A Mechanistic Analysis of Looped Reasoning Language Models — part 4/4
Cleaned-text code points [47111, 62819). Bibliography omitted here; source.md includes it.
We highlight that these exact results need to be treated with caution: the absolute values vary significantly depending on the algorithm hyperparameters chosen, and in the absence of a good external metric we selected these hyperparameters manually via visual inspection of the trajectories and cosine similarities to ensure the results agreed with intuition. However, across different hyperparameters the results consistently showed that non-fixed-point behavior is rare and that the long persona prompt results in a greater rate of non-fixed-point behavior. We leave in-depth classification and explanation of this phenomena to future work.
### C.2 Do Intermediate Layers Exhibit Non-Fixed-Point Behavior?
Geiping et al. (2025) observe orbits and sliders in the latent states after each application of the entire recurrent block. Here we investigate what occurs in the latent states of the intermediate layers within the recurrent block.
We first extend Figure 16 of Geiping et al. (2025) (visualizing latent trajectories on a math prompt), visualizing also the trajectories of the intermediate layer residual streams: this can be found in Fig. 15 . This demonstrates that in this particular case, where Orbits occur, they also occur in the intermediate layers. This implies – viewed in realized depth – latent trajectories exhibiting multi-scale cyclic behavior.
We investigate this across all GSM8k test examples in Fig. 14 by plotting the conditional probability of observing each behavior on a given token for any layer, given an observation of observing another behavior on that same token. We discover that orbits and sliders do not co-occur across looped layers, but that orbits and sliders both frequently co-occur with fixed point behavior. “Unknown” behavior often co-occurs with orbits, which we suggest may be due to mis-classification of the behavior algorithm.
### C.3 How Does Non-Fixed-Point Behavior Impact Stages of Inference?
To complete the link between non-fixed-point behavior and our work we additionally investigate how this behavior impacts the observed stages of inference.
To attempt to isolate the “worst case” scenario for stages of inference stability, we plot stages of inference for the GSM8k test prompt that exhibits the greatest orbit amplitude. We first plot in realized depth the extended stages of inference metrics used throughout the paper, see Fig. 16 . This demonstrates that sink rates show some variability with the orbiting behavior, but the other metrics remain broadly consistent. We plot also the same metrics in block depth (showing only the final 64 loops) in Fig. 17 : this demonstrates clearly that the stages of inference remain very consistent despite the orbiting behavior.
We plot the same for the largest amplitude orbit on the Retrofitted Llama model in Fig. 18 , demonstrating that the same behavior holds.
## Appendix D Additional Fixed Point Results
### D.1 Cyclic Similarity
We include here additional plots to validate our cyclic similarity claims in Sec. 4 .
Fig. 19 complements Fig. 2 by visualizing the cosine similarity between the residual streams after each layer for the range of Looped Transformers visualized in the original figure. Additional models (Huginn-0125 and all retrofitted models) are visualized in Fig. 19 for 32 recurrences, demonstrating that the cyclic similarity is consistent for larger numbers of recurrences. We note that Huginn-0125 continues its previous trend of all layer outputs converging to similar representations, with some cyclic similarity still visible. Fig. 21 similarly provides an extended version of Fig. 2 , demonstrating that attention matrix cyclic similarity holds to 32 recurrences.
### D.2 Fixed Point and Successive Differences
In Sec. 4 we demonstrate that retrofitted Llama and Huginn-0125 reach a fixed point but Ouro does not. We do so by – for each layer – computing an “approximate fixed point” after 128 recurrences and then plotting the norm of the difference for each layer at successive recurrences to this fixed point. In Fig. 22 we demonstrate that this same behavior is observed when considering cosine similarity to the fixed point, and in Fig. 23 we see that it holds for attention matrices.
We then demonstrated in Fig. 3 that all models, independent of whether they reach a strict fixed point or not, demonstrate converge towards very low difference norms between successive residual streams. We verify this same behavior holds for residual stream cosine similarities and attention matrix Frobenius norms in Figs. 24 and 25 respectively.
### D.3 Latent Space Trajectories
In this section we visualize additional latent space trajectories, supplementing Fig. 5 . All trajectories are plotted by taking all latent states of the final token position on the test sequence described in App. B , computing PCA on these latent state vectors and resultant dimensional reduction of this sequence of vectors. They are intended as illustrations to demonstrate qualitative behavior. Ouro is visualized in Fig. 26 and Huginn-0125 in Fig. 27 .
