Compositional Discrete Forecasting with Orthogonal Frequency-Band Codebooks

Long-term time-series forecasting must extrapolate recurring local structures whose frequency components reappear in different combinations. Continuous representations compose these factors flexibly but do not expose a finite vocabulary of reusable patterns. Existing discrete methods typically bind complete patches to single codes, tokenize transform coefficients, or retain continuous residual paths, leaving compositional codebook-based forecasting unresolved. We propose PatchVQFormer, which factorizes each patch into orthogonal frequency bands and represents it as a composition of frozen, band-specific waveform prototypes. Training-set energy and predictability determine the band partition, while band-dependent codebook capacities are selected on validation data. A spatial-then-temporal backbone uses masked future queries to predict all future band distributions in parallel. Training combines hard token classification with differentiable sequence reconstruction, whereas inference uses parameter-free soft expectations over frozen waveform prototypes. Across eight multivariate benchmarks and four horizons, PatchVQFormer achieves an average MSE/MAE of $0.291/0.342$ and ranks first in $11/32$ MSE and $8/32$ MAE comparisons. On the four ETT benchmarks at $H\in \{336,720\}$, it improves average MSE over the strongest listed competitor by $1.8\%/1.0\%$. In controlled ETT ablations, frequency-band factorization reduces average MSE over the no-Fourier variant by $4.5\%$. These results establish compositional frequency-band vocabularies as an accurate and structured approach for long-horizon forecasting without a continuous residual decoder.