Empirical Dynamical Centers of Transformer Generation: A Geometric Framework and Its Implications for Alignment

We present SERC (Structure–Energy–Resonance–Coherence), a geometric framework for monitoring language model generation dynamics in real time. The system state is mapped to the standard 3-simplex via four components derived from hidden states. A relational tension functional Ω measures distance from the simplex center P₀. We prove that P₀ is the unique globally asymptotically stable equilibrium of the projected gradient flow, with exponential relaxation rate e^{−32t}. A series of eight experiments (NB1–NB8) across three models yields four empirical contributions: (1) R-dominance correlations of 0.84–0.94; (2) C as the primary diagnostic dimension; (3) a guardrail achieving TP=1, FP=0 with 89.5% token reduction — the first content-free pathology detection based solely on generation geometry; (4) a fundamental finding: geometric P₀ is not the natural center of any tested model's dynamics. Empirical barycenters lie far from (0.25, 0.25, 0.25, 0.25), are stable across prompt sets, and constitute a measurable property of model architecture and training. RLHF training moves the barycenter further from geometric P₀ — not closer — by systematically increasing C-dominance. This raises the question: is the empirical barycenter a computable characterization of a model, and can it predict alignment behavior?