Browse › Mathematical physics › 263 The ionization and generalized ionization conjectures
Uniform excess charge for Coulomb molecules and the outer radius of neutral atoms
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Abstract
We prove that a molecule with M fixed nuclei of real charges at least one and total nuclear charge Z strictly binds at most $Z+CM$ electrons, where C is universal. The result holds for the full nonrelativistic Coulomb Hamiltonian with two spin states and arbitrary distinct nuclear positions. For neutral atoms with integer nuclear charge Z ≥ 1, the first ionization energy and, for every ground state, the radius outside which one half of an electron remains are each bounded above and below by positive universal constants.
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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-M528-Ram-Coulomb-Excess-Charge (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; 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
- Yao Xu (cadamcat), atomic-ionization-lean4 (GitHub, Lean 4/Mathlib). A third-party Lean 4 development whose main statement includes the uniform two-sided first-ionization bound and half-electron radius bound (Theorem 1.2 and Corollary 1.3 here), together with the energy asymptotic of the companion paper; the README says it has no
sorryor added axioms, was "Developed with AI assistance", and reuses OpenAI'sCoulombRadii/CoulombIonizationlibraries unchanged. [AI-generated per source] [family-level] - OpenAI math release, lean/docs/263.md: The ionization and generalized ionization conjectures (Lean scope notes, 6 October 2026). The family's Lean scope notes cover only the two companion papers (energy joint limit and radius iterated limit); this paper's Theorems 1.1–1.2 and Corollary 1.3 are not stated there as main Lean results. [family-level]
Possibly relevant literature (found by the agent, not checked)
- P. T. Nam, The ionization problem in quantum mechanics (arXiv:2206.15393, 2022; in the EMS volume for E. H. Lieb's 90th birthday). A survey of the ionization conjecture and its history, which provides background for the excess-charge and ionization-energy claims (Theorem 1.1, Corollary 1.3).
- R. L. Frank, P. T. Nam, H. Van Den Bosch, The ionization conjecture in Thomas-Fermi-Dirac-von Weizsäcker theory (arXiv:1606.07355; Comm. Pure Appl. Math.). Proves an atomic excess-charge bound $N\le Z+C$ and a radius bound in TFDW theory through comparison with Thomas–Fermi and exterior-region control; this is related to the screening strategy behind Theorems 1.1–1.2.
- Y. Goto, The maximal excess charge in reduced Hartree-Fock molecule (arXiv:2002.11864; Rev. Math. Phys. 33 (2021) 2150008). Proves a $Z+C$ excess-charge bound for molecules in reduced Hartree–Fock theory, a variational-model antecedent of the molecular bound $n\le Z+CM$ in Theorem 1.1.
- L. Chen, Y. Wu, Asymptotic Neutrality for the Ionization Conjecture in the Two-Center Thomas-Fermi-Dirac-von Weizsäcker Theory (arXiv:2610.09997, 7 October 2026; revised 9 October 2026). Proves a position-uniform bound $N\le Z+CZ^{153/154}(1+\log Z)^{3/8}$ for two-center TFDW molecules, the molecular setting of Theorem 1.1 in an approximate model; it does not mention the OpenAI paper.
Digests and expositions
- Ying Xie, What the Machine Invented: Mining New Mathematical Instruments from OpenAI's October 2026 Release of AI-Generated Mathematics (arXiv:2610.17090, October 2026). Section 9.2.2 summarizes this paper's claims and identifies noisy-observation conditioning of electron positions as its main new instrument; it notes that the full quantitative bookkeeping was not checked and that the headline strict-binding result is not formalized. [family-level]
- Latent Seminar, Coulomb ionization: excess charge, energy costs, and outer radii (7 October 2026). Explains all three manuscripts through conditional screening, priced-sector energy comparisons and expected exterior mass, explicitly limiting its review to statements and selected proof mechanisms. [family-level]
Elsewhere
- VibeMathed: Problem record
- Emergent Mind: family 263
Independent sites; their content is not reviewed here.
Cite
@misc{OAI:Uniform-excess-charge-for-Coulomb-molecules-and-the-outer-radius-of-neutral-atoms-September-24-2026,
author = {{OpenAI}},
title = {{Uniform excess charge for Coulomb molecules and the outer radius of neutral atoms}},
howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/Uniform-excess-charge-for-Coulomb-molecules-and-the-outer-radius-of-neutral-atoms-September-24-2026/paper.pdf}{OAI:Uniform-excess-charge-for-Coulomb-molecules-and-the-outer-radius-of-neutral-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.
Other papers in this family:
- Generalized ionization energies for full Coulomb atoms · September 24, 2026
- Generalized outer-electron radii of neutral Coulomb atoms · September 24, 2026