Theorem 1.1. For every potential v as above there are constants ρ_*(v)>0 and c_*(v)>0 with the following property. For every 0<ρ<ρ_*(v) there is L_0(ρ,v)<∞ such that, for every L≥L_0(ρ,v) , every integer N≥1 satisfying ρ/2≤N/L^3≤2ρ , and …

Theorem 1.1. For every potential v as above there are constants ρ_*(v)>0 and c_*(v)>0 with the following property. For every 0<ρ<ρ_*(v) there is L_0(ρ,v)<∞ such that, for every L≥L_0(ρ,v) , every integer N≥1 satisfying ρ/2≤N/L^3≤2ρ , and every 0≤T≤ρ^2 , ⟨u_0,γ^ (1) _ N,L,T u_0⟩ N ≥c_*(v). Consequently, for each such fixed density ρ and fixed 0≤T≤ρ^2 , every sequence with L→∞ and N/L^3→ρ has lower limiting condensate fraction at least c_*(v) .