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 small F16 trees — often cheaper than the equations that define them. Whether those trees are the cheapest possible is not shown.
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equations across 12+ domains — from 1-node trivialities up (the 2037-node Reed-Solomon ceiling this post once gave is withdrawn). 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.
sin(x) costs 101 nodes as an 8-term Taylor series by April's BEST-routing count, and 1 complex EML node through Euler's formula. The accuracy figures first printed beside the Taylor and Fourier counts are not reproduced.
23 elementary functions classified by EML depth, each row with what its depth rests on: a Lean theorem, a construction, an argument, or nothing yet.
Built from 1, EML values stay real through depth 4. At depth 5 every non-real value has Im = −π exactly; at depth 6 there are 13,600 distinct imaginary parts. 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.