Note for the P versus NP problem (II)

The P versus NP problem is a fundamental question in computer science. It asks whether problems whose solutions can be quickly verified can also be quickly solved. Here, "quickly" refers to computational time that grows proportionally to the size of the input (polynomial time). While the problem's roots trace back to a 1955 letter from John Nash, its formalization is attributed to Stephen Cook and Leonid Levin. Despite extensive research, a definitive answer remains elusive. Closely tied to this is the concept of NP-completeness. If a single NP-complete problem could be solved efficiently, it would imply that all problems in NP can be solved efficiently, proving that P equals NP. Our work presents a polynomial-time algorithm for the CLOSURE problem. We implemented a Python-based solution for this polynomial time algorithm, which is available on GitHub under the username "frankvegadelgado". In addition, we propose that CLOSURE is actually an NP-complete problem, which would imply that P equals NP. This work is an expansion and refinement of the article "Note for the P versus NP problem", published in IPI Letters.