monogate.dev
The Identity
Build a tree that computes x. Just x. It's harder than you think.
Every node computes eml(a, b) = exp(a) − ln(b)
Your leaves are x and 1. Build a tree whose output equals its input.
The depth-4 proof (click each step)
The EML Identity Theorem: the simplest function (x → x) requires the non-trivial composition eml(1, eml(eml(1, eml(x,1)), 1)). This was discovered computationally during EML Fourier analysis and proved algebraically in four steps. Depth 4 is minimal — no EML tree of depth 1, 2, or 3 equals the identity function. The proof uses only eml(a,b) = exp(a) − ln(b), the fact that ln(1) = 0, and the fact that ln(exp(z)) = z.