Finite Resolution and Pair-Admissibility Reduction of Rough Composites on the 6k ±1 Candidate Axis

A reproduction-sealed finite experiment studies whether the remaining second-order structure contains an additional stable mode after successive reductions. The tested data show systematic endpoint reduction of fixed-lag coefficients while reciprocal-density-scale windows remain non-flat. Prime-by-prime localization places the principal finite transition inside the exact pure-semiprime regime 2 < u < 3, excluding finite k = 3 availabilityas the direct threshold explanation. At fixed representative window scales the residual localizes further to LPF-layer interaction and structured nonzero candidate-index lags. The measured off-lag LPF half-kernel is strongly aligned with a finite small-prime pair-admissibility regressor; after projection, the residual shape fraction and Fejér-weighted load are small over the tested pure-semiprime grid. Restoring k= 3 produces a nonzero cross-order coupling, but the full 27-cell control does not reveal a stable signed additional cross-order load residual. The internal k = 3 LPF shape remains the principal finite unresolved sector.