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