Large Language Models are widely used to generate and adapt cultural texts, yet the depth of their cultural representation remains poorly quantified. Intuitively, as a narrative text expands in length, the diversity of cultural words should scale proportionately. To formally test this, we evaluate the FairyTaleQA dataset, adapted by three models and introduce our primary contribution: the Contextual Stereotype Amplification Index (CSAI), an evaluation framework combining LLM-as-a-judge extraction, embedding-based cliché anchoring, and Natural Language Inference (NLI) congruence validation. By mapping the frequency of extracted Culture Specific Items (CSIs) against narrative length using Heaps’ Law (V = k ⋅ T𝛽), we present empirical evidence of a systematic limitation in current systems: they struggle to scale cultural diversity even under explicit cultural prompting. Models rapidly hit a "Cultural Vocabulary Ceiling," constrained to a fixed set of hyper-stereotypical terms. Furthermore, we demonstrate that merely optimizing for higher CSI frequency as done in prior works rewards logically broken tokenism. Our CSAI formulation actively penalizes such gratuitous stereotyping, offering a more principled approach to measuring and evaluating cultural homogenization in generative AI systems.