The original C.2 error restricted to five small canonical factors

The two remaining supported modulus factors are removed by an exact finite partition and the same-mask analytic payment. The original signed coefficients, low-omega conditions, beta-family SW constants, and frequency cutoff are retained. No estimate of the remaining IV.3 oscillatory sum is asserted.

theorem MathlibNt.AnalyticNumberTheory.LargeSieve.LiLiuPrereqFouvry.wMaskedTruncated_eq_modulus_support_small_add_large (M Y : ℝ) (H : ℕ → ℕ → ℕ) (N Q : Finset ℕ) (β c : ℕ → ℝ) (a : ℤ) (P : WOriginalTuple → Prop) : wMaskedTruncated M H N Q β c a P = (wMaskedTruncated M H N Q β c a fun (t : WOriginalTuple) => P t ∧ ↑(wGCDTuple t).δ₁ ≤ Y ∧ ↑(wGCDTuple t).δ₂ ≤ Y) + wMaskedTruncated M H N Q β c a fun (t : WOriginalTuple) => P t ∧ (Y < ↑(wGCDTuple t).δ₁ ∨ Y < ↑(wGCDTuple t).δ₂)

An exact subdivision inside any previous mask, with overlap of the two large supported-modulus conditions counted only once.

theorem MathlibNt.AnalyticNumberTheory.LargeSieve.LiLiuPrereqFouvry.signedError_sq_le_lowOmega_five_small_truncated_c2 {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)) {ε η : ℝ} (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 / 10) → 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| ≤ 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) ∧ ↑(t.1.1.gcd t.1.2) ≤ x ^ η) ∧ ↑(wGCDTuple t).d₁ ≤ x ^ η) ∧ ↑(wGCDTuple t).δ₁ ≤ x ^ η ∧ ↑(wGCDTuple t).δ₂ ≤ x ^ η) + x ^ 2 / Real.log x ^ A

The original error, now with all five canonical coordinates small. Constants of the original coprime-SW family remain uniform before its changing index, residue, and all scales. All fixed divisor orders may be zero.