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 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 Dsim N^d (d is spatial dimension) or Dsim 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.