The EML Operator and the Complexity Structure of the Riemann Explicit Formula

We study the expression complexity of the Riemann explicit formula from the perspective of grammar-based symbolic regression. Starting from the EML operator eml (a, b) = exp (a) − log (b), we show that this operator fails to produce oscillatory terms and therefore cannot encode the explicit-formula structure. We then derive a replacement operator emlᵦeta (g, t) = (cos (γt) + 2γ sin (γt) ) / (¼ + γ²) directly from the decomposition of x^ρ/ρ on the critical line Re (ρ) = ½. This operator encodes the constraint Re (ρ) = ½ algebraically, via the ¼ = (½) ² in its denominator, rather than requiring it to be discovered empirically. Using ordinary least-squares fits, we demonstrate that emlᵦeta achieves statistically identical fit quality to standard trigonometric grammar while using 3. 1–3. 7× fewer expression nodes across all tested approximation orders. The operator connects to the Clausen function family and to the polylogarithm on the unit circle at the specific index forced by the ζ functional equation. We report full experimental provenance, including a negative result on the original EML operator.