WriteSAE: Sparse Autoencoders for Recurrent State

Abstract: We introduce WriteSAE, the first sparse autoencoder that decomposes and edits the matrix cache write of state-space and hybrid recurrent LLMs, where residual SAEs cannot reach. Existing SAEs read residual streams, but Gated DeltaNet, Mamba-2, and RWKV-7 write to a $d_k \times d_v$ cache through rank-1 updates $k_t v_t^\top$ that no vector atom can replace. WriteSAE factors each decoder atom into the native write shape, exposes a closed form for the per-token logit shift, and trains under matched Frobenius norm so atoms swap one cache slot at a time. Atom substitution beats matched-norm ablation on 92.4% of $n=4{,}851$ firings at Qwen3.5-0.8B L9 H4, the 87-atom population test holds at 89.8%, the closed form predicts measured effects at $R^2=0.98$, and Mamba-2-370M substitutes at 88.1% over 2,500 firings. Sustained three-position installs at $3\times$ lift midrank target-in-continuation from 33.3% to 100% under greedy decoding, the first behavioral install at the matrix-recurrent write site.