Controlled Temporal Rank of Quantum Processes: Memory Factorization and Collision-Model Transfer Spectra

Instrument-specific quantum Markov order records the length of a successfulhistory-blocking sequence. It need not specify how many independent memorymodes can be carried into later observations. To resolve that structure, wedefine a controlled temporal response map from a process tensor, a real spanof centered contrasts among admissible past operations, and a chosen futuretester space. The rank is independent of operational coordinates. In everycompatible finite sequential realization the map factors through thetraceless Hermitian operator space of the retained memory, giving\(ρ_nᶜᵗʳˡ≤ d_M^2-1\)and hence an architecture-relative lower bound on latent-memory dimension. For a finite write--read collision delay line, restricted to one-body latentperturbations, fixed blank future writes, and nonadaptive informationallycomplete product probes, the response map decomposes into localmemory-to-probe transfer maps:\[ ρn,L(1,prod) =min\{n+1,L\}\,q(U).\]The same process has exact instrument-specific Markov order $L+1$ under afixed accessible replacement block for every admissible history and futureinstrument. Blocking length and transferred-mode structure are thereforeindependent within this collision family. An operator-Schmidt construction expresses $q(U)$ as the rank of aninteraction transfer matrix. For two-qubit interactions, all three transfersingular values are obtained in Cartan coordinates. The partial-SWAP identityestablishes full transfer rank in equal dimensions, and a real-analytic-minorargument gives maximal rank for Haar-almost-every equal-dimensionalbipartite unitary. Rank-one and rank-three two-qubit interactions may have thesame scalar local transfer norm. The paper also gives finite-sample rankcertificates and a finite-library intervention-design problem. Symbolic andnumerical validation files accompany the manuscript.