The Anatomy of a Truth Direction: Knowledge-Dependent Dimensionality, a Relational Law, and a Convergent Category Geometry in Small Language Models

The evolution of NLP models from early transformers like BERT and GPT to recent large-scale architectures has significantly improved language understanding. However, understanding how these models internally represent factual knowledge remains a challenge. Previous work, such as Bürger et al. (2024), identified that truth resides in a multidimensional subspace, but lacked tools for unsupervised, low-cost analysis. Most methods relied on supervised probes or feature dictionaries, which are costly and less scalable. Recent studies on layer-wise dynamics and attention specialization provided insights but did not fully elucidate the geometric structure of truth representations. This research builds on these foundations, proposing a simple yet powerful SVD-based approach to extract a linear truth axis, revealing knowledge-dependent dimensionality and relational laws across models, including the Gemma series.

The core problem is to understand whether the internal representation of factual truth in language models can be characterized by a simple geometric structure, and how this structure varies with the model's knowledge level. Existing methods either depend on supervision, are computationally expensive, or lack interpretability. Moreover, the stability of these geometric structures, especially the sign of the truth axes, remains problematic. Addressing these issues is crucial for interpretability, robustness, and knowledge extraction. The challenge lies in developing a low-cost, unsupervised method that can reliably identify and compare truth representations across different models and categories, and in establishing the underlying relational laws that govern their interactions.

The main innovations include: 1) an unsupervised, training-free extraction of the truth direction using SVD on minimal sentence pairs, avoiding label dependence; 2) revealing that truth dimensionality is knowledge-dependent, with signals concentrating on a single axis for known facts; 3) establishing a set of relational laws linking attention, FFN opposition, and decay mechanisms, supported by spectral consensus correction to address sign instability; 4) identifying category-specific truth axes with semantic sign arrangements that converge across models, indicating a shared geometric organization. These advances provide a new, scalable way to analyze internal knowledge structures without supervision or feature dictionaries.

• �� Collect sentence pairs differing in a single content word, with known truth labels. • Construct difference matrices of hidden states at the last token across pairs. • Apply SVD to these matrices to identify the dominant singular vector as the truth axis. • Use permutation nulls to assess statistical significance and nullify label influence. • Extend analysis across multiple models and categories, verifying stability and semantic sign patterns. • Introduce spectral consensus to correct sign instability, ensuring consistent geometry. • Analyze attention, feed-forward, and SwiGLU contributions to establish relational laws. • Validate findings on Gemma-2-2b, adapting tools for sandwich normalization, confirming universality.

The experiments utilize curated fact pairs, CounterFact, and TruthfulQA datasets to evaluate truth signal extraction. The primary metric is the AUC of the truth axis at various layers, with cross-validation and permutation nulls to ensure statistical significance. The analysis compares single-axis, multi-axis, and polarity-based models, examining how knowledge level influences the signal. Stress tests involve category expansion, pair density variation, and re-sampling to assess geometric stability. Cross-model validation includes Qwen, Llama, and Gemma series, demonstrating the universality of the relational laws and the effectiveness of spectral correction. The experiments confirm that the truth signal is real, knowledge-dependent, and governed by identifiable geometric laws.

The truth axis in Qwen2.5-1.5B achieves an AUC of 0.938 on held-out facts, significantly above chance (0.5), confirming a genuine geometric signal. Knowledge level influences the concentration: well-known facts produce a sharp, single-axis signal, while unknown facts diffuse, reducing the signal's clarity. Cross-model analysis reveals consistent relational laws: attention propagates truth frames, FFN opposes them, SwiGLU contributes to post-peak decay. Category-specific axes form a semantically signed arrangement, converging across models. Spectral consensus effectively stabilizes sign and enhances geometric robustness. These findings generalize to Gemma-2-2b, validating the universality of the laws across architectures and training regimes.

The insights enable low-cost, unsupervised interpretability tools for internal knowledge analysis, aiding debugging, fact verification, and robustness in NLP applications. They facilitate understanding of how factual knowledge is geometrically organized, supporting model transparency. Additionally, the methods can inform architecture design by highlighting the roles of attention and FFN in truth propagation. Long-term, these geometric principles could guide the development of models with more stable, interpretable internal representations, improving factual consistency and reasoning capabilities in complex AI systems.

The current analysis is limited to sentence-level facts and small to medium models; scalability to larger, multi-modal models remains untested. The spectral consensus correction, while effective, may not fully address sign instability in all scenarios, especially under distribution shifts. The geometric interpretation relies on linear assumptions, which may oversimplify nonlinear internal dynamics. Further research is needed to extend these insights to multi-task, multi-modal settings and to understand causal mechanisms behind the observed laws.