  Our main result is the proof of an inequality between the spectral numbers of
a Lagrangian and the spectral numbers of its reductions, in the opposite
direction to the classical inequality (see e.g [Vit92]). This has applications
to the "Geometrically bounded Lagrangians are spectrally bounded" conjecture
from [Vit08], to the structure of elements in the $\gamma$-completion of the
set of exact Lagrangians. We also investigate the local path-connectedness of
the set of Hamiltonian diffeomorphisms with the spectral metric.
