The Topological Trouble With Transformers

Abstract: Transformers encode structure in sequences via an expanding contextual history. However, their purely feedforward architecture fundamentally limits dynamic state tracking. State tracking -- the iterative updating of latent variables reflecting an evolving environment -- involves inherently sequential dependencies that feedforward networks struggle to maintain. Consequently, feedforward models push evolving state representations deeper into their layer stack with each new input step, rendering information inaccessible in shallow layers and ultimately exhausting the model's depth. While this depth limit can be bypassed by dynamic depth models and by explicit or latent thinking that externalizes state representations, these solutions are computationally and memory inefficient. In this article, we argue that temporally extended cognition requires refocusing from explicit thought traces to implicit activation dynamics via recurrent architectures. We introduce a taxonomy of recurrent and continuous-thought transformer architectures, categorizing them by their recurrence axis (depth versus step) and their ratio of input tokens to recurrence steps. Finally, we outline promising research directions, including enhanced state-space models and coarse-grained recurrence, to better integrate state tracking into modern foundation models.

• The paper identifies state maintenance limitations in transformers due to vertical drift, leading to context-incoherent outputs.
• It demonstrates that the feedforward architecture prevents efficient iterative updates of latent states, impeding dynamic sequential reasoning.
• The study categorizes recurrent and hybrid designs as potential remedies to overcome depth-induced state saturation in transformer models.

The Topological Trouble With Transformers

Introduction

The expressivity and inductive bias of transformer-based architectures have positioned them as the dominant paradigms for sequence modeling in natural language processing. However, the fundamental topological constraints imposed by their feedforward design undermine their ability to perform dynamic state tracking—a core cognitive operation required for temporally extended reasoning and world modeling. This paper elucidates intrinsic architectural limitations in transformers regarding state maintenance, provides mechanistic diagnostics and failure examples, and systematizes the rapidly diversifying design space of recurrent and hybrid models, outlining theoretical and practical directions for remedying the serial shortcomings of modern foundation models (2604.17121).

Architectural Topology and State Tracking Deficit

Transformers manipulate input sequences through deep, feedforward stacks with cross-token interaction effected by multi-head self-attention. Although this enables expansive context windows and flexible associative retrieval, the architecture does not natively support the iterative update of a compact latent state $s_t = f(s_{t-1}, x_t)$ underlying recurrent neural computation. As new tokens are processed, transformers push updated state representations deeper into the stack. Over the course of a sequence, this “vertical drift” eventually saturates the available depth, leading to state representations that become inaccessible to subsequent shallow-layer computations.

Figure 1: Schematic depiction of the transformer stack as a barrier to progressive state propagation and illustration of the fundamental depth-induced state tracking limitation.

For sequential reasoning tasks, such as belief state maintenance in dialogue or context-sensitive word sense disambiguation, this topology precipitates failures. Downstream layers only partially inherit contextually disambiguated signals, causing the model to flip-flop between inconsistent world models. Even when sufficient context is available to resolve ambiguity, delayed convergence to the correct latent state can lead to erroneous output during rapid inference (see also analysis in [lepori2025]).

Figure 2: State representations resolved deep within the stack are unavailable to earlier layers during subsequent inference steps, causing context-incoherence in output.

Furthermore, the paper emphasizes that while theoretically, $\log n$-depth transformers suffice for certain regular language recognition tasks, these results are constructivist and do not imply that the necessary representations are easy to learn or robustly maintained across diverse, real-world tasks [merrill2025]. Empirical evidence points to practical fragility in belief updating and unreliable integration of temporally extended information [biran2024, sawyer2025, venhoff2025].

Workarounds and Explicit Thinking

A common mitigation strategy is to externalize interim computations as explicit "thought" tokens (chain-of-thought prompting [wei2022chain]), making inferred states re-accessible as inputs. This approach can alleviate depth bottlenecks in discrete steps of reasoning and support multi-hop inference.

However, the paper critiques this as a structurally inefficient hack, since basic semantic bookkeeping (such as semantic frame updating or referent tracking) should be implicit, rapid, and automatic. Resorting to language-like self-talk for microcognitive state is wasteful and can flood the context window with redundant content.

