Causal interventions (steering)

methods / Causal Validation / Causal interventions (steering)

Techniqueintermediate

Directly adds or ablates a fixed direction in activation space via vector arithmetic and observes the effect on model output — no second forward pass needed, unlike activation patching.

Used in (153 observations)

55 harmfulness-subconcept probes collapse to one dominant directionmeasured

structure: Linear Subspace · models: Llama-3.1-8B-Instruct, Qwen2-7B-Instruct · paper: Death by a Thousand Directions: Exploring the Geometry of Harmfulness in LLMs through Subconcept Probing

A 1-layer RNN solves modular addition via six Fourier frequenciesmeasured

structure: Lissajous Curves · models: Modular-Arithmetic Vanilla RNN (1 layer, tanh, d_h=256, mod 113) · paper: Modular Addition in Recurrent Neural Networks Requires Low-Rank Fourier Circuits

Refusal is a causally-minimal SAE-latent set with hydra redundancymeasured

structure: Linear Subspace · models: Gemma-2-2B-it, Llama-3.1-8B-Instruct · paper: Beyond I'm Sorry, I Can't: Dissecting Large Language Model Refusal

A moral-framework probe decodes and steers mid-reasoning ethicsmeasured

structure: Linear Direction, Linear Separability · models: Llama 3.3 70B Instruct, Qwen2.5-72B · paper: Understanding Moral Reasoning Trajectories in LLMs: Toward Probing-Based Explainability

An eval/deploy probe generalizes but its steering direction is inertmeasured

structure: Linear Direction, Linear Separability · models: Llama 3.3 70B Instruct · paper: Probing and Steering Evaluation Awareness of Language Models

A latent-context direction drives in-context hierarchy trackingmeasured

structure: Linear Direction · models: Qwen2.5-0.5B, Qwen2.5-1.5B, Qwen2.5-3B, Gemma-2-2B, Llama-3.2-3B, SmolLM3-3B, Qwen3-0.6B · paper: A Circuit for Predicting Hierarchical Structure In-Context in Large Language Models

A grid-organized cell subspace supports relational binding in LLMsmeasured

structure: Linear Subspace · models: Llama-3-8B-Instruct, Qwen3-8B · paper: Cell-Based Representation of Relational Binding in Language Models

A PCA 'Ordering ID' direction localizes and steers bindingmeasured

structure: Linear Subspace, Linear Direction · models: Llama-2-7B, Llama-3-8B, Qwen1.5-7B, Pythia-6.9B, Float-7B (code fine-tuned LM) · paper: Representational Analysis of Binding in Language Models

A fitted linear projection (PISCO) disentangles style from contentmeasured

structure: Linear Subspace · models: ResNet-18 (supervised, ImageNet), ResNet-18 (SimCLR contrastive pretraining, CIFAR-10) · paper: Simple Disentanglement of Style and Content in Visual Representations

A memory RNN forms a line attractor without any bifurcationmeasured

structure: Line Attractor · models: Piecewise-linear RNN (N=40, delayed-addition short-term-memory task) · paper: Understanding and Controlling the Geometry of Memory Organization in RNNs

An SAE on IndexTTS2 enables causal latent emotion steeringmeasured

structure: Linear Direction · models: IndexTTS2 · paper: Sparse Autoencoders for Interpretable Emotion Control in Text-to-Speech

AlphaZero's novel chess concepts are causally steerable and teachablemeasured

structure: Linear Direction · models: AlphaZero (chess, ResNet policy/value network, self-play) · paper: Bridging the Human-AI Knowledge Gap: Concept Discovery and Transfer in AlphaZero

A model's self-report tracks a steerable emotive-state directionmeasured

structure: Linear Direction · models: Llama-3.2-3B-Instruct, Llama-3.2-1B-Instruct, Llama-3.1-8B-Instruct, Gemma 3 4B Instruct, Qwen2.5-7B-Instruct · paper: Quantitative Introspection in Language Models: Tracking Emotive States Across Conversation

Twelve trait directions form stable antisocial and prosocial clustersmeasured

structure: Linear Direction · models: Llama-3.1-8B, Qwen2.5-14B, Qwen-2.5-7B, Qwen2.5-0.5B, Llama-3.2-1B · paper: Intrinsic Guardrails: How Semantic Geometry of Personality Interacts with Emergent Misalignment in LLMs

Grid-like codes emerge in an LSTM path-integrator and are causally load-bearingmeasured

structure: Torus · models: LSTM path-integration module + A3C navigation agent · paper: Vector-based Navigation Using Grid-like Representations in Artificial Agents

Toy transformers build an orthogonal hypothesis frame for Bayesian updatingmeasured

structure: Linear Subspace, Linear Direction · models: Bayesian Wind Tunnel Transformer (bijection task, 6 layers, 6 heads, d_model=192), Bayesian Wind Tunnel Transformer (HMM filtering task, 9 layers, 8 heads, d_model=256) · paper: The Bayesian Geometry of Transformer Attention

SAE latents in Gemma-2-9B show preliminary simplex-structured belief geometrymeasured

structure: Belief State Geometry Hypothesis (Mixed-State Presentation), Polytope (Simplex) · models: Gemma-2-9B · paper: Finding Belief Geometries with Sparse Autoencoders

Beta-VAE axis-alignment emerges from information-bottleneck pressuremeasured

structure: Linear Direction · models: β-VAE (convolutional encoder/decoder, dSprites/colored-dSprites/CelebA/3D-Chairs) · paper: Understanding disentangling in β-VAE

TabICLv2 predicts by a literal class centroid; TabPFNv2 votesmeasured

structure: Linear Direction · models: TabPFN v2 (tabular in-context-learning foundation model), Mitra, TabICLv2 · paper: A Mechanistic Study of Tabular Foundation Models

A PCA gender direction in word2vec is causally debiasablemeasured

structure: Linear Direction · models: word2vec (Google News, 300d) · paper: Man is to Computer Programmer as Woman is to Homemaker? Debiasing Word Embeddings

