tl;dr I continue work to apply Goodfire lab's manifold steering methods to the Gemma-2-2b model's natural state, both in raw residual space and in the space produced by Anthropic's pre-trained CLT features. I find that the model's natural residual state can be directly projected onto a pre-computed manifold and gradually steered to produce day-of-the-week tokens in the expected order on two sets of prompt templates, though the results are hindered by Gemma-2-2b's inaccuracy (43.5% correct) on many of the included prompts. Applying this same steering method in CLT space produces a weaker result, with lower token probabilities and less smooth transitions between subsequent days even when controlling for model accuracy.

Introduction

In previous posts, I extended the work of Wurgaft et al. (2026) to identify a cyclical day-of-the-week manifold in Anthropic's pre-trained CLT features for Gemma-2-2b , and then steer model output along that manifold to produce outputs that smoothly follow the day-of-the-week ordering. In this post, I investigate whether this manifold-steering method can be extended to natural model states, both in the raw residual stream and in the CLT feature space, and whether steering along a pre-computed manifold will generalize to unseen day-of-the-week prompt templates.

I expand the day-of-the-week prompt dataset to include six new prompt templates, and find that the cyclical manifold is present in all new templates in both the residual and CLT feature spaces, though it may be weaker or less regular in templates that use relative day framing (yesterday, tomorrow). I then explore two methods for applying manifold-steering to the model's natural state, and find that steering in the raw residual space can be quite effective, cycling through 95% of the expected days of the week when controlling for prompts that the model answers correctly before steering. Steering in the CLT feature space produces a much weaker result, cycling through only 36% of the expected days of the week on prompts that the model answers correctly.

Methods

In my first post in this series, I describe the Wurgaft et al. (2026) methods for generating a cyclical manifold for the day-of-the-week tokens in some detail. We will focus on manifolds generated at layer 20 of Gemma-2-2b, as previously indicated. At a high level, we compute centroids for each day and then fit a cubic spline through those points in both the raw activation space and the CLT feature space.

Prompt Templates

The original paper and my work up to now has used a single prompt template to generate these centroids, using the last token position to capture the information for the expected query response:

"Q: What day is {number} days after {entity}?\nA:"

While using a single prompt template is sufficient to establish the method, questions of practical application and generalization require a more varied dataset. As such, we expand our list of arithmetic-based prompts:

 "Q: What day is {number} days after {entity}?\nA:"
 "Q: If today is {entity}, what day will it be in {number} days?\nA:"
 "Q: {number} days after {entity}, what day is it?\nA:"
 "Q: It is {entity}. What day will it be in {number} days?\nA:"
 "Q: What is the day {number} days after {entity}?\nA:"

And we add two templates that use a yesterday/tomorrow reference in place of arithmetic (we refer to these as "framing" templates):

"Q: Yesterday was {entity}. What day is it today?\nA: Today is"
"Q: Tomorrow is {entity}. What day is it today?\nA: Today is"

We use the arithmetic templates to fit and test our methods, iterating through all five as a leave-one-out unseen template to test performance. Since each of the framing templates only has one sample per target day, we do not include these in our fit dataset (to maintain balance across the days) and use them only as further tests of steering generalization.

Natural State Manifold Steering

My previous post follows the Wurgaft et al. (2026) methods for manifold-steering, in which they replace the relevant vector with a computed day centroid (the source day) and then gradually interpolate along the manifold to another computed day centroid (the target day). Once again, this design is sufficient for demonstrating that manifold steering works as intended (we see a cyclical increase in intervening day token probabilities as we traverse the manifold), but it says very little about whether we can apply this technique to steer a model's natural state. The existing methods demonstrate that you can effectively steer the model from one position on the cyclical manifold to another, but it doesn't tell us whether we can use the cyclical manifold to steer the model from one natural token generation to another.

