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structures / Manifolds / Cone

Cone

OBJECTgeometric-objectsubsetnon-manifoldapex-singularconvexintermediatehow it's classified →

The topological cone construction over a base space B — rays from an apex through every point of B, scaled by a shared non-negative 'magnitude' coordinate. Two instances: a polyhedral cone over a finite set of directions, and a circular cone over S¹ (angle × magnitude).

Replicationcomputed from the corpus — never hand-assigned
10 papers · no shared authors3 architecture classes · across papers2 domains · across papers10 model families · across papers
Filled = two or more values reported by papers that share no author — replication. Outlined = two or more values, but all from a single study — breadth, not replication. Grey = a single value. Derived from paper authorship and each model's architecture class, domain and family; it updates itself when a paper is added.

Definition

The cone over a base space BB is the set of rays from a fixed apex through every point of BB, parametrised by a non-negative scalar λ≥0\lambda \geq 0:

Cone(B)={ λ⋅b  :  b∈B, λ≥0 }\mathrm{Cone}(B) = \{\, \lambda \cdot b \;:\; b \in B,\ \lambda \geq 0 \,\}

Two special cases matter here:

  • B={b1,…,bN}B = \{b_1,\ldots,b_N\}, a finite set of directions — a polyhedral convex cone, RN={∑iλibi:λi≥0}R_N = \{\sum_i \lambda_i b_i : \lambda_i \geq 0\}, a wedge/pyramid shape.
  • B=S1B = S^1, a circle — a classical circular (ice-cream) cone: angular coordinate θ\theta times a shared radial magnitude r≥0r \geq 0, i.e. points (rcos⁡θ,rsin⁡θ)(r\cos\theta, r\sin\theta) for r≥0r \geq 0.

Intuition

A single ray from the origin is a 1D half-line. A polyhedral cone is what several rays sweep out when they must combine with the same sign (non-negative coefficients only — no cancellation allowed). A circular cone is an old-fashioned lampshade: the angle around the shade encodes one variable, and how far from the tip you are (the radius) encodes something else entirely, independent of angle.

Properties

  • Non-negativity is the defining restriction. Unlike a linear subspace (any real coefficients, including negative — closed under x↦−xx \mapsto -x) or an affine subspace (offset, no origin-anchoring), a cone’s coefficients must be ≥0\geq 0: the apex is a genuine corner, not a point you can pass smoothly through in every direction.
  • Not a manifold at the apex. Every point of Cone(B)∖{0}\mathrm{Cone}(B) \setminus \{0\} has a neighbourhood homeomorphic to R×B\mathbb{R}\times B locally, but the apex itself (where all rays meet) has no Euclidean neighbourhood when BB has more than one point — it is a genuine singular point.
  • Polyhedral case: NN generators, causal independence is a separate question from linear independence. NN linearly independent directions span an NN-dimensional cone, but two directions being orthogonal only guarantees they don’t interfere under linear superposition — it does not guarantee that intervening on one leaves the other’s causal effect intact. Distinguishing ”NN orthogonal directions” from ”NN causally-independent generators” requires an intervention, not just a geometric check (e.g. ablate one candidate direction, verify the others still reproduce their effect).
  • Circular case: collapsing rr recovers the circle. Projecting Cone(S1)\mathrm{Cone}(S^1) onto the unit circle (x↦x/∥x∥x \mapsto x/\|x\|) exactly recovers Circle — the cone is the circle’s angular structure with one extra, independent radial degree of freedom.
  • Convexity. A polyhedral cone RNR_N is always convex: if u,v∈RNu,v \in R_N then u+v∈RNu+v \in R_N (coefficients add and stay non-negative), and αu∈RN\alpha u \in R_N for α≥0\alpha \geq 0.

Relative to concept-lattice

The polyhedral case above is given in V-representation (non-negative combinations of generator rays bib_i). The exact same object class also has an H-representation — the intersection of finitely many origin-passing half-spaces, {v:v⋅dm≥0 ∀m}\{\mathbf v : \mathbf v\cdot\mathbf d_m \geq 0\ \forall m\} — which is how Concept lattice (Formal Concept Analysis half-space model) builds a “concept” out of several linear attribute directions simultaneously. What’s new there isn’t the single-cone shape itself, but the order-theoretic system of many such cones connected by a Galois connection into a complete lattice with meet/join operations.

Key evidence

Alper & Averbuch-Elor (2024) find off-the-shelf CLIP/OpenCLIP/ALIGN embedding spaces exhibit an emergent Euclidean relaxation of this cone structure: anchoring the apex at the model’s own embedding of the empty string (“entailment root” r\mathbf r), radial distance dr(e)=∥e−r∥d_{\mathbf r}(\mathbf e)=\|\mathbf e-\mathbf r\| measures concept genericity and an exterior-angle metric at r\mathbf r tests entailment direction between two concepts — order-consistency with true hierarchical ordering reaches Kendall’s τd=0.89\tau_d=0.89 zero-shot on their HierarCaps benchmark, and causal fine-tuning targeting this exact geometry raises it to τd=0.99\tau_d=0.99 while leaving standard task performance unchanged. See vlm-embeddings-exhibit-emergent-radial-entailment-hierarchy-improvable-by-fine-tuning via Radial Entailment Embedding.

Baek, Li & Tegmark (2024) find a second, structurally distinct instance of this shape: a discrete-tree cone rather than a continuous-hierarchy one. Fitting a coordinate-wise order-embedding score (the cone probe) on top of a 10-dimensional PCA-reduced residual-stream subspace, across five instruction-tuned LLMs answering in-context genealogy questions (“is X a descendant of Y?”), they find the fitted cone subspace is not merely correlationally present but causally implicated: projected activation patching restricted to this subspace produces an effect on the model’s answer (measured via logit difference) comparable to or larger than patching a same-rank top-PCA-component subspace, with an unrestricted full-layer patch producing a still larger effect — evidence of real but incomplete localization of the tree structure to the fitted cone subspace. See baek-etal-2024-cone-probe-isolates-a-tree-like-subspace-for-genealogy-descendant-relations-causally-validated-by-projected-activation-patching.