A per-head Fisher-ratio score selects steering that nearly matches fine-tuningmeasured

structure: Linear Separability · models: InternVL3-8B · paper: Steering and Rectifying Latent Representation Manifolds in Frozen Multi-modal LLMs for Video Anomaly Detection

Feature absorption is decoder-direction composition, causally isolablemeasured

structure: Linear Direction · models: Gemma-2-2B, Qwen2-0.5B, Llama-3.2-1B · paper: A is for Absorption: Studying Feature Splitting and Absorption in Sparse Autoencoders

A real trained dot-product recommender's item embeddings carry a rank-one popularity-aligned direction that a prior-separation intervention removesmeasured

structure: Linear Direction · models: 128-dim dot-product matrix-factorization recommender, trained on Alibaba/Tianchi mobile-recommendation logs (Cheng) · paper: A Rank-One Popularity Component in Dot-Product Recommender Scores: Population Theory and Prior-Separation Evidence

Group-composition networks implement a representation-theoretic Fourier algorithmmeasured

structure: Circle · models: Small MLP/Transformer trained on finite-group composition tasks · paper: A Toy Model of Universality: Reverse Engineering How Networks Learn Group Operations

Clamping SAE batch-effect features corrects single-cell batch integrationmeasured

structure: Linear Direction · models: scGPT (whole-human pretrained checkpoint), scFoundation (pretrained) · paper: Sparse Autoencoders Reveal Interpretable Features in Single-Cell Foundation Models

Claude Sonnet 4.5 encodes 171 emotion directions that steer safety behaviormeasured

structure: Linear Direction · models: Claude Sonnet 4.5 · paper: Emotion Concepts and their Function in a Large Language Model

CLIP's modality gap is a provable, tunable cone effectmeasured

structure: Anisotropy, Linear Direction · models: CLIP ViT-B/32, CLIP ViT-B/16 · paper: Mind the Gap: Understanding the Modality Gap in Multi-modal Contrastive Representation Learning

CLIP/SigLIP covariance splits into signal and shared-noise subspacesmeasured

structure: Dimensional collapse · models: CLIP ViT-B/16, CLIP ViT-B/32, CLIP ViT-L/14, SigLIP, SigLIP 2 · paper: Your CLIP has 164 dimensions of noise: Exploring the embeddings covariance eigenspectrum of contrastively pretrained vision-language transformers

Audio-text embeddings split into shared head and private tailmeasured

structure: Linear Subspace, Anisotropy · models: CLAP (HTSAT-BERT-ZS, trained on WavCaps), DRCap's CLAP (trained on WavCaps + SoundVECaps) · paper: COMET: Concept Space Dissection of the Modality Gap in Audio-Text Multimodal Contrastive Embeddings

Concept directions anti-concentrate in the unembedding's low-variance spectral tailmeasured

structure: Linear Direction · models: Llama-3.2-1B, Llama-3.2-3B, Llama-3.1-8B, Qwen2.5-0.5B, Qwen2.5-1.5B, Qwen2.5-3B, Qwen-2.5-7B, Qwen2.5-14B, Qwen3-0.6B, Qwen3-1.7B, Qwen3-4B, Qwen3-8B, Gemma-2-2B, Gemma-2-9B, Mistral-7B-v0.3, SmolLM2-1.7B, OLMoE-1B-7B, JetMoE-8B · paper: Concepts Whisper While Syntax Shouts: Spectral Anti-Concentration and the Dual Geometry of Transformer Representations

Linear Artificial Tomography reads and controls concepts across LLMsmeasured

structure: Linear Direction · models: Llama-2-7B-Chat, Llama-2-13B-Chat, Llama-2-70B-Chat, Vicuna-13B, Vicuna-33B-Uncensored, DeBERTa-xxlarge-v2-MNLI · paper: Representation Engineering: A Top-Down Approach to AI Transparency

Conformal normalization provably yields a torus, only favoring hexagonal gridsmeasured

structure: Torus · models: Conformal-normalization linear path-integration RNN, Conformal-normalization nonlinear (CANN-like) path-integration RNN · paper: Emergence of Grid-like Representations by Training Recurrent Networks with Conformal Normalization

Contrastive reflection-intent directions steer reflective behaviormeasured

structure: Linear Direction · models: Qwen2.5-3B, Gemma 3 4B Instruct · paper: Unveiling the Latent Directions of Reflection in Large Language Models

CosyVoice2's LM module has a clean composable emotion subspacemeasured

structure: Linear Subspace · models: CosyVoice2 · paper: A Geometric Perspective on Composable Emotion Steering in Text-to-Speech Models

After mean-centering, 88 languages share one XLM-R subspacemeasured

structure: Linear Subspace · models: XLM-RoBERTa base · paper: The Geometry of Multilingual Language Model Representations

Same concept, different SAE directions across modalitiesmeasured

structure: Linear Direction, Anisotropy · models: CLIP ViT-B/32, MetaCLIP ViT-B/32, OpenCLIP ViT-B/32, SigLIP 2 · paper: Same Concept, Different Directions: Cross-Modal Feature Heterogeneity in Sparse Autoencoders

Sinusoidal number geometry recurs across eight LLMs and is causalmeasured

structure: Lissajous Curves · models: OLMo 2 1B, OLMo 2 7B, OLMo 2 13B, Llama-3.2-1B, Llama-3.2-3B, Llama-3-8B, Llama-3.1-8B, Phi-4 (15B) · paper: Unravelling the Mechanisms of Manipulating Numbers in Language Models

Cultural-value steering axes are entangled, mirroring real value correlationsmeasured

structure: Linear Direction, Linear Subspace · models: Llama-3.2-3B-Instruct, Qwen3-4B-Instruct, Gemma 3 4B Instruct · paper: Scenario-based Probing and Steering Cultural Values in Large Language Models

