arXiv:2604.26018v1 Announce Type: cross Abstract: We introduce QERNEL, a foundational neural wavefunction that variationally solves families of parameterized many-electron Hamiltonians and captures their ground states throughout parameter space within a single model. QERNEL combines FiLM-based parameter conditioning with scale-efficient architectural elements -- mixture of experts and grouped-query attention, substantially improving expressivity at low computational cost. We apply QERNEL to interacting electrons in semiconductor moir\'e heterobilayers, training a single weight-shared model for systems of up to 150 electrons. By solving the many-electron Schr\"odinger equation conditioned on moir\'e potential depth, QERNEL captures both quantum liquid and crystal states and discovers the sharp phase transition between them, marked by abrupt changes in interaction energy and charge density. Our work establishes a foundation model for moir\'e quantum materials and a scalable architecture toward a Large Electron Model for solids.