tl;dr I continue work to extend the Goodfire lab's cyclical manifold methods to Anthropic's pre-trained CLT feature space on Gemma-2-2B. I find that manifold steering produces the expected cyclical transition between days of the week in both activation space and CLT feature space, but that CLT features provide a weaker steering signal, suggesting that the MLP sublayers in which they originate only hold a portion of the full day-of-the-week signal. Colab notebook implementation can be found here.
Introduction
In a previous post, I applied methods from Wurgaft et al. (2026) to identify and analyze the presence of a cyclical day-of-the-week manifold in Gemma-2-2B. Following their work, I confirmed that this manifold structure exists in both the raw residual stream and what they term "behavior space", and extended their methods to the pre-trained CLT feature space released by Anthropic. I found that the cyclical manifold exists among these extracted features in a moderately "cleaner" form than in the raw residuals.
In this post, I complete the extension of the Wurgaft et al. methods by performing manifold steering along the day-of-the-week manifold in Gemma-2-2B's raw residual space and CLT feature space. My results show that the manifold-steering methods work as expected in both residual and CLT feature space, with gradual perturbations moving smoothly through the ordered days, and traditional linear-steering methods "transporting" from start day to the end day without cleanly passing through the intermediate days. However, while the overall pattern holds true in CLT space, the steered token probabilities are weaker than in the raw residual space, which I suggest can be attributed to the more limited scope of the CLT feature steering vectors.
Methods
In my last post, I describe the methods for identifying the day-of-the-week cyclical manifold in more detail. At a high level, we compute centroids for each day and then fit a cubic spline through those points. We generate three manifolds using the raw activations, the pre-trained CLT features, and the model's output probability distribution (converted into Hellinger space)--the 2D PCA projections of these manifolds are shown below. In this work, we perform manifold-steering experiments on the activation and CLT spaces, and use the behavior space to evaluate our steering results.
Steering Interventions
For the steering experiment, we follow the Wurgaft et al. methods to steer the model output directly from one day centroid to another, both with a more typical linear steering method and along the cyclical manifold. We take a static sample of 16 prompts drawn from the full 49 day-of-the-week prompts (e.g. "Q: What day is two days after Monday? A:"), and fully replace the intervention-specific vector for the last token position with the computed centroid for the source day. We then move that replacement vector in 50 increments to the computed day centroid for the target day. For the linear steering method, this is simple linear interpolation, and for the manifold steering method, this involves interpolating the angular coordinate θ along the shortest arc between the source and target centroids and decoding each angle through the cubic spline.
With and as the computed day centroids, as the cubic spline, and as the angular coordinate of those centroids on , and over 50 steps, the two steering vectors are:
Concretely, this means that Mon Thurs steering on the prompt "Q: What day is two days after Monday? A:" replaces the vector for the final token (which would otherwise produce the model's answer) with the computed centroid for Monday. This is the starting point, and is already an intervention rather than the model's natural state. We then gradually move that vector (linearly or along the geodesic) to the computed centroid for Thursday, the end point.
In the activation space, linear steering replaces the entire activation vector at a given layer, and manifold steering replaces only the the 48 PCA components used to fit the manifold, leaving the rest of the prompt's activations unchanged. In CLT feature space, the replaced vector for both linear and manifold steering is the set of CLT features used to compute the centroids, consisting of the top 64 features in a layer, chosen based on a between-day F-statistic. Since features are read from the residual stream but not stored in it directly, the steering is applied by writing the difference between the steering vector and the prompt's natural feature vector through the each feature's decoder into the MLP output.
This same intervention is written through the CLT decoders for all downstream MLP blocks, since a cross-layer feature can write to subsequent layers. By adding only the decoded difference in the selected feature activations, we leave unselected features and transcoder reconstruction error out of the steered output.
Intervention Location
In order to find the best layer for our steering experiments, we follow the methods used in the Goodfire lab's demo notebook. Over all layers, we compute the day-of-the-week centroids and apply our steering intervention at (which is the full computed centroid). We then compare the output probability of the target day token and select the layer at which this probability peaks--this should be roughly equivalent to the point at which the day-of-the-week generation signal is strongest.
CLT Feature Details
I won't go deeply into how the CLT features are generated (see Ameisen et al. 2025 for those details), but I do want to cover a few concepts that are helpful in interpreting the CLT steering results seen here.
