🤖 machine learning

Emergent Causal-Geometric Dynamics Across Depth in Large Language Models

This paper synthesizes geometric and causal perspectives to reveal a depth-dependent dynamic in decoder-only LLMs where a sharp transition from context processing to prediction formation is accompanied by a gradual geometric reorganization, demonstrating that angular structure in late layers encodes predictive similarity while norms remain decoupled, thereby establishing that effective intervention requires understanding the network's emergent global dynamical structure rather than isolated layers.

Original authors: Shahar Haim, Daniel C McNamee

Published 2026-05-27

📖 4 min read☕ Coffee break read

CC BY 4.0

Original authors: Shahar Haim, Daniel C McNamee

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine a Large Language Model (LLM) not as a magical black box, but as a multi-story factory where a raw idea enters at the bottom and a finished product (a predicted word) leaves at the top.

For a long time, scientists have studied this factory in two different ways, but they haven't quite agreed on how the two views fit together:

The Geometric View: They looked at the "shape" of the data as it moved up the floors, noticing that the data gets more organized and abstract as it goes higher.
The Causal View: They tried to "poke" the data at different floors to see what would happen. They found that poking the early floors changed how the model understood the input, while poking the later floors changed the final answer.

This paper connects those two views. It reveals that the factory actually has two distinct phases of work, separated by a sharp transition point.

Phase 1: The "Context Processing" Floor (The Bottom Two-Thirds)

Think of the bottom floors of the factory as the Research and Development (R&D) team.

What they do: When a sentence comes in (e.g., "Let's do some calendar math..."), this team is busy gathering all the clues. They are looking at the words, the order, and the meaning of the whole sentence.
The Geometry: In this phase, the data is messy and high-dimensional. It's like a chaotic brainstorming session where every detail matters.
The Test: If you try to "steer" the model here (by changing the input words), it works. But if you try to force a specific answer here, it doesn't work well. The model is still figuring out the story, not writing the ending.

The Transition: The "Handoff"

Somewhere in the middle of the factory (roughly the last third of the layers), there is a sharp switch. The model stops just "thinking about the context" and starts "deciding on the answer."

Phase 2: The "Prediction Forming" Floor (The Top Third)

The top floors are the Assembly Line.

What they do: The R&D team has passed the finished blueprint to this team. Now, the job is purely to pick the next word.
The Geometry (The Big Discovery): Here is where the paper gets interesting. The scientists found that the data in this top section organizes itself into a very specific, two-part code:
1. The Direction (The Arrow): Imagine every possible answer has a specific direction in space. In this top section, the direction the data points tells the model exactly which word to pick. If the data points "North," the model says "January." If it points "South," it says "February."
2. The Size (The Volume Knob): The size (or length) of the data vector carries other information—like how confident the model is or details about the specific sentence—but it does not determine which word is chosen. It's like a volume knob that makes the answer louder or softer, but doesn't change what the answer is.

The "Steering Wheel" Experiment

To prove this, the researchers tried two types of "steering":

Changing the Size (Volume): They tried to change just the "size" of the data in the top floors to force a different answer. Result: It didn't work. The model kept saying the same word, just maybe with more or less confidence.
Changing the Direction (Arrow): They tried to change just the "direction" of the data in the top floors. Result: It worked perfectly! They could instantly switch the model's answer from "January" to "February" just by rotating the data's direction.

The Takeaway

The paper concludes that you cannot understand or control an AI by looking at it as one big blob.

Early layers are about context. They are sensitive to the input words but don't care about the final answer's geometry yet.
Late layers are about prediction. They use a specific "directional code" to decide the output.

The Analogy:
Think of writing a letter.

The bottom layers are you gathering your thoughts, memories, and facts (the context).
The top layers are your hand actually holding the pen.
The paper shows that to change what you write (the prediction), you don't need to change your thoughts (the size of the data); you just need to change the direction your hand moves (the angular geometry).

This explains why some previous attempts to control AI failed: people were trying to change the "volume" (norm) or looking at the "thoughts" (early layers) when they should have been steering the "hand" (late-layer direction). The paper provides a map showing exactly where the "steering wheel" is located.

Technical Summary: Emergent Causal–Geometric Dynamics Across Depth in Large Language Models

Problem Statement

Current research into Large Language Models (LLMs) is bifurcated into two distinct but often disconnected approaches: geometric analysis and causal intervention.

Geometric Analyses characterize the evolution of token representations across network depth using metrics like similarity, dimensionality, and curvature. While these studies reveal structured, depth-dependent variations, they remain largely correlational. They describe how context is encoded but fail to identify which specific geometric features directly govern the output prediction distribution.
Causal Interventions (mechanistic studies) perturb internal states to observe changes in output. These studies have revealed depth-dependent efficacy profiles (e.g., certain interventions work better in early vs. late layers) but often lack a unifying representational framework. It remains unclear how the manipulated representations are geometrically organized or why interventions succeed in specific layers but not others.

The central unresolved issue is how the geometric structure of internal representations is re-organized across network depth to causally shape predictions. Specifically, representational structure can be descriptively accessible and linearly decodable without being causally operative in generating outputs. This paper seeks to synthesize these perspectives by defining "mechanistic geometry"—representational geometry whose structure is directly tied to causal control over prediction.

