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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
Periodic barrier — Lean, MachLib
Every real EML tree defined for large x eventually keeps one sign.
sin(x) never does — it keeps returning to 0 and to 1.
Therefore: no finite real EML tree equals sin(x).
sin(x) never does — it keeps returning to 0 and to 1.
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, once the leaf ix is given and Im or Re is read off outside the operator.
Periodic barrier (Lean, MachLib): every real EML tree defined for large x eventually keeps one sign — 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, given ix, read through Im; built entirely from eml, starting from 1 and x: at most 243 nodes (ledger).