The Laughlin spectral gap conjecture for the V1 pseudopotential on the sphere

Theorem 1.1: for every $0<\gamma<\gamma_*=4616733319001/10^{14}>1/25$ there is $Q_\gamma$ with $H_Q^2\ge\gamma H_Q$ on the full lowest-Landau-level fermionic Fock space for $Q\ge Q_\gamma$, independently of particle number. Corollary 1.2: $H_{N,3(N-1)}\ge\frac1{25}(1-P_{\mathrm L,N})$ for $N\ge N_0$. Section 7 proves the planar analogue at $\gamma_*$, the fermionic cubic case of Rougerie's Conjecture A.1. The companion adds stability under bounded projected scalar potentials. Thresholds are existential; bosonic and other fillings are not treated.

Theorem 1.1: for every $0<\gamma<\gamma_*=4616733319001/10^{14}>1/25$ there is $Q_\gamma$ with $H_Q^2\ge\gamma H_Q$ on the full lowest-Landau-level fermionic Fock space for $Q\ge Q_\gamma$, independently of particle number. Corollary 1.2: $H_{N,3(N-1)}\ge\frac1{25}(1-P_{\mathrm L,N})$ for $N\ge N_0$. Section 7 proves the planar analogue at $\gamma_*$, the fermionic cubic case of Rougerie's Conjecture A.1. The companion adds stability under bounded projected scalar potentials. Thresholds are existential; bosonic and other fillings are not treated.