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 = Eᵣesponse - Equestion) 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 (Spearman rho = 0. 86-0. 96, all p < 0. 001) ; that human coupling signatures are geometrically distinguishable with p = 0. 000 across all three models, with convergent evidence from iMessage data identifying a romantic dyad as rank 1 of 902 possible contact pairings without any metadata; and that the method generalizes beyond language, confirmed via chess games (541 games, 19, 000+ move pairs, p = 0. 000) and extended cross-domain validation across email, therapy, and acoustic music. We introduce the concepts of refraction vectors, coupling topology, coupling drift, and stereoscopic validation. The method is substrate-independent: the same arithmetic detects coupling topology in language, messaging, chess, email, therapeutic dialogue, and acoustic music.