Mixed partition functions are exactly the graph parameters of exponentially bounded edge-connection rank

We prove a conjecture of Regts and Sevenster: a complex-valued graph parameter $f$ with $f(\emptyset)=1$ has exponentially bounded edge-connection rank if and only if it is a mixed partition function. The bound is exact: for a real number $R\ge 1$, the connection ranks satisfy $\operatorname{rk} M_{