Polymorphic Combinatorial Frameworks (PCF): Guiding the Design of Mathematically-Grounded, Adaptive AI Agents

The Polymorphic Combinatorial Framework (PCF) leverages Large Language Models (LLMs) and mathematical frameworks to design adaptive AI agents for complex, dynamic environments. Grounded in combinatorial logic, topos theory, and rough fuzzy set theory, PCF defines a multidimensional SPARK parameter space (Skills, Personalities, Approaches, Resources, Knowledge) to capture agent behaviors. This paper demonstrates how LLMs can parameterize complex spaces, estimate likely parameter values, and model variabilities. Using PCF, we parameterized a mock café domain, estimated variables/variabilities, and conducted over 1.25 million Monte Carlo simulations. The results revealed trends in agent adaptability and performance across five complexity tiers, with diminishing returns at higher complexity levels highlighting thresholds for scalable designs. PCF enables self-optimizing agents for scenarios while maintaining logical consistency. This framework supports scalable, dynamic, explainable, and ethical AI applications in domains like customer service, healthcare, robotics, and collaborative systems, paving the way for adaptable and cooperative next-generation polymorphic agents.