Time series forecasting (TSF) plays a vital role across various domains such as finance, energy, healthcare, and meteorology. Currently, most deep learning based TSF methods typically operate with a fixed lookback window. This approach comes from the high compute and memory costs of long contexts, as well as the standard practice of using sliding windows. This creates a trade-off. Making the window larger reduces the number of training samples, which can harm stability and generalization. However, keeping the window small prevents the model from using long history during inference. We propose an inference-only streaming autoregressive framework that replaces repeated full-context recomputation with a one-time context warmup and incremental decoding, enabling efficient long-history forecasting without retraining. While straightforward caching attentions is brittle for time series due to distribution shifts and noisy or redundant histories, we address these issues with cache-consistent normalization and selective memory under a fixed cache budget. Across diverse benchmarks, our approach substantially reduces inference latency with no or marginal accuracy loss, and often improves performance when longer lookbacks are beneficial.