A minimal and universal representation of fermionic wavefunctions (fermions = bosons + one)

Representing fermionic wavefunctions efficiently is a central problem in quantum physics, chemistry and materials science. In this work, we introduce a universal and exact representation of continuous antisymmetric functions by lifting them to continuous symmetric functions defined on an enlarged space. Building on this lifting, we obtain a \emph{parity-graded representation} of fermionic wavefunctions, expressed in terms of symmetric feature variables that encode particle configuration and antisymmetric feature variables that encode exchange statistics. This representation is both exact and minimal: the number of required features scales as $D\sim N^d$ ($d$ is spatial dimension) or $D\sim N$ depending on the symmetric feature maps employed. Our results provide a rigorous mathematical foundation for efficient representations of fermionic wavefunctions and enable scalable and systematically improvable neural network solvers for many-electron systems.