Ontological Trajectory Forecasting via Finite Semigroup Iteration and Lie Algebra Approximation in Geopolitical Knowledge Graphs

Researchers introduce EL-DRUIN, an ontological reasoning system that uses finite semigroup algebra and Lie algebra to forecast geopolitical relationship trajectories rather than relying on LLM pattern matching. The system models political dynamics as composable states, identifies convergence points (attractors), and provides calibrated probability estimates for long-term geopolitical outcomes, with applications to scenarios like US-China technology decoupling.

EL-DRUIN represents a methodological departure from current AI-driven geopolitical analysis by grounding predictions in formal mathematical structures rather than language model heuristics. The system treats geopolitical relationships as discrete states within a finite semigroup algebra, where pattern compositions follow an explicit rule set. By embedding these patterns in an 8-dimensional Lie algebra space and iterating forward through time, the framework identifies stable end-states (idempotent absorbing states) that function as long-run attractors. This approach addresses a critical limitation of LLM-based systems: their outputs remain constrained by training data patterns and lack interpretable mathematical foundations.

The introduction of Bayesian posterior weighting—combining ontology-derived confidence priors with Lie similarity measures—enables probabilistic forecasting without relying on self-reported model confidence. The system explicitly identifies bifurcation points where multiple competing attractors have comparable posterior probability, flagging scenarios of genuine uncertainty.

For geopolitical analysts, defense strategists, and policy researchers, this architecture offers transparency absent from black-box LLM systems. The open-source Streamlit interface exposing full computation traces allows external verification and calibration. The demonstrated applications to US-China technology decoupling and Taiwan Strait coercion suggest relevance to high-stakes strategic forecasting.

This framework may influence how intelligence agencies and think tanks approach scenario planning, though adoption faces barriers: domain experts must validate the composition tables, and the system's predictive accuracy remains unvalidated against historical geopolitical trajectories. Future work should focus on out-of-sample testing and comparison against human expert forecasts.

• →EL-DRUIN replaces LLM pattern-matching with formal semigroup algebra and Lie algebra mathematics for geopolitical forecasting.
• →The system identifies stable long-term attractors and bifurcation points, highlighting genuine strategic uncertainty rather than false confidence.
• →Bayesian posterior weighting provides calibrated, interpretable probabilities grounded in mathematical similarity rather than model self-reporting.
• →Open-source architecture with full computation transparency enables external validation and reduces reliance on proprietary AI black boxes.
• →Applications to US-China decoupling and Taiwan scenarios demonstrate relevance to strategic intelligence and defense policy planning.