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/emlₛtar/blob/main/paper/emlₛtarfinalᵥ5. 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/emlₛtar Software: https: //github. com/antparis/oxieml-star (DOI: 10. 5281/zenodo. 20238531)