16 operators, one structural rule: exp(+x) with no domain restriction implies exactly complete. exp(-x) implies incomplete. -exp(x) implies approximately complete. The Exponential Position Theorem explains all 16 classifications at once.
The only operation in SuperBEST costing more than 3 nodes was general-domain addition at 11n. It now costs 2 nodes. The table is complete.
One formula predicts the SuperBEST node cost of any scientific equation. Proved, validated on 187 equations, and open-sourced.
Four structural classes, the cost decomposition theorem (T38), complexity classes O(1)/O(N)/O(N²), and the Linear Ceiling Conjecture (T39): a complete theory of how many EML nodes any standard scientific equation requires.
Every function has a minimum node count. We now know the complete depth spectrum: 1, 2, 3, ∞ — and why depth-4 exists but contains no standard functions. Plus: multiplication drops to 2 nodes.
ELSb(0, x) = exp(0 − ln(x)) = 1/x. One node. SuperBEST v4: 18 nodes total, 75.3% savings. The reciprocal was never a division problem.
A single transcendence fact about tan(1) is the root cause behind three separate EML results: the multiplication lower bound, the depth-3 ceiling for standard functions, and the complex density behavior.
Three completeness classes for exp-ln operators: exactly complete (EML), approximately complete (EMN), and incomplete (all others). Two new theorems prove EMN's exact limits and approximate power.
f(x) = exp(x) − ln(x) satisfies f(x) > x for all real x > 0. The gap is minimized at x ≈ 1.31 where f(x) − x ≥ 1.648. This is a theorem about the operator's self-interaction — and it separates EML from every other operator in the family.
sin(x) costs 101 nodes as a Taylor series in BEST routing. The same function is 1 complex EML node using Fourier. This 100x gap validates the lab's sound design and reveals a deep structural fact about the operator.
The BEST router's mul entry drops to 3 nodes via exl(ln(x), exp(y)) = x·y. The lower bound is 3n, confirmed tight by exhaustive search. Gap fully closed.
We applied the DEML incompleteness template to seven exp-ln operators. Six are incomplete. One is open. One surprise: a gate with the identity function built in.
We proved that a depth-k EML tree has at most 2k+2 real zeros, and verified computationally that the true bound may be as low as 2 for all k ≥ 3. This strengthens the Infinite Zeros Barrier from qualitative to quantitative.
deml(x,y) = exp(−x) − ln(y) cannot construct exp(+x) or neg(x). Structural proof + exhaustive search over 861,952 trees.
Why sin(x) cannot be expressed as a finite real EML tree. Every real EML tree is real-analytic with finitely many zeros.
Under strict principal-branch semantics, i is unreachable from {eml, 1}. Lean-verified. Depth-6 closest approach: 0.99999524.
EML trees are dense in C([a,b]). Any continuous function approximable to arbitrary precision — but not always exactly representable.