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Generalized ionization energies for full Coulomb atoms

OpenAI · September 24, 2026

Lean

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Abstract

We prove the energy part of the generalized ionization conjecture for full nonrelativistic Coulomb atoms with two electron spin states. The energy needed to remove m electrons is asymptotic to $a_{\mathrm{TF}}m^{7/3}$ whenever $m\to\infty$ and $Z/m\to\infty$, with the Thomas–Fermi constant in atomic units. We also obtain both conjectured iterated limits.

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Notes

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work in progress

Collected automatically by AI tools (Claude Opus 5.5 and GPT-6 Astra, 2026-10-21) and not checked by a person. May be wrong or incomplete. Report an error.

Follow-up works

  • muralisubbarao2 (GitHub), TR-M529-Ram-Coulomb-Ionization-Energies (GitHub repository "Improved solutions to 239 of the 722 math problems posted by OpenAI", undated). A technical report whose readable outline and links point to this preprint and its Lean files; the repository README calls its reports "a first draft prepared with the help of GPT 6 Astra" that has "not been verified by humans yet". [AI-generated per source]

Formalization

  • OpenAI math release, lean/docs/263.md: The ionization and generalized ionization conjectures (Lean scope notes, 6 October 2026). States that the formalization covers $I_m(Z)/m^{7/3}\to a_{\mathrm{TF}}$ as $m\to\infty$, $Z/m\to\infty$, with the iterated limits, excluding fixed-$m$ convergence and ground-state attainment; the comparator statement is CoulombIonization.lean. [family-level]
  • Yao Xu (cadamcat), atomic-ionization-lean4 (GitHub, Lean 4/Mathlib). A third-party Lean 4 development whose main statement includes this paper's Theorem 1.1 together with the companion's first-ionization and radius bounds; the README says it was "Developed with AI assistance" and reuses OpenAI's CoulombIonization library unchanged, with some proofs taking different routes. [AI-generated per source] [family-level]

Possibly relevant literature (found by the agent, not checked)

Digests and expositions

Elsewhere

Independent sites; their content is not reviewed here.

Cite

@misc{OAI:Generalized-ionization-energies-for-full-Coulomb-atoms-September-24-2026,
  author = {{OpenAI}},
  title = {{Generalized ionization energies for full Coulomb atoms}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Generalized-ionization-energies-for-full-Coulomb-atoms-September-24-2026/paper.pdf}{OAI:Generalized-ionization-energies-for-full-Coulomb-atoms-September-24-2026}},
  year = {2026}
}

Lean formalization

OpenAI provides Lean files for this family: lean/docs/263.md. The Lean statements have not been checked by people against the paper's claims.

Family 263: The ionization and generalized ionization conjectures

For the full nonrelativistic Coulomb model with two electron spin states, proves that a molecule with M fixed nuclei of charges at least one and total charge Z strictly binds at most $Z+CM$ electrons. Neutral-atom first ionization energies and radii containing all but an expected half-electron have universal positive upper and lower bounds. The energy cost of removing m electrons has Thomas–Fermi asymptotics as $m\to\infty$ and $Z/m\to\infty$; neutral-atom outer radii have the corresponding iterated-limit asymptotics, taking $Z\to\infty$ first.

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