# Displacement Geometry Captures Platonic Shared Reality Across Models and Modalities

arXiv: https://arxiv.org/abs/2609.24209 · HTML: https://mlsanity.com/paper/2609.24209/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 27 of its 33 references that are in the map.
- First posted on arXiv 21 Sep 2026.

## Builds on (most cited first, 27 of 27)

- [Distilling the Knowledge in a Neural Network](https://mlsanity.com/paper/1503.02531/) (2015)
- [Emerging Properties in Self-Supervised Vision Transformers](https://mlsanity.com/paper/2104.14294/) (2021)
- [GLU Variants Improve Transformer](https://mlsanity.com/paper/2002.05202/) (2020)
- [Toy Models of Superposition](https://mlsanity.com/paper/2209.10652/) (2022)
- [Eyes Wide Shut? Exploring the Visual Shortcomings of Multimodal LLMs](https://mlsanity.com/paper/2401.06209/) (2024)
- [The Linear Representation Hypothesis and the Geometry of Large Language Models](https://mlsanity.com/paper/2311.03658/) (2023)
- [Similarity-Preserving Knowledge Distillation](https://mlsanity.com/paper/1907.09682/) (2019)
- [LLM2Vec: Large Language Models Are Secretly Powerful Text Encoders](https://mlsanity.com/paper/2404.05961/) (2024)
- [Emergent Linear Representations in World Models of Self-Supervised Sequence Models](https://mlsanity.com/paper/2309.00941/) (2023)
- [Relative representations enable zero-shot latent space communication](https://mlsanity.com/paper/2209.15430/) (2022)
- [The Geometry of Categorical and Hierarchical Concepts in Large Language Models](https://mlsanity.com/paper/2406.01506/) (2024)
- [Getting aligned on representational alignment](https://mlsanity.com/paper/2310.13018/) (2023)
- [DreamLIP: Language-Image Pre-training with Long Captions](https://mlsanity.com/paper/2403.17007/) (2024)
- [Latent Space Translation via Semantic Alignment](https://mlsanity.com/paper/2311.00664/) (2023)
- [ASIF: Coupled Data Turns Unimodal Models to Multimodal Without Training](https://mlsanity.com/paper/2210.01738/) (2022)
- [Revisiting the Platonic Representation Hypothesis: An Aristotelian View](https://mlsanity.com/paper/2602.14486/) (2026)
- [Assessing and Learning Alignment of Unimodal Vision and Language Models](https://mlsanity.com/paper/2412.04616/) (2024)
- [Understanding the Emergence of Multimodal Representation Alignment](https://mlsanity.com/paper/2502.16282/) (2025)
- [Do Vision and Language Models Share Concepts? A Vector Space Alignment Study](https://mlsanity.com/paper/2302.06555/) (2023)
- [The Universal Weight Subspace Hypothesis](https://mlsanity.com/paper/2512.05117/) (2025)
- [Fantastic Gains and Where to Find Them: On the Existence and Prospect of General Knowledge Transfer between Any Pretrained Model](https://mlsanity.com/paper/2310.17653/) (2023)
- [Back into Plato's Cave: Examining Cross-modal Representational Convergence at Scale](https://mlsanity.com/paper/2604.18572/) (2026)
- [Words That Make Language Models Perceive](https://mlsanity.com/paper/2510.02425/) (2025)
- [Better Language Models Exhibit Higher Visual Alignment](https://mlsanity.com/paper/2410.07173/) (2024)
- [Compositional Generalization Requires Linear, Orthogonal Representations in Vision Embedding Models](https://mlsanity.com/paper/2602.24264/) (2026)
- [LLM2Vec-Gen: Generative Embeddings from Large Language Models](https://mlsanity.com/paper/2603.10913/) (2026)
- [Modulating Cross-Modal Convergence with Single-Stimulus, Intra-Modal Dispersion](https://mlsanity.com/paper/2604.21836/) (2026)

## Abstract (arXiv)

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.