The ionization and generalized ionization conjectures

What OpenAI claims

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 → ∞ and Z/m → ∞; neutral-atom outer radii have the corresponding iterated-limit asymptotics, taking Z → ∞ first. (Lean)

Papers (3)

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

   24 September 2026 · Preprint, not peer reviewed · PDF

   OpenAI's 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.

2. Generalized ionization energies for full Coulomb atoms

   24 September 2026 · Preprint, not peer reviewed · PDF

   OpenAI's 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_TFm^7/3 whenever m → ∞ and Z/m → ∞, with the Thomas–Fermi constant in atomic units. We also obtain both conjectured iterated limits.

3. Generalized outer-electron radii of neutral Coulomb atoms

   24 September 2026 · Preprint, not peer reviewed · PDF

   OpenAI's abstract 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 as m tends to infinity.

Status

Preprint. The papers have not been independently reviewed. Treat the result as OpenAI's claim until other mathematicians have checked it. If a video exists, it illustrates the statement; it is not the proof.