At a high level, transcoder features are generated through an encoder-decoder architecture that takes an input, produces a latent representation of that input (through the encoder), and then aims to reproduce a known output from that representation (through the decoder). In this particular application, the transcoder has been trained to generate a sparse representation of the information in the MLP sublayer (often producing interpretable semantic features), using the residual stream entering the MLP sublayer as its input and learning to reproduce the MLP output. Importantly, this means the features are trained to capture only the information the MLP uses, and to reproduce only what the MLP writes--we don't expect that content in the residual stream that the MLP doesn't use, either from the attention sublayer or from previous layers, would be encoded in the CLT features.
The cross-layer mechanism in the CLT feature generation was adopted in recognition that semantic concepts are unlikely to be processed solely within a single layer, and that allowing a feature to exist across multiple layers would be likely to produce simpler and more complete features. Functionally, CLT features are extracted by the transcoder at layer and contribute to the decoded outputs at all layers (stated differently, layer 's MLP output is reconstructed jointly by all active features from layers ). During training, this means that a semantic concept can be concentrated into a single feature read at layer , rather than having independent, partially explanatory features spread across many different layers.
Results and Discussion
Day-of-the-Week Location
As seen in the plot below, our exploration of steering location for the day-of-the-week tokens clearly indicates that layer 20 is where the day-of-the-week selection signal is located in the Gemma-2-2B model. In both the raw residual stream intervention and the CLT feature intervention, there is a sharp increase in target day token probability at layer 20.
Also clearly visible on this plot is the difference in "steerability" between the two intervention methods. We point to the cross-layer component of the CLT features to explain the isolated steerability in that space when compared to the persistently elevated token probability in the raw residual space. A CLT feature read at layer consolidates the signal that would otherwise be spread across subsequent layers, since it writes to all layers . As such, later layers have nothing additional to encode about that feature and the steering signal disappears[1]. The residual stream, on the other hand, is cumulative, so once the day-of-the-week signal is introduced in layer 20, it remains present in the residual stream throughout the remainder of the model.
We discuss the difference in overall token probability between the two methods more below.
Cyclical Steering Performance
The plots in this section show the probability of each of the day-of-the-week tokens (and an "other" token that holds the probability mass for all other tokens in the vocabulary) as we perform our centroid-to-centroid steering intervention from Monday to Thursday. As a reminder, we replace the relevant vector with the computed Monday centroid for each method, then gradually interpolate (either linearly or along the cyclical manifold) to the centroid for Thursday.
As seen in this first set of plots, steering on the raw activation space produces the same clean results reported in Wurgaft et al. (2026)--steering along the cyclical manifold transitions through each of the computed centroids and produces a corresponding peak in token probability, while linear steering shows a "transportation" effect between the target and source days without cleanly passing through the intervening days.
Looking at the CLT feature plots, we see a similar pattern, but with lower token probabilities overall and the final intervention position not fully recovering Thursday as the most probable day-of-the-week token. For the CLT steering intervention, we found an average p(target) of 0.27 compared to 0.54 for activation steering. Experiments that included increasing the CLT intervention scale and including features from multiple layers did not effectively improve steering performance, suggesting that this isn't an issue of a "weak" steering vector [2] or of CLT features at multiple layers driving day-of-the-week selection [3].
We suggest that the difference in steered token probability can be attributed to the gap in signal between the full residual stream and the MLP sublayers. Since the CLT features are trained only to capture the information necessary to reproduce the MLP output, we don't expect other information in the residual stream from the attention sublayers or previous layers to be well-represented in the CLT features. The relatively weak overall token probability, then, suggests that a substantial proportion of the day-of-the-week signal is carried outside of the MLP sublayer[4]. Unexpectedly, this result ties directly back to another project of mine, in which I find that refusal behavior is shared between the MLP-sublayer and other model components.
Steering "Naturalness"
Another method that Wurgaft et al. (2026) used to evaluate the manifold steering effect was to compare the steered probability distribution to the "natural" day-of-the-week manifold in behavior space. This manifold is generated based on a Hellinger projection of the model's output probabilities, so if manifold steering in either activation or CLT feature space is fully capturing the day-of-the-week signal from the model, we would expect the manifold produced by the steered outputs to closely follow that of the unsteered model.
The behavior space projection below provides a different view of the cyclical token probabilities shown above, where we once again see that the manifold steering more closely follows the output distribution than linear steering for both the residual stream and CLT feature interventions. We also see the difference in overall probability magnitude, with full activation steering slightly overshooting the "natural" probability distribution, and the CLT steering having much lower probabilities.
