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The periodic spin-one Haldane gap

OpenAI · September 24, 2026

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Abstract

We prove a uniform positive spectral gap for the pure antiferromagnetic spin-one Heisenberg chain on even periodic rings, establishing the positive even-periodic formulation of the spin-one Haldane conjecture.

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Comments and discussion

  • Hal Tasaki, OpenAI's proof of the Haldane gap in the S=1 antiferromagnetic Heisenberg chain (author's website, October 2026). Tasaki reports carefully studying the periodic paper's analytic argument and describes it favorably, explicitly stating that he has not independently checked the computer-assisted estimates.
  • Masaki Oshikawa (@MasakiOshikawa), post on X (6 Oct 2026). In Japanese, Oshikawa asks Hal Tasaki whether the results should be understood as proving the existence of the Haldane gap and that the S=1 Haldane phase is SPT. [family-level]
  • Hal Tasaki (@Hal_Tasaki), post on X (11 Oct 2026). In Japanese, Tasaki says he is carefully working through OpenAI's Haldane-gap proof and rewriting it in his own words, calling the paper's writing very poor. [family-level]
  • MC (@dualm), post on X (10 Oct 2026). Reports reading the proof and finding "an interesting new idea (or 'trick') that I have not seen before, followed by a huge, almost brute-force calculation." [family-level]
  • Rudy Pei (@PeaBrane10), post on X (8 Oct 2026). Notes that the uniform gap for even rings with L ≥ 60, combined with Hastings–Koma / Nachtergaele–Sims, gives exponential decay of ground-state correlations uniformly in L, "the other half of Haldane's prediction".
  • Ryui Kaneko (@ryuikaneko), post on X (7 Oct 2026). In Japanese, Kaneko remarks that the proof seems to yield upper as well as lower bounds and wonders whether similar estimates could bound the gap of the S=1/2 kagome-lattice Heisenberg antiferromagnet.
  • Marcus Bintz (@spurious_bows), post on X (12 Oct 2026; also on Bluesky). Points out that Lemma 4.1 and Proposition 4.3 of this paper coincide with Lemma 2.1 and Proposition 2.2 of the release's Sharp mass bounds for the two-dimensional O(4) model.

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Cite

@misc{OAI:The-periodic-spin-one-Haldane-gap-September-24-2026,
  author = {{OpenAI}},
  title = {{The periodic spin-one Haldane gap}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/The-periodic-spin-one-Haldane-gap-September-24-2026/paper.pdf}{OAI:The-periodic-spin-one-Haldane-gap-September-24-2026}},
  year = {2026}
}

From OpenAI's README for this paper:

Verification

The finite verifier requires Python 3, NumPy, and SciPy with assertions enabled. For a complete fresh recomputation, run from this paper directory with a new explicit output directory:

env -u PYTHONOPTIMIZE python3 -B verification/verify.py \
  --output "$PWD/.verification-output/full-001"

Choose a new output directory for each run. A successful full run writes verification.json with success: true and full_recomputation: true. The existing --only selections and --workers option are unchanged; see verification/computations/README.md for tested versions, commands, runtimes, and exact scope. Partial selections explicitly label remaining certificate records as frozen evidence. These finite checks do not replace the analytic uniform-gap argument.

Family 268: The spin-one Haldane gap

Proves the spin-one Haldane gap conjecture for the pure antiferromagnetic Heisenberg chain on even periodic rings: the spectral gap stays uniformly positive as the chain grows. A companion establishes a gap for odd open chains with endpoint field $h=3/5$ and gives boundary-selected infinite-volume states with topological index −1.

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