Hamilton decompositions of all directed tori at odd modulus

Abstract: Let $D_d(m) = \operatorname{Cay}((\mathbb{Z}/m\mathbb{Z})^d, {e_0, \ldots, e_{d-1}})$ be the directed Cartesian product of $d$ directed $m$-cycles. We prove that $D_d(m)$ admits a directed Hamilton decomposition for every dimension $d \geq 2$ and every odd modulus $m \geq 3$. The proof combines two new closure mechanisms with a small set of base dimensions. The high-modulus count branch handles every odd $d \geq 5$ and every odd $m \geq d$ via triangular prefix coordinates and a primitivity criterion controlled by gcd conditions on symbol counts. The base-tail modular-trade branch handles the complementary range $m < d$ by decomposing a base multigraph into cylinders and scheduling active tail residues by local symbol trades; it yields the successor closure $b \mapsto 2b+1$ for $b \geq 5$. Together with multiplicative product closure, these reduce the all-dimensions theorem to the four base dimensions $d \in {2, 3, 5, 7}$. Dimensions $2$ and $3$ are proved here; dimensions $5$ and $7$ are imported from companion arXiv preprints. A Lean 4 formalization records the same all-dimensions endpoint. As an independent consequence, the dimensions $2$ and $3$ alone solve every odd $d \geq 29$, by a dyadic-triadic interval-hitting argument.