AI 中文总结
本文证明仅用幂和对数即可构造固定表达式模拟任意标签系统，并由此得出可达性不可判定性，且该机制可表示所有Kalmár初等函数。
AI 中文摘要
我证明任何标签系统的一个精确步骤可以写成仅使用幂和对数的单个固定表达式。标签状态被表示为一个整数；计算该表达式得到下一配置的整数编码，并将答案反馈回同一表达式，从而逐步跟随计算过程。更新的离散部分来自利用主复对数中的回绕，这有效地起到了与类似方法中的取整、取模或正弦相同的作用。迭代表达式仅使用$x^y$和$\log_b a$：结果表明这两个二元运算就足够了，无需常数。由此可知，对于单个固定表达式，低于固定阈值的可达性是不可判定的。作为另一个应用，同一机制表示加法、$2^x$以及采用约定$x\bmod0=x$的余数，从而表示每个Kalmár初等函数。
英文摘要
I show that one exact step of any tag system can be written as a single fixed expression using only powers and logarithms. A tag state is represented as an integer; evaluating the expression yields the integer encoding of the next configuration, and feeding the answer back into the same expression follows the computation step by step. The discrete part of the update comes from leveraging the wraparound in the principal complex logarithm, which effectively serves the same purpose as floor, mod, or sin in comparable approaches. The iterating expressions use only $x^y$ and $\log_b a$: it turns out that these two binary operations are sufficient, with no need for constants. It follows that reachability below a fixed threshold is undecidable for one fixed expression. As a separate application, the same mechanism represents addition, $2^x$, and remainder with the convention $x\bmod0=x$, and hence every Kalmár elementary function.
Comments11 pages. Added Appendix C giving a constant-free, 23-operation expression for the shortcut Collatz map
