  We derive the 3D energy critical quintic NLS from quantum many-body dynamics
with 3-body interaction in the T^3 (periodic) setting. Due to the known
complexity of the energy critical setting, previous progress was limited in
comparison to the 2-body interaction case yielding energy subcritical cubic
NLS. Previously, the only result for the 3D energy critical case was HTX, which
proved the uniqueness part of the argument in the case of small solutions. In
the main part of this paper, we develop methods to prove the convergence of the
BBGKY hierarchy to the infinite Gross-Pitaevskii (GP) hierarchy, and
separately, the uniqueness of large GP solutions. Since the trace estimate used
in the previous proofs of convergence is the false sharp trace estimate in our
setting, we instead introduce a new frequency interaction analysis and apply
the finite dimensional quantum de Finetti theorem. For the large solution
uniqueness argument, we discover the new HUFL (hierarchical uniform frequency
localization) property for the GP hierarchy and use it to prove a new type of
uniqueness theorem. The HUFL property reduces to a new statement even for NLS.
With the help of CKSTT,IP which proved the global well-posedness for the
quintic NLS, this new uniqueness theorem establishes global uniqueness.
