theorem MathlibNt.AnalyticNumberTheory.LargeSieve.LiLiuPrereqFouvry.factorLowOmega_signedError_difference_payment_kscale (i k u v A : ℕ) {Cscale ε : ℝ} (hCscale : 1 ≤ Cscale) (hε : 0 < ε) : ∀ᶠ (x : ℝ) in Filter.atTop, ∀ (M T L : ℝ), 1 ≤ M → 1 ≤ T → 1 ≤ L → x = 4 * M * T → x ^ ε ≤ T → L ≤ x ^ (5 / 9) → ∀ (S N Q : Finset ℕ), (∀ m ∈ S, M ≤ ↑m ∧ ↑m ≤ 2 * M) → (∀ n ∈ N, T ≤ ↑n ∧ ↑n ≤ 2 * T) → Q ⊆ Finset.Ioc 0 ⌊L⌋₊ → ∀ (α β γ ζ : ℕ → ℝ), (∀ m ∈ S, |α m| ≤ ↑((fouvryTau i) m)) → (∀ n ∈ N, |β n| ≤ ↑((fouvryTau k) n)) → (∀ (r : ℕ), |γ r| ≤ ↑((fouvryTau u) r)) → (∀ (s : ℕ), |ζ s| ≤ ↑((fouvryTau v) s)) → ∀ (a : ℤ), a ≠ 0 → |↑a| ≤ Cscale * x → |signedError S N Q α β (factorConvolution γ ζ) a - signedError S N Q α β (factorConvolution γ (betaLowOmega ζ (highOmegaCutoff x))) a| ≤ x / Real.log x ^ A