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

UPDATE v6 (May 16, 2026):

New — Galaxy rotation curve application (125 SPARC + 23 LITTLE THINGS galaxies):

• eml★ characterization battery (H1-M2): confirmed as non-holomorphicity detector
• Vortex topology interpretation REFUTED (fake vortex negative control: 5/5)
• Luminosity is the best predictor of eml★ activation (rho=-0.273, p=0.004)
• DM fraction not significant (p=0.81), MOND not significant (p=0.35)
• LITTLE THINGS independent replication: 43.5% eml★ activation rate
• Paper draft: "Symbolic regression with anti-holomorphic operators reveals non-holomorphic structure in low-luminosity galaxy rotation curves"

Software: OxiEML-Star v0.3.0 — https://zenodo.org/records/20238531 Full v5 PDF: https://github.com/antparis/eml_star/blob/main/paper/eml_star_final_v5.pdf Related: Garmaev et al. (2026), Complex Equation Learner, arXiv:2605.03841

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)

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.

v6 additions: Galaxy rotation curves — eml★ applied to 148 galaxies via genetic programming. Establishes eml★ as a non-holomorphicity detector with luminosity as the primary physical driver. Vortex topology interpretation refuted by controlled experiments.

GitHub: https://github.com/antparis/eml_star Software: https://github.com/antparis/oxieml-star (DOI: 10.5281/zenodo.20238531)