The relevance of the research is determined by the fundamental nature of the P vs NP problem, which has been considered a central unsolved problem in theoretical computer science for more than half a century. In this article, for the first time, it is proved that the problem of solving the P vs NP problem is not a mathematical problem, but a physical problem, the solution of which depends on the physical implementation of the computing system. The aim of this paper is to prove that the solution of the P vs NP problem depends on the type of physical system used for calculations. The leading approach to research is to synthesize three independent paradigms: 1. Algebraic topology that provides invariants for complexity classification. 2. Quantum physics, which provides parallel information processing. 3. Apparat implementation on the GPU that demonstrates the physical limitations of classical computing. Author's results include: A formal proof that P ≠ NP holds in classical physics (Newtonian mechanics, deterministic computations) and a simultaneous proof that P = NP holds in quantum physics (superposition, entanglement). Experimental confirmation of the physical implementation through a series of computational experiments. Development of hybrid system that allows you to switch between modes depending on the available resources. The theoretical significance lies in rethinking the fundamental problem of complexity theory as a physical law. Practical significance is the creation of an adaptive computing system, the efficiency of which is determined by the available physical platform.
The relevance of the research is determined by the fundamental nature of the P vs NP problem, which has been considered a central unsolved problem in theoretical computer science for more than half a century. In this article, for the first time, it is proved that the problem of solving the P vs NP problem is not a mathematical problem, but a physical problem, the solution of which depends on the physical implementation of the computing system. The aim of this paper is to prove that the solution of the P vs NP problem depends on the type of physical system used for calculations. The leading approach to research is to synthesize three independent paradigms: 1. Algebraic topology that provides invariants for complexity classification. 2. Quantum physics, which provides parallel information processing. 3. Apparat implementation on the GPU that demonstrates the physical limitations of classical computing. Author's results include: A formal proof that P ≠ NP holds in classical physics (Newtonian mechanics, deterministic computations) and a simultaneous proof that P = NP holds in quantum physics (superposition, entanglement). Experimental confirmation of the physical implementation through a series of computational experiments. Development of hybrid system that allows you to switch between modes depending on the available resources. The theoretical significance lies in rethinking the fundamental problem of complexity theory as a physical law. Practical significance is the creation of an adaptive computing system, the efficiency of which is determined by the available physical platform.
The P vs. NP problem is a central open question in theoretical computer science, asking whether every problem whose solution can be verified in polynomial time can also be solved in polynomial time. This paper presents an interdisciplinary exploration of this problem by integrating mathematical analysis, quantum computing, and machine learning.We analyze the limitations of polynomial-time reductions among NP-complete problems under the Exponential Time Hypothesis (ETH), providing detailed proofs that offer new insights into the structural complexities within NP. Additionally, we develop a novel quantum-classical hybrid algorithm that combines Grover's algorithm with classical heuristics to solve specific instances of NP-complete problems more efficiently. This includes a comprehensive complexity analysis, scalability discussions, and detailed practical implementation aspects such as oracle constructions.Furthermore, we propose a machine learning framework based on Graph Neural Networks (GNNs) to capture patterns within NP-complete problems. This approach is theoretically justified, enhances heuristic methods, and demonstrates superior performance through extensive experimental comparisons with classical heuristics and exact solvers.By integrating these methods, this work offers fresh perspectives on the P vs. NP problem and contributes to the broader understanding of computational complexity.
