When Common Optimal Projections Exist: A Trace/Operator-Norm Fork for PSD Families

For a finite family of positive semidefinite operators on Rᵈ, we compare two notions of common rank-r optimality: the Ky Fan trace score tr (PM) and the operator-norm score lambdaₘax (PKP). For the trace score, a common optimal projection exists exactly when the family has a shared dominant r-dimensional invariant subspace, called r-coherence. For the operator-norm score, a common optimal projection exists exactly when one r-plane intersects every leading eigenspace, called r-coverability. These conditions differ: r-coherence implies r-coverability, but the converse fails, as shown by a 3 x 3 witness. When coverability fails, we introduce an eigenvalue-weighted Ky Fan surrogate and prove coarse and instance-dependent regret bounds. The results clarify how the existence of a common optimal low-rank projection depends on the scoring rule.