A Unified Approximation Theory of Scaled Cognition: From Neural Hierarchies to World Models and Strategic Reasoning

This preprint presents a unified approximation theory for scaled cognition across neural, cognitive, and artificial systems. Building on renormalization-based approximation principles, the work formalizes how full computational models—ranging from multimodal temporal cortex hierarchies and perceptual categorization to recursive cognition, developmental scaling, world-model construction, and long-horizon strategic reasoning—are systematically reduced to finite, resource-limited approximations. The framework introduces three approximation errors, a unified fidelity criterion, and a scale-dependent renormalization flow that governs transitions between levels of abstraction. By applying this structure across six major domains, the paper reveals a universal mathematical form underlying sensory hierarchies, category formation, recursive reasoning, LLM behavior, hippocampal updating, and strategic planning. The result is a general theoretical foundation for understanding how cognition emerges from the interaction between full models and their scaled approximations.