Hamilton decompositions of the directed 5-torus for odd modulus

Abstract

We prove that the directed five-dimensional torus $D_5(m)=\mathrm{Cay}(({\mathbb{Z}}_m{)}^5,\{e_0,e_1,e_2,e_3,e_4\})$ has a Hamilton decomposition for every odd integer $m\ge 3$. 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 $m\ge 5$, 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.