  Assuming the Riemann hypothesis (RH) and the linear independence conjecture
(LI), we show that the weighted count of primes in multiple disjoint short
intervals has a multivariate Gaussian logarithmic limiting distribution with
weak negative correlation. As a consequence, we derive short-interval
counterparts for many important works in the literature of the Shanks-R\'enyi
prime number race, including the sharp phase transition from all races being
asymptotically unbiased to the existence of biased races. Our result remains
novel, even for primes in a single moving interval, especially under a
quantitative formulation of the linear independence conjecture (QLI).