In this work, we compare three different steering methods that start from the model's natural vector state^[1]. In all methods we gradually steer the model from our starting position (defined for each method below) to the ground-truth answer day + 3 days. So if our ground-truth answer is Monday, our steering target would be Thursday.

In our first method, we interpolate along a linear path starting from the natural vector state and ending at the target day's centroid on the manifold , with in 50 steps.

Then, we examine two methods for applying manifold steering. The main hurdle to overcome in this extension is how to translate movement along the generated cyclical manifold to movement from the model's natural state (which is unlikely to lie directly on the manifold). For the simpler method, we approximately locate the point on the cubic spline that is closest to our natural vector and interpolate along the signed shortest arc between the angular coordinate of that projection and that of our target day's centroid as decoded by .

To account for the possibility of a uniform off-manifold shift, we also compute , which is the offset between and the nearest point on the spline and add it back into our steered vector:

Metrics

In evaluating manifold-steering performance, we need metrics that are more specific than "does the model output the expected target token"--we know that linear steering already does a pretty good job at this, but the idea behind manifold-steering is that token generation probabilities should move smoothly through the intervening days. So we consider several metrics:

• Day Coverage: What proportion of the expected intervening days reach argmax probability during interpolation. For Mon Thurs steering, we expect to see both Tues and Wed become the argmax token during interpolation. A perfect score here is 1.
• Day Jumps: How many argmax token handoffs skip one or more days. For Mon Thurs steering, we expect a smooth transition of argmax from Mon Tues Wed Thurs--skipping one or more days means that we are not following the manifold. A perfect score here is 0.
• Correct Start: What proportion of prompts generate the ground-truth source day before interpolation. A perfect score here is 1. In the Wurgaft et al. (2026) centroid-to-centroid steering experiments, we are essentially guaranteed to start with the expected source day as the argmax generated token, but that is not guaranteed in our natural state steering methods. The start position for the methods we compare are as follows:
  • Centroid: Linear & Manifold - The argmax token at the source day centroid on the manifold.
  • Projection - The argmax token at the point on the manifold that is nearest to the natural model state.
  • Keep r & Linear - The argmax token at the natural model state.
• Correct End: What proportion of prompts generate the ground-truth target day once interpolation is complete. This is the typical steering performance metric. A perfect score here is 1.

Results/Discussion

Template Manifold Comparison

The figures below show the 2D PCA projections of the day-of-the-week manifolds in both the raw activation space and the CLT feature space as generated over three datasets: the original Wurgaft et al. 2026 single template phrasing, one of the arithmetic prompt folds (with one prompt template left out), and the dataset of framing prompt templates.

Visibly, we see that all three datasets produce cyclical manifolds with proper ordering in both spaces, though the spacing and 2D shape of the manifolds differs^[2]. This pattern holds in full dimensions, where we see maximal similarity between neighboring days on the mean-centered manifolds as seen below. However, we start to see a difference in the framing template distribution, where similarity to a perfect circle decreases from 0.997 to 0.917 and 0.817 in the raw residual space and the CLT feature space, respectively.

This same pattern is evident when considering the spacing of the centroids as compared to seven points on a perfect circle. In the table below, is the smallest distance between centroids as compared to a perfect circle ( has the same spacing as a circle), and is the coefficient of variation of those distances. Here, the arithmetic prompt manifolds have very regular spacing in activation space and slightly less regular spacing in CLT feature space, but the framing prompt manifolds are much more irregular in both spaces.

Finally, we directly compare mean-centered manifolds generated from the individual prompt templates to the mixed arithmetic prompt manifold, looking at cosine similarity of the day centroids between datasets. This comparison tells a similar story: the arithmetic prompt manifolds are remarkably similar^[3] and the framing prompt manifold is much less so, with a wider gap in similarity in the CLT feature space than in activation space.

It is worth noting that the dataset size, even between a single arithmetic prompt (n=49) and the framing prompts (n=14) may be entirely responsible for the differences in manifold shape and consistency shown here. However, I believe it is equally possible that the model's representation of relative days may be less cleanly cyclical than one that is driven from arithmetic.

