  The Maxwell-Calladine index theorem plays a central role in our current
understanding of the mechanical rigidity of discrete materials. By considering
the geometric constraints each material component imposes on a set of
underlying degrees of freedom, the theorem relates the emergence of rigidity to
constraint counting arguments. However, the Maxwell-Calladine paradigm is
significantly limited -- its exclusive reliance on the geometric relationships
between constraints and degrees of freedom completely neglects the actual
energetic costs of deforming individual components. To address this limitation,
we derive a generalization of the Maxwell-Calladine index theorem based on
susceptibilities that naturally incorporate local energetic properties such as
stiffness and prestress. Using this extended framework, we investigate how
local energetics modify the classical constraint counting picture to capture
the relationship between deformations and external forces. We then combine this
formalism with group representation theory to design mechanical metamaterials
where differences in symmetry between local energy costs and structural
geometry are exploited to control responses to external forces.
