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

B"urger et al. (2024) demonstrated that truth representations in large language models are universal across statement polarity but reside within a multidimensional subspace. The truth value of a statement is linearly readable from a residual stream of language model, but it is not clear how much of that representation fits on a single direction, which component builds it, or what it is made of. We conducted a study based on these questions, with one instrument: a training-free axis, the dominant direction of the singular value decomposition (SVD) of hidden-state differences over true/false minimal pairs, identified without labels up to one global sign. Extensive evaluation across 14 models from 6 diverse architectural families (including MoE), read and extract at cost O(d) per token. We close with a pre-registered prediction on whether the arrangement extends to categories whose truth is computed rather than retrieved.