Exclusivity Without Marginal Loss: A Coupled-Rounding Primitive and Full-LP Obstruction for Chromatic Correlation Clustering

Chromatic Correlation Clustering (CCC) assigns each cluster one color and penalizes positive edges unless they are co-clustered under their input color. The standard vertex-pair LP already implies cross-color pairwise packing. We turn this property into disjoint potential color claims using one uniform draw per pivot-pair, preserving each selected-color admission marginal. When multiple colors have positive affinity, a claim-respecting projection of independent claims cannot preserve those marginals and exclusivity simultaneously. For the resulting independent pivot-pair algorithm, a feasible three-vertex LP solution makes the true-color third-pivot loss unbounded relative to the positive edge's LP charge; a corrected accounting also includes wrong-color single capture. We analyze a shared-threshold variant that correlates admissions across endpoints: it minimizes the immediate positive-edge loss at fixed marginals, but maximizes negative-edge co-admission. On the witness, every within-pivot coupling preserving the pivot and color laws and the canonical marginals incurs an immediate third-pivot loss of at least one-third of the positive-edge LP charge, exceeding the $0.06$ residual budget of the considered $2.06$ proof route. The barrier extends to every $L\ge 3$ with all colors active. A compact pivot-consistency LP excludes the witness and bounds this immediate loss for the shared-threshold rule; it does not control continuation or yield a new approximation ratio. Alongside a synthetic mechanism check, we report a clustering-quality audit against existing rounding and certified small-instance optima. No improved worst-case approximation factor is claimed.