  In this paper, we prove a new non-renormalization theorem which arises from
UV/IR mixing. This theorem and its corollaries are relevant for all
four-dimensional tachyon-free closed string theories which can be realized from
higher-dimensional theories via geometric compactifications. As such, they
therefore apply regardless of the presence or absence of spacetime
supersymmetry and regardless of the gauge symmetries or matter content
involved. This theorem resolves a hidden clash between modular invariance and
the process of decompactification, and enables us to uncover a number of
surprising phenomenological properties of these theories. Chief among these is
the fact that certain physical quantities within such theories cannot exhibit
logarithmic or power-law running and instead enter an effective fixed-point
regime above the compactification scale. This cessation of running occurs as
the result of the UV/IR mixing inherent in the theory. These effects apply not
only for gauge couplings but also for the Higgs mass and other quantities of
phenomenological interest, thereby eliminating the logarithmic and/or power-law
running that might have otherwise appeared for such quantities. These results
illustrate the power of UV/IR mixing to tame divergences -- even without
supersymmetry -- and reinforce the notion that UV/IR mixing may play a vital
role in resolving hierarchy problems without supersymmetry.
