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A Fock-space inequality and the Laughlin spectral gap

OpenAI · September 24, 2026

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Abstract

We prove the spherical fermionic Laughlin spectral-gap conjecture for the full V1 interaction. With coefficient one for each pair projector, the gap above the Laughlin state at filling 1/3 on the round sphere is at least 1/25 for all sufficiently large systems. More generally, $H_Q^2\ge\gamma H_Q$ for some fixed $\gamma\gt 1/25$ on the entire lowest-Landau-level Fock space for all sufficiently large flux Q, independently of particle number.

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work in progress

Collected automatically by AI tools (Claude Opus 5.5 and GPT-6 Astra, 2026-10-23) and not checked by a person. May be wrong or incomplete. Report an error.

Follow-up works

  • Qiang Gao, Improving the lower bound on the Laughlin gap: An SDP search (arXiv:2610.12700, 2026-10-08). Keeps this paper's architecture, refines its comparison estimates and optimizes its auxiliary operators by an SDP search certified in exact rational arithmetic, reporting $H_Q^2\ge\gamma H_Q$ for every $\gamma<0.2190037845063196\ldots$ at sufficiently large flux and hence a spherical Laughlin gap of at least $1/5$; the paper says the proof and search were completed by Claude Opus 5.5 under the author's direction. [AI-generated per source]
  • Qiang Gao and Taige Wang, Thermodynamic-limit bootstrap: Solving fractional quantum Hall ground states (arXiv:2610.21317, 2026-10-15). Cites this paper's Fock-space positivity and rotation averaging as antecedents to a finite-orbital-window bootstrap for thermodynamic ground-state energies and correlations.
  • Xin Shen (curator), The Bosonic Laughlin Spectral Gap: A Candidate Proof (GitHub, v0.1, 7 October 2026). Adapts this paper's Fock-space comparison strategy to the bosonic V0 model and claims $H_Q^2\ge H_Q/3$ for $Q\ge3088$; the repository states that independent review remains open. [AI-generated per source]
  • GitHub user Yuchenian, vortexable-laughlin-gap (GitHub repository, 2026). Self-described AI-assisted research draft that transfers a reference Laughlin spectral inequality to changed vortexable one-particle subspaces, taking Gao's refinement of this paper (arXiv:2610.12700) as its input. [AI-generated per source]

Comments and discussion

  • Harry Stoltz (@HarryStoltz1), post on X (2026-10-07). In a two-post thread, describes the result as a stopping rule for algorithms preparing a Laughlin state, which "can stop when the energy gets below a certain cutoff".

Formalization

  • OpenAI, Uniform Laughlin gap and stability under bounded scalar disorder: Lean scope notes (OpenAI Math Release, lean/docs/269.md, 2026). Describes Lean statements for the $1/100$ finite spherical bound, the Fock-space inequality $H_Q^2\ge\gamma H_Q$ for $\gamma<\gamma_*$, the $1/25$ gap inequality and the planar endpoint inequality, with links to the comparator files; stability under projected one-body potentials is explicitly excluded. [family-level]
  • OpenAI, Lean development OAI/Analysis/LaughlinFock (openai/math repository). The release's Lean sources for the Fock-space inequality, including the certificate-row files; Gao's arXiv:2610.12700 cites this directory as the formalization of the original constant.

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Cite

@misc{OAI:A-Fock-space-inequality-and-the-Laughlin-spectral-gap-September-24-2026,
  author = {{OpenAI}},
  title = {{A Fock-space inequality and the Laughlin spectral gap}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/A-Fock-space-inequality-and-the-Laughlin-spectral-gap-September-24-2026/A-Fock-space-inequality-and-the-Laughlin-spectral-gap-September-24-2026.pdf}{OAI:A-Fock-space-inequality-and-the-Laughlin-spectral-gap-September-24-2026}},
  year = {2026}
}

Lean formalization

OpenAI provides Lean files for this family: lean/docs/269.md. The Lean statements have not been checked by people against the paper's claims.

Family 269: Uniform Laughlin gap and stability under bounded scalar disorder

Proves the fermionic Laughlin spectral-gap conjecture for the full V1 interaction at filling 1/3 on the round sphere. The unique ground state remains uniformly gapped under sufficiently weak bounded real scalar one-body potentials projected to the lowest Landau level. Both the gap and disorder threshold are uniform over all sufficiently large particle numbers and all normalized potential profiles.

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