How position-dependent cell fate acquisition occurs during embryogenesis is a central question in developmental biology. To study this process, we developed a defined, high-throughput assay to induce peri-gastrulation-associated patterning in geometrically confined human pluripotent stem cell (hPSC) colonies. We observed that, upon BMP4 treatment, phosphorylated SMAD1 (pSMAD1) activity in the colonies organized into a radial gradient. We developed a reaction-diffusion (RD)-based computational model and observed that the self-organization of pSMAD1 signaling was consistent with the RD principle. Consequent fate acquisition occurred as a function of both pSMAD1 signaling strength and duration of induction, consistent with the positional-information (PI) paradigm. We propose that the self-organized peri-gastrulation-like fate patterning in BMP4-treated geometrically confined hPSC colonies arises via a stepwise model of RD followed by PI. This two-step model predicted experimental responses to perturbations of key parameters such as colony size and BMP4 dose. Furthermore, it also predicted experimental conditions that resulted in RD-like periodic patterning in large hPSC colonies, and rescued peri-gastrulation-like patterning in colony sizes previously thought to be reticent to this behavior.