Pairwise closure of power-law resistor networks and sharp three-terminal elimination defects

We characterize when eliminating the interior vertices of a finite positive-weight power-law resistor network preserves an energy given by a sum of pairwise terms with the same exponent. For every fixed 1<p<∞, p≠2, this holds exactly when each connected component of the interior graph meets at most two boundary vertices. The criterion is independent of the positive weights. The proof treats zero voltage drops directly: second differences handle p<2, while an active quadratic energy and a zero-drop energy of order p handle p>2. For a three-terminal star with sorted weights A≥B≥C>0, the best uniform pairwise approximation error on boundary data of oscillation at most one is comparable, with constants depending only on p, to C(B/A)^(1/(p−1)). For fixed positive weights it has a strictly positive first-order coefficient in |p−2|. We also give first-order weak-arm limits and transfer the estimates to stars whose arms are arbitrary two-terminal networks.