Spectral gap of the unperturbed spherical Laughlin Hamiltonian

For fermions at filling 1/3 on the round sphere with flux 3(N−1), does the unperturbed V₁ Laughlin parent Hamiltonian have a positive spectral gap bounded away from zero as N grows?

Why it matters and what counts as progress

Why it matters: A uniform gap would control low-energy excitations in the growing-particle spherical model.

What counts as progress: A positive N-independent gap for every sufficiently large N in this exact unperturbed model, or a sequence showing gap closure; disorder stability is a separate question.

How the problem developed

1983 · Model [1]
F. Duncan M. Haldane Formulated the Laughlin hierarchy in spherical geometry, the setting used by later studies of the spherical pseudopotential Hamiltonian.
24 Sept 2026 · Proof claim [2]
OpenAI A manuscript claims a uniform spectral gap for the unperturbed fermionic V₁ Hamiltonian at filling 1/3 on the sphere for all sufficiently large particle numbers.

AI contribution

A Fock-space inequality and the Laughlin spectral gap

OpenAI

For the fermionic Laughlin state at filling 1/3 on the round sphere and flux 3(N−1), the unperturbed V1 Hamiltonian has a uniform gap for all sufficiently large particle numbers N. The released Lean scope gives a 1/25 Fock-space distance estimate and a 1/100 finite spherical gap.

AI involvementAI-led

ImportanceMajor

Technical verificationFormal (Lean 4 kernel and Comparator)

Human understandingFew