour (green quark)
QUARK3: TWIST[ω | grade 2→3] -- third colour (blue quark)
LEPTON: MERGE[ω∧ω† | singlet projection] -- colourless lepton (neutrino/electron)
READOUT: FLIP[Y | hypercharge eigenvalue] -- measure U(1) charge
The nilpotency ω³ = 0 means a fourth TWIST would annihilate the state — there is no fourth colour. The programme terminates naturally after three TWIST steps.
Three generations: applying the programme to three independent minimal left ideals (one per generation) and then BIND-ing them gives the full three-generation lepton-quark structure. The BIND opcode here encodes the CKM/PMNS mixing — the entanglement between generation-labelled ideals.
Computable output
- Quantum numbers of one generation: after running the programme, FLIP yields the hypercharge Y = −1/3 (quarks, per grade) and Y = +1 (lepton). The SU(3) colour representation is 3 (quarks) + 1 (lepton), matching the Standard Model exactly.
- Nilpotency = generation count: ω³ = 0 gives exactly 3 quark colours. This is a prediction of the octonionic structure, not an input.
- G₂ as the symmetry group of BIND at H²: the 14-dimensional Lie algebra of G₂ = the automorphism group of 𝕆 = the symmetry group of the BIND opcode’s Frobenius algebra at H². This connects Furey’s derivation of SU(3) colour to the ISA’s statement that H² BIND has G₂ symmetry.
Connection to the ISA framework
Furey’s ω is TWIST at H². The crossing-change interpretation:
- Each TWIST step applies one Reidemeister move to the octonionic knot living on the Hopf torus T² ⊂ S¹⁵. Grade k corresponds to a torus knot T(k+1, 3−k) on the octonionic Hopf torus.
- Three TWIST steps exhaust the three non-trivial grades (k = 0 → 1 → 2 → 3), after which ω³ = 0 kills further progression. The lepton is the grade-3 state where the knot has been fully unknotted — the colourless singlet.
- The SU(3) symmetry = the group of Reidemeister moves that preserve the knot type within each grade. Colour = knot label within H².
Relation to other H² zoo entries:
| Entry | H² object | BIND computation |
|---|---|---|
| G01 (YM instanton) | c₂ ∈ ℤ of SU(2) bundle | ∫tr(F∧F) over S⁴ |
| G02 (Amplituhedron) | Volume form on Gr(k,n) | BCFW recursion = BIND tree |
| G03 (Higgs mechanism) | Goldstone sector of G/H | SNAP↑ removes LINK |
| OC01 (Furey ladder) | Minimal ideal of Cℓ(6) | Three TWIST steps within 𝕆-fibre |
All four are H² computations; they differ in which aspect of the octonionic Hopf bundle they access.
Validation
- One-generation spectrum: Furey (2018) derives the correct U(1) × SU(3) quantum numbers for one generation of quarks and leptons from the minimal left ideal of Cℓ(6). No free parameters.
- Nilpotency → 3 colours: ω³ = 0 follows from the octonion multiplication table, not from SU(3) representation theory. That it gives exactly 3 is a derivation, not an assumption.
- G₂ ⊃ SU(3) ⊃ U(1): the exceptional Lie group G₂ = Aut(𝕆) contains SU(3) as the subgroup preserving a chosen octonion unit — a classical result (Cartan 1914). Furey’s SU(3) colour = this subgroup acting on the minimal ideal.
- Open: three-generation structure (why ω acts three times on distinct ideals) and mass spectrum require additional structure beyond the basic octonionic construction. The BIND extension (two-ideal entanglement) is the natural ISA framework for this open problem.
Part of the ISA Zoo. See also: G01 — Yang-Mills Instantons, [§9 of the Hopf ISA paper — three Hopf fibrations and the ISA tier ladder].