🤖 machine learning

Semantic Sections: An Atlas-Native Feature Ontology for Obstructed Representation Spaces

This paper proposes "semantic sections" as a new ontology for interpreting AI features in obstructed representation spaces, demonstrating that locally coherent meanings often fail to form globally consistent vectors and require an atlas-based framework for accurate discovery and identity recovery.

Original authors: Hossein Javidnia

Published 2026-03-24

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Hossein Javidnia

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

The Big Idea: Why "One Global Truth" Fails in AI

Imagine you are trying to describe a specific concept (like "justice" or "a cat") inside a massive, complex machine (a Large Language Model).

For a long time, researchers believed that every concept in these machines was like a single, giant lighthouse beam. They thought: "If we can find the one specific direction in the machine's brain that points to 'justice,' we can find it everywhere, no matter what the machine is saying."

This paper argues that this view is wrong.

The author, Hossein Javidnia, suggests that in many parts of these AI brains, concepts don't exist as one giant beam. Instead, they exist as a patchwork quilt of local meanings that change slightly depending on where you are in the machine. Trying to force them into one single beam is like trying to flatten a globe into a flat map without tearing it—it just doesn't work.

The Core Metaphor: The Traveler's Atlas

To understand this, imagine a traveler exploring a strange, shifting world.

The Old Way (Global Vector): The traveler assumes the world is flat. They draw one map. They think, "North is always North." But when they travel, they realize that "North" on their map points in a different direction than "North" in the next town. The map is broken.
The New Way (Semantic Section): The traveler realizes the world is made of many overlapping local maps (an Atlas).
- In Town A, "North" points one way.
- In Town B, "North" points slightly differently.
- In Town C, it points yet another way.
- The Insight: The concept of "North" isn't one single arrow. It is a family of local arrows that are connected by rules on how to translate between towns.

The paper calls this family of local arrows a "Semantic Section." It's a feature that is consistent locally (in each town) but might twist or rotate as you move from town to town.

The Three Types of "Features"

The paper discovers that these features fall into three categories, like different types of travelers:

1. The Tree-Local Traveler (The Safe Path)

Imagine a traveler walking down a path that never loops back. They start at a tree, walk to a river, then to a mountain. Because they never circle back, they never have to check if their map is consistent.

What it means: The AI feature is clear and consistent, but only because it hasn't been tested by a complex loop. It's a "local truth" that hasn't been challenged.

2. The Globalizable Traveler (The Perfect Loop)

Imagine a traveler who walks in a perfect circle and returns to the start. When they get back, their map says "North" is exactly where they left it.

What it means: The feature is so strong that it survives the journey around the loop. It is a "true global feature" that works everywhere. The paper found these, but they are rare.

3. The Twisted Traveler (The Möbius Strip)

This is the most interesting discovery. Imagine a traveler walking in a circle. When they return to the start, they realize they have been flipped upside down! The map says "North," but now it actually points "South."

What it means: The feature is locally coherent (it makes sense in every single town), but globally broken. If you try to force it into one single "North" direction, it fails. The paper calls these "Twisted Sections." They are real, meaningful concepts, but they are "twisted" by the geometry of the AI's brain.

The Big Experiment: Proving the Old Way is Broken

The author tested this on three famous AI models: Llama, Qwen, and Gemma.

The Test:
They took a "Semantic Section" (a patchwork of local meanings) and asked: "If we just look at the raw numbers (vectors) of these local meanings, do they look like the same thing?"

The Result:

The Old Way (Raw Similarity): If you just compare the numbers, the answer is NO. The "local North" in Town A looks completely different from the "local North" in Town B. A standard computer program would say, "These are two totally different things!"
The New Way (Semantic Sections): When you use the "Atlas" method (checking how they translate between towns), the answer is YES. They are clearly the same concept, just expressed differently in different contexts.

The Analogy:
Think of a word like "Bank."

In a river town, "Bank" means the side of a river.
In a city, "Bank" means a place for money.
If you just look at the letters, they are the same word. But if you look at the meaning in context, they are different.
The Twist: In these AI models, the "meaning" of a feature is like a word that changes its definition slightly as you move through the sentence, but it's still the same underlying idea. The old methods missed this because they were looking for a single, unchanging definition.

Why Does This Matter?

Better AI Understanding: If we keep trying to force AI features into single "global vectors," we are going to miss a huge amount of how the AI actually thinks. We will think features are broken or non-existent when they are actually just "twisted."
Safety and Debugging: If we want to fix an AI (e.g., stop it from being racist or lying), we need to know exactly how its internal concepts work. If we use the wrong map (the global vector), we might fix the wrong thing or break something else.
A New Language for AI: The paper proposes a new way to talk about AI features. Instead of asking, "Where is the 'Cat' vector?", we should ask, "How does the 'Cat' concept travel through the different layers of the brain, and does it twist along the way?"

Summary in One Sentence

This paper argues that AI concepts aren't single, static arrows pointing in one direction; they are traveling families of local meanings that can twist and turn as they move through the AI's brain, and we need a new "Atlas" to map them correctly.

1. Problem Statement

Current mechanistic interpretability largely relies on a global-feature ontology, which assumes that a semantic feature corresponds to a single, globally consistent vector, direction, or dictionary atom across all contexts in a neural network.

However, recent work on atlas-based representation spaces (Javidnia, 2026) has demonstrated that neural representations can be "obstructed." In these spaces:

Local feature structures are coherent within specific contexts (charts) but fail to assemble into a single global vector due to proxy shearing and nontrivial holonomy.
A candidate meaning may appear consistent locally but "twists" or rotates when transported around a loop in the context graph, preventing it from being defined as a single global object.
The Gap: Existing methods fail to identify these features because they search for global vectors. If a feature is locally coherent but globally obstructed, standard global-vector similarity metrics fail to recognize it as a single semantic entity.

