Breaking the KV Cache Bottleneck: Fan Duality Model Achieves O(1) Decode Memory with Superior Associative Recall

We present FDM (Fan Duality Model), a linear sequence architecture that resolves the fundamental tension between memory efficiency and associative recall in sequence modeling. FDM separates sequence processing into two components: a wave component (recurrent scan via phase-preserving Givens rotations) that compresses long-range patterns into a fixed-size complex hidden state, and a particle component (local-global cache) that retrieves specific tokens via learned associative addressing with W+K=272 slots independent of sequence length N. This yields strictly O(1) decode memory: 867 MB fixed across all prompt lengths 128-8,192 tokens, versus Transformer's 853-4,247 MB (4.9x reduction at N=8,192). Beyond the architecture, we discover that jointly training the wave and particle components leads to suboptimal convergence. We propose Freeze-Scan, a two-phase training strategy that freezes the recurrent scan and optimizes the cache jointly with embeddings, achieving PPL=64.9 on WikiText-103 in 44K steps -- a 7.5x improvement over full fine-tuning (PPL=487). On Multi-Query Associative Recall (MQAR), FDM achieves 0.966 accuracy, surpassing Transformer (0.606) by 59.5%, while pure scan without cache scores only 0.011, confirming the necessity of the particle component. Finally, we introduce Holographic Reference Beam Decoding, interpreting the complex hidden state h_t as a holographic plate encoding the entire temporal history. Using the current input x_t as a reference beam to modulate h_t reduces PPL by up to 2.13 points (PPL=62.79) with a 4-head orthogonal reference beam using only 1.3M additional parameters, providing empirical support for the holographic interpretation. Code and pretrained weights: https://github.com/YasongFan/FDM

MIPT-SSM: Scaling Language Models with $O(1)$ Inference Cache via Phase Transitions

We present MIPT-SSM, a neural sequence architecture built on the physics of Measurement-Induced Phase Transitions (MIPT). The central idea is a learned measurement rate $p_{t}\in(0,1)$ that routes computation between two regimes: wave phase $(p_{t}\rightarrow0)$, where information propagates as distributed complex-phase interference; and particle phase $(p_{t}\rightarrow1)$ where the state collapses onto the current token, enabling precise local storage. These two regimes are provably incompatible in a single linear operator one of the few "no-go theorems" in sequence modeling and $p_{t}$ is our way around it. The model is predicted to exhibit a phase transition at critical sequence length $N^{*}\approx1024$, where the information density ratio $N/D$ crosses unity, consistent with our memory scaling observations. On AG News (four-class classification), MIPT achieves 0.905 accuracy versus Transformer's 0.736 (+16.6%), stable across 3 seeds. At $N=8192$ MIPT requires 810 MB versus Transformer's 34,651 MB a 42.8x memory reduction. On exact-recall ("needle-in-a-haystack"), our causal sparse KV cache achieves 0.968 accuracy. Remarkably, under unbounded cache capacity, the $p_{t}$ gate autonomously learns to store only the single critical token (averaging $1.0/512$ slots used), filtering out all noise and achieving a 99.8% sparsity rate. On language modeling (WikiText-103, 31M parameters), MIPT-LM with $K=64$ cache reaches PPL 92.1 versus Transformer's 90.5 (gap: 1.8%) while inference KV cache shrinks from $O(N)$ to $O(64)$.