theorem MathlibNt.SieveTheory.LiLiuOnePlusOneNine.GoldbachBig.goldbachS1_alphaFourFiftyThree_normalized_lower (δ ε : ℝ) (hδ : 0 < δ) (hε : 0 < ε) (hε1 : ε < 1) : ∃ (N₀ : ℕ), 4 ≤ N₀ ∧ ∀ (N : ℕ), N₀ ≤ N → Even N → (53 / 2 * Real.exp (-Real.eulerMascheroniConstant) * (1 - ε) * SwitchingPrinciple.dimensionOneLowerLinearSieveFactor 6 - δ) * (SingularSeries.liuSingularSeries N * ↑N / Real.log ↑N ^ 2) ≤ ↑(goldbachS1 (goldbachDifferenceCarrier N ε) N (↑N ^ (4 / 53)))