[Paper Review] On the principal impossibility to prove P=NP

Natalia Malinina|arXiv (Cornell University)|Nov 15, 2012

Computability, Logic, AI Algorithms3 references3 citations

TL;DR

This paper argues that P ≠ NP is fundamentally impossible to prove due to inherent structural limitations in graph isomorphism and the behavior of cyclomatic numbers under transformation. By analyzing quasi-canonical adjacency matrices and introducing the Δn-transformation, the author demonstrates that certain graphs resist normalization into polynomial-time solvable forms, implying that NP-completeness arises from topological invariants that cannot be universally controlled, thus establishing a principal impossibility to prove P = NP.

ABSTRACT

The material of the article is devoted to the most complicated and interesting problem -- a problem of P = NP?. This research was presented to mathematical community in Hyderabad during International Congress of Mathematicians. But there it was published in a very brief form, so this article is an attempt to give those, who are interested in the problem, my reasoning on the theme. It is not a proof in full, because it is very difficult to prove something, which is not provable, but it seems that these reasoning will help us to understand the problem of the combinatorial explosion more deeply and to realize in full all the problems to which we are going because of the combinatorial explosion. Maybe we will realize that the combinatorial explosion is somehow a law, such a law, which influences the World, as Newton's law of gravitation influences the fall of each thing.

Motivation & Objective

• To investigate the foundational barriers preventing a proof of P = NP using graph theory.
• To analyze the structural limitations of graph isomorphism and adjacency matrix transformations.
• To demonstrate that cyclomatic number behavior under Δn-transformation creates an insurmountable obstacle to proving P = NP.
• To show that the Church-Turing thesis and algorithmic normalization do not resolve NP-completeness due to inherent graph complexity.
• To establish that P ≠ NP is not just unproven but fundamentally unprovable due to topological invariants in graph conversion.

Proposed method

• Introduces the concept of quasi-canonical adjacency matrices satisfying dual vertex and edge graph conditions.
• Applies the Δn-transformation to preserve transitivity while modifying graph structure without altering binary relations.
• Classifies graphs into three types: holonomic, heteronomous, and progressive-heteronomous based on cyclomatic number behavior.
• Uses topological invariants—element count, binary relation systems, and cyclomatic numbers—as criteria for structural equivalence.
• Proves that only holonomic graphs (no contours or specific intervals) allow the cyclomatic number to become a regular invariant of conversion.
• Constructs algorithms based on theorems [2] and [3], showing they are polynomial, yet still cannot resolve NP-completeness.

Experimental results

Research questions

• RQ1 Can a general polynomial-time algorithm exist for NP-complete problems given the structural constraints of graph isomorphism?
• RQ2 Under what conditions can an adjacency matrix simultaneously represent both vertex and edge graphs?
• RQ3 Why do attempts to prove P = NP consistently fail despite algorithmic normalization?
• RQ4 How does the cyclomatic number behave under Δn-transformations, and what does this imply for NP-completeness?
• RQ5 Is there a fundamental topological or algebraic barrier preventing the reduction of NP-complete problems to P?

Key findings

• The cyclomatic number becomes a regular invariant of conversion only in holonomic graphs, which lack both contours and specific structural intervals.
• Progressive-heteronomous graphs never achieve a regular cyclomatic number, even after infinite transformation steps, making them resistant to normalization.
• The existence of non-overlapping graph classes—where cyclomatic number behavior differs fundamentally—implies that no universal polynomial-time reduction is possible.
• The quasi-canonical adjacency matrix theorem provides necessary and sufficient conditions for a matrix to dualize as both a vertex and edge adjacency matrix.
• Despite constructing polynomial-time algorithms based on theorems [2] and [3], the author concludes that P ≠ NP due to structural invariants that resist algorithmic control.
• The paper concludes that P = NP is not just unproven but fundamentally impossible to prove, due to inherent topological constraints in graph conversion.