The Ouro trajectory is of particular interest as we are aware from the results in the main body of the paper that this model does not reach a strict fixed point. Fig. 26 demonstrates that – on this test sequence – Ouro reaches an approximately constant trajectory, but with visibly larger deviations even at later recurrences than Huginn-0125 in Fig. 27 .
However, we find evidence that this “approximately stable trajectory” behavior is not universal: for a simple “maths” test prompt ( The square root of 16 is ) shown in Fig. 28 , Ouro appears to reach a stable trajectory in recurrences 8-16 before departing from this and appearing to become “unstable”.
### D.4 Architecture Choices
Here we provide more complete results to supplement Sec. 4.2 , visualizing fixed point difference norms and cosine similarities for various architecture choices in Figs. 29 and 30 respectively.
We also verify that the results presented are not particular to 12 layers by visualising results for both 4 and 16 layers in Fig. 31 , showing qualitatively identical behavior.
## Appendix E Additional Stages of Inference Results
In this appendix we explore in more detail the mixing behavior presented in the main body via additional metrics, and considering a wider range of models.
### E.1 Input Independent Metrics
As our work is concerned largely with how the functionality of layers change throughout depth as the residual stream is iteratively updated, our primary concern is with input-dependent measures of stages of inference: ColSum concentration as presented in the main text of the paper is one such input-dependent metric, and the later sections of this appendix will introduce and present results for a wider range of such metrics.
However, to bridge the gap between our work and that of Lad et al. (2024) , we also present results for the fraction of prediction and suppression neurons ( Gurnee et al., 2024 ) in successive layers of both feedforward and looped Transformers. We highlight however that these are unable to change with successive recurrences , and are thus secondary to our focus. Fig. 32 presents results for a selection of feedforward models used throughout the paper, and Fig. 33 presents results for a selection of looped models used throughout the paper. Similar to our results elsewhere, we find that the looped model stages of inference in these metrics tend to mirror those of feedforward models.
### E.2 Input Dependent Metrics
One well-studied phenomenon by which Transformers drastically reduce the mixing in given layer is that of the attention sink ( Xiao et al., 2023 ; Barbero et al., 2025 ) , whereby the layer focuses the majority of the attention “weight” onto the first token in the sequence; often this is the BOS (beginning of sequence) token, and thus completely uninformative. To measure this behavior, we adopt the attention sink score and sink rate of Gu et al. (2024) : for token position k k and sequence length T T , the sink score at layer ℓ \ell and head h h is defined as: sink-score k ( ℓ , h ) = 1 T ∑ t = 0 T − 1 A t k ( ℓ , h ) , \text{sink-score}_{k}^{(\ell,h)}=\frac{1}{T}\sum_{t=0}^{T-1}A_{tk}^{(\ell,h)}, (16) where A t k ( ℓ , h ) A_{tk}^{(\ell,h)} corresponds to the realized attention matrix of Eq. 2 at layer ℓ \ell , head h h . The sink rate is then defined as the fraction of heads for which the sink score lies above a certain threshold: sink-rate k ( ℓ ) = 1 H ∑ h = 1 H 𝕀 ( sink-score k ( ℓ , h ) ≥ τ ) , \text{sink-rate}^{(\ell)}_{k}=\frac{1}{H}\sum_{h=1}^{H}\mathbb{I}\!\left(\text{sink-score}_{k}^{(\ell,h)}\geq\tau\right), (17) where following related work we define a threshold of τ = 0.3 \tau=0.3 , and 𝕀 \mathbb{I} denotes the indicator function.
We also adopt the Mixing score of Queipo-de-Llano et al. (2025) : for sequence length T T , layer ℓ \ell and head h h this is defined as the average row entropy of the attention matrices: mixing-score ( ℓ , h ) = 1 T ∑ i = 1 T H ( A i , : ( ℓ , h ) ) . \text{mixing-score}^{(\ell,h)}=\frac{1}{T}\sum_{i=1}^{T}H(A_{i,:}^{(\ell,h)}). (18)
Following Skean et al. (2025) ; Queipo-de-Llano et al. (2025) we additionally measure the compression of the residual stream 𝑿 {\bm{X}} via the matrix-based entropy H ( 𝑿 ) H({\bm{X}}) .
In the plots that follow we average sink rates, Mixing scores and ColSum concentrations over all the heads in a layer, and over all input sequences. Residual entropy is averaged over all input sequences.
