OpenAI's proof of the Haldane gap in the S = 1 antiferromagnetic Heisenberg chain

On October 6, 2026, OpenAI released a collection of several hundred mathematical manuscripts reporting results obtained by an internal model. Among them are two papers on the Haldane gap, a problem on which I have also worked:

1. The periodic spin-one Haldane gap.
2. A boundary-field gap for the spin-one Heisenberg chain.

I carefully studied the analytic argument in the first paper, which proves the Haldane conjecture for the S = 1 antiferromagnetic Heisenberg chain in the standard periodic-chain formulation. I found the argument remarkably simple and elegant, and wanted to share it with others interested in the problem.

I have prepared two versions of expository notes on the analytic part of the proof. The complete proof also uses computer-assisted estimates, which I have not independently checked. I hope you will enjoy reading one of the two versions.

Hal Tasaki

• Continuous-time version (requires a little functional analysis; follows OpenAI's original transfer-operator construction)
  Download PDF (34 pages, 3 figures)
  Published: October 18, 2026; last revised: October 19, 2026.
• Discrete-time version (the analytic criterion requires just freshman calculus and linear algebra—no functional analysis!)
  Download PDF (34 pages, 3 figures)
  Published: October 18, 2026; last revised: October 19, 2026.

You may also be interested in this catalogue of OpenAI papers related to mathematical physics.