  We introduce how to pushforward shifted symplectic fibrations along base
changes. This is achieved by considering symplectic forms that are closed in a
stronger sense. Examples include: symplectic zero loci and symplectic
quotients. Observing that twisted cotangent bundles are symplectic
pushforwards, we obtain an equivalence between symplectic fibrations and
Lagrangians to critical loci.
  We provide two local structure theorems for symplectic fibrations: a smooth
local structure theorem for higher stacks via symplectic zero loci and twisted
cotangents, and an {\'e}tale local structure theorem for $1$-stacks with
reductive stabilizers via symplectic quotients of the smooth local models.
  We resolve deformation invariance issue in Donaldson-Thomas theory of
Calabi-Yau $4$-folds. Abstractly, we associate virtual Lagrangian cycles for
oriented $(-2)$-symplectic fibrations as unique functorial bivariant classes
over the exact loci. For moduli of perfect complexes, we show that the exact
loci consist of deformations for which the $(0,4)$-Hodge pieces of the second
Chern characters remain zero.