Winninger (2026) extracts a fourth account of refusal’s cone-like structure, cheaper than the constrained-optimization search used to fit the polyhedral cone above: training a Recursive Feature Machine as a target-vs-rest refusal classifier directly on Qwen3 (1.7B-14B) and Qwen2.5-7B-Instruct residual-stream activations, then taking the ranked top-k eigenvectors of the resulting Average Gradient Outer Product matrix (RFM/AGOP Subspace Extraction (LLM activations)) as the subspace. Cumulative ablation shows attack-success rate rises monotonically with the number of ablated eigenvector dimensions, and the dimensionality needed to cross a 50% attack-success threshold scales with model size — single-direction ablation suffices for smaller models, but Qwen3-8B/14B need 3 or more dimensions — with steering along the top eigenvector inducing refusal and random-direction controls of matched rank confirming the effect is not noise. See rfm-agop-extracted-refusal-subspace-dimensionality-required-for-full-suppression-scales-with-model-size.

Exercises

Base

  1. Is the union of two opposite rays, {λv:λ≥0}∪{−λv:λ≥0}\{\lambda v : \lambda \geq 0\} \cup \{-\lambda v : \lambda \geq 0\} for a fixed vector vv, a cone in the sense defined above, and is it the same set as the 1D linear subspace span{v}\text{span}\{v\}?
Solution

Yes to both. As a set, {λv:λ≥0}∪{−λv:λ≥0}={μv:μ∈R}=span{v}\{\lambda v : \lambda \geq 0\}\cup\{-\lambda v:\lambda\geq0\} = \{\mu v : \mu \in \mathbb{R}\} = \text{span}\{v\} — allowing both signs of the scalar recovers the full line. This is the cone over the 2-point base B={v,−v}B = \{v, -v\} (or equivalently Cone(S0)\text{Cone}(S^0), the cone over the 0-sphere), and it happens to coincide with a linear subspace precisely because the base is symmetric under negation. A generic finite base (not closed under negation) gives a proper wedge, not a full subspace.

  1. Give a base BB for which Cone(B)\mathrm{Cone}(B) is exactly a half-plane in R2\mathbb{R}^2.
Solution

Take BB to be a semicircle, e.g. B={(cos⁡θ,sin⁡θ):θ∈[0,π]}B = \{(\cos\theta,\sin\theta) : \theta \in [0,\pi]\}. Then Cone(B)={r(cos⁡θ,sin⁡θ):r≥0, θ∈[0,π]}\mathrm{Cone}(B) = \{r(\cos\theta,\sin\theta) : r\geq0,\ \theta\in[0,\pi]\} is exactly the closed upper half-plane {(x,y):y≥0}\{(x,y): y \geq 0\}.

Middle

  1. Prove that a polyhedral cone RN={∑iλibi:λi≥0}R_N = \{\sum_i \lambda_i b_i : \lambda_i \geq 0\} is convex.
Solution

Let u=∑iλibiu = \sum_i \lambda_i b_i and v=∑iμibiv = \sum_i \mu_i b_i be in RNR_N (so all λi,μi≥0\lambda_i,\mu_i \geq 0), and let t∈[0,1]t \in [0,1]. Then tu+(1−t)v=∑i(tλi+(1−t)μi) bitu + (1-t)v = \sum_i (t\lambda_i + (1-t)\mu_i)\, b_i. Each coefficient tλi+(1−t)μi≥0t\lambda_i+(1-t)\mu_i \geq 0 since t,(1−t),λi,μi≥0t,(1-t),\lambda_i,\mu_i \geq 0, so tu+(1−t)v∈RNtu+(1-t)v \in R_N. Hence RNR_N is convex.

  1. For the circular cone Cone(S1)={(rcos⁡θ,rsin⁡θ):r≥0, θ∈[0,2π)}\mathrm{Cone}(S^1) = \{(r\cos\theta, r\sin\theta) : r\geq 0,\ \theta \in [0,2\pi)\}, what is this set, concretely, as a subset of R2\mathbb{R}^2? Is the map (r,θ)↦(rcos⁡θ,rsin⁡θ)(r,\theta)\mapsto(r\cos\theta,r\sin\theta) a homeomorphism from [0,∞)×S1[0,\infty)\times S^1 onto its image?
Solution

The set is all of R2\mathbb{R}^2 (every point has some polar representation). The map is continuous and surjective onto R2\mathbb{R}^2, but it is not a homeomorphism from [0,∞)×S1[0,\infty)\times S^1: at r=0r=0, every value of θ\theta maps to the same point (the origin), so the map is not injective there, and [0,∞)×S1[0,\infty)\times S^1 (a half-infinite cylinder) is not homeomorphic to R2\mathbb{R}^2 near that collapsed circle. This is exactly the sense in which the apex is a genuine topological singularity — the cone construction pinches the {0}×S1\{0\}\times S^1 boundary circle of the cylinder down to a single point.

Pro

  1. Suppose a behavior is mediated by a polyhedral cone with N=3N=3 generators b1,b2,b3∈Rdb_1,b_2,b_3 \in \mathbb{R}^d that are pairwise orthogonal. An experimenter ablates the component of activations along b1b_1 (projects it out) and finds the behavior is fully eliminated on every test example. Does this refute the 3-generator cone account in favor of a single-direction account? What must additionally be true of b2,b3b_2, b_3‘s coefficients across the test examples for the two accounts to be distinguishable?
Solution

Not necessarily — full elimination after ablating b1b_1 is also consistent with a genuine 3-generator cone if, on that particular test set, the coefficients λ2,λ3\lambda_2,\lambda_3 happen to be zero (or small enough that the behavior’s threshold isn’t met without λ1b1\lambda_1 b_1‘s contribution) for every example. The two accounts (single direction vs. 3-generator cone) are only distinguishable on examples where λ2\lambda_2 or λ3\lambda_3 is large enough to independently cross the behavior’s activation threshold — i.e. the experimenter needs a test set where each generator, in isolation (with the others ablated), still triggers the behavior on some examples. Only if ablating b1b_1 eliminates the behavior on every example, including ones specifically chosen to have large λ2,λ3\lambda_2,\lambda_3, does the data support a true single-direction account over the polyhedral cone.

