发表机构
Warsaw University of Technology（华沙理工大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本研究通过计算机辅助实验，在不同可微性类别下确定了洗牌方阵的尖锐判据和阈值，并刻画了态射类及删除距离等性质。
AI 中文摘要
对二元游程可微性的实验得出了关于洗牌方阵的尖锐计算机辅助判据。每个长度大于16的$C^3$-词，当且仅当两个字母的重数均为偶数时，才是洗牌方阵。所有34个非空偶Parikh例外都是平滑的，并且在每个更高的可微性类别中持续存在。对于$C^2$，尖锐阈值为48，有212个非空例外。在$C^1$级别，不存在全局奇偶阈值，但每个长度至少为36的偶Parikh非洗牌方阵都有适当的非空洗牌方阵前缀和后缀。恰好有230个非空偶Parikh $C^1$-词没有非空洗牌方阵前缀。避免此类前缀的完整树最终由422条周期射线组成。因此，非空Kolakoski前缀是洗牌方阵，当且仅当两个重数均为偶数，除了长度为4和8的情况。我们还刻画了反映洗牌方阵的态射类。对于双二元词，我们确定了精确的删除距离和最大孪生词，并证明了单次局部修复的尖锐界限。精确递推、残差状态检查和独立的Python程序使有限计算可复现。
英文摘要
Experiments on binary run-length differentiability lead to sharp computer-assisted criteria for shuffle squares. Every $C^3$-word of length greater than $16$ is a shuffle square exactly when both letter multiplicities are even. All $34$ nonempty even-Parikh exceptions are smooth and persist in every higher differentiability class. For $C^2$ the sharp threshold is $48$, with $212$ nonempty exceptions. At level $C^1$ no global parity threshold exists, but every even-Parikh non-shuffle-square of length at least $36$ has proper nonempty shuffle-square prefixes and suffixes. Exactly $230$ nonempty even-Parikh $C^1$-words have no nonempty shuffle-square prefix. The full tree avoiding such prefixes eventually consists of $422$ periodic rays. Consequently, a nonempty Kolakoski prefix is a shuffle square exactly when both multiplicities are even, apart from lengths $4$ and $8$. We also characterize classes of morphisms reflecting shuffle squares. For doubly binary words, we determine the exact deletion distance and largest twins, and prove sharp bounds for single local repairs. Exact recurrences, residual-state checks, and separate Python programs make the finite computations reproducible.
Comments35 pages, 8 listings
