P Equals NP: Core-Based Smart Sampling Framework

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 smart random samples, 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.