Visual concept directions split into localized-entity vs distributed-abstract encodingmeasured

structure: Linear Direction · models: Qwen2.5-VL 7B Instruct, Qwen3-VL 8B, Qwen3-VL 32B, LLaVA-OneVision-1.5 8B, Gemma 3 4B Instruct, Gemma 3 27B Instruct · paper: Causal Probing for Internal Visual Representations in Multimodal Large Language Models

DAS finds number- and unit-specific quantity-comparison subspacesmeasured

structure: Linear Subspace, Linear Direction · models: Qwen3-4B, Qwen3-8B, OLMo 3 7B · paper: Language Models Compare Quantities Using Number-specific and Unit-specific Heuristics

A model-free Sokoban agent linearly encodes a causal planmeasured

structure: Linear Direction · models: DRC(3,3) Sokoban Agent · paper: Interpreting Emergent Planning in Model-Free Reinforcement Learning

MLLM visual tokens split into sink, dead, alive clustersmeasured

structure: Anisotropy · models: LLaVA-1.5-7B, LLaVA-1.5-13B, InternVL3-8B · paper: What Do Visual Tokens Really Encode? Uncovering Sparsity and Redundancy in Multimodal Large Language Models

Fourier-circuit synchronization completes before grokking, causally advanceablemeasured

structure: Circle · models: Grokking Modular-Arithmetic Transformer (2 layers, 4 heads, pre-LN, d_model=128, mod 113/149/197) · paper: Circuit Synchronization Precedes Generalization: A Causal Precursor to Grokking

Multimodal function vectors causally raise visual-relation accuracymeasured

structure: Linear Direction · models: OpenFlamingo-4B · paper: Multimodal Function Vectors for Visual Relations

A function vector decomposes additively into causal per-shot sub-vectorsmeasured

structure: Linear Direction · models: Gemma-2-2B, Gemma-2-9B, Gemma-2-27B, Llama-3.2-1B, Llama-3.2-3B, Llama-3.1-8B-Instruct · paper: How Few-Shot Examples Add Up: A Causal Decomposition of Function Vectors in In-Context Learning

Leela Chess Zero's correct moves appear early, then get overriddenmeasured

structure: Linear Subspace, Linear Direction · models: Leela Chess Zero (policy network) · paper: The Algorithm Is Not the Behavior: Learned Priors Override Look-Ahead in a Chess-Playing Neural Network

GANSpace PCA finds near-independent, unimodal, content-style-separated directionsmeasured

structure: Linear Direction · models: StyleGAN2 (trained on FFHQ, 1024x1024), BigGAN-deep (512px) · paper: GANSpace: Discovering Interpretable GAN Controls

A Gemma-2-2B probe direction detects and bidirectionally steers hallucinationmeasured

structure: Linear Direction · models: Gemma-2-2B, Gemma-2-2B-it · paper: A Single Direction of Truth: An Observer Model's Linear Residual Probe Exposes and Steers Contextual Hallucinations

Fine-tuned GPT-2-XL solves addition with sparse Fourier featuresmeasured

structure: Lissajous Curves · models: GPT-2-XL · paper: Pre-trained Large Language Models Use Fourier Features to Compute Addition

Concept-direction local ID tracks generality and restores distilled-diffusion diversitymeasured

structure: Intrinsic-dimension profile across depth · models: Stable Diffusion XL, SDXL-DMD (4-step distilled) · paper: ELROND: Exploring and Decomposing Intrinsic Capabilities of Diffusion Models

Transcoder-decoded planning directions causally control future tokensmeasured

structure: Linear Direction · models: Qwen3-0.6B, Qwen3-1.7B, Qwen3-4B, Qwen3-8B, Qwen3-14B · paper: Latent Planning Emerges with Scale

Sparse logit-lens directions in VLA FFN value vectors steer robot actionsmeasured

structure: Linear Direction · models: OpenVLA-7B, pi0 (PaliGemma VLA backbone, predecessor of pi0.5) · paper: Mechanistic Interpretability for Steering Vision-Language-Action Models

SAE features selected by CoT-minus-direct trigger a reasoning modemeasured

structure: Linear Direction · models: Llama-3.1-8B-Instruct, Llama 3.3 70B Instruct, Qwen3-0.6B, Qwen3-4B, Gemma 3 4B Instruct, Gemma 3 12B Instruct · paper: Reasoning Beyond Chain-of-Thought: A Latent Computational Mode in Large Language Models

LLM hidden states warp categorically at digit-count boundariesmeasured

structure: Decision boundary (as a codimension-1 hypersurface), Linear Direction · models: Llama-3-8B-Instruct, Mistral-7B-Instruct-v0.3, Gemma-2-9B-it, Qwen2.5-7B-Instruct, Phi-3.5-mini-instruct, Llama-3-8B · paper: Categorical Perception in Large Language Model Hidden States: Structural Warping at Digit-Count Boundaries

In-context exemplars reorganize concept geometry to a specified graphmeasured

structure: Circle, 2D Grid (Square Lattice) · models: Llama-3.1-8B, Llama-3.2-1B, Llama-3.1-8B-Instruct, Gemma-2-2B, Gemma-2-9B · paper: In-Context Learning of Representations

Inference-time causal probing separates causal from correlational directionsmeasured

structure: Linear Direction · models: Llama-3-8B-Instruct, Pythia-70M · paper: Inference-Time Causal Probing in LLMs

InterFaceGAN SVM boundary normals are linear, editable, and partially entangledmeasured

structure: Linear Direction · models: PGGAN (trained on CelebA-HQ, 1024x1024), StyleGAN (trained on FFHQ, 1024x1024) · paper: InterFaceGAN: Interpreting the Disentangled Face Representation Learned by GANs

Iso-energy SAE isolates a bimodal cross-modal subspacemeasured

structure: Linear Subspace, Anisotropy · models: CLIP ViT-B/32, CLIP ViT-L/14, OpenCLIP ViT-B/32, OpenCLIP ViT-L/14, SigLIP, SigLIP 2 · paper: Cross-Modal Redundancy and the Geometry of Vision-Language Embeddings

