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Sharp Ionization Bounds for Helium and Lithium

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

Contributed by Alec Shelley ↗, Charles Yang ↗, Daniel Tartakovsky ↗

Version 1 / Nov 12, 2026 / CC BY 4.0

Abstract

How many electrons can a nucleus bind? In the non-relativistic Coulomb model, Lieb proved in 1984 that a nucleus of charge ZZ binds fewer than 2Z+12Z+1 electrons. While this bound has been improved, the sharp upper bound has remained unknown for any nucleus other than hydrogen. We solve this problem for Z=2Z=2 and Z=3Z=3 by proving that helium binds two or fewer electrons and lithium binds four or fewer. Both bounds are sharp. Our main tool is an adaptation of Lieb's argument with a quadratic weight, which reduces the nonbinding of doubly negative ions to finitely many inequalities that we verify in a computer-assisted proof.

Provenance statement

This work builds off of a previous result by the authors on a computer-assisted proof that dianionic helium does not exist (as well as the work of many others). The ideas were primarily human generated, while the proofs in this version were primarily AI generated. Fable, Astra, and the Pharos harness were all used.

Tools used

OpenAI
ChatGPT
Frenzymath
Pharos
Anthropic
Claude Fable
Anthropic
Claude Code

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Version history

  1. v1 · 88 pagesSubmitted by Alec ShelleyInitial depositCurrentNov 11, 2026