The directions nearest a model's prediction decide what kind of answer you get. The next ones out decide where it goes — about five tokens later.

Preprint: https://arxiv.org/abs/2608.12447; Supplementary materials; Code.

What do we mean when we say that a transformer model has privileged geometry? I honestly wasn't sure about that when I started down this rabbit hole, because that wasn't the initial point of the work.

If you want to jump right to the most unexpected finding, scroll down to #8 where I show temporal stratification of output behavior in residual stream geometry.

I began the work that led to this paper with the intention of trying to better understand the undifferentiated bulk of the residual stream that is not directly connected to the prediction mechanism. This was my original motivation for looking at the geometry after removing the prediction direction. Around the same time, I learned about thin-shell geometry, and realized there was likely a connection with residual stream geometry, and the rabbit hole suddenly got deeper.

Much of the work in the paper was born from a process of getting experimental findings that don't align with expectations and then figuring out the next experiment to better understand the data. It was only after I had run nearly all the experiments that I began to think about the term privileged. I didn't make the decision to use that term lightly, but I believe the data warrants it. More importantly, while deciding whether to use that term I realized that there doesn't appear to be a community consensus about what exactly privileged should mean with respect to residual stream geometry. Consequently, part of my motivation for this post is to start that discussion. I will lay out what I've found and give you access to the code so there can be a discussion about this topic that may have broad ramifications.

What has already been called privileged?

Elhage et al (2023) showed that transformer models care about the orientation of their geometry. You can't just rotate the residual stream vectors and rotate the weights the same way, even though theory says you should be able to do that (van Nierop, 2024). Elhage and his group traced most of the effect to normalization done by the Adam optimizer and called it a privileged basis.

Then in June 2026 a group of researchers from Anthropic came out with fascinating work that identified a narrow set of directions that they believe act as the intermediary between surface level prediction and deep computation in large language models (Gurnee et al. 2026). They also labeled this structure as privileged. It is interesting that Gurnee et al.'s work is also derived from a logit-lens based methodology, but the differences between their methods and mine are vast and so the results are not directly comparable.

What I am calling privileged.

The results of my work show that transformer residual streams contain a tightly stratified structuring of geometry that aligns with variance proximity to the model's current prediction direction. The stratification affects prompt discrimination in unexpected ways, it orders manifold complexity and it has functional and behavioral consequences when it is intervened upon. This set of qualities makes the geometry I found clearly different than either of the geometries laid out by Elhage et al. and Gurnee et al. I made the decision to call it privileged, because the geometry I found is distinct from the rest of the residual stream and is consistent across 18 models from six architectures including base and instruction tuned models with a 2.8x range in model size.

I am not putting all the results from the preprint in this post. The paper (Guda 2026) contains much more data and analysis. I am making all my code available so the community can run the same experiments on other models to see if these findings really are universal to transformer architectures as they seem to be from my experiments. I am particularly interested in seeing whether we see the same pattern on larger models.

Here is the rabbit hole I'm going to take you down:

1. What is the prediction direction;
2. How I decompose the residual stream;
3. Why rotation is more important than magnitude;
4. The prediction is almost orthogonal to the principal variance axes;
5. The dimensionality of the directions that are high variance relative to the prediction is surprisingly narrow;
6. Manifold complexity is higher nearer the prediction;
7. Discrimination declines and then reverses as you move away from the prediction;
8. Model behavior stratifies temporally as you move away from the prediction;
9. Is this privilege?
10. The end of the rabbit hole?
11. A few open questions.

1. What is the prediction direction?

My experiments are based on the logit lens methodology (nostalgebraist, 2020; Belrose et al., 2023) to view the residual stream from the perspective of the token that the model is going to predict given greedy generation. The prediction direction is the unembedding row for that token, but I use it as an analytical anchor for assessing the residual variance after removing the direction with a principal components analysis. The prediction direction itself is not a content carrying direction within the residual stream. The direction of the unembedding row for the token "Paris" is not the direction that carries the concept of Paris in a model which gets surfaced during computation. A token wins over other competing tokens by having more of the residual stream pointing at it, but this happens as a cumulation of influence rather than a single content direction of high magnitude.