Wurgaft et al. 2026 also compute the cumulative output energy between the steered outputs and the computed day-of-the-week manifold in behavior space to quantify the "naturalness" of the steered outputs. This energy metric includes all 42 combinations of source and target day, while the previous visualizations only captured the MonThurs combination. With cumulative energy of 0.60 and 0.79 for manifold and linear steering in activation space, compared to 2.53 and 2.92 in the CLT feature space, we see once again that the residual stream intervention produces outputs that are more similar to the unsteered model[5], but that manifold steering improves that similarity in both activation and CLT feature space.
Conclusion
The results in this post confirm the viability of the manifold steering methods introduced by Wurgaft et al. (2026) when applied to both the full residual stream and the CLT feature space of Gemma-2-2B. The steering behavior also provides some interesting insights into the structure of the CLT features, particularly the cross-layer consolidation of feature steering signal, and the gap between MLP-sublayer and full residual stream steering signal.
The work in this post, and in the Wurgaft et al. paper, is limited to a steering demonstration between two "unnatural" vectors--from one computed day-of-the-week centroid to another. In future work, I hope to further extend the manifold steering work to demonstrate whether it can be used to gradually steer a model's natural output along a computed trajectory. This would present an alternative to the current linear steering methods that would more accurately reflect the expected nonlinear geometry of semantic concepts in LLM latent spaces.
AI Use Disclosure
Claude Code was used heavily in developing the methods in the Colab notebook associated with this post and was consulted when interpreting results and designing figures for analysis. All writing is the author's own work with revision suggestions from Claude, and all results, analyses, and conclusions were verified by the author.
We note that another explanation for this isolated steerability would be that the MLP sublayers after block 20 do not contribute any signal about the day-of-the-week. Distinguishing these two theories would require analysis of how the selected CLT feature decoders contribute to later layer MLP blocks.
Steering interventions in Ameisen et al. 2025 are applied with various scaling coefficients for maximum steering effect. We applied our CLT steering vectors multiplied by and only marginally increases average p(target) from 0.27 to 0.32 (and then back down to 0.30 at ). When compared to the full residual stream steering p(target) of 0.54, this suggests that steering strength is not responsible for the gap alone.
In Ameisen et al. 2025 features are grouped into "supernodes" before steering interventions are applied, suggesting that multiple related feature may be necessary for effective CLT-based steering. We performed a naive version of this by including features from layers with the four highest p(target) from our intervention location sweep and found only very minimal increase in average p(target), from 0.27 to 0.31. This supports the cross-layer feature consolidation that we discussed previously, suggesting that the CLT features at layer 20 are responsible for most of the day-of-the-week signal captured by the MLP sublayers.
In this experiment, we are comparing steering effect from one intervention location to another (from the computed Monday centroid to the Thursday centroid), so the low token probabilities on the plot reflect CLT-steered outputs, even at .
It's worth noting that the cumulative energy metric used here is computing the Bhattacharyya distance from the steered output curve to the closest point on the "natural" behavior space manifold. This says nothing about whether the output distribution is being steered "correctly", just whether the output distribution is similar to one found on the computed manifold. Thus, linear steering on the full residual stream here is considered more "natural" than the CLT-feature manifold steering, even though it doesn't produce the expected cyclical token probabilities.
tl;dr I continue work to extend the Goodfire lab's cyclical manifold methods to Anthropic's pre-trained CLT feature space on Gemma-2-2B. I find that manifold steering produces the expected cyclical transition between days of the week in both activation space and CLT feature space, but that CLT features provide a weaker steering signal, suggesting that the MLP sublayers in which they originate only hold a portion of the full day-of-the-week signal. Colab notebook implementation can be found here.
Introduction
In a previous post, I applied methods from Wurgaft et al. (2026) to identify and analyze the presence of a cyclical day-of-the-week manifold in Gemma-2-2B. Following their work, I confirmed that this manifold structure exists in both the raw residual stream and what they term "behavior space", and extended their methods to the pre-trained CLT feature space released by Anthropic. I found that the cyclical manifold exists among these extracted features in a moderately "cleaner" form than in the raw residuals.
In this post, I complete the extension of the Wurgaft et al. methods by performing manifold steering along the day-of-the-week manifold in Gemma-2-2B's raw residual space and CLT feature space. My results show that the manifold-steering methods work as expected in both residual and CLT feature space, with gradual perturbations moving smoothly through the ordered days, and traditional linear-steering methods "transporting" from start day to the end day without cleanly passing through the intermediate days. However, while the overall pattern holds true in CLT space, the steered token probabilities are weaker than in the raw residual space, which I suggest can be attributed to the more limited scope of the CLT feature steering vectors.
