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Solovej's generalized ionization conjecture for neutral Coulomb atoms (energies and radii)

For the full nonrelativistic Coulomb atom with nuclear charge ZZ and two spin states, let Im(Z)=EZ(Z−m)−EZ(Z)I_m(Z)=E_Z(Z-m)-E_Z(Z) be the energy needed to remove mm electrons from the neutral atom, and let Rm(Z)R_m(Z) be the radius outside which an expected mm electrons of a neutral ground state lie. Thomas-Fermi theory predicts ImTF∼aTFm7/3I_m^{TF}\sim a_{TF}m^{7/3} and radii ∼(81π2/2)1/3m−1/3\sim(81\pi^2/2)^{1/3}m^{-1/3} as Z→∞Z\to\infty and then m→∞m\to\infty. Solovej (2016) conjectured that the same asymptotics hold for the true many-body atom: lim⁡m→∞m−7/3lim sup⁡Z→∞Im(Z)=lim⁡m→∞m−7/3lim inf⁡Z→∞Im(Z)=aTF\lim_{m\to\infty}m^{-7/3}\limsup_{Z\to\infty}I_m(Z)=\lim_{m\to\infty}m^{-7/3}\liminf_{Z\to\infty}I_m(Z)=a_{TF}, and likewise m1/3Rm(Z)→(81π2/2)1/3m^{1/3}R_m(Z)\to(81\pi^2/2)^{1/3} for the upper and lower large-ZZ limits. He had proved the analogues in Hartree-Fock theory. Do these iterated limits hold for the full Schrodinger atom?

Result
Proved(see note)
Status
Candidate (review pending)
AI contribution
AI-discovered
Method
Argument
Field
Many-body quantum mechanics; Thomas-Fermi theory
Posed by
Jan Philip Solovej
Year posed
2016
Years open
10y
Solved
2026-09-24
Model
Unreleased internal OpenAI model
Vendor
OpenAI
Collaborators
—
Verification
Lean-checked, statement unaudited
Publication
Announced
Collection
OpenAI math release (October 2026), version adc7f12
Significance
32 / 100
Disclosed cost
—
Wikipedia
No dedicated article

What was actually shown

Energy (principal, Theorem 1.1): Im(Z)/m7/3→aTF>0I_m(Z)/m^{7/3}\to a_{TF}>0 whenever m→∞m\to\infty and Z/m→∞Z/m\to\infty, with aTFa_{TF} the Thomas-Fermi constant; this is stronger than and implies Solovej's iterated limits, which are stated as (1.4)-(1.5). Radius (companion, Theorem 1.1): for every choice of neutral ground states, m1/3lim sup⁡ZRm(Z)m^{1/3}\limsup_Z R_m(Z) and m1/3lim inf⁡ZRm(Z)m^{1/3}\liminf_Z R_m(Z) both tend to (81π2/2)1/3(81\pi^2/2)^{1/3}, with Z→∞Z\to\infty first. Not shown: convergence of Im(Z)I_m(Z) or Rm(Z)R_m(Z) at fixed mm, a joint limit for radii, or relativistic and molecular versions.

What the AI did

The release README says the results were produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier model results. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author. The energy and radius manuscripts both rely on the screening estimates of the companion Uniform excess charge for Coulomb molecules and the outer radius of neutral atoms (same family, same date).

Verification

No independent mathematician has checked this yet. Checked here: Theorem 1.1 of the energy manuscript and Theorem 1.1 of the radius manuscript were read against Solovej's Equation (1). The proofs were not refereed. Energy: lean/formalization.yaml lists comparator CoulombIonization, declaration OAI.CoulombAtom.generalized_ionization. Its statement, read here: there is a>0a>0 with the Thomas-Fermi weak-ionization characterization, the joint limit Im(Z)/m7/3→aI_m(Z)/m^{7/3}\to a whenever m→∞m\to\infty and Z/m→∞Z/m\to\infty, and both iterated limits, with energies defined as infima of the Coulomb form over antisymmetric two-spin Sobolev states. Radius: ComparatorChallenges/CoulombRadii.json exists with solution module OAI.Analysis.CoulombRadii.Main present at the pinned commit, but the challenge is not in the formalization catalogue; its statement, read here, gives both upper and lower iterated radius limits equal to (81π2/2)1/3(81\pi^2/2)^{1/3} for every sequence of normalized ground states, with existence of ground states taken as a hypothesis. Both state the headline claims. Neither was rebuilt here.

Sources

Changelog1 change

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