When Independent Kraus Padding Is Suboptimal: Representation-Invariant Gap Criteria and Robust Sequential Embeddings

Independent Kraus padding completes residual quantum instruments specified only on a reachable subspace, but its cost can depend on Kraus gauge. Optimization over outcome-compatible representations yields an exact first-crossing formula for the representation-invariant padding width. At capacity saturation, zero defect is equivalent to projectivity of the induced measurement and vanishing state-update decomposition excess. Effect-rank, Schmidt-number, determinantal, and Stiefel certificates provide progressively stronger tests for positive defects. A universal wrapper lifts every capacity-saturated local defect to a globally optimal two-step separation, robust below one-half normalized diamond error. A nondisturbing uniform-label instrument attains the asymptotic factor-two endpoint.