  1. Consider Cone(S1)\mathrm{Cone}(S^1) collapsed by the map π(x)=x/∥x∥\pi(x) = x/\|x\| for x≠0x \neq 0. Show that π\pi restricted to Cone(S1)∖{0}\mathrm{Cone}(S^1)\setminus\{0\} is a well-defined continuous surjection onto S1S^1, and explain in what sense this operation “recovers the circle” from the cone while discarding exactly one real degree of freedom.
Solution

For x=(rcos⁡θ,rsin⁡θ)x = (r\cos\theta, r\sin\theta) with r>0r > 0, π(x)=(cos⁡θ,sin⁡θ)∈S1\pi(x) = (\cos\theta,\sin\theta) \in S^1, which is well defined (division by ∥x∥=r≠0\|x\|=r\neq 0 is legal) and continuous as a composition of continuous operations. It is surjective since every θ\theta is attained (e.g. by any r>0r>0). The fibers π−1(cos⁡θ,sin⁡θ)={(rcos⁡θ,rsin⁡θ):r>0}\pi^{-1}(\cos\theta,\sin\theta) = \{(r\cos\theta,r\sin\theta): r>0\} are exactly the rays of the cone, each parametrised by the single remaining coordinate r∈(0,∞)r \in (0,\infty) — so π\pi quotients out exactly the radial degree of freedom, leaving only the angular one, which is precisely S1S^1. This is the formal version of “cosine-normalizing every point recovers the pure circle as a cross-section.”

Found in (7 observations · 10 families)

Llama

Investigating Representation Universality: Case Study on Genealogical Representations (2024)measured

A cone probe isolates a causal tree-like subspace for genealogy

Details

Baek et al. define a differentiable order-embedding cone probe (provably transitive and antisymmetric for a tree's descendant-of relation) and fit it on a 15-node family tree, then on 10-dimensional PCA-reduced activations of five instruction-tuned LLMs [baek-etal-2024-cone-probe-genealogical-representation-universality] Activation patching restricted to the fitted cone subspace produces a causal effect on genealogy Q&A comparable to or larger than a same-rank top-PCA subspace, but smaller than an unrestricted full-layer patch [baek-etal-2024-cone-probe-genealogical-representation-universality] The cone subspace thus captures real but incomplete causal structure, and a context-shuffle control degrades performance, consistent with reliance on depth-ordered structure [baek-etal-2024-cone-probe-genealogical-representation-universality] A complementary model-stitching experiment (OPT/Pythia/Mistral/Llama, 410M-8B) finds low next-token-loss degradation when splicing early/mid layers between independently-trained models [baek-etal-2024-cone-probe-genealogical-representation-universality] The authors explicitly hedge the universality conclusion given the lack of ground-truth graph representations [baek-etal-2024-cone-probe-genealogical-representation-universality]

models: Llama-3.1-8B-Instruct, Llama-3-8B-Instruct, Llama-3.2-3B-Instruct · method: Cone probe (order-embedding for descendant relations), Activation patching
The Lattice Representation Hypothesis of Large Language Models (2026)measured

LLM attribute half-spaces intersect into a complete concept lattice

Details

- The Lattice Representation Hypothesis models each binary attribute (e.g. "can fly") as a linear direction with a threshold, so a concept defined by co-occurring attributes is the convex polyhedral cone formed by intersecting their origin-passing half-spaces. [xiong-2026] - Theorem 1 proves the Galois closure of the softened, thresholded object-attribute relation is a complete lattice, and Proposition 1 absorbs per-attribute thresholds into one global embedding shift (a canonical origin-passing form). [xiong-2026] - Across five WordNet sub-hierarchies in LLaMA-3.1-8B, Gemma-7B and Mistral-7B, LDA/Fisher attribute directions recover the ground-truth formal context at F1 above 78% on physical domains (best Gemma-7B 83.2% on Animal) versus 53-68% mean-embedding and 45-48% random baselines. [xiong-2026] - Projection-profile subsumption scoring recovers WordNet's hypernym partial order (up to F1 77.1, LLaMA on Animal), and soft meet/join operators beat baselines at retrieving the true lowest-common-subconcept / least-common-hypernym (Table 3). [xiong-2026] - A scaling comparison across LLaMA-3 models (3B, 8B, 70B) shows scaling helps abstract domains far more than physical ones; observational, no intervention. [xiong-2026]

models: Llama-3.1-8B, LLaMA-3 3B, Llama-3-8B, Llama-3-70B · method: Linear probing, Formal Concept Analysis (FCA) lattice construction
The Geometry of Refusal in Large Language Models: Concept Cones and Representational Independence (2025)measured

Refusal is mediated by a multi-dimensional concept cone

Details

Wollschlager et al. find refusal in Gemma 2, Qwen 2.5, and Llama 3 8B is mediated by a polyhedral convex cone of up to 5 causally-independent directions, found via Refusal Cone Optimization (non-negative coefficients, re-orthogonalized) [wollschlager-etal-2025] Orthogonality between directions is not sufficient for independence; only ablating one and testing whether the others still reproduce the behavior establishes it [wollschlager-etal-2025] Cone dimensionality varies with model and scale, plateauing around 4D for Gemma 2 2B [wollschlager-etal-2025] This is the most geometrically elaborate of three competing accounts of refusal, versus the single-direction and affine-function accounts [wollschlager-etal-2025]

models: Llama-3-8B-Instruct · method: Difference-in-means direction, Causal interventions (steering), Supervised direction search (gradient-based optimization)
Symmetry in Language Statistics Shapes the Geometry of Model Representations (2026), Not All Language Model Features Are One-Dimensionally Linear (2024), Shape Happens: Automatic Feature Manifold Discovery in LLMs via Supervised Multi-Dimensional Scaling (2025), Do Sparse Autoencoders Capture Concept Manifolds? (2026)measured

