Sharp marginal-erasure bounds and positive completion of nonunital quantum channels

We determine the exact Frobenius conditioning of deleting the traceless part of one local matrix algebra on C^m tensor C^k. For 1<=r<m, the inverse constants are sqrt(m/(m-r)) for two states of rank at most r, and sqrt((m+r)/(m-r)) for one such state against an arbitrary density matrix. Both are attained, independently of k>=2; at rank m exact collisions occur. The one-sided estimate follows from a scalar optimization combining positivity, trace control and the partial-trace norm. It yields a noise and spectral-tail bound for convex positive completion. For normalized Choi matrices, the retained coordinates are the action of a trace-preserving channel on traceless inputs. Consequently this action identifies every channel of Kraus rank below its input dimension among all channels, without a unitality assumption. For square channels we express the same conclusion through a complete exact unitary 2-design of average gate fidelities. The rank threshold is sharp, whereas optimal conditioning within the trace-preserving subclass is left open. The fidelity frame identities and general kernel/inertia method are prior work; no compressed-sampling or new randomized-benchmarking noise guarantee is asserted.