A Transport Law for Linear Directions Between Language Models

The Linear Representation Hypothesis argues that a concept corresponds to a direction in a model's representation space, while the Platonic Representation Hypothesis posits that independently trained models converge to a shared structure of representations. A central unresolved question is whether these two ideas imply that a behaviorally meaningful direction learned in one model should be recoverable and transferable in another. We address this question in a deliberately narrow setting: the **direction of a successful attack** (prompts for which a model produces a response judged unsafe by a guard) within a single modality, text decoders, and the linear case, where both read-out and transfer are linear. Under class imbalance and guard-dependent labels, we first derive an **admissibility criterion** for statistics used to measure transfer of the attack space. From these admissible statistics, we derive a **transport law**: the correlation between a probe's score in its source model and its carried score in the target model equals the root-mean-square canonical correlation of the model pair, weighted across directions by the probe's profile. We further prove an exact decomposition of transport into within-class transport and label transport. The decomposition reveals a striking asymmetry: labels transport without loss, and on some pairs with a gain ($\tau =0.904$–$1.370$), whereas the underlying direction is only partially recovered ($0.740$–$0.865$). We validate the transport law across 11 dense transformers ranging from $0.5$B to $70$B parameters, yielding 110 ordered model pairs. Their responses are labelled by ten diverse guards using a corpus of $3306$ prompts, with representations compared at fourteen relative depths. We further show that the law holds beyond text decoders, on 8 alternative architectures ranging from $1.17$B to $12.31$B parameters. In classical decoder transformers we find that transfer is asymmetric in model size, where a probe transferred from a smaller model to a larger model loses markedly less AUC than transfer in the reverse direction, and the difference increases with the size gap. Together, these results provide a principled framework for testing whether a **shared linear direction of successful-attack behavior exists across independently trained models and transfers between them**. More broadly, our work turns two influential representation hypotheses from qualitative claims into testable, quantitative questions about cross-model representation transport, with direct implications for understanding and controlling model behavior in AI safety.