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May 15, 2026Open Access

eml★ — Minimal Anti-Holomorphic Extension of the EML Sheffer Operator

Key Points

This research aims to establish the eml★ operator as a minimal anti-holomorphic extension of the EML Sheffer operator, addressing structural limitations of the existing model.
Verified 60 physical systems across 50+ domains with exact results
Proven completeness of the {eml, eml★} basis
Conducted numerical experiments and ablation studies to compare performance metrics.
eml★ achieved an MSE of 5.89 × 10⁻³³, outperforming eml with an MSE of 12.97
Causal GP experiments with eml★ achieved an MSE of approximately 1.5 × 10⁻³¹
Exact recovery of 25–100% in various anti-holomorphic targets validated improvement.

Abstract

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/emlₛtar/blob/main/paper/emlₛtarfinalᵥ5. 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/emlₛtar Software: https: //github. com/antparis/oxieml-star

eml★ — Minimal Anti-Holomorphic Extension of the EML Sheffer Operator

Key Points

Abstract

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