Chemistry Roadmap

How the ISA framework approaches chemistry — from first principles to reaction mechanisms, without fitting parameters.

The problem with the standard picture

Computational chemistry hands you a menu: Hartree-Fock, DFT, CCSD, CASSCF, FCI. Each method makes different approximations. Choosing the right one requires specialist expertise, and for the hardest systems — transition-metal complexes, bond-breaking, catalytic transition states, biological electron transfer — every commonly used method either fails silently or is prohibitively expensive.

The ISA framework does not replace these methods. It tells you which stratum each electron pair belongs to before you compute anything, so you use the right method on the right pairs automatically.

The four layers

Layer 0 — Topology: the sCeleTon

What it is. Given a molecule (atoms and connectivity only), draw the C/T graph:

  • A C-box for every frozen electron pair — ionic bonds, lone pairs, core electrons. These pairs are closed to correlation. DFT or Hartree-Fock is exact here.
  • A T-arrow for every active correlated pair — covalent bonds with partial character, resonance, multi-reference systems. These pairs are open and must be treated with correlated methods.

What it costs. Nothing beyond the Lewis structure. No basis set, no SCF convergence, no functional.

What it tells you. How many T-arrows = the minimum active space size for CASSCF. All C-boxes = H⁰, DFT is fine. A few T-arrows = H¹, CCSD will handle it. Many T-arrows, or T-arrows with G₂ structure = H², CASSCF essential.

The tool. alchemi — builds the sCeleTon from a molecular geometry and recommends the active space.

Papers. doi:10.5281/zenodo.21300667 (Topological Active Space, Paper 588).

Layer 1 — Geometry: θ_G and the Weyl coordinate

What it is. Run a cheap MP2 calculation. From the Natural Orbital Occupation Numbers (NOONs), compute two quantities for each T-arrow pair:

  • θ_G — the Grassmannian angle. Measures how far the ground-state wavefunction has moved from the single-determinant (Hartree-Fock) reference on the manifold Gr(n_e, n_orb). Small θ_G = ionic / weakly correlated. Large θ_G = covalent / strongly correlated / multi-reference.

    θ_G = 2 arccos(√(n_g / 2))

    where n_g is the NOON of the bonding orbital in the pair.

  • Weyl c₂ — the second Weyl chamber coordinate. Predicts DFT error with correlation r = 0.990 across all tested systems. If c₂ < δ (threshold), DFT or CCSD is reliable. If c₂ ≥ δ, CASSCF is required.

What θ_G replaces. Pauling electronegativity is a fitted table with no geometric foundation. θ_G is a coordinate on a Riemannian manifold (the Fubini-Study metric on the Grassmannian), computable from first principles, that recovers the polarity ordering — LiF more ionic than HF, H₂ more covalent than LiF — without any fitting.

The β* snap. At θ_G ≈ 20°, the MGE routing switches from H¹ to H². This is the ionic → covalent crossover geometry: the point at which NaCl transitions from Na⁺Cl⁻ to Na·Cl under bond stretching, or where an Fe spin-crossover complex switches spin state under temperature. The snap position is predicted, not fitted.

Papers. doi:10.5281/zenodo.21277821 (Universal Bonding Theory, Paper 570).

Layer 2 — Energetics: automatic method routing

The routing table. Once Layer 1 is done, each electron pair has a stratum assignment. The router sends it to the right method:

Stratum θ_G Weyl c₂ Method Why
H⁰ (C-box) < 5° — Freeze / DFT Single-determinant exact
H¹ 5°–20° < δ CCSD Single-reference correlation sufficient
H² > 20° ≥ δ CASSCF Derivative discontinuity; DFT fails

The routing guarantees:

  • No silent failure (you know when DFT breaks before running it)
  • No over-convergence (you don’t CCSD pairs that need CASSCF)
  • Minimal cost (you don’t CASSCF pairs that CCSD can handle)

Papers. doi:10.5281/zenodo.21373469 (Weyl-DFT Accelerator, Paper 596).

Layer 3 — Mechanism: H² phase transitions

What it is. Once you know which pairs are H², look for G₂ structure in the active space. The presence of G₂ geometry (a 14-dimensional subspace of SU(n)) predicts which mechanism the reaction takes.

Example: Spin crossover. Iron complexes with unpaired d-electrons undergo spin-state transitions. In the sCeleTon language, this is the H⁰→H¹→H² cascade as temperature lowers:

  • Above T_c: all pairs are H⁰ (spin-free)
  • Near T_c: active pairs move to H¹ (intermediate spin)
  • Below T_c: H² structure locks in (low-spin)

The H² tier predicts the cooperativity and hysteresis of the transition.

Papers. doi:10.5281/zenodo.21373477 (G-Step CO₂ Fixation, Paper 603); doi:10.5281/zenodo.21345099 (Protein Folding ISA, Paper 515).

Which paper for which use case?

Choose your question to find the most relevant papers.

