Causal Echoes: Task Precision Through Physical Cancellation

Control actions can improve the measurements used to learn a system. For unknown linear systems with signal-dependent sensing, paired inputs nearly cancel in the plant and leave a small residual with small absolute readout noise. Repeating this physical cancellation sharpens the information needed for a future control task. Under radial relative noise and zero additive floors, $K$ actionable packets yield minimax orders ${\sigma }^K$ for the learning-terminal state and ${\sigma }^{K+1}$ for a future command using a final upload; full physical calibration remains at order $\sigma$ in dimension at least two. Matching lower bounds allow adaptive packet timing and silence. A calibrated-drift gain family gives a finite separation from every noninteractive acquisition, including randomized designs, under nonzero readout and deployment floors. Held-out matrix results are consistent with the predicted orders and show how conditioning and noise floors limit the useful refinement depth.