Recurrent Architectures: Expanding the Taxonomy

To circumvent topological limitations, the authors systematize both established and emerging “recurrent” transformer architectures:

• Depth Recurrence: Layers are re-applied over the same input (looped/universal transformers). While this increases effective depth and supports trainable multi-step reasoning, updated state representations still move upward and remain depth-limited.
• Step Recurrence: Within-layer recurrency permits information propagation along the sequential axis (as in SSM-based models like MAMBA [gu2024], canon layers [allenzhu2025], or RWKV [peng2025rwkv]), enabling unbounded state velocity horizontally.
• Depth + Step Recurrence: Complex hybrids involve recurrency both in layer and time axes (e.g., feedback transformers [fan2021], blockwise recurrent models, or latent-thought loops [hao2025coconut]). These designs can implement attractor dynamics or multi-step fixed-point solvers, further decoupling the bottleneck between representation depth and time.

  Figure 3: Equivalence of recurrent and feedforward neural topologies via unrolling; motivates how recurrence can be introduced into transformer stacks.

  

  Figure 4: Unrolling various recurrent transformer designs, highlighting how hybrid recurrence axes can in principle support unbounded state propagation.

  

  Figure 5: Illustration of latent-thought models, where multiple autoregressive steps operate internally before the next external token is ingested.

  

  Figure 6: State-space models propagate recurrence laterally within layers, directly linking representations across time steps.

This expanded taxonomy enables a structured analysis of which designs are capable of indefinite state tracking and which fundamentally inherit feedforward limitations.

Mechanistic and Empirical Evidence

Mechanistic interpretability (e.g., causal patching and Patchscope interventions [ghandeharioun2024, lepori2025]) demonstrates that delayed state resolution and shallow-layer inaccessibility frequently cause context-incoherent generations. The paper draws on both synthetic and natural examples, including ambiguity resolution and rule-based games, to highlight how depth-bound architectures produce inference failures that are resistant to shallow architectural tweaks.

Examples such as the inconsistent handling of polysemous terms (“bank” as riverbank vs. financial institution) empirically substantiate the impact of architectural constraints on real-world language understanding. The paper also reviews limitations in efficient training and the dangers of parallelized approximate “workarounds” (e.g., composition shortcuts, as discussed in [prakash2026, li2025, piotrowski2025, hu2025]).

Implications and Future Directions

Theoretical Implications

• Serial Scaling Hypothesis: The paper notes that massively parallel architectures (transformers) are mismatched to inherently serial tasks, supporting arguments from formal computational theory that establish parallelism/serialism expressivity tradeoffs [merrill-sabharwal-2023-parallelism, liu2026serialscalinghypothesis, Strobl2024].
• Topological Rigidity: Feedforward-only models, regardless of width and context window size, cannot express arbitrary state dynamics without explicit recurrence. Attempts to “hack” recurrence into feedforward stacks via depth, lookback, or shortcut solutions are inevitably limited by the fundamental information flow topology.

Practical Directions

The authors suggest several architectural and training innovations:

• Hybrid and Enhanced SSMs: Incorporation of SSM modules with nonlinear/gated updates [yang2025path, yang2025gated, peng2025rwkv, allenzhu2025]. The promise is parallel training efficiency with improved statefulness.
• Coarse Recurrence: Recurrence at thought/sentence granularity (rather than per-token) can allow for more tractable, linguistically meaningful memory and longer-term state propagation [borazjanizadeh2026].
• Variable-Depth and Adaptive Models: Dynamic depth/looping allows efficient tradeoffs between compute and reasoning power [yang2024, alabdulmohsin2025recursive, bae2025, rodkin2025].
• Efficient Training Paradigms: Multi-stage and kernelized training regimes can combine efficient pretraining with delayed, targeted recurrent fine-tuning. Advanced optimization methods (e.g., Neumann recurrent backpropagation [liao2018]) are required to scale attractor or fixed-point recurrent models.
• Representational Alignment: Leveraging the natural alignment properties induced by residual pathways and canon layers to facilitate gradual, robust state handoff between time and depth axes.

Conclusion

This work rigorously characterizes the limitations of transformer architectures for truly dynamic, temporally coherent cognition. The analysis demonstrates that state tracking deficiencies are a direct consequence of the feedforward stack topology, causing unavoidable tradeoffs in deep inference and serial reasoning. By categorizing and motivating a variety of recurrent, hybrid, and SSM-based models, the paper provides a structured guide for architectural innovation. Long-term, bridging the gap to efficient, flexible, and robust dynamical representation is necessary to overcome the fundamental bottlenecks outlined herein and to support foundation models capable of human-like, temporally extended cognition (2604.17121).