  We introduce and prove the "root theorem", which establishes a condition for
families of operators to annihilate all root states associated with zero modes
of a given positive semi-definite $k$-body Hamiltonian chosen from a large
class. This class is motivated by fractional quantum Hall and related problems,
and features generally long-ranged, one-dimensional, dipole-conserving terms.
Our theorem streamlines analysis of zero-modes in contexts where "generalized"
or "entangled" Pauli principles apply. One major application of the theorem is
to parent Hamiltonians for mixed Landau-level wave functions, such as
unprojected composite fermion or parton-like states that were recently
discussed in the literature, where it is difficult to rigorously establish a
complete set of zero modes with traditional polynomial techniques. As a simple
application we show that a modified $V_1$ pseudo-potential, obtained via
retention of only half the terms, stabilizes the $\nu=1/2$ Tao-Thouless state
as the unique densest ground state.
