  We show that there is an emergent lattice description for the continuous
fractional quantum Hall (FQH) systems, with a generalised set of few-body
coherent states. In particular, model Hamiltonians of the FQH effect are
equivalent to the real space von Neumann lattice of local projection operators
imposed on a continuous system in the thermodynamic limit. It can be
analytically derived that tuning local one-body potentials in such lattices
amounts to the tuning of individual two or few-body pseudopotentials. For some
cases, we can realise pure few-body pseudopotentials important for stabilising
exotic non-Abelian topological phases. This new approach can thus potentially
lead to experimental realisation of coveted non-Abelian quantum fluids
including the Moore-Read state and the Fibonacci state. The reformulation of
the FQHE as a sum of local projections opens up new path for rigorously proving
the incompressibility of microscopic Hamiltonians in the thermodynamic limit.
