Large Language Models (LLMs) exhibit remarkable capabilities, yet it remains unclear to what extent these reflect sophisticated recall or genuine reasoning ability. We introduce chess as a controlled testbed aimed at disentangling these faculties. Leveraging the game's structure and scalable engine evaluations, we construct a taxonomy of positions varying in density of relevant priors -- ranging from common states solvable by memorization to completely novel ones requiring generalization. Crucially, our approach achieves this distinction without requiring explicit knowledge of the models' training data. Applying this taxonomy, we combine a longitudinal analysis of the GPT lineage with a rigorous evaluation of contemporary models, including Claude Opus and Gemini. Our analysis reveals a steep gradient: performance consistently degrades as the density of relevant priors decreases. Notably, for tasks with few relevant priors, base model performance regresses to the random-play baseline, and all models propose illegal moves at sharply increasing rates, indicating that they fail to reliably apply the fixed rules of the game in unfamiliar positions. While newer models improve, progress slows significantly for tasks with sparse priors. Furthermore, while reasoning-augmented inference improves performance, its relative marginal benefit per token decreases in the absence of relevant priors. These results suggest limitations in systematic generalization, highlighting the need for mechanisms beyond scale to achieve robust performance when deprived of relevant priors.

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