Fewer than 25 register neurons with consistent directions cause ViT outlier tokensmeasured

structure: Linear Direction · models: OpenCLIP ViT-B/16 (LAION-2B), DINOv2 ViT-L/14, LLaVA-Llama-3-8B · paper: Vision Transformers Don't Need Trained Registers

RoPE massive values concentrate at fixed Q/K dimensionsmeasured

structure: Anisotropy · models: Llama-3-8B-Instruct, Gemma-2-9B-it, Qwen2.5-7B-Instruct, Llama-2-7B-Chat, GPT-2-small, OPT-350M, OPT-2.7B · paper: Massive Values in Self-Attention Modules are the Key to Contextual Knowledge Understanding

A delta-loss crosscoder isolates 1-2 causal directions per fine-tuning organismmeasured

structure: Linear Direction · models: Llama-3.1-8B-Instruct, Gemma-2-9B-it, Qwen2.5-7B · paper: Delta-Crosscoder: Robust Crosscoder Model Diffing in Narrow Fine-Tuning Regimes

Trajectory-subspace curvature perturbations causally modulate next-token entropymeasured

structure: Intrinsic-dimension profile across depth · models: GPT-2 XL, Pythia-2.8B, Custom 12-layer GPT-2-Small (trained from scratch, 100M tokens) · paper: Representational Curvature Modulates Behavioral Uncertainty in Large Language Models

SAE feature modality composition shifts by depth in a TTS LM and steers speechmeasured

structure: Linear Direction · models: CosyVoice3 (Qwen2.5-0.5B backbone) · paper: Interpreting and Steering a Text-to-Speech Language Model with Sparse Autoencoders

A language difference-of-means direction causally steers target-language outputmeasured

structure: Linear Direction · models: XLM-RoBERTa base · paper: The Geometry of Multilingual Language Model Representations

Layer intrinsic-dimension peak aligns with the best brain-predicting layermeasured

structure: Intrinsic-dimension profile across depth · models: OPT-125M, OPT-1.3B, OPT-13B, Pythia-160M, Pythia-410M, Pythia-6.9B, WavLM-base-plus, WavLM Large, Whisper large · paper: Abstraction Induces the Brain Alignment of Language and Speech Models

Nonlinear (small-MLP) probes reliably decode the full Othello board state from an 8-layer, 512-dim GPT-style transformer's internal activations at every move, and activation-patching interventions that flip the decoded board-state representation causally change the model's subsequent move predictionsmeasured

structure: Nonlinear World-Model Decodability · models: OthelloGPT · paper: Emergent World Representations: Exploring a Sequence Model Trained on a Synthetic Task

Splicing probed entity-state representations causally shifts generationmeasured

structure: Linear Direction, Linear Subspace · models: BART-base, T5-Base · paper: Implicit Representations of Meaning in Neural Language Models

Lightweight VLMs linearly encode object presence but never orientationmeasured

structure: Linear Separability · models: Ovis2.5-2B, InternVL3.5-2B, VST-3B · paper: Probing Visual Concepts in Lightweight Vision-Language Models for Automated Driving

Explicit unit-circle and neighbor losses impose a 2D timbre spacemeasured

structure: Circle · models: pGESAM (pitch-conditioned Generative Sample Map), EnCodec · paper: Pitch-Conditioned Instrument Sound Synthesis from an Interactive Timbre Latent Space

Linear probes decode future Blocksworld planning steps from one passmeasured

structure: Linear Separability, Linear Direction · models: Llama-2-7B-Chat, Vicuna-7B · paper: Unlocking the Future: Look-Ahead Planning Mechanistic Interpretability in LLMs

LLMs encode each decimal digit as its own circlemeasured

structure: Circle · models: Llama-3-8B, Mistral-7B · paper: Language Models Encode Numbers Using Digit Representations in Base 10

LLaMA-3.1-8B linearly encodes grid position; action-correctness units are causalmeasured

structure: Linear Subspace · models: Llama-3.2-1B-Instruct, Llama-3.2-3B-Instruct, Llama-3.1-8B-Instruct, Llama-3.1-70B-Instruct · paper: From Text to Space: Mapping Abstract Spatial Models in LLMs during a Grid-World Navigation Task

Oversized vision-token norms suppress VLM spatial awarenessmeasured

structure: Anisotropy · models: LLaVA-1.5-7B · paper: Beyond Semantics: Rediscovering Spatial Awareness in Vision-Language Models

LLMs consolidate entity-state tracking into a local binary subspacemeasured

structure: Linear Subspace, Linear Direction · models: CodeLlama-13B, Gemma-2-2B, Llama 3.1 70B · paper: Do Language Models Track Entities Across State Changes?

Bottleneck-localized steering beats full-network steering in music diffusionmeasured

structure: Linear Direction · models: AudioLDM2, Stable Audio Open, Ace-Step · paper: TADA! Tuning Audio Diffusion Models Through Activation Steering

Numbers lie on a logarithmically-compressed 1D number linemeasured

structure: 1D continuum manifold · models: Pythia-2.8B, Llama-2-7B, Llama-3.1-8B, Llama-3.2-1B, GPT-2-Large, Mistral-7B, Llama-3.1-8B-Instruct, Llama-3.2-1B-Instruct, Qwen2.5-3B-Instruct, Qwen2.5-3B, Llama-3.2-3B-Instruct, Llama-3.2-3B, Gemma-2-2B-it, Gemma-2-2B, Llama-3.1-70B-Instruct, Llama-3-8B-Instruct, Llama-3-8B, Mistral-7B-Instruct-v0.3, Qwen2.5-7B-Instruct · paper: Number Representations in LLMs: A Computational Parallel to Human Perception, Shape Happens: Automatic Feature Manifold Discovery in LLMs via Supervised Multi-Dimensional Scaling, Weber's Law in Transformer Magnitude Representations: Efficient Coding, Representational Geometry, and Psychophysical Laws in Language Models, LLMs Know More About Numbers than They Can Say

