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

We prove a periodic long-time Boltzmann--Grad limit for hard spheres in every fixed spatial dimension $d\ge 4$. The componentwise long-bond estimate used in dimensions two and three does not yield the required arbitrary-dimensional power. Our replacement keeps two connected time sublayers as a single positive-kernel packet and selects the first two lower collision atoms as landing roots. For disjoint landing edges, the two incidence frames combine directly. For overlapping edges, the particle line first appearing at the second landing supplies a collision-free chord; eliminating this chord produces a nonnegative Jacobi index form, while the remaining one-speed frame cannot focus under the rooted upper collision word. The resulting cellwise estimate is valid for an arbitrary nonnegative joint test function and gives the relative factor $\varepsilon^{2(d-1)}\varepsilon_*^{-2d}$, with the unique full-component $\varepsilon^{-(d-1)}$ normalization counted once. A bounded-degree finite-fibre coarea argument controls all global recovery branches. We also replace the Euclidean no-double-overlap inference by a periodic first-failure construction with permanent incidence labels. Inserting these two geometric inputs into the long-time cumulant expansion yields, uniformly for $0\le t\le t_{\rm fin}$ and $1\le s\le |\log\varepsilon|$, the rate $\varepsilon^{1/(400d)}$. The proof also isolates a fixed-word multi-landing principle in arbitrary dimension; an optional mixed velocity--time exterior condition gives a sharper local loss but is not used in the main theorem.