AI 中文总结
本文推广了有限有序集的 $A^A$ 问题，证明对变量出现次数至多十二次的幂项 t，$t(A)\cong t(B)$ 蕴含 $A\cong B$，并给出覆盖所有括号化的构造与证明。
AI 中文摘要
设 X 和 Y 为有限有序集，$X^Y$ 表示从 Y 到 X 的保序映射构成的有序集。设 t 为由单个变量通过幂运算形成的项。我们证明，对于每个变量出现次数至多十二次的项 t 以及任意有限有序集 A 和 B，$t(A)\cong t(B)$ 蕴含 $A\cong B$。首先发展了一般构造和重构原理；第二部分给出了逐次出现的证明。在十二次出现时，证明覆盖了 4,766 个交换正规形，代表所有 58,786 种二元括号化，包括 40 个例外形式。此外还包含了两个额外的无界重构族、完整的有限审计数据以及 457 个精确高度比较证书。
英文摘要
Let X and Y be finite ordered sets and let $X^Y$ denote the ordered set of order-preserving maps from $Y$ to $X$. Let $t$ be a~term formed from a~single variable by exponentiation. We prove that, for every term $t$ with at most twelve variable occurrences and arbitrary finite ordered sets $A,B$, $t(A)\cong t(B)$ implies that $A\cong B$. General constructions and reconstruction principles are developed first; the second part gives the occurrence-by-occurrence proofs. At twelve occurrences the proof covers 4,766 interchange normal forms, representing all 58,786 binary parenthesizations, including 40 exceptional forms. Two additional unbounded reconstruction families, complete finite audit data, and 457 exact height-comparison certificates are included.