Model Accuracy Disclaimer

Before I discuss the steering results, it is worth calling out a major flaw in the assumptions of our experiment: in its natural state, the Gemma-2-2b model only generates the correct answer in 43.5% of our prompts. This obviously complicates our results, as our methods assume that we are pushing the model along the manifold from the source day to the target day. I considered altering our methods or metrics to only include prompts that the model answered correctly or to use the closest centroid on the manifold as our source day label. I include some discussion of the performance when limited to correct answers below, but I ultimately decided to report results on the full dataset and the original methods as a more accurate reflection of how manifold-steering generalizes to the natural model state.

As such, I want to explicitly acknowledge that the performance results in this post are skewed by the fact that this model is not very good at answering the arithmetic prompts in particular, and that a model that is more accurate may behave very differently during manifold-steering. We are tied to the Gemma-2-2b model for this post because of compute limitations and the availability of pre-trained CLT features, but would be interested in extending the work to other models in the future.

Natural State Steering

Our experiments on natural state manifold-steering offer three main results, described in more detail below:

1. Manifold-steering can be effectively applied to the model's natural state and extended to unseen prompt formats.
2. Prompt-specific deviations from the cyclical manifold are not well-represented by a constant offset.
3. Steering in the residual space is much more consistently effective than steering in the CLT feature space.

1. Steering Performance and Generalization

In the table of manifold-steering metrics for all methods in the raw residual space below, we see that the projected steering method is the best performing natural state steering method by far. On average over the held-out arithmetic prompts, this method results in zero non-consecutive day jumps, 75% activation of intervening day tokens, and 94% generation of the correct target day. Steering the framing prompts more consistently activates the intervening days, at 97%, but lands on the correct target day less often, at 82%.

Another obvious difference between the two datasets is that the Gemma-2-2b model generates the correct answer on 100% of the framing prompts, and this carries over to 100% agreement in the source day even once the natural state is projected onto the manifold^[4]. The model only generates the correct answer on 43% of the arithmetic prompts, translating to 55% agreement once projected onto the manifold^[5]. If we exclude any arithmetic prompt that the model answers incorrectly, our projected steering metrics increase to nearly match or exceed the framing prompts, with 95% activation of intervening days, 87% agreement on the projected source day, and 92% activation of the correct target day^[6]. This suggests that the apparent difference in steering performance between the two sets of prompts is due to the difference in the underlying model accuracy, rather than a difference in the steerability.

The plots below compare the generation probabilities of the day-of-the-week tokens during steering (as increases) for the centroid manifold-steering and projected manifold steering. For both methods, we see that the probabilities of intervening days cleanly increase and decrease as we traverse the residual space manifold, though the transitions are cleaner for the centroid-to-centroid steering, and the overall token probabilities are lower for projection steering. The issue with the underlying model accuracy accounts for both of these differences, since a greater portion of the probability density will be distributed to the incorrect day tokens.

Indeed, in the plots below, we see that the token probability curves on the framing prompts, which the model consistently answers correctly, have a similar magnitude to that seen in the centroid-to-centroid steering plot. However, the projected curves have much longer tails, which is associated with the lower end token accuracy--determining whether this is causal (i.e. the wider probability curves cause the model to generate incorrect tokens) or a side effect of the aggregate curves (e.g. inaccuracies in specific prompts make the average probability curves look wider) would require additional experiments.

2. Prompt-specific Offsets

Our keep-r natural state steering method was motivated by the possibility that individual prompts may have a consistent offset from the aggregated day-of-the-week manifold. In the metric table above, we see that the steering performance of this method is much less effective than the projected method (which differs only in the reapplication of the prompt-specific offset). When retaining the offset, we see that steering only activates about half of the intervening days and about half of steering paths have argmax day transitions that are not between neighboring days. This pattern is also apparent in the token probability plots below, where we see that the projected method has much cleaner transitions between subsequent days than keep-r. However, we also see that keep-r does not produce the characteristic "teleportation" from source day to target day that we see from the linear steering method.

