A hard chemistry problem, proved at last?
Two AI preprints claim QMA-hardness of the electronic ground energy, with no basis set
Covers result 275 in OpenAI’s catalog: all 2 papers. See its connections to other results and the work it cites.
3:42. English captions available. The video loads from YouTube when you press play.
Chapters
The claim, in plain words
Chemistry’s central calculation: given fixed atomic nuclei, find the lowest energy of the electrons around them. For hydrogen it is exactly −1/2 hartree, the atomic unit of energy. For many electrons this is expected to be hard. QMA, the quantum version of NP that Kitaev studied, captures such problems, and lattice and spin versions were proved QMA-hard years ago. In 2021, O’Gorman, Irani, Whitfield and Fefferman proved hardness when the electrons are restricted to a given basis of orbitals, and asked whether it stays hard in the full continuum, with only positively charged nuclei. The catch: restricting electrons to a basis can only raise the energy, so a hard problem in a basis says nothing about whether some exotic continuum state sneaks lower.
Result 275 is two OpenAI preprints, 69 pages, both dated September 24, 2026. They claim that deciding whether this lowest energy is below one threshold or above another is QMA-hard, over all electron states in three dimensions with fixed nuclei, with thresholds at least one hartree apart. One paper uses nuclei with large charges written in binary; the other uses only unit charges, like protons, at rational positions.
What it doesn’t claim
That the problem lies in QMA, so not that it is QMA-complete; anything about real molecules or practical chemistry; or other versions of the question, with a bounded basis error. The hard instances are monsters: in the binary construction a 3×3 grid of spins becomes 66 nuclei and 33 electrons, but each charge has thousands of digits; in the unit construction even toy settings of its formulas give about 1021 protons for only a few hundred electrons, and a one-hartree gap is tiny beside their total energies.
Where it stands
- Posted as preprints
- OpenAI, two papers, 69 pages, dated September 24, 2026
- Peer reviewed
- Not yet
- Formalized in Lean
- Both theorems (docs)
Last checked October 19, 2026. The preprints have not been revised. Both theorems are formalized in Lean, about 220,000 lines with no unproved steps, which we did not build; Lean even proves the hardness of the starting magnet, though it derives the binary theorem from the unit one. OSVerify has not reviewed them, and we found no comment from specialists.
How the proofs go
- A hard magnet made of electrons. The proof starts from a quantum magnet known to be hard and builds it out of electrons. Each spin is an electron in its own well, and the magnetic coupling comes from an electron briefly hopping to a neighbour and back, Anderson’s superexchange; in the binary paper’s formula, hydrogen’s on-site repulsion is exactly 5/8 hartree. Signed couplings are made positive with mediator spins.
- The whole continuum. The last step is the new one: a lower bound over every continuum state, not only those in a basis. The unit paper also designs smooth wells, each with in-plane energy exactly −1/2, and chops a positive charge cloud into single protons.
The numbers in the video
- Ours: hydrogen at −0.4999999 hartree; the splitting between the two lowest levels of H2+ shrinks exponentially with proton distance.
- Ours: the mediator gadget’s error shrinks like 1/√Δ, about nine times inside the paper’s bound.
- Ours: the sizes of the constructions, from 66 nuclei with charges of thousands of digits to about 1021 unit nuclei at toy settings.
Read the source
- QMA-Hardness of Continuum Coulomb Energy with Unit Nuclear Charges (PDF), 22 pages.
- Continuum Coulomb Hardness with Binary Nuclear Charges (PDF), 47 pages.
- The Lean formalization notes.
- The collection’s catalog, where this is result 275.
The 2 papers in this result
OpenAI groups related papers, such as alternative proofs, supporting results and consequences, into one result; we label the first paper it lists the main paper and the others companions.
Transcript
Read the full script
0:00The question
Chemistry’s central calculation: given fixed atomic nuclei, find the lowest energy of the electrons around them. For hydrogen it is exactly −1/2 hartree, the atomic unit of energy, and our computer reproduces that. For two protons sharing one electron, the bonding state sits lower, and the splitting between the two lowest levels shrinks exponentially with distance. For many electrons this is expected to be hard. QMA, the quantum version of NP that Kitaev studied, captures such problems, and lattice and spin versions were proved QMA-hard years ago. In 2021, O’Gorman, Irani, Whitfield and Fefferman proved hardness when the electrons are restricted to a given basis of orbitals, and asked whether it stays hard in the full continuum, with only positively charged nuclei. The catch: restricting electrons to a basis can only raise the energy, so a hard problem in a basis says nothing about whether some exotic continuum state sneaks lower.
1:03The claims
Result 275 is two OpenAI preprints, 69 pages, both dated September 24, 2026. They claim that deciding whether this lowest energy is below one threshold or above another is QMA-hard, over all electron states in three dimensions with fixed nuclei, with thresholds at least one hartree apart. One paper uses nuclei with large charges written in binary; the other uses only unit charges, like protons, at rational positions. The hard instances are monsters. In the binary construction, a 3×3 grid of spins becomes 66 nuclei and 33 electrons, but each charge has thousands of digits. In the unit construction, even toy settings of its formulas give about 10²¹ protons, for only a few hundred electrons, and a 1-hartree gap is tiny beside their total energies. Not claimed: that the problem lies in QMA, so not QMA-complete; anything about real molecules or practical chemistry; or other versions of the question, with a bounded basis error.
2:13How the proofs go
The proof starts from a quantum magnet known to be hard and builds it out of electrons. Each spin is an electron in its own well, and the magnetic coupling comes from an electron briefly hopping to a neighbour and back, Anderson’s superexchange, with hydrogen’s on-site repulsion, exactly 5/8 hartree, in the binary paper’s formula. Signed couplings are made positive with mediator spins; our computer checks that the gadget’s error shrinks like 1/√(energy scale), well inside the paper’s bound. The last step is the new one: a lower bound over every continuum state, not only those in a basis. The unit paper also designs smooth wells, each with in-plane energy exactly −1/2, and chops a positive charge cloud into single protons.
3:03Where it stands
The preprints are unrefereed and unchanged since release. Both theorems are formalized in Lean, about 220,000 lines with no unproved steps, which we did not build; Lean even proves the hardness of the starting magnet, though it derives the binary theorem from the unit one. OSVerify has not reviewed them, and we found no comment from specialists. If they hold, the many-electron problem is as hard as anything in QMA even when all you are given is a list of nuclei, though the hard cases found are systems no chemist will ever meet.
Spot a mistake?
This episode simplifies on purpose, but it shouldn’t be wrong. If something is, open an issue on GitHub or leave a comment on YouTube. Corrections are noted here with a date.