AI 中文总结
该研究证明单个连通多立方体平移平铺三维欧氏空间的问题不可判定，通过归约Wang平铺问题完成证明，且三维是该类问题可判定性的最优维度。
AI 中文摘要
我们证明以下问题是co-RE完全的，因此是不可判定的：给定单个（连通的）多立方体，判断它是否能通过平移平铺三维欧氏空间。我们利用Greenfeld和Tao的带装饰的双素数数独构造，以及适配自OpenAI的三维非周期瓦片的循环编码，从Wang平铺问题进行归约，并应用Kim的归约使原型瓦片通过面连通。三维是最优的：已知在ℤ²中，以及在ℝ²中对于单个（可能不连通的）多格，平移单平铺是可判定的。
英文摘要
We prove co-RE-completeness, and thus undecidability, of the following problem: given a single (connected) polycube, decide whether it tiles 3D Euclidean space by translations. We reduce from Wang tiling using the decorated two-prime Sudoku construction of Greenfeld and Tao and a cyclic encoding adapted from OpenAI's 3D aperiodic tile, and apply a reduction of Kim to make the prototile connected (via faces). Dimension three is optimal: translational monotiling is known to be decidable in $\mathbb{Z}^2$ and for a single (possibly disconnected) polyomino in $\mathbb{R}^2$.
Comments7 pages, 1 figure, 1 table