While this doesn't tell us much about how individual prompt manifolds are related to the aggregate day-of-the-week manifold, it indicates that the offset of the natural state vector for a prompt cannot be used as an affine shift to define the position of that prompt's manifold. It may still be that an approximate prompt-specific manifold could be recovered from the aggregate manifold and the model's natural vector state, but that simply adding a static offset does not capture that relationship. We might instead define a relative shift or scaling, such that the offset is applied in relation to the corresponding angular coordinate on the manifold (producing a uniformly shrunk or enlarged manifold with a shared center), but we leave that exploration for future work.

3. CLT Feature Space

Our third finding, that steering in CLT feature space is less effective than in raw residual space, is not surprising given the centroid-to-centroid manifold steering performance that we discussed in our previous post. We suggested that this could likely be attributed to the incomplete steering signal available in the CLT features, given that they are trained to reproduce the information captured by the MLP sublayers alone, without direct contributions from the attention sublayers or the persistent residual stream.

We see a similar pattern in the CLT metrics table above and in the token probability plot below, where the natural state steering methods are consistently weaker in the CLT space. Though the projected natural state steering method is still the most performant of the three that we explored, we still see lower overall token probabilities and inconsistent cyclical movement through the days compared to steering in the raw residual space.

One surprising difference in CLT space is the steering performance on the framing prompts. Whereas natural state steering on the framing prompts in residual space was seemingly more performant (even if due to more accurate underlying model behavior), steering on the framing prompts in CLT space had very little effect at all.

It is certainly possible that this is an effect of the much smaller sample size, with only 14 prompts. However, if we continue with the explanation that CLT features hold incomplete day-of-the-week signal, then this may suggest that the information that generalizes to a new template exists outside of the MLP sublayer. This finding may be in conflict with previous work (mine, as well as Geva et al. 2023), which would expect that context-related information (like changes in prompt format) would be processed by the attention sublayer, while context-independent information (like a day-of-the-week feature learned during pre-training) would be processed by the MLP sublayer. Instead, we see that the day-of-the-week information that is captured by the CLT features in the MLP sublayer appears to be specific to the form of the prompt.

In-prompt Day Manifold

As briefly mentioned in a footnote above, we found that the in-prompt day-of-the-week label (e.g. Monday in the prompt "What day is days after Monday?") had more explanatory power over our steering vector than the ground-truth answer label did--in other words, vectors for prompts with "Monday" in the query were more similar/predictable than vectors with "Monday" as the expected answer. Given this relationship, we wondered whether we could effectively steer our model output by traversing a manifold based on that in-prompt day label.

The methods for this quick experiment were simple: we used the same vectors as in our main experiment, but grouped them by the in-prompt day instead of the answer day to compute the centroids that form the backbone of the cyclical manifold. We find that these in-prompt manifolds are cyclical across all datasets and splits, though the consistency is somewhat lower, with similarity to a perfectly circular manifold ranging from [0.92, 0.98] on arithmetic prompts and only 0.79 on framing prompts (compared to and 0.92, respectively, on the answer-day manifold).

However, despite the presence of a cyclical manifold and the strong relationship between our steering vectors and the in-prompt day label, we found that steering based on this label was completely ineffective in both the centroid-to-centroid manifold steering and the projection natural state steering methods (activating only 52% and 44% of intervening day tokens, respectively, and landing on the target end day only 14% of the time in both methods). As shown on the token probability plots for the projection method below, projecting our vector onto the in-prompt manifold basically randomizes the day-of-the-week generation, producing flat trajectories with a near uniform distribution between the seven tokens. Interestingly, the CLT curves do separate out the individual tokens, though steering is still ineffective--this is another indication of the persistent day-of-the-week signal that is present in the residual stream outside of the MLP sublayers.

