AI 中文总结
该研究针对字及其卷曲数变换的重复避免问题，通过Thue-Morse形态构造与穷举搜索得出：4是允许同时实现无限无重叠性的最小字母表大小，相关结果经Walnut验证。
AI 中文摘要
我们研究字${\f w}$及其卷曲数变换$C({\f w})$中的重复避免问题。针对大小为2、3和4的字母表，我们使用基于Thue-Morse的形态构造方法与穷举有限搜索。存在长度不超过84的三元字，使得${\f w}$和$C({\f w})$均为无重叠字；而在四元字母表上存在无限例子。因此，4是允许同时实现无限无重叠性的最小字母表大小。无限构造在Walnut中验证，有限最大值通过穷举广度优先搜索获得并经独立检查。
英文摘要
We study repetition avoidance in a word ${\bf w}$ and its curling-number transform $C({\bf w})$. For alphabets of sizes $2$, $3$, and $4$, we use Thue-Morse-based morphic constructions and exhaustive finite searches. A ternary word for which both ${\bf w}$ and $C({\bf w})$ are overlap-free has length at most $84$, whereas over four letters an infinite example exists. Hence $4$ is the smallest alphabet size admitting simultaneous infinite overlap-freeness. The infinite constructions are verified in Walnut; the finite maxima are obtained by exhaustive breadth-first search and checked independently.
