Selective Certificates for Reduced-Space Robust Schatten-p Optimization

Reduced-space optimization can be stationary in its working coordinates while a finite-amplitude direction outside that space still decreases the objective. We turn this failure mode into a selective certificate interface for robust Schatten-$p$ self-representation. The exact split $r_Z^2=a^2+e^2$ separates in-space from omitted original-space error, while a bounded top-$k$ normal scan tests finite-amplitude escapes. The controller returns a certificate for the declared $({\Phi }_{\beta },\mathcal{D},[0,T],\eta )$ object or a typed abstention when coverage, rank, amplitude, or resources are insufficient. We prove the residual split, a finite nonnegative-Huber tail closure with optional Lipschitz tightening, a strict grid-error bound, and separate descent and compute ledgers. An incomplete tail scan remains an explicitly scoped prefix. The image panel reports both $\Delta \Phi$ and $\Delta F$, their envelope gap, majorant acceptance diagnostics, and full/prefix certificate counts. This separation is the point: compressed computation should expose exactly what it certifies and abstain when the requested original-space condition has not been covered.