Note for the P versus NP Problem

We use the 1-in-3 3sat problem

We created a Github repository with the software solution in Python

We show that the Monotone Weighted Xor 2-satisfiability problem ($MWX2SAT$) is $NP$-complete and $P$ at the same time.

We improved the Lemma 1

We changed the abstract and content

We improved the content.

In the previous version, we create two ways to prove that MWX2SAT is in P. In this version, we remove the one that is flawed and we keep the single one that is correct.

We improved the Lemma 1 and added the Discussion section

We replaced in the definition of the K-CLOSURE the content "($u \in V'$ or $v \in V'$) or ($u \notin V'$ or $v \notin V'$)" by "($u \in V'$ and $v \in V'$) or ($u \notin V'$ and $v \notin V'$)"