Shrinking Squares.An empirical exploration

Start with a sequence $S=(a,b,c,d$ of positive integers and find the derived sequence $S_{1}=T(S)=(|a-b|,|b-c|,|c-d|,|d-a|$ .Define a sequence $S,S_{1},S_{2}=T(S_{1}),S_{3}=T(S_{2}),...$ .

Let $S_{i}$ minimizes the sum of the four elements.What is the sum of the four elements of $S_{i}$ ?