  Classical magnets exhibit exotic ground state properties such as spin liquids
and fractionalization, promising a manifestation of superposition and
projective symmetry construction in classical theory. While system-specific
spin-ice or soft-spin models exist, a formal theory for general classical
magnets remains elusive. Here, we introduce a generic symmetry group
construction built from a vector field in a plaquette of classical spins,
demonstrating how classical spins superpose in irreducible representations
(irreps) of the symmetry group. The corresponding probability amplitudes serve
as order parameters and local spins as fragmented excitations. The formalism
offers a many-body vector field representation of diverse ground states,
including spin liquids and fragmented phases described as degenerate ensembles
of irreps. We apply the theory to a frustrated square Kagome lattice, where
spin-ice or soft spin rules are inapt, to describe spin liquids and fragmented
phases, all validated through irreps ensembles and unbiased Monte Carlo
simulation. Our generic theory sheds light on previously unknown aspects of
spin-liquid phases and fragmentation and broadens their applications to other
branches of field theory.
