DepShift: Measuring What Cross-Channel Modeling Buys, and What It Costs Under Dependency Shift

A recent method proposes channel shuffling as a certificate that a multivariate forecaster does not over-rely on cross-channel structure. It certifies less than it is taken to. Shuffling is a channel permutation, a relabeling of a coupling that stays intact; the disturbance a deployed model faces is a rewiring, the coupling changing with channel identities fixed. A relabeling also permutes the per-channel marginals unless the channels it swaps share one law, whereas a change in the coupling's lag leaves every single-channel law exactly unchanged. We build DepShift, a generator on which the lag change is the pure cross-channel disturbance and a relabeling is not, and an excess-fragility contrast against a channel-independent control. Permutation-equivariance does not confer robustness to rewiring: the same model is more fragile to a rewiring than to a relabeling at 11 of 12 coupling configurations, null in the factor topology, and the flip holds for CPiRi's released code at 5 of 5, uncorrected by its own shuffling regulariser. Along a dose axis, and replicated on six fresh seeds, the equivariant model's response to a lag change grows far faster than its response to a relabeling. What permutation degradation it does show is its evaluation pipeline's, and the same normalisation switch leaves the fragility of all three identity-storing architectures we put through it in place. The two classes also separate out of sample at full channel count. To read a null one must know what was achievable, so we compute a closed-form ceiling for the cross-channel increment: against it the equivariant model pays a tenth to a quarter of the optimal predictor's price under a lag change and under a twentieth under relabeling. We report where this fails: the ceiling does not transfer to real data, 3 pre-registered attempts to attribute the real-data ordering to coupling movement failed, the real-data test is met on one of three datasets, and a ratio-metric artifact produced a retired hypothesis's predicted sign at p=2e-59 inside a protocol written by authors trying not to be fooled.