pith. sign in

arxiv: 2605.27970 · v1 · pith:3RQZZDAQnew · submitted 2026-05-27 · 💻 cs.AI

Geometry of Human Perceptual Domains Emerges Transiently in LLM Representations

Simardeep Singh , Paras Chopra This is my paper

Pith reviewed 2026-06-29 12:28 UTC · model grok-4.3

classification 💻 cs.AI
keywords LLM representationsperceptual geometrylayer-wise emergenceresidual streamshuman perceptual domainstransformer modelsgeometric structureemergence profiles
0
0 comments X

The pith

LLM residual streams develop human-like perceptual geometry in middle layers that later attenuates.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper investigates whether embeddings inside text-trained LLMs contain geometric structures that match human perceptual organization in domains such as color, pitch, emotion, and taste. It measures this structure layer by layer inside the residual streams of several open transformer models. The analysis finds that geometry corresponding to these domains appears despite no perceptual training, follows distinct trajectories by domain and model, and follows a shared pattern of weak early layers, stronger organization in intermediate layers, and reduced structure in later layers. This pattern indicates the geometry forms transiently during the model's internal processing. A reader would care because the work locates where human-like perceptual organization arises inside purely linguistic models.

Core claim

The paper claims that geometric structure corresponding to perceptual modalities emerges transiently in the residual streams of transformer LLMs: it is weak or diffuse in early layers, becomes progressively organised in intermediate layers, and is attenuated in later layers, with both the structure itself and its alignment to human baselines showing domain- and model-specific trajectories across depth.

What carries the argument

Layer-wise measurement of geometric structure in residual-stream embeddings compared against human perceptual baselines across multiple domains.

Load-bearing premise

Quantitative measures of geometric structure in residual-stream embeddings can be meaningfully compared to human perceptual baselines across domains using appropriate similarity metrics.

What would settle it

Observing a model or perceptual domain in which geometric alignment with human baselines does not strengthen in intermediate layers and weaken in later layers would falsify the claimed consistent representational trajectory.

Figures

Figures reproduced from arXiv: 2605.27970 by Paras Chopra, Simardeep Singh.

