Universal-Clique Amplification and Hardness of Assignment-Optimal Edge Clique Covers

An edge clique cover has assignment cost equal to the sum of its clique sizes. Unless $P=NP$, this cost admits no polynomial-time $O(N^{1/2-\eta})$-approximation for any fixed $0<\eta<1/2$, even on connected $N$-vertex graphs. A universal-clique join transfers coloring hardness without the additive obstruction in earlier reductions. Its exact residual formula also exhibits a change of optimal cover when a universal twin is added. Explicit counterexamples refute two proposed twin-extension rules, while weighted compression of true-twin classes preserves the optimum.