Methods
In my last post, I describe the methods for identifying the day-of-the-week cyclical manifold in more detail. At a high level, we compute centroids for each day and then fit a cubic spline through those points. We generate three manifolds using the raw activations, the pre-trained CLT features, and the model's output probability distribution (converted into Hellinger space)--the 2D PCA projections of these manifolds are shown below. In this work, we perform manifold-steering experiments on the activation and CLT spaces, and use the behavior space to evaluate our steering results.
Steering Interventions
For the steering experiment, we follow the Wurgaft et al. methods to steer the model output directly from one day centroid to another, both with a more typical linear steering method and along the cyclical manifold. We take a static sample of 16 prompts drawn from the full 49 day-of-the-week prompts (e.g. "Q: What day is two days after Monday? A:"), and fully replace the intervention-specific vector for the last token position with the computed centroid for the source day. We then move that replacement vector in 50 increments to the computed day centroid for the target day. For the linear steering method, this is simple linear interpolation, and for the manifold steering method, this involves interpolating the angular coordinate θ along the shortest arc between the source and target centroids and decoding each angle through the cubic spline.
With
Concretely, this means that Mon
In the activation space, linear steering replaces the entire activation vector at a given layer, and manifold steering replaces only the the 48 PCA components used to fit the manifold, leaving the rest of the prompt's activations unchanged. In CLT feature space, the replaced vector for both linear and manifold steering is the set of CLT features
This same intervention is written through the CLT decoders for all downstream MLP blocks, since a cross-layer feature can write to subsequent layers. By adding only the decoded difference in the selected feature activations, we leave unselected features and transcoder reconstruction error out of the steered output.
Intervention Location
In order to find the best layer for our steering experiments, we follow the methods used in the Goodfire lab's demo notebook. Over all layers, we compute the day-of-the-week centroids and apply our steering intervention at
CLT Feature Details
I won't go deeply into how the CLT features are generated (see Ameisen et al. 2025 for those details), but I do want to cover a few concepts that are helpful in interpreting the CLT steering results seen here.
At a high level, transcoder features are generated through an encoder-decoder architecture that takes an input, produces a latent representation of that input (through the encoder), and then aims to reproduce a known output from that representation (through the decoder). In this particular application, the transcoder has been trained to generate a sparse representation of the information in the MLP sublayer (often producing interpretable semantic features), using the residual stream entering the MLP sublayer as its input and learning to reproduce the MLP output. Importantly, this means the features are trained to capture only the information the MLP uses, and to reproduce only what the MLP writes--we don't expect that content in the residual stream that the MLP doesn't use, either from the attention sublayer or from previous layers, would be encoded in the CLT features.
The cross-layer mechanism in the CLT feature generation was adopted in recognition that semantic concepts are unlikely to be processed solely within a single layer, and that allowing a feature to exist across multiple layers would be likely to produce simpler and more complete features. Functionally, CLT features are extracted by the transcoder at layer
Results and Discussion
Day-of-the-Week Location
As seen in the plot below, our exploration of steering location for the day-of-the-week tokens clearly indicates that layer 20 is where the day-of-the-week selection signal is located in the Gemma-2-2B model. In both the raw residual stream intervention and the CLT feature intervention, there is a sharp increase in target day token probability at layer 20.
Also clearly visible on this plot is the difference in "steerability" between the two intervention methods. We point to the cross-layer component of the CLT features to explain the isolated steerability in that space when compared to the persistently elevated token probability in the raw residual space. A CLT feature read at layer
We discuss the difference in overall token probability between the two methods more below.
Cyclical Steering Performance
The plots in this section show the probability of each of the day-of-the-week tokens (and an "other" token that holds the probability mass for all other tokens in the vocabulary) as we perform our centroid-to-centroid steering intervention from Monday to Thursday. As a reminder, we replace the relevant vector with the computed Monday centroid for each method, then gradually interpolate (either linearly or along the cyclical manifold) to the centroid for Thursday.
As seen in this first set of plots, steering on the raw activation space produces the same clean results reported in Wurgaft et al. (2026)--steering along the cyclical manifold transitions through each of the computed centroids and produces a corresponding peak in token probability, while linear steering shows a "transportation" effect between the target and source days without cleanly passing through the intervening days.
