theorem G12FlexibleWF.exists_rectangle_paid_C2_sieve : ∃ (K : ℝ) (C : ℝ), 1 < K ∧ 0 < C ∧ ∀ (η : ℝ), 0 < η → η < 1 / 8 → ∀ (Cscale ζ : ℝ), 1 ≤ Cscale → 0 < ζ → ζ ≤ 1 / 10 → ∀ (A : ℕ), ∀ᶠ (x : ℝ) in Filter.atTop, ∀ (M U T V : ℕ), 1 ≤ M → M ≤ U → U ≤ 2 * M → 1 ≤ T → T ≤ V → V ≤ 2 * T → ∀ (ν : ℝ), 4 * ↑M * ↑T = x → ζ ≤ ν → ν ≤ 1 / 10 + ζ / 10 → ↑T = x ^ ν → ∀ (N : ℕ), 0 < N → ↑N ≤ Cscale * x → Even N → ↑N ^ (4 / 53) ≤ ↑T → ↑V < ↑N ^ (1 / 10) → ∀ (ε Z : ℝ), 2 ≤ Z → Z ≤ √(x ^ ((5 - 5 * ν) / 9 - ζ)) → x ^ ((5 - 5 * ν) / 9 - ζ) ≤ ↑N → 2 ≤ MathlibNt.SieveTheory.LiLiuPrereqWF.externalInternalLevel (x ^ ((5 - 5 * ν) / 9 - ζ)) η → have Q := x ^ ((5 - 5 * ν) / 9 - ζ); have P := MathlibNt.SieveTheory.LiLiuOnePlusOneNine.GoldbachBig.goldbachB10SiftingPrimes N Z; have D := MathlibNt.SieveTheory.LiLiuPrereqWF.externalInternalLevel Q η; have S := MathlibNt.SieveTheory.LiLiuPrereqWF.externalTags true P D η Z; have B := G12FlexibleRectangle.rectangle N ε M U T V; have Euler := ∏ p ∈ P, (1 - AnalyticNumberTheory.Sieve.goldbachNu p); have E := C * (η + (η ^ 8)⁻¹ * Real.exp (6 * K + 2) * Real.log Q ^ (-(1 / 3))); (400 * ∑ p ∈ B, MathlibNt.SieveTheory.LiLiuOnePlusOneNine.GoldbachBig.goldbachG12NormalizedCoefficient N p.1 * if Nat.Prime (N - p.2 * p.1) then 1 else 0) ≤ 400 * mass N B * Euler * (MathlibNt.SieveTheory.JurkatRichert1965ChenGammaOneQOne.jr1965F (Real.log Q / Real.log Z) + E) + 400 * ↑S.card * (x / Real.log x ^ A) + 400 * ↑S.card * correctionBudget N Q η + 8000 * ↑⌈Z⌉₊

An actual flexible-cell C2 sieve: no discrepancy, gate, or outside residual is carried as an unproved premise. Only numerical level/geometry admissibility remains.