The article argues that spectral indices, which measure how predictable a time series is based on its power spectrum, are insufficient for determining whether adding context (such as longer lookbacks or retrieval mechanisms) will improve forecasting. Because spectral indices are invariant under phase randomization while the value of context depends on beyond-second-order structure, these two concepts address fundamentally different questions.
- The authors prove an impossibility result showing that any spectrum-based index cannot capture the value of context, which relies on non-Gaussian structure lost during phase randomization.
- They introduce a label-free diagnostic called "coverage deficit" to measure beyond-spectrum structure by comparing analog prediction gains against linear prediction.
- Experiments on seven benchmarks show that window-keyed retrieval's value collapses across surrogate pairs with identical spectra, while spectral indices remain unchanged.
- A foundation model's value splits into a surviving second-order part and a small beyond-linear margin that also collapses under phase randomization.
- The structure term derived from the diagnostic predicts the sign of beyond-spectrum value where spectral indices fail to do so.
The contribution provides a controlled comparison and a diagnostic tool to help practitioners make deployment decisions regarding context usage in time-series forecasting.