State Space Language Models

Mamba-2 compresses all context into a fixed-size recurrent state that dominates decoding memory and bandwidth. Existing compressors quantize state values or prune state coordinates, scoring each channel independently in the native basis. We examine trained Mamba-2 states and find that energy is almost rank-one, but the output does not read the energetic directions: energy concentrates in a bias-like direction accumulated by the recurrence, while the content the output reads lives in a low-dimensional subspace that is not aligned with the state's coordinates. Even output-aware scores are computed coordinate by coordinate, so they miss the directions that determine the input--output map. Compression should therefore target the write--read subspace rather than individual coordinates. We propose \method{}, a post-training transform that instantiates balanced truncation with full controllability and observability Gramians and applies the rotation exactly via Mamba-2's scalar-identity transition. Because the useful rank varies widely across layers, we further introduce HARP, which allocates ranks per layer from the Hankel spectrum and protects retrieval-sensitive layers, improving both perplexity and retrieval over uniform budgets. Calibration takes a single forward pass, an order of magnitude faster than GHOST's released protocol, with lower peak memory. With HARP, \method{} halves the state dimension at an average WikiText-2 perplexity increase of $0.3$ points ($1.8\%$) across 130M--2.7B, versus $1.6$ ($9.3\%$) for GHOST, and the gap widens at 25% retention.

Long-context models often use answer-bearing evidence less effectively when it appears away from a prompt boundary. We compare this position bias across Transformer, state-space, and hybrid Mamba architectures by moving one gold document through a fixed set of nine distractors. The evaluated Transformers show a beginning advantage that the evaluated state-space models lack, while both families favor evidence near the end. A matched pure-versus-hybrid Mamba comparison points in the same direction but remains statistically uncertain. The family difference appears among models that can answer the task. Transformer primacy reappears on synthetic retrieval, where state-space ceiling prevents a family comparison. A corpus swap within one state-space architecture changes overall accuracy without a detectable change in curve shape. Prompt changes can create or nearly remove a beginning advantage. Position remains decodable across architectures even when response curves differ, and attention concentration near initial tokens covaries with beginning sensitivity in the tested Transformer family. These exploratory results show that effective context use depends on the evaluated architecture family, capability, task, and prompt, without isolating the sequence mixer as the sole cause.

State space models (SSMs) combine fast inference and fixed state size with competitive language-modeling performance. Recently, state souping has been proposed as a procedure whereby an SSM independently encodes chunks of tokens like documents, tool definitions, or memory records. The resulting state encodings may be cached and recombined to cheaply construct a context for downstream queries. However, both state souping and vanilla SSMs deteriorate on long, crowded contexts, limiting their utility for long-horizon domains. To address this issue, we propose weighted state souping, a simple method that learns query-dependent weights for composing cached states. The resulting models improve long-context performance substantially across both single- and multi-hop document question answering, needle-in-a-haystack retrieval, variable tracking, and tool use. Our method can be further adapted into an efficient sparse variant, yielding near-constant encoding cost with respect to context length while preserving (and sometimes improving) performance. We also present simple theoretical arguments that illustrate why souping works and when it may fail, and discuss informally the advantages of weighted souping. Weighted state souping thus combines reusable context representations with selective retrieval, offering an efficient approach to language modeling over large collections of documents, tools, and other forms of structured context that increasingly dominate modern agentic workloads.

State space models provide efficient linear-complexity processing for long sequences, but their performance frequently degrades as context length grows. We propose SuPER-Mamba, a training-free framework that uses Surprisal as a model-intrinsic signal for long-range state routing in pretrained Mamba models. SuPER-Mamba computes position-wise Surprisal from next-token predictions, aggregates neighboring responses through Local Pooling, and uses the resulting scores to prioritize Global-Channel updates without query-dependent scoring or additional training. We evaluate SuPER-Mamba on three state-space models across the LongBench-E benchmark. The results show that SuPER-Mamba improves the average score from 8.06 to 15.69, corresponding to a 94.7% relative gain over the Base Models and a 13.1% improvement over the state-of-the-art prior baseline. Ablation and length-wise analyses further support the effectiveness of the proposed routing framework.

