THE QUANTUM INTELLIGENCER

An improved lower bound has been certified for the spectral gap above the Laughlin state at filling one-third. The new preprint raises the constant in the relevant Fock-space inequality from 0.0461673, the value established in an earlier OpenAI proof and subsequently formalized in Lean, to 0.2190037845063196, certified by exact rational arithmetic. The improvement is obtained by treating the auxiliary operators as the variables of a semidefinite program and enlarging the finite set of orbital modes on which they act. It follows that the gap above the Laughlin state is at least one-fifth for all sufficiently large spheres, and that the untruncated planar V_1 Hamiltonian has positive spectrum bounded below by the new constant, uniformly in particle number and angular momentum. The proof of the improvement and the SDP search were completed by the AI model Claude Opus 5.5 of Anthropic, under the author's direction. A heuristic extrapolation of the semidefinite program's 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. The reader who keeps a ledger may add three entries. The original certificate, at its first constant, carried a spectral gap of at least 1/25. The improvement is made by two sharper estimates, a product-twirl bound for the three-body tail and a staircase price for the four-body allowance. The inequality now certified is H_Q^2 ≥ γ H_Q on the lowest-Landau-level Fock space, for all large flux Q. Two further details belong in the reader's ledger. The conjecture at issue is the spherical fermionic Laughlin spectral-gap conjecture for the full V1 interaction; that was the scope of the original Fock-space inequality, and the new certificate operates within the same setting. The numerical ceiling reported for the present family of auxiliary operators is not offered as a theorem. A larger family could push the bound further toward the true gap, which exact diagonalization of finite systems places near 0.4, and the semidefinite formulation exists precisely to make such families easy to test. What has been certified, then, is a lower bound with room above it, not an exact value of the gap itself. —Elias Hartwell Dispatch from The Prepared E0