We present \textbf{EGPF} (Equilibrium-Guided Personalization Framework), a mathematically rigorous architecture unifying Bayesian game theory, category theory, information theory, and generative AI for hyper-personalized physician engagement in the pharmaceutical domain. Our framework models the pharma--physician interaction as an incomplete-information Bayesian game where physician behavioral types are inferred via functorial mappings from observational categories, equilibrium strategies guide content generation through large language models (LLMs), and information-theoretic feedback loops ensure adaptive recalibration. We formalize behavior composition through category-theoretic functors, natural transformations, and monoidal structures, enabling modular, composable physician archetypes that respect structural invariants under domain shift. We introduce a novel \textit{Rate-Distortion Equilibrium} (RDE) criterion that bounds the personalization--privacy tradeoff, an \textit{Evolutionary Game Dynamics} layer for population-level behavior modeling, a \textit{Mechanism Design} module for incentive-compatible engagement, and a \textit{Sheaf-Theoretic} extension for multi-scale behavioral consistency. We prove convergence of our iterative belief-update mechanism at rate $O(\frac{K\log K}{t \cdot C_{\min}})$ and establish finite-sample regret bounds. Extensive experiments on synthetic pharma datasets and a real-world HCP engagement pilot demonstrate a 34\% improvement in engagement prediction (AUC) and 28\% lift in content relevance scores compared to state-of-the-art methods.

翻译：我们提出**EGPF**（均衡引导式个性化框架），这是一个融合贝叶斯博弈论、范畴论、信息论与生成式人工智能的数学严谨架构，用于医药领域高度个性化的医生互动。该框架将医药企业-医生互动建模为不完全信息贝叶斯博弈：医生行为类型通过观测类别的函子映射进行推断，均衡策略引导大语言模型生成内容，信息论反馈循环确保自适应校准。我们利用范畴论中的函子、自然变换与幺半群结构对行为组合进行形式化建模，构建模块化、可组合的医生原型，使其在领域迁移下保持结构不变性。我们引入新型**率失真均衡**准则以约束个性化-隐私权衡，设计**演化博弈动力学**层建模群体行为，提出**机制设计**模块实现激励相容互动，并建立**层论拓展**确保多尺度行为一致性。我们证明迭代信念更新机制以$O(\frac{K\log K}{t \cdot C_{\min}})$速率收敛，并给出有限样本遗憾界。在合成医药数据集及真实医疗专业人员互动试点中的大量实验表明，相较于现有最优方法，我们的方法在互动预测（AUC）上提升34%，内容相关性评分提升28%。