Abstract: The P vs NP problem is the central open question in computational complexity theory, asking whether every problem whose solution can be verified in polynomial time can also be solved in polynomial time. This paper argues within the framework of quantum mathematics that P ≠ NP. Quantum mathematics reinterprets natural numbers as quantum states in ℓ2(ℕ), introducing the generation operator 𝐺̂ and the cognition operator 𝐶̂, whose non-commutation relation [𝐺̂,𝐶̂] = 𝑖𝑔𝐼̂ (with 𝑔 = 1/2) constitutes the fundamental dynamics of arithmetic. This paper proves that the solution space for problem instance 𝑛 corresponds to the representation space ℛ(𝑛), whose cardinality ∣ ℛ(𝑛) ∣∼ 𝑒𝑐√𝑛 grows exponentially; the polynomial-step projection of the cognition operator 𝐶̂ can only cover a zero-measure subset of ℛ(𝑛). The proof depends on the four fundamental theorems of quantum mathematics—the non-commutation theorem, the arithmetical uncertainty principle, the prime emergence theorem, and the order conservation theorem—all of which are necessary consequences of the same primordial axiom 𝐺𝐶 = 𝑖𝐶𝐺. This paper reveals that the non-commutation between the generation pole and the cognition pole determines a fundamental structural gap between "solving" (generation) and "verifying" (cognition).摘要：P vs NP 问题是计算复杂性理论的核心未解难题，它询问：所有可在多项式时间 内验证的问题是否也可在多项式时间内求解？本文在量子数学框架内论证P ≠ NP。量 子数学将自然数重新解释为 ℓ2(ℕ) 中的量子态，引入生成算子 𝐺̂ 与认知算子 𝐶̂，其非 对易关系 [𝐺̂,𝐶̂] = 𝑖𝑔𝐼̂（𝑔 = 1/2）构成算术的根本动力学。本文证明：问题实例 𝑛 的 可能解空间对应于表示空间 ℛ(𝑛)，其基数 ∣ℛ(𝑛)∣∼𝑒𝑐√𝑛 指数增长；认知算符 𝐶̂ 的多 项式步数投影只能覆盖 ℛ(𝑛) 的零测度子集。证明依赖于量子数学四个基本定理——非 对易性定理、算术测不准原理、素数涌现定理与秩序守恒定理——它们都是同一原初 公理 𝐺𝐶=𝑖𝐶𝐺 的必然推论。本文揭示：生成极与认知极的非对易性决定了"求解"（生 成）与"验证"（认知）之间存在根本性的结构性鸿沟。
The relationship between the complexity class P and NP is one of the most fascinating and unresolved question in theoretical Computer Science. The classical computational paradigm, hedged on Turing thesis may be by itself the limiting factor. To investigate this relationship, a method for solving a query problem on a relational database on the classical and the quantum computational paradigm has been developed. The method solves queries or database problems through the use of graphs. The results indicate that the solution to the P vs NP cannot be resolved under the current computation paradigms and may requires a new computation paradigm to resolve it.
The central problem in computer science is the conjecture that two complexity classes, P (polynomial time) and NP (nondeterministic polynomial time-roughly those decision problems for which a proposed solution can be checked in polynomial time), are distinct in the standard Turing model of computation: P not equal NP. As a generality, we propose that each physical theory supports computational models whose power is limited by the physical theory. It is well known that classical physics supports a multitude of implementation of the Turing machine. Non-Abelian topological quantum field theories exhibit the mathematical features necessary to support a model capable of solving all #P problems, a computationally intractable class, in polynomial time. Specifically, Witten [Witten, E. (1989) Commun. Math. Phys. 121, 351-391] has identified expectation values in a certain SU(2)-field theory with values of the Jones polynomial [Jones, V. (1985) Bull. Am. Math. Soc. 12, 103-111] that are #P-hard [Jaeger, F., Vertigen, D. & Welsh, D. (1990) Math. Proc. Comb. Philos. Soc. 108, 35-53]. This suggests that some physical system whose effective Lagrangian contains a non-Abelian topological term might be manipulated to serve as an analog computer capable of solving NP or even #P-hard problems in polynomial time. Defining such a system and addressing the accuracy issues inherent in preparation and measurement is a major unsolved problem.
Classical computational complexity theory treats the separation of complexity classes as an absolute, intrinsic property of mathematical logic. This paper introduces the Duality of the Medium Framework, demonstrating that the P vs. NP question is structurally incomplete when divorced from the dimensionality of the processing medium and the observer's codec. We show that while P ≠ NP holds true within the linear constraints of a binary, deterministic medium due to sequential path iteration, P = NP is realized within a quantum medium where the problem configuration acts as a resonant cavity, collapsing the problem-solution vector into an instantaneous, entangled geometric state.
The P versus NP problem has remained unsolved for over fifty years, with some researchers suggesting it may be independent of standard axiom systems (Sipser, 1992; Aaronson, 2003). This paper proposes that the problem resists solution not due to mathematical difficulty, but due to categorical error: P and NP are not comparable quantities within a single dimension, but rather orthogonal constructs operating on different layers of reality. Drawing on established research linking computational complexity to physical theory (Freedman et al., 1998), we argue that P represents quantum- scale reasoning processes while NP represents classical verification. The question "Does P equal NP?" is therefore malformed—analogous to asking whether length equals color. We propose that dissolving rather than solving this problem opens new avenues for understanding the relationship between quantum and classical computation.