monogate.dev / challenges / Tight EML Pumping Lemma
Tight EML Pumping Lemma
OPEN
The formal Pumping Lemma gives a 2^k upper bound on zeros of depth-k EML trees. Empirically the maximum observed is 0-1 zeros regardless of depth. Prove or disprove: the tight bound is O(k) (linear in depth), not O(2^k). Specifically, either (a) construct a depth-k EML tree with Omega(k) zeros on [-8,8], or (b) prove all depth-k EML trees on any compact interval have at most k+1 zeros.
Grammar
Allowed
✓eml(x, y) = exp(x) − ln(y) — the single operator
✓Terminal: constant 1
✓Terminal: variable x (for function challenges)
✓Compositions: any finite binary tree over the above
Forbidden
✗ln(0) — undefined, throws RangeError
✗Math.sin, Math.cos, Math.PI, or any Math.* shortcut
✗Any function outside the EML grammar
✗Extended-reals convention (ln(0) = −∞) — not this grammar
Under extended-reals grammar (pveierland/eml-eval), i is solved at K=75. This challenge is the strict grammar variant where ln(0) throws.
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