OpenAI · 263-01 · 2026-09-24 · Mathematical physics
Uniform excess charge for Coulomb molecules and the outer radius of neutral atoms
preprints/Uniform-excess-charge-for-Coulomb-molecules-and-the-outer-radius-of-neutral-atoms-September-24-2026
Family 263. The ionization and generalized ionization conjectures.
The authors are OpenAI. This companion does not verify the proof and does not adopt it as a theorem of Ngo Cheung’s series. It is not a premise. O.1 stays NOT_CLOSED. Anti-de Sitter results are not a de Sitter state, not Em02-E’s relational lapse, not XVI-P’s QdS scheme, and not XVI-Q’s sigma-image record. No quantum coefficient is admitted. No compilation is asserted.
Read against the standing problems
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
For the full nonrelativistic Coulomb model with two electron spin states, proves that a molecule with fixed nuclei of charges at least one and total charge strictly binds at most 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 electrons has Thomas–Fermi asymptotics as and ; neutral-atom outer radii have the corresponding iterated-limit asymptotics, taking first.
We prove that a molecule with M fixed nuclei of real charges at least one and total nuclear charge Z strictly binds at most 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.
Preprint source and Lean scope
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A Lean scope note is stored for family 263, but it does not name this preprint. It names 263-02, 263-03. This page still has the full preprint source. No compilation is asserted. O.1 stays NOT_CLOSED.