Spatial-FWP Mamba

A Spatiotemporally Decoupled Memory Architecture with a Causal Trajectory Prior and Multiscale Frequency Forgetting

Abstract

Sequence models compress history in token order, but sequence order does not always agree with physical proximity: two observations separated by many steps may come from the same location, whereas consecutive observations may be separated by a large spatial displacement. For scanning tasks with physical coordinates in two or three dimensions, we propose Spatial-FWP Mamba, which maintains temporal-semantic memory separately from coordinate-addressed spatial memory. The Mamba branch models temporal dependencies and semantic representations, while the Spatial-FWP branch writes observation summaries into a fixed-size associative matrix and uses the current position key to recall historical information from the same or nearby locations.

The spatial branch relies on an explicit assumption: the scan remains inside a bounded workspace known in advance, and after coordinate normalization and frequency configuration, multiscale sinusoidal position keys exhibit reliable local similarity within that domain. Under this condition, position-triggered recall aggregates relevant history with one matrix-vector product, without constructing a sliding window or storing and traversing all previous coordinates. To regulate updates to the fixed-capacity memory, we introduce a causal trajectory-prior matrix, referred to as the Prior Hologram (PH), that records historical coordinates only. Querying PH with the current position produces a multiscale summary of the historical position distribution around that point. This query result and the current displacement are processed by a lightweight geometry-only gating network to produce a pair-shared diagonal forgetting gate and a scalar write gate. Observation content is used only as the value stored in memory and does not participate in spatial gating. Each spatial update applies forgetting first, then reads the historical state, and finally writes the current observation. The recurrence can be implemented as a PH prefix scan followed by a diagonal-affine FWP prefix scan. We present the formulation, complexity, applicability boundaries, and an evaluation protocol without reporting unverified performance figures.