Transcriptional control ensures genes are expressed in the right amounts at the correct times and locations. Understanding quantitatively how regulatory systems convert input signals to appropriate outputs remains a challenge. For the first time, we successfully model even skipped (eve) stripes 2 and 3+7 across the entire fly embryo at cellular resolution. A straightforward statistical relationship explains how transcription factor (TF) concentrations define eve’s complex spatial expression, without the need for pairwise interactions or cross-regulatory dynamics. Simulating thousands of TF combinations, we recover known regulators and suggest new candidates. Finally, we accurately predict the intricate effects of perturbations including TF mutations and misexpression. Our approach imposes minimal assumptions about regulatory function; instead we infer underlying mechanisms from models that best fit the data, like the lack of TF-specific thresholds and the positional value of homotypic interactions. Our study provides a general and quantitative method for elucidating the regulation of diverse biological systems.