OpenAI · 268-01 · 2026-09-24 · Mathematical physics
The periodic spin-one Haldane gap
preprints/The-periodic-spin-one-Haldane-gap-September-24-2026
Family 268. The spin-one Haldane gap.
The authors are OpenAI. This companion does not verify the proof and does not adopt it as a theorem of Ngo Cheung’s series. It is not a premise. O.1 stays NOT_CLOSED. Anti-de Sitter results are not a de Sitter state, not Em02-E’s relational lapse, not XVI-P’s QdS scheme, and not XVI-Q’s sigma-image record. No quantum coefficient is admitted. No compilation is asserted.
Read against the standing problems
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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 and gives boundary-selected infinite-volume states with topological index .
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.
Preprint source and Lean scope
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No Lean scope note is in the repository for this family. The preprint source below is still the full manuscript, not only the abstract. No compilation is asserted.