Hamilton decompositions of all directed tori at odd modulus

Let $D_d(m)=\mathrm{C}\mathrm{a}\mathrm{y}((\mathbb{Z}\mathrm{/}m\mathbb{Z}{)}^d,e_0,\dots ,e_{d-1})$ denote the directed Cayley graph on the positive coordinate basis, equivalently the Cartesian product of $d$ directed cycles of length $m$. The equal side directed Hamilton decomposition problem asks when the arc set of $D_d(m)$ partitions into $d$ directed Hamilton cycles. We prove that such a decomposition exists for every $d\ge 2$ and every odd $m\ge 3$, settling the equal side directed Hamilton decomposition problem at all odd moduli. The proof combines root flat certificate theorem, a prefix count primitivity criterion, and a modular trade lifting theorem with two closure principles: the Cartesian product and the successor step $b\mapsto 2b+1$. Together these propagate the small base dimensions $d\in 2,3,5,7$ to all $d\ge 2$. The boundary cases $D_7(3)$ and $D_7(5)$, where the prefix-count family saturates its zero symbol budget, are handled by explicit non prefix zero set root flat certificates whose zero set compiler. An accompanying Lean 4 formalization verifies the main theorem and the finite certificate predicates.