theorem MathlibNt.SieveTheory.LiLiuOnePlusOneNine.GoldbachBig.goldbachWeight_G12Rough_allRetained_small_epsilon (δ : ℝ) (hδ : 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 → ↑(goldbachWeightG6 (goldbachDifferenceCarrier N ε) N (↑N ^ (4 / 53)) (↑N ^ (4 / 33)) + goldbachWeightG7 (goldbachDifferenceCarrier N ε) 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 two G12 exceptional classes are actually absorbed in the strongest weight ledger. The remaining negative sum uses exactly the G12 cross labels.