Every molecule runs a Galois programme

The spin state, valence, and catalytic barrier of any transition-metal complex are determined by G-orbit walks on its molecular G-set — computable in O(1), without DFT, at room temperature.

The claim

Molecules compute. A transition-metal complex is not a passive arrangement of atoms — it is a computing device that evaluates a programme determined by its symmetry group G acting on its set of electron configurations. The output is the spin state, the bond order, the reaction rate.

The programme is a G-orbit walk: a sequence of group-orbit closures on the molecular G-set. Each step costs one ISA opcode. The total programme length is the number of opcodes; the ground state is the orbit that closes first.

Three quantities are computable in O(1) from the orbit structure alone, without solving the Schrödinger equation:

Spin state — which orbit in the d-electron manifold is closed determines whether the complex is high-spin or low-spin. This is the spin-crossover (SCO) threshold as a discrete orbit condition.
Valence — the number of open orbits in the bonding manifold equals the bond order. Aufbau, Hund’s rules, and the Taube electron-transfer rules are theorems, not empirical generalisations.
Catalytic barrier — the barrier to a reaction at a transition-metal centre is the cost of the TWIST opcode at the spin-state crossing between reactant and product orbit configurations. The Brønsted-Evans-Polanyi (BEP) linear scaling relation is the ISA opcode cost scaling law.

Why it matters

DFT has failed for 50 years to deliver a better Haber-Bosch catalyst. The global ammonia synthesis process (N₂ + 3H₂ → 2NH₃) consumes ~1.5% of global energy, feeds half the world’s population, and has run on the same promoted-iron catalyst since 1909. DFT (GGA-PBE) predicts the N₂ dissociation barrier with an error of 0.3–0.5 eV — equivalent to a factor of 10⁵ in reaction rate at 500 K via the Arrhenius equation. This is why DFT alone has not found a better catalyst despite enormous investment.

The ISA diagnosis: the N₂ dissociation step on Fe(111) is a spin-state crossing — a TWIST opcode. DFT fails here for exactly the same reason it fails for molecular spin-crossover complexes: the derivative discontinuity at the spin-state transition point. The ISA identifies this crossing as a tropical vertex and provides the barrier energy from orbit-stability theory, without DFT+U parameter fitting.

The commercial consequence: a catalyst design rule that predicts, without fitting, which Fe surface promoter loadings and which secondary metals optimise the N₂ dissociation barrier would be immediately commercialisable. The global ammonia market exceeds $70B/year. This is the target of Paper 562.

Beyond catalysis: Fe(II) spin-crossover complexes are candidates for molecular memory and switching devices. The ISA’s O(1) prediction of the SCO threshold — validated at 90% accuracy on the Paper 488 benchmark — means rational design of SCO materials without DFT ensemble runs.

The evidence

Paper	What it shows
Paper 488	G-walk Chemistry: Fe(II) SCO as a TWIST gate; 90% accuracy on L0 benchmark without DFT
Paper 489	Galois Computing: G-orbit walks as the 4th computing paradigm; 300 K / decoherence-immune
Paper 490	Galois Protein Design: Aufbau/Hund/Taube as ISA theorems; RNR 5/5, PSII 2/4, Hb Hill n_H = 3.29
Paper 491	Galois Chemistry = Tropical DFT: Wigner vertex theorem; TS diagrams = tropical varieties; derivative discontinuity = tropical singularity; 20/20 on SCO benchmark
Paper 492	Langlands for Galois Chemistry: G-local system on molecular graph; BEP slope = Whittaker function
Paper 562	Haber-Bosch ISA: Fe B₅ site spin-state crossing = TWIST opcode; barrier = tropical vertex energy; promoter loading = β* condition

Key experiment: x562a (running July 2026) — Fe₅ cluster model of the Fe(111) B₅ active site. UHF→UMP2→CASSCF(ISA) at three geometries along the N₂ dissociation path. The NOON spectrum is predicted to collapse from H¹ (clean surface, high-spin) to deep H² (transition state) — confirming the TWIST opcode fires at the barrier.

The Grassmannian connection

The ISA active space selection (x560c, Paper 560) is itself a Grassmannian computation: the occupied-orbital manifold is a point in $\mathrm{Gr}(N_e, N_\mathrm{orb})$, and the NOON stratification identifies the geodesic distance from the Hartree-Fock reference. The TWIST opcode fires when this distance crosses the β* threshold θ_G ≈ 20° — the same threshold that separates single-reference from multi-reference in molecular bonding (Paper 570), Mott transitions in lattice models (Paper 563), and QEC threshold failure (Paper 577).

The Haber-Bosch N₂ dissociation is the heterogeneous catalysis member of this universal Grassmannian family. The β* snap at the B₅ site is the same geometric event as the bond-breaking snap in H₂ at R ≈ 1.5 Å and the Mott transition at U/t ≈ 1.8.

What would falsify it

The G-orbit claim is falsified if:

A transition-metal complex is found whose spin state cannot be determined from the orbit structure of its d-electron manifold — i.e., the spin state changes without any change in orbit occupancy.
The Haber-Bosch CASSCF(ISA) barrier (x562a) is more than 0.2 eV from the experimental value, after the cluster model finite-size effects are accounted for — which would mean the TWIST opcode cost is not a reliable barrier predictor.
DFT+U with a carefully fitted U parameter is shown to give systematically better barriers than CASSCF(ISA) across the SCO benchmark of Paper 491 — which would mean the O(1) ISA prediction adds nothing beyond empirical fitting.

Open questions

Can the BEP slope be derived analytically from ISA opcode cost scaling? The linear BEP relation (barrier ∝ adsorption energy) is empirical across all transition metals. If it follows from the TWIST opcode cost being a linear function of the orbit-occupancy change, that is a fundamental derivation of a century-old empirical law.
What is the optimal promoter loading for Fe-based HB catalysts? The ISA prediction (Paper 562, x562b): the β* optimal K loading maximises the number of H² Fe 3d orbitals at the B₅ site. Can this be verified experimentally?
Does the G-orbit walk generalise to heterogeneous alloy surfaces? Fe-Co, Fe-Ru, Fe-Re alloys have mixed coordination environments. The ISA opcode cost (TWIST cost at the B₅-equivalent site) should still predict the activity ordering — but the orbit structure of a multi-metal site is richer.