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arxiv: 2605.14125 · v2 · pith:4SJQMRSFnew · submitted 2026-05-13 · 💻 cs.CL

Polar probe linearly decodes semantic structures from LLMs

Pablo J. Diego-Sim\'on , Pierre Orhan , Emmanuel Chemla , Yair Lakretz , Jean-R\'emi King This is my paper

Pith reviewed 2026-05-20 20:20 UTC · model grok-4.3

classification 💻 cs.CL
keywords semantic structureslarge language modelspolar probelinear decodingrelation embeddingsconcept bindingneural codesactivations
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The pith

Large language models encode semantic relations between entities as distances and directions between their embeddings.

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

The paper tests whether LLMs represent relations in semantic structures through a simple geometrical rule in their internal activations. Distance between entity embeddings signals that a relation exists, while direction signals its type. The authors introduce a Polar Probe that linearly extracts these structures from a subspace of activations in models processing short task descriptions across arithmetic, visual scenes, family trees, metro maps, and social interactions. This representation appears strongest in middle layers, improves as models get better at the tasks, generalizes to new entities and relation types, and predicts how well models answer questions about the structures. The findings point to a basic binding mechanism that LLMs may use to build complex meaning from concepts.

Core claim

We propose a simple neural code, whereby the existence and the type of relations between entities are represented by the distance and the direction between their embeddings, respectively. We test this hypothesis in a variety of Large Language Models (LLMs), each input with natural-language descriptions of minimalist tasks from five different domains: arithmetic, visual scenes, family trees, metro maps and social interactions. Results show that the true semantic structures can be linearly recovered with a Polar Probe targeting a subspace of LLMs' layer activations. This code emerges mostly in middle layers and improves with LLM performance. These Polar Probes successfully generalize to new实体s

What carries the argument

The Polar Probe, a linear decoder applied to a subspace of LLM layer activations to recover the polar geometry of semantic structures where distance indicates relation existence and direction indicates relation type.

If this is right

  • The geometrical code for binding relations appears primarily in middle layers rather than early or late ones.
  • Higher-quality polar representations track better overall performance of the LLM on the semantic tasks.
  • The probes generalize to new entities and relation types within the tested domains.
  • Recovery accuracy drops as the number of entities and relations in a structure increases.
  • Stronger polar decoding in activations corresponds to better model performance on questions about the semantic structures.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • This binding principle could be tested by checking whether models lose semantic understanding when activations are perturbed along the polar directions.
  • The same probe approach might reveal whether non-LLM neural networks use similar distance-direction codes for relations.
  • If the code holds in open text, it would suggest LLMs build meaning without needing explicit symbolic structures.
  • Larger structures degrading the signal points to a possible limit on how complex a representation one layer subspace can hold.

Load-bearing premise

The minimalist natural-language descriptions of tasks in five domains are sufficient to reveal the general mechanism by which LLMs bind concepts into semantic structures in open-ended text.

What would settle it

If a Polar Probe trained on activations from these five minimalist domains fails to recover accurate semantic structures when applied to the same models processing longer, open-ended natural language text.

Figures

Figures reproduced from arXiv: 2605.14125 by Emmanuel Chemla, Jean-R\'emi King, Pablo J. Diego-Sim\'on, Pierre Orhan, Yair Lakretz.