2. Prediction anchoring methodology

The method I developed to look at the residual stream is based on looking at the variance in residual stream geometry among structured sets of prompts. It uses principal component analysis to remove the effect of the prediction direction thus separating the remaining variance as it relates to that direction. I call the method PDSF (Prediction-anchored Decomposition into Functional Subspaces). After each PC analysis, I create bins by determining the participation ratio (PR) of the residual, which gives a measure of effective dimensionality or how many variations carry the majority of variance. Using this method we can decompose the hidden state h into four pairwise-orthogonal components that sum exactly to h: P (the rank-1 prediction direction), D (dominant variance orthogonal to P), S (the next variance layer), and F (the remainder — thousands of ambient dimensions, >99% of the stream, except the two GPT-OSS models at 98.99%). Importantly, D, S and F are measurement bins only, and I'll show later that they are cuts through a continuum rather than model boundaries. For full details on the method see Guda (2026).

Two important notes:

• The PDSF method is relatively simple and only imposes two properties on the decomposition: Orthogonality to P and sequential variance ordering. So D>S>F in variance and all are orthogonal to P and to each other.
• I use the term proximity in this paper to refer to proximity along that variance ordering. So geometry that is "near" P is geometry whose variance is high relative to P.

I used three sets of prompts in this work to assess how models architect the residual stream in response to variation in semantic structure at different scales and across different structural groupings.

• SpecA is a set of 224 short questions whose meaning and single-token answer are held constant. The prompts are in 14 groups of 16 variants (arithmetic; logic, factual recall, and linguistics) in a 2⁴ full factorial over four surface factors (paraphrase, answer-constraint phrasing, a clutter preamble, and output-format instruction).
• SpecB consists of 96 open-ended narrative continuation prompts in 12 semantic categories. However in the behavioral experiments outlined below used an 80 prompt standard tier from this set.
• "Diverse" is a prompt set that contains 84 prompts across 21 groups that span eight processing regimes (English narrative, English analytical, Romance-language narrative, East-Asian language narratives, code completion, formal mathematics, structured instructions, and unusual register).

3. There is a geometric reason to focus on direction rather than magnitude

High dimensional geometries do not behave the way we expect them to based on our experience with two or three dimensions. Specifically, as the number of dimensions increase, the standard deviation decreases relative to the radius and variation spreads in a shell around the mean rather than a cloud. The thickness of that shell then stays roughly constant while the radius grows with the square root of the number of dimensions. So by the time you reach 1000 dimensions, the SD is roughly 2.2% of the radius, compared to 50% in two dimensions. It is called the thin shell theorem, because as dimensionality gets very high radial variation diminishes such that variation of any one dimension becomes more and more confined to the shell. The math for this isn't terribly complex, but it is entirely counterintuitive. For a more complete explanation see Vershynin (2018) and appendix B.1 of Guda 2026.

When I first came across thin shell theory I was surprised that I couldn't find any references to it in the residual stream literature. This seemed important to understand given that I was looking at how variance is distributed in residual stream directions. It was not a given that residual streams follow this, because thin shell theorem expects a normal distribution. However it seemed likely given that the model applies a norm scaling to the residual stream at every layer.

I tested the geometry on a subset of seven models from my testbed and found that they all have residual stream variance patterns that fall within the expected boundaries of a normal thin shell expectation (full results in Appendix B.1, Guda 2026). This table below is a slightly different way of looking at the numbers in Appendix B.1, that gives a little extra information. The models I tested fall within the range of thin-shell expectations, but they are all on the low side. That is, residual streams do have thin shell geometry, but it is a thicker shell than would be expected if their dimensions were normally distributed. The "Shell-implied effective d" shows mean number of dimensions you would expect given the width of the model's shell. In short, residual streams have some latitude for variation in magnitude, but not much. Most of the variation happens in direction.

I also ran causal and behavioral tests to ascertain that the difference between angular variation and magnitude are meaningful, and they are (results in Section 5 of the preprint). In other words, the residual stream has far more room to vary in angular direction than it does in magnitude. The difference shows up in both KL divergence at the output and in behavior. This finding may seem tangential (no pun intended), but it sets the stage for much of what I report on here and in the preprint.

