Result 263, Mathematical physics

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+CMZ+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→∞m\to\infty and Z/m→∞Z/m\to\infty; neutral-atom outer radii have the corresponding iterated-limit asymptotics, taking Z→∞Z\to\infty first.

Lean formalization Proof

The bigger picture

Why it matters

How many extra electrons can a molecule hold, and how tightly does an atom retain its outer electrons? The manuscripts report universal limits and precise scaling laws for these questions in a quantum model.

What changes?

The manuscripts bound strictly bound electron counts by Z + C M in the full nonrelativistic Coulomb model with two spin states. The M fixed nuclei have arbitrary distinct positions and real charges at least one, totaling Z; C is universal. For neutral atoms with integer Z at least one, the first ionization energy and, for every ground state, the radius enclosing all but an expected half-electron have universal positive upper and lower bounds.

What does that help mathematicians do?

Removing m electrons from a neutral atom costs asymptotically the Thomas-Fermi constant times m to the seven-thirds power, in atomic units, as m and Z/m both diverge. For neutral-atom radii leaving expected count m outside, every ground-state choice has upper and lower large-Z limits asymptotic to (81 pi squared / 2) to the one-third power times m to the minus-one-third power as m grows. Taking Z first matters: this does not establish unrestricted simultaneous-limit behavior.

Are there practical applications?

The immediate value is foundational for mathematical atomic physics. The molecular bound limits excess electrons by the number of nuclei, not their total charge. The atomic bounds prevent first-removal energies and outer size from vanishing or diverging with Z. The asymptotics connect the full quantum model to Thomas-Fermi predictions, rather than supplying a practical computation method.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

3 manuscripts

Uniform excess charge for Coulomb molecules and the outer radius of neutral atoms

September 24, 2026 84 pages

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+CMZ+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.

PDF LaTeX source
Cite (BibTeX) 
@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}
}

Generalized ionization energies for full Coulomb atoms

September 24, 2026 33 pages Main result formalized in Lean

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 aTFm7/3a_{\mathrm{TF}}m^{7/3} whenever m→∞m\to\infty and Z/m→∞Z/m\to\infty, with the Thomas–Fermi constant in atomic units. We also obtain both conjectured iterated limits.

PDF LaTeX source
Cite (BibTeX) 
@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}
}

Generalized outer-electron radii of neutral Coulomb atoms

September 24, 2026 17 pages

We prove the radius part of the generalized ionization conjecture for neutral nonrelativistic Coulomb atoms with two spin states. For every choice of ground states, the upper and lower large-nuclear-charge limits of the radius defined by an expected exterior electron mass m are both asymptotic to (81π2/2)1/3m−1/3(81\pi^2/2)^{1/3}m^{-1/3} as m tends to infinity.

PDF LaTeX source
Cite (BibTeX) 
@misc{OAI:Generalized-outer-electron-radii-of-neutral-Coulomb-atoms-September-24-2026,
  author = {{OpenAI}},
  title = {{Generalized outer-electron radii of neutral Coulomb atoms}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Generalized-outer-electron-radii-of-neutral-Coulomb-atoms-September-24-2026/paper.pdf}{OAI:Generalized-outer-electron-radii-of-neutral-Coulomb-atoms-September-24-2026}},
  year = {2026}
}

Lean formalization

OpenAI's note on what the formalization covers, from lean/docs/263.md.

The ionization and generalized ionization conjectures

The following describes the scope of the Lean formalization related to the following accompanying paper(s):

Scope

The formalization gives the large-ionization asymptotics for the full nonrelativistic two-spin Coulomb atom. Let Im(Z)I_m(Z) be the energy needed to remove mm electrons from a neutral atom of nuclear charge ZZ. There is a positive coefficient aTFa_{\mathrm{TF}} with the Thomas–Fermi variational characterization such that Im(Z)/m7/3→aTFI_m(Z)/m^{7/3}\to a_{\mathrm{TF}} whenever m→∞m\to\infty and Z/m→∞Z/m\to\infty, with integers Z>m≥1Z>m\ge1. The corresponding iterated limsup and liminf limits hold with Z→∞Z\to\infty first. The energy uses the full antisymmetric Sobolev form domain; fixed-mm convergence and ground-state attainment are outside this statement.

For neutral Coulomb atoms with two electron spin states, the formalization proves the asymptotic law for generalized outer-electron radii defined by an expected exterior electron mass mm. For every choice of normalized ground states, both the upper and lower large-nuclear-charge radius limits satisfy m1/3Rm⟶(81π2/2)1/3m^{1/3}R_m\longrightarrow (81\pi^2/2)^{1/3} as m→∞m\to\infty. The large-nuclear-charge limit is taken first. Ground states are inputs to the statement; their existence is not asserted separately.

Comparator links

Result Comparator statement
Generalized Coulomb ionization-energy asymptotics CoulombIonization.lean
Asymptotic outer-electron radii of neutral atoms CoulombRadii.lean

Data from github.com/openai/math at commit adc7f12, committed October 6, 2026 at 21:58 UTC, last checked for changes about 8 hours ago. Titles, subjects, summaries, abstracts and Lean notes are OpenAI's; page counts are read from the PDFs. The map, related results, search, kinds of results and the named-problem index are Emergent Mind's, built with text embeddings and an LLM, and may contain errors.

An Emergent Mind Labs project. Emergent Mind is not affiliated with OpenAI. None of these results has been peer reviewed. Cite the manuscripts themselves, using the BibTeX on each result's page.