Cyclical time is represented as a circle, replicated and causally validated

Details

Cyclical temporal concepts (days of the week, months) are each independently represented as points on a circle in the residual stream, linearly decodable from activations [karkada-etal-2026] Despite the "toroidal" id, the confirmed finding is a single circle per cyclic variable, not a joint torus; no cited paper shows two such circles coexisting as orthogonal factors of one representation [karkada-etal-2026] Karkada et al. independently derive the same single-circle geometry analytically from translation-symmetric co-occurrence statistics [karkada-etal-2026] Engels et al. discover the same circles in GPT-2-small (layer 7) and Mistral-7B by clustering SAE dictionary elements, and causally validate them via activation patching on Mistral-7B and Llama-3-8B, where patching to a rotated point changes a day/month arithmetic answer [engels-etal-2024] Tiblias et al. recover the same circular date/month structure via a supervised distance-fit (sMDS) across Qwen2.5-3B, Llama-3.2-3B, and Gemma-2-2B, persisting at 3B/8B/70B Llama scale [tiblias-etal-2025] The circle is task-dependent: the identical date context collapses into a non-cyclic clustered or linear manifold when the completion cue asks for season or temperature rather than recency [tiblias-etal-2025] It is causally necessary: noise confined to the located 2D subspace degrades temporal-reasoning accuracy as much as noise across the full residual stream, while an equally-sized random subspace has negligible effect [tiblias-etal-2025] Bhalla et al. reproduce the days-of-week circle in Llama-3.1-8B (layer 19) via PCA and causally steer along its principal components to shift the predicted day-of-week token smoothly [bhalla-etal-2026]

models: Llama-3-8B, Llama-3.2-3B-Instruct, Llama-3.2-3B, Llama-3.1-8B-Instruct, Llama-3.1-70B-Instruct, Llama-3.1-8B · method: Linear probing, PCA, Activation patching, Sparse Autoencoders (SAE), Supervised Multi-Dimensional Scaling (SMDS), Subspace noise perturbation, Activation Steering (Addition)

Gemma

Investigating Representation Universality: Case Study on Genealogical Representations (2024)measured

A cone probe isolates a causal tree-like subspace for genealogy

Details

Baek et al. define a differentiable order-embedding cone probe (provably transitive and antisymmetric for a tree's descendant-of relation) and fit it on a 15-node family tree, then on 10-dimensional PCA-reduced activations of five instruction-tuned LLMs [baek-etal-2024-cone-probe-genealogical-representation-universality] Activation patching restricted to the fitted cone subspace produces a causal effect on genealogy Q&A comparable to or larger than a same-rank top-PCA subspace, but smaller than an unrestricted full-layer patch [baek-etal-2024-cone-probe-genealogical-representation-universality] The cone subspace thus captures real but incomplete causal structure, and a context-shuffle control degrades performance, consistent with reliance on depth-ordered structure [baek-etal-2024-cone-probe-genealogical-representation-universality] A complementary model-stitching experiment (OPT/Pythia/Mistral/Llama, 410M-8B) finds low next-token-loss degradation when splicing early/mid layers between independently-trained models [baek-etal-2024-cone-probe-genealogical-representation-universality] The authors explicitly hedge the universality conclusion given the lack of ground-truth graph representations [baek-etal-2024-cone-probe-genealogical-representation-universality]

models: Gemma-2-2B-it, Gemma-2-9B-it · method: Cone probe (order-embedding for descendant relations), Activation patching
The Lattice Representation Hypothesis of Large Language Models (2026)measured

LLM attribute half-spaces intersect into a complete concept lattice

Details

- The Lattice Representation Hypothesis models each binary attribute (e.g. "can fly") as a linear direction with a threshold, so a concept defined by co-occurring attributes is the convex polyhedral cone formed by intersecting their origin-passing half-spaces. [xiong-2026] - Theorem 1 proves the Galois closure of the softened, thresholded object-attribute relation is a complete lattice, and Proposition 1 absorbs per-attribute thresholds into one global embedding shift (a canonical origin-passing form). [xiong-2026] - Across five WordNet sub-hierarchies in LLaMA-3.1-8B, Gemma-7B and Mistral-7B, LDA/Fisher attribute directions recover the ground-truth formal context at F1 above 78% on physical domains (best Gemma-7B 83.2% on Animal) versus 53-68% mean-embedding and 45-48% random baselines. [xiong-2026] - Projection-profile subsumption scoring recovers WordNet's hypernym partial order (up to F1 77.1, LLaMA on Animal), and soft meet/join operators beat baselines at retrieving the true lowest-common-subconcept / least-common-hypernym (Table 3). [xiong-2026] - A scaling comparison across LLaMA-3 models (3B, 8B, 70B) shows scaling helps abstract domains far more than physical ones; observational, no intervention. [xiong-2026]

models: Gemma-7B · method: Linear probing, Formal Concept Analysis (FCA) lattice construction
The Geometry of Refusal in Large Language Models: Concept Cones and Representational Independence (2025)measured

Refusal is mediated by a multi-dimensional concept cone

Details

Wollschlager et al. find refusal in Gemma 2, Qwen 2.5, and Llama 3 8B is mediated by a polyhedral convex cone of up to 5 causally-independent directions, found via Refusal Cone Optimization (non-negative coefficients, re-orthogonalized) [wollschlager-etal-2025] Orthogonality between directions is not sufficient for independence; only ablating one and testing whether the others still reproduce the behavior establishes it [wollschlager-etal-2025] Cone dimensionality varies with model and scale, plateauing around 4D for Gemma 2 2B [wollschlager-etal-2025] This is the most geometrically elaborate of three competing accounts of refusal, versus the single-direction and affine-function accounts [wollschlager-etal-2025]

models: Gemma-2-2B-it, Gemma-2-9B-it · method: Difference-in-means direction, Causal interventions (steering), Supervised direction search (gradient-based optimization)
Symmetry in Language Statistics Shapes the Geometry of Model Representations (2026), Not All Language Model Features Are One-Dimensionally Linear (2024), Shape Happens: Automatic Feature Manifold Discovery in LLMs via Supervised Multi-Dimensional Scaling (2025), Do Sparse Autoencoders Capture Concept Manifolds? (2026)measured

