Improving the lower bound on the Laughlin gap: An SDP search

A recent preprint by OpenAI resolved the spherical fermionic Laughlin spectral-gap conjecture for the full $V_1$ interaction by a Fock-space inequality: $H_Q^2\ge \gamma H_Q$ on the lowest-Landau-level Fock space for all large flux $Q$ and every $\gamma$ below $0.0461673\dots$, so that the gap above the Laughlin state at filling $1\mathrm{/}3$ is at least $1\mathrm{/}25$. That proof, with its original constant, has also been formalized in Lean. We show that the same argument supports a much larger constant. Two sharper estimates, a product-twirl bound for the three-body tail and a staircase price for the four-body allowance, raise the constant above $0.0601$ without changing the original certificate. Treating the auxiliary operators as the variables of a semidefinite program (SDP) and enlarging the finite set of orbital modes on which they act then gives $H_Q^2\ge \gamma H_Q$ for every $\gamma <{\gamma }_{\ast }=0.2190037845063196\dots$, certified by exact rational arithmetic. Consequently the gap above the Laughlin state is at least $1\mathrm{/}5$ for all sufficiently large spheres, and the untruncated planar $V_1$ Hamiltonian has positive spectrum bounded below by ${\gamma }_{\ast }$, uniformly in particle number and angular momentum. A heuristic extrapolation of the SDP optima suggests that auxiliary operators of the present form can give at most about $0.222$, while exact diagonalization of finite systems places the true gap near $0.4$. A larger family of auxiliary operators could push the bound further towards the true gap, and the SDP formulation makes such families easy to test. The proof of the improvement and the SDP search were completed by the AI model Claude Opus 5.5 (Anthropic) under the author’s direction.