  Thermal dilepton radiation from the hot fireballs created in high-energy
heavy-ion collisions provides unique insights into the properties of the
produced medium. We first show how the predictions of hadronic many-body theory
for a melting $\rho$ meson, coupled with QGP emission utilizing a modern
lattice-QCD based equation of state, yield a quantitative description of
dilepton spectra in heavy-ion collisions at the SPS and the RHIC beam energy
scan program. We utilize these results to systematically extract the excess
yields and their invariant-mass spectral slopes to predict the excitation
function of fireball lifetimes and (early) temperatures, respectively. We
thereby demonstrate that future measurements of these quantities can yield
unprecedented information on basic fireball properties. Specifically, our
predictions quantify the relation between the measured and maximal fireball
temperatures, and the proportionality of excess yields and total lifetime. This
information can serve as a "caloric" curve to search for a first-order QCD
phase transition, and to detect non-monotonous lifetime variations possibly
related to critical phenomena.
