Researchers crack the mathematical code for generating all elementary functions
A new mathematical framework has identified a minimal set of two operators that can generate every standard elementary function through finite composition. The discovery, verified across 60 physical systems with extreme precision, could streamline how engineers and scientists build computational models—potentially reducing complexity in everything from physics simulations to machine learning algorithms.
Originaltitel: eml★ — Minimal Anti-Holomorphic Extension of the EML Sheffer Operator
UPDATE v5 (May 13, 2026): New sections added: - Rust implementation (OxiEML-Star fork, 13 files, 0 errors, Theorem 3.1 verified MSE 7.2e-33) - 60 physical systems verified across 50+ domains (all EXACT, MSE ≤ 10⁻³¹) - 6 formulas rediscovered blindly from raw data (all EXACT) - 549 new algebraic identities in the {eml, eml★} framework - Completeness proof: {eml, eml★} is the minimal complete basis - Ramanujan mock theta function shadow components verified (Zwegers completion) Full v5 PDF: https://github.com/antparis/eml_star/blob/main/paper/eml_star_final_v5.pdf Software: https://github.com/antparis/oxieml-star (DOI: 10.5281/zenodo.20152988) Related: Garmaev et al. (2026), Complex Equation Learner, arXiv:2605.03841 --- Odrzywołek (2026) showed that eml(x, y) = exp(x) − ln(y), together with the constant 1, generates all standard elementary functions via finite composition. We identify a structural limitation: eml is holomorphic, so complex conjugation, and real and imaginary parts are not reachable by finite eml-compositions. We introduce the companion operator eml★(x, y) = exp(x) − ln(conj(y)), which acts as a mirror reflecting the imaginary axis. We prove: (i) conj(z) = 1 − eml★(0, eml(z, 1)) at depth 2, conditional on Im(z) in [−π, π); (ii) {eml, eml★, 1} is dense in C(K, C) for every compact K by Stone–Weierstrass; (iii) the exact branch limitation is Im(z) in [−π, π). A direct numerical experiment confirms Theorem 3.1 to machine precision: eml★ achieves MSE = 5.89 × 10⁻³³ vs. 12.97 for eml alone — a ratio of 2.2 × 10³³. A causal GP experiment (50 runs, depth 8) with eml★ achieves factual MSE ≈ 1.5 × 10⁻³¹. An ablation study (23 runs, depth 12, eml★ removed) yields mean MSE ≈ 2.44, confirming that eml★ is an optimal expressive compressor — not a structural necessity — for anti-holomorphic targets. v4 additions: Seeded GP experiments across 5 anti-holomorphic targets (conj(z), |z|², Re(z), |ψ₁+ψ₂|² interference, |ψ|² quantum probability) with exact recovery in 25–100% of runs. Head-to-head control with native numpy.conj on |exp(z)|² confirms identical causal signal (MSE without conjugation ≈ 1.8 × 10⁴ in both frameworks). numpy is 4× faster; eml★ provides a unified algebraic framework covering the entire anti-holomorphic family. Complete LaTeX source included. v5 additions: Rust implementation (OxiEML-Star), 60 physics systems verified, blind formula discovery (6/6 exact), 549 new mathematical identities, Ramanujan mock theta function verification, completeness proof for {eml, eml★}. Citation: Garmaev et al. (2026), Complex Equation Learner, arXiv:2605.03841 — CEQL uses complex weights but lacks conjugation as primitive; eml★ fills this gap. GitHub: https://github.com/antparis/eml_star Software: https://github.com/antparis/oxieml-star