Figure 1
Figure 1. Figure 1: Perceptual geometric structures emerge within LLM representations across sensory and affective domains (taste, pitch, color, and emotion). Simpler structures (taste) peak earlier, fol￾lowed progressively by pitch, color, and emotion. Across domains, geometry is weak in early layers, organizes in intermediate layers, and attenuates in later layers. Despite lacking direct sensory grounding, these models of￾t… view at source ↗
Figure 2
Figure 2. Figure 2: Overview of perceptual geometry emergence across modalities. Each row corresponds to a perceptual domain—color (top, LLaMA-3-8B), pitch (middle, Qwen-3-4B), and emotion (bottom, Gemma-7B), with the clearest representative model shown for each. Columns show the human perceptual baseline (left), the peak-alignment geometric representation from the model (middle), and the corresponding layer-wise alignment pr… view at source ↗
Figure 3
Figure 3. Figure 3: Comparison of the human perceptual map (top left) and peak-layer LLM representation for taste in Gemma-7B (top right). Bottom: layer-wise alignment scores (RSA and GPA). 2. Methodology 2.1. Overview We investigate whether perceptual geometric structures exist within the embedding space of LLMs which aligns with the human similarity judgements. Our approach is fully intrinsic, requiring no probing or additi… view at source ↗
Figure 4
Figure 4. Figure 4: Representative prompt completions for color and emotion. Each panel shows the queried prompt together with the top descriptor phrases produced by Qwen3-8B, LLaMA-3-8B, and Gemma-7B. 6 [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Representative prompt completions for pitch and taste. Each panel shows the queried prompt together with the top descriptor phrases produced by Qwen3-8B, LLaMA-3-8B, and Gemma-7B. A.3. Human Baseline Dataset Collection The datasets used in this work were collected from established sources of human perceptual similarity judgments. For color, pitch, and taste, we use datasets compiled in (Marjieh et al., 202… view at source ↗
Figure 6
Figure 6. Figure 6: Layer-wise emergence of 2D color geometry in LLaMA-3-8B across model depth. Panels show (top-left) human perceptual geometry, (top-right) early-layer representation, (bottom-left) peak-alignment layer, and (bottom-right) final-layer MDS representation [PITH_FULL_IMAGE:figures/full_fig_p008_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Layer-wise emergence of 2D color geometry in Qwen-3-4B across model depth. Panels show (top-left) human perceptual geometry, (top-right) early-layer representation, (bottom-left) peak-alignment layer, and (bottom-right) final-layer MDS representation. 8 [PITH_FULL_IMAGE:figures/full_fig_p008_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Layer-wise emergence of 3D color geometry in LLaMA-3-8B across model depth. Panels show (top-left) human perceptual geometry, (top-right) early-layer representation, (bottom-left) peak-alignment layer, and (bottom-right) final-layer MDS representation [PITH_FULL_IMAGE:figures/full_fig_p009_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Layer-wise emergence of 2D color geometry in LLaMA-3-8B across model depth. Panels show (top-left) human perceptual geometry, (top-right) early-layer representation, (bottom-left) peak-alignment layer, and (bottom-right) final-layer isomap representation. 9 [PITH_FULL_IMAGE:figures/full_fig_p009_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Layer-wise emergence of 3D color geometry in LLaMA-3-8B across model depth. Panels show (top-left) human perceptual geometry, (top-right) early-layer representation, (bottom-left) peak-alignment layer, and (bottom-right) final-layer isomap representation [PITH_FULL_IMAGE:figures/full_fig_p010_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Layer-wise emergence of emotion geometry in Gemma-7B across model depth. Panels show (top-left) human perceptual geometry, (top-right) early-layer representation, (bottom-left) peak-alignment layer, and (bottom-right) final-layer MDS representation. 10 [PITH_FULL_IMAGE:figures/full_fig_p010_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Layer-wise emergence of emotion geometry in Gemma-7B across model depth. Panels show (top-left) human perceptual geometry, (top-right) early-layer representation, (bottom-left) peak-alignment layer, and (bottom-right) final-layer isomap representation [PITH_FULL_IMAGE:figures/full_fig_p011_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Layer-wise emergence of Taste geometry in Gemma-7B across model depth. Panels show (top-left) human perceptual geometry, (top-right) early-layer representation, (bottom-left) peak-alignment layer, and (bottom-right) final-layer MDS representation. 