Paper

Published and in-progress work on the EML operator. ← monogate.org

Main paper

Published · 2026

Universal Elementary Function Representation via a Single Binary Operator

Andrzej Odrzywołek · arXiv:2603.21852 · CC BY 4.0

Proves that the single operator eml(x, y) = exp(x) − ln(y) generates every elementary function as a finite binary tree, with explicit constructions for all standard functions. Introduces the EML depth hierarchy: five strata from arithmetic (depth 0) through oscillatory (depth 3), with a gap — there is no depth 4.

Read on arXiv →

In preparation

Draft · STATUS: DRAFT — not ready for arXiv

Near-misses and transcendental obstructions in EML tree closure

Monogate Research · 2026

Studies the EML operator eml(x,y) = exp(x) − ln(y) and the smallest set EML₁ containing 1 and closed under eml. Central question: whether i ∈ EML₁. Proves depth-5 values have Im(z) ≤ 0, and that depth-6 values can reach Im ≈ 0.99999524 (gap 4.76×10⁻⁶). The obstruction: exact Im = 1 requires Re(y) = π/tan(1), which is transcendental (Hermite–Lindemann).

Read the blog post →

Cite

@misc{odrzywołek2026eml,
  title  = {Universal Elementary Function Representation via a Single Binary Operator},
  author = {Odrzywołek, Andrzej},
  year   = {2026},
  url    = {https://arxiv.org/abs/2603.21852}
}