Note: The idea that direction is more important than magnitude is not novel. A number of other researchers have already found that direction is more important than magnitude including Vardhan & Sai Teja (2026) who found that angular perturbations to hidden states damage language modeling far more than magnitude-matched ones. However, to my knowledge nobody has yet explained why that is. This is likely the explanation.

4. The prediction direction is almost orthogonal to the principal variance axes

The finding that residual stream geometry operates on a thin shell agrees with the hypothesis that token choice happens because a given logit has more directions pointed at it than competing logits, not because it receives support from one direction with a higher magnitude.

However, the thin shell finding does not predict what you see when you look at how the prediction direction compares to the rest of the residual stream. My measurements show that in all models and prompt sets, the prediction direction sits roughly 84–87° from the principal variance axes, averaged over depth. If this doesn't strike you as odd, consider that the entire residual stream feeds into the unembedding matrix to develop the distribution from which the next token is chosen. The fact that the prediction direction is nearly orthogonal to the principal variance axes means that the directions carrying the largest cross-prompt variance point largely away from the prediction. The implications of this are quite broad, and I go into many of them in the preprint.

The main thing tells us that is important for this post is that prediction anchoring does not simply rediscover the geometry that PCA would find. Instead, it reveals an organization that standard variance ranking alone does not recover.

5. The geometry near prediction stays narrow as models widen, but expands with the diversity of what they are asked to distinguish

If you run the PDSF code on a different model, the first thing that you should notice is that the effective dimensionality (widths) of the D and S subspaces near the prediction are constrained. Across the models in the testbed, there was no relationship between subspace width and model size. However, there were differences between prompt sets.

As models grow in dimensionality they do not allocate those dimensions to variance near the prediction direction. Instead, the dimensionality is absorbed into the high dimensional, prediction distal part of the residual stream. On SpecA, across the ten instruction-tuned models, the effective dimensionality of D is only 3.97 ± 0.75, and S is 10.5 ± 1.5. Figure 2 shows the lack of relationship between effective dimensionality and model size.

On the much more heterogeneous SpecB prompt set, the underlying P-removed effective dimensionality is 13.6 with a broader range 5.9–23.4. At first I interpreted this as an effect of prompt complexity, but further experiments since I wrote up the preprint suggest something more specific. Because these measurements are done on a population of prompts, the effective dimensionality is a measure of the diversity across prompts. When you decompose the SpecB variance into within and between group components, the SpecB measurements end up with the roughly same narrow dimensionality as SpecA. This raises a question about how diversity/complexity change prediction proximal variance within individual prompts that I was not able to answer in this study. Nevertheless, the finding holds that the region of high variance near the prediction is narrow and unchanging across a 2.8x increase in model width.

6. The closer we get to the prediction, the more dimensional folding exists

Given that the high variance region near the prediction direction is narrow and that the variation is largely directional, one might reasonably expect to find a higher information density per dimension. I would expect these regions to encode a wide range of information that must be packed into relatively few dimensions, and such packing pressure would predict a more folded or locally structured manifold geometry. I tested this hypothesis by assessing the manifold complexity of the subspaces with a kNN/global distance ratio, measured as the ratio of the mean Euclidean distance from a point to its k-th nearest neighbor vs the mean pairwise Euclidean distance across all points in the cloud. Measured this way, values near 1 indicate a flat, uniform geometry while lower values indicate tightly folded structure with points that are locally clustered relative to their global spread.

My measurements showed this to be the case. In all cases the D bins are more complex than S, which are more complex than F. I also looked at whether residual streams have a similar manifold complexity gradient when ordered by overall variance without removing P, by constructing equivalently sized bins drawn ordered from a straight PCA analysis. I also tested a shuffled control. The results showed that both the control and the P-anchored bins have manifold complexity gradients, but the gradient across the PDSF bins is steeper for all models. Figure 3 shows the manifold complexity gradient across model depth for all prompt sets and the comparison of the slope of manifold complexity between the PDSF bins and the controls.

There is another manifold complexity result that is interesting. As prompts become more heterogeneous the difference in complexity becomes larger, not smaller. The variance-ranked control recovers about 80% of the prediction-anchored manifold complexity gradient on the simplest prompt set, 71% on the semantically broader SpecB prompt set, and only 46% on the most diverse prompt set that contains prompts of entirely different regimes like math and East Asian languages. On the most heterogeneous set the prediction-anchored gradient runs 1.6 to 8.1 times steeper than the control.

