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

Bürger et al. (2024) demonstrated that truth representations in large language models are universal across statement polarity but reside within a multidimensional subspace. We extend that framework along three questions: how the dimensionality of the subspace depends on the model’s knowledge, which architectural component builds the truth direction, and what the direction is a mixture of. In Part I (one model), a training-free directional probe derived from the SVD of hidden-state minimal pairs shows that the dimensionality of truth is knowledge-dependent: the signal is concentrated on a single axis for behaviorally known facts (held-out AUC 0.938) and becomes diffuse as knowledge decreases or material heterogeneity increases; seven falsification experiments sustain the one-dimensional reading, and the same decomposition recovers the supervised polarity direction of Bürger et al. at cosine 0.959 without polarity labels. In Part II (four base models, two families, pre-registered predictions), a relational law emerges: attention propagates truth frames it did not write, the feed-forward network opposes the frame of its current block, and the genuine post-peak decay is causally attributed on all four models to the SwiGLU value stream; per-category truth axes form a semantically signed arrangement that contradicts the lexical surface and converges across families (Mantel p = 0.0009). Stress tests at three scales (33 categories, 888 pairs per relation, six sampling seeds) expose a sign instability of the per-category orientation, repair it with a declared spectral consensus gauge, and sharpen the convergence into a knowledge-gated law, strengthening from +0.42 on dense mixed-knowledge material to +0.74 on knowledge-restricted categories: the geometry of the mixture belongs to the knowledge domain rather than to the architecture.