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578 expressions, 50 Lean theorems, 5 PyPI packages, an npm port, a HuggingFace dataset, three websites, four interactive demos. Two weeks. One human. Here's what actually worked, what failed, and what the audit system caught before it reached the public.
Hand a damped-oscillator equation to a computer and it can tell you, without knowing any physics, that there's one oscillation and one decay inside it. Across 193 expressions and 12 domains, this counter holds at ρ = +0.885.
The best-selling synthesizer in history runs on a Bessel function. The Gibbs phenomenon's 9% overshoot is a theorem you can hear. Three interactive demos at 1op.io let you turn structural complexity into sound.
The NAND gate of continuous math. A single binary operation eml(x, y) = exp(x) − ln(y) generates every elementary function — and the structural fingerprint of an expression turns out to predict where it came from.
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.
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.
A thought experiment: if tan(1) could be built from EML trees, what would follow? The conditional chain connects to Schanuel's conjecture and would collapse the depth hierarchy.
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.
Depth-k trees have at most 2^k zeros (proved). Observed maximum is O(k). The gap suggests a tighter bound — still open.
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.
exp(−x) is blocked for 1-node EML across five operator families. DEML breaks through: deml(x,1) = exp(−x) exactly.