Abstract
Linked energy-economy models can iterate between a top-down model of the economy and a bottom-up model of the energy system. This process continues, with each model exchanging relevant quantities, until their results converge. Although a recent coordination algorithm lowers the cost of each round by predicting bottom-up solutions from stored bases, the alternation remains a fixed-point iteration. It converges only where the round map is locally contracting, its stopping rule bounds no distance to a fixed point, and with several equilibria it reports whichever one the starting point selects. This paper addresses all three shortcomings for linkages that exchange a single scalar in each direction. The linked equilibrium becomes the root of a scalar function, located by bracketing with an exact derivative recovered from the stored basis inverse, and every candidate carries either a certificate that a true equilibrium exists nearby and is locally unique, or an explicit failure report. Extending the pointwise basis test to its closed-form validity interval makes the bottom-up response exactly piecewise rational, so all equilibria in a stated range along the tracked top-down branch can be enumerated and classified. The scalar solver reproduces the published equilibria, each unique in that range, with half the top-down solves and slightly more bottom-up solves. A parametric root finder removes almost all bottom-up work. The gain at which the corner equilibrium is born has a closed form built from the validity intervals. An amplified demand link illustrates the structure beyond that threshold, including a repelling equilibrium that no damped alternation can reach.