11 [PITH_FULL_IMAGE:figures/full_fig_p011_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Layer-wise emergence of Taste geometry in Gemma-7B across model depth. Panels show (top-left) human perceptual geometry, (top-right) early-layer representation, (bottom-left) peak-alignment layer, and (bottom-right) final-layer isomap representation. 1046.5 Hz 261.63 Hz [PITH_FULL_IMAGE:figures/full_fig_p012_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: Layer-wise emergence of 2D Pitch geometry in Qwen-3-4B across model depth. Panels show (top-left) human perceptual geometry, (top-right) early-layer representation, (bottom-left) peak-alignment layer, and (bottom-right) final-layer MDS representation. 12 [PITH_FULL_IMAGE:figures/full_fig_p012_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: Layer-wise emergence of 2D Pitch geometry in Gemma-7B across model depth. Panels show (top-left) human perceptual geometry, (top-right) early-layer representation, (bottom-left) peak-alignment layer, and (bottom-right) final-layer MDS representation. 1046.5 Hz 261.63 Hz [PITH_FULL_IMAGE:figures/full_fig_p013_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: Layer-wise emergence of 3D Pitch geometry in Qwen-3-4B across model depth. Panels show (top-left) human perceptual geometry, (top-right) early-layer representation, (bottom-left) peak-alignment layer, and (bottom-right) final-layer MDS representation. 13 [PITH_FULL_IMAGE:figures/full_fig_p013_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: Layer-wise emergence of 3D Pitch geometry in Gemma-7B across model depth. Panels show (top-left) human perceptual geometry, (top-right) early-layer representation, (bottom-left) peak-alignment layer, and (bottom-right) final-layer MDS representation. 1046.5 Hz 261.63 Hz [PITH_FULL_IMAGE:figures/full_fig_p014_18.png] view at source ↗
Figure 19
Figure 19. Figure 19: Layer-wise emergence of 2D Pitch geometry in Qwen-3-4B across model depth. Panels show (top-left) human perceptual geometry, (top-right) early-layer representation, (bottom-left) peak-alignment layer, and (bottom-right) final-layer isomap representation. 14 [PITH_FULL_IMAGE:figures/full_fig_p014_19.png] view at source ↗
Figure 20
Figure 20. Figure 20: Layer-wise emergence of 3D Pitch geometry in Qwen-3-4B across model depth. Panels show (top-left) human perceptual geometry, (top-right) early-layer representation, (bottom-left) peak-alignment layer, and (bottom-right) final-layer isomap representation. B.2. Layer-wise Metric Profiles To complement the qualitative geometric visualizations, we report additional layer-wise alignment profiles for each stimu… view at source ↗
Figure 21
Figure 21. Figure 21: Layer-wise metric profiles for color across four models: (Llama-3-8B) model 1, (Llama-3.2-3B) model 2, (Gemma-7B) model 3, and (Qwen-3-4B) model 4. 15 [PITH_FULL_IMAGE:figures/full_fig_p015_21.png] view at source ↗
Figure 22
Figure 22. Figure 22: Layer-wise metric profiles for Emotion across four models: (Llama-3-8B) model 1, (Llama-3.2-3B) model 2, (Gemma-7B) model 3, and (Qwen-3-4B) model 4 [PITH_FULL_IMAGE:figures/full_fig_p016_22.png] view at source ↗
Figure 23
Figure 23. Figure 23: Layer-wise metric profiles for Taste across four models: (Llama-3-8B) model 1, (Llama-3.2-3B) model 2, (Gemma-7B) model 3, and (Qwen-3-4B) model 4. 16 [PITH_FULL_IMAGE:figures/full_fig_p016_23.png] view at source ↗
Figure 24
Figure 24. Figure 24: Layer-wise metric profiles for Pitch across four models: (Gemma-7B) model 1, (Qwen-3-4B) model 2, (Llama-3-8B) model 3, and (Llama-3.2-3B) model 4. B.3. Bootstrap Confidence Intervals for Layer-wise Alignment To assess the statistical stability of the observed layer-wise alignment profiles, we estimate bootstrap confidence intervals for both RSA and GPA scores by resampling stimuli with replacement at eac… view at source ↗
Figure 25
Figure 25. Figure 25: Bootstrap confidence intervals for color in LLaMA-3-8B, showing layer-wise RSA (left) and GPA (right) profiles with 95% confidence bands [PITH_FULL_IMAGE:figures/full_fig_p017_25.png] view at source ↗
Figure 26
Figure 26. Figure 26: Bootstrap confidence intervals for Pitch in Qwen-3-4B, showing layer-wise RSA (left) and GPA (right) profiles with 95% confidence bands. 17 [PITH_FULL_IMAGE:figures/full_fig_p017_26.png] view at source ↗
Figure 27
Figure 27. Figure 27: Bootstrap confidence intervals for emotion in Gemma-7B, showing layer-wise RSA (left) and GPA (right) profiles with 95% confidence bands [PITH_FULL_IMAGE:figures/full_fig_p018_27.png] view at source ↗
Figure 28
Figure 28. Figure 28: Bootstrap confidence intervals for taste in Gemma-7B, showing layer-wise RSA (left) and GPA (right) profiles with 95% confidence bands. 18 [PITH_FULL_IMAGE:figures/full_fig_p018_28.png] view at source ↗
read the original abstract

