Quantum Processes with Recovery: Normal Form, Stability, and Memory Certification

We characterize two-step quantum processes with a prescribed first channel whose unperturbed final evolution exactly recovers the input. For a system of dimension d and a first channel of Kraus rank k > 1, the compatible family is affinely bijective to the set of channels from dimension d(k−1) to dimension d. For qubit dephasing with residual coherence r, an intermediate X intervention not only reconstructs the parametrizing channel but also attains the optimal operational distance, equal to (1−r²) times the diamond distance between the parametrizing channels. We determine the classical subfamily, its minimum register cardinality, and the maximal temporal negativity (1−r²)/2. For imperfect recovery, we prove O(√δ) proximity to the exact family while preserving the first channel, with dimension-independent constants. An explicit two-qubit example shows that the exponent 1/2 cannot be uniformly improved. Finally, we derive a Pauli-cycle witness whose exact classically attainable region is characterized by a quadratic inequality involving the cyclic coherence and two return probabilities. The witness requires neither population preservation nor perfect recovery. A linear version uses twelve experimental settings and admits explicit finite-sample significance bounds with calibrated control errors. Analytical proofs are accompanied by reproducible code. No experimental data or departures from standard quantum mechanics are claimed.