4D Semantic Coupling: A Mathematical Framework for Measuring Cognitive Complexity in AI Dialogue

Contemporary evaluation methods for large language models (LLMs) fail to quantify cognitive complexity in extended dialogue. Existing metrics such as perplexity and BLEU focus on surface-level text features and overlook the evolving semantic structures that characterize deep reasoning. We propose a novel four-dimensional (4D) mathematical framework that models dialogue as a topological space composed of sequential (D₁), contextual (D₂), embedded semantic (D₃), and temporal-loop (D₄) dimensions. This space is operationalized via three core metrics: Semantic Curvature (Δκ), Fractal Similarity (α), and Entropy Shift (ΔH), which together capture non-linear reasoning, self-organizing structure, and semantic reconfiguration. A 64session longitudinal study under the Semantic Pressure Architecture (SPA) protocol demonstrates significant increases in all metrics (Δκ = 0.42, α = 0.79, ΔH = 0.18; all p 0.83). This indicates that each model encodes a unique geometric lens on meaning, establishing our framework as both a complexity measure and a meta-analytic tool for probing LLM semantics. We provide an open-source implementation to enable reproducible and architecture-agnostic evaluation.