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

Qihang Wu

cs.AIApr 11, 2026

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

Authors: Qihang Wu

Organizations: Department of Industrial and Systems Engineering, The Hong Kong Polytechnic University

Abstract

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.

Figures & tables

Pattern Name	Domain	Mechanism Class	Prior	Inverse Pattern
Hegemonic Sanctions	geopolitics	coercive_leverage	0.78	Sanctions Relief / Normalisation
Entity-List Technology Blockade	technology	tech_denial	0.80	Technology Licence / Unblocking
Interstate Military Conflict	military	kinetic_escalation	0.75	Ceasefire / Peace Agreement
Great-Power Coercion/Deterrence	geopolitics	coercive_leverage	0.70	Diplomatic Concession / De-escalation
Multilateral Alliance Sanctions	geopolitics	multilateral_pressure	0.73	Multilateral Sanctions Relief
Tech Decoupling / Technology Iron Curtain	technology	tech_decoupling	0.76	Technology Cooperation Reintegration

Table 1: Selected Dynamic Patterns (10 of 18 shown). Full registry available in the open-source repository.

Scenario	Primary Attractor	$P(\alpha)$	Secondary Attractor	Bifurcation?	$c_{\mathrm{final}}$
US-China tech decoupling	Technology Standards Leadership	0.51	Tech Decoupling / Technology Iron Curtain	Step 2	0.34
US-China trade war	Hegemonic Sanctions	0.58	Trade War / Decoupling	No	0.37
US-China financial isolation	Hegemonic Sanctions	0.62	Financial Reintegration	No	0.39
China-Taiwan military coercion	Multilateral Alliance Sanctions	0.54	Non-State Armed Proxy Conflict	Step 3	0.31
China-Taiwan invasion	Multilateral Alliance Sanctions	0.66	Non-State Armed Proxy Conflict	No	0.29
Russia-Ukraine war trajectory	Multilateral Alliance Sanctions	0.61	Resource Dependency / Energy Weaponisation	Step 4	0.30

Table 2: Forecasting results across six geopolitical scenarios.

P(\alpha)

is the primary attractor’s normalised Bayesian posterior.

c_{\mathrm{final}}=c_{0}\cdot\lambda^{T}

is the calibrated final confidence.

Internal key (code)	English display name	Domain
Hegemonic Sanctions	Hegemonic Sanctions	geopolitics
Entity-List Technology Blockade	Entity-List Technology Blockade	technology
Interstate Military Conflict	Interstate Military Conflict	military
Great-Power Coercion/Deterrence	Great-Power Coercion/Deterrence	geopolitics
Multilateral Alliance Sanctions	Multilateral Alliance Sanctions	geopolitics
Tech Decoupling / Technology Iron Curtain	Tech Decoupling / Technology Iron Curtain	technology

Table 3: Pattern glossary. Internal keys are used in the codebase; English display names are used in all user-facing outputs and in this paper.

Metric	EL-DRUIN	GPT-4 ( $N=5$ )
Confidence source	Ontology priors $\times$ Bayesian posterior (deterministic)	Self-reported by LLM (no auditable derivation)
Confidence verifiable	Yes — compute trace anchors every value	No — free-text assertion
Stability ( $\sigma$ )	0.000 (deterministic)	$>0$ (stochastic variance)
Traceability	Full compute trace (prior $\times$ similarity $\times$ decay)	None (no computable trace)
Entity invention guard	Enforced (invented proper nouns trigger fallback)	Not enforced by default
Numeric consistency	Locked by deterministic pipeline	Self-reported; may vary

Table 4: Baseline comparison metrics. EL-DRUIN vs GPT-4 (

N=5

\sigma

denotes standard deviation across runs.

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Abstract

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