PRISM: Patch-based Regression with an Interpretable Shape–Scale–Level Model for Time-Series Forecasting

Long-horizon time-series forecasting requires representations that preserve multi-scale temporal structure while adapting to changes in amplitude and level across windows. We study PRISM, a lightweight, channel-independent forecaster that composes established components — a multi-scale patch encoder with a shared U-Net, a learned-query direct (non-autoregressive) decoder, and an explicit output head y_hat = shape·scale + level. None of these components is individually novel. Rather than claim state-of-the-art accuracy, we ask a set of answerable questions: does the explicit decomposition mean what its names say, does multi-scale patching help, where does the model help on real data, and does controlled synthetic evidence support the mechanism. Our load-bearing result is a controlled, three-seed mechanism study: under a gauge-aware analysis on synthetic data with known generative factors, the shape·scale+level head recovers the generative factors — the scale head tracks the true amplitude where amplitude varies (across-sample R^2 = 0.73, mean of 3 seeds), the level head tracks the true offset where offset varies (R^2 = 0.99; 0.94 under joint regime shifts), and both stay near zero where their factor is absent. This recovery is a property of the head, not the multi-scale encoder: a single-patch control sharing the same head attains near-identical recovery (R^2 = 0.67/0.98/0.93), so it is backbone-robust. This is our only multi-seed result and we treat it as the headline; we also disclose that the per-head raw recovery is weaker than the gauge-aware R^2 (e.g. S2 scale regression slopes are 0.36–0.46, not about 1). All real-data results use a single seed (seed = 1) and carry no statistical-significance claim; every WIN/LOSS verdict is indicative only. Under this caveat, PRISM matches or beats the best of six strong baselines on a characterizable high-trend, low-volatility niche (notably the ETTm2 family, 3.5–13.9% better than the best baseline — though these ETT-family wins carry a train-split asterisk, 0.8 vs the baselines' 0.75, not fully resolved), while being competitive-or-better on 12/44 cells overall (7 wins, 5 competitive, 32 losses; 27%) and not state-of-the-art in aggregate. We report negative results plainly: multi-scale patching does not beat matched single-patch on real data, added capacity barely helps, and gating, learnable-Fourier, period-folding, and channel-mixing are inert or harmful. Beyond long horizons, a mask-aware member of the same family, PRISM-Masked, beats the official M4 competition winner ES-RNN on five of six frequency groups and on the weighted total (0.8125 vs 0.821 OWA, single model averaged over three seeds; within roughly 0.015 of N-BEATS at 0.797), though it is not state of the art. Our contribution is new knowledge, not a new component: a verifiably faithful decomposition under controlled factors, an honest account of when a small patch forecaster helps, a short-horizon generalization check on M4, and a set of negative results.