theorem MathlibNt.SieveTheory.LiLiuOnePlusOneNine.GoldbachBig.goldbachS1_beta_normalized_lower (δ : ℝ) (hδ : 0 < δ) (ε : ℝ) : 0 < ε → ε < 1 → ∃ (N₀ : ℕ), 4 ≤ N₀ ∧ ∀ (N : ℕ), N₀ ≤ N → Even N → (33 / 2 * Real.exp (-Real.eulerMascheroniConstant) * (1 - ε) * SwitchingPrinciple.dimensionOneLowerLinearSieveFactor (33 / 8) - δ) * SingularSeries.liuSingularSeries N * ↑N / Real.log ↑N ^ 2 ≤ ↑(goldbachS1 (goldbachDifferenceCarrier N ε) N (↑N ^ (4 / 33)))

The beta lower bound at the true endpoint 33 / 8 is obtained by first choosing a fixed admissible s < 33 / 8 from continuity of the genuine lower sieve factor and then invoking the fixed-s normalized lower bound.