To understand why it makes such a difference to view the geometry from a different angle, consider a simple three-dimensional analogy of how a mountain looks viewed from above versus from the side. From above we see little to none of the elevational change that we use to identify a mountain. Looking down from a plane you only recognize elevational features from their shadows or clues like snowpack. Without those clues, you have to be at the correct angle to truly see the elevational change and recognize the mountain. When we anchor our analysis on the prediction direction it doesn't give us a “correct” view of residual geometry, but my results show that it gives us a functionally informative one. Structure that is difficult to see when directions are ranked by overall variance becomes much more pronounced when they are organized relative to prediction.

This was the first data that made me consider the possibility that the structure I was measuring was privileged compared to the rest of the residual stream geometry.

7. The discrimination surprise

When language models predict a token, they must discriminate between different types of content. All the prompt sets I used have within and between group structure so that I could measure how models separate prompts geometrically based on these types of differences. The PDSF method defines the D, S and F bins using a nested PCA analysis inside the P-orthogonal residual by finding the highest variance differences first and then the next highest and so on. Because of this I assumed there would be an a priori expectation of a gradient from highly discriminatory near the prediction to less or no discrimination at the distal end. Prompts from the same group should tend to be more similar in D, less so in S, and eventually become indistinguishable from other groups as we move into F, the subspace with the least cross-prompt variance per dimension.

I used a Cohen's d test to assess how discrimination changes along the variance spectrum based on differences in angle between the geometries of each subspace across the population of each prompt set. A high Cohen's d number means the geometry separates the prompts by their groups and a low number means within and between group differences are not greatly different.

The gradient exists as I predicted, but the unexpected finding was that it crosses zero such that F anti-discriminates among prompts. Anti-discrimination means that prompts are more similar across groups than within groups (Figure 4).

Nothing in my experiments explains this anti-discrimination, only that it exists. One possibility, that I developed in the paper, is that F carries prompt-specific configuration rather than group-level organization. Another interpretation is that it may be related to the model pointing non-predicting directions away from the prediction. It is impossible to know what this means without further work. Regardless of the reasoning, the empirical result is clear. As we move away from the prediction direction, prompt-group discrimination doesn’t just fade, it reverses.

8. Model behavior stratifies temporally as you move away from the prediction

Everything up to here has been about geometry. How do models architect the geometry of the residual stream in response to variation in prompts. As interesting as that is, the important question is whether or not the differences play a functional role in the model. To test this I created an experiment in which models generate a response to the open-ended prompts of SpecB, and during the generation I intervene on each different subspaces using a fixed Haar-random orthogonal rotation applied to the target subspace's coefficients at every generation step, at roughly 70% of network depth. This rotational intervention preserves rank, norm and within-subspace energy while scrambling the directional organization. I chose to focus on rotational interventions, because the results of the thin shell assessment demonstrated that model variation lives in direction, not magnitude. I had the intervention apply continuously so there is no clean pass in which the model can put things back together (In the preprint I give results showing that models can reconstruct the D and S subspaces after a rotation intervention in early layers).

Because the F subspace was too large to easily apply a rotation at every pass, I extended the PDSF methodology to create an analytical bin of the highest variance directions within F, called that bin F-topK. I also ran a control experiment with a bin of 12 randomly chosen directions from the residual (statistically most likely to fall in F). Each intervention was compared against the baseline generation done with no intervention under greedy generation. The full details for the experiment are laid out in the preprint.

Two very clear results were immediately apparent. The first result was in timing of when the model output diverged from the baseline (Figure 5, panel A). Intervening on the D bin causes the model to change the output immediately 64% of the time (mean first-divergence token of 1.17 ± 0.39 ). In contrast, when S is intervened on the model produces the baseline response verbatim for several tokens before it diverges. The divergence response to scrambling F-topK was statistically indistinguishable from S-scramble at the model level, and the random control diverged far later than either D, S or F-topK with a mean divergence of 22.86 ± 5.03.