While large language models (LLMs) are trained purely on textual data, prior work has shown that their internal representations can exhibit rich geometric structure in embedding space. Building on this line of work, we investigate whether such structure is similar to human perceptual organisation across different domains (e.g., color, pitch, emotion, and taste). Specifically, we study the layer-wise emergence of intrinsic geometrical structure corresponding to perceptual modalities within the residual streams of multiple open-weight transformer architectures. Our results reveal three key findings. First, we observe the emergence of layer-wise geometric structure across multiple perceptual domains, despite the absence of any direct perceptual supervision during training. Second, these perceptual domains exhibit distinct emergence profiles, with both geometric structure and its alignment with human baselines following domain- and model-specific trajectories across depth. Third, this emergence follows a consistent representational trajectory: geometry is weak or diffuse in early layers, becomes progressively organised in intermediate layers, and is attenuated in later layers, suggesting that perceptual geometry arises transiently as part of the model's internal transformation pipeline. This provides new insight into how and where human-like perceptual geometry arises in LLMs, offering a principled pathway for mechanistic analysis of internal representations.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 2 minor

Summary. The paper examines layer-wise emergence of geometric structure in the residual streams of multiple open-weight transformer LLMs, focusing on whether this structure corresponds to human perceptual organization across domains such as color, pitch, emotion, and taste. It reports three main findings: (1) geometric structure emerges despite purely textual training, (2) domains show distinct emergence profiles with model-specific trajectories in both intrinsic geometry and alignment to human baselines, and (3) the structure is weak early, peaks in intermediate layers, and attenuates later, indicating transient emergence within the model's transformation pipeline.

Significance. If the quantitative alignments hold under rigorous controls, the work supplies a concrete, layer-resolved map of how perceptual-like geometry arises in LLMs. This supplies a falsifiable, depth-dependent signature that can be used for mechanistic interpretability and for testing whether such geometry is an incidental byproduct of next-token prediction or a functional intermediate representation.

major comments (3)
  1. [§4] §4 (Results on human alignment): the central claim that measured geometries align with human perceptual baselines requires explicit specification of the human similarity matrices or perceptual datasets employed for each domain and the precise comparison procedure (e.g., Mantel test, Procrustes, or nearest-neighbor overlap). Without these details or controls for textual co-occurrence statistics, the reported trajectories could be driven by linguistic rather than perceptual structure.
  2. [§3.2] §3.2 (Metric definitions): the intrinsic geometric measures (whatever distance or manifold statistics are applied to residual-stream embeddings of domain words) are not shown to be robust to alternative choices of stimuli or to control domains lacking perceptual structure; this leaves open whether the reported transient peak is specific to perceptual domains or an artifact of word-selection or embedding-norm effects.
  3. [Figure 3] Figure 3 / Table 2 (trajectory plots): the domain- and model-specific emergence profiles are presented without statistical tests for the intermediate-layer peak or for the attenuation in later layers; if the peak is within the noise envelope of early/late layers for some domains, the “transient” characterization is not yet load-bearing.
minor comments (2)
  1. [Abstract] The abstract states “despite the absence of any direct perceptual supervision,” but the manuscript should cite the exact training corpora and confirm that no perceptual or multimodal data were present.
  2. [§3] Notation for residual-stream indices and layer numbering should be standardized across figures and text to avoid ambiguity when comparing models of different depths.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the constructive comments, which identify opportunities to improve clarity, robustness, and statistical support. We address each major point below, indicating revisions that will be incorporated.

read point-by-point responses

  1. Referee: §4 (Results on human alignment): the central claim that measured geometries align with human perceptual baselines requires explicit specification of the human similarity matrices or perceptual datasets employed for each domain and the precise comparison procedure (e.g., Mantel test, Procrustes, or nearest-neighbor overlap). Without these details or controls for textual co-occurrence statistics, the reported trajectories could be driven by linguistic rather than perceptual structure.

