  We study the geometric change of Chow cohomology classes in projective toric
varieties under the Weil-McMullen dual of the intersection product with a
Lefschetz element. Based on this, we introduce toric chordality, a
generalization of graph chordality to higher skeleta of simplicial complexes
with a coordinatization over characteristic 0, leading us to a far-reaching
generalization of Kalai's work on applications of rigidity of frameworks to
polytope theory. In contrast to "homological" chordality, the notion that is
usually studied as a higher-dimensional analogue of graph chordality, we will
show that toric chordality has several advantageous properties and
applications. -- Most strikingly, we will see that toric chordality allows us
to introduce a higher version of Dirac's propagation principle. -- Aside from
the propagation theorem, we also study the interplay with the geometric
properties of the simplicial chain complex of the underlying simplicial
complex, culminating in a quantified version of the Stanley--Murai--Nevo
generalized lower bound theorem. -- Finally, we apply our technique to give a
simple proof of the generalized lower bound theorem in polytope theory and --
prove the balanced generalized lower bound conjecture of Klee and Novik.
