A single two-input gate suffices for all of Boolean logic in digital hardware. No comparable primitive has been known for continuous mathematics: computing elementary functions such as sin, cos, sqrt, and log has always required multiple distinct operations. Here I show that a single binary operator, eml(x,y)=exp(x)-ln(y), together with the constant 1, generates the standard repertoire of a scientific calculator. This includes constants such as $e$, $π$, and $i$; arithmetic operations including $+$, $-$, $\times$, $/$, and exponentiation as well as the usual transcendental and algebraic functions. For example, $e^x=\operatorname{eml}(x,1)$, $\ln x=\operatorname{eml}(1,\operatorname{eml}(\operatorname{eml}(1,x),1))$, and likewise for all other operations. That such an operator exists was not anticipated; I found it by systematic exhaustive search and established constructively that it suffices for the concrete scientific-calculator basis. In EML (Exp-Minus-Log) form, every such expression becomes a binary tree of identical nodes, yielding a grammar as simple as $S \to 1 \mid \operatorname{eml}(S,S)$. This uniform structure also enables gradient-based symbolic regression: using EML trees as trainable circuits with standard optimizers (Adam), I demonstrate the feasibility of exact recovery of closed-form elementary functions from numerical data at shallow tree depths up to 4. The same architecture can fit arbitrary data, but when the generating law is elementary, it may recover the exact formula.

翻译：单个二输入门即可实现数字硬件中的所有布尔逻辑。然而，连续数学领域长期缺乏类似的基元：计算初等函数（如 sin、cos、sqrt 和 log）始终需要多种不同运算。本文证明，单个二元算子 eml(x,y)=exp(x)-ln(y) 与常数 1 相结合，即可生成科学计算器的标准函数库。这包括 $e$、$\pi$、$i$ 等常数；$+$、$-$、$\times$、$/$ 及乘方等算术运算，以及常见的超越函数与代数函数。例如，$e^x=\operatorname{eml}(x,1)$、$\ln x=\operatorname{eml}(1,\operatorname{eml}(\operatorname{eml}(1,x),1))$，其他所有操作亦如此。此类算子的存在此前未被预期；我通过系统性穷举搜索发现它，并建设性地证明其足以构成具体科学计算器的基础。在 EML（指数减对数）形式下，每个表达式均成为同质节点的二叉树，从而产生 $S \to 1 \mid \operatorname{eml}(S,S)$ 这般简单的文法。这种统一结构还能实现基于梯度的符号回归：利用 EML 树作为可训练电路（采用 Adam 等标准优化器），我证明了在深度不超过 4 的浅层树中，可从数值数据精确恢复闭式初等函数的可行性。同一架构可拟合任意数据，但当生成律为初等函数时，它可能恢复精确公式。