A Periodic Long-Time Boltzmann–Grad Limit in Every Fixed Dimension d ≥ 4 via Two-Landing-Root Packets

We prove a long-time Boltzmann–Grad limit for hard spheres on the periodic torus in every fixed spatial dimension d ≥ 4. The componentwise long-bond estimate used in dimensions two and three does not provide the power required in higher dimensions. We replace it by a joint two-sublayer construction: two connected time sublayers are treated as a single positive-kernel packet, and the first two lower collision atoms are selected as landing roots. For disjoint landing edges, the corresponding incidence frames combine directly. For overlapping edges, the particle line first appearing at the second landing contains a collision-free chord; retaining its radial relative speed produces an exact rank-(d − 1) positive Schur factor, while the remaining one-time wedge cannot focus under the rooted upper collision word. The resulting estimate is valid for arbitrary nonnegative joint kernels and gives the relative packet factor ε^(2(d−1)) ε_*^(−2(d−1)), with the unique full-component normalization ε^(−(d−1)) counted exactly once. A bounded-degree finite-fibre coarea argument accounts for all global recovery branches, while a periodic first-failure construction with permanent incidence labels replaces the Euclidean no-double-overlap inference. Inserted into the long-time cumulant expansion, these inputs show that the rescaled s-particle correlations converge in L¹ to the corresponding tensor-product Boltzmann profile at rate ε^(1/(400d)), uniformly for 0 ≤ t ≤ t_fin and 1 ≤ s ≤ |log ε|. The earlier full-chord estimate with loss ε_*^(−2d) remains an independent coarse fallback. Beyond the k = 2 mechanism used in the main theorem, we establish, for every fixed k ≥ 1, a birth-flag multi-landing frame theorem for regular connected full two-sublayer packets in arbitrary fixed dimension. Whenever the lower word contains at least 2k − 1 physical lines, its canonical birth flag yields the operator factor ε^((d−1)(k−1)) ε_*^(−k(d−1)); k = 2 is the first gain-producing case.