A weak-perturbation response direction stays near-orthogonal across every layermeasured

structure: Linear Direction · models: Gemma-2-2B, Llama-3.2-3B-Instruct, GPT-2-XL · paper: Universal Response and Emergence of Induction in LLMs

Mahalanobis cosine between truth directions predicts cross-domain generalizationmeasured

structure: Linear Direction · models: Llama 3.3 70B Instruct, Llama-3.1-8B-Instruct, Llama-3.2-3B-Instruct, Qwen2.5-14B-Instruct, Qwen2.5-7B-Instruct · paper: The Truthfulness Spectrum Hypothesis

An invariant reasoning subspace stabilizes latent chain-of-thought refinementmeasured

structure: Linear Subspace · models: GPT-2 Small, GPT-2 Medium, Qwen2.5-Math-1.5B · paper: Invariant Reasoning Directions in Latent Trajectories of Language Models

Positional encodings anchor index-based spatial structure in ViTsmeasured

structure: Anisotropy · models: ViT-S/16 (12 layers, 8 heads, dim 384, trained from scratch on ImageNet-100) · paper: Positional Encodings Anchor Spatial Structure in Vision Transformers: A Geometric Perspective on Robustness

Massive activations provably cause compression valleys synced with attention sinksmeasured

structure: Dimensional collapse · models: Pythia-410M, Pythia-6.9B, GPT-OSS-20B, Gemma-7B · paper: Attention Sinks and Compression Valleys in LLMs Are Two Sides of the Same Coin

Timestep-conditioned transcoder directions give exact FLUX attribution graphsmeasured

structure: Linear Direction · models: FLUX.1 [schnell] · paper: DifFRACT: Diffusion Feature Reconstruction and Attribution for Circuit Tracing

LLMs implement word2vec-style relational vector offsets, causally patchablemeasured

structure: Linear Direction · models: GPT-J-6B, GPT-2 Small, GPT-2 Medium, GPT-2 Large, GPT-2 XL, BLOOM-176B · paper: Language Models Implement Simple Word2Vec-style Vector Arithmetic

Latent Scaling corrects crosscoder sparsity artifacts and isolates chat latentsmeasured

structure: Linear Direction · models: Gemma-2-2B, Gemma-2-2B-it · paper: Overcoming Sparsity Artifacts in Crosscoders to Interpret Chat-Tuning

A per-token mean-difference direction fingerprints narrow fine-tuningmeasured

structure: Linear Direction · models: Qwen3-1.7B, Llama-3.1-8B-Instruct, Gemma 3 1B IT, Qwen2.5-7B-Instruct · paper: Narrow Finetuning Leaves Clearly Readable Traces in Activation Differences

Modular-addition networks universally learn a torus projecting to a discmeasured

structure: Torus, Circle · models: Grokking Modular-Arithmetic Transformer (1 layer, 4 heads, d_model=128, mod 113), Clock/Pizza Transformer, Model A (1 layer, constant attention alpha=0, width 128, mod 59), Clock/Pizza Transformer, Model B (1 layer, normal attention alpha=1, width 128, mod 59) · paper: Progress Measures for Grokking via Mechanistic Interpretability, The Clock and the Pizza: Two Stories in Mechanistic Explanation of Neural Networks, On the Geometry and Topology of Representations: The Manifolds of Modular Addition

Word embeddings share a mean plus top-PC anisotropymeasured

structure: Anisotropy · models: word2vec (Google News, 300d), GloVe (840B token Common Crawl) · paper: All-but-the-Top: Simple and Effective Postprocessing for Word Representations

Negation is a single linear direction decodable by layer 4measured

structure: Linear Direction · models: Llama-3.1-8B, Mistral-7B-v0.1 · paper: How Language Models Process Negation

A single SAE latent causally mediates refusal with a dose-response tradeoffmeasured

structure: Linear Direction · models: Phi-3-mini, Llama-3.1-8B-Instruct · paper: Steering Language Model Refusal with Sparse Autoencoder Features

Off-the-shelf persona vectors are near-orthogonal to sycophancy yet reduce itmeasured

structure: Linear Direction · models: Gemma 2 27B Instruct, Qwen3-32B · paper: Playing Devil's Advocate: Off-the-Shelf Persona Vectors Rival Targeted Steering for Sycophancy

A mean-difference binding-ID direction in Q/K space redirects attentionmeasured

structure: Linear Direction · models: Gemma 2 9B IT, Gemma 3 12B IT, Llama-3.2-3B-Instruct, Llama-3.1-8B-Instruct · paper: A Retrieval-Conditioned Rebinding Circuit for Dynamic Entity Tracking in Large Language Models

SAE decoder cosine reveals feature splitting in dense embeddings and steers searchmeasured

structure: Linear Direction · models: text-embedding-3-small · paper: Disentangling Dense Embeddings with Sparse Autoencoders

SAE probe directions decode a pitch>loudness>timbre hierarchy and steer audiomeasured

structure: Linear Direction · models: DiffRhythm, EnCodec, WavTokenizer, Stable Audio Open · paper: Learning Interpretable Features in Audio Latent Spaces via Sparse Autoencoders

A participation-ratio spectral signal finds attention circuits label-free across scalemeasured

structure: Intrinsic-dimension profile across depth · models: TS-51M (custom 8-layer x 512d x 16-head transformer, TinyStories, 6 pretraining seeds), nanoGPT-style GPT-2 124M (FineWeb-10B), Pythia-160M, Pythia-410M, Pythia-1B, OLMo-1B, OLMoE-1B-7B · paper: Spectral Probe-Circuits: A Three-Step Recipe for Identifying Attention-Head Circuits in Pretrained Transformers

A GPT-2 trained on grid navigation forms an orthogonal xy coordinate mapmeasured

structure: Linear Subspace · models: Custom GPT-2-Small (trained from scratch on synthetic grid-navigation token sequences) · paper: Cognitive Maps in Language Models: A Mechanistic Analysis of Spatial Planning

