Global stability and transition maximizers for ternary Boolean compression

We study how far a nearly hardest ternary Boolean response can lie from the set of hardest responses with the same total cost. The approximation class consists of two independent hidden stars, with nonnegative unary and pair costs. For unrestricted source networks and for sources of treewidth at most two, we determine the complete uniform-distance stability curves. The curves have several linear regimes, plateaus and jumps. We classify every maximizing response at five unrestricted-source transitions, including a transition from isolated points to triangles and one with two cubes of maximizers. We also identify all terminal maximizers and prove a sharp quantitative rigidity bound near that endpoint. Short analytical arguments recover the complete hardest-input families and their interval projections. Both global value curves have analytical capacity proofs. Selected equality and transition classifications retain finite rational proof covers with separately supplied exact readers. The exact distance and width-two recognition theorems from the companion papers are explicit premises. The results are presented as candidate theorems supported by written proofs and exact checks. The accompanying material provides the sources, software, assurance record, verification and reproduction instructions.