Autoregressive Transformer KV caches grow linearly with context length; sliding-window caching bounds memory but discards evicted tokens entirely, so relevant evidence outside the window becomes inaccessible. We introduce \emph{Tensor Cache}, a two-level cache that pairs sliding-window softmax attention as a first-level cache (L1) with a fixed-size outer-product fast-weight memory as a second-level cache (L2) fed by KV pairs evicted from the window. Recent tokens remain in exact local attention; evicted pairs are compressed into a per-layer matrix
A and read by future queries through a single matrix multiplication, exploiting the linear-attention identity
qt(ki⊗vi)=⟨qt,ki⟩vi. A learned scalar gate fuses the L1 and L2 outputs, and per-head decay and write-rate parameters are trained end-to-end. The outer-product memory and the read identity are well-known; our contribution is their use as an L2 cache fed exclusively by sliding-window evictions, plus identifying that the common chunked-mean training shortcut
A←λA+η(kˉ⊗vˉ) silently introduces
C2−C spurious cross-token outer products per chunk, and closing the gap with a parallel weighted-sum scan equivalent to per-token writes within float32 epsilon. Across systems scaling, controlled associative recall, long-context language modeling, and memory-capacity diagnostics, Tensor Cache improves the memory--quality frontier over bounded-state baselines.