Result 269, Mathematical physics

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.

Lean formalization Proof

The bigger picture

Why it matters

A spectral gap is the energy needed to excite a system out of its ground state. These manuscripts claim that a spherical Laughlin model keeps this protective separation as it grows, even under weak scalar disorder.

What changes?

For fermions at filling one-third on round spheres, the manuscripts report a gap of at least 1/25 for the full V1 pairwise interaction, with coefficient one per pair projector and flux q = 3(N-1) for N particles. They also report that sufficiently weak bounded real scalar one-body potentials, projected to the lowest Landau level, preserve a unique ground state and a gap. The disorder threshold and resulting gap are uniform over sufficiently large N and potential profiles of supremum norm at most one.

What does that help mathematicians do?

The underlying claim extends beyond the Laughlin particle number: throughout the lowest-Landau-level Fock space, which includes all particle-number sectors, the squared energy operator is bounded below by gamma times that operator, with a fixed gamma greater than 1/25 for sufficiently large flux. This excludes arbitrarily small positive energies uniformly across those sectors. The stability manuscript uses this stronger control to rule out gap closure under the stated weak disorder, even as particle number grows.

Are there practical applications?

The immediate value is foundational for the mathematical description of the fractional quantum Hall effect: the claimed stability makes the spherical model robust to bounded scalar imperfections, rather than only an idealized disorder-free system. The stability constants are existential, not explicit tolerances. The result does not by itself establish comparable robustness for other geometries, interactions, or experimental conditions.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

2 manuscripts

Uniform Stability of the Spherical Laughlin Gap

October 5, 2026 69 pages

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)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.

PDF LaTeX source
Cite (BibTeX) 
@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}
}

A Fock-space inequality and the Laughlin spectral gap

September 24, 2026 22 pages Main result formalized in Lean

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, HQ2≥γHQH_Q^2\ge\gamma H_Q for some fixed γ>1/25\gamma\gt 1/25 on the entire lowest-Landau-level Fock space for all sufficiently large flux Q, independently of particle number.

PDF LaTeX source
Cite (BibTeX) 
@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's note on what the formalization covers, from lean/docs/269.md.

Uniform Laughlin gap and stability under bounded scalar disorder

The following describes the scope of the Lean formalization related to the following accompanying paper(s):

Scope

The linked formalization proves the uniform unperturbed gap estimate used in the paper's stability argument for the fermionic Laughlin state at filling 1/31/3 on the sphere. At flux q=3(N−1)q=3(N-1) and all sufficiently large particle numbers NN, every antisymmetric state has V1V_1 energy at least 1/251/25 times its squared distance from the Laughlin ground-state line.

This selected statement is the unperturbed Fock-space inequality. Stability under projected one-body potentials and uniqueness of the perturbed ground state are outside it.

The Laughlin spectral-gap problem asks for a positive gap that remains uniform as the system grows. The formalization includes the finite spherical V1V_1 bound of 1/1001/100 above the Laughlin state at flux 3(N−1)3(N-1) for all sufficiently large particle numbers NN. It also proves the stronger Fock-space inequality HQ2≥γHQH_Q^2\ge\gamma H_Q for every fixed 0<γ<γ∗0<\gamma<\gamma_* and all sufficiently large fluxes QQ, independently of particle number, where γ∗=4616733319001/1014>1/25\gamma_*=4616733319001/10^{14}>1/25.

For the untruncated planar model, the formalization proves the corresponding inequality at the endpoint γ∗\gamma_* on every homogeneous particle sector. These statements use coefficient one for each pair projector.

Comparator links

Result Comparator statement
Uniform unperturbed spherical Laughlin gap inequality LaughlinGap.lean
Uniform Laughlin V1V_1 spectral gap Laughlin.lean
Uniform Fock-space spectral-gap inequality LaughlinFock.lean
Planar spectral-gap inequality LaughlinPlanar.lean

Data from github.com/openai/math at commit adc7f12, committed October 6, 2026 at 21:58 UTC, last checked for changes about 8 hours ago. Titles, subjects, summaries, abstracts and Lean notes are OpenAI's; page counts are read from the PDFs. The map, related results, search, kinds of results and the named-problem index are Emergent Mind's, built with text embeddings and an LLM, and may contain errors.

An Emergent Mind Labs project. Emergent Mind is not affiliated with OpenAI. None of these results has been peer reviewed. Cite the manuscripts themselves, using the BibTeX on each result's page.