OpenAI claim summary for Phys-67
physicsmathematicsTop Open Problemscontributionliterature_reportClaim: literature report
Summary of a claim published in openai/math; not a claim by this platform.
The spherical fermionic Laughlin $V_1$ model is claimed uniformly gapped at filling $1/3$, with a stronger lower bound on the square of the interaction across the entire lowest-level Fock space. The question retains these same parameters and conventions. Within these assumptions, this is a claimed affirmative answer to the stated question.
Limitations: The linked openai/math material has not been examined or verified by the platform.
Manuscript
Contents
A size-independent spectral gap above the spherical Laughlin line
OpenAI's claim
OpenAI presents a proof in the selected manuscript. The spherical fermionic Laughlin $V_1$ model is claimed uniformly gapped at filling $1/3$, with a stronger lower bound on the square of the interaction across the entire lowest-level Fock space.Mathematical statement of the claimed result
Result 1.
For the lowest-Landau-level pseudopotential $H_{N,Q}=\sum_{i<j}P_{ij}^{(1)}$ on $\bigwedge^NU_Q$, with pair total-spin projector $Q-1$ and coefficient one, the source claims an integer $N_0$ such that \[ H_{N,3(N-1)}\ge\frac1{25}(I-P_{\mathrm L,N})\qquad(N\ge N_0). \] Here $P_{\mathrm L,N}$ projects onto the line spanned by $\prod_{i<j}(u_iv_j-u_jv_i)^3$. Thus the spherical Laughlin line is a unique zero-energy ground space with a size-independent positive excitation gap.Outline of the argument
Exterior-algebra identities expand the interaction square into spin coefficients. Uniform asymptotic control of these coefficients bounds cross terms against the original interaction, giving a Fock-space inequality independent of particle number and hence the Laughlin gap.Formal verification
For family 269, the documented entries give unperturbed spherical Laughlin and stronger Fock-space gap inequalities. Stability against scalar one-body disorder and uniqueness of the perturbed ground state are not included. The listed Comparator files areLaughlin.lean, LaughlinFock.lean, LaughlinGap.lean, LaughlinPlanar.lean. Named declarations include OAI.Laughlin.mainTarget_proved. This site has not executed the Lean code. The scope descriptions do not certify the remaining manuscript arguments.Status caveat
OpenAI attributes these manuscripts to work by an internal model. Its release cautions that arguments without formalization can contain mistakes. This catalogue has not independently checked the proofs; the labels report the mathematical claims attributed to OpenAI.Sources
- OpenAI, A Fock-space inequality and the Laughlin spectral gap, 2026-09-24; citation key
OAI:A-Fock-space-inequality-and-the-Laughlin-spectral-gap-September-24-2026. PDF (GitHub); PDF (direct); Manuscript page and citation; Latest version; Statement source: thm:main. - Family 269: Lean scope.
- Comparator Laughlin.json.
- Comparator Laughlin.lean.
- Comparator LaughlinFock.json.
- Comparator LaughlinFock.lean.
- Comparator LaughlinGap.json.
- Comparator LaughlinGap.lean.
- Comparator LaughlinPlanar.json.
- Comparator LaughlinPlanar.lean.
- OpenAI catalogue entry.