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Uniform Stability of the Spherical Laughlin Gap
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Abstract
We prove that the fermionic Laughlin V1 Hamiltonian at filling 1/3 on expanding round spheres retains a unique ground state and a uniform spectral gap under sufficiently weak bounded real scalar one-body potentials projected to the lowest Landau level. With coefficient one for each pair projector and Laughlin flux $q=3(N-1)$, the gap and perturbation threshold are uniform for all sufficiently large particle numbers and all potential profiles of supremum norm at most one. The result uses the uniform unperturbed Fock-space gap and gives existential constants.
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work in progress
Collected automatically by AI tools (Claude Opus 5.5 and GPT-6 Astra, 2026-10-21) 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). Reports an SDP-based improvement of the family's unperturbed spherical gap bound from $1/25$ to $1/5$ for sufficiently large systems, strengthening the type of unperturbed input used in the stability paper. [AI-generated per source] [family-level]
Formalization
- OpenAI, Uniform Laughlin gap and stability under bounded scalar disorder: Lean scope notes (OpenAI Math Release, lean/docs/269.md, 2026). Describes formalizations of the unperturbed spherical gap used as input here, the particle-number-uniform Fock-space inequality and the planar endpoint inequality, explicitly excluding stability under projected one-body potentials and uniqueness of the perturbed ground state. [family-level]
Related prior literature mentioned by others
- Nicolas Rougerie, On the Laughlin function and its perturbations (arXiv:1906.11656, 2019-06-27). Gao's follow-up cites Appendix A as a formulation of the spectral-gap conjecture addressed by this family's unperturbed result; the survey also discusses the response of Laughlin correlations to perturbations. [family-level]
Possibly relevant literature (found by the agent, not checked)
- N. Rougerie, On the stability of Laughlin's fractional quantum Hall phase (arXiv:2208.02619, 2022). Review of the stability, rigidity and incompressibility of the Laughlin phase from a mathematical-physics viewpoint; relevant to the stability question in the introduction, and distinct from the 2019 Rougerie survey that is cited.
- M. B. Hastings and S. Michalakis, Quantization of Hall Conductance For Interacting Electrons on a Torus (arXiv:1306.1258, Commun. Math. Phys. 334 (2015)). Uses quasi-adiabatic evolution for gapped interacting charged systems in a Hall setting; related to the paper's use of quasiadiabatic continuation to transport the ground line.
- S. Bachmann, S. Schraven, J. Shapiro and S. Warzel, Fractional Hall Conductance of Laughlin States from a Topological Index (arXiv:2608.27767, 27 August 2026). Argues that Laughlin states form topologically stable fractional quantum Hall phases, with rigorous results on thin cylinders; it bears on the stability of the Laughlin phase treated here.
News and blogs
- Carlos Toledo, OpenAI's math release: what it claims beyond what it checks (self-published audit, 2026-10-10). Entries F-541 and F-542 report exact-arithmetic checks of both manuscripts' seven-row certificates, including 13 three-body margins and 23 four-body certificates, with certification limited to these finite components. [family-level]
Elsewhere
- Emergent Mind: family 269
Independent sites; their content is not reviewed here.
Cite
@misc{OAI:Uniform-Stability-of-the-Spherical-Laughlin-Gap-October-5-2026,
author = {{OpenAI}},
title = {{Uniform Stability of the Spherical Laughlin Gap}},
howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/Uniform-Stability-of-the-Spherical-Laughlin-Gap-October-5-2026/uniform-stability-spherical-laughlin-gap.pdf}{OAI:Uniform-Stability-of-the-Spherical-Laughlin-Gap-October-5-2026}},
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.
Other papers in this family:
- A Fock-space inequality and the Laughlin spectral gap · September 24, 2026