State Space Models (SSMs) compress sequence history into a bounded recurrent state, making the resulting memory law a central architectural choice for long-context performance. Most modern SSMs rely on ODE-based dynamics that lead to exponential forgetting, limiting their ability to retain information over broad temporal ranges. We introduce FRAC, a selective SSM architecture derived from fractional dynamics that replaces this exponential decay with power-law long memory. To make fractional dynamics practical, FRAC approximates the heavy-tailed target kernel with a finite-state, log-spaced sum of exponential modes. This construction turns fractional memory into an efficient recurrent module with parallel training and prefill, while retaining bounded-state autoregressive decoding. Extensive experiments, including 1.3B-parameter language modeling, demonstrate that FRAC consistently improves long-context performance over state-of-the-art SSM baselines while staying competitive on short-context. These results show that fractional dynamics provide a practical and effective prior for long-context SSMs.

Can the global-workspace account of transformer representations extend to state-space language models? We fit Jacobian lenses to the residual streams and recurrent states of Mamba-1, Mamba-2 and Mamba-3, using the original 1000-prompt recipe. Joint residual--state readouts improve recovery of known intermediate concepts over the residual lens on at least five of six task families in every tested Mamba checkpoint. On Mamba-2, state alone exceeds residual and logit lenses on all six families; a normalised joint readout improves on both components on five. Temporal maps and word-list experiments show earlier content remaining state-readable as residual visibility changes. We also propose sign-guarded steering, which improves target top-five success over coordinate exchange on matched verbal-report trials in five models. Recurrent state alone supports this verbal access. These gains do not extend consistently to relational answers: guarded edits often output the edited concept itself, and Mamba-3's joint edits can disrupt successful state-only redirection. Recurrent state thus provides a complementary carrier of workspace content, whose recovery, persistence and causal uses require separate measurements.

We study the problem of demonstration selection, which involves selecting a subset of examples for prepending to a query to a language model. This problem is closely related to in-context learning and language model inference. Since the inference cost of a transformer model scales quadratically with sequence length, the selection problem becomes especially challenging in a long-context scenario. In this paper, we tackle this problem by building on state space models (SSMs), which require only linear inference time given the input. Our approach involves two algorithms. The first learns a small set of SSMs through distillation of a (trained) transformer model. We partition all the layers into consecutive groups. Then for each group, we estimate a separate state space model to replicate the input-output behavior within the adjacent layers. Second, we map the distilled model outputs to a small set of tokens, and apply these embeddings for demonstration selection in downstream applications. We perform extensive experiments in both synthetic and real-world datasets to validate our approach. We demonstrate that the distilled SSMs only incur an approximation error of less than $0.7\%$ relative to the true output. In downstream evaluation, we show that on several text classification and reasoning tasks, our approach reduces FLOPs by $14.2\times$ and improves accuracy by $6.48\%$ relative to baseline demonstration selection methods.

Transformer-based language models rely on self-attention, whose computation is permutation-equivariant and therefore lacks an intrinsic mechanism for representing token order. Existing architectures address this limitation by explicitly incorporating positional information through learned positional embeddings or hand-crafted positional encodings, such as rotary positional encoding (RoPE), treating positional information as an architecturally acquired capability rather than an inherent property of the model. Motivated by the pursuit of positional-encoding-free architectures, this work explores a language model architecture that integrates causal state-space equations to implicitly encode positional information before attention computation. Specifically, each model block applies a causal state-space equation before self-attention, allowing recurrent state dynamics to encode sequential information into token representations. Consequently, subsequent attention layers operate on position-aware representations without requiring explicit positional encodings while retaining the expressive modeling capacity of self-attention. We present ZetaGPT, a compact hybrid language model designed for research, rapid prototyping, algorithm verification, and educational applications. In addition to the proposed architecture, ZetaGPT provides a fully open-source, end-to-end training pipeline encompassing dataset construction, tokenizer training, pretraining, supervised fine-tuning, reinforcement learning from human feedback (RLHF), and chain-of-thought (CoT) reasoning via pure reinforcement learning. To the best of our knowledge, ZetaGPT is the first open-source small language model without explicit positional encoding and establishes a compact, reproducible reference implementation for the development and empirical study of positional-encoding-free language models.

