theorem G12FineGrid.original_total_grid_partition {ρ : ℝ} (hρ : 1 < ρ) {N : ℕ} (hN : 2 ≤ N) (ε : ℝ) : ↑(MathlibNt.SieveTheory.LiLiuOnePlusOneNine.GoldbachBig.goldbachG12ProductPrimeTotal N ε (↑N ^ (4 / 53)) (↑N ^ (4 / 33)) (↑N ^ (3 / 11))) = (G12LowHighOutput.outputCount N (G12LowHighOutput.high N ε) + 400 * ∑ p ∈ cutoffSlice N ε, MathlibNt.SieveTheory.LiLiuOnePlusOneNine.GoldbachBig.goldbachG12NormalizedCoefficient N p.1 * if Nat.Prime (N - p.2 * p.1) then 1 else 0) + 400 * ∑ k ∈ indices ρ N, ((∑ p ∈ safe ρ N ε k, MathlibNt.SieveTheory.LiLiuOnePlusOneNine.GoldbachBig.goldbachG12NormalizedCoefficient N p.1 * if Nat.Prime (N - p.2 * p.1) then 1 else 0) + ∑ p ∈ boundaryCell ρ N ε k, MathlibNt.SieveTheory.LiLiuOnePlusOneNine.GoldbachBig.goldbachG12NormalizedCoefficient N p.1 * if Nat.Prime (N - p.2 * p.1) then 1 else 0)

The original complete product-prime count, not merely the low mother, retains the high output, the cutoff slice and every unsafe boundary atom.