Fractal Knowledge Trees for Provability-Guided Multi-Agent Learning

We introduce Fractal Information AI, a proof-of-concept framework for structured knowledge representation and selective multi-agent learning. Knowledge is represented by recursively nested tree structures whose nodes carry uncertainty, entropy-like information measures, and a model-defined provability score. A Learning Efficiency Measure (LEM) combines these quantities to prioritize knowledge updates, while a cross-agent flux term represents exchange of information between agents. The framework is formalized through node utility, network-level learning-efficiency measures, stochastic update equations, and decision-impact functions. Synthetic simulations illustrate the resulting dynamics, including selective node prioritization and the appearance of highly connected or influential nodes. These simulations should be interpreted as demonstrations of the internal behavior of the proposed model rather than as evidence of superiority over neural, reinforcement-learning, or knowledge-graph baselines. A central clarification of this audited formulation is that “provability” denotes a model-level score unless a node is explicitly linked to a formal language, axiom system, and proof procedure. Gödel incompleteness therefore motivates caution about formal closure but does not itself provide a learning metric or establish limits for contemporary statistical AI. The paper identifies the formal definitions, benchmark comparisons, ablation studies, calibration procedures, and energy measurements required to convert the present conceptual architecture into a validated learning method.