Low-rank adaptation introduces a static learned update applied identically to every input. The update provides task-level adaptation but does not explicitly represent token-level or instance-level state variation. A family of adapters is proposed that introduces selective state-space control at two complementary granularities. At the token level, MaLoRA (Mamba-modulated low-rank adaptation) makes the adapter's scaling factor a dynamic input-dependent function with recurrent state across tokens, in contrast to the stateless modulators of prior work. The token-level adapter improves over low-rank adaptation. On the other hand, it differentiates tokens by structural role but not by contextual relevance, which motivates placing evidence selection at the context level. At the context level, MaRA (Mamba Retrieval Adapter) tracks cross-segment reasoning state and selects the segments most relevant to the query. State-space controlled retrieval of approximately three million parameters exceeds an eight-billion-parameter dense retriever on supporting-paragraph recall. Although base models perform poorly on the task without adaptation (14 to 25 F1), MaRA recovers the evidence relevance latent in their representations. Across three frozen backbones and two multi-hop reasoning benchmarks, the end-to-end family improves reasoning accuracy on every cell of the 3-by-2 grid, by +6.4 F1 (+10.0% relative) on average over the LoRA baseline.

Recent work on looped language models suggests that many reasoning problems benefit from greater computational depth rather than from additional independent parameters. Existing studies, however, focus almost exclusively on Transformer backbones, leaving open whether this principle also applies to state-space language models. We investigate Looped Mamba and Looped Hybrid Mamba-Transformer architectures, which repeatedly apply a shared Mamba (or hybrid) block to introduce explicit finite-depth recurrent computation. On two controlled reasoning tasks-Mano (modular-arithmetic manipulation) and p-hop induction-Looped Mamba consistently outperforms parameter-matched non-looped baselines and, in several settings, matches or exceeds non-looped models of equal effective depth. We then extend the study to language model pre-training under matched iso-parameter and iso-FLOPs protocols, which jointly disentangle the effects of parameter sharing and effective depth: looped models remain competitive on downstream benchmarks with substantially fewer distinct parameters, although deeper non-looped models retain an advantage in validation perplexity under strict iso-FLOPs comparisons. Finally, we adapt Ouro's two-stage exit gate to Looped Mamba for threshold-controlled selection among recurrent-step outputs. Executing such exits on a state-space backbone, however, leaves the recurrent state without its deeper updates, and validation perplexity then degrades severely. We therefore introduce a cache-hole adaptation that aligns continued training with skipped-state inference. At the scales studied, the adapted model keeps perplexity close to full computation and matches or exceeds full-compute exit-state selection on downstream benchmarks while executing roughly half of the recurrent steps, which translates into measured inference speedups once the prefill is compute-bound.

We present Harmonic, a hierarchical state space model (SSM) for language modeling. The architecture stacks three recurrent levels at progressively slower timescales; each level receives the prediction error of the level below as input, rather than its raw hidden state. On enwiki8 with equal token budgets, Harmonic outperforms a comparable Transformer (28M params) by +1.4% at 1K tokens, +6.7% at 8K tokens, and +11.4% at 32K tokens (bpt, lower is better). It also outperforms Mamba at every tested length by 0.7--1.8%. At 64K tokens, both Mamba and Transformer run out of memory on an 80GB H100; Harmonic trains successfully, reaching 6.169 bpt. Results replicate on WikiText-103 (H-TF gap +1.7% to +7.2% across 1K--32K). At 1B parameter scale, replacing all attention layers in TinyLlama 1.1B with HarmonicBlock eliminates the RoPE positional encoding limit: the resulting Hallamonic model maintains stable loss across sequence lengths 1K--8K on two independent clean benchmarks (Lambada and fineweb-edu held-out), while TinyLlama degrades catastrophically past its 2K-token RoPE limit (gap: +9.4 bpt at seq=8K on Lambada). Compute is O(L) per forward pass vs. O(L^2) for attention. Logs: https://github.com/Omibranch/harmonic-logs.

Manifold-Constrained Hyper-Connections (mHC) introduce a stability-motivated variant of multi stream residual mixing by constraining residual stream mixing matrices to the manifold of doubly stochastic matrices via Sinkhorn-Knopp projection. In his work, we study whether mHC-style constrained multi-stream residual topology transfers effectively to state space model (SSM) language modeling. We implement a static mHC mechanism around an SSM block by expanding the residual stream into multiple parallel streams, aggregating streams into a single SSM input through simplex-constrained pre-mixing, scattering the SSM output back to streams through simplex-constrained post-mixing, and applying Sinkhorn-projected residual stream mixing at each layer. We further introduce stream-specialized adapters that add lightweight stream-specific capacity through a shared bottleneck with per-stream scaling, applied both before stream aggregation and after the SSM output prior to scattering. We evaluate baseline single-stream SSM, static mHC SSM, and mHC SSM with adapters on WikiText-2 using identical training settings and report checkpoint-based validation loss, perplexity, throughput, and peak GPU memory. Under the reported fair checkpoint evaluation, static mHC improves validation loss from 6.3507 to 6.2448 and reduces perplexity from 572.91 to 515.35, while mHC with adapters further improves validation loss to 6.1353 and perplexity to 461.88. These gains are accompanied by modest throughput reductions from 1025.52 to 964.81 and 938.90 tokens per second, and increased peak memory from 2365 MB to 2568 MB and 3092 MB. The results suggest that mHC-inspired constrained multi-stream residual mixing can yield measurable quality improvements in SSM language models and that stream-specialized adapter capacity can further enhance performance with predictable efficiency tradeoffs.

