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 · softplus(x) = 1 node via LEAd
• Depth 2 — logarithmic · negation = 2 nodes · ln(x) = 1 node
• Depth 3 — oscillatory (sin, cos via complex bypass) · mul/sub/add = 3 nodes
• Depth ∞ — non-constructible over ℝ

There is no depth 4. The jump is direct. Full depth atlas → · Theorem catalog →

The completeness theorem

Among all 16 exp-ln operators, exactly 8 are completely expressive: EML, EAL, EXL, EDL, EPL, LEAd, ELAd, ELSb. The structural rule is simple — if the operator has exp(+x) with no domain restriction, it is exactly complete. If it has exp(−x), it is incomplete. One rule explains all 16 cases (T26–T28). Why exp(+x) means complete →

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

• Why exp(+x) Means Complete: The Structural Theorem new
• SuperBEST: All Operations Proved Optimal
• The Infinite Zeros Barrier
• 0.99999524: The Near-Miss
• Euler's Formula Is the EML Operator
• All 44 posts →

30 theorems proved · depth atlas · monogate.dev · Paper: arXiv:2603.21852 · pip install monogate