Electrons that won’t squeeze, proved at last?
Two AI preprints claim the Laughlin gap, and its stability under weak disorder
Covers result 269 in OpenAI’s catalog: all 2 papers. See its connections to other results and the work it cites.
4:31. English captions available. The video loads from YouTube when you press play.
Chapters
The claim, in plain words
Put electrons in a thin layer under a strong magnetic field, and at one third filling of the lowest energy level they form a strange liquid: the fractional quantum Hall effect, discovered in 1982. Laughlin explained it in 1983 with a wave function in which every pair of electrons keeps its distance. Haldane then put the electrons on a sphere and found a simple model with Laughlin’s state as its exact ground state: every pair of electrons in relative angular momentum one, the closest two electrons can get, pays one unit of energy.
The liquid’s resistance to compression rests on a gap: every other state should cost at least a fixed amount of energy, however many electrons there are. Physicists have believed in this gap for forty years. Rougerie’s 2019 survey called the conjecture a major open mathematical problem of the field; earlier rigorous results covered only truncated interactions on thin cylinders and tori.
Result 269 is two OpenAI preprints, 91 pages, dated September 24 and October 5, 2026. The first claims that for Haldane’s model on the sphere, once there are enough electrons, every state orthogonal to Laughlin’s costs at least 1/25. It also claims a squared-energy inequality for every particle number at once, and, as a corollary, the planar version of the conjecture as Rougerie wrote it down, with an explicit constant (about 0.046) for every number of electrons. The second claims the gap survives weak disorder: add any bounded real potential, projected to the lowest level, at small enough strength, and the ground state stays unique with a gap above a fixed amount, for every large number of electrons. That is subtle, because the total size of the perturbation grows with the number of electrons.
What it doesn’t claim
The real Coulomb interaction, other fillings, bosons, strong disorder, or explicit thresholds: the numbers of electrons and the disorder strength from which the results hold are shown to exist but not computed. The proven 1/25 is about ten times below the numerical value, about 0.4.
Where it stands
- Posted as preprints
- OpenAI, two papers, 91 pages, dated September 24 and October 5, 2026
- Peer reviewed
- Not yet
- Formalized in Lean
- In part: the gap results, not the disorder paper (docs)
Last checked October 19, 2026. The preprints have not been revised. The gap results, including 1/25 and the planar version, are formalized in Lean: four statements, about 229,000 lines with no unproved steps, which we did not build. The disorder paper, the longer of the two, is not formalized, and OSVerify has not reviewed either. Within two weeks, a paper by Gao, its proof completed by Anthropic’s Claude under the author’s direction, raised the constant to at least 1/5. A Harvard group, Scheer and Khalaf, posted an independent proof of at least 1/100, reporting that they had their exact certificate in August.
How the proofs go
- Squaring the energy. The square of the energy splits into two-, three- and four-body parts. The troublesome parts are beaten by a sum of squares built from seven rows of integers on nine orbitals, averaged over all rotations of the sphere. Exact fractions certify the finite inequalities, and a change of variables carries them to every large sphere, with nothing growing with the number of electrons.
- Disorder. The second paper shows that in zero-energy states, missing electrons disturb local quantities at most in proportion to their number, and builds stability from there.
The numbers in the video
- Ours: Haldane’s model on spheres of 3 to 10 electrons: one zero-energy ground state; gap 0.79 at 3 electrons, 0.50 at 10.
- Ours: with opposite potentials on the two halves of the sphere, the naive safe disorder strength shrinks from 0.117 to 0.045 between 4 and 7 electrons, while the actual drop in the gap barely changes.
- Ours: the paper’s certificate replayed in exact fractions; every printed number comes out. Most of the 13 three-body checks pass by about 0.00025; the budget is 93.5% for the squares, about 2% for a tail and a safety allowance, and 4.6% left over as the constant.
Read the source
- A Fock-Space Inequality and the Laughlin Spectral Gap (PDF), 22 pages.
- Uniform Stability of the Spherical Laughlin Gap (PDF), 69 pages.
- The Lean formalization notes.
- The collection’s catalog, where this is result 269.
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
Put electrons in a thin layer under a strong magnetic field, and at one third filling of the lowest energy level they form a strange liquid: the fractional quantum Hall effect, discovered in 1982. Laughlin explained it in 1983 with a wave function in which every pair of electrons keeps its distance. Haldane then put the electrons on a sphere and found a simple model with Laughlin’s state as its exact ground state: every pair of electrons in relative angular momentum one, the closest that two electrons can get, pays one unit of energy. The liquid’s resistance to compression rests on a gap: every other state should cost at least a fixed amount of energy, however many electrons there are. Our computer shows the model on spheres of up to 10 electrons: a single zero-energy ground state, and a gap falling from 0.79 at 3 electrons to 0.50 at 10, apparently levelling off near 0.4. Physicists have believed in this gap for 40 years, but no proof existed. Rougerie’s 2019 survey called it a major open mathematical problem of the field. Earlier rigorous results covered only truncated interactions on thin cylinders and tori.
1:18The claims
Result 269 is two OpenAI preprints, 91 pages, dated September 24 and October 5, 2026. The first claims that for Haldane’s model on the sphere, once there are enough electrons, every state at right angles to Laughlin’s costs at least 1/25. It also claims a squared-energy inequality for every particle number at once, and, as a corollary, the planar version of the conjecture as Rougerie wrote it down, with an explicit constant for every number of electrons. The second claims the gap survives weak disorder: add any bounded real potential, projected to the lowest level, at small enough strength, and the ground state stays unique with a gap above a fixed amount, for every large number of electrons. That is subtle, because the total size of the perturbation grows with the number of electrons. Our computer shows it on a sphere with opposite potentials on its two halves: the naive safe strength shrinks by more than half from 4 to 7 electrons, while the actual drop in the gap barely changes. Not claimed: the real Coulomb interaction, other fillings, bosons, strong disorder, or explicit thresholds. The proven 1/25 is about 10 times below the numerical value.
2:38How the proofs go
The proof squares the energy. The square splits into two-, three- and four-body parts, and the troublesome parts are beaten by a sum of squares built from 7 rows of integers on 9 orbitals, averaged over all rotations of the sphere. Exact fractions certify the finite inequalities, and a change of variables carries them to every large sphere, with nothing growing with the number of electrons. The disorder paper then shows that in zero-energy states, missing electrons disturb local quantities at most in proportion to their number, and builds stability from there. Our check: we replayed the paper’s certificate in exact fractions, and every printed number comes out. Most of the 13 three-body checks pass by only about 0.00025. Of the energy budget, about 93.5% pays for the squares, about 2% covers a tail and a safety allowance, and the 4.6% left over is the constant.
3:36Where it stands
The preprints are unrefereed and unchanged since release. The gap results, including the 1/25 and the planar version, are formalized in Lean, about 229,000 lines with no unproved steps, which we did not build. The disorder paper, the longer of the two, is not formalized, and OSVerify has not reviewed either. Within two weeks, a paper by Gao, its proof completed by Anthropic’s Claude under the author’s direction, raised the constant to at least 1/5. A Harvard group, Scheer and Khalaf, posted an independent proof of at least 1/100, reporting that they had their exact certificate in August. If they hold, the gap above Laughlin’s state in Haldane’s model is a theorem: it never closes as electrons are added, even if the proven size is 10 times too small.
Spot a mistake?
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