Displacement Geometry Captures Platonic Shared Reality Across Models and Modalities
The Platonic Representation Hypothesis (PRH) claims that independently trained models converge on a shared statistical model of reality, yet recent work finds only weak pointwise similarity between models. In this paper, we show that what models share is not the location of samples in representation space, but the directions (displacement vectors) between them. Under a single orthogonal alignment--rotation and reflection only--these displacement vectors are substantially preserved across 44 independently trained vision and language encoders spanning modalities and asymmetric capability pairs, consistent with the PRH evidence. The samples' absolute positions are not, consistent with recent counter-evidence. Both arise from a single decomposition: representations split into a shared semantic component that is linearly aligned across models, and a private capability component that is not. We trace this geometry to concept-level structure: within a model, parent concepts are orthogonal to their child variation vectors; across models, concept displacements are parallel. Our theory falsifiably predicts (and experiments confirm) that fine-tuning preserves pointwise similarity but collapses displacement, and that relational distillation does the opposite. A major implication is that, because semantics align linearly but capabilities do not, capabilities can be imported from one model to another using a single cached forward pass through the source. We call this Shadow Casting. As a proof of concept, our SHADOWCLIP instantiation outperforms strong fine-tuned baselines at orders of magnitude less compute. A cache can be released alongside open model weights, letting one model's capabilities be downloaded and imported into any number of other models without fine-tuning.