  The dynamical large deviations principle for the three-dimensional
incompressible Landau-Lifschitz-Navier-Stokes equations is shown, in the joint
scaling regime of vanishing noise intensity and correlation length. This proves
the consistency of the large deviations in lattice gas models \cite{QY}, with
Landau-Lifschitz fluctuating hydrodynamics \cite{LL87}. Secondly, in the course
of the proof, we unveil a novel relation between the validity of the
deterministic energy equality for the deterministic forced Navier-Stokes
equations and matching large deviations upper and lower bounds. In particular,
we conclude that time-reversible uniqueness to the forced Navier-Stokes
equations implies the validity of the energy equality, thus generalising the
classical Lions-Ladyzhenskaya result. Thirdly, we prove that no non-trivial
large deviations result can be true for local-in-time strong solutions.