There were also significant differences in the kind of change in the output. I classified the outputs into four categories: Equivalent, Variant (same task framing, different content), Frame-shift (the baseline's framing is disrupted and a different kind of response comes out), and Failure. Scrambling D produced Frame-shift results in 42.3% of the outputs, far more than any of the other treatments. In contrast the S, F-topK and random treatments were dominated by Variant responses (Figure 5, panel B).

Below is one clear example of the type of frame shift vs variant we see in scrambling D vs S. The output from the D scramble is relevant to the prompt, but it is an entirely different type of response. The output from the S scramble is identical to the baseline for the first 30 tokens and then changes the content a little. You can explore the actual responses for three of the models in the interactive Figure 6 further down the page.

Example from Llama 8B:

Prompt: "The sleek spacecraft emerged from hyperspace, its hull scarred from the journey."

Baseline:

As the ship broke through the veil of hyperspace, the stars ahead snapped into focus, a brilliant tapestry of light and color that stretched out before the vessel like an endless canvas. The ship's hull, once a gleaming silver, now bore the scars of its journey, a testament to the unforgiving nature

D-scramble (first divergence token 0):

I'm happy to continue the story, but I want to make sure I understand the context. It seems like we're starting in the middle of a space opera or science fiction story. Here's a possible continuation:

The sleek spacecraft emerged from hyperspace, its hull scarred from the journey. The ship's engines,

S-scramble (first divergence token 31):

As the ship broke through the veil of hyperspace, the stars ahead snapped into focus, a brilliant tapestry of light and color that stretched out before the crew like a celestial canvas. The ship's instruments began to hum and whine as they adjusted to the sudden return to normal space, the gentle thrum of the

It's worth mentioning that almost none of the interventions produced garbage (D-scramble Failure was 6.1% and S-scramble 4.0%). So the functional contrast the intervention reveals isn't about the coherence of the output, it is about when the output changes and what kind of change it makes.The biggest takeaway in my opinion is the temporal dissociation. The fact that the high variance directions closest to the prediction seem to control immediate output while the directions in the next variance cut down change the output further out seems to have broad implications. The most immediately apparent implication is that the prediction direction alone does not hold the prediction. If it did, scrambling D would not change the first token.

Up to this point, my reading of the geometry results led me to believe that the ordering I'd found in manifold complexity and discrimination were measurements of an underlying gradient. However, the behavioral results gave some concrete evidence of a gradient versus actual subspaces. Scrambling F-topK produced behavior that was statistically indistinguishable from scrambling S at the model level in both divergence timing and content. This is despite the fact that F-topK is much more F in all the geometric measures. It has lower variance, it is anti-discriminatory, and it is also strictly orthogonal to S. To me this is clear evidence that the PDSF method is creating measurement cuts through a continuum rather than functional boundaries. If the S/F boundary marked a real functional division, crossing it should change the behavioral regime. It doesn’t. The major behavioral transition occurs earlier, between D and S, even though D, S and F-topK are simply successive cuts through the same ordered residual geometry.

Figure 6. Interactive figure to explore behavioral outputs

9. What does privileged mean, and what are the consequences of these results?

So, is privileged the right word? What I found is not a defined geometric space, and it is not a direction. The structure appears in every model I tested – small to large, base and instruction, dense and MoE. Here are the features again:

1. Stratified by variance and discrimination;
2. Dimensionally narrow regardless of model width;
3. More highly folded ( higher manifold complexity) than expected from the surrounding geometry;
4. Has clear functional implications related to the model's response framing, the next token and the tokens that come after it.

What are the consequences of this finding? While I hesitate to draw too many conclusions about the potential implications, at a minimum I think two of the findings are potentially relevant to the legibility of large language models. The structuring of high variance directions relative to the prediction is extremely narrow, and the majority of the residual stream points away from both the prediction and from this high variance region are both findings that may have ramifications that touch on our ability to interpret models via their output.

Many researchers have expressed concern about the future of model legibility. Korbak et al. (2025) warned that chain-of-thought monitorability is fragile, and Casper et al. (2024) argued that black-box access is not sufficient for rigorous auditing. Lindsey et al.'s (2025) direct attempts to trace circuits inside a frontier model reported insight on roughly a quarter of the prompts investigated. So legibility is a real concern, and it's possible that these results offer some perspective from a geometric perspective. Is it possible that there is a geometric barrier to understanding deep computation from output-based measures? Maybe.