Long chain-of-thought reasoning and agentic tool-calling produce traces spanning tens of thousands of tokens, yet Transformer KV caches grow linearly with sequence length, creating a memory bottleneck on commodity hardware. State-space models offer constant-memory recurrence but suffer a memory cliff: retrieval accuracy collapses once the gap between a stored fact and its query exceeds the effective horizon of the recurrent state. We introduce Echo, a KV-cache-free associative recall architecture built around Spectral Koopman Attention (SKA); a drop-in replacement for attention layers that augments SSM blocks with a closed-form dynamical operator whose sufficient statistics are accumulated in constant memory with no KV cache. Echo fits a spectral linear system to the key and value history via kernel ridge regression and retrieves through a learned power-iterated filter, all from $O(r^{2})$ streaming state where $r$ is a small projection rank. On the Multi-Query Associative Recall benchmark, a pure Mamba-2 SSM fails to exceed chance accuracy (${\sim}3\%$) across all gap lengths and KV-pair counts, while at the 50M parameter scale SKA-augmented models achieve $100\%$ retrieval accuracy on every configuration tested, including distractor gaps of $4{,}096$ tokens with $32$ KV pairs. Across five additional transfer benchmarks including needle-in-a-haystack, tool-trace, and multi-hop retrieval, SKA consistently outperforms both pure SSM and SSM+Attention hybrids while maintaining constant inference memory. Ablations confirm that the spectral operator, not the prefix masking strategy, drives the retrieval gain.

Modern sequence modeling is dominated by two families: Transformers, whose self-attention can access arbitrary elements of the visible sequence, and structured state-space models, which propagate information through an explicit recurrent state. These mechanisms face different limitations on long contexts: when attention is diffuse, the influence of individual tokens is diluted across the effective support, while recurrent state propagation can lose long-range sensitivity unless information is actively preserved. As a result, both mechanisms face challenges in preserving and selectively retrieving information over long contexts. We propose Sessa, a decoder that places attention inside a recurrent feedback path. This creates many attention-based paths through which past tokens can influence future states, rather than relying on a single attention read or a single recurrent chain. We prove that, under explicit assumptions and matched regimes, Sessa admits power-law memory tails $O(\ell^{-β})$ for $0 < β< 1$, with slower decay than in the corresponding Transformer and Mamba-style baselines. We further give an explicit construction that achieves this power-law rate. Under the same assumptions, Sessa is the only model class among those considered that realizes flexible selective retrieval, including profiles whose influence does not decay with distance. Consistent with this theoretical advantage, across matched experiments, Sessa achieves the strongest performance on long-context benchmarks while remaining competitive with Transformer and Mamba-style baselines on short-context language modeling.

Language models cannot be random. This paper introduces Entropic Deviation (ED), the normalised KL divergence between a model's token distribution and the uniform distribution, and measures it systematically across 31,200 generations spanning seven models, two architectures (transformer and state space), nine prompt categories, three temperatures, and five languages. Under semantically neutral prompts (empty strings, random characters, nonsense syllables) transformers still exhibit ED of approximately 0.30, meaning that 88-93% of the non-randomness observed under semantic prompts is intrinsic to the learned weights rather than induced by context. Three transformer families (Gemma, Llama, Qwen) converge on nearly identical ED values despite different training data and vocabularies. A state space model (Mamba2) reveals a qualitatively different regime: twice the ED, three times lower within-sequence variance, and massive sensitivity to temperature (r = -0.78) where transformers are nearly immune (r < 0.05). Cross-lingual experiments with Qwen-32B show a stable gradient across five languages (English, Japanese, Chinese, Polish, Arabic) that does not correlate with token fertility and persists when two languages sharing an identical tokeniser subset are compared. These findings establish a structural lower bound on randomness in pretrained language models, characterise how this bound differs across architectures, and demonstrate that language itself modulates the bound independently of tokenisation.