Figure 1
Figure 1. Figure 1: Polar probes linearly read out semantic structures from LLM activations. A: A natural-language description specifies a set of entities and their typed relations (illustrated here for spatial layout , where entities are objects and relations are spatial predicates (left of/right, top of/ below). B: The description corresponds to a semantic structure, formal￾ized as a relational graph whose nodes are entitie… view at source ↗
Figure 2
Figure 2. Figure 2: Polar probe geometry mirrors the gold semantic structure. Top: Expected polar probe geometry for semantic structures from every domain. Bottom: 2D PCA of probe-space entity representations from 10 different descriptions of a semantic structure in the test set; large markers denote entity centroids and lines indicate gold relations. The projections tend to follow the polar code: direction encodes relation t… view at source ↗
Figure 3
Figure 3. Figure 3: Semantic structures are most linearly decodable in the middle layers, only in pretrained LLMs. Spearman’s ρ for relation existence (blue) and type (orange) decoded by a polar probe from Llama3-8B across layers in five domains. In pretrained models (solid), decoding peaks around layers 12–15 and remains high in late layers. In randomly initialized models (dashed), both scores remain close to chance across a… view at source ↗
Figure 4
Figure 4. Figure 4: Polar probe performance grows with pretraining, falls with the number of entities in the relational graph, and degrades with out-of-distribution (OOD) entities and relation surface forms. Top: Spearman’s ρ for relation existence (blue) and type (orange) vs. pretraining steps at the best layer of OLMo-7B. Middle: Polar probe performance vs. number of entities in the graph at the best layer of Llama3.1-8B. B… view at source ↗
Figure 5
Figure 5. Figure 5 [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Polar probe prototypes steer LLM predictions: Probability of a correct answer under steering at layer 11 of Llama3-8B. LLM size. To evaluate whether the capacity of the LLMs influenced the geometry of semantic struc￾tures, we trained and evaluated polar probes on models from the Pythia suite spanning 14M–6.9B parameters (Biderman et al., 2023). Polar probe per￾formance increases with model size, for each o… view at source ↗
Figure 7
Figure 7. Figure 7: Semantic domain subspaces are largely disjoint, with a spatial–ordinal overlap. Cross-domain alignment at the best layer of Llama3-8B, quantified via the principal angles in LLM space (higher = more overlap). 5.2 Correlation with downstream predictions To determine whether polar representations are merely epiphenomenal or instead reflect representations used by the LLM, we conduct a representation–behavior… view at source ↗
Figure 7
Figure 7. Figure 7: Semantic domain subspaces are largely disjoint, with a spatial–ordinal overlap. Cross-domain alignment at the best layer of Llama3-8B, quantified via the principal angles in LLM space (higher = more overlap). 6.2 Correlation with downstream predictions To determine whether polar representations are merely epiphenomenal or instead reflect representations used by the LLM, we conduct a representation–behavior… view at source ↗
Figure 8
Figure 8. Figure 8: Type probe errors predict LLM’s downstream performance. Layerwise Spearman correlation between existence (blue) and type (orange) probe-space errors and logit of the correct answer on a Question-Answering task over semantic structures. 5.3 Additional baselines [PITH_FULL_IMAGE:figures/full_fig_p016_8.png] view at source ↗
Figure 8
Figure 8. Figure 8: Type probe errors predict LLM’s downstream performance. Layerwise Spearman correlation between existence (blue) and type (orange) probe-space errors and logit of the correct answer on a Question-Answering task over semantic structures. 6.3 Additional baselines [PITH_FULL_IMAGE:figures/full_fig_p017_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Semantic structures are encoded in linear subspaces of LLM activations. Training prototype vectors with a fixed identity probe yields Existence scores close to chance. Type scores are high in the spatial layout and family tree domains but remain near chance elsewhere. Finally, the linear baseline, shown as a horizontal line, performs very close to chance across all domains. 5.4 Naturalistic evaluation We e… view at source ↗
Figure 9
Figure 9. Figure 9: Semantic structures are encoded in linear subspaces of LLM activations. Training prototype vectors with a fixed identity probe yields Existence scores close to chance. Type scores are high in the spatial layout and family tree domains but remain near chance elsewhere. Finally, the linear baseline, shown as a horizontal line, performs very close to chance across all domains. 6.4 Naturalistic evaluation We e… view at source ↗
Figure 10
Figure 10. Figure 10: Polar probes trained on the Spatial Layout controlled dataset generalize to a naturalistic and multilingual sentences. Polar probes are trained on the controlled Spatial Layout dataset and evaluated on an LLM-generated naturalistic dataset within the same semantic domain. Probe performance substantially exceeds both chance level and the untrained baseline. 5.5 Causal interventions 0 10 20 30 40 50 0.06 0.… view at source ↗
Figure 10
Figure 10. Figure 10: Polar probes trained on the Spatial Layout controlled dataset generalize to a naturalistic and multilingual sentences. Polar probes are trained on the controlled Spatial Layout dataset and evaluated on an LLM-generated naturalistic dataset within the same semantic domain. Probe performance substantially exceeds both chance level and the untrained baseline. 6.5 Causal interventions 0 10 20 30 40 50 0.06 0.… view at source ↗
Figure 11
Figure 11. Figure 11: Interventions along polar probe directions causally modulate model predic￾tions, with the strongest effects in middle layers.. Probability of the correct token under positive and negative direction steering. In middle layers, positive-prototype interventions reliably increase the probability and negative-prototype interventions decrease it. 0 5 10 15 20 25 30 Layer 0.00 0.01 0.02 0.03 0.04 0.05 Steering e… view at source ↗
Figure 11
Figure 11. Figure 11: Interventions along polar probe directions causally modulate model predic￾tions, with the strongest effects in middle layers.. Probability of the correct token under positive and negative direction steering. In middle layers, positive-prototype interventions reliably increase the probability and negative-prototype interventions decrease it. 0 5 10 15 20 25 30 Layer 0.00 0.01 0.02 0.03 0.04 0.05 Steering e… view at source ↗
Figure 12
Figure 12. Figure 12: Middle layers show maximal response to causal interventions. Layerwise mean difference in probability between positive-signed and negative-signed prototypes. 17 [PITH_FULL_IMAGE:figures/full_fig_p017_12.png] view at source ↗
Figure 12
Figure 12. Figure 12: Middle layers show maximal response to causal interventions. Layerwise mean difference in probability between positive-signed and negative-signed prototypes. 18 [PITH_FULL_IMAGE:figures/full_fig_p018_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Semantic domain subspaces and uncontextualized embeddings are disjoint. Cross-domain alignment at the best layer of Llama3-8B, quantified via the principal angles in LLM space (higher = more overlap) 5.7 Predicted vs.Ground-truth distances 1 2 3 4 5 6 Semantic graph distance 0.001 0.002 0.003 0.004 0.005 Probe distance =0.83 [PITH_FULL_IMAGE:figures/full_fig_p018_13.png] view at source ↗
Figure 13
Figure 13. Figure 13: Semantic domain subspaces and uncontextualized embeddings are disjoint. Cross-domain alignment at the best layer of Llama3-8B, quantified via the principal angles in LLM space (higher = more overlap) 6.7 Predicted vs.Ground-truth distances 1 2 3 4 5 6 Semantic graph distance 0.001 0.002 0.003 0.004 0.005 Probe distance =0.83 [PITH_FULL_IMAGE:figures/full_fig_p019_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Semantic distance and Probe distance used to calculate Spearman’s ρ 18 [PITH_FULL_IMAGE:figures/full_fig_p018_14.png] view at source ↗
Figure 14
Figure 14. Figure 14: Semantic distance and Probe distance used to calculate Spearman’s ρ 19 [PITH_FULL_IMAGE:figures/full_fig_p019_14.png] view at source ↗
read the original abstract

