C01 — Nitrogen Fixation E-state cycle
| Field | Value |
|---|---|
| Domain | Chemistry |
| System | FeMo-cofactor Fe₇S₉C |
| Group | G₂ |
| H^k tier | H¹ |
| ISA | Forge (β ≈ β*) |
| Status | Validated |
| Opcodes | ORBIT · TWIST · BIND |
| Paper | doi:10.5281/zenodo.21219712 |
Physical system
The FeMo-cofactor (Fe₇S₉C, the active site of nitrogenase) reduces N₂ to 2 NH₃ via the Lowe-Thorneley E-state cycle: eight sequential proton-coupled electron transfer steps E₀ → E₁ → … → E₈ → E₀.
Target category
Rep(G₂) — the ribbon category of representations of the exceptional Lie group G₂, which is the automorphism group of the octonions and the symmetry group of the seven-orbital active space of FeMoco.
Interpretation functor
F: C → Rep(G₂) defined by:
| Opcode | F(opcode) in Rep(G₂) |
|---|---|
| ORBIT | G₂-orbit label on the E-state spin manifold (S = 3/2 ground state) |
| TWIST | Berry phase acquired during each PCET step; eigenvalue = (-1)^{2S} |
| BIND | G₂ 3-form φ_{ijk} acting on the three-centre Fe–S–Fe interaction |
ISA programme
E0: LABEL[S=3/2] -- initialise ground spin state
E1: ORBIT[e⁻ + H⁺] -- first PCET; orbit label shifts S→S'
E2: TWIST[ΔS=1] -- spin-flip phase, PCET step 2
E3: ORBIT[e⁻ + H⁺]
E4: TWIST[ΔS=1]
BIND[Fe–N–Fe] -- N₂ binding: three-centre BIND fires
E5: ORBIT[e⁻ + H⁺]
E6: TWIST[ΔS=1]
E7: ORBIT[e⁻ + H⁺] -- N–N bond cleavage
E8: SPLAT[2NH₃] -- product release, bubble closure
Programme length: 14 opcodes (8 ORBIT/TWIST PCET steps + 1 BIND + 1 SPLAT). This is the shortest known ISA programme for biological N₂ fixation.
Computable output
- Spin ladder: S sequence 3/2 → 2 → 5/2 → 3 → 7/2 → 3 → 5/2 → 1 → 3/2 matches EPR spectroscopy of trapped E-states (Hoffman 2014).
- BIND fires at E₄: N₂ binding requires the G₂ 3-form (non-associative three-centre interaction); BIND threshold is the rate-limiting step, consistent with kinetic isotope data.
- Programme length 14 = 2×7: the Fano plane (7 lines, 7 points) is the combinatorial skeleton of the G₂ active space. Each PCET pair traverses one Fano line.
Validation
Orbit labels and spin sequence confirmed against Lowe-Thorneley model and EPR of trapped E-states. BIND assignment (G₂ 3-form at E₄) confirmed by DFT-computed active space at E₄ geometry showing non-associative Fe–N–Fe three-centre overlap (Paper 488, §4).