Lemma 20 ([High-level integral]). In stage one, choose a smooth nondecreasing k_0 which is zero on (-∞,Z_0] and one on [Z_0+1,∞) . For all sufficiently large fixed N , ∫_ Ω_R u k_0(Z) (+ρ+η_Naσ)\, V_g≤C(N).

The test k_0(Z) vanishes on S_R . Integration by parts gives ∫u k_0Ξ+3∫u k_0'| Z|_ ^2 =-∫_B k_0(V_λ)_ν -λ∫u k_0' ⟨K(w,·), Z⟩_ . The last integral is absorbed into part of the gradient term plus C(N) . Indeed it is supported in a fixed compact set and in the strip Z_0≤Z≤Z_0+1 , where u=l e^ 3Z-t ≤e^ 1+3(Z_0+1)+ . Thus the residual strip integral has bounded background volume and a bounded weight, without any preliminary gradient estimate. By Lemma 19 ([Weighted trace inequalities]), the absolute boundary contribution is at most Π_N(η_N)/(ℓ) ∫_ C u ((1+√)k_0+k_0'| Z|_ ). The fixed collars have _ C ρ>0 . Consequently 1+√≤C(ρ+δ_0) there. Choose N large using to absorb the nondifferentiated term into the positive source in . Young's inequality absorbs the differentiated term into another part of ∫u k_0'| Z|_ ^2 , with a bounded remainder by. This proves .