定向五维 torus 的 Hamilton 分解
Hamilton decompositions of the directed 5-torus for odd modulus
AI总结：
本文证明了当 m 为奇数且 m≥3 时，定向五维 torus D_5(m) 存在 Hamilton 分解，通过构造方法和零集拉丁表证明了其分解性。
摘要：
我们证明了定向五维 torus D_5(m) = Cay((Z_m)^5, {e_0, e_1, e_2, e_3, e_4}) 对于每个奇数 m ≥ 3 都存在 Hamilton 分解。这是首次在更高维情况下，返回映射方法需要真正的零集选择器而非 odometer 类修正。构造方法通过一个循环层计划分配五个外出生成器，其中一层由零集拉丁表确定；显式有限精确覆盖证书证明该层是匹配。通过循环对称性，所有颜色类的 Hamiltonicity 减少到单个归一化返回映射。对于 m ≥ 5，显式首次返回计算在 p = 2 截面给出一个诱导循环，其行程长度总和为 m^4。剩余的 m = 3 情况通过打印的有限循环证书解决。一个配套的 Lean 4 格式化提供对 Cayley 声明和有限证书的独立机器验证；源代码、审计脚本和辅助搜索代码可在 https://github.com/aria1th/Torus-Hamilton-Decomposition-Program 获取。
英文摘要：
We prove that the directed five-dimensional torus $D_5(m) = \operatorname{Cay}((\mathbb{Z}_m)^5, \{e_0, e_1, e_2, e_3, e_4\})$ has a Hamilton decomposition for every odd integer $m \geq 3$. This is the first higher-dimensional case in which the return-map method requires a genuine zero-set selector rather than an odometer-type correction. The construction assigns the five outgoing generators by a cyclic layer schedule with one non-constant layer determined by a zero-set Latin table; an explicit finite exact-cover certificate proves that this layer is a matching. By cyclic symmetry, Hamiltonicity of all color classes reduces to a single normalized return map. For $m \geq 5$, an explicit first-return calculation on the section $p = 2$ gives one induced cycle whose excursion lengths sum to $m^4$. The remaining modulus $m = 3$ is settled by a printed finite cycle certificate. A companion Lean 4 formalization provides an independent machine verification of the Cayley statement and the finite certificates; source, audit scripts, and ancillary search code are available at https://github.com/aria1th/Torus-Hamilton-Decomposition-Program.
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