Cyclical time is represented as a circle, replicated and causally validated

Details

Cyclical temporal concepts (days of the week, months) are each independently represented as points on a circle in the residual stream, linearly decodable from activations [karkada-etal-2026] Despite the "toroidal" id, the confirmed finding is a single circle per cyclic variable, not a joint torus; no cited paper shows two such circles coexisting as orthogonal factors of one representation [karkada-etal-2026] Karkada et al. independently derive the same single-circle geometry analytically from translation-symmetric co-occurrence statistics [karkada-etal-2026] Engels et al. discover the same circles in GPT-2-small (layer 7) and Mistral-7B by clustering SAE dictionary elements, and causally validate them via activation patching on Mistral-7B and Llama-3-8B, where patching to a rotated point changes a day/month arithmetic answer [engels-etal-2024] Tiblias et al. recover the same circular date/month structure via a supervised distance-fit (sMDS) across Qwen2.5-3B, Llama-3.2-3B, and Gemma-2-2B, persisting at 3B/8B/70B Llama scale [tiblias-etal-2025] The circle is task-dependent: the identical date context collapses into a non-cyclic clustered or linear manifold when the completion cue asks for season or temperature rather than recency [tiblias-etal-2025] It is causally necessary: noise confined to the located 2D subspace degrades temporal-reasoning accuracy as much as noise across the full residual stream, while an equally-sized random subspace has negligible effect [tiblias-etal-2025] Bhalla et al. reproduce the days-of-week circle in Llama-3.1-8B (layer 19) via PCA and causally steer along its principal components to shift the predicted day-of-week token smoothly [bhalla-etal-2026]

models: Gemma-2-2B, EmbeddingGemma, Gemma-2-2B-it · method: Linear probing, PCA, Activation patching, Sparse Autoencoders (SAE), Supervised Multi-Dimensional Scaling (SMDS), Subspace noise perturbation, Activation Steering (Addition)
From Directions to Cones: Exploring Multidimensional Representations of Propositional Facts in LLMs (2025)measured

Propositional truth is mediated by a multi-dimensional concept cone

Details

Yu et al. extend the concept-cone framework from refusal to truth, finding truth judgments in Qwen2.5 and Gemma-2 are causally mediated by a polyhedral cone of up to 5 non-negative-combination directions [yu-etal-2025] Larger models (Qwen-7B, Gemma-9B) sustain near-100% Answer Switching Rate across all 5 tested dimensions while smaller models degrade beyond 2-3 [yu-etal-2025] Only the first cone axis aligns with the classic difference-in-means truth direction (cosine ~0.1-0.2), so difference-in-means captures just one facet of a multi-dimensional truth subspace [yu-etal-2025] Cone-based interventions preserve unrelated behavior (mean KL divergence 0.03-0.05 on Alpaca instructions) [yu-etal-2025]

models: Gemma-2-2B-it, Gemma-2-9B-it · method: Difference-in-means direction, Causal interventions (steering), Supervised direction search (gradient-based optimization), Geometric analysis, PCA

OPT

Investigating Representation Universality: Case Study on Genealogical Representations (2024)measured

A cone probe isolates a causal tree-like subspace for genealogy

Details

Baek et al. define a differentiable order-embedding cone probe (provably transitive and antisymmetric for a tree's descendant-of relation) and fit it on a 15-node family tree, then on 10-dimensional PCA-reduced activations of five instruction-tuned LLMs [baek-etal-2024-cone-probe-genealogical-representation-universality] Activation patching restricted to the fitted cone subspace produces a causal effect on genealogy Q&A comparable to or larger than a same-rank top-PCA subspace, but smaller than an unrestricted full-layer patch [baek-etal-2024-cone-probe-genealogical-representation-universality] The cone subspace thus captures real but incomplete causal structure, and a context-shuffle control degrades performance, consistent with reliance on depth-ordered structure [baek-etal-2024-cone-probe-genealogical-representation-universality] A complementary model-stitching experiment (OPT/Pythia/Mistral/Llama, 410M-8B) finds low next-token-loss degradation when splicing early/mid layers between independently-trained models [baek-etal-2024-cone-probe-genealogical-representation-universality] The authors explicitly hedge the universality conclusion given the lack of ground-truth graph representations [baek-etal-2024-cone-probe-genealogical-representation-universality]

models: OPT-1.3B, OPT-2.7B, OPT-6.7B · method: Cone probe (order-embedding for descendant relations), Activation patching

Pythia

Investigating Representation Universality: Case Study on Genealogical Representations (2024)measured

A cone probe isolates a causal tree-like subspace for genealogy

Details

Baek et al. define a differentiable order-embedding cone probe (provably transitive and antisymmetric for a tree's descendant-of relation) and fit it on a 15-node family tree, then on 10-dimensional PCA-reduced activations of five instruction-tuned LLMs [baek-etal-2024-cone-probe-genealogical-representation-universality] Activation patching restricted to the fitted cone subspace produces a causal effect on genealogy Q&A comparable to or larger than a same-rank top-PCA subspace, but smaller than an unrestricted full-layer patch [baek-etal-2024-cone-probe-genealogical-representation-universality] The cone subspace thus captures real but incomplete causal structure, and a context-shuffle control degrades performance, consistent with reliance on depth-ordered structure [baek-etal-2024-cone-probe-genealogical-representation-universality] A complementary model-stitching experiment (OPT/Pythia/Mistral/Llama, 410M-8B) finds low next-token-loss degradation when splicing early/mid layers between independently-trained models [baek-etal-2024-cone-probe-genealogical-representation-universality] The authors explicitly hedge the universality conclusion given the lack of ground-truth graph representations [baek-etal-2024-cone-probe-genealogical-representation-universality]

models: Pythia-410M, Pythia-1.4B, Pythia-2.8B · method: Cone probe (order-embedding for descendant relations), Activation patching

