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

I 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. These scalars are domain-general, operating on any sequence of vectors in a metric space equipped with a differentiable time parameter. I combine this framework with knowledge graph path-derived reward signals to create a unified system for both trajectory analysis and model training. I evaluate on three domains: physical motion (simulated kinematics), conversational reasoning (20,000 dialogue turns from 164 conversations, expanded to 308 sessions from 19,000 prompts), and knowledge graph traversal (199 multi-hop paths from a graph kernel). The key finding is that transition pressure, defined as the rate of commitment increase minus the rate of uncertainty decrease (dC/dt - dU/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. In an expanded evaluation, inscription-derived features encoding conversational dynamics as sigil probability distributions achieve 79.5% accuracy via gradient boosting on the original 39 sessions (z = 3.68, p < 0.001), a +7.7pp improvement over the transition pressure baseline. On knowledge graph paths, KG-path rewards achieve 100% pairwise ranking accuracy (Cohen's d = 11.17) on a seeded synthetic evaluation (seed=42, n=15 gold/silver/bronze triplets), a structural guarantee arising from chain continuity construction rather than an empirical surprise. I do not claim state-of-the-art performance on any single task. I show that a single geometric framework, with no task-specific training, produces significant signal across domains, suggesting that trajectory geometry captures a general property of reasoning.