theorem MathlibNt.SieveTheory.LiLiuOnePlusOneNine.GoldbachBig.goldbachWeight_finiteMainLedger : ∃ (B : ℝ), 0 ≤ B ∧ ∀ (δ : ℝ), 0 < δ → ∃ (ε₀ : ℝ), 0 < ε₀ ∧ ε₀ ≤ 2 / 15 ∧ ∀ (ε : ℝ), 0 < ε → ε < ε₀ → ∃ (N₀ : ℕ), 4 ≤ N₀ ∧ ∀ (N : ℕ), N₀ ≤ N → Even N → goldbachPairLowerMain N ε B (goldbachG6Pairs N (↑N ^ (4 / 53)) (↑N ^ (4 / 33))) + goldbachPairLowerMain N ε B (goldbachG7Pairs N (↑N ^ (4 / 53)) (↑N ^ (4 / 33)) (↑N ^ (3 / 11))) - ↑(∑ v ∈ goldbachG12Labels N (↑N ^ (4 / 53)) (↑N ^ (4 / 33)) (↑N ^ (3 / 11)), goldbachG11RoughCount N ε v) + (124341093 / 200000000 - goldbachB9PaperSplitIntegral - 10385101 / 100000000 - δ) * (SingularSeries.liuSingularSeries N * ↑N / Real.log ↑N ^ 2) ≤ 4 * ↑(D19 N)

The actual weight ledger after G6/G7 BV payment and G12 exception payment. Its two positive terms are the original finite signed Rosser main sums; its negative term is the original cross rough count. No integral estimate is assumed.