Accurate single-object tracking and short-term motion forecasting remain challenging under occlusion, scale variation, and temporal drift, which disrupt the temporal coherence required for real-time perception. We introduce \textbf{SOTFormer}, a minimal constant-memory temporal transformer that unifies object detection, tracking, and short-horizon trajectory prediction within a single end-to-end framework. Unlike prior models with recurrent or stacked temporal encoders, SOTFormer achieves stable identity propagation through a ground-truth-primed memory and a burn-in anchor loss that explicitly stabilizes initialization. A single lightweight temporal-attention layer refines embeddings across frames, enabling real-time inference with fixed GPU memory. On the Mini-LaSOT (20%) benchmark, SOTFormer attains 76.3 AUC and 53.7 FPS (AMP, 4.3 GB VRAM), outperforming transformer baselines such as TrackFormer and MOTRv2 under fast motion, scale change, and occlusion.