theorem MathlibNt.AnalyticNumberTheory.LargeSieve.LiLiuPrereqFouvry.betaClean_wMaskedTruncated_largeGCD_dyadic_kscale (i k j : ℕ) (A : ℝ) {η Cscale : ℝ} (hη : 0 < η) (hCscale : 1 ≤ Cscale) : ∀ᶠ (x : ℝ) in Filter.atTop, ∀ (M T L : ℝ), 1 ≤ M → 1 ≤ T → 1 ≤ L → 4 * M * T = x → 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| ≤ Cscale * x → ∀ (P : WOriginalTuple → Prop), (∀ t ∈ wOriginalTuples N Q a, P t → x ^ η < ↑(t.2.1.gcd t.2.2)) → |wMaskedTruncated M (wUniformCutoff M (x ^ η)) N Q (betaClean β a) c a P| ≤ 12 * M * T ^ 2 * x ^ (-η / 8) ∧ (∑ m ∈ S, α m ^ 2) * |wMaskedTruncated M (wUniformCutoff M (x ^ η)) N Q (betaClean β a) c a P| ≤ x ^ 2 / Real.log x ^ A

Physical application of the accepted first-gcd estimate to betaClean. The factor 12 is 3 * (2*T)^2 / T^2; the beta endpoint is not confused with its lower dyadic scale. No beta nondivisibility assumption remains.