We prove that the uniform random non-separable planar map with n edges admits the Brownian sphere as Gromov-Hausdorff-Prokhorov scaling limit as n tends to infinity. Our proof introduces a non-bijective ``common-core transfer method'' that constitutes a novel and universal proof strategy for scaling limits of random discrete structures. As an application, we complete the phase diagram for limiting shapes of block-weighted planar maps by Stufler (2019). We describe phases with limits given by the Brownian sphere, stable trees, and Brownian sphere decorated stable trees recently introduced by Senizergues, Stefansson and Stufler (2023).