Learning directionality between variables is crucial yet challenging, especially for mechanistic relationships without a priori ordering assumptions. We propose a coefficient of asymmetry to quantify directional asymmetry using Shannon's entropy within a generative exposure mapping (GEM) framework. GEMs arise from experiments where a generative function $g$ maps exposure $X$ to outcome $Y$ through $Y = g(X)$, extended to noise-perturbed GEMs as $Y = g(X) + ε$. Our approach considers a rich class of generative functions while providing statistical inference for uncertainty quantification - a gap in existing bivariate causal discovery techniques. We establish large-sample theoretical guarantees through data-splitting and cross-fitting techniques, implementing fast Fourier transformation-based density estimation to avoid parameter tuning. The methodology accommodates contamination in outcome measurements. Extensive simulations demonstrate superior performance compared to competing causal discovery methods. Applied to epigenetic data examining DNA methylation and blood pressure relationships, our method unveils novel pathways for cardiovascular disease genes \emph{FGF5} and \emph{HSD11B2}. This framework serves as a discovery tool for improving scientific research rigor, with GEM-induced asymmetry representing a low-dimensional imprint of underlying causality