Is spectral predictability enough? Testing Ω-matching for synthetic pretraining under distribution shift

Synthetic pretraining under distribution shift requires deciding what synthetic data to use. A natural criterion is to match the spectral predictability $\mathrm{\Omega }$ of the source to the target, but is $\mathrm{\Omega }$ sufficient? We test this hypothesis, which we call predictability-aware pretraining, across three architectures (LSTM, PatchTST, TCN) and three seeds. Under ideal synthetic-to-synthetic conditions, $\mathrm{\Omega }$-matched pretraining can outperform even an oracle baseline (+46.8% MSE reduction, LSTM), though the benefit is already architecture-dependent. On the real-world ETTm1 benchmark, however, this relationship does not hold: sine-plus-noise pretraining at matched $\mathrm{\Omega }$ yields only a marginal mean improvement +2.3% averaged over all three architectures, up to +65.5% on individual channels), whereas $\mathrm{\Omega }$ matched chaotic ODE trajectories systematically degrade performance -13.2%). The two sources agree on the direction of transfer for only 53% of configurations across all three architectures (Pearson r=-0.02), indicating that $\mathrm{\Omega }$ carries little information about which source will transfer beneficially. We further observe an architecture-dependent asymmetry in the response to the direction of the $\mathrm{\Omega }$ gap. We conclude that $\mathrm{\Omega }$ matching can help in controlled settings but is not sufficient to predict transfer across sources, and we discuss additional signal properties that a source-selection criterion may need to consider.