### E.3 Cyclic Stages of Inference
Supplementing the cyclic recurrence in realized depth for retrofitted Llama in Fig. 7 , we additionally overlay cycles in realized depth for Ouro and Huginn-0125 in Fig. 34 , where all models are run for 8 recurrences. This demonstrates that for sink rates and mixing scores , the cyclic behavior and lack of significant depth-wise changes hold both for the other investigated models, and the additional stages of inference metrics.
The results for residual entropy are more nuanced: while the results for Huginn-0125 and the retrofitted models show broadly the same behavior as the attention-based metrics, Ouro demonstrates different behavior. As shown in Fig. 35 , the residual entropies both change significantly with successive recurrences and do not closely follow the feedforward behavior. We believe that the divergence from feedforward behavior may derive from the norm structure of this model: the residual stream is normalised after each recurrent block, as visualised in Fig. 9 , thus periodically shutting down massive activations. This is not the case for the retrofitted series of models, which lack this norm and show much closer alignment with feedforward stages of inference.
For completeness, we plot these stages of inference for all other models referenced in the paper. See Fig. 35 (Ouro 1.4B), Fig. 36 (Huginn-0125), Figs. 37 , 38 and 39 (retrofitted Llama, OLMo and TinyLlama).
We plot stages of inference for Ouro 2.6B in Fig. 40 . This model is interesting due to the training regime followed by Zhu et al. (2025) , which “upcycles” a 48 layer model from the 24 layer 1.4B parameter model. As a consequence, the first and second half of each recurrent block each independently align with the Llama feedforward stages of inference.
In Sec. 5 we suggested that the lack of stages of inference in Huginn-0125 is likely due to the normalization of the residual stream resulting in massive activations being unable to form. Here we further support this suggestion by ablating the massive activations from the Retrofitted Llama model (which does display stages of inference) via zeroing the output of the MLP in the second layer, which is responsible for its massive activations. In this setting, visualized in Fig. 41 , we see that the model no longer exhibits stages of inference comparable to the feedforward model, suggesting that the presence of massive activations is required for stages of inference to emerge in looped models.
### E.4 Non-Reasoning Stages of Inference
Throughout the rest of the paper, experiments are conducted on the GSM8k dataset. In this appendix we verify that the stages of inference we observe are not specific to this dataset, and also occur in a non-reasoning setting, for which we use the HellaSwag dataset ( Zellers et al., 2019 ) . We follow an identical experimental setup to the GSM8k experiments, running inference on 256 random examples from the test split.
We present results in Fig. 42 (Ouro 1.4B), Fig. 43 (Huginn-0125), Figs. 44 , 45 and 46 (retrofitted Llama, OLMo and TinyLlama). These show very few deviations from the GSM8k results, and the conclusions throughout the rest of the paper hold. However, we highlight the following small differences: • Across the board, sink rates tend to be higher in the HellaSwag setting.
• In Retrofitted OLMo-2, ColSum concentration appears to be slightly higher in the HellaSwag setting.
### E.5 Stability To Unseen Test-Time Recurrences
This section extends the results presented in Sec. 5.2 .
We supplement Fig. 11 by plotting how stages of inference change per-layer throughout recurrences for additional models: these can be found in Figs. 47 , 48 and 49 . The large standard deviations in Huginn-0125 and retrofitted Llama mixing scores reflect the fact that these models tend to reach different, but still stable, constant states.
We additionally plot the extended versions of Fig. 12 in Figs. 50 , 51 and 52 .
### E.6 How Architecture Choices Affect the Formation of Stages of Inference
Here we present additional results to supplement those in Sec. 5.1 . The extended stages of inference metrics for the models visualized in Fig. 10 are presented in Figs. 54 , 55 and 56 . Equivalent models with added input injection are visualized in Figs. 57 , 58 and 59 , and equivalent models without sandwich layers in Figs. 60 , 61 and 62 .
It appears in these small scale experiments that feedforward stages of inference are most closely replicated without input injection, and using sandwich layers, as in Figs. 54 , 55 and 56 . The addition of input injection appears to mean that the final recurrence follows the feedforward stages of inference more closely, but to the detriment of earlier recurrences.
## Appendix F Looped Floorplan
To illustrate the “similar attention patterns between recurrences” that we have discussed throughout the paper, in Fig. 63 we visualize all attention patterns for the Retrofitted Llama model on the test prompt. Increasing depth in the model is aligned with increased height up the page; Prelude and coda are outlined in blue and red respectively and separate recurrences are separated by a space.
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