Parameter-efficient linearization turns a pretrained autoregressive Transformer into an efficient model by hybridizing sliding-window attention (SWA) with a constant-size recurrent memory. But does every layer's recurrent state contribute to long-context retrieval? Investigating an SWA and Test-Time Training (TTT) hybrid, we find that retrieval is sparse across depth, concentrated in a few anchor layers, and that the remaining states hold similar information expressed through layer-specific transformations.

We exploit both observations in Anchor: TTT memory placed only at the anchor layers, and a cross-layer recurrent state written once and read by many layers, each through its own learned read map that realizes a layer-specific stretch and rotation, along with demonstrated generalization to other hybrid attentions. On Llama-3.1-8B and Mistral-7B, Anchor delivers 2.3-3.1x higher recall and 6.7-8.9 points higher MMLU accuracy than LoLCATs with a 10x smaller recurrent state, retains 91-95% of the teacher's MMLU accuracy, and cuts end-to-end latency by up to 4.3x relative to softmax attention.