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The Laughlin spectral gap conjecture for the V1 pseudopotential on the sphere

On the sphere with QQ flux quanta, NN fermions in the lowest Landau level interact through Haldane's V1V_1 pseudopotential HN,Q=∑i<jPij(1)H_{N,Q}=\sum_{i<j}P^{(1)}_{ij}, which penalizes each pair in relative angular momentum one. At Q=3(N−1)Q=3(N-1) the Laughlin state ∏i<j(uivj−ujvi)3\prod_{i<j}(u_iv_j-u_jv_i)^3 is its unique zero mode. Numerics since Haldane and Rezayi (1985) indicate a gap above it, and the folklore spectral-gap conjecture is recorded in Rougerie's survey (Conjecture A.1); uniform gaps were proved only for truncated pseudopotentials on thin cylinders and tori (Nachtergaele-Warzel-Young, Warzel-Young). Is there γ>0\gamma>0 with HN,3(N−1)≥γ(1−PL)H_{N,3(N-1)}\ge\gamma(1-P_{\mathrm L}) for all sufficiently large NN?

Result
Proved(see note)
Status
Candidate (review pending)
AI contribution
AI-discovered
Method
Argument
Field
Fractional quantum Hall effect; spectral gaps
Posed by
Folklore since Haldane (1983) and Haldane-Rezayi (1985); recorded as Conjecture A.1 in N. Rougerie's survey, as cited by the manuscript
Year posed
1985
Years open
41y
Solved
2026-09-24
Model
Unreleased internal OpenAI model
Vendor
OpenAI
Collaborators
—
Verification
Lean-checked, statement unaudited
Publication
Announced
Collection
OpenAI math release (October 2026), version adc7f12
Significance
38 / 100
Disclosed cost
—
Wikipedia
No dedicated article

What was actually shown

Theorem 1.1: for every 0<γ<γ∗=4616733319001/1014>1/250<\gamma<\gamma_*=4616733319001/10^{14}>1/25 there is QγQ_\gamma with HQ2≥γHQH_Q^2\ge\gamma H_Q on the full lowest-Landau-level fermionic Fock space for Q≥QγQ\ge Q_\gamma, independently of particle number. Corollary 1.2: HN,3(N−1)≥125(1−PL,N)H_{N,3(N-1)}\ge\frac1{25}(1-P_{\mathrm L,N}) for N≥N0N\ge N_0. Section 7 proves the planar analogue at γ∗\gamma_*, the fermionic cubic case of Rougerie's Conjecture A.1. The companion adds stability under bounded projected scalar potentials. Thresholds are existential; bosonic and other fillings are not treated.

What the AI did

The release README says every result in openai/math was produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result. This result is not one of the README's two exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author. The family's later manuscript, 'Uniform Stability of the Spherical Laughlin Gap' (October 5, 2026), extends the gap to weak bounded scalar disorder and is a link here. Both manuscripts ship verification directories.

Verification

No independent mathematician has checked this yet. Checked here: Theorem 1.1 and Corollary 1.2 of the Fock-space manuscript were read against the conjecture: HQ2≥γHQH_Q^2\ge\gamma H_Q on the whole Fock space for every γ<γ∗≈0.0462\gamma<\gamma_*\approx0.0462 and large QQ, hence a gap of 1/251/25 above the Laughlin state at Q=3(N−1)Q=3(N-1) for large NN. lean/formalization.yaml lists Laughlin.json (declaration OAI.Laughlin.mainTarget_proved, file OAI/Analysis/Laughlin/Main.lean). The statement Laughlin.lean was read here; not rebuilt here. It encodes antisymmetric states in a number basis, the V1V_1 energy through explicit pair coefficients and the Laughlin vector through its polynomial, and states energy at least 1/1001/100 times squared distance to the Laughlin line for all N≥N0N\ge N_0 at flux 3(N−1)3(N-1): a uniform positive gap, the headline (with a weaker constant than the paper's 1/25, which LaughlinGap.lean states). Fidelity of the pair-coefficient encoding to P(1)P^{(1)} was not audited. Permitted axioms: propext, Quot.sound, Classical.choice.

Sources

Changelog1 change

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