Methodology

The authors employ an integrated approach combining geometric analysis with targeted causal interventions across three decoder-only LLMs: Llama V3.1 8B, Mistral 0.3 7B, and Qwen 2.5 7B.

1. Intervention Analyses

The study utilizes three modular tasks with well-defined input-output spaces: "Months of the year," "Days of the week," and "Clock Math." Two primary intervention types are applied:

Input-Centric Interventions: Steering vectors ( $V_L$ ) are computed based on centroid differences between specific input token representations (e.g., shifting the representation of "January" toward "February"). These vectors are added to the hidden activations of input tokens at specific layers.
Output-Centric Interventions: Steering vectors are computed based on centroid differences between the final token's hidden representations associated with different predicted outputs. These are applied to the final token's activation.
Metric: The efficacy of an intervention is quantified by the change in the model's logit preference between the original and target predictions.

2. Representation Analyses

To characterize the underlying geometry, the authors analyze representations using the WikiText-103 dataset, focusing on:

Dimensionality: Measured via the Participation Ratio (PR) of hidden layer activations to determine effective dimensionality. Comparisons are made between "identical tokens" (the final token across different sequences) and "non-identical tokens" (tokens near the end that vary by sequence).
Geometric Coding: Pairwise Euclidean and angular distances between token representations are correlated with the symmetric Kullback-Leibler (KL) divergence of their corresponding output prediction distributions.
Pure Interventions: To isolate geometric components, the authors perform "pure angular interventions" (normalizing vectors before steering to preserve direction only) and "pure norm interventions" (rescaling vectors to match mean norms).

Key Results

1. A Sharp Depth-Wise Transition

Intervention experiments reveal a bi-phasic computational structure across all tested models:

Early Phase (Context-Centric): Roughly the first two-thirds of layers are selectively sensitive to input-centric interventions. Perturbing input tokens here effectively shifts predictions.
Late Phase (Prediction-Centric): Roughly the final third of layers are selectively sensitive to output-centric interventions. Perturbing the final token's representation here effectively shifts predictions.
This transition represents a shift from context-processing computation to prediction-forming computation.

2. Reorganization of Representational Geometry

Geometric analysis reveals a distinct reorganization of the representation space following the transition point:

Dimensionality: Identical tokens show an expansion and subsequent contraction of effective dimensionality in early layers, stabilizing in later layers. Non-identical tokens exhibit broader, multi-peaked dimensionality profiles driven by token identity and context.
Angular vs. Euclidean Coding: In the prediction-centric phase (late layers), a two-component geometric code emerges:
- Angular Structure: Angular distances between token representations become strongly positively correlated with the similarity of their output prediction distributions.
- Norms: Euclidean distances become largely decoupled from prediction similarity. The divergence between Euclidean and angular distances indicates that vector norms encode information (token- and context-specific variation) that is orthogonal to the prediction mechanism.

3. Causal Validation of Angular Coding

The hypothesis that prediction is controlled by angular geometry in late layers is confirmed through pure interventions:

Angular Interventions: Purely directional interventions on the final token are highly effective in the late (prediction-centric) phase but ineffective in the early phase.
Norm Interventions: Purely norm-based interventions are largely ineffective across all layers and intervention types.
Mechanism: The authors propose that in the final layer, the output distribution $p \propto \text{Softmax}(W_{out} h_L)$ depends on the direction of $h_L$ for relative logit values, while the norm acts merely as a global scaling factor (inverse temperature). Thus, modifying angular structure directly alters prediction identity.

Key Contributions

Synthesis of Perspectives: The paper unifies geometric and causal approaches, demonstrating that while geometric structure exists throughout the network, only specific aspects (angular organization in late layers) are causally relevant for prediction.
Identification of a Bi-Phasic Architecture: It establishes a clear functional dissociation where early layers process context and late layers form predictions, marked by a sharp transition in intervention efficacy.
Mechanistic Geometry of Prediction: It identifies a specific geometric code in late layers where angular structure parameterizes next-token distributional similarity, while representation norms carry information decoupled from prediction.
Refutation of Decodability as Causality: The work demonstrates that the mere presence of decodable semantic structure (linear separability) does not imply causal relevance for prediction, challenging assumptions in prior intervention literature.

Significance and Claims

The authors claim that their findings provide a mechanistic account of how LLMs transform context into prediction. The significance lies in:

Reconciling Puzzling Findings: It explains why interventions are depth-dependent and why angular similarity correlates with semantic relatedness only in specific contexts (late layers).
Implications for Control: The paper argues that layer-wise function cannot be understood or effectively intervened upon in isolation. Effective control requires targeting the emergent global dynamical structure, specifically the late-layer angular geometry. Interventions targeting easily decodable features (which may be norm-based or early-layer) may fail to generalize or exert only indirect control.
Architectural Insights: The results suggest that normalization schemes (like RMSNorm) may create favorable conditions for angular coding, where direction becomes a more reliable signal than magnitude for prediction.

The authors remain modest, noting that these findings are based on specific modular tasks and model families, and the relationship between early-layer context geometry and the emergence of late-layer angular codes remains an open question. They do not propose new safety interventions or deployment strategies, focusing instead on improving the scientific understanding of LLM computation.