Mathematical Framework for Characterizing Emotional Individuality in Large Language Models: Temperature Control, Fuzzy Entropy, and Persona-Based Diversity Analysis

Evaluating emotional understanding in Large Language Models (LLMs) is challenging because assessments are subjective, ambiguous, multidimensional, and sensitive to controllable generation parameters. We developed a unified mathematical framework for characterizing LLM “emotional individuality” that integrates softmax sampling–temperature control (the decoding-time temperature parameter exposed by the API and typically used to modulate output randomness during token generation), fuzzy set theory with Shannon-type fuzzy entropy, and persona-based cognitive diversity analysis. We evaluated 36 API-accessible LLMs from seven major vendors on Japanese literary texts, using four personas each assigned a sampling temperature (T∈{0.1,0.4,0.7,0.9}), yielding 4227/4320 trial responses (97.8% coverage), of which 4067/4227 contained valid numeric emotion scores (96.2%). Temperature controllability varied approximately 25-fold (κM∈[0.039,0.982]) with both positive and negative temperature–variance relationships across models. Because each sampling temperature is deterministically assigned to a persona in our design, κM should be interpreted as an operational temperature–variance association across persona conditions rather than an isolated causal temperature effect. The model-level mean fuzzy entropy ranged from approximately 0.40 to 0.66, and the numerical stability consistency scores ranged from approximately 0.548 to 0.780. We also observed text-dependent structure, including genre-specific variation in the Interest–Sadness relationship. For practitioners, the framework is most directly useful as a benchmark-design and model-screening template for structured emotion-scoring tasks; its empirical conclusions remain limited to the present Japanese literary, text-only setting.