Summary: Let \(K\) be a complete, algebraically closed nonarchimedean valued field, and let \(\varphi(z)\in K(z)\) have degree \(d\geq 2\). We show there is a canonical way to assign nonnegative integer weights \(w_\varphi(P)\) to points of the Berkovich projective line over \(K\) in such a way that \(\sum_Pw_\varphi(P)=d-1\). When \(\varphi\) has bad reduction, the set of points with nonzero weight forms a distributed analogue of the unique point which occurs when \(\varphi\) has potential good reduction. Using this, we characterize the minimal resultant locus of \(\varphi\) in analytic and moduli-theoretic terms: analytically, it is the barycenter of the weight-measure associated to \(\varphi\); moduli-theoretically, it is the closure of the set of points where \(\varphi\) has semistable reduction, in the sense of geometric invariant theory.