Scaling inference-time compute has emerged as an important driver of LLM performance, making inference efficiency a central focus of model design alongside model quality. While the current Transformer-based models deliver strong model quality, their quadratic compute and linear memory make inference expensive. This has spurred the development of sub-quadratic models with reduced linear compute and constant memory requirements. However, many recent linear models trade off model quality and capability for algorithmic efficiency, failing on tasks such as state tracking. Moreover, their theoretically linear inference remains hardware-inefficient in practice. Guided by an inference-first perspective, we introduce three core methodological improvements inspired by the state space model (SSM) viewpoint of linear models. We combine: (1) a more expressive recurrence derived from SSM discretization, (2) a complex-valued state update rule that enables richer state tracking, and (3) a multi-input, multi-output (MIMO) formulation for better model performance without increasing decode latency. Together with architectural refinements, our Mamba-3 model achieves significant gains across retrieval, state-tracking, and downstream language modeling tasks. At the 1.5B scale, Mamba-3 improves average downstream accuracy by 0.6 percentage points compared to the next best model (Gated DeltaNet), with Mamba-3's MIMO variant further improving accuracy by another 1.2 points for a total 1.8 point gain. Across state-size experiments, Mamba-3 achieves comparable perplexity to Mamba-2 despite using half of its predecessor's state size. Our evaluations demonstrate Mamba-3's ability to advance the performance-efficiency Pareto frontier.

State-space models (SSMs) have emerged as an efficient strategy for building powerful language models, avoiding the quadratic complexity of computing attention in transformers. Despite their promise, the interpretability and steerability of modern SSMs remain relatively underexplored. We take a major step in this direction by identifying activation subspace bottlenecks in the Mamba family of SSM models using tools from mechanistic interpretability. We then introduce a test-time steering intervention that simply multiplies the activations of the identified bottlenecks by a scalar. Across 7 SSMs and 6 diverse benchmarks, this intervention improves performance by an average of 8.27%, without requiring any task-specific tuning. Finally, we validate that the identified bottlenecks are indeed hindering performance by modifying them to yield an architecture we call Stable-Mamba, which achieves long-context performance gains when retrained from scratch.

State space models (SSMs) like Mamba offer efficient alternatives to Transformer-based language models, with linear time complexity. Yet, their adversarial robustness remains critically unexplored. This paper studies the phenomenon whereby specific short input phrases induce a partial amnesia effect in such models, by irreversibly overwriting information in their hidden states, referred to as a Hidden State Poisoning Attack (HiSPA). Our benchmark RoBench-25 allows evaluating a model's information retrieval capabilities when subject to HiSPAs, and confirms the vulnerability of SSMs against such attacks. Even the recent Jamba-1.7-Mini SSM--Transformer (a 52B hybrid model) collapses on RoBench-25 under some HiSPA triggers, whereas pure Transformers do not. We also observe that HiSPA triggers significantly weaken the Jamba model on the popular Open-Prompt-Injections benchmark, unlike pure Transformers. We further show that the theoretical and empirical findings extend to Mamba-2, and also analyse a Mamba-2-based hybrid (Nemotron-3-Nano). Finally, our interpretability study reveals patterns in Mamba's hidden layers during HiSPAs that could be used to build a HiSPA mitigation system. The full code and data to reproduce the experiments can be found at https://github.com/TortueSagace/hispa.

State Space Models (SSMs) have emerged as efficient, linear-time alternatives to Transformer-Based Models (TBMs) for long-context processing. However, their different mechanisms raise an important question: how does contextual representation flow across depth differ between these model families and their hybrids? We examine this question across twenty-one pretrained models, combining token-wise spectral diversity, layer-wise variation on seven datasets, and answer-location probing on two retrieval tasks. This joint analysis reveals that TBMs retain most of their token diversity at the output, whereas SSMs reach richer intermediate diversity but retain less of it on average, with late-layer compression. Comparisons with reinitialized weights reveal that these profiles depend on pretrained weights, while hybrids combine strong answer-location accessibility with architecture-dependent geometry, showing that high accessibility need not require the steadiest representation flow. Finally, we conduct theoretical case studies connecting these contrasts to mechanism-level differences: query-time token mixing in TBMs, repeated recurrent updates in SSMs, and preserved attention updates in hybrids.