How do artificial neural networks bind concepts to form complex semantic structures? Here, we propose a simple neural code, whereby the existence and the type of relations between entities are represented by the distance and the direction between their embeddings, respectively. We test this hypothesis in a variety of Large Language Models (LLMs), each input with natural-language descriptions of minimalist tasks from five different domains: arithmetic, visual scenes, family trees, metro maps and social interactions. Results show that the true semantic structures can be linearly recovered with a Polar Probe targeting a subspace of LLMs' layer activations. Second, this code emerges mostly in middle layers and improves with LLM performance. Third, these Polar Probes successfully generalize to new entities and relation types, but degrades with the size of the semantic structure. Finally, the quality of the polar representation correlates with the LLM's ability to answer questions about the semantic structure. Together, these findings suggest that LLMs learn to build complex semantic structures by binding representations with a simple geometrical principle.

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 proposes that LLMs bind concepts into semantic structures via a simple polar code in which distance between entity embeddings encodes relation existence and direction encodes relation type. It introduces a linear 'Polar Probe' applied to subspaces of layer activations and tests recovery of ground-truth structures from explicit natural-language task descriptions across five domains (arithmetic, visual scenes, family trees, metro maps, social interactions). Key results include emergence of the code mainly in middle layers, improvement with model performance, successful generalization to new entities/relations (with degradation for larger structures), and correlation between probe quality and the LLM's ability to answer questions about the structures.

Significance. If the central results hold after addressing methodological gaps, the work would provide concrete evidence for a geometrical mechanism of semantic binding in LLMs, with implications for interpretability research. Strengths include the multi-domain evaluation, explicit generalization tests, and correlation with downstream task performance; these elements make the findings more falsifiable than single-domain probe studies. The approach also supplies a concrete, testable hypothesis (polar geometry) rather than purely post-hoc interpretations.

