Interaction Budgets for Gated Wavelet Denoising: An EMPR Theory of Operator- Versus Correlation-Induced Coupling

Modern non-stationary signal denoisers increasingly replace a global wavelet threshold by a \emph{detect-then-act} (gated) rule that first localises an artefact in the time--scale plane and then suppresses it selectively. Such gates couple sub-bands, and it has been unclear how to quantify, in a basis-intrinsic and interpretable way, \emph{how much} of the denoising performance is produced by that coupling. We introduce the \emph{interaction budget} $I=1-\sum_j S_j$ and prove that it equals the normalised $L^2$-distance of the performance functional to the space of band-additive functions; hence $I=0$ for any band-diagonal operator and $I>0$ exactly when the gate couples bands. We then (i) give an exact pairwise identity for a binary gate, a two-point lower bound, and a sharp iff-condition on the gate; (ii) identify a coupling-strength functional with the leading-order law $I(\kappa)=(\mathcal C/V_0)\kappa^2+o(\kappa^2)$; and (iii) bound the second-order HDMR/EMPR truncation error, which vanishes for single-trigger gates. Finally, for coloured noise---where the sub-band factors are correlated and $I$ alone conflates the two effects---an EMPR product-support decomposition \emph{separates} operator-induced from correlation-induced interaction, with a correlation-invariant operator signature and a permutation-based estimator. Controlled experiments confirm all results.