theorem MathlibNt.SieveTheory.LiLiuOnePlusOneNine.GoldbachBig.goldbachWeightG11PaidBase_eq_actual (N : ℕ) (ε : ℝ) : goldbachWeightG11PaidBase N ε = goldbachWeightRemainingFour (goldbachDifferenceCarrier N ε) N (↑N ^ (4 / 53)) (↑N ^ (4 / 33)) (↑N ^ (3 / 11)) + goldbachWeightG11 (goldbachDifferenceCarrier N ε) N (↑N ^ (4 / 53)) (↑N ^ (4 / 33))

Exact cancellation of the intermediate rough and switched bookkeeping counts.

theorem MathlibNt.SieveTheory.LiLiuOnePlusOneNine.GoldbachBig.goldbachWeight_uniformRational_consumed_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 → ∀ (Z : ℝ), 1 ≤ Z → Z ≤ √↑N → ↑(goldbachWeightG11PaidBase N ε) - ↑(goldbachB9LowPositivePrefixSiftedCount N ε Z) + (62033529 / 100000000 - 392796161 / 100000000 - 10385101 / 100000000 - δ) * (SingularSeries.liuSingularSeries N * ↑N / Real.log ↑N ^ 2) ≤ 4 * ↑(D19 N)

All coefficients evaluated here come from accepted actual producers. The signed paid base and the strict-low positive-prefix count remain unestimated.