Coupled iterative linear solvers that recompute stochastic feature matrices at each step arise in quantum reservoir computing, tensor decomposition, and federated learning, yet a quantitative theory for how noise propagates through their iterations has been lacking. We develop such a theory and validate it experimentally within a four-equation encode–decode protocol built on a quantum reservoir, using a full XYZ Hamiltonian (10~data qubits, feature dimension~76, 16~random Hamiltonian realizations, 48~runs per condition). Under shot noise (1,000~shots) and depolarizing noise ($p=0.005$), three noise amplification mechanisms—train–evaluate mismatch, cross-iteration drift, and shared-target coupling—place the mean-squared error (MSE) at ${10}^{-3}$–${10}^{-1}$. We derive a closed-form expression, $MSE\propto {\sigma }^2{\sum }_k{s}_k^2/(s_k^2+\lambda {)}^2$, that reproduces the scaling and order of magnitude of the observed noise floor; it is not a quantitative prediction, since it omits a training-noise bias that we measure to be the larger contribution. Asymmetric resource allocation (10~encoding shots, ${10}^5$~decoding shots) exploits the structural separability of noise channels, yielding a 102-fold MSE improvement. Comparison with six baselines and single-body operator experiments ($d=31$) confirms that the iterative solver structure—not the feature dimension—is the dominant noise bottleneck, as single-shot solvers retain machine precision under identical noise. We further show that the machine-precision reconstruction obtained under ideal conditions (MSE $\sim {10}^{-17}$ for data lengths up to~30) must be read as evidence of information transport through the protocol’s latent code: the code remains at its random initialization, and substituting for it a vector unrelated to the input, or a constant, leaves the reconstruction unchanged. That value is the interpolation residual of an underdetermined readout, and it disappears at the encoder rank threshold $\Nc = d$. We therefore withdraw the reversibility claim associated with this protocol and identify the controls that distinguish information transport from interpolation in reservoir readouts.