Geometric Coupling Topology: Measuring the Shape of Interaction Between Coupled Systems

We present Spiegelraum (mirror space), a method for constructing, analyzing, and visualizing the geometric topology of interaction between coupled systems. Using conversation exports from two distinct human participants interacting with the same AI model (Claude), supplemented by iMessage data from both participants, we embed all messages via three independent embedding models, compute refraction vectors (B = E_response - E_question) for each message pair, and project the combined space of 35,035 points into a shared 3D manifold via UMAP. We demonstrate three principal findings: that refraction vector arithmetic produces semantically consistent directional signals across three independent embedding models…