AI 中文总结
本文否定回答了加法、整数除法与固定底指数构成Kalmár初等函数基的问题，证明其即使加乘法也不完备，但变底指数$x^y$可构成基，且该不完整类对真值完备。
AI 中文摘要
Prunescu、Sauras-Altuzarra和Shunia曾问，加法、全化整数除法以及以二为底的指数运算是否构成Kalmár初等函数的代入基。我们对此问题给出否定回答，即使加入乘法后亦然。在二的独立幂上，由加法、乘法和整数除法构建的每个固定项在给定界以下仅取多对数多个不同值，且同样的限制在任意嵌套$2^x$后仍然成立。没有这样的项能从$2^n$恢复$n$，因此$\floor{\log_2 x}$和整数余数均不可表示。该障碍并非以二为底所特有。对每个固定整数$b\geq 2$，同样的稀疏界成立，且常数与$b$无关，$\langle x+y,\lfloor x/y\rfloor,b^x\rangle$仍是初等函数的真子类。变基指数运算则反转结论：加法、整数除法和$x^y$确实构成代入基，即$$\langle x+y,\\ \lfloor x/y\rfloor,\\ x^y\rangle=\mathcal{E}.$$一旦底数可依赖于输入，整数余数和截断减法变得可表示。不完整的以二为底的类对真值而言仍是完备的。每个初等特征函数都属于该类，每个具有固定有限值域的初等函数亦如此。
英文摘要
Prunescu, Sauras-Altuzarra, and Shunia asked whether addition, totalized integer division, and base-two exponentiation form a substitution basis for the Kalmár elementary functions. We answer this problem negatively, even after adjoining multiplication. On independent powers of two, every fixed term built from addition, multiplication, and integer division takes only polylogarithmically many distinct values below a given bound, and the same restriction survives arbitrary nesting of $2^x$. No such term recovers $n$ from $2^n$, so neither $\lfloor\log_2 x\rfloor$ nor integer remainder is representable. The obstruction is not special to the base two. For every fixed integer $b\geq 2$, the same sparsity bound holds with constants independent of $b$, and $\langle x+y,\lfloor x/y\rfloor,b^x\rangle$ remains a proper subclass of the elementary functions. Variable exponentiation reverses the conclusion: addition, integer division, and $x^y$ do form a substitution basis, $$ \langle x+y,\ \lfloor x/y\rfloor,\ x^y\rangle=\mathcal{E} .$$ Once the base may depend on the input, integer remainder and truncated subtraction become representable. The incomplete base-two class is nevertheless complete for truth values. Every elementary characteristic function belongs to it, as does every elementary function with a fixed finite range.
