  We show that short-range pair correlations in a strongly interacting Fermi
gas follow a simple universal law described by Tan's relations. This is
achieved through measurements of the static structure factor which displays a
universal scaling proportional to the ratio of Tan's contact to the momentum
$C/q$. Bragg spectroscopy of ultracold $^6$Li atoms from a periodic optical
potential is used to measure the structure factor for a wide range of momenta
and interaction strengths, providing broad confirmation of this universal law.
We calibrate our Bragg spectra using the $f$-sum rule, which is found to
improve the accuracy of the structure factor measurement.
