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. That is not yet a contradiction, since sin is real-analytic with isolated zeros too: the Infinite Zeros Barrier needs a bound on how many zeros a finite tree can have, and that bound is still open. MachLib rules sin out another way, by periodicity. Read it →

The bypass

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

Complexity strata

Two different counts get called depth. Atlas depth, used in the list below and on /atlas, counts nested exp/ln applications and treats arithmetic as free. EML tree depth counts eml nodes along the longest path from the root to a leaf, with nothing free. The node counts in the list are a third measure: F16 nodes in the SuperBEST table.

• Atlas depth 0 — arithmetic (polynomials, rationals)
• Atlas depth 1 — exponential · softplus(x) = 1 F16 node via LEAd
• Atlas depth 2 — logarithmic · negation = 2 F16 nodes · ln(x) = 2 F16 nodes (1 with the census operator EXL)
• Atlas depth 3 — oscillatory (sin, cos via complex bypass) · add = 2, sub = 2, mul = 1 (x, y > 0) or 3 F16 nodes
• Atlas depth ∞ — non-constructible over ℝ

In EML tree depth the counts are larger, since arithmetic is not free: ln x needs exactly 3 and x + 1 exactly 4 on (0, ∞). Full depth atlas → · Theorem catalog →

The completeness conjecture

Among all 16 exp-ln operators, 8 are conjectured to be completely expressive over ℂ: EML, EAL, EXL, EDL, EPL, LEAd, ELAd, ELSb. For EML it is the published result; over ℝ, EAL is not, since every real EAL tree is nondecreasing. The proposed 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 would explain all 16 cases, but T26–T28 have no proofs. Does exp(+x) mean complete? →

The open problem

Can you construct i from {1}? Under strict principal-branch ln, no: every defined value is real (T17). Under the complex principal branch it is open. The depth-6 value closest to i is 4.76×10⁻⁶ away, and nothing shows that gap cannot close; the tan(1) argument once given here for that is not a proof. The near-miss →

Research blog

• Does exp(+x) Mean Complete? A Structural Conjecture new
• The SuperBEST Table, v1 (corrected)
• The Infinite Zeros Barrier
• 0.99999524: The Near-Miss
• Euler's Formula Is the EML Operator
• All 44 posts →

17 THEOREM-tier results in the catalog · depth atlas · monogate.dev · Paper: arXiv:2603.21852 · pip install monogate