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Kohn-Sham ensemble v-representability of Coulomb ground-state densities by potentials in L^{3/2}+L^infinity

Kohn-Sham density functional theory replaces an interacting electronic ground state by noninteracting electrons in a common local potential that reproduces its density. Levy and Lieb separated the existence of states with a prescribed density from the existence of a potential for which such states are ground states; ensembles of degenerate noninteracting ground states enlarge the class (Ullrich-Kohn). Explicit non-representable model densities and numerical violations are known, but not for an actual Coulomb system. Is the ground-state density of every finite Coulomb molecule the density of a ground-state ensemble of noninteracting electrons in some real local potential v∈L3/2(R3)+L∞(R3)v\in L^{3/2}(\mathbb R^3)+L^\infty(\mathbb R^3)?

Result
Disproved(see note)
Status
Candidate (review pending)
AI contribution
AI-discovered
Method
Construction
Field
Density functional theory; noninteracting v-representability
Posed by
Assumed by Kohn and Sham (1965); posed as the v-representability problem by Mel Levy (1979) and Elliott H. Lieb (Density functionals for Coulomb systems, 1983); reviewed by Penz et al. (2023)
Year posed
1983
Years open
43y
Solved
2026-09-25
Model
Unreleased internal OpenAI model
Vendor
OpenAI
Collaborators
—
Verification
Unreviewed
Publication
Announced
Collection
OpenAI math release (October 2026), version adc7f12
Significance
22 / 100
Disclosed cost
—
Wikipedia
No dedicated article

What was actually shown

Theorem 1.1: there is a finite positive integer ZZ such that the three-electron Coulomb Hamiltonian with two nuclei of charge ZZ at ±(D/Z)ez\pm(D/Z)e_z, D=10200D=10^{200}, has a normalizable absolute ground state Ψ\Psi (sector Sz=1/2S_z=1/2) whose spin-summed density is not the density of any ground-state ensemble of Hs(v)H_s(v) for any real v∈L3/2+L∞v\in L^{3/2}+L^\infty. The mechanism is the Gori-Giorgi-Gal-Baerends nodal-plane asymptotics, made rigorous, plus a kinetic certificate against all ensembles. A corollary rules out a local-potential first variation of the ensemble Hartree-exchange-correlation functional at that density. Not covered: larger potential classes, atoms, or physically realistic charges.

What the AI did

Produced by an unreleased internal OpenAI model as part of OpenAI's openai/math release. The release README says the results used one fixed procedure averaging about three hours of ChatGPT Pro thinking compute per result; this family is not among the README's stated exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscripts are authored as OpenAI with no human author named.

Verification

No independent mathematician has checked this yet. Checked here: abstract, introduction and Theorem 1.1, read against the representability question as the manuscript frames it from Hohenberg-Kohn, Kohn-Sham, Levy, Lieb and Ullrich-Kohn; proof not refereed. No Lean formalization exists for this family. Scope limits stated by the paper: the counterexample uses nuclear separation D=10200D=10^{200} in scaled units and a charge ZZ given by an exact but non-numerical formula with no effective bound; it excludes only real spin-independent local potentials in L3/2+L∞L^{3/2}+L^\infty, and says nothing about distributional, spin-dependent or nonlocal potentials.

Sources

Changelog1 change

Discussion