The Stability of Probabilistic Manifolds in Curved Spacetime

This work analyzes the stability of probabilistic manifolds embedded in curved spacetime, extending earlier formulations in which gravity emerges from the localization of probability. By introducing an affine probabilistic connection that preserves normalization, the framework shows that curvature induces a natural correction to coherence transport on the underlying state manifold. The resulting curvature‑dependent coherence equation yields a stability criterion in which positive curvature enhances localization and suppresses divergence between probability configurations, while negative curvature drives decoherence growth. A generalized divergence metric and curvature‑modulated coherence gate unify gravitational localization, quantum coherence limits, and vacuum stability within a single geometric principle: the self‑consistency of probability under curvature. The analysis predicts curvature‑dependent decoherence rates, suppression of false‑vacuum tunneling in regions of positive curvature, and cosmological thresholds associated with coherence instability. This establishes a dynamic feedback loop between probability, coherence, and geometry, offering a unified interpretation of gravitational behavior and quantum stability.