Monosemantic Whisper SAE latents encode phonetic/lexical features and steer transcriptsmeasured

structure: Linear Direction · models: Whisper base · paper: Mechanistic Interpretability of ASR models using Sparse Autoencoders

Fitted part-of-speech CLIP subspaces disentangle content from appearancemeasured

structure: Linear Subspace · models: CLIP ViT-B/32 · paper: Parts of Speech-Grounded Subspaces in Vision-Language Models

Linguistic categories occupy nested low-dimensional subspacesmeasured

structure: Linear Subspace · models: ELMo (5.5B-word pretrained, 2-layer biLSTM), BERT-base-cased · paper: The Low-Dimensional Linear Geometry of Contextualized Word Representations

Cross-modally correlated soft-robot VAE axes are individually manipulable primitivesmeasured

structure: Linear Direction · models: ProSoRo Multi-modal Variational Autoencoder (motion/force/shape latent proprioception) · paper: Anchoring Morphological Representations Unlocks Latent Proprioception in Soft Robots

DCGAN latent vector arithmetic yields a smiling man from face-attribute offsetsmeasured

structure: Linear Direction · models: DCGAN (trained on aligned & cropped celebrity faces) · paper: Unsupervised Representation Learning with Deep Convolutional Generative Adversarial Networks

A single sentiment neuron in a byte-level mLSTM linearly encodes and steers sentimentmeasured

structure: Linear Direction · models: Byte mLSTM (4096-unit, trained on ~82M Amazon reviews) · paper: Learning to Generate Reviews and Discovering Sentiment

Refusal is better described as an affine function than a directionmeasured

structure: Affine Subspace · models: Llama-3-8B-Instruct, Llama-3-70B-Instruct, Hermes Eagle RWKV v5 (7B) · paper: Refusal in LLMs is an Affine Function

Refusal is mediated by a multi-dimensional concept conemeasured

structure: Cone · models: Gemma-2-2B-it, Gemma-2-9B-it, Qwen2.5-1.5B-Instruct, Qwen2.5-14B-Instruct, Llama-3-8B-Instruct · paper: The Geometry of Refusal in Large Language Models: Concept Cones and Representational Independence

Refusal directions are approximately parallel across 14 languages and transfermeasured

structure: Linear Direction · models: Gemma-2-9B-it, Llama-3.1-8B-Instruct, Qwen2.5-7B-Instruct · paper: Refusal Direction is Universal Across Safety-Aligned Languages

Refusal is mediated by a single direction across modelsmeasured

structure: Linear Direction · models: Llama-3-70B-Instruct, Gemma-7B-it, Qwen-1.8B-Chat, Vicuna-13B, Qwen1.5-1.8B-Chat, Qwen1.5-32B-Chat, Llama-2-13B-Chat, Llama-3.1-8B-Instruct, NeuralDaredevil-8B-abliterated, Hermes-2-Pro-Llama-3-8B, OLMo-7B-SFT, Zephyr-7B-Beta, H2O-Danube3-4B-Chat, Gemma-2-9B-it, Qwen2.5-7B-Instruct · paper: Refusal in Language Models Is Mediated by a Single Direction, Representation Engineering: A Top-Down Approach to AI Transparency, Programming Refusal with Conditional Activation Steering, Refusal Direction is Universal Across Safety-Aligned Languages

Linear relational concepts invert an LRE into an editable object directionmeasured

structure: Linear Direction · models: Llama-2-7B, GPT-J-6B · paper: Identifying Linear Relational Concepts in Large Language Models

RFM/AGOP directions from frozen MusicGen enable real-time multi-attribute steeringmeasured

structure: Linear Direction · models: MusicGen-Large · paper: Steering Autoregressive Music Generation with Recursive Feature Machines

Refusal-subspace dimensionality for full suppression scales with model sizemeasured

structure: Cone · models: Qwen3-1.7B, Qwen3-4B, Qwen3-8B, Qwen3-14B, Qwen2.5-7B-Instruct · paper: Fast Multi-dimensional Refusal Subspaces via RFM-AGOP

A few rogue dimensions drive transformer anisotropymeasured

structure: Anisotropy · models: BERT-base-cased, RoBERTa-base, GPT-2-small, XLNet-base-cased, word2vec (Google News, 300d) · paper: All Bark and No Bite: Rogue Dimensions in Transformer Language Models Obscure Representational Quality

RSA-localized concept vectors are surface-form invariant and causally guide behaviormeasured

structure: Linear Direction · models: Llama-3.1-8B, Llama 3.1 70B · paper: Analogical Reasoning Inside Large Language Models: Concept Vectors and the Limits of Abstraction

Shared SAE morphosyntactic directions are causally necessary and sufficient across languagesmeasured

structure: Linear Direction · models: Llama-3-8B, Aya-23-8B · paper: Large Language Models Share Representations of Latent Grammatical Concepts Across Typologically Diverse Languages

SD concept directions show composition, style, and texture set at successive denoising stepsmeasured

structure: Linear Direction · models: Stable Diffusion v1.4 · paper: Emergence and Evolution of Interpretable Concepts in Diffusion Models

Hierarchical concepts occupy domain-specific 150-250D subspaces, steerablemeasured

structure: Linear Subspace · models: Llama-3.2-3B, Llama-3.1-8B, Qwen3-8B, Qwen3-14B · paper: Linear Representations of Hierarchical Concepts in Language Models

A single linear ViT direction encodes depth and causally drives depth estimationmeasured

structure: Linear Direction · models: ViT-B/16 (ImageNet-21k, supervised) · paper: From Edges to Depth: Probing the Spatial Hierarchy in Vision Transformers

A Backpack model sense-vector topology partially transfers across languagesmeasured

structure: Linear Subspace · models: GPT-2-small, GPT-2-Medium, GPT-2-Large · paper: Multilinguality as Sense Adaptation

