One Operator

How a single equation generates all elementary functions — and what it reveals about mathematical complexity.

In March 2026, Andrzej Odrzywołek published a proof that a single binary operator generates every elementary function as a finite binary tree:

Exponentials. Logarithms. Trigonometry. Polynomials. All of them — as finite compositions of this one operator with the constant 1. Not approximately. Exactly.

The simplest case

Set y = 1. Since ln(1) = 0, the equation becomes eml(x, 1) = exp(x). One application. The entire exponential function from a single node in the tree.

The barrier

Now try sin(x) over the reals. You can't build it. A finite EML tree is real-analytic, and a non-zero real-analytic function has isolated zeros. But sin(x) zeros at every multiple of π. Contradiction. This is the Infinite Zeros Barrier — provable in two lines. Read it →

The bypass

Over ℂ, eml(ix, 1) = exp(ix) = cos(x) + i·sin(x). One node. Depth ∞ over ℝ vs. depth 1 over ℂ. Same function, different domain, different complexity. Why →

Complexity strata

• Depth 0 — arithmetic (polynomials, rationals)
• Depth 1 — exponential
• Depth 2 — logarithmic
• Depth 3 — oscillatory (sin, cos via complex bypass)
• Depth ∞ — non-constructible over ℝ

There is no depth 4. The jump is direct. Full theorem catalog →

The open problem

Can you construct i from {1} under strict principal-branch ln? Depth-6 values reach Im ≈ 0.99999524. The gap of 4.76×10⁻⁶ is structural — tan(1) is transcendental (Lindemann–Weierstrass). The near-miss →

Research blog

• The Infinite Zeros Barrier
• 0.99999524: The Near-Miss
• Euler's Formula Is the EML Operator
• What BEST Routing Saves
• Auditing 1200 Sessions

Library: monogate.dev · Paper: arXiv:2603.21852 · pip install monogate