Shared Phase and Retention Control for Efficient Adaptive Spectral Recurrence

As new evidence arrives, a sequence model must update what it remembers and how memory influences predictions. While Transformers incur computation and cache costs scaling with context length, fixed-state recurrent models offer constant-memory inference. However, linear and spectral recurrences traditionally rely on static transitions, failing to dynamically revise how stored representations decay or rotate. While recent selective architectures introduce input-dependent transitions, they assign independent controls to every memory mode, coupling control cost to state capacity. We show that high-dimensional spectral memory does not require high-dimensional control, and introduce Shared Phase and Retention Control for Efficient Adaptive Spectral Recurrence (SPARC). SPARC employs just two input-dependent scalar signals to coordinate memory retention and phase rotation across heterogeneous complex modes, while preserving mode-specific baseline timescales and frequencies. Its diagonal affine recurrence supports parallel associative scans for sequence-level BPTT as well as exact structured Real-Time Recurrent Learning (RTRL) for online credit assignment. Across partially observable continuous control, POPGym, and sequence classification, SPARC achieves a 9.09% relative return improvement on Walker-P and a 1.36% relative accuracy gain on FordA over second-best methods. On an NVIDIA Blackwell GPU, our implementation reduces recurrent-mixer training latency by 18.2%-34.2% in fixed-token workloads and accelerates scans by 3.1x-4.7x over an optimized RG-LRU baseline. These results show that two shared control signals can efficiently govern adaptive spectral memory across online and full-sequence settings. Code is available at https://github.com/Botwwt/sparc.