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The Barrier
sin(x) is unreachable in real EML. Collect the approximation shadows — then find the complex bypass.
0/9 shadow fragments collected
Select a barrier function above to explore its shadows
Shadow census
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fragments collected
real approx0
ℂ bypasses0
T01 — Infinite Zeros Barrier
Any finite real EML tree is real-analytic with finitely many zeros.
sin(x) has infinitely many zeros.
Therefore: no finite real EML tree equals sin(x).
sin(x) has infinitely many zeros.
Therefore: no finite real EML tree equals sin(x).
sin(x) ∈ EML-∞(ℝ)
T03 — locked
Collect a ℂ bypass shadow to unlock the Euler Gateway theorem.
What you're seeing
The functions above the horizon are EML-∞ over ℝ — no finite real tree reaches them. Below: the best approximation shadows at each search depth. MSE decreases, but never reaches zero. The golden zone shows what's possible over ℂ: exact solutions in 1 node.
T01 (§12): Any finite real EML tree is real-analytic with finitely many zeros — sin(x) is excluded. N=12 search (§35): 1,704,034,304 trees confirmed zero sin(x) candidates. T03 (§18): sin(x) = Im(eml(ix,1)). Over ℂ, 1 node. The real-domain barrier is structural, not fundamental.