A Fock-space inequality and the Laughlin spectral gap

We prove the spherical fermionic Laughlin spectral-gap conjecture for the full V${}_1$ interaction. With coefficient one for each pair projector, the gap above the Laughlin state at filling 1/3 on the round sphere is at least 1/25 for all sufficiently large systems. More generally, $H_Q^2\ge \gamma H_Q$ for some fixed $\gamma >1/25$ on the entire lowest-Landau-level Fock space for all sufficiently large flux Q, independently of particle number.