theorem MathlibNt.SieveTheory.LiLiuOnePlusOneNine.GoldbachBig.goldbachS1_levelSix_lowerDensitySix (ρ : ℝ) (hρ : 0 < ρ) : ∃ (N₀ : ℕ), 2 ≤ N₀ ∧ ∀ (N : ℕ), N₀ ≤ N → ∀ (hEven : Even N) (ε : ℝ), 0 < ε → ε < 1 → (SwitchingPrinciple.dimensionOneLowerLinearSieveFactor 6 - ρ) * AnalyticNumberTheory.Sieve.sieveProductPrimeFactors (goldbachS1BoundingSieve N hEven ε (↑N ^ (4 / 53))) ≤ BoundingSieve.mainSum (LinearSieve.lowerRosserWeight (goldbachS1ProdPrimes N (↑N ^ (4 / 53))) (S1LevelSixD N))

Fixed s = 6 lower-density producer for the actual S1 sieve at the original α = 4 / 53 cutoff. The source-relative Suzuki theorem supplies the main term, the exact bridge identifies it with the genuine lower Rosser main sum, and the adaptive-depth error is absorbed uniformly in the finite carrier.