We prove that the fermionic Laughlin V₁ Hamiltonian at filling 1/3 on expanding round spheres retains a unique ground state and a uniform spectral gap under sufficiently weak bounded real scalar one-body potentials projected to the lowest Landau level. With coefficient one for each pair projector and Laughlin flux $q=3(N-1)$, the gap and perturbation threshold are uniform for all sufficiently large particle numbers and all potential profiles of supremum norm at most one. The result uses the uniform unperturbed Fock-space gap and gives existential constants.

We prove the spherical fermionic Laughlin spectral-gap conjecture for the full V₁ 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 $\gamma\gt 1/25$ on the entire lowest-Landau-level Fock space for all sufficiently large flux Q, independently of particle number.