In digital logic, NAND suffices. Every Boolean function — AND, OR, NOT, XOR — can be built from NAND gates alone. The entire edifice of digital computation rests on a single binary operation.
Odrzywołek (arXiv:2603.21852) finds the continuous analog. A single binary operator, eml(x, y) = exp(x) − ln(y), combined with the constant 1, generates every function on a scientific calculator. Addition, subtraction, multiplication, division, exponentiation, logarithms, trigonometric functions, square roots, and the constants e, π, and i — all expressible as compositions of eml and 1.
The grammar is minimal: S → 1 | eml(S, S). Every expression is a binary tree where every internal node performs the same operation and every leaf is 1. The tree depth determines the complexity of the function. At depth 1, eml(1, 1) = e − 0 = e. From there, through nesting, the full repertoire unfolds.
The discovery came from exhaustive search over binary operators of the form f(x, y) = a · g(x) + b · h(y), where g and h are chosen from {x, exp(x), ln(x)}. Most combinations fail — they generate only a subset of elementary functions, or they require additional constants beyond 1. The eml operator is the one that works.
The practical application: symbolic regression. Given numerical data, fit an EML tree by treating each leaf's constant as a trainable parameter and using standard optimizers (Adam). At tree depth 4, the method recovers exact closed-form elementary functions from data. The search space is much smaller than general symbolic regression because every node performs the same operation — only the tree structure and the leaf values vary. The expressiveness is universal but the search is constrained.
The parallel to NAND is precise. NAND is functionally complete for Boolean algebra. EML is functionally complete for elementary analysis. Neither is the most natural way to express the functions it generates — no one writes computer programs entirely in NAND, and no one would write equations entirely in EML. But the existence of the single gate reveals the structure: all elementary functions are compositions of one primitive, the way all Boolean functions are compositions of another.