We present an emergent hydrodynamic framework for fermionic statistics within a 3 + 1-dimensional quantized superfluid effective field theory (EFT). While classi-cal vortices in irrotational fluids are bosonic, we demonstrate that quantized vortex loops generically acquire an effective internal framing arising from the anisotropicgeometry and phase-gradient structure of their cores. These framed vortex loops inhabit a configuration space C ∼= (T 3 × SO(3))N /SN . We define the Berry con-nection on the Hilbert bundle of condensate states and show that the adiabatic exchange of two such defects corresponds to a non-contractible loop in SO(3). Dueto π1(SO(3)) ∼= Z2, the resulting holonomy yields a topological phase π, producing the fermionic exchange sign eiπ = −1. This provides a concrete mechanical real-ization of the Finkelstein–Rubinstein constraint and demonstrates the possibility that half-integer spin and fermionic exchange statistics can emerge as topologicalproperties of a superfluid vacuum.
Alex Smith (Mon,) studied this question.