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

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 causal GP experiment (100 runs) confirms eml★ is structurally necessary: Median ATE = 11.71, 95% CI [9.64, 12.57]. GitHub: https://github.com/antparis/eml_star