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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 discover what 1 complex node does.
0 real trees searched · best MSE: —
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T01 — Infinite Zeros Barrier
Any finite real EML tree is a composition of exp and ln — both real-analytic. Real-analytic functions have at most finitely many zeros on any bounded interval (unless identically zero). sin(x) has infinitely many zeros: nπ for all n ∈ ℤ. Contradiction. QED.
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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. T01 (Infinite Zeros Barrier, §14) provides the structural proof: finite real EML trees are real-analytic, and real-analytic functions have finitely many zeros. sin(x) has infinitely many. T03 (Euler Gateway, §16) provides the resolution: over ℂ, sin(x) = Im(eml(ix, 1)) in exactly one node. The gap between real and complex EML is where the barrier lives.