Anticipation Geometry: Domain-General Trajectory Characterization with Knowledge Graph-Grounded Rewards

We present Anticipation Geometry, a mathematical framework that characterizes trajectories through arbitrary state spaces using seven geometric scalars: commitment, uncertainty, transition pressure, recovery margin, phase stiffness, novelty, and stability. Originally developed for physical motion capture in the Comp-Core system, we prove these scalars are domain-general, operating on any sequence of vectors in a metric space equipped with a differentiable time parameter. We combine this framework with knowledge graph path-derived reward signals, extending the domain-specific superintelligence (DSS) architecture proposed by Belova et al. (2026), to create a unified system for both trajectory analysis and model training. We evaluate on three domains: physical motion (simulated kinematics), conversational reasoning (20,000 dialogue turns from 164 conversations embedded with MiniLM and e5-large), and knowledge graph traversal (199 multi-hop paths from a production graph kernel). Our key finding is that transition pressure, defined as $\frac{d(\text{commitment})}{dt} - \frac{d(\text{uncertainty})}{dt}$, is a statistically significant predictor of reasoning convergence: its sign predicts conversation convergence at 71.8% accuracy (z = 2.72, p < 0.007), and its standard deviation achieves 69.8% accuracy (+8.1pp over baseline) as a single feature on higher-dimensional embeddings. On knowledge graph paths, anticipation-augmented rewards discriminate valid from hard-negative paths with 81.0% pairwise accuracy (Cohen's d = 2.23). We do not claim state-of-the-art performance on any single task. We demonstrate that a single geometric framework, with no task-specific training, produces significant signal across domains, suggesting that trajectory geometry captures a fundamental property of reasoning that transcends domain boundaries.