P not equal to NP

TL;DR

The paper tackles the classic $P$ vs $NP$ problem and aims to exhibit a concrete $NP$-only problem using a novel tile-based system called Debilandia. It defines a generation-based 4x4-tile game, proves Debilandia is Turing complete, and embeds a decision problem whose resolution would separate $NP$ from $P$. It argues the problem is not in $P$ via reduction to the Halting Problem while providing a polynomial-time verifier to place it in $NP$, with a specific bound $f(N)=2ET+2E+3P+18T+10$ and $N=P+E+T+4$. The authors conclude that there exists at least one problem in $NP$ that is not in $P$, reinforcing $P\neq NP$, though the presentation intermingles contradictory statements and leaves room for scrutiny of rigor.

Abstract

This article finds the answer to the question: for any problem from which a non-deterministic algorithm can be derived which verifies whether an answer is correct or not in polynomial time (complexity class NP), is it possible to create an algorithm that finds the right answer to the problem in polynomial time (complexity class P)? For this purpose, this article shows a decision problem and analyzes it to demonstrate that this problem does not belong to the complexity class P, but it belongs to the class NP; doing so it will be proved that it exists at least one problem that belongs to class NP but not to class P, which means that this article will prove that not all NP problems are P.

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