The Laughlin spectral gap

The Laughlin spectral gap problem asks whether the parent Hamiltonian of Laughlin's fractional quantum Hall state has a spectral gap above its ground state that stays bounded away from zero as the system grows. Two manuscripts in OpenAI's public mathematics collection (github.com/openai/math), dated 24 September and 5 October 2026, claim a uniform gap for the fermionic state at filling 1/3 on the round sphere with the full V₁ interaction, and its stability under weak one-body potentials. This page summarizes the published claims and lists publicly available follow-up work. Statements are the respective authors' claims; the publication of a paper, code or a formal proof is not presented here as independent verification.

Background

As recounted in the introduction of the 24 September manuscript:
- The model. In Haldane's spherical geometry the pseudopotential Hamiltonian charges every pair of particles energy one in relative angular momentum one and zero otherwise; the Laughlin state at filling 1/3 is its exact zero mode. Haldane and Rezayi's finite-size study of 1985 gave numerical evidence for a gap and distinguished it from a proof. Rougerie's survey records the spectral-gap conjecture, including a planar formulation as its Conjecture A.1.
- Earlier rigorous results. Nachtergaele, Warzel and Young proved uniform gaps for a truncated fermionic pseudopotential, and Warzel and Young obtained thin-torus bounds; Lemm, Nachtergaele, Warzel and Young derived recursions between particle sectors; Lieb, Rougerie and Yngvason proved density rigidity for fully correlated Laughlin states. The manuscript distinguishes its result from these by the full spherical pair interaction, which is not truncated.

The original results

Fock-space inequality (24 September 2026). With coefficient one for each pair projector, Theorem 1.1 of the first manuscript states that for every γ below γ_* = 4616733319001/10¹⁴, about 0.0462, there is a flux threshold Q_γ, independent of the particle number, such that H_Q² ≥ γH_Q on the whole lowest-Landau-level Fock space for every flux Q ≥ Q_γ. Its Corollary 1.2 states that at the Laughlin flux 3(N − 1) the gap above the Laughlin state is at least 1/25 for all sufficiently large N. The thresholds are existential. The manuscript also proves the planar inequality at the endpoint γ_*, uniformly in the particle number and the total degree, and states that this settles the fermionic cubic case of Conjecture A.1 in Rougerie's survey.
- Method, as described: positivity of H² − γH in the manner of Knabe. The normal-ordered expansion of H_Q² is compared with a rotation-averaged sum of squares of seven auxiliary operators acting on finitely many orbitals; the interaction itself is never truncated.
Uniform stability (5 October 2026). The second manuscript states that on expanding round spheres the same Hamiltonian at filling 1/3 keeps a unique ground state and a uniform gap under sufficiently weak bounded real scalar one-body potentials projected to the lowest Landau level. The gap and the perturbation threshold are uniform over all sufficiently large particle numbers and all potential profiles of supremum norm at most one; the constants are existential.
Sources. Commit 1581d4c of the openai/math repository, folders A Fock-space inequality and the Laughlin spectral gap and Uniform Stability of the Spherical Laughlin Gap.
Formalization, as described by the repository. The scope note lean/docs/269.md states that the formalization proves the unperturbed inequality used in the stability argument: at flux 3(N − 1) and all sufficiently large N, every antisymmetric state has V₁ energy at least 1/25 times its squared distance from the Laughlin ground-state line. It also includes a finite spherical bound of 1/100 above the Laughlin state, the Fock-space inequality for every γ below γ_*, and the planar inequality at the endpoint. Stability under one-body potentials and uniqueness of the perturbed ground state are outside it. The Comparator statements are LaughlinGap, Laughlin, LaughlinFock and LaughlinPlanar.

Public follow-up work

Date	Work	Relation	Public claim	Public status (checked 17 October 2026)
2026-10-07 (version 0.1; latest commit `fda5da1`, 8 October 2026)	Xin Shen, bosonic-laughlin-spectral-gap (GitHub)	Extension (candidate proof)	The bosonic analogue on the sphere: for the V₀ Hamiltonian at filling 1/2, H_Q² ≥ H_Q/3 in every particle sector for flux Q ≥ 3088, which would give a gap of at least 1/3 above the Laughlin zero mode for N ≥ 1545. It adapts the Fock-space comparison strategy of the 24 September manuscript; later commits add Lean proofs of parts of the argument.	The README describes the work as an AI-generated candidate proof and states that independent review of the argument and its implementation remains open.
2026-10-08	Qiang Gao, arXiv:2610.12700 (v1), titled Improving the lower bound on the Laughlin gap: An SDP search	Extension	Two sharper estimates raise the constant of the same argument above 0.0601. Treating the auxiliary operators as the variables of a semidefinite program on more orbital modes then gives H_Q² ≥ γH_Q for every γ below 0.2190037845…, certified in exact rational arithmetic, so the gap above the Laughlin state is at least 1/5 on all sufficiently large spheres, and the planar Hamiltonian's positive spectrum is bounded below by the same constant. The abstract estimates that auxiliary operators of this form give at most about 0.222, while exact diagonalization of finite systems places the true gap near 0.4.	arXiv v1. The abstract states that the proof of the improvement and the search were completed by the AI model Claude Opus 5.5 under the author's direction.
2026-10-13	M. G. Scheer, E. Khalaf, arXiv:2610.18854 (v1), titled A hierarchy of exact lower bounds on the Laughlin spectral gap	Parallel work	A bootstrap hierarchy of finite semidefinite programs gives rigorous lower bounds on the gap of the first Haldane pseudopotential on the sphere, for every flux and in the thermodynamic limit. The paper reports a numerical thermodynamic-limit bound of 0.0693 and states that an exact rational certificate establishing a gap of at least 1/100 will be released with it, in the paper's normalization. Its note added dates the numerical bound to June and the certificate to August 2026, before the OpenAI release, and compares the two approaches.	arXiv v1. The acknowledgements state that AI tools assisted with literature search, coding and some calculations and did not contribute in a substantial way to the main ideas or the writing.
2026-10-16 (latest commit `03da87c`)	certainpineapple/Laughlin-torus-gap (GitHub; the README names Xiang-Jian Hou as preparer, with advice from Ying-Hai Wu)	Extension (candidate proof)	Uniform Laughlin gaps on large flat tori, for the bosonic V₀ interaction at filling 1/2 and the fermionic V₁ interaction at filling 1/3, under a hypothesis on the shortest nonzero lattice vector. It cites the two OpenAI manuscripts and the bosonic repository above as method antecedents and states that no torus gap theorem is imported from them.	The README states that AI agents made substantive contributions. It records kernel acceptance of two exported Lean endpoints whose source is not released, so that acceptance cannot be reproduced from the repository.

Notes

The list covers follow-ups found in public sources up to the date above; it is not a priority or completeness claim.