Let $D_d(m) = \mathrm{Cay}((\mathbb{Z}/m\mathbb{Z})^d, {e_0, \ldots, e_{d-1}})$ 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 $d \geq 2$ and every odd $m \geq 3$, 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 $d \geq 2$. 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.

翻译：设 $D_d(m) = \mathrm{Cay}((\mathbb{Z}/m\mathbb{Z})^d, {e_0, \ldots, e_{d-1}})$ 表示正坐标基上的有向Cayley图，等价于 $d$ 个长度为 $m$ 的有向圈的笛卡尔积。等边有向哈密顿分解问题询问何时 $D_d(m)$ 的弧集可划分为 $d$ 个有向哈密顿圈。我们证明，对于每个 $d \geq 2$ 和每个奇数 $m \geq 3$，这样的分解均存在，从而解决了所有奇数模数下的等边有向哈密顿分解问题。该证明结合了根平坦证书定理、前缀计数本原性准则、模数提升定理以及两个闭包原理：笛卡尔积与后继步骤 $b \mapsto 2b+1$。这些方法将小基维数 $d \in {2, 3, 5, 7}$ 传播至所有 $d \geq 2$。对于边界情况 $D_7(3)$ 和 $D_7(5)$（此时前缀计数族耗尽零符号预算），我们通过显式非前缀零集根平坦证书处理，其零集编译器已实现。随附的Lean 4形式化验证了主要定理及有限证书谓词。