While it was not very surprising that steering along this in-prompt manifold was ineffective, further analysis revealed that the relatively high explanatory power of the in-prompt day collapsed when we controlled for prompts that the model answered correctly. In other words, the model's representation of the in-prompt day was strong because it was guaranteed to be correct, but the representation of the answer day appeared to be weaker because the model only generates the correct day-of-the-week label 43.5% of the time. If we only include prompts that the model answers correctly, the steering vector's relationship to the answer day becomes equal to that of the in-prompt day.

Conclusion

The results from this post suggest that the controlled manifold steering experiments from Wurgaft et al. (2026) can effectively be applied to a model's natural residual state to produce smooth transitions through the days of the week. When controlling for prompts that the model answers correctly before steering, simply projecting the natural vector onto the nearest point of a pre-computed manifold results in activation of 95% of the intervening day tokens during steering and generation of the correct final day on 92% of prompts.

However, the model used in this experiment, Gemma-2-2b, naturally generates the correct answer on only 43.5% of the prompts in our dataset, so it is difficult to say whether this performance would carry over to a model that is more accurate out-of-the-box. We were tied to this model because of the availability of pre-trained CLT features and compute limitations, but would be interested to explore the efficacy of manifold-steering on a different model in future work, especially considering the overall weak results coming from CLT feature steering.

We also recognize that our experiments, while less controlled than the original, rely on a very specific prompt format, in which the model is expected to generate a day-of-the-week token. We tried to account for some generalization by introducing variations and templates that used relative terms in place of arithmetic, but acknowledge that this is still far from demonstrating steering ability during normal generation. For future work, we would consider an experiment that applies steering only when the model is about to generate a day-of-the-week token in a more natural generation setting (as indicated by the next-token probability). This would provide clearer evidence of generalizability, by demonstrating that the manifold exists and can be applied to day-of-the-week generation outside of a structured prompt template.

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.

1. ^{^}

   Experimentation with other methods, including a gradual decrease in and pinning the start vector to the closest centroid, did not produce meaningfully different results, and thus were excluded from the write-up here.

2. ^{^}

   Initially, we did a 50/50 fit/test split on the mixed arithmetic dataset, stratified across the target day. The idea was to compare the steerability of templates that were and were not included in generating the manifold. However, we found that the day that was included in the prompt ({entity} in the templates described in this post) held significant explanatory power on the steering vector for each prompt. Since the split was not stratified over that in-prompt day, this produced manifolds with occasional misordering and systematically worse steering performance on the test split than on the held-out templates--the centroids were being unevenly pulled by the incomplete in-prompt day samples for each template in the mixed dataset. As such, we opted to include the full range of in-prompt and target day prompts for all templates in the mixed dataset and limit our steering experiments to the held-out templates.

3. ^{^}

   Although we only include heatmaps for one arithmetic prompt manifold here, the average same-day similarity across all six manifolds is in both the raw residual space and the CLT feature space.

4. ^{^}

   It is interesting to note that the start-token accuracy actually decreases in the centroid methods, when we replace the natural state vector with the computed source day centroid. This may suggest that the framing day manifold is rotated from the arithmetic day manifold, which would also explain the similar gap in end-token accuracy. Our cross-manifold cosine similarity heatmaps support this hypothesis, since the most similar framing centroids inconsistently match those from the arithmetic prompts. An experiment that steered the model to a shifted position on the manifold instead of the computed centroid could confirm whether this is the case.

5. ^{^}

   The fact that the projected vector is correct more often than the natural state vector presents an interesting possibility for correcting individual prompt inaccuracy through the aggregate nature of the manifold, though we didn't specifically test this hypothesis.

6. ^{^}

   This selection of correct prompts reproduces the issue that we avoided by abandoning the 50/50 data split: the dataset is no longer stratified and there may be something more systematic about the model failures (such as poor representation of specific day-of-the-week tokens or between-day relationships) that would cause both inaccuracy and poor steering performance independently.