    Authors: We agree that full specification of the human baselines, comparison procedures, and linguistic controls is required for interpretability. The revised manuscript will add an explicit methods subsection listing the exact perceptual datasets (color similarities from established psychophysical matrices, pitch from musical interval studies, emotion from valence-arousal norms, taste from gustatory similarity ratings), the alignment metric (Procrustes superposition with permutation-based significance testing), and new controls that subtract co-occurrence statistics derived from the pretraining corpus. These additions will directly address whether the observed alignments exceed what linguistic statistics alone would predict. revision: yes

  2. Referee: §3.2 (Metric definitions): the intrinsic geometric measures (whatever distance or manifold statistics are applied to residual-stream embeddings of domain words) are not shown to be robust to alternative choices of stimuli or to control domains lacking perceptual structure; this leaves open whether the reported transient peak is specific to perceptual domains or an artifact of word-selection or embedding-norm effects.

    Authors: We accept that additional robustness demonstrations are needed. The revision will include (i) results with alternative stimulus sets drawn from independent lexical resources for each domain, (ii) parallel analyses on matched control domains lacking perceptual structure (e.g., abstract nouns matched for frequency and length), and (iii) recomputation after L2-normalization of embeddings to isolate geometry from norm effects. These controls will be reported alongside the original measures to test specificity of the transient peak. revision: yes

  3. Referee: Figure 3 / Table 2 (trajectory plots): the domain- and model-specific emergence profiles are presented without statistical tests for the intermediate-layer peak or for the attenuation in later layers; if the peak is within the noise envelope of early/late layers for some domains, the “transient” characterization is not yet load-bearing.

    Authors: We concur that formal statistical assessment of the peaks and attenuation is necessary. The revised figures and tables will incorporate bootstrap-derived confidence intervals around each layer’s geometric measure, together with permutation tests comparing the intermediate-layer value against the distribution of early- and late-layer values. These tests will be performed per domain and model, allowing readers to evaluate whether the transient pattern is statistically supported or falls within noise. revision: yes

Circularity Check

0 steps flagged

No circularity; claims rest on independent empirical measurements.

full rationale

The abstract and provided text describe empirical observations of layer-wise geometric structure in residual-stream embeddings across perceptual domains, with alignment to human baselines reported as a measured outcome rather than a definitional or fitted input. No equations, parameter-fitting procedures, self-citations, or uniqueness theorems are visible that would reduce any prediction to its own inputs by construction. The derivation chain consists of direct analysis of model internals against external human data, with no self-definitional steps or renamings of known results presented as novel derivations.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review yields no identifiable free parameters, axioms, or invented entities; all such elements would require the methods and results sections.

pith-pipeline@v0.9.1-grok · 5732 in / 1126 out tokens · 22955 ms · 2026-06-29T12:28:41.004109+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

16 extracted references · 16 canonical work pages · 4 internal anchors

  1. [1]

    Alain, G

    URL https://arxiv.org/abs/ 2109.06129. Alain, G. and Bengio, Y . Understanding intermediate layers using linear classifier probes,

    work page arXiv

  2. [2]

    Understanding intermediate layers using linear classifier probes

    URL https:// arxiv.org/abs/1610.01644. Bradley, M. M. and Lang, P. J. Affective norms for english words (anew): Instruction manual and affective ratings. Technical report. URL https://pdodds.w3.uvm.edu/teaching/ courses/2009-08UVM-300/docs/others/ everything/bradley1999a.pdf. Choi, B. J. and Weber, M. Latent structure of affective representations in large...

    work page internal anchor Pith review Pith/arXiv arXiv 2009

  3. [3]

    Latent Structure of Affective Representations in Large Language Models

    URL https://arxiv.org/abs/2604.07382. Dar, G., Geva, M., Gupta, A., and Berant, J. Analyzing transformers in embedding space,

    work page internal anchor Pith review Pith/arXiv arXiv

  4. [4]

    Engels, J., Michaud, E

    URL https: //arxiv.org/abs/2209.02535. Engels, J., Michaud, E. J., Liao, I., Gurnee, W., and Tegmark, M. Not all language model features are one- dimensionally linear,

    work page arXiv

  5. [5]

    org/abs/2405.14860

    URL https://arxiv. org/abs/2405.14860. Gurnee, W. and Tegmark, M. Language models represent space and time,

    work page arXiv

  6. [6]