Mistral

Investigating Representation Universality: Case Study on Genealogical Representations (2024)measured

A cone probe isolates a causal tree-like subspace for genealogy

Details

Baek et al. define a differentiable order-embedding cone probe (provably transitive and antisymmetric for a tree's descendant-of relation) and fit it on a 15-node family tree, then on 10-dimensional PCA-reduced activations of five instruction-tuned LLMs [baek-etal-2024-cone-probe-genealogical-representation-universality] Activation patching restricted to the fitted cone subspace produces a causal effect on genealogy Q&A comparable to or larger than a same-rank top-PCA subspace, but smaller than an unrestricted full-layer patch [baek-etal-2024-cone-probe-genealogical-representation-universality] The cone subspace thus captures real but incomplete causal structure, and a context-shuffle control degrades performance, consistent with reliance on depth-ordered structure [baek-etal-2024-cone-probe-genealogical-representation-universality] A complementary model-stitching experiment (OPT/Pythia/Mistral/Llama, 410M-8B) finds low next-token-loss degradation when splicing early/mid layers between independently-trained models [baek-etal-2024-cone-probe-genealogical-representation-universality] The authors explicitly hedge the universality conclusion given the lack of ground-truth graph representations [baek-etal-2024-cone-probe-genealogical-representation-universality]

models: Mistral-7B-Instruct-v0.1 · method: Cone probe (order-embedding for descendant relations), Activation patching
The Lattice Representation Hypothesis of Large Language Models (2026)measured

LLM attribute half-spaces intersect into a complete concept lattice

Details

- The Lattice Representation Hypothesis models each binary attribute (e.g. "can fly") as a linear direction with a threshold, so a concept defined by co-occurring attributes is the convex polyhedral cone formed by intersecting their origin-passing half-spaces. [xiong-2026] - Theorem 1 proves the Galois closure of the softened, thresholded object-attribute relation is a complete lattice, and Proposition 1 absorbs per-attribute thresholds into one global embedding shift (a canonical origin-passing form). [xiong-2026] - Across five WordNet sub-hierarchies in LLaMA-3.1-8B, Gemma-7B and Mistral-7B, LDA/Fisher attribute directions recover the ground-truth formal context at F1 above 78% on physical domains (best Gemma-7B 83.2% on Animal) versus 53-68% mean-embedding and 45-48% random baselines. [xiong-2026] - Projection-profile subsumption scoring recovers WordNet's hypernym partial order (up to F1 77.1, LLaMA on Animal), and soft meet/join operators beat baselines at retrieving the true lowest-common-subconcept / least-common-hypernym (Table 3). [xiong-2026] - A scaling comparison across LLaMA-3 models (3B, 8B, 70B) shows scaling helps abstract domains far more than physical ones; observational, no intervention. [xiong-2026]

models: Mistral-7B · method: Linear probing, Formal Concept Analysis (FCA) lattice construction
Symmetry in Language Statistics Shapes the Geometry of Model Representations (2026), Not All Language Model Features Are One-Dimensionally Linear (2024), Shape Happens: Automatic Feature Manifold Discovery in LLMs via Supervised Multi-Dimensional Scaling (2025), Do Sparse Autoencoders Capture Concept Manifolds? (2026)measured

Cyclical time is represented as a circle, replicated and causally validated

Details

Cyclical temporal concepts (days of the week, months) are each independently represented as points on a circle in the residual stream, linearly decodable from activations [karkada-etal-2026] Despite the "toroidal" id, the confirmed finding is a single circle per cyclic variable, not a joint torus; no cited paper shows two such circles coexisting as orthogonal factors of one representation [karkada-etal-2026] Karkada et al. independently derive the same single-circle geometry analytically from translation-symmetric co-occurrence statistics [karkada-etal-2026] Engels et al. discover the same circles in GPT-2-small (layer 7) and Mistral-7B by clustering SAE dictionary elements, and causally validate them via activation patching on Mistral-7B and Llama-3-8B, where patching to a rotated point changes a day/month arithmetic answer [engels-etal-2024] Tiblias et al. recover the same circular date/month structure via a supervised distance-fit (sMDS) across Qwen2.5-3B, Llama-3.2-3B, and Gemma-2-2B, persisting at 3B/8B/70B Llama scale [tiblias-etal-2025] The circle is task-dependent: the identical date context collapses into a non-cyclic clustered or linear manifold when the completion cue asks for season or temperature rather than recency [tiblias-etal-2025] It is causally necessary: noise confined to the located 2D subspace degrades temporal-reasoning accuracy as much as noise across the full residual stream, while an equally-sized random subspace has negligible effect [tiblias-etal-2025] Bhalla et al. reproduce the days-of-week circle in Llama-3.1-8B (layer 19) via PCA and causally steer along its principal components to shift the predicted day-of-week token smoothly [bhalla-etal-2026]

models: Mistral-7B · method: Linear probing, PCA, Activation patching, Sparse Autoencoders (SAE), Supervised Multi-Dimensional Scaling (SMDS), Subspace noise perturbation, Activation Steering (Addition)

Qwen

The Geometry of Refusal in Large Language Models: Concept Cones and Representational Independence (2025)measured

Refusal is mediated by a multi-dimensional concept cone

Details

Wollschlager et al. find refusal in Gemma 2, Qwen 2.5, and Llama 3 8B is mediated by a polyhedral convex cone of up to 5 causally-independent directions, found via Refusal Cone Optimization (non-negative coefficients, re-orthogonalized) [wollschlager-etal-2025] Orthogonality between directions is not sufficient for independence; only ablating one and testing whether the others still reproduce the behavior establishes it [wollschlager-etal-2025] Cone dimensionality varies with model and scale, plateauing around 4D for Gemma 2 2B [wollschlager-etal-2025] This is the most geometrically elaborate of three competing accounts of refusal, versus the single-direction and affine-function accounts [wollschlager-etal-2025]

models: Qwen2.5-1.5B-Instruct, Qwen2.5-14B-Instruct · method: Difference-in-means direction, Causal interventions (steering), Supervised direction search (gradient-based optimization)
Fast Multi-dimensional Refusal Subspaces via RFM-AGOP (2026)measured

