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1.7 Billion Trees
N=1 through N=12 exhaustive search for sin(x). Watch the MSE shrink — but never reach zero. Then see what one complex eml node does, given the leaf i·x.
0 real trees searched · best MSE: —
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Periodic barrier — Lean, MachLib
Every real EML tree defined for large x eventually keeps one sign: past some point it is always positive, always negative, or always zero. sin(x) returns to 0 and to 1 arbitrarily far out, so it never settles. Contradiction. QED. (Counting zeros does not get there alone: sin, like every tree that is not identically zero, has finitely many zeros on any bounded interval.)
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real trees searched
0/12 real levels
§35 of the EML paper: 1,704,034,304 real EML trees were searched exhaustively up to depth N=12. Zero candidates for sin(x) were found. The periodic barrier provides the proof (Lean, MachLib): every real EML tree defined for large x eventually keeps one sign, and sin(x) never does. T03 (Euler Gateway, §16) is the route around it: over ℂ, sin(x) = Im(eml(ix, 1)) — 1 node with the leaf ix given and Im read off; built entirely from eml, its cost: at most 243 nodes. The gap between real and complex EML is where the barrier lives.