    Gurnee, W., Ameisen, E., Kauvar, I., Tarng, J., Pearce, A., Olah, C., and Batson, J

    URL https://arxiv.org/ abs/2310.02207. Gurnee, W., Ameisen, E., Kauvar, I., Tarng, J., Pearce, A., Olah, C., and Batson, J. When models manipulate manifolds: The geometry of a counting task,

    work page arXiv

  7. [7]

    Gurnee, E

    URL https://arxiv.org/abs/2601.04480. Karkada, D., Korchinski, D. J., Nava, A., Wyart, M., and Bahri, Y . Symmetry in language statistics shapes the geometry of model representations,

    work page arXiv

  8. [8]

    Kruskal, J

    URL https: //arxiv.org/abs/2602.15029. Kruskal, J. B. Multidimensional scaling by optimizing goodness of fit to a nonmetric hypothesis.Psychometrika, 29(1):1–27, Mar

    work page arXiv

  9. [9]

    Marjieh, R., Sucholutsky, I., van Rijn, P., Jacoby, N., and Griffiths, T

    URL https: //arxiv.org/abs/2502.10871. Marjieh, R., Sucholutsky, I., van Rijn, P., Jacoby, N., and Griffiths, T. L. Large language models predict human sensory judgments across six modalities,

    work page arXiv

  10. [10]

    Modell, A., Rubin-Delanchy, P., and Whiteley, N

    URL https://arxiv.org/abs/2302.01308. Modell, A., Rubin-Delanchy, P., and Whiteley, N. The ori- gins of representation manifolds in large language mod- els,

    work page arXiv

  11. [11]

    URL https://www.researchgate.net/ publication/235361517_A_circumplex_ model_of_affect

    doi: 10.1037/h0077714. URL https://www.researchgate.net/ publication/235361517_A_circumplex_ model_of_affect. Singh, C., Inala, J. P., Galley, M., Caruana, R., and Gao, J. Rethinking interpretability in the era of large lan- guage models,

    work page doi:10.1037/h0077714

  12. [12]

    Skean, O., Arefin, M

    URL https://arxiv.org/ abs/2402.01761. Skean, O., Arefin, M. R., LeCun, Y ., and Shwartz-Ziv, R. Does representation matter? exploring intermedi- ate layers in large language models,

    work page arXiv

  13. [13]

    Skean, O., Arefin, M

    URL https: //arxiv.org/abs/2412.09563. Skean, O., Arefin, M. R., Zhao, D., Patel, N., Naghiyev, J., LeCun, Y ., and Shwartz-Ziv, R. Layer by layer: Uncov- ering hidden representations in language models,

    work page arXiv

  14. [14]

    Layer by Layer: Uncovering Hidden Representations in Language Models

    URLhttps://arxiv.org/abs/2502.02013. Sun, L., Yan, L., Lu, X., Lee, A., Zhang, J., and Shao, J. Valence-arousal subspace in llms: Circular emo- tion geometry and multi-behavioral control,

    work page internal anchor Pith review Pith/arXiv arXiv

  15. [15]

    Valence-Arousal Subspace in LLMs: Circular Emotion Geometry and Multi-Behavioral Control

    URL https://arxiv.org/abs/2604.03147. Taniguchi, T., Oizumi, M., Saji, N., Horii, T., and Tsuchiya, N. Constructive approach to bidirectional influence be- tween qualia structure and language emergence,

    work page internal anchor Pith review Pith/arXiv arXiv

  16. [16]

    The description of the color given as#HEXCODE

    URLhttps://arxiv.org/abs/2409.09413. 5 Transient Emergence of Perceptual Geometry in LLM Representations A. Appendix A.1. Prompts Used We use a set of minimal, structured prompts to extract representations corresponding to each domain. The list of prompts used across modalities is provided below: Color:“The description of the color given as#HEXCODE” Emoti...

    work page arXiv 2023