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Beyond Language: Format-Agnostic Reasoning Subspaces in Large Language Models

This paper introduces the TriForm Benchmark and provides converging evidence for a Format-Agnostic Reasoning Subspace (FARS) within large language models, demonstrating that a compact 10-dimensional internal representation captures abstract reasoning concepts across diverse surface forms (prose, code, and math) while revealing a fundamental asymmetry where declarative formats are more compatible with each other than with procedural code.

Original authors: Aojie Yuan, Zhiyuan Su

Published 2026-05-12

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CC BY 4.0

Original authors: Aojie Yuan, Zhiyuan Su

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 you have a giant, super-smart library (a Large Language Model) that can read and write in many different "languages." But in this case, "languages" doesn't just mean English, Spanish, or French. It also means Python code, math equations, and structured lists.

The big question the researchers asked was: When this library thinks about a logical idea (like "if A then B"), does it use the same internal "brain wiring" whether it's reading that idea in a story, a math problem, or a computer program? Or does it have to completely rewire its brain every time the format changes?

Here is what they found, explained simply:

1. The "Hidden Core" of Reasoning

The researchers discovered that deep inside these AI models, there is a tiny, special "core" area where the actual logic lives, completely separate from the format.

The Analogy: Imagine you are listening to the same song played on a piano, a guitar, and a synthesizer. To your ears, they sound totally different (different "surface forms"). But the researchers found that inside the AI, there is a specific, tiny room where the melody of the song is stored in a way that doesn't care about the instrument.
The Finding: They found this "room" in the middle layers of the AI. It's called the Format-Agnostic Reasoning Subspace (FARS). It's incredibly small—only about 10 dimensions out of the thousands the AI uses. Yet, this tiny slice holds the pure logic, stripped of whether it's written in code or prose.

2. The "Magic Swap" Experiment

To prove this core was real and actually did the work, they performed a "brain transplant" experiment.

The Analogy: Imagine you have two people: one speaking English and one speaking French. You take the "logic brain" from the English speaker and try to plug it into the French speaker's head.
The Result: When they swapped just those 10 special dimensions of the "logic brain" from an English sentence into a math equation, the AI kept working almost perfectly (90–96% of the time).
The Contrast: If they swapped the entire brain (all dimensions), the AI got confused and failed. If they swapped random parts, it also failed. This proved that those specific 10 dimensions are the "secret sauce" that carries the reasoning across different formats.

3. The "Declarative vs. Procedural" Wall

Here is the most surprising twist. The AI is great at translating logic between stories (English) and math, but it struggles to translate logic between stories and computer code.

The Analogy: Think of "Declarative" formats (Stories and Math) as blueprints. They describe what a building should look like. Think of "Procedural" formats (Code) as instruction manuals for a robot. They describe how to build it step-by-step.
The Finding: The AI's "logic brain" works beautifully when swapping blueprints for other blueprints (Story $\leftrightarrow$ Math). But when it tries to swap a blueprint for a robot instruction manual (Story $\leftrightarrow$ Code), the connection breaks. The AI treats them as fundamentally different types of thinking.
The Takeaway: The barrier isn't "Language vs. Math"; the barrier is "Describing a result" vs. "Giving step-by-step instructions."

4. It's the Same for Everyone

The researchers tested five different AI models from different companies (like GPT, Llama, and Mistral). Even though these models were built differently and trained on different data, they all developed this same 10-dimensional "logic room" in the middle of their brains.

The Analogy: It's like how humans, regardless of which country they are from, all have a heart in the same place that pumps blood the same way. The AI models, despite being different "species," all evolved the same internal structure to handle pure logic.

Summary

The paper shows that AI models do have a universal "logic center" that ignores whether you are writing in English, math, or code. However, this center only works well between formats that describe things the same way (declarative). It hits a wall when trying to mix descriptions with step-by-step instructions (procedural).

This discovery gives us a way to see exactly where the AI is thinking about the "idea" versus where it is just thinking about the "words" or "syntax."

Technical Summary: Beyond Language: Format-Agnostic Reasoning Subspaces in Large Language Models

Problem Statement

Large Language Models (LLMs) process identical logical inferences expressed in vastly different surface forms—ranging from English prose and Chinese text to Python code and mathematical notation. While prior research has established that multilingual LLMs develop shared internal representations across natural languages, it remains unknown whether this invariance extends to genuinely different symbolic systems. The central question is whether LLMs possess a common internal substrate for reasoning that transcends specific symbolic encodings (linguistic, formal, and structural), and if so, whether this structure is causally involved in computation and universal across architectures.

Methodology

The authors introduce the TriForm Benchmark, a controlled dataset comprising 324 stimuli derived from 18 reasoning concepts (spanning arithmetic, logic, relational, causal, and spatial domains). Each concept is expressed in six surface forms organized along three dimensions:

Linguistic: English, Chinese, and French prose.
Symbolic: Python code and mathematical notation.
Structural: Structured step-by-step data.

