Constant-Memory Recall: Learned Associations in a Fixed Matrix State

Fixed-state recurrences--linear attention and state-space models--are reported to lag behind attention on associative recall, but whole-architecture comparisons cannot say which ingredient is responsible. We decompose masked multi-query recall at a fixed state budget along three single-knob axes: a short causal convolution, the transition structure (rank-1 delta rule vs. diagonal), and decay. The convolution dominates (~+0.5 recall in both families under matched training): comparisons that pit convolution-free cells against a convolution-equipped Mamba measure the missing convolution, not the recurrence. The rank-1 transition beats its diagonal ablation by +0.19/+0.32 at 16/32 pairs, but the margin shrinks to +0.03 once both cells carry the convolution, and a state-matched Mamba-2 ties the unarmed rank-1 cell: no class claim survives. Cells that solve 32-pair recall degrade gracefully with load yet fall to chance retrieving 4 pairs from a distractor haystack--flat across lengths and transitions. Interference under sparse supervision, not capacity: a distance curriculum takes the unchanged architecture from 0.021 to 1.000. Training is a lock-in lottery--a seed either locks in or does not--and the curriculum is the lever. Lock-in rises from 1/10 to 7/10 (p=0.02); dense supervision adds nothing; at L=256 a shaped ramp reopens a boundary the uniform curriculum cannot (4/5 vs. 0/9); and at L=512, where the ramp collapses (0/6), gating it on measured accuracy locks in 6/6 (p=0.001). Bidirectional denoiser cells, reading the query before the haystack, show no measurable advantage over causal training (ten seeds), and collision-key retrieval needs two layers. Arming for recall is free on an S_5 state-tracking guardrail--the armed cell is significantly better at every depth (p<=0.0044). These replace "recurrent models are bad at recall" with a measured decomposition and two cheap interventions.

Attention provides direct access to past representations, but retaining an ever-growing history is costly. Recurrent models bound persistent state, yet must preserve selected information while processing subsequent inputs. We introduce SchemaMem, an attention-based recurrent memory architecture combining chunk-local attention with a persistent, schema-indexed phase state. Learned schema embeddings provide a shared representational reference for reading and writing. Reads use the current state, whereas writes use the layer input and static schema embeddings, excluding direct feedback from that layer's own state. Chunk-boundary commits aggregate bounded phase increments through forward computation. The same parameters also support full-history attention training before and during recurrent training. We studied selective updates, preservation, and delayed retrieval in a controlled address--value task, comparing three-layer models with approximately matched parameter counts and persistent-state dimensions. Across nine address/value settings and three training seeds, SchemaMem has higher mean written-value retention at four times the maximum training delay than both baselines, which are trained toward a higher in-range accuracy target. Updated-value recovery favors SchemaMem in all nine settings against Mamba-3 and seven against Gated DeltaNet. Defaults consistently favor Gated DeltaNet over SchemaMem at that delay, and SchemaMem requires substantially more optimization steps. These results identify a promising retention--optimization trade-off in schema-indexed recurrence.

Test-time memorization of sequential information is a foundational capability for any intelligent system that needs to rapidly store and retrieve data, as demonstrated by the attention mechanism in large language models. However, Transformers and linear attention variants are trained via backpropagation through time (BPTT), which has drawbacks such as its enormous memory requirements. Predictive coding (PC), an algorithm that optimizes a prediction-error energy function with local updates, offers a compelling solution but still lacks test-time memory for sequential data. We develop a solution to this problem by treating the per-step memory update $ΔS_t$ as a latent variable, as opposed to a slow weight to be learned. By taking the log probability of a standard PC-based probabilistic model, we derive a PC-native solution for verbatim sequence memory, and relate our update rule to common choices in the linear attention literature (e.g., DeltaNet). We present an extensive set of empirical tests to probe our system's properties, including its scaling capacity, memory footprint, and ability to infer a context from evidence (e.g., bank (river) vs. bank (money)). We show how DeltaNet-PC can be used for storing single sequences of data, multiple sequences of data, and key-value associative memory pairs with perfect recall accuracy.