We present BISTRO Survey 850 μm dust emission polarization observations of the L1495A-B10 region of the Taurus molecular cloud, taken at the James Clerk Maxwell Telescope (JCMT). We observe a roughly triangular network of dense filaments. We detect nine of the dense starless cores embedded within these filaments in polarization, finding that the plane-of-sky orientation of the core-scale magnetic field lies roughly perpendicular to the filaments in almost all cases. We also find that the large-scale magnetic field orientation measured by Planck is not correlated with any of the core or filament structures, except in the case of the lowest-density core. We propose a scenario for early prestellar evolution that is both an extension to, and consistent with, previous models, introducing an additional evolutionary transitional stage between field-dominated and matter-dominated evolution, observed here for the first time. In this scenario, the cloud collapses first to a sheet-like structure. Uniquely, we appear to be seeing this sheet almost face on. The sheet fragments into filaments, which in turn form cores. However, the material must reach a certain critical density before the evolution changes from being field dominated to being matter dominated. We measure the sheet surface density and the magnetic field strength at that transition for the first time and show consistency with an analytical prediction that had previously gone untested for over 50 yr.