ICL separability rises first, then alignment with the unembeddingmeasured

structure: Linear Direction, Linear Separability · models: Llama-2-7B, Llama-2-13B, Llama-2-70B, Llama-3-8B, Llama-3-70B, Gemma-2B, Gemma-7B · paper: Unifying Attention Heads and Task Vectors via Hidden State Geometry in In-Context Learning

A Top-k SAE on sentence embeddings gives near-orthogonal, steerable latentsmeasured

structure: Linear Subspace · models: E5-large-v2 · paper: Aligning Sentence Embeddings to Human Concepts via Sparse Autoencoders

A continuous-time attention operator makes fixed LLMs behave continuouslymeasured

structure: Continuous causal attention operator (CCT) · models: Llama-2-13B-Chat, Llama-3-8B, Phi-3-Medium-4k-Instruct, Gemma-7B, Gemma-2-9B, Mistral-7B, GPT-2 · paper: Language Models Are Implicitly Continuous

A Sokoban RNN's plan representation predicts 50 steps ahead and generalizes OODmeasured

structure: Linear Direction · models: DRC(3,3) Sokoban Agent · paper: Planning in a Recurrent Neural Network That Plays Sokoban

A diff-in-means misalignment direction transfers across Qwen-14B fine-tunesmeasured

structure: Linear Direction · models: Qwen2.5-14B-Instruct · paper: Convergent Linear Representations of Emergent Misalignment

Sparse nonnegative concept decomposition enables causal CLIP editingmeasured

structure: Linear Direction, Anisotropy · models: OpenCLIP ViT-B/32, CLIP ResNet-50 · paper: Interpreting CLIP with Sparse Linear Concept Embeddings (SpLiCE)

A diff-in-means language-identity direction steers output language without accuracy lossmeasured

structure: Linear Direction · models: Llama-3.1-8B-Instruct, Qwen2.5-7B-Instruct, Gemma-2-2B-it · paper: ReCoVeR the Target Language: Language Steering Without Sacrificing Task Performance

StyleGAN2's W space is globally warped; local bases avoid traversal collapsemeasured

structure: Affine Subspace, Linear Subspace · models: StyleGAN2 (trained on FFHQ, 1024x1024) · paper: Do Not Escape From the Manifold: Discovering the Local Coordinates on the Latent Space of GANs

Chain-of-thought traverses step-specific subspaces that sharpen with depthmeasured

structure: Linear Separability · models: Llama-3.1-8B, Llama-3.1-8B-Instruct, DeepSeek-R1-Distill-Llama-8B · paper: LLM Reasoning as Trajectories: Step-Specific Representation Geometry and Correctness Signals

SVD-extracted ViT-L task subspaces recover most full-feature performancemeasured

structure: Linear Subspace · models: DINOv2 ViT-L/14, MAE ViT-Large (Masked Autoencoder), iBOT ViT-L/16 · paper: Understanding Geometric Representations in Self-Supervised Vision Transformers via Subspace Intervention

SwiGLU spikes collapse keys into a sink-forming subspacemeasured

structure: Dimensional collapse · models: Custom controlled 7B Transformer (trained from scratch for architecture/normalization ablations), Llama-2-7B · paper: The Spike, the Sparse and the Sink: Anatomy of Massive Activations and Attention Sinks

A causally-validated Diffusion Steering Lens tracks real head-ablation effects in a ViTmeasured

structure: Linear Direction · models: CLIP ViT-bigG/14 (Kandinsky 2.2 image encoder) · paper: Decoding Vision Transformers: The Diffusion Steering Lens

Task-vector subspace implements Bayesian retrieval; OOD generalization is near-orthogonalmeasured

structure: Linear Direction · models: Mini-ICL synthetic RoPE transformer (6L/2H/d128 for dice and Markov-chain tasks; 16L for linear regression), Qwen-2.5-7B · paper: Task Vector Geometry Underlies Dual Modes of Task Inference in Transformers

Natural task vectors emerge weakly in deep models; a TVP loss forces a layer to carry the taskmeasured

structure: Linear Direction · models: Synthetic GPT-2-style ICL transformer (3-8 layers, trained from scratch on regression/token-offset/GINC/RegBench tasks) · paper: Task Vectors in In-Context Learning: Emergence, Formation, and Benefit

Temporal knowledge drift is a linear direction orthogonal to correctness and uncertaintymeasured

structure: Linear Direction · models: Llama-2-7B-Chat, Mistral-7B-Instruct-v0.1, Llama-3.1-8B-Instruct, Qwen2.5-7B-Instruct, Gemma-2-9B-it, Gemma-2-2B-it · paper: The Geometry of Forgetting: Temporal Knowledge Drift as an Independent Axis in LLM Representations

A temporal-preference subgraph collapses continuous horizon geometry into a binary preferencemeasured

structure: Linear Direction · models: Qwen3-4B-Instruct-2507 · paper: Temporal Preference Concepts and Their Functions in a Large Language Model

Modality gap converges to angle between collapsed hyperplanesmeasured

structure: Anisotropy, Linear Subspace · models: CLIP ViT-B/32 · paper: Decipher the Modality Gap in Multimodal Contrastive Learning: From Convergent Representations to Pairwise Alignment

Low-temperature InfoNCE is the modality gap's causemeasured

structure: Anisotropy, Linear Direction · models: CLIP ViT-B/16 · paper: On the Modality Gap and the Contrastive Loss in Multi-modal Representation Learning

Torus-cluster membership, not gridness, is causally load-bearing for path integrationmeasured

structure: Torus · models: CARNN trained on path integration across multiple (1-50) environments · paper: Coherently Remapping Toroidal Cells But Not Grid Cells are Responsible for Path Integration in Virtual Agents

