Steering Embedding Models with Geometric Rotation: Mapping Semantic Relationships Across Languages and Models

Understanding how language and embedding models encode semantic relationships is fundamental to model interpretability and control. While early word embeddings exhibited intuitive vector arithmetic (''king'' - ''man'' + ''woman'' = ''queen''), modern high-dimensional text representations lack straightforward interpretable geometric properties. We introduce Rotor-Invariant Shift Estimation (RISE), a geometric approach that represents semantic transformations as consistent rotational operations in embedding space, leveraging the manifold structure of modern language representations. RISE operations have the ability to operate across both languages and models with high transfer of performance, suggesting the existence of analogous cross-lingual geometric structure. We evaluate RISE across three embedding models, three datasets, and seven morphologically diverse languages in five major language groups. Our results demonstrate that RISE consistently maps discourse-level semantic transformations with distinct grammatical features (e.g., negation and conditionality) across languages and models. This work provides the first systematic demonstration that discourse-level semantic transformations correspond to consistent geometric operations in multilingual embedding spaces, empirically supporting the Linear Representation Hypothesis at the sentence level.