No valid real EML tree with 4 or 5 nodes evaluates to $2$

Proved

EmlComplexity.not_attains_two_four_five

by WillR · Sep 7, 2026 · Mathlib 0df444a (Lean v4.33.1)

elementary-functionseml-complexityexpression-complexitylower-bound

For $m = 4$ and $m = 5$ there is no valid closed EML tree with exactly $m$ nodes whose real-branch value is $2$ . This is a finite check over the $56$ trees of sizes $4$ and $5$ : valid trees are separated from $2$ by certified rational enclosures of nested exponentials and logarithms, and degenerate trees are ruled out exactly. It is one band of the full lower bound below $9$ nodes.

Preamble

import Definitions.Def_EmlComplexity

Formal statement

namespace EmlComplexity
theorem not_attains_two_four_five : ∀ m : ℕ, 4 ≤ m → m ≤ 5 → ¬ Attains (2 : ℝ) m := by sorry
end EmlComplexity

Source

Odrzywolek, All elementary functions from a single operator, arXiv:2603.21852 (2026), Section 4.1 and Table 4; witness trees from the enumeration in oaustegard/eml-sr, benchmarks/eml_complexity.md (2026-09-04)

View graph

No valid real EML tree with 4 or 5 nodes evaluates to 222

No valid real EML tree with 4 or 5 nodes evaluates to $2$