S³‑Delta: Spectral‑Stratum SpatioTemporal Delta

A Scale-Reliable Delta Operator for Long-Horizon Fuzzy Spatial Memory

Abstract

Coordinate-bearing streams obey two orders: observations arrive in time, whereas memories should be reactivated by physical location. We study fixed-capacity memory for agents that scan two- or three-dimensional spaces and must fuzzily recall previously observed values near the current position. Standard fast weights support one-matrix key-value recall, but uniform forgetting cannot identify reliable spatial scales, while the conventional Delta rule treats all Fourier components as one estimate.

We propose Spectral-Stratum SpatioTemporal Delta (S3-Delta). Each sine-cosine pair of a fixed Fourier key defines a wavelength stratum of one shared fast-weight matrix. Before each write, wavelength-conditioned continuous-time decay advances the state and the complete matrix produces one causal recall. After the current observation arrives, removing each stratum counterfactually measures whether it reduced or increased the recall error. Accumulated signal and noise evidence forms a reliability gate for later readout, while one normalized Delta residual updates the complete matrix and preserves a shared target across scales.

Across variable-density two-dimensional scans from 16K to 262K events, S3-Delta outperforms tuned dense Delta in all 30 seed-level comparisons over static and slowly changing environments, reducing mean error by 4.73% to 8.99%. It also improves by 4.46% to 8.65% over an additionally trained causal Gated-Delta control. At 262K, ablations identify counterfactual reliability as the main source of improvement and wavelength-conditioned decay as a smaller, consistent contribution. With 32 Fourier pairs, the spatial kernel attains 93.4% radial monotonicity. The main state remains O(d_k d_v) and is independent of sequence length. These results support S3-Delta as an interpretable, fixed-capacity operator for long-horizon fuzzy spatial memory.