  Discrete Scale Relativity proposes a new symmetry principle called discrete
cosmological self-similarity which relates each class of systems and phenomena
on a given Scale of nature's discrete cosmological hierarchy to the equivalent
class of analogue systems and phenomena on any other Scale. The new symmetry
principle can be understood in terms of discrete scale invariance involving the
spatial, temporal and dynamic parameters of all systems and phenomena. This new
paradigm predicts a rigorous discrete self-similarity between Stellar Scale
variable stars and Atomic Scale excited atoms undergoing energy-level
transitions and sub-threshold oscillations. Previously, methods for
demonstrating and testing the proposed symmetry principle have been applied to
RR Lyrae, Delta Scuti and ZZ Ceti variable stars. In the present paper we apply
the same analytical methods and diagnostic tests to a new class of variable
stars: SX Phoenicis variables. Double-mode pulsators are shown to provide an
especially useful means of testing the uniqueness and rigor of the conceptual
principles and discrete self-similar scaling of Discrete Scale Relativity.