Beyond that, I believe that the demonstration that residual stream geometry operates on a thin shell is useful and should become part of the way we approach work on these geometric spaces. Knowing that the majority of variation is angular or directional is important. I also think that the anti-discrimination finding is very significant in of itself. I have been pushing a little further on this idea, and the things that I am finding are interesting.

10. Beyond the prediction axis

That was one deep rabbit hole, but I've left one more dark corner to explore. This work changed my own feelings about what privileged geometry might be, and I'm now convinced that residual streams may contain various structures that operate differently than the rest of the geometry. The last part of my paper talks about one more finding that may be exactly that, although what I found is really just a teaser.

All the geometry we've talked about in this paper is static. I knew this while I was doing the work and did a brief couple of experiments to look at how these spaces change from token to token – call it the temporal axis – and what I found was extremely intriguing.

Specifically, what I did was to decompose the residual stream temporally from one token to the next to see what changes and what doesn't, specifically in the prediction distal part of the geometry. What I found is that the residual stream separates into two components: a near-static scaffold that remains more or less the same over successive passes, and a dynamic envelope of directions that change with every new generation and is essential for autoregressive generation. When the dynamic subspace is intervened on with the same persistent rotation that I used in the behavioral test above, it causes the model to fail 92.3% of the time. If it is frozen (using the same state from the previous token) the model collapses. This division of the prediction distal geometry and the effects were the same in every model I tested.

Is this privileged? That's a good question. It is clearly essential.

11. Open questions

I made all the code for this research open so that people can test my findings on other models and put all these assertions to the test. Below are some of the assertions that should still be tested. I'd love to work with another researcher who is interested in any of these questions or one of the many others that I haven't listed.

1. All of this work uses the known predicted token as an anchor from which to view the residual stream. It is important to what we see if we instead use a different token as a lens. The important ones to check would be a competing alternative token, a null token and a clearly wrong token.
2. The PCA control against which the PDSF manifold complexity is compared is not a true control, because it is not anchored on anything. This could be easily tested against other tokens and null tokens to see if anchoring the geometry recreates the same manifold complexity gradient. I'd like to know the answer to this, but I haven't gotten around to it yet.
3. GPT-OSS 120B and Mixtral both have behaviors that differ from the rest of the testbed in the scramble test. GPT-OSS 120B reverses the D/S immediate-divergence ordering (S 36.2% > D 28.8%) even though the mean-token ordering is preserved. Mixtral's F-topK Frame-shift reaches 15.0%, which is comparable to its own S-scramble rate. Why is this?
4. All the behavioral tests I report here are at one depth (~70%). More tests should be done at different depths to see if the behavioral gradient holds.
5. The scaling claim rests on a 2.8× range in residual width across models up to 120B parameters. I want to know what happens in bigger models.

References

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Casper, S., Ezell, C., Siegmann, C., Kolt, N., Curtis, T. L., Bucknall, B., et al. (2024). Black-box access is insufficient for rigorous AI audits. FAccT '24. arXiv:2401.14446.

Elhage, N., Lasenby, R., & Olah, C. (2023). Privileged bases in the transformer residual stream. Transformer Circuits Thread.

Guda, N (2026). Geometric and Behavioral Stratification in Transformer Residual Streams. arXiv:2608.12447 (https://arxiv.org/abs/2608.12447)

Gurnee, W., Sofroniew, N., Pearce, A., et al. (2026). Verbalizable Representations Form a Global Workspace in Language Models. Transformer Circuits Thread.

Korbak, T., Balesni, M., Barnes, E., Bengio, Y., Benton, J., Bloom, J., Chen, M., et al. (2025). Chain of thought monitorability: A new and fragile opportunity for AI safety. arXiv:2507.11473.

Lindsey, J., Gurnee, W., Ameisen, E., Chen, B., Pearce, A., et al. (2025). On the biology of a large language model. Transformer Circuits Thread.

Nikolaou et al. (2025). Language models are injective and hence invertible. arXiv:2510.15511v3.

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Vardhan, M. S., & Sai Teja, L. (2026). Disentangling direction and magnitude in transformer representations: A double dissociation through L2-matched perturbation analysis. arXiv:2602.11169.

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