Virtual Monopoles in the FeMo-Cofactor: $G_2$ Symmetry, Non-Associative Bond Breaking, and Nitrogenase as a Pachner-Move Computer

The FeMo-cofactor ($\mathrm{Fe_7MoS_9C}$) of nitrogenase is the most catalytically potent metal cluster in biology, yet its mechanism has resisted explanation for fifty years. Three anomalies persist: broken-symmetry DFT consistently misassigns the $S=3/2$ resting-state spin, the central $\mu_6$-carbide is essential for activity but chemically inert during turnover, and the enzyme consumes sixteen ATP per $\mathrm{N_2}$ fixed — four times the thermodynamic minimum.

We propose that these anomalies share a single topological explanation. The six-iron trigonal prism core of the cofactor, together with the central carbide, constitutes a seven-node system whose exchange topology realises the Fano plane $\mathrm{PG}(2,2)$. The automorphism group of the Fano plane is $G_2 = \mathrm{Aut}(\mathbb{O})$, and we conjecture that the electronic Hamiltonian carries $G_2$ symmetry rather than the crystallographic $D_3$. A $G_2$-symmetric Heisenberg model with ferromagnetic carbide lines and antiferromagnetic iron lines reproduces the $S \approx 3/2$ ground state without broken symmetry.

The frontier orbitals of $\mathrm{N_2}$ independently realise a second copy of $\mathrm{PG}(2,2)$: five of the seven Fano lines couple bonding to antibonding orbitals, encoding the Dewar–Chatt–Duncanson mechanism as a geometric identity rather than a perturbative approximation. Nitrogen binding glues the two Fano planes via lattice surgery, and bond cleavage is identified with the Fano-Line Closure operation — an unfactorisable projection that simultaneously couples $\sigma$ and $\pi^*$ in a single non-associative step.

The full catalytic cycle maps onto a fifteen-instruction program in the 731-RPU Origami instruction set architecture, with each elementary step corresponding to a bistellar (Pachner) move on the electronic simplicial complex. The 15% ATP overhead above the thermodynamic minimum is identified with the Fano-line closure energy cost.

Part of the Adelic Simplicial Architecture (ASA) research programme.