Supported-modulus exclusions at the actual C.2 scale

For x = 4MT, T ≤ x^(1/9) and L ≤ x^(5/9) imply L ≤ M eventually. This pays the actual progression endpoints and preserves the same mask in the original sum, exact zero mode, and uniformly truncated tail.

theorem MathlibNt.AnalyticNumberTheory.LargeSieve.LiLiuPrereqFouvry.betaClean_wMaskedTruncated_modulus_support_dyadic_kscale (Cscale : ℝ) (hCscale : 1 ≤ Cscale) (i k j A : ℕ) {η : ℝ} (hη : 0 < η) : ∀ᶠ (x : ℝ) in Filter.atTop, ∀ (M T L : ℝ), 1 ≤ M → 1 ≤ T → 1 ≤ L → 4 * M * T = x → T ≤ x ^ (1 / 9) → 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 < ↑(wGCDTuple t).δ₁ ∨ x ^ η < ↑(wGCDTuple t).δ₂) → (∑ 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

Both supported modulus factors are paid together, even inside an arbitrary further submask. All fixed orders may be zero; no SW or nondivisibility premise is imposed on the original beta coefficients.