theorem MathlibNt.SieveTheory.LiLiuOnePlusOneNine.GoldbachBig.goldbachComposite_moving_count_lower (B ρ : ℝ) (hB : 0 ≤ B) (hρ : 0 < ρ) : ∃ (N₀ : ℕ), 4 ≤ N₀ ∧ ∀ (N : ℕ), N₀ ≤ N → ∀ (hEven : Even N), ∀ ε < 2 / 15, ∀ (m : ℕ), 0 < m → ↑N ^ (8 / 53) ≤ ↑m → ↑m ≤ ↑N ^ (13 / 33) → have S := goldbachS3BoundingSieve N hEven ε (↑N ^ (4 / 53)) m; have D := LiuWeight.panModulusCutoff N B / m + 1; have t := Real.log ↑D / Real.log (↑N ^ (4 / 53)); S.totalMass * AnalyticNumberTheory.Sieve.sieveProductPrimeFactors S * max 0 (JurkatRichert1965ChenGammaOneQOne.jr1965f t - ρ) - LinearSieve.lowerErrSum S D (LinearSieve.lowerRosserWeight S.prodPrimes D) ≤ ↑(literalH (goldbachDifferenceCarrier N ε) N m (↑N ^ (4 / 53)))

The actual count on every moving composite layer, including t < 2. Clipping is justified by count nonnegativity, not signed-density nonnegativity. The literal Rosser remainder remains available for the proved full BV sum.