Refusal-subspace dimensionality for full suppression scales with model size

Details

Winninger trains a Recursive Feature Machine refusal classifier on Qwen3 (1.7B-14B) and Qwen2.5-7B-Instruct activations, taking the top-k eigenvectors of its Average Gradient Outer Product as a refusal subspace [winninger-2026-fast-multi-dimensional-refusal-subspaces-via-rfm-agop] Cumulative ablation raises attack-success rate monotonically with the number of ablated dimensions, and the count needed to cross 50% scales with model size: k=1 suffices for smaller models while Qwen3-8B and 14B need 3 or more [winninger-2026-fast-multi-dimensional-refusal-subspaces-via-rfm-agop] Steering along the top eigenvector alone induces refusal, with lower-ranked eigenvectors progressively less effective, and matched-rank random-direction controls confirm the effect is not from ablating an arbitrary subspace [winninger-2026-fast-multi-dimensional-refusal-subspaces-via-rfm-agop]

models: Qwen3-1.7B, Qwen3-4B, Qwen3-8B, Qwen3-14B, Qwen2.5-7B-Instruct · method: RFM/AGOP Subspace Extraction (LLM activations), Difference-in-means direction, Causal interventions (steering)
Symmetry in Language Statistics Shapes the Geometry of Model Representations (2026), Not All Language Model Features Are One-Dimensionally Linear (2024), Shape Happens: Automatic Feature Manifold Discovery in LLMs via Supervised Multi-Dimensional Scaling (2025), Do Sparse Autoencoders Capture Concept Manifolds? (2026)measured

Cyclical time is represented as a circle, replicated and causally validated

Details

Cyclical temporal concepts (days of the week, months) are each independently represented as points on a circle in the residual stream, linearly decodable from activations [karkada-etal-2026] Despite the "toroidal" id, the confirmed finding is a single circle per cyclic variable, not a joint torus; no cited paper shows two such circles coexisting as orthogonal factors of one representation [karkada-etal-2026] Karkada et al. independently derive the same single-circle geometry analytically from translation-symmetric co-occurrence statistics [karkada-etal-2026] Engels et al. discover the same circles in GPT-2-small (layer 7) and Mistral-7B by clustering SAE dictionary elements, and causally validate them via activation patching on Mistral-7B and Llama-3-8B, where patching to a rotated point changes a day/month arithmetic answer [engels-etal-2024] Tiblias et al. recover the same circular date/month structure via a supervised distance-fit (sMDS) across Qwen2.5-3B, Llama-3.2-3B, and Gemma-2-2B, persisting at 3B/8B/70B Llama scale [tiblias-etal-2025] The circle is task-dependent: the identical date context collapses into a non-cyclic clustered or linear manifold when the completion cue asks for season or temperature rather than recency [tiblias-etal-2025] It is causally necessary: noise confined to the located 2D subspace degrades temporal-reasoning accuracy as much as noise across the full residual stream, while an equally-sized random subspace has negligible effect [tiblias-etal-2025] Bhalla et al. reproduce the days-of-week circle in Llama-3.1-8B (layer 19) via PCA and causally steer along its principal components to shift the predicted day-of-week token smoothly [bhalla-etal-2026]

models: Qwen2.5-3B-Instruct, Qwen2.5-3B · method: Linear probing, PCA, Activation patching, Sparse Autoencoders (SAE), Supervised Multi-Dimensional Scaling (SMDS), Subspace noise perturbation, Activation Steering (Addition)
From Directions to Cones: Exploring Multidimensional Representations of Propositional Facts in LLMs (2025)measured

Propositional truth is mediated by a multi-dimensional concept cone

Details

Yu et al. extend the concept-cone framework from refusal to truth, finding truth judgments in Qwen2.5 and Gemma-2 are causally mediated by a polyhedral cone of up to 5 non-negative-combination directions [yu-etal-2025] Larger models (Qwen-7B, Gemma-9B) sustain near-100% Answer Switching Rate across all 5 tested dimensions while smaller models degrade beyond 2-3 [yu-etal-2025] Only the first cone axis aligns with the classic difference-in-means truth direction (cosine ~0.1-0.2), so difference-in-means captures just one facet of a multi-dimensional truth subspace [yu-etal-2025] Cone-based interventions preserve unrelated behavior (mean KL divergence 0.03-0.05 on Alpaca instructions) [yu-etal-2025]

models: Qwen2.5-3B-Instruct, Qwen2.5-7B-Instruct, Qwen2.5-14B-Instruct · method: Difference-in-means direction, Causal interventions (steering), Supervised direction search (gradient-based optimization), Geometric analysis, PCA

GPT

Symmetry in Language Statistics Shapes the Geometry of Model Representations (2026), Not All Language Model Features Are One-Dimensionally Linear (2024), Shape Happens: Automatic Feature Manifold Discovery in LLMs via Supervised Multi-Dimensional Scaling (2025), Do Sparse Autoencoders Capture Concept Manifolds? (2026)measured

