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, pi, and i; arithmetic operations including addition, subtraction, multiplication, division, and exponentiation as well as the usual transcendental and algebraic functions. For example, exp(x)=eml(x,1), ln(x)=eml(1,eml(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 -> 1 | 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：数字硬件中，一个两输入逻辑门即可实现全部布尔逻辑。但在连续数学领域，从未发现类似的原始算子：计算sin、cos、sqrt、log等初等函数始终需要多个不同的运算操作。本文证明，单一二元算子eml(x,y)=exp(x)-ln(y)与常数1结合，即可生成科学计算器的标准函数库。这包括e、π、i等常数；涵盖加法、减法、乘法、除法、乘方等算术运算，以及常见的超越函数和代数函数。例如exp(x)=eml(x,1)，ln(x)=eml(1,eml(eml(1,x),1))，其他所有运算均可类推。此类算子的存在性此前未被预见；我通过系统性穷举搜索发现该算子，并以构造性方式证明其足以构成科学计算器的基础运算集。在EML（指数减对数）形式下，每个表达式都化为同构节点的二叉树，从而形成与文法S -> 1 | eml(S,S)同样简洁的语法结构。这种统一结构还支持梯度驱动的符号回归：将EML树作为可训练电路并使用标准优化器（Adam），我证明了在不超过4层的浅树深度下，可从数值数据中精确恢复闭式初等函数的可行性。该架构可拟合任意数据，但当生成规律为初等函数时，它可能恢复出精确的公式形式。