Ainta's weighted n-point simple-zero refinement at n = 8 (seven gaps) on the AM-form window v(s) = cos(√2 s) + Σ_{j=1}^{12} c_j cos(2πjs) of AMTOPA's 13-term family (zeta-exact-pressure), with the twelve corrections re-optimised for the eight-point bound and the 28 pair weights and 7 pressures re-solved by LP (B = 404350/10⁸). Theorem D's window-dependent parts are re-proved in-file (admissibility, zero and prime sides, the a/b/J limits, the exact constant H ≥ 0.67216841). The block bound uses the band profile min((q+1)/q, E_band) ≤ defect (AMTOPA, MIT; as presented in Gebendorfer, "Zeta Geometry and Class Cap", Lemma 3.3), proved in-file for any Hermitian G and injective rank labelling, so at q = 7 the condition c(m−7) ≤ 1 relaxes to c(m−7) ≤ 8/7 and m = 148. The finite certificate PC8CL proves c = 805003/10⁸ ≤ F_w on g₀ ≤ g₆ (the rest by reflection) with 5,651 leaves over 32 root walks. Its cells are pair-distance polytopes: each split adds a difference constraint on the prefix sums
κ = (6734812 : ℝ) / 10000000