Cyclical time is represented as a circle, replicated and causally validated

Details

Cyclical temporal concepts (days of the week, months) are each independently represented as points on a circle in the residual stream, linearly decodable from activations [karkada-etal-2026] Despite the "toroidal" id, the confirmed finding is a single circle per cyclic variable, not a joint torus; no cited paper shows two such circles coexisting as orthogonal factors of one representation [karkada-etal-2026] Karkada et al. independently derive the same single-circle geometry analytically from translation-symmetric co-occurrence statistics [karkada-etal-2026] Engels et al. discover the same circles in GPT-2-small (layer 7) and Mistral-7B by clustering SAE dictionary elements, and causally validate them via activation patching on Mistral-7B and Llama-3-8B, where patching to a rotated point changes a day/month arithmetic answer [engels-etal-2024] Tiblias et al. recover the same circular date/month structure via a supervised distance-fit (sMDS) across Qwen2.5-3B, Llama-3.2-3B, and Gemma-2-2B, persisting at 3B/8B/70B Llama scale [tiblias-etal-2025] The circle is task-dependent: the identical date context collapses into a non-cyclic clustered or linear manifold when the completion cue asks for season or temperature rather than recency [tiblias-etal-2025] It is causally necessary: noise confined to the located 2D subspace degrades temporal-reasoning accuracy as much as noise across the full residual stream, while an equally-sized random subspace has negligible effect [tiblias-etal-2025] Bhalla et al. reproduce the days-of-week circle in Llama-3.1-8B (layer 19) via PCA and causally steer along its principal components to shift the predicted day-of-week token smoothly [bhalla-etal-2026]

models: GPT-2-small · method: Linear probing, PCA, Activation patching, Sparse Autoencoders (SAE), Supervised Multi-Dimensional Scaling (SMDS), Subspace noise perturbation, Activation Steering (Addition)

word2vec

Symmetry in Language Statistics Shapes the Geometry of Model Representations (2026), Not All Language Model Features Are One-Dimensionally Linear (2024), Shape Happens: Automatic Feature Manifold Discovery in LLMs via Supervised Multi-Dimensional Scaling (2025), Do Sparse Autoencoders Capture Concept Manifolds? (2026)measured

Cyclical time is represented as a circle, replicated and causally validated

Details

Cyclical temporal concepts (days of the week, months) are each independently represented as points on a circle in the residual stream, linearly decodable from activations [karkada-etal-2026] Despite the "toroidal" id, the confirmed finding is a single circle per cyclic variable, not a joint torus; no cited paper shows two such circles coexisting as orthogonal factors of one representation [karkada-etal-2026] Karkada et al. independently derive the same single-circle geometry analytically from translation-symmetric co-occurrence statistics [karkada-etal-2026] Engels et al. discover the same circles in GPT-2-small (layer 7) and Mistral-7B by clustering SAE dictionary elements, and causally validate them via activation patching on Mistral-7B and Llama-3-8B, where patching to a rotated point changes a day/month arithmetic answer [engels-etal-2024] Tiblias et al. recover the same circular date/month structure via a supervised distance-fit (sMDS) across Qwen2.5-3B, Llama-3.2-3B, and Gemma-2-2B, persisting at 3B/8B/70B Llama scale [tiblias-etal-2025] The circle is task-dependent: the identical date context collapses into a non-cyclic clustered or linear manifold when the completion cue asks for season or temperature rather than recency [tiblias-etal-2025] It is causally necessary: noise confined to the located 2D subspace degrades temporal-reasoning accuracy as much as noise across the full residual stream, while an equally-sized random subspace has negligible effect [tiblias-etal-2025] Bhalla et al. reproduce the days-of-week circle in Llama-3.1-8B (layer 19) via PCA and causally steer along its principal components to shift the predicted day-of-week token smoothly [bhalla-etal-2026]

models: word2vec (trained on Wikipedia) · method: Linear probing, PCA, Activation patching, Sparse Autoencoders (SAE), Supervised Multi-Dimensional Scaling (SMDS), Subspace noise perturbation, Activation Steering (Addition)

CLIP (Contrastive Language-Image Pretraining)

Emergent Visual-Semantic Hierarchies in Image-Text Representations (2024)measured

VLM embeddings show an emergent radial entailment hierarchy, improvable by fine-tuning

Details

Alper & Averbuch-Elor propose Radial Embedding, treating the empty-string embedding as an entailment root and concept genericity as Euclidean distance from it, an explicit relaxation of hyperbolic entailment cones [alper-averbuch-elor-2024-radial-embedding] Zero-shot, CLIP-Base reaches order-consistency tau_d=0.89 and OpenCLIP-H tau_d=0.83 on their HierarCaps benchmark, surpassing prior purpose-built models [alper-averbuch-elor-2024-radial-embedding] Fine-tuning only the text encoder with a contrastive exterior-angle loss raises tau_d to 0.99 (CLIP-Base) and 0.97 (OpenCLIP-H) and HyperLex correlation from 0.06 to 0.37 [alper-averbuch-elor-2024-radial-embedding] The fine-tuning leaves standard multimodal tasks (COCO retrieval, CIFAR classification) near-unchanged, ruling out a general-capability side effect [alper-averbuch-elor-2024-radial-embedding]

models: CLIP ViT-B/32 · method: Radial Entailment Embedding

OpenCLIP

Emergent Visual-Semantic Hierarchies in Image-Text Representations (2024)measured

VLM embeddings show an emergent radial entailment hierarchy, improvable by fine-tuning

Details

Alper & Averbuch-Elor propose Radial Embedding, treating the empty-string embedding as an entailment root and concept genericity as Euclidean distance from it, an explicit relaxation of hyperbolic entailment cones [alper-averbuch-elor-2024-radial-embedding] Zero-shot, CLIP-Base reaches order-consistency tau_d=0.89 and OpenCLIP-H tau_d=0.83 on their HierarCaps benchmark, surpassing prior purpose-built models [alper-averbuch-elor-2024-radial-embedding] Fine-tuning only the text encoder with a contrastive exterior-angle loss raises tau_d to 0.99 (CLIP-Base) and 0.97 (OpenCLIP-H) and HyperLex correlation from 0.06 to 0.37 [alper-averbuch-elor-2024-radial-embedding] The fine-tuning leaves standard multimodal tasks (COCO retrieval, CIFAR classification) near-unchanged, ruling out a general-capability side effect [alper-averbuch-elor-2024-radial-embedding]

models: OpenCLIP ViT-H/14 · method: Radial Entailment Embedding