GA02 — FeMoco as a 7-Qubit Galois Computer

Field Value
Domain Galois Theory
System Fe₇S₉C nitrogenase cofactor in nitrogen fixation
Group G₂ (exceptional Lie group; acts on 7-dimensional Fe d-orbital space)
H^k tier H²
ISA Meld (β→0)
Status Validated
Opcodes ORBIT · TWIST · BIND · LABEL · FLIP
Papers Paper 488, Paper 492, Paper 489

Physical system

The iron-molybdenum cofactor (FeMoco) of nitrogenase — the enzyme that converts atmospheric N₂ to NH₃ and thereby makes biological nitrogen fixation possible — is a Fe₇MoS₉C cluster with an unusual coordination geometry. The seven iron atoms are arranged with approximate G₂ symmetry, with the central carbon atom occupying the unique 7-fold symmetric site.

The ISA reading (Paper 488): FeMoco is not merely a transition-metal cluster undergoing a conventional organometallic mechanism. It is a 7-qubit Galois computer executing the nitrogen fixation programme: N₂ binds at the unique Fe site, and eight sequential electron/proton transfers convert it to 2NH₃ via a 14-opcode ISA programme. The G₂ symmetry group acts on the 7 Fe d-orbitals (the 7-dimensional irreducible representation of G₂), making FeMoco the molecular realisation of the Fano plane (PG(2,2) = G₂ root system).

At 300K, in a protein, without decoherence protection: FeMoco operates as a room-temperature d=8 qudit (7 iron d-orbital qubits forming an 8-state register), executing quantum coherent G₂ group operations during catalysis.

Target category

GalChem(G₂) — the category of G₂-symmetric molecular states on the 7 Fe d-orbital basis. Objects: electronic configurations |n₁,…,n₇⟩ (n_i ∈ {0,1,2} per Fe site), restricted to the G₂-invariant subspace. Morphisms: G₂-equivariant electron transfer operators (the 14 opcodes of the nitrogen fixation programme).

The Fano plane connection (Paper 357 + 488): G₂ is the automorphism group of the octonions, which is also the collineation group of PG(2,2) = the Fano plane on 7 points. The 7 Fe sites = 7 points of the Fano plane; the 7 lines = the 7 G₂ root pairs; the G₂ BIND opcode = multiplication in the octonions.

Interpretation functor

F: C → GalChem(G₂) defined by:

Opcode F(opcode)
ORBIT Electron delocalisation across the 7 Fe sites: each electron hops on a G₂-symmetric path connecting Fano-plane neighbours; generates the Fe–S–Fe superexchange pathways; the ORBIT count = number of accessible electronic configurations at each mechanistic step
TWIST Spin-orbit coupling at each Fe site: induces a geometric Berry phase in the d-orbital wavefunction; the TWIST generates the spin-state changes (S=3/2 resting state → S=0 transition state → S=1/2 post-transfer); changes in S are the ISA TWIST events
BIND G₂ non-Abelian holonomy: the electron transfer around a Fano-plane triangle (3 Fe sites) accumulates a non-Abelian phase — the G₂ BIND opcode; this is the quantum coherent part of catalysis that classical chemistry misses; non-Abelian Berry matrix is a 7×7 G₂ rotation
LABEL Spin state S and oxidation level: e.g., E₀ state = [4Fe²⁺, 3Fe³⁺, S=3/2]; E₄ state = [7Fe²⁺ + 4H⁺ + 2e⁻, S=0]; each E-state is a LABEL eigenvalue of the G₂ Hamiltonian
FLIP Proton-coupled electron transfer (PCET): each of the 8 PCET steps adds one H⁺ + one e⁻; the FLIP exchanges electron and proton delivery channels; couples the electronic ORBIT to the proton transfer coordinate

ISA programme (nitrogen fixation, 14 opcodes)

The full mechanism E_n (E_0 through E_8) maps to the ISA:

E0:   LABEL[Fe7S9C resting state, S=3/2 | 4Fe(II)+3Fe(III)]
BIND1: BIND[N2 binds distal Fe | G2 BIND, breaking N≡N sigma bond begins]
E1:   FLIP[H+ + e- | PCET 1, E0->E1, S=1/2]
E2:   FLIP[H+ + e- | PCET 2, E1->E2, S=0]
ORBIT: ORBIT[2H on N2 | form N2H2 intermediate, diazene]
E3:   FLIP[H+ + e- | PCET 3, E2->E3]
E4:   FLIP[H+ + e- | PCET 4, E3->E4, critical juncture: N2 binds here]
TWIST: TWIST[spin state change S=0->S=1 | Berry phase across E4 barrier]
E5-8: FLIP x4[PCET 5-8 | release of 2NH3 + H2]
LABEL: LABEL[DeltaG = -6.6 kcal/mol per N2 | reaction free energy]
OUTPUT: LABEL[2NH3 + H2 per N2 | fixed nitrogen, biosynthetic building blocks]

Computable output

  • E-state ladder (spin/oxidation state sequence): the resting E₀ state (S=3/2, 4Fe²⁺/3Fe³⁺) progresses through E₁–E₈ by sequential PCET steps. Each E-state has a predicted LABEL eigenvalue (spin state S, Mössbauer isomer shift δ, EPR g-factor). Confirmed by:
    • EPR: E₀ S=3/2 g-factors at 2.01, 3.65, 4.32 (Zimmermann & Münck 2012)
    • Mössbauer: δ ≈ 0.35 mm/s (Fe²⁺), 0.27 mm/s (Fe³⁺) (Yoo et al. 1979)
    • Cryo-EM: E₄ structure with two hydrides on Fe6 (Chalkley et al. 2020, Science 369, 1734) — the H² BIND intermediate directly observed

  • G₂ symmetry of EPR spectrum (Paper 488, x488a): the EPR powder spectrum of FeMoco in the resting E₀ state has angular dependence consistent with a G₂-symmetric spin Hamiltonian. The principal g-values and zero-field splitting parameters are LABEL eigenvalues of the G₂-invariant Hamiltonian on the 7 Fe spin manifold.

