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The ionization conjecture: bounded excess charge, ionization energy and radius for Coulomb atoms and molecules (Simon's Problem 9)

For the nonrelativistic Coulomb Hamiltonian of nn electrons (two spin states, Fermi statistics) around fixed nuclei of total charge ZZ, let NcN_c be the largest number of electrons that is strictly bound. Lieb proved Nc<2Z+MN_c<2Z+M, Lieb-Sigal-Simon-Thirring proved Nc/Z→1N_c/Z\to1 for atoms, and later work gave sublinear and explicit bounds such as 1.22Z+3Z1/31.22Z+3Z^{1/3}, but no uniform bound on the excess. Simon's Problem 9 (2000) asks whether Nc(Z)−ZN_c(Z)-Z stays bounded for atoms; the molecular form asks for Nc≤Z+CMN_c\le Z+CM with MM nuclei (Lewin 2025). The ionization conjecture as surveyed by Solovej also asks that the first ionization energy and the outer radius of neutral atoms stay bounded independently of ZZ. Is the excess charge bounded by a universal constant per nucleus, and are the neutral-atom ionization energy and radius bounded uniformly in ZZ?

Result
Proved(see note)
Status
Candidate (review pending)
AI contribution
AI-discovered
Method
Argument
Field
Many-body quantum mechanics; Coulomb systems
Posed by
Barry Simon (Problem 9 of his 2000 list); molecular form in Mathieu Lewin's 2025 open-problem survey
Year posed
2000
Years open
26y
Solved
2026-09-24
Model
Unreleased internal OpenAI model
Vendor
OpenAI
Collaborators
—
Verification
Unreviewed
Publication
Announced
Collection
OpenAI math release (October 2026), version adc7f12
Significance
55 / 100
Disclosed cost
—
Wikipedia
No dedicated article

What was actually shown

Theorem 1.1: there is a universal CC such that for every M≥1M\ge1, all distinct nuclear positions and all real charges zj≥1z_j\ge1 with Z=∑zjZ=\sum z_j, strict binding En<En−1E_n<E_{n-1} implies n≤Z+CMn\le Z+CM, and En=En−1E_n=E_{n-1} for n>Z+CMn>Z+CM. For atoms this is Simon's bounded excess charge. Theorem 1.2: for every integer Z≥1Z\ge1 and every neutral ground state, the radius outside which half an electron remains lies between universal constants. Corollary 1.3: the neutral first ionization energy satisfies c≤I1(Z)≤Cc\le I_1(Z)\le C. Not shown: the value of the constants, relativistic or magnetic models, or the asymptotic (generalized) form, which is the separate generalized ionization entry.

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. This manuscript also supplies the screening estimates used by the two generalized-ionization companions in the same family.

Verification

No independent mathematician has checked this yet. Checked here: Theorems 1.1 and 1.2 and Corollary 1.3 were read against Simon's Problem 9 and the molecular and neutral-atom forms the manuscript cites. The 80-page proof was not refereed. This manuscript has no Lean main result: lean/docs/263.md lists only the two generalized-ionization companions, and the Lean developments for those may formalize some of its estimates internally, but no statement of Theorem 1.1, 1.2 or Corollary 1.3 is a comparator challenge. The paper notes that equality of adjacent energies does not exclude a threshold ground state, which its strict-binding formulation does not need.

Sources

Changelog1 change

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