16 operators, one proposed structural rule: exp(+x) with no domain restriction implies exactly complete, exp(-x) incomplete, -exp(x) approximately complete. The Exponential Position Theorem would explain all 16 classifications at once, but none of its directions has a proof, and over ℝ it fails for EAL.
A thought experiment: if tan(1) could be built from EML trees, what would follow? The conditional chain connects to Schanuel's conjecture. The collapse of the depth hierarchy it once predicted rested on a Depth Stability Theorem that is withdrawn.
Every function has a minimum node count. The depth spectrum claimed here, 1, 2, 3, ∞ with no standard function at depth 4, is wrong: x + 1 has depth exactly 4. Plus: multiplication in 2 nodes for x > 0.
A transcendence fact about tan(1) was offered as the root cause of three EML results, through a 'Depth Stability Theorem'. That theorem is withdrawn: sin has no real EML tree of any depth, but over ℂ it is one node. None of the three results uses tan(1).
Three completeness classes for exp-ln operators: exactly complete (EML), approximately complete (EMN), and incomplete (all others). Two conjectures about EMN's exact limits and approximate power, with sketches that stop short of proofs.
We enumerated EML trees over the constant 1 with up to 7 nodes and tracked how close they get to π. The density conjecture is open, and one piece of evidence first given for it, an imaginary part equal to π, was a floating-point artifact.
We applied the DEML incompleteness template to seven exp-ln operators. It suggests six are incomplete and leaves one open, but the template has a gap. One surprise: a gate with the identity function built in.
A conjectured bound of 2k+2 real zeros for an EML tree with k internal nodes, an induction for it that does not hold, and zero counts for all 862,118 trees with at most 8 internal nodes: at most 3 sign changes on [−2, 2].
Why deml(x,y) = exp(−x) − ln(y) should not construct exp(+x) or neg(x): an argument on paper with a gap, no Lean proof, and an exhaustive search over 862,118 trees.
Do depth-k trees have at most 2^k zeros? The proof offered does not work. Up to 8 internal nodes the most sign changes found was 3, too few sizes to show a growth rate.
The claim that EML trees are dense in C([a,b]), and a four-step argument for it that has a gap; no Lean proof.