Volume 42 - KS XOR Frontier Family with Localized Top-Pair Fine-Order Bifurcation under Sigma_41

Volume 42 certifies a frontier-family object for the KS XOR instrument lineage Sigma_41 in a deterministic two-dimensional KS-family substrate. The study object is the seed-indexed frontier anchor depth observable n*(alpha | seed) over the declared seed set {0,1,2,3,4} and the resolved alpha band emitted by the production notebook. Within the certified band, the KS XOR frontier forms a monotonically widening seed-fiber family: the width atlas increases as alpha decreases. The certified order atlas shows stable coarse partition0 < 3 < 4 < {1,2}throughout the resolved band. The only observed fine-order change is localized to the upper pair {1,2}. To isolate that event, this volume definesg21(alpha) = n*(alpha | 1) - n*(alpha | 2)and certifies a sign-change bracket together with a local secant-based root estimate. The production value isalpha_* ≈ 9.6475818,reported as the localized root of the top-pair inversion condition g21(alpha)=0. The volume also certifies an affine qualification bound: a four-slice affine family remains admissible for coarse centering but is not order-safe in the localized bifurcation zone. Evidence for the certified claim set consists of a locked production notebook, explicit verification and synthesis gates, machine-readable export records, manuscript-ready results emission, supporting tables, and SHA-256 manifesting of the final bundle. This volume does not certify a closed-form bifurcation law, a global affine continuation rule, cross-resolution invariance, cross-family transport, or behavior outside the resolved alpha band and declared sign-change bracket. It certifies only the bounded frontier-family object stated above under the inherited Sigma_41 instrument lineage.