HEXAGONbeta

A Bound Helium Dianion Does Not Exist

A mix of human-written and AI-generated textHuman understanding: some partsmath-ph — Mathematical Physicsmath.SP — Spectral Theory

Contributed by Alec Shelley ↗, Daniel Tartakovsky ↗

Version 1 / Nov 03, 2026 / CC BY 4.0

Abstract

For a non-relativistic atom with ZZ protons and NN bound electrons, Lieb proved in 1984 that N<2Z+1N<2Z+1. Several improved bounds have been developed, but they all leave open the question of whether a bound helium dianion can exist. We prove that the answer is no. The ground-state energies of the four- and three-electron helium Hamiltonians coincide, E4(Z=2)=E3(Z=2)E_4(Z{=}2)=E_3(Z{=}2), so a helium nucleus binds at most three electrons. A weighted form of Lieb's identity reduces the problem to the nonnegativity of an explicit two-electron Schrodinger operator mm, and Barta's inequality further reduces it to the pointwise nonnegativity of an explicit function of three variables. We prove the nonnegativity of this function by a combination of analytic estimates and a computer-assisted certification in interval arithmetic, and we publish the certificate with the paper.

Provenance statement

The mathematical approach was developed by the authors in conversation with Anthropic’s Claude (Claude Fable), building on Lieb’s 1984 identity and the author's own unpublished improvements to it. Claude wrote the code for the computer-assisted certification and drafted the manuscript; a second Claude instance ran the computation on Stanford’s Sherlock cluster (about 560 CPU core-hours). The manuscript was written in parts by a human and in parts by Fable. The certificate and code are archived at Zenodo (10.5281/zenodo.23753889, 10.5281/zenodo.23754394).

Tools used

Anthropic
Claude Fable

References

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Version history

  1. v1Submitted by Alec ShelleyInitial depositCurrentNov 03, 2026