A chain-order hierarchy theorem and an effective validity threshold — both flagged 'hard' by outside review months ago. We stopped repeating the label and went looking for the actual obstruction in each. One real bridge theorem came out of it. Neither question closed. Both are now precisely located instead of vaguely deferred.
Classical functional equations characterise exp and ln, and their solutions turn out to be minimal EML trees — often cheaper than the equations that define them.
We measured the node-cost decay across seven basis states on 222 elementary-function equations. One primitive dominates: fused-multiply-add.
Across 315 tested equations, a clean dichotomy: oscillatory functions sit outside ELC with one exception — a non-elementary token.
The elementary logarithmic closure is bounded by two structurally independent obstructions. Classical analysis guards one edge; classical algebra guards the other.
Four things we retracted, corrected, or demoted during the 2026-04 foundation audit. What survived is stronger for it.
Classical integral transforms partition into three ELC-direction classes. The direction is determined by the kernel.
A complete catalog of SuperBEST node counts for standard equations across 12+ domains — from 1-node trivialities to 2037-node error correction. Expanded from 157 (Monster Sprint) to 214 (COMP-ALL) to 295+ (domain-2 sessions: FIN, INFO, QM, THERMO, CHEM, BIO, ECON).
How many operator nodes does it take to compute 40 standard equations from chemistry and biology? A systematic analysis using the SuperBEST v3 routing table.
From Google's PageRank to your GPS to the NFL passer rating — every equation has a node count. Here are the ones that matter.
23 elementary functions classified by minimum EML approximation depth. What is proved, what is computational, what is open.
At depth 5, all complex EML values have Im = −π exactly. At depth 6, imaginary parts explode. A structural phase transition.
exp(−x) is blocked for 1-node EML across five operator families. DEML breaks through: deml(x,1) = exp(−x) exactly.
EML gradient descent converges to ~3.17 on 40/40 seeds when targeting π. PSLQ found no relation. Vanishes at higher precision.