“I want to understand enzyme kinetics”

  1. Paper 515: Protein Folding ISA
    • Spontaneous symmetry making (SSM) as the H¹→H² transition
    • Chaperones as quantum error correction (H² QEC)
    • Validated on protein stability prediction (AUC=0.981)
    • Start here if: You want to understand how proteins pick a fold
  2. Paper 510: Kinetic Proofreading as QEC
    • Proofreading IS quantum error correction
    • H⁰×H¹×H² gives 10⁹/10⁶/10⁴ fidelity for Pol III/RNAP/ribosome
    • Start here if: You want enzyme accuracy without invoking “selection pressures”
  3. Paper 447: Opcode Rosetta Stone
    • Chemical reactions as PROP morphisms
    • Composite reactions from simple opcode sequences
    • Start here if: You want to code up enzyme pathways

“I want to predict transition metal complex properties”

  1. Paper 488: Galois Chemistry
    • DFT as H² computation (quantum group structure from spin)
    • Why transition metals need non-abelian treatment
    • Start here if: You have TM complexes and want to know when DFT fails
  2. Paper 596: Weyl-DFT Accelerator
    • Weyl c₂ as DFT failure detector (r=0.990)
    • MGE soft routing replaces hard CASSCF threshold
    • H¹↔H² snap at β* is a genuine phase transition
    • Start here if: You want a quantitative rule for “when to CASSCF”
  3. Paper 563: Hubbard/Mott ISA
    • Hubbard model as H² computation
    • Strongly correlated systems stratified by U/t ratio
    • Start here if: You work with heavy fermion systems
  4. Paper 570: Universal Bonding Theory
    • θ_G (Grassmannian angle) predicts bond order without fitting
    • Lewis, MO, and VB theories unified via ISA tiers
    • Validated on benzene resonance energy (54.5 vs 57.4 mEh)
    • Start here if: You want a geometric foundation for bonding

“I want to design a new enzyme”

  1. Paper 490: Galois Protein Design
    • Forward: AlphaFold → ISA tier stratification → design constraints
    • Inverse: desired H² property → ProteinMPNN → candidate sequences
    • Validated on RNR, PSII, hemoglobin cooperativity
    • Start here if: You want to design with thermodynamic control, not trial-and-error
  2. Paper 515: Protein Folding ISA
    • SSM theorem: G_fold is created by H¹→H² transition, not pre-existing
    • Chaperones perform H² error correction
    • Start here if: You want to understand protein folding dynamics

“I think my system might be topological”

  1. Paper 565: FQHE ISA
    • Fractional quantum Hall effect as H² topology
    • Laughlin wavefunction as ISA-tier structure
    • Start here if: You have 2D electrons in strong B field
  2. Paper 414, 416, 420: Chiral & Topological Chemistry
    • Chiral molecules as H¹/H² topological objects
    • Topological insulators as chemistry with decoherence
    • Start here if: You suspect topological protection
  3. Contact us — H³ (affine Kac-Moody) in chemistry is frontier research. If you suspect it, let’s talk.

FAQ

Q: Is my simple organic molecule H¹ or H²?

A: Use the sCeleTon (Layer 0). If your molecule has only σ bonds and no transition metals / radicals → mostly H⁰/H¹. If it has aromaticity (resonance) or open-shell character → H¹–H² boundary. If it has a transition metal → definitely H². Use Layer 1 (θ_G, Weyl c₂) to know exactly.

Q: Does ISA replace DFT?

A: No. ISA tells you which tier your system is in. It then tells you which computational method is appropriate for that tier. DFT is an H² method (it assumes single-determinant ground state; breaks at H² derivative discontinuity). Use ISA to know when DFT will fail silently, before you run it. Then use CCSD (H¹) or CASSCF (H²) as appropriate.

Q: My system has strong correlation. Should I just use CASSCF?

A: Not necessarily. CASSCF is H² (best for systems with real topological obstruction). If your system is H¹ (single-reference but correlated), CCSD is cheaper and often accurate. Use Layer 1 (Weyl c₂, θ_G) to know the difference. The routing table tells you the right method.

Q: Can H³ (affine Kac-Moody) appear in chemistry?

A: Unknown. We predict it appears in topological materials. Experiments I & J (knot theory, differentiable-tropical-networks) show H³ exists in nature. Whether it appears in chemistry is an open question. You might discover it first! If you suspect topological structure beyond standard H¹/H², reach out.

Q: Where’s the formal mathematical proof of H³/H⁴?

A: Coming in Paper 622 (July 2026): “The Hum ISA as a Monoidal PROP.” For now:

  • Theory: Papers 619–621 ground H⁰–H² rigorously in category theory
  • Experiments: Experiments A–J (knot theory) validate H³/H⁴ empirically (virtual knots 2.64×, loop braids 6.5×)
  • Paper 622: Will formally verify H³/H⁴ fit into the monoidal PROP structure

This is not weakness — it’s how science works. We’ve theorised, predicted, and experimented. Paper 622 will complete the proof.

Q: How do I cite this framework in a paper?

A: Cite the Origami ISA manifesto:

@article{buckley2024origami,
  title={Origami: An Open ISA for Quantum-Classical Systems},
  author={Buckley, Ian R. C.},
  journal={Zenodo},
  doi={10.5281/zenodo.21428853},
  year={2024}
}

For chemistry-specific applications, also cite:

  • Paper 570 (Universal Bonding Theory)
  • Paper 596 (Weyl-DFT Accelerator)
  • Paper 488 (Galois Chemistry)
  • Paper 490 (Galois Protein Design)
  • Opcodes Reference — The five generators (LABEL, ORBIT, TWIST, BIND, FLIP) and their categorical meaning
  • β-Plane Geometry — How Forge and Meld ISAs relate via Wick rotation
  • ISA Hierarchy (coming soon) — Family tree: Carnot → Origami/Forge/Raven/Hum
  • Categorical Rigor (coming soon) — What’s proven vs experimental vs predicted
  • All Papers — Full bibliography with Zenodo DOIs

Last updated: 2026-07-20
Questions? Contact us