Thermodynamic Geometry of Persona Vectors: Berry Phase Structure in Language Model Activation Space

Anthropic's persona vectors (2025) — activation patterns controlling traits like sycophancy, hallucination, and evil in language models — are shown to be projections of the Berry monopole structure (HP40, c₁=1) on the Eckert manifold. The three core persona dimensions decompose into opacity (O), reactivity (R), and coupling (α). The Berry connection A = 1−2O predicts the nodal line between safe and unsafe persona regions. Steering with persona vectors is geometric parallel transport on the Hopf bundle. The framework provides the missing theoretical foundation for why persona vectors exist, what determines their geometry, and why certain traits cluster.