major comments (3)
  1. [Methods] Methods section: The abstract and results claim successful linear recovery and generalization but supply no statistical details (e.g., p-values, confidence intervals), no controls for probe complexity (e.g., comparison to random or linear baselines of matched dimensionality), and no description of how subspaces were selected. These omissions make it impossible to judge whether the reported performance is specific to the polar hypothesis or could arise from any sufficiently expressive linear readout.
  2. [Results] Results (domain experiments): All inputs consist of explicit natural-language descriptions that already enumerate the full set of entities and relations. This design leaves open whether the linearly decodable polar code reflects the model's internal binding mechanism or simply its encoding of surface structure already stated in the prompt. A control condition with implicit or open-ended text (where relations must be inferred) is needed to support the broader claim about how LLMs construct semantic structures.
  3. [Generalization experiments] Generalization experiments: The reported degradation with semantic-structure size is interesting, but without an analysis of how probe dimensionality or regularization scales with structure size, it is unclear whether the degradation is a property of the hypothesized polar code or an artifact of probe capacity.
minor comments (2)
  1. [Abstract] Abstract: The term 'Polar Probe' is used without a one-sentence definition or reference to its mathematical formulation, which reduces accessibility for readers outside the immediate subfield.
  2. [Introduction] Notation: The distinction between 'distance' (existence) and 'direction' (type) is central yet introduced without an explicit equation or diagram in the early sections; adding a compact formal statement would improve clarity.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for their constructive and detailed feedback. We address each major comment below and indicate where revisions will be made to strengthen the manuscript.

read point-by-point responses

  1. Referee: [Methods] Methods section: The abstract and results claim successful linear recovery and generalization but supply no statistical details (e.g., p-values, confidence intervals), no controls for probe complexity (e.g., comparison to random or linear baselines of matched dimensionality), and no description of how subspaces were selected. These omissions make it impossible to judge whether the reported performance is specific to the polar hypothesis or could arise from any sufficiently expressive linear readout.

    Authors: We agree that these details were insufficient. The revised manuscript will report p-values and confidence intervals for all probe results. We will add controls comparing the Polar Probe against random baselines and linear readouts of matched dimensionality. We will also describe the subspace selection process, which selected middle-layer subspaces based on preliminary layer-wise probe performance. revision: yes

  2. Referee: [Results] Results (domain experiments): All inputs consist of explicit natural-language descriptions that already enumerate the full set of entities and relations. This design leaves open whether the linearly decodable polar code reflects the model's internal binding mechanism or simply its encoding of surface structure already stated in the prompt. A control condition with implicit or open-ended text (where relations must be inferred) is needed to support the broader claim about how LLMs construct semantic structures.

    Authors: The explicit descriptions were chosen to supply verifiable ground-truth structures for quantitative probe evaluation across domains. The probe recovers geometry from post-prompt activations, and successful generalization to novel entities (absent from the original prompt) provides evidence that the representation is not limited to surface copying. We will add an explicit discussion of this limitation and note that implicit-prompt controls are a valuable direction for future work. revision: partial

  3. Referee: [Generalization experiments] Generalization experiments: The reported degradation with semantic-structure size is interesting, but without an analysis of how probe dimensionality or regularization scales with structure size, it is unclear whether the degradation is a property of the hypothesized polar code or an artifact of probe capacity.

    Authors: We will incorporate an analysis of probe dimensionality and regularization strength as functions of semantic-structure size. This will help determine whether the observed degradation arises from properties of the polar code itself or from capacity limits of the linear probe. revision: yes

Circularity Check

0 steps flagged

No significant circularity; empirical probe results are independent of inputs

full rationale

The paper advances a hypothesis that LLMs bind semantic structures via distance/direction in embedding space and tests it by training Polar Probes on layer activations to recover structures explicitly described in the input prompts across five domains. No load-bearing step reduces by construction to the inputs: the probe is a fitted linear decoder measuring decodability rather than a self-defined quantity, the ground-truth structures are external to the model's equations, and no self-citation chains or ansatzes are invoked to force the outcome. The reported emergence in middle layers, generalization, and correlation with question-answering are measured outcomes, not tautological renamings or fitted predictions. The derivation is therefore self-contained against the experimental benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The work rests on the untested premise that a simple geometric relation in embedding space suffices to represent arbitrary semantic binding; the Polar Probe itself is introduced as a new analysis tool without independent prior validation.

axioms (1)
  • domain assumption LLM embeddings form a space in which linear subspaces can isolate relational information
    Core premise required for the Polar Probe to succeed; invoked when the probe is applied to layer activations.
invented entities (1)
  • Polar Probe no independent evidence
    purpose: Linear decoder that targets a subspace of activations to recover distance and direction encoding of relations
    New analysis construct introduced to test the geometric hypothesis; no independent evidence supplied outside the current experiments.

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