S³‑Delta: Spectral‑Stratum SpatioTemporal Delta A Scale‑Reliable Operator for Long‑Horizon Fuzzy Spatial Memory

Abstract

Long sequences with coordinates obey two different orders: observations arrive in time, while memories should be reactivated by physical location. For an agent that continuously scans a two- or three-dimensional space, a useful fixed-capacity memory should fuzzily recall nearby history at the current position. Spatially closer memories should usually respond more strongly, older memories should gradually fade, and a useful memory should become easier to retrieve again after a revisit. Standard fast weights can recall key-value associations with one matrix multiplication, but uniform decay cannot determine which spatial scales remain reliable. The conventional Delta rule corrects the total error at the current key, yet treats all frequency components as one indivisible estimate. We propose Spectral-Stratum SpatioTemporal Delta (S³-Delta, written S3-Delta in plain text). Its three "S" terms denote Spectral, Stratum, and SpatioTemporal. Each sine-cosine pair in a fixed Fourier positional key is treated as a wavelength stratum with its own 2 × d_v fast-weight state. At a new position, all strata first decay according to the actual elapsed time and jointly recall history. Once the complete current observation becomes available, the operator asks a counterfactual question for every stratum: would removing this stratum increase or decrease the error?. The answer accumulates as signal and noise evidence and forms an analytic reliability gate for future readout. The strata do not independently guess the write target. Instead, one shared raw Delta residual coordinates all writes and retains an explicit fitting property at the current address. The resulting operator closes the loop of recall, verification, and consolidation without an additional learned gating network. Across variable-density two-dimensional scans from 16K to 262K events, we compare S³-Delta with a tuned dense Delta baseline. The proposed operator wins all 30 paired tests across static and slowly changing environments. Mean error is reduced by 3.85%-7.23%, and the lower bound of every five-seed bootstrap confidence interval remains positive. Ablations at 262K show that counterfactual scale reliability is the main source of improvement while wavelength-conditioned lifetimes provide a smaller but consistent contribution. With 32 Fourier pairs, the spatial kernel reaches 93.4% radial monotonicity and a near-to-far response ratio of 7.00. Continuous-time decay is invariant to regrouping irregular steps up to errors of about 10^{-14}. State capacity is independent of sequence length: the main state remains O(d_k d_v) and reliability adds only d_k scalars.