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

arXiv: https://arxiv.org/abs/2604.10087 · HTML: https://mlsanity.com/paper/2604.10087/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Placed in Test-time compute and efficient thinking, one of 17 directions in LLM reasoning.
- Builds on 3 of its 10 references that are in the map.
- First posted on arXiv 11 Apr 2026.

## Placed in

- LLM reasoning / [Test-time compute and efficient thinking](https://mlsanity.com/lr/test-time-compute-and-efficient-thinking/)

## Builds on (most cited first, 3 of 3)

- [Group Equivariant Convolutional Networks](https://mlsanity.com/paper/1602.07576/) (2016)
- [Agent-OM: Leveraging LLM Agents for Ontology Matching](https://mlsanity.com/paper/2312.00326/) (2023)
- [LLMs4OM: Matching Ontologies with Large Language Models](https://mlsanity.com/paper/2404.10317/) (2024)

## Abstract (arXiv)

We present EL-DRUIN, an ontological reasoning system for geopolitical intelligence analysis that combines formal ontology, finite semigroup algebra, and Lie algebra approximation to forecast long-run relationship trajectories. Current LLM-based political analysis systems operate as summarisation engines, producing outputs bounded by textual pattern matching. EL-DRUIN departs from this paradigm by modelling geopolitical relationships as states in a finite set of named Dynamic Patterns, composing patterns via a semigroup operation whose structure constants are defined by an explicit composition table, and embedding each pattern as a vector in an 8-dimensional semantic Lie algebra space. Forward simulation iterates this semigroup operation, yielding reachable pattern sets at each discrete timestep; convergence to idempotent absorbing states (fixed points of the composition) constitutes the predicted long-run attractor. Bayesian posterior weights combine ontology-derived confidence priors with a Lie similarity term measuring the cosine similarity between the vector sum of composing patterns and the target pattern vector, providing interpretable, calibrated probabilities that are not self-reported by a language model. Bifurcation points -- steps at which two candidate attractors have near-equal posterior mass -- are detected and exposed to downstream analysis. We demonstrate the framework on six geopolitical scenarios including US-China technology decoupling and the Taiwan Strait military coercion trajectory. The architecture is publicly available as an open-source system with a Streamlit frontend exposing full computation traces, Bayesian posterior breakdowns, and 8D ontological state vectors.