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Closure
Every elementary function that oscillates or flows lives at EML depth 3. Try to escape. You can't.
Select two objects to combine. Everything stays at depth 3.
Combinations tried
0
all returned depth 3
Who's in here with you
sin(x), cos(x), tanh(x)
erf(x), lgamma(x)
Fourier series
wave equation, Onsager solution
ψ = Ae^{ikx}, L-functions
All EML-3. All closed under eml().
EML-4 does not exist. Six proofs.
EML-3 is the deepest finite stratum. The N=3 Singularity (§37) shows a 1.4-million-fold MSE improvement at depth 3 — not gradual, a phase transition. The EML Closure Theorem (§41) proves the stratum is algebraically sealed: no EML composition of depth-3 functions can exceed depth 3. EML-4 is a proved gap, not a conjecture. The only exit is categorification — EML-∞, the infinite-node limit.