<b>P Equals NP: Core-Based Smart Sampling Framework</b>

This paper proposes a comprehensive framework for addressing the P vs NP problem by integrating three original hypotheses—Double Negation, n-1, and Language Hypotheses—with the concepts of smart sampling, core (nucleus) identification, and temporal analysis. The framework introduces <b>smart random samples</b>, intersection-based reductions, and negation operations to reduce the solution space efficiently within polynomial bounds. The nucleus is defined as the minimal critical component of the problem, whose discovery significantly accelerates the solution process. Examples from computational theory and Arabic language grammar illustrate the practical applicability.本文提出了一套针对P与NP问题的综合研究框架，通过将三项原创性假设——双重否定假设（Double Negation）、n-1假设（n-1）与语言假设（Language Hypotheses）——与智能采样、核心（核）识别以及时序分析等概念相结合。该框架引入了智能随机样本（smart random samples）、基于交集的归约方法与取反操作，得以在多项式界内高效缩减问题的解空间。研究将核心（核）定义为问题的最小关键组分，对该组分的发现可显著加速问题求解进程。来自计算理论与阿拉伯语语法的实例验证了该框架的实际应用可行性。