Kohn–Sham can’t fake it, proved at last?
An AI preprint claims a Coulomb ground-state density with no Kohn–Sham ensemble representation
Covers result 278 in OpenAI’s catalog: its one paper. See its connections to other results and the work it cites.
3:34. English captions available. The video loads from YouTube when you press play.
Chapters
The claim, in plain words
A many-electron wave function has three coordinates per electron, far too many to compute. Density functional theory works with the electron density instead, three coordinates in total; Hohenberg and Kohn showed in 1964 that the ground-state density determines everything. In 1965 Kohn and Sham added the practical trick: replace the interacting electrons by noninteracting ones moving in a single effective potential that reproduces the same density.
That trick quietly assumes such a potential exists. Mixtures of states, called ensembles, patch some failures. Densities that even ensembles cannot reproduce were known since 1983, but they were prescribed by hand, not computed from Coulomb molecules. For continuous three-dimensional systems, a 2026 paper called the question completely open.
Result 278 is one OpenAI preprint, 45 pages, dated September 25, 2026. It constructs a molecule with two equal nuclei of positive integer charge and three electrons whose lowest-energy density cannot be reproduced by any ensemble of noninteracting ground states, in any single real, spin-independent potential of the standard class used in the rigorous theory (L3/2 + L∞). The molecule is a mathematical object: the nuclei sit 2 × 10200 divided by their charge atomic units apart, and the charge is defined by a formula that has never been evaluated, with no upper bound given. The paper credits the physical mechanism to Gori-Giorgi, Gál and Baerends.
What it doesn’t claim
That density functional theory fails in practice, anything about spin-dependent, distributional or nonlocal potentials, or that no potential exists at all: a formal one does, it just falls outside the standard class.
Where it stands
- Posted as a preprint
- OpenAI, 45 pages, dated September 25, 2026
- Peer reviewed
- Not yet
- Checked in Lean
- No
Last checked October 19, 2026. The preprint has not been revised. There is no Lean formalization, it is not listed on OSVerify, and we found no comment from specialists.
How the proof goes
- A faint trace on the mirror plane. Rescaled, the electron repulsion becomes weak, and the ground state is close to two electrons in a symmetric orbital and one in an antisymmetric orbital, which vanishes on the mirror plane between the nuclei. Far out on that plane, the true density still fades at the same exponential rate, only slightly weaker, because the departing electron polarizes the ion left behind. The only candidate noninteracting state would fade faster there in any admissible potential, so none works.
- Where it works, and where it fails. In our one-dimensional toy on a finite grid, the paper’s recipe for the only candidate potential reproduces the interacting density to 12 digits; the obstruction needs a plane reaching to infinity, which the toy lacks. In our model of the paper’s asymptotics, the mismatch distance grows like one over the tiny level splitting: about 1,300 units when the splitting is 0.01.
The numbers in the video
- Ours: in a two-dimensional well, an ensemble reaches a round density that no single real determinant in that well can.
- Ours: a 1D three-electron toy on a finite grid where the recipe works, and a model of the mismatch distance.
Read the source
- A Coulomb Ground-State Density without Kohn–Sham Ensemble Representation (PDF), 45 pages.
- The collection’s catalog, where this is result 278.
The paper
OpenAI groups related papers into one result; this one has a single paper.
Transcript
Read the full script
0:00The question
A many-electron wave function has three coordinates per electron, far too many to compute. Density functional theory works with the electron density instead, three coordinates in total; Hohenberg and Kohn showed in 1964 that the ground-state density determines everything. In 1965 Kohn and Sham added the practical trick: replace the interacting electrons by noninteracting ones moving in a single effective potential that reproduces the same density. That trick quietly assumes such a potential exists. Mixtures of states, called ensembles, patch some failures: in our two-dimensional toy, an ensemble reaches a round, rotation-invariant density that no single real determinant in that well can. Densities that even ensembles cannot reproduce were known since 1983, but they were prescribed by hand, not computed from Coulomb molecules. For continuous three-dimensional systems, a 2026 paper called the question completely open.
1:04The claims
Result 278 is one OpenAI preprint, 45 pages, dated September 25, 2026. It constructs a molecule with two equal nuclei of positive integer charge and three electrons whose lowest-energy density cannot be reproduced by any ensemble of noninteracting ground states, in any single real, spin-independent potential of the standard class used in the rigorous theory. The molecule is a mathematical object. The nuclei sit 2 × 10²⁰⁰ divided by their charge, atomic units apart, and the charge is defined by a formula that has never been evaluated, with no upper bound given. The paper credits the physical mechanism to Gori-Giorgi, Gál and Baerends. Not claimed: that density functional theory fails in practice, anything about spin-dependent, distributional or nonlocal potentials, or that no potential exists at all: a formal one does, it just falls outside the standard class.
2:08How the proof goes
Rescaled, the electron repulsion becomes weak, and the ground state is close to two electrons in a symmetric orbital and one in an antisymmetric orbital, which vanishes on the mirror plane between the nuclei. Far out on that plane, the true density still fades at the same exponential rate, only slightly weaker, because the departing electron polarizes the ion left behind. The only candidate noninteracting state would fade faster there in any admissible potential, so none works. The paper’s recipe for the only candidate potential is concrete: in our one-dimensional toy on a finite grid it reproduces the interacting density to 12 digits; the obstruction needs a plane reaching to infinity, which the toy lacks. In our model of the paper’s asymptotics, the mismatch distance grows like 1/(level splitting): about 1,300 units when the splitting is 0.01.
3:06Where it stands
The preprint is unrefereed and unchanged since release. There is no Lean formalization, OSVerify has not reviewed it, and we found no comment from specialists. If it holds, Kohn and Sham’s stand-in electrons, in the standard class of potentials, cannot mimic every true ground-state density, even with ensembles, though the molecule lies far beyond real chemistry.
Spot a mistake?
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