PaCo: Evidence-Supported Phase Coordinates for Alignment-Aware Time Series Forecasting

Long-term time series forecasting is challenging because recurring temporal patterns may reappear at different temporal positions across observation windows. Such alignment variation changes local observations while preserving the underlying pattern, making forecasting representations prone to mixing pattern content with alignment state. Existing temporal and spectral forecasting models have improved pattern extraction, and recent phase-aware methods further introduce spectral phase into temporal modeling. However, using phase is not equivalent to representing it as an alignment state. Spectral phase is a circular variable associated with temporal shifts rather than an unconstrained scalar, while the stability of an observed phase direction is related to the amplitude of the corresponding Fourier coefficient. In this work, we propose PaCo, an evidence-supported phase coordinate conditioning framework for time series forecasting. PaCo constructs spectral phase coordinates that preserve circular phase direction and incorporate amplitude-derived evidence for the observed direction. These coordinates serve as local alignment states to condition temporal representations and guide variable-level context modeling. Experiments on public long-term forecasting benchmarks and an industrial Wastewater Treatment Aeration (WWTA) dataset show that PaCo achieves competitive forecasting performance, while ablation and phase-response analyses support the importance of the proposed evidence-supported phase coordinate design. The code is available at https://anonymous.4open.science/r/Anonymization-68A4/.