  The recent link discovered between generalized Legendre transforms and
non-dually flat statistical manifolds suggests a fundamental reason behind the
ubiquity of R\'{e}nyi's divergence and entropy in a wide range of physical
phenomena. However, these early findings still provide little intuition on the
nature of this relationship and its implications for physical systems. Here we
shed new light on the Legendre transform by revealing the consequences of its
deformation via symplectic geometry and complexification. These findings reveal
a novel common framework that leads to a principled and unified understanding
of physical systems that are not well-described by classic
information-theoretic quantities.
