The Gamma Vector: A Three-Axis Framework for Measuring Cognitive Rigidity Across LLM Architectures

We present the Gamma Vector (Γ), a three-dimensional metric for operationalizing cognitive rigidity in Large Language Models. The framework decomposes LLM response behavior into three orthogonal axes: γ₁ (Belief Inertia), γ₂ (Counterfactual Openness), and γ₃ (Identity Threat Response). Across six experiments (>1,700 trials, three major LLM families), we establish four key findings: distinct cognitive signatures per model family, dose-dependent rigidity reduction through targeted prompting, model-specific vulnerability operators, and a Three-Genus Taxonomy linked to training methodology (RLHF, reasoning, base). Results have direct implications for AI alignment evaluation, benchmark methodology, and multi-agent system design.