When Olympiad Problems Produce EML Trees
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.
observation 13 posts
FMA Is the Only Primitive That Matters
We measured the node-cost decay across seven basis states on 222 elementary-function equations. One primitive dominates: fused-multiply-add.
The Oscillation Boundary
Across 315 tested equations, a clean dichotomy: oscillatory functions sit outside ELC with one exception — a non-elementary token.
Two Boundaries of ELC
The elementary logarithmic closure is bounded by two structurally independent obstructions. Classical analysis guards one edge; classical algebra guards the other.
What We Got Wrong
Four things we retracted, corrected, or demoted during the 2026-04 foundation audit. What survived is stronger for it.
Which Way Does the Transform Go?
Classical integral transforms partition into three ELC-direction classes. The direction is determined by the kernel.
214 Equations: The SuperBEST Cost of Science
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).
SuperBEST Node Costs: Chemistry and Biology
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.
The SuperBEST Cost of Everything
From Google's PageRank to your GPS to the NFL passer rating — every equation has a node count. Here are the ones that matter.
The EML Complexity Census
23 elementary functions classified by minimum EML approximation depth. What is proved, what is computational, what is open.
The Depth-6 Phase Transition
At depth 5, all complex EML values have Im = −π exactly. At depth 6, imaginary parts explode. A structural phase transition.
exp(−x) and the Five-Operator Barrier
exp(−x) is blocked for 1-node EML across five operator families. DEML breaks through: deml(x,1) = exp(−x) exactly.
The Phantom Attractor Is Not Real
EML gradient descent converges to ~3.17 on 40/40 seeds when targeting π. PSLQ found no relation. Vanishes at higher precision.