Laughlin gap lower bound raised about fivefold: the certificate is now searched, not hand-picked

The certified lower bound on the gap above the Laughlin state is raised to about five times the previous constant, without changing the proof architecture. The earlier proof produced a smaller constant, and the auxiliary operators in its certificate were hand-picked rather than optimized. The new constant is certified in exact rational arithmetic, the certificate passes at least two independently written exact checkers, and those checkers reproduce the old constant exactly from the original seven rows.

The gain comes from treating the auxiliary operators as the variables of a semidefinite program and enlarging the finite set of orbital modes on which they act. The original proof fixed a small set of auxiliary operators and only improved the individual constants entering the final bound. Along the diagonal the SDP optimum rises from the original size to larger sizes with steadily shrinking increments, and at fixed size relaxing the bound on the created mode is the most effective single change.

The work gives a lower bound on the gap above the Laughlin state on the sphere and a positive lower bound on the spectrum of the untruncated planar Hamiltonian, both uniform in particle number and angular momentum. It is aimed at researchers concerned with quantitative lower bounds on the fractional quantum Hall gap, and the certificates and verification scripts are distributed so the constants can be rechecked in exact rational arithmetic. The author notes that a larger family of auxiliary operators could push the bound further toward the true gap, and that the SDP formulation makes such families easy to test.

The author calls the geometric extrapolation of the diagonal SDP increments a heuristic extrapolation rather than a bound, and estimates a ceiling of roughly a certain range for auxiliary operators of the present form. The exact diagonalization that places the true gap is not part of the proof. The author also states that the improved bound and its Lean formalization are still in progress and will appear in a later version.