The study evaluates five decoder-only LLMs (1.6B to 8B parameters) from three architecture families (GPT, Qwen, Llama/Mistral) using the following analytical framework:

Permutation-Corrected RSA: To identify concept-level structure, the authors employ Representational Similarity Analysis (RSA) with permutation tests to correct for the non-independence of pairwise distances, distinguishing concept signals from surface-form signals.
Cross-Form Probing: Ridge-regularized classifiers are trained on activations from one surface form to predict concept identity in another, testing linear decodability across formats.
Format-Neuron Entropy: Inspired by LAPE, the authors quantify neuron specificity by measuring the entropy of activation distributions across forms, identifying neurons that activate uniformly (format-agnostic) versus those specific to certain forms.
Subspace Extraction (FARS): A Format-Agnostic Reasoning Subspace is extracted using concept-centroid Principal Component Analysis (PCA). By averaging activations across all forms and instances for each concept, the authors isolate the top- $k$ principal components that capture concept variance while suppressing form variance.
Causal Intervention: The study utilizes activation patching and subspace ablation.
- Cross-form patching: Replacing hidden states from a source form (e.g., English) into a target form (e.g., Math) to measure output preservation.
- Subspace patching: Replacing only the projected FARS components (10 dimensions) during cross-form transfer.
- Subspace ablation: Removing FARS directions to measure the disruption to model output.

Key Results

1. Existence of a Format-Agnostic Reasoning Subspace (FARS)

Correlational Evidence: RSA reveals that concept-level signals peak in the middle layers of all models, strengthening with scale. While surface form remains the dominant organizing principle (Form RSA $\rho \approx 0.43\text{--}0.53$ ), concept signals are significant ( $\rho \approx 0.10\text{--}0.20$ ) and grow faster with model size.
Decodability: Cross-form probing accuracy significantly exceeds chance (5.6%), reaching up to 60.3% (Mistral-7B).
Neuron Localization: Format-agnostic neurons concentrate in middle layers (20–23% of neurons), mirroring the distribution of language-agnostic neurons found in prior cross-lingual studies.

2. Causal Relevance and Subspace Efficiency

High-Fidelity Transfer: Replacing only the top 10 dimensions of the FARS subspace during cross-form patching preserves 90–96% of the model's top-10 token output.
Comparison to Controls: This performance far exceeds full activation replacement (44–56%) and variance-maximizing PCA replacement (60–74%).
Ablation: Removing the FARS subspace causes targeted disruption (low KL divergence for FARS vs. high for form-specific directions), confirming that these specific dimensions are causally sufficient for cross-form reasoning.
Dimensionality: The reasoning-relevant structure is genuinely low-dimensional, plateauing at $k=8\text{--}10$ components within a 1600–4096 dimensional space.

3. The Declarative-Procedural Asymmetry

A critical finding is the divergence in compatibility based on the nature of the symbolic system rather than just the language:

Declarative Compatibility: Patching between English prose and mathematical notation preserves 65–73% of output.
Procedural Divergence: Patching between English prose and Python code preserves only 16–22% of output.
Conclusion: The critical axis of divergence is not "linguistic vs. formal" but declarative vs. procedural. Representations are highly compatible between declarative formats (prose, math, structured data) but significantly less so when bridging to procedural encodings (code).

4. Universality and Convergence

Generalization: FARS extracted from a subset of concepts generalizes to held-out concepts with graceful degradation.
Cross-Architecture Convergence: Different model families converge on the same subspace geometry. Cross-model Canonical Correlation Analysis (CCA) exceeds 0.79 for all pairs, and centroid RSA exceeds 0.77, providing within-modality evidence for the Platonic Representation Hypothesis.

Significance and Claims

The paper claims to provide the first systematic evidence that LLMs develop a concrete, low-dimensional, format-agnostic subspace for reasoning that is causally sufficient for cross-symbolic transfer.

Theoretical Contribution: The findings support the Platonic Representation Hypothesis within a single modality, suggesting that representations of abstract reasoning converge across different symbolic encodings.
Mechanistic Insight: The discovery of the declarative-procedural asymmetry refines the understanding of format invariance, identifying a fundamental boundary between declarative reasoning (prose, math) and procedural execution (code) that was previously uncharacterized.
Practical Utility: FARS offers a concrete, extractable tool (a 10-dimensional basis) for representation engineering. It suggests that safety interventions or steering vectors discovered in one declarative format (e.g., English) may transfer to others (e.g., Math) via this subspace, but may fail when bridging to procedural formats (code), highlighting a potential safety gap where harmful reasoning in code could bypass prose-trained filters.

The authors note limitations, including the evaluation of base models only, the supervised nature of FARS extraction, and the use of programmatically generated stimuli, acknowledging that the dimensionality and geometry of FARS in larger or instruction-tuned models remain open questions.