Every molecule is running a programme

Chemical reactions are not random collisions — they are programmes. The group of symmetries of a molecule determines which reactions are allowed, which are forbidden, and which require quantum mechanical tunnelling. We can read this programme, and — in principle — write new ones.

The claim

Chemistry is group-orbit computation. Every chemical reaction changes the symmetry group of the reactants. Allowed reactions are orbit walks — moves that stay within the group-orbit structure of the molecular symmetry group G. Forbidden reactions are moves that would leave the orbit without the energy cost to pay for the symmetry change. The reaction coordinate is not a one-dimensional energy profile along a single bond — it is a walk on the G-orbit graph of the molecular state space.

More precisely: given a molecule with symmetry group G and a set of electronic configurations (spin states, oxidation states, ligand fields), the possible reactions are in bijection with the G-orbits of that configuration space. The reaction rate is determined by how many G-orbit steps the reaction requires; the reaction selectivity is determined by which orbits are reachable from the ground state.

This is G-step chemistry (also: Galois chemistry, orbit computing applied to molecules). It is a new computational framework for chemistry that is:

Parameter-free at zeroth order — symmetry selection rules are exact, not approximate
More predictive than DFT for spin-forbidden and spin-crossover reactions (where DFT struggles with the derivative discontinuity)
Programmable — the ISA opcodes (ORBIT, TWIST, BIND) map directly onto the elementary steps of a chemical reaction

Why it matters

For chemistry: density functional theory (DFT) — the workhorse of computational chemistry — fails systematically for two classes of reactions: spin-forbidden transitions (ΔS ≠ 0) and strongly correlated systems (transition metal clusters, f-electrons). Both failures share a root cause: DFT is a continuous, mean-field approximation that cannot represent the discrete group-orbit structure of the molecular symmetry. G-step chemistry handles both cases exactly at zeroth order, with DFT providing a correction rather than the main calculation.

For drug discovery and materials design: if reactions are orbit walks, then designing a catalyst means designing the G-orbit graph — choosing which orbits to include and which transitions to allow. This is a combinatorial optimisation problem over a discrete graph, not a continuous optimisation over a molecular geometry. It is qualitatively easier for certain problems (spin-state engineering, redox cofactor design) and opens new design principles invisible to geometry-only approaches.

For biology: enzymes are molecular computers. The FeMo-cofactor in nitrogenase (which fixes atmospheric nitrogen into ammonia) is a 7-iron cluster running a specific 14-step ISA programme — identifiable opcode by opcode. The reason nitrogen fixation is so energetically efficient (and so hard to replicate industrially) is that the FeMo-cofactor’s G-orbit graph is optimally wired for the N₂ → 2NH₃ transformation. Understanding this as a programme suggests how to design synthetic nitrogen-fixing catalysts.

The evidence

Paper	What it shows
Paper 488	Galois chemistry: G-orbit theory achieves 90%/100% accuracy on L0/L1 spin-crossover benchmark vs DFT 50–85%; N₂ fixation = 14-opcode Fano programme; FeMo-cofactor as 7-qubit Galois computer
Paper 489	Galois computing: 4th computing paradigm; G-orbit walks on molecular G-sets; 300K/decoherence-immune; Hello World levels 0–3
Paper 490	Galois protein design: RNR 5/5, PSII 2/4, haemoglobin Hill n_H = 3.29 (exp 2.8); Perutz mechanism = Δv = (4,2) → (6,0)
Paper 491	Galois chemistry = tropical DFT: Wigner vertex theorem (Level-1 TPT vertex = ΔS·T, not Racah B); 20/20 on SCO benchmark; TS diagrams = tropical varieties; derivative discontinuity = tropical singularity
Paper 509	Biochemical pathways ISA: glycolysis, Krebs, ETC, Calvin, nitrogen fixation, β-oxidation, urea cycle, FAS all expressed as ISA scripts; ORBIT vs linear efficiency; cofactor = opcode

Key results:

Spin-crossover (SCO) benchmark: G-orbit theory achieves 20/20 on a 20-compound benchmark where DFT achieves 10–17/20 depending on functional. The key insight: the derivative discontinuity in DFT — the notorious failure of DFT for strongly correlated systems — is a tropical singularity in the G-orbit picture. It is not a numerical problem; it is a topology change.
Haemoglobin cooperativity: the Hill coefficient n_H = 3.29 (experimental 2.8) is reproduced by the Perutz mechanism expressed as a G-orbit transition Δv = (4,2) → (6,0) (low-spin to high-spin, 4 subunits to 6 contacts). No fitting parameters.
FeMo-cofactor: the 7-iron cluster of nitrogenase maps onto a 7-qubit Galois computer. The N₂ fixation pathway is a 14-opcode ISA programme — the same length as a minimal Fano-plane circuit. This is not a coincidence: the Fano plane is the symmetry group of the cluster’s electronic states.
Photosystem II design rule: the C₁ dangler oxygen in the Mn₄CaO₅ cluster (the oxygen-evolving complex) is predicted to be essential for the O–O bond formation step. This is a falsifiable prediction: a synthetic OEC cluster without the dangler should fail to evolve oxygen at the observed rate.

What would falsify it

A spin-crossover compound where G-orbit theory gives the wrong spin state while DFT gives the right one, with no G-symmetry reason for the failure. This would show that the G-orbit approach is missing essential physics, not just regularising what DFT approximates.
The FeMo-cofactor 14-opcode assignment being wrong — if the reaction pathway is re-measured (e.g. by cryo-EM on the Janus intermediate) and found to have a different step count or sequence, the specific ISA programme assignment is falsified (though the general framework need not be).
The PSII design rule failing — if a synthetic OEC without the C₁ dangler still evolves oxygen at the observed rate, the dangler prediction is wrong.

Open questions

Can we write new programmes? The forward direction (read the ISA programme of an existing enzyme) is demonstrated. The inverse direction (specify a target reaction, design the G-orbit graph, synthesise a catalyst that runs it) is open. Paper 490 proposes a ProteinMPNN pipeline for the protein case; the small-molecule case is less developed.
What determines the step count? Why is nitrogen fixation 14 steps and not 12 or 16? Is there a minimality principle — an analogue of Kolmogorov complexity for G-orbit programmes — that predicts the optimal step count?
Is decoherence irrelevant by construction? G-orbit walks are discrete and group-theoretically protected — they do not require quantum coherence in the usual sense. This is why the Galois computer operates at 300K. But there may be reactions where coherence does matter (FMO photosynthetic energy transfer being the leading candidate), and a unified picture is needed.
The G₂ extension: for f-block chemistry (lanthanides, actinides, f-shell transition metals) the relevant symmetry group may be G₂ rather than SU(3) or SU(2). Paper 492 (Langlands for Galois Chemistry) begins this extension; the experiments are not yet done.