Kohn–Sham ensemble representability of Coulomb densities

Kohn–Sham ensemble representability of Coulomb densities concerns whether the electron density of an interacting molecular ground state can be reproduced by noninteracting electrons moving in one common local potential, as Kohn–Sham density functional theory assumes. A manuscript in OpenAI's public mathematics collection, dated 25 September 2026, claims a three-electron molecule with two equal nuclear charges whose ground-state density has no such representation by any ground-state ensemble, for real spin-independent potentials in the standard class L^3/2 + L^∞. The collection has no Lean formalization of this result.

Background

As recounted in the manuscript's introduction:
- Representability. Hohenberg and Kohn introduced the density-functional viewpoint; Kohn and Sham sought to reproduce an interacting ground-state density by noninteracting electrons in a common local potential. The constrained-search formulations of Levy and Lieb separate the existence of states with a given density from the existence of a potential for which such states minimize the energy. Penz and coauthors review the distinctions between uniqueness of a density–potential map and existence of a representing potential.
- Ensembles. Mixing several degenerate noninteracting ground states can give densities that no single determinant gives; Ullrich and Kohn make this distinction explicit for singlet ground-state ensembles. A failure of one orbital ansatz therefore does not settle the ensemble question.
- Earlier examples and settings. Wagner and collaborators discuss a prescribed two-electron density whose interior cusp forces its inverse potential out of L^3/2 + L^∞, but it is not constructed as a Coulomb ground-state density. Trushin, Erhard and Görling report numerical indications of failures from atomic density inversions. Existence results hold in other settings: Chayes, Chayes and Ruskai for quantum lattice systems, Corso for spinless fermions on a bounded interval and Sutter and collaborators on a one-dimensional torus, the last two with distributional potentials.
- The mechanism. Gori-Giorgi, Gál and Baerends studied the asymptotic density when the leading removal amplitude has a nodal plane: coupling to other ionic channels can give the same exponential decay on and off the plane, with an extra polynomial suppression on it, which makes a regular Kohn–Sham potential difficult.

The original result

Claim (25 September 2026). The main theorem of A Coulomb ground-state density without Kohn–Sham ensemble representation sets D = 10²⁰⁰ and places two nuclei of equal positive integer charge Z at distance D/Z on either side of the origin. For a finite Z given by an exact formula, the three-electron Coulomb Hamiltonian has a normalizable absolute ground state in the sector S_z = 1/2, but no potential v in L^3/2 + L^∞ and no ground-state ensemble of the noninteracting Hamiltonian with potential v has the same spin-summed density almost everywhere. A corollary states that the same density has no local-potential first variation of the ensemble Hartree–exchange–correlation functional along every segment of its finite-energy domain.
Scope, as stated: the representing ground space may have any degeneracy and the ensemble may contain complex spinful states. The specification of Z is nonnumerical, and no effective numerical bound for the charge is asserted. The manuscript makes no assertion about distributional potentials, spin-dependent potentials or nonlocal one-particle operators.
Method, as described: after scaling by 1/Z, the two lowest orbitals are an even orbital and an odd orbital that vanishes on the median plane between the nuclei, and the ground state continues the determinant with two electrons in the even and one in the odd orbital. A channel analysis shows that far out on the median plane the density keeps a nonzero tail of lower polynomial order. An angle minimization splits the density into two orbitals, and a weighted spectral estimate shows that their determinant minimizes the kinetic energy among all ensembles with this density, so any representing potential must equal one candidate up to a constant. Translated limits of the orbital equations then show that this candidate cannot lie in L^3/2 + L^∞, because the plane tail would have to decay faster than it does.
Sources. Commit adc7f12 of the openai/math repository: A Coulomb ground-state density without Kohn–Sham ensemble representation.
Formalization. The collection lists no Lean formalization or Comparator challenge for this family.
Within the collection, other manuscripts on Coulomb systems include those described on The ionization and generalized ionization conjectures, QMA-hardness of continuum Coulomb energy and The Monge ansatz for the three-marginal Coulomb cost.

Public follow-up work

Date	Work	Relation	Public claim	Public status (checked 31 October 2026)
2026-10-07	A Coulomb density without Kohn–Sham ensemble representation, an article of the Latent Seminar blog at lowrank.github.io	Reading, exposition	Explains family 278: densities and noninteracting representation; the finite Coulomb counterexample; a slow tail on the median plane; two orbitals from an angle minimization; why arbitrary ensembles are excluded; translated potentials and the trace contradiction; exact parameters and a functional consequence; proof outlines, including why the first nonzero planar tail occurs at order four.	Blog article; it states that its review covers statements, introductions and selected proof mechanisms and is not an independent verification of the complete proofs. Read on 31 October 2026.
2026-10-10	Cody Haycraft, A Kohn–Sham Hamiltonian for the OpenAI three-electron density: a note on representability beyond L^3/2 + L^∞, a working draft in a public GitHub repository, commit `caee7f1`	Response, extension	Keeps the preprint's exclusion in L^3/2 + L^∞ and observes that the preprint's own estimates make the operator −½Δ + V nonnegative on smooth compactly supported functions, where V is the preprint's only candidate potential, Δg divided by 2g. Its Theorem A states that the Friedrichs extension is a self-adjoint Kohn–Sham Hamiltonian whose three-electron Aufbau determinant is a ground state with density n, and that any representing potential in the KLMN class equals V up to a constant. Its Theorem B, form domain H¹ and the KLMN form sum, is conditional on an open uniform strip estimate, Estimate 5.5. A one-dimensional strip model solved numerically suggests that V carries an attractive sheet along the nodal plane; the statements for the molecule are given as conjectures.	Working draft of 10 October 2026, not refereed; the README records seven review rounds, lists Estimate 5.5 as open and states that the auxiliary Lean lemmas check only elementary steps, not the analytic content. Found through the unreviewed agent-collected notes of the community revision mathcommunity/oai-math-revised and read at the source on 31 October 2026.
2026-10-18	AI Proved What?, episode 278 Kohn–Sham can't fake it, proved at last?, remade version of 23 October 2026	Reading, exposition	Video explainer of the 45-page preprint: a molecule with two equal nuclei of positive integer charge and three electrons whose lowest-energy density no ensemble of noninteracting ground states reproduces in any single real spin-independent potential of the standard class. It notes that the charge is defined by a formula that has never been evaluated, that the paper credits the physical mechanism to Gori-Giorgi, Gál and Baerends, and that the preprint does not claim a practical failure of density functional theory or anything about spin-dependent, distributional or nonlocal potentials. It adds its own toy computations, including a one-dimensional finite-grid model, which lacks the infinite plane the obstruction needs.	Episode status, last checked 30 October 2026: posted as a preprint, not revised; under expert review, not yet; peer reviewed, not yet; checked in Lean, no. The site states that it is made with Anthropic's Claude. Read on 31 October 2026.

Notes

The background summarizes the manuscript's own account of earlier work.
The list covers follow-ups found in public sources up to the date above; it is not a priority or completeness claim.