theorem MathlibNt.SieveTheory.LiLiuOnePlusOneNine.GoldbachBig.goldbachS3_oneThird_normalized_upper (δ ε : ℝ) (hδ : 0 < δ) (hε : 0 < ε) (hεu : ε < 2 / 15) : ∃ (N₀ : ℕ), 4 ≤ N₀ ∧ ∀ (N : ℕ), N₀ ≤ N → Even N → ↑(goldbachS3Closed (goldbachDifferenceCarrier N ε) N (↑N ^ (4 / 53)) (↑N ^ (1 / 3))) ≤ ((1 - ε) * (53 / 2) * Real.exp (-Real.eulerMascheroniConstant) * goldbachS3_primeKernelIntegral (1 / 3) + δ) * (SingularSeries.liuSingularSeries N * ↑N / Real.log ↑N ^ 2)

theorem MathlibNt.SieveTheory.LiLiuOnePlusOneNine.GoldbachBig.goldbachS3_threeElevenths_normalized_upper (δ ε : ℝ) (hδ : 0 < δ) (hε : 0 < ε) (hεu : ε < 2 / 15) : ∃ (N₀ : ℕ), 4 ≤ N₀ ∧ ∀ (N : ℕ), N₀ ≤ N → Even N → ↑(goldbachS3Closed (goldbachDifferenceCarrier N ε) N (↑N ^ (4 / 53)) (↑N ^ (3 / 11))) ≤ ((1 - ε) * (53 / 2) * Real.exp (-Real.eulerMascheroniConstant) * goldbachS3_primeKernelIntegral (3 / 11) + δ) * (SingularSeries.liuSingularSeries N * ↑N / Real.log ↑N ^ 2)