  From the data on baryon production in heavy-ion collisions it is shown that a
set of measurable functions exist, one for each baryon type, that depend
exponentially on transverse momenta up to the maximum values detected for all
collision energies ranging from the low end of RHIC BES to the high end of CERN
LHC. The implied temperatures satisfy a scaling law in collision energy with a
universal exponent for all baryon types which is the novel discovery. A
self-similar thermal source is implied. Furthermore, it is shown how the
scaling behavior depends on centralities. Those features in the data are
related to simple properties of light and strange quarks by use of the
recombination model. Prediction of $\phi$ meson production is made and then
verified by existing data.
