theorem MathlibNt.SieveTheory.LiLiuOnePlusOneNine.GoldbachBig.goldbachBV_logCube_normalized (C δ : ℝ) (hδ : 0 < δ) : ∀ᶠ (N : ℕ) in Filter.atTop, C * ↑N / Real.log ↑N ^ 3 ≤ δ * (SingularSeries.liuSingularSeries N * ↑N / Real.log ↑N ^ 2)

One extra logarithm absorbs a fixed BV constant into the actual singular-series-normalized error, uniformly in the ambient integer.

theorem MathlibNt.SieveTheory.LiLiuOnePlusOneNine.GoldbachBig.goldbachG67_lowerRosser_normalized : ∃ (B : ℝ), 0 ≤ B ∧ ∀ (δ : ℝ), 0 < δ → ∀ (ε : ℝ), 0 < ε → ε < 2 / 15 → ∃ (N₀ : ℕ), 4 ≤ N₀ ∧ ∀ (N : ℕ), N₀ ≤ N → Even N → goldbachPairLowerMain N ε B (goldbachG6Pairs N (↑N ^ (4 / 53)) (↑N ^ (4 / 33))) - δ * (SingularSeries.liuSingularSeries N * ↑N / Real.log ↑N ^ 2) ≤ ↑(goldbachWeightG6 (goldbachDifferenceCarrier N ε) N (↑N ^ (4 / 53)) (↑N ^ (4 / 33))) ∧ goldbachPairLowerMain N ε B (goldbachG7Pairs N (↑N ^ (4 / 53)) (↑N ^ (4 / 33)) (↑N ^ (3 / 11))) - δ * (SingularSeries.liuSingularSeries N * ↑N / Real.log ↑N ^ 2) ≤ ↑(goldbachWeightG7 (goldbachDifferenceCarrier N ε) N (↑N ^ (4 / 53)) (↑N ^ (4 / 33)) (↑N ^ (3 / 11)))

Both original positive terms, with arbitrary relative BV error. The signed finite main sums are retained; no density or integral estimate is assumed.