Attraction and repulsion along extremal Einstein--two-Maxwell--dilaton branches

We study extremal mass branches in four-dimensional Einstein gravity coupled to one dilaton and two Maxwell fields with opposite-sign dilaton couplings. For generic couplings the extremal mass is governed by a nonlinear first-order equation with no known closed-form solution. We prove that this equation has a unique positive $C^2$ solution on the full real charge-ratio line and that the solution is real analytic, so positivity on the entire positive charge cone selects a unique global mass branch. Along this branch, any two nonproportional positive charge vectors attract for $ab>-1$, have vanishing leading force at $ab=-1$, and repel for $ab<-1$. The same transition fixes the transverse convexity of the extremal mass surface and the sign of the mass difference under arbitrary finite positive-charge partitions. The sign theorem extends to higher dimensions through the known dimensional rescaling. A radial reconstruction then shows that every point of the mass branch defines an asymptotically flat exterior with a finite-area inner endpoint, while smooth or analytic horizon extension is a separate condition that restricts the couplings and scalar flow.