2. Methodology

A. Theoretical Framework: Semantic Sections

The paper proposes replacing the "global vector" with a Semantic Section.

Definition: A semantic section is a transport-compatible family of local feature representatives defined over a context atlas (a graph of overlapping charts). Instead of one vector, a feature is a set of local vectors $\{s_c\}$ , one per chart, related by transport maps $T_{vu}$ between overlapping charts.
Coherence: A section is $\epsilon$ -coherent if the transported representative from one chart aligns closely with the local representative in the next chart (up to sign and scaling).
Classification Regimes:
1. Tree-local: Coherent along paths in a tree structure but not tested against loops.
2. Globalizable: Coherent on edges and cycle-consistent. Transporting a representative around a loop returns the same vector (up to sign), meaning the feature is globally definable.
3. Twisted: Locally coherent on edges but cycle-inconsistent. Transporting around a loop results in a rotated vector (nontrivial holonomy). These are locally stable meanings that cannot be globalized into a single vector.

B. Discovery and Certification Pipeline

The authors developed a pipeline to discover these sections in frozen Large Language Models (LLMs):

Seeded Propagation: Start with a seed representative in a chart and propagate it through the chart graph using learned transport maps.
Synchronization: When a chart is reached via multiple paths, aggregate the incoming representatives (aligning signs) to create a synchronized local representative.
Defect-Based Pruning: Remove edges/charts where the transport defect (misalignment) exceeds a threshold. This isolates coherent "cores" from diffuse noise.
Cycle-Aware Taxonomy: Classify the resulting support graph based on its Betti number (cycle count) and holonomy defect:
- Globalizable: Low edge defect + Low holonomy defect + Cycles present.
- Twisted: Low edge defect + High holonomy defect + Cycles present.
- Tree-local: Low edge defect + No cycles.
Deduplication: Merge rediscovered sections based on support overlap (Jaccard index) and geometric similarity to count unique semantic objects rather than seed instances.

C. Empirical Protocol

Models: Llama 3.2 3B Instruct, Qwen 2.5 3B Instruct, and Gemma 2 2B IT.
Layer: Primarily Layer 16, with robustness checks on Layers 12 and 20 for Llama.
Metric: Comparison of Section-based Identity Recovery (perfect by definition) vs. Raw Global-Vector Similarity (signed cosine similarity between local representatives).

3. Key Contributions

Ontological Shift: Introduces Semantic Sections as the correct primitive for features in obstructed representation spaces, replacing the global-vector assumption.
Formal Theory: Proves that tree-supported propagation is always pathwise realizable, but genuine globalization requires cycle consistency. It formalizes holonomy as an obstruction to single-valued semantic identity.
Discovery Pipeline: A practical, conservative algorithm for discovering, certifying, and classifying semantic sections (Globalizable, Twisted, Tree-local) in frozen LLMs.
Empirical Census: Provides the first cross-model evidence that semantic sections exist and are structurally heterogeneous.
Identity Recovery Proof: Demonstrates that raw global-vector similarity fails to recover semantic identity even for "globalizable" sections, whereas section-based recovery is perfect.

4. Key Results

A. Cross-Model Census (Layer 16)

After deduplication, the models exhibit distinct regimes:

Gemma 2 2B IT: Exhibits the richest structure, with 14 globalizable sections, 4 twisted sections, and 8 tree-local sections. It shows substantial local-to-global semantic structure.
Llama 3.2 3B Instruct: Supports a sparse but complete taxonomy (1 globalizable, 1 twisted, 2 tree-local).
Qwen 2.5 3B Instruct: Dominated by tree-local coherence (5 tree-local vs. 1 globalizable). This serves as a control regime where features are locally coherent but rarely globalize.

B. Failure of Global-Vector Similarity

This is the most critical empirical finding. Even for certified globalizable sections:

Low Cosine Similarity: The raw signed cosine similarity between local representatives of the same semantic section across different charts is typically very low (often < 0.3).
Identity Recovery Collapse:
- For Gemma's 14 globalizable sections (1,221 true within-section pairs), raw similarity at threshold 0.5 recovers only 2.62% of pairs. At threshold 0.7, it recovers 0%.
- In contrast, section-based identity recovery is 100% on certified supports.
Conclusion: A global-vector ontology would fragment a single semantic object into many unrelated vectors. The semantic section formalism correctly identifies them as one object.

C. Robustness

Layer Variation: In Llama, cycle-supported structure is strongest at Layer 16, weaker at Layer 12, and nearly absent at Layer 20, suggesting depth-dependent semantic organization.
Threshold Sensitivity: The qualitative census (presence of twisted/globalizable sections) remains stable across variations in defect and deduplication thresholds.

5. Significance

Theoretical Impact: The paper moves interpretability from a "search for global directions" to a "local-to-global transport problem." It redefines holonomy not just as a geometric curiosity, but as a fundamental obstruction to defining features as single vectors.
Practical Impact: It explains why standard feature discovery methods (like SAEs or global clustering) might fail in certain regimes. It provides a new framework for identifying features that are locally meaningful but globally "twisted."
Future Directions: Suggests that future interpretability tools must handle transport compatibility and cycle consistency. It opens new questions about how to edit, intervene, or align these "twisted" sections, which cannot be treated as simple vector manipulations.

In summary, the paper argues that in obstructed representation spaces, a feature is not a vector, but a transport-compatible family of local vectors. The "Semantic Section" is the necessary mathematical object to capture semantic identity where global vectors fail.