Trained recurrent (RNN/GRU/Mamba) policies' hidden states spontaneously converge to a stable, low-dimensional closed-orbit limit cycle that is contractive under finite-time Lyapunov analysis, and selectively randomizing only the top CCA-aligned dimensions between neural state and behavior causally collapses navigation performance back to baseline while randomizing all other dimensions preserves itmeasured

structure: Limit cycle (stable periodic attractor) · models: GRU policy (PPO, Procgen Jumper), Mamba policy (PPO, Procgen Jumper) · paper: Unraveling the Hidden Dynamical Structure in Recurrent Neural Policies

Transformer Jacobians grow non-normal with depth, collapsing effective rankmeasured

structure: Jacobian non-normality (Schur decomposition of per-layer operators) · models: Llama-3.1-8B, Gemma 4 E4B, OLMo 3 7B · paper: Dynamics of the Transformer Residual Stream: Coupling Spectral Geometry to Network Topology

Crosscoder pre-'wait' feature directions set which reasoning pattern followsmeasured

structure: Linear Direction · models: DeepSeek-R1-Distill-Llama-8B, Llama-3.1-8B · paper: Internal states before "wait" modulate reasoning patterns

Propositional truth is mediated by a multi-dimensional concept conemeasured

structure: Cone · models: Qwen2.5-3B-Instruct, Qwen2.5-7B-Instruct, Qwen2.5-14B-Instruct, Gemma-2-2B-it, Gemma-2-9B-it · paper: From Directions to Cones: Exploring Multidimensional Representations of Propositional Facts in LLMs

A linear truth direction generalizes across datasets and is causally effectivemeasured

structure: Linear Direction · models: Llama-2-7B, Llama-2-13B, Llama-2-70B, Llama-2-7B-Chat, Llama-2-13B-Chat, Mistral-7B-v0.1 · paper: The Geometry of Truth: Emergent Linear Structure in LLM Representations of True/False Datasets, On the Universal Truthfulness Hyperplane Inside LLMs

The truth direction rotates and rescales under added context and steers labelsmeasured

structure: Linear Direction · models: Llama-3.1-8B-Instruct, Mistral NeMo 12B Instruct (2407), Qwen3-4B-Instruct, SmolLM3-3B · paper: How Context Shapes Truth: Geometric Transformations of Statement-level Truth Representations in LLMs

Truthfulness is linearly decodable per attention head, and ITI steers itmeasured

structure: Linear Direction · models: LLaMA-7B, Alpaca-7B, Vicuna-7B · paper: Inference-Time Intervention: Eliciting Truthful Answers from a Language Model

Negative and positive valence localize to distinct depths; a negative-locus direction steers valencemeasured

structure: Linear Direction · models: Llama-3.2-1B-Instruct, Qwen2.5-1.5B-Instruct, Qwen2.5-3B-Instruct · paper: Negative Before Positive: Asymmetric Valence Processing in Large Language Models

Language units are script-conditioned; typology decodes with depthmeasured

structure: Linear Separability · models: Llama-3.2-1B, Gemma-2-2B · paper: Multilingual Language Models Encode Script Over Linguistic Structure

Frozen Mamba collapses into an extreme anisotropy conemeasured

structure: Anisotropy · models: Mamba-130m (state-spaces/mamba-130m-hf), RoBERTa-base · paper: Lost in State Space: Probing Frozen Mamba Representations

Digit basins stratified by carry-potential fibers predict off-by-one errorsmeasured

structure: Stratified Manifold with Continuous Fibers · models: Qwen3-4B, Qwen3-8B, Gemma 3 4B Instruct · paper: The Shape of Addition: Geometric Structures of Arithmetic in Large Language Models

A Sudoku transformer builds substructure-level linear constraint directionsmeasured

structure: Linear Direction, Constraint-Algebra Basis Hypothesis · models: Sudoku Transformer (8 layers, 8 heads, d_model=576) · paper: Transformers Linearly Represent Highly Structured World Models

Self-supervised NaviNeRF finds dimension-restricted semantic directions in StyleNeRF W+measured

structure: Linear Direction · models: StyleNeRF (FFHQ, 256x256) · paper: NaviNeRF: NeRF-based 3D Representation Disentanglement by Latent Semantic Navigation

Position-aware SAE safety steering generalizes across domains within a diffusion modelmeasured

structure: Linear Direction · models: FLUX.1, Stable Diffusion 3.5 · paper: Robust and Generalizable Safety Steering for Text-to-Image Diffusion Transformers

Agentic-trait probe directions share R^2 yet are orthogonal, revealing one agency axismeasured

structure: Linear Direction · models: Qwen3.5-35B-A3B (MoE, ~3B active/token) · paper: Behavioral Steering in a 35B MoE Language Model via SAE-Decoded Probe Vectors: One Agency Axis, Not Five Traits

Residualizing adjacent denoising timesteps yields directionally-stable SD feature trajectoriesmeasured

structure: Linear Direction · models: Stable Diffusion v1.5 · paper: Residualized Temporal Sparse Autoencoders for Interpreting Diffusion Models

Back-projecting the refusal direction identifies attack-suppressed vs attack-robust headsmeasured

structure: Linear Direction · models: Llama-3-8B-Instruct, Llama-2-7B-Chat · paper: Robust Harmful Features Under Jailbreak Attacks: Mechanistic Evidence from Attention Head Specialization in Large Language Models

Function-vector heads drive ICL; induction heads matter little and FV heads evolve from themmeasured

structure: Linear Direction · models: Pythia-70M, Pythia-160M, Pythia-410M, Pythia-1B, Pythia-1.4B, Pythia-2.8B, Pythia-6.9B, GPT-2-small, GPT-2-Medium, GPT-2-Large, GPT-2-XL, Llama-2-7B · paper: Which Attention Heads Matter for In-Context Learning?

Game-state variables are near-linearly decodable with architecture-dependent layer profilesmeasured

structure: Linear Direction · models: IRIS (Atari 100k, Breakout/Pong), DIAMOND (Atari 100k, Breakout/Pong) · paper: What Do World Models Learn in RL? Probing Latent Representations in Learned Environment Simulators