Construct the constant π from terminal {1} under strict principal-branch grammar.

Construct i = sqrt(-1) using extended EML grammar (complex inputs allowed). Open problem: can i be reached as an EML value over C?

Nearest miss: Im = 0.99999524 at depth 6 — gap = 4.76e-6

Proved impossible under the strict real grammar (principal-branch ln, ln(0) throws). Theorem T17, machine-verified in Lean 4: no finite real EML tree from {1} evaluates to i = sqrt(-1). Under the extended-reals grammar the problem remains open.

Proved impossible by T01 (Infinite Zeros Barrier). cos(x) has infinitely many zeros on the real line; every finite-depth EML tree has at most finitely many zeros. No EML expression can equal cos(x).

Proved impossible by T01 (Infinite Zeros Barrier). Every depth-k real EML tree has at most 2^k zeros on any compact interval. sin(x) has infinitely many zeros, so no finite-depth EML tree can equal sin(x) exactly.