theorem MathlibNt.AnalyticNumberTheory.LargeSieve.LiLiuPrereqFouvry.highOmega_original_signedError_log_payment_kscale (i k j 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 → 4 * M * T = x → 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 c : ℕ → ℝ), (∀ m ∈ S, |α m| ≤ ↑((fouvryTau i) m)) → (∀ n ∈ N, |β n| ≤ ↑((fouvryTau k) n)) → (∀ q ∈ Q, |c q| ≤ ↑((fouvryTau j) q)) → ∀ (a : ℤ), a ≠ 0 → |↑a| ≤ Cscale * x → |signedError S N Q α (betaHighOmega β (highOmegaCutoff x)) c a| ≤ x / Real.log x ^ A ∧ |signedError S N Q α β c a - signedError S N Q α (betaLowOmega β (highOmegaCutoff x)) c a| ≤ x / Real.log x ^ A

theorem MathlibNt.AnalyticNumberTheory.LargeSieve.LiLiuPrereqFouvry.signedError_sq_le_lowOmega_firstGCD_truncated_kscale {ι : Type u_1} {κ k i j : ℕ} (A : ℕ) {T : ι → ℝ} {N : ι → Finset ℕ} {β : ι → ℕ → ℝ} (hSW : BetaCoprimeSWFamily κ T N β) (hT : ∀ (z : ι), 1 ≤ T z) (hN : ∀ (z : ι), ∀ n ∈ N z, T z ≤ ↑n ∧ ↑n ≤ 2 * T z) (hβ : ∀ (z : ι), ∀ n ∈ N z, |β z n| ≤ ↑((fouvryTau k) n)) {Cscale ε η : ℝ} (hCscale : 1 ≤ Cscale) (hε : 0 < ε) (hη : 0 < η) : ∀ᶠ (x : ℝ) in Filter.atTop, ∀ (z : ι) (M L : ℝ), 1 ≤ M → 1 ≤ L → 4 * M * T z = x → x ^ ε ≤ T z → T z ≤ x ^ (1 / 9) → L ≤ x ^ (5 / 9) → ∀ (S Q : Finset ℕ), (∀ m ∈ S, M ≤ ↑m ∧ ↑m ≤ 2 * M) → Q ⊆ Finset.Ioc 0 ⌊L⌋₊ → ∀ (α c : ℕ → ℝ), (∀ m ∈ S, |α m| ≤ ↑((fouvryTau i) m)) → (∀ q ∈ Q, |c q| ≤ ↑((fouvryTau j) q)) → ∀ (a : ℤ), a ≠ 0 → |↑a| ≤ Cscale * x → signedError S (N z) Q α (β z) c a ^ 2 ≤ ((4 * ∑ m ∈ S, α m ^ 2) * wMaskedTruncated M (wUniformCutoff M (x ^ η)) (N z) Q (betaClean (β z) a) c a fun (t : WOriginalTuple) => ↑(t.2.1.gcd t.2.2) ≤ x ^ η ∧ ↑t.2.1.primeFactors.card ≤ highOmegaCutoff x ∧ ↑t.2.2.primeFactors.card ≤ highOmegaCutoff x) + x ^ 2 / Real.log x ^ A