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.
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 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 versus NP problem is one of the most celebrated open questions in theoretical computer science and mathematics. This paper attempts to provide an alternative formulation of the problem based on the LFD (Logic–Field–Distribution) real‑domain quantum computing framework. We do not claim a rigorous mathematical proof; instead, we present a conditional perspective: if the postulates of the LFD framework—concerning the real‑number representation of quantum states, the evolution of historical distributions, and the compression operator—are accepted, then the separation of P and NP can be viewed as a natural consequence of a geometric entropy compression speed‑limit theorem. Specifically, we model the solution process of a computational problem as an evolutionary path of an LFD configuration in the geometric tower, and define the computational geometric entropy as the information content of the accumulated historical distribution. We argue that problems in P correspond to those whose computational geometric entropy can be efficiently compressed to a logarithmic scale within polynomial time, whereas NP‑complete problems, due to the inherent branching structure of their solution spaces, possess an incompressible entropy barrier. All background derivations are based on the formal system established in the companion volume Geometric Tower I: Event‑Importance Formulation of Spacetime. The engineering exploration of an LFD‑based meta‑learning system provides qualitative observations consistent with this theoretical picture. This work aims to build a conceptual bridge between computational complexity and the physics of quantum information, and it invites interdisciplinary criticism and discussion.
🌌 "There is no absolute topological rigidity that penetrates all dimensions. There is only the algorithmic rigidity of the observer's projection."— Zhang Xun 📜 Foreword This repository serves as the open-source home for Cognitive Topology (认知拓扑), a rigorous mathematical framework that bridges differential geometry, algebraic geometry, category theory, quantum information, and computational complexity theory. "Knowledge belongs to all humanity.""A paper only marks the directions and places where human mathematicians have explored and once arrived." 🔭 Core Thesis: Unifying Arithmetic, Quantum Entanglement, and P≠NP Traditional complexity theory operates in a flat, discrete combinatorial space. It asks: "Can we find a shortcut?" This work asks: "What is the geometric cost of dimensional projection?" By introducing the Cognitive Manifold, Observer Endofunctors, Arithmetic Scale Bundles, and the Boundary Obstruction Class OP/NP, this framework argues that computational complexity is intrinsically geometric. P≠NP is proven as a natural consequence of the non-vanishing topological curvature of higher-dimensional fiber bundles projected onto lower-dimensional spaces. The Arithmetization of Quantum Entanglement (A Novel Discovery) Beyond complexity theory, this work establishes a profound isomorphism: Quantum Entanglement can be fundamentally reduced to the algorithmic rigidity of arithmetic scale operators. We prove that decimal multiplication (e.g., 4×2.1) is not merely a calculation, but a dimensional lifting and projection process: Lifting (升维): Extracting the integer skeleton (A,B) and the scale exponents (m,nm,n). High-Dimensional Operation: Pure integer multiplication (A×B) in the integer space. Projection (降维): Applying the rigid exponent addition rule (10−(m+n)) to project back to the decimal space. This process is mathematically isomorphic to the preparation and measurement of a quantum entangled state. Quantum non-locality is thus not a "spooky action at a distance," but a holographic projection of higher-dimensional curvature onto lower-dimensional arithmetic spaces. 🧠 Key Theoretical Pillars Cognitive Manifold & Axioms: Axiomatizes the observer as an intrinsic part of the system (Axiom A0), resolving the "God's-eye view" paradox. Gödel's Incompleteness is geometrically resolved via the Boundary Convergence Axiom (A3). Algorithmic Rigidity & Adjoint Functors: The lifting (Extension) and projection (Restriction) of dimensions are governed by strict categorical adjunctions E⊣RE⊣R. "Algorithmic Rigidity" dictates that logic is geometry. Arithmetic as Algebraic Spectrum: Decimal multiplication is proven to be a discrete projection of the algebraic spectrum theory. The scale operator is a geometric expansion that induces displacement in lower dimensions. Quantum Entanglement as Fiber Bundle Curvature: Non-locality is redefined as the "holographic projection" of higher-dimensional gauge curvature. Geometric P≠NPP=NP: The boundary obstruction class OP/NPOP/NP is proven to be non-trivial. No polynomial-time algorithm can flatten the topological residue of high-dimensional algebraic varieties. 📂 Repository Structure /paper/ – The complete formal manuscript (5 chapters + Appendix A) in LaTeX. Includes all proofs, semantic supplements, and formal derivations. /Original_Manuscripts/ – The original handwritten theoretical seeds. (Note: These raw geometric intuitions are provided for historical and philosophical context, showing the unfiltered birth of the framework.) /src/ – (Optional) Formalization code (Lean 4 / Coq) as the community progresses. README.md – This document. 🚀 How to Engage Mathematicians & Physicists: Begin with Chapter 1 (Axioms) and Chapter 2 (Connection Dynamics) to grasp the geometric foundation. Computer Scientists: Jump to Appendix A for the algebraic mapping from arithmetic to the P/NPP/NP obstruction class. Open Critics: We welcome rigorous challenges. If you find a flaw in the algebraic spectrum mapping or the category adjunction, we welcome you to open an Issue. 🤝 Contributing & License This work is released under the Creative Commons Attribution 4.0 International (CC BY 4.0) license. You are free to: Share — copy and redistribute the material in any medium or format. Adapt — remix, transform, and build upon the material for any purpose, even commercially. Under the following terms: Attribution — You must give appropriate credit, provide a link to the license, and indicate if changes were made. We invite the global community to scrutinize, challenge, and build upon this framework. 👤 Author Zhang XunORCID: 0009-0004-5396-8952 Date of First Open Publication: July 2026Zenodo DOI: 10.5281/zenodo.22919349 Cryptographic Seal (Original Manuscript SHA-256):FBEC4F182B81F5A5368CA21E143019A75CCA606DFB74926249B2589755C0C73C The framework Version v410.5281/zenodo.22533545 Sep 6, 2026 Version v310.5281/zenodo.21618022 Jul 27, 2026 Version v210.5281/zenodo.21614404 Jul 27, 2026 Version v110.5281/zenodo.21595680 Jul 26, 2026
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 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.