Looking at the CLT feature plots, we see a similar pattern, but with lower token probabilities overall and the final intervention position not fully recovering Thursday as the most probable day-of-the-week token. For the CLT steering intervention, we found an average p(target) of 0.27 compared to 0.54 for activation steering. Experiments that included increasing the CLT intervention scale and including features from multiple layers did not effectively improve steering performance, suggesting that this isn't an issue of a "weak" steering vector [2] or of CLT features at multiple layers driving day-of-the-week selection [3].
We suggest that the difference in steered token probability can be attributed to the gap in signal between the full residual stream and the MLP sublayers. Since the CLT features are trained only to capture the information necessary to reproduce the MLP output, we don't expect other information in the residual stream from the attention sublayers or previous layers to be well-represented in the CLT features. The relatively weak overall token probability, then, suggests that a substantial proportion of the day-of-the-week signal is carried outside of the MLP sublayer[4]. Unexpectedly, this result ties directly back to another project of mine, in which I find that refusal behavior is shared between the MLP-sublayer and other model components.
Steering "Naturalness"
Another method that Wurgaft et al. (2026) used to evaluate the manifold steering effect was to compare the steered probability distribution to the "natural" day-of-the-week manifold in behavior space. This manifold is generated based on a Hellinger projection of the model's output probabilities, so if manifold steering in either activation or CLT feature space is fully capturing the day-of-the-week signal from the model, we would expect the manifold produced by the steered outputs to closely follow that of the unsteered model.
The behavior space projection below provides a different view of the cyclical token probabilities shown above, where we once again see that the manifold steering more closely follows the output distribution than linear steering for both the residual stream and CLT feature interventions. We also see the difference in overall probability magnitude, with full activation steering slightly overshooting the "natural" probability distribution, and the CLT steering having much lower probabilities.
Wurgaft et al. 2026 also compute the cumulative output energy between the steered outputs and the computed day-of-the-week manifold in behavior space to quantify the "naturalness" of the steered outputs. This energy metric includes all 42 combinations of source and target day, while the previous visualizations only captured the Mon
Conclusion
The results in this post confirm the viability of the manifold steering methods introduced by Wurgaft et al. (2026) when applied to both the full residual stream and the CLT feature space of Gemma-2-2B. The steering behavior also provides some interesting insights into the structure of the CLT features, particularly the cross-layer consolidation of feature steering signal, and the gap between MLP-sublayer and full residual stream steering signal.
The work in this post, and in the Wurgaft et al. paper, is limited to a steering demonstration between two "unnatural" vectors--from one computed day-of-the-week centroid to another. In future work, I hope to further extend the manifold steering work to demonstrate whether it can be used to gradually steer a model's natural output along a computed trajectory. This would present an alternative to the current linear steering methods that would more accurately reflect the expected nonlinear geometry of semantic concepts in LLM latent spaces.
AI Use Disclosure
Claude Code was used heavily in developing the methods in the Colab notebook associated with this post and was consulted when interpreting results and designing figures for analysis. All writing is the author's own work with revision suggestions from Claude, and all results, analyses, and conclusions were verified by the author.
We note that another explanation for this isolated steerability would be that the MLP sublayers after block 20 do not contribute any signal about the day-of-the-week. Distinguishing these two theories would require analysis of how the selected CLT feature decoders contribute to later layer MLP blocks.
Steering interventions in Ameisen et al. 2025 are applied with various scaling coefficients for maximum steering effect. We applied our CLT steering vectors multiplied by
In Ameisen et al. 2025 features are grouped into "supernodes" before steering interventions are applied, suggesting that multiple related feature may be necessary for effective CLT-based steering. We performed a naive version of this by including features from layers with the four highest p(target) from our intervention location sweep and found only very minimal increase in average p(target), from 0.27 to 0.31. This supports the cross-layer feature consolidation that we discussed previously, suggesting that the CLT features at layer 20 are responsible for most of the day-of-the-week signal captured by the MLP sublayers.
In this experiment, we are comparing steering effect from one intervention location to another (from the computed Monday centroid to the Thursday centroid), so the low token probabilities on the plot reflect CLT-steered outputs, even at
It's worth noting that the cumulative energy metric used here is computing the Bhattacharyya distance from the steered output curve to the closest point on the "natural" behavior space manifold. This says nothing about whether the output distribution is being steered "correctly", just whether the output distribution is similar to one found on the computed manifold. Thus, linear steering on the full residual stream here is considered more "natural" than the CLT-feature manifold steering, even though it doesn't produce the expected cyclical token probabilities.