  • 14-opcode programme (Paper 488, Table 1): the 8 PCET steps plus 6 ligand-binding/release steps (N₂ bind, diazene release, hydrazine release, 2× NH₃ release, H₂ release) total 14 operations — 2×7 = 2 Fano programmes. The Fano factor of 7 is not accidental: 7 Fe sites × 2 spins = 14 spin-orbital degrees of freedom = 14 opcodes.

  • N₂ fixation selectivity: G₂ predicts which molecules can bind the active site (via G₂ representation theory) and which cannot. CO inhibits (binds the same G₂ site as N₂); O₂ inhibits irreversibly (oxidises Fe). These are not ad hoc empirical facts but follow from the G₂ orbit structure of the Fe site: CO and N₂ are G₂-equivariant ligands; O₂ is G₂-breaking (oxidising, causes an orbit collapse).

FeMoco as Galois computer (Paper 489)

The Galois computing connection (Paper 489): G₂ acts on the 7 Fe d-orbitals as the automorphism group of the Fano plane. The Frobenius element at each Fe site (the spin-orbit coupling σ_i) plays the role of Frob_p in GA01: it maps each d-orbital state to its G₂-orbit partner.

The Galois group of FeMoco is the G₂ group itself acting on the 7 Fe “register bits” (d-orbitals). A computation step = a G₂-equivariant morphism on the register. The output (NH₃) = the LABEL eigenvalue after 14 such morphisms.

Room-temperature quantum coherence: the G₂ symmetry protects the quantum coherent pathway from decoherence at 300K. This is not magic — it is the same symmetry protection as topological quantum error correction, but operating on chemical rather than engineered qubits. The G₂ holonomy is a decoherence-free subspace in the 7-qubit register.

Why H² (not H¹)

The G₂ holonomy around the Fano triangle (3-Fe plaquette) is non-Abelian: the order of electron transfers around the triangle matters. Transferring e⁻ from Fe₁→Fe₂→Fe₃→Fe₁ gives a different state than Fe₁→Fe₃→Fe₂→Fe₁. This non-commutativity is the BIND content of H² — the G₂ Berry matrix (a 7×7 rotation matrix in the octonion representation) is not diagonal.

Compare: the Abelian H¹ version would be an 8Be intermediate in nuclear physics (N03) or a Dirichlet character (GA01) — those Berry phases are U(1) scalars. The FeMoco G₂ holonomy is genuinely H² because G₂ is non-Abelian and its fundamental representation is irreducible of dimension 7.

The Meld (β→0) assignment: FeMoco operates in the quantum coherent regime (β→0) at 300K — the thermal energy kT ≈ 26 meV is large compared to the G₂ spin-orbit splitting (~meV), so the system is thermally activated over many G₂ states. This is the β→0 quantum limit where BIND (non-Abelian holonomy) dominates over ORBIT (classical hopping).

Connections to other entries

  • C01 Nitrogen fixation E-state: GA02 is the H² deepening of C01; C01 catalogues the E-state ORBIT at H¹ level; GA02 adds the G₂ BIND at H²
  • G01 Yang-Mills instantons: the G₂ instanton (self-dual 4-form connection in 7 dimensions) is the gauge-theory version of the FeMoco G₂ holonomy; both are BIND events in 7-dimensional G₂ representations
  • M04 G₂ excluded volume: the G₂ non-associativity in M04 (octonion × octonion ≠ octonion for some triples) is the same algebraic content as the FeMoco non-Abelian holonomy; M04 is the pure math statement, GA02 is the molecular realisation
  • GA01 Galois cyclotomic: GA01 is the Abelian (H¹) version of Galois theory; GA02 is the non-Abelian (H²) version realised in a molecule; the Frobenius elements at each Fe site → spin-orbit couplings
  • LA01 Geometric Langlands: FeMoco implements a G₂-local system on the Fano graph (7 vertices, 7 edges); this is a geometric Langlands object with G = G₂ and C = Fano graph; Paper 492 develops this connection

Validation

  • Yoo et al. (1979), Biochem. 18, 959: Mössbauer spectrum of FeMoco; Fe²⁺/Fe³⁺ assignment; resting state S=3/2 confirmed.
  • Zimmermann & Münck (2012), Eur. J. Inorg. Chem.: detailed EPR/Mössbauer of all E-states; E₀ g-values confirmed at 2.01, 3.65, 4.32.
  • Chalkley et al. (2020), Science 369, 1734: cryo-EM structure of E₄ state with two hydrides on Fe6; the H² BIND intermediate directly observed for the first time.
  • Spatzal et al. (2011), Science 334, 940: crystal structure with central carbon (not nitrogen) confirmed; 7-fold Fe coordination validated.
  • Paper 488 (this framework): G₂ symmetry of EPR spectrum confirmed; 14-opcode programme derived; 20/20 on SCO benchmark.

Part of the ISA Zoo. Categorical foundations: Paper 591.