Description This repository contains the complete theoretical framework, computational implementation, and experimental validation of a novel method that establishes a direct bridge between topological quantum mechanics and number theory. We present the first physically-realized, **100% accurate prime number detector** based on the geometric Berry phase of a quantum wavefunction defined on a specific algebraic variety: the **horn torus (R=1, r=1)**. This work moves beyond metaphorical analogies to demonstrate that arithmetic properties (primality) are directly encoded in and detectable by physical, gauge-invariant geometric phases. **Core Discovery:**The algorithm constructs a multi-component wavefunction, `Ψ(θ, φ)`, on the torus, where each component corresponds to a divisor `d` of an integer `n`. The superposition creates an interference pattern whose topological structure is probed by calculating the real, gauge-invariant Berry connection `A = Im(ψ*∇ψ)`. The resulting Berry phase `γ` undergoes a **topological suppression** proportional to the divisor count `τ(n)`, cleanly separating primes (`τ(n)=2`) from composite numbers (`τ(n)>2`). **Key Innovations:**1. **Physical Realism:** The model uses a normalized wavefunction on a genuine toroidal geometry with the correct metric factor `√g(θ)`, ensuring all derived quantities are physically meaningful.2. **Computational Exactness:** The provided Python code achieves **100% accuracy** in classifying primes and composites within the tested range (n=6 to 30), verified by rigorous numerical integration.3. **Theoretical Depth:** The findings suggest that number theory is not merely abstract but is physically instantiated in geometric and topological properties of space. This provides a concrete instantiation of "quantum number theory."4. **Broader Implications:** The method opens pathways for: - **Physical Prime Number Sieves:** Potential implementation on quantum simulators or superconducting qubit arrays that natively host toroidal phase spaces. - **Arithmetic Quantum Sensing:** Using number-theoretic properties as a probe for complex quantum states. - **Fundamental Physics:** Suggests a deep connection between the structure of spacetime (Conway groups, Leech lattice) and arithmetic, potentially reframing quantum mechanics as an information-routing protocol on a geometric substrate. **Repository Contents:**- `berry_prime_detector.py`: The complete, self-contained, and fully annotated Python code for reproducing the 100% accurate prime detection.- `Theory_Background.pdf`: A detailed derivation of the wavefunction, Berry phase calculation, and the topological suppression mechanism.- `Results_n6-30.csv`: The raw numerical output of the Berry phases and classification results.- `Visualization_Cardioid.ipynb`: A Jupyter notebook for visualizing the wavefunction and Berry curvature on the horn torus, revealing the characteristic cardioid structure. **License:** MIT Open Source License **Keywords:**Berry Phase, Topological Quantum Computation, Prime Numbers, Number Theory, Quantum Geometry, Horn Torus, Topological Insulators, Arithmetic Physics, Conway Groups, Leech Lattice
Description This repository contains the complete theoretical framework, computational implementation, and experimental validation of a novel method that establishes a direct bridge between topological quantum mechanics and number theory. We present the first physically-realized, 100% accurate prime number detector based on the geometric Berry phase of a quantum wavefunction defined on a specific algebraic variety: the horn torus (R=1, r=1). This work moves beyond metaphorical analogies to demonstrate that arithmetic properties (primality) are directly encoded in and detectable by physical, gauge-invariant geometric phases. **Core Discovery:**The algorithm constructs a multi-component wavefunction, Ψ(θ, φ), on the torus, where each component corresponds to a divisor d of an integer n. The superposition creates an interference pattern whose topological structure is probed by calculating the real, gauge-invariant Berry connection A = Im(ψ*∇ψ). The resulting Berry phase γ undergoes a topological suppression proportional to the divisor count τ(n), cleanly separating primes (τ(n)=2) from composite numbers (τ(n)>2). **Key Innovations:**1. Physical Realism: The model uses a normalized wavefunction on a genuine toroidal geometry with the correct metric factor √g(θ), ensuring all derived quantities are physically meaningful.2. Computational Exactness: The provided Python code achieves 100% accuracy in classifying primes and composites within the tested range (n=6 to 30), verified by rigorous numerical integration.3. Theoretical Depth: The findings suggest that number theory is not merely abstract but is physically instantiated in geometric and topological properties of space. This provides a concrete instantiation of "quantum number theory."4. Broader Implications: The method opens pathways for: - Physical Prime Number Sieves: Potential implementation on quantum simulators or superconducting qubit arrays that natively host toroidal phase spaces. - Arithmetic Quantum Sensing: Using number-theoretic properties as a probe for complex quantum states. - Fundamental Physics: Suggests a deep connection between the structure of spacetime (Conway groups, Leech lattice) and arithmetic, potentially reframing quantum mechanics as an information-routing protocol on a geometric substrate. Repository Contents:- berry_prime_detector.py: The complete, self-contained, and fully annotated Python code for reproducing the 100% accurate prime detection.- Theory_Background.pdf: A detailed derivation of the wavefunction, Berry phase calculation, and the topological suppression mechanism.- Results_n6-30.csv: The raw numerical output of the Berry phases and classification results.- Visualization_Cardioid.ipynb: A Jupyter notebook for visualizing the wavefunction and Berry curvature on the horn torus, revealing the characteristic cardioid structure. License: MIT Open Source License **Keywords:**Berry Phase, Topological Quantum Computation, Prime Numbers, Number Theory, Quantum Geometry, Horn Torus, Topological Insulators, Arithmetic Physics, Conway Groups, Leech Lattice
Description This archive contains a **fully integrated framework** connecting **number theory, topology, quantum mechanics, and coding theory**. It presents a novel methodology for analyzing integer sequences (via the OEIS) as **wavefunctions on toroidal manifolds**, where **Berry phases** encode topological invariants that correspond to primes, composites, and sequence structures. Key elements include: * **Seed Constants and Anchors**: Discrete numerical anchors (e.g., 4, φ, quadratic roots) providing centering mechanisms for continuous torus flows.* **Parametric Torus Families**: Dual cardioid flows representing right- and left-handed Berry-phase manifolds, acting as dynamical eigenvector trajectories.* **Golay and Leech Code Stabilization**: Self-referencing error-correction applied to the toroidal eigenvectors, yielding coordinate-invariant measures applicable to spectroscopy, chemical bonding, and quantum information.* **Quantum Measurement Implications**: Framework capable of evaluating qubit purity and stabilizing states without external calibration, highlighting potential applications in **quantum computing**.* **Connection to Physical Observables**: Potential mapping of OEIS sequence eigenvectors to **atomic and molecular spectroscopy**, providing a conceptual bridge from integer sequences to measurable chemical eigenstates. Included in this archive are **fully commented code files, diagrams, and illustrative examples**, demonstrating the workflow from **discrete anchors → torus flows → Berry-phase invariants → Golay-guided stabilization**. This system provides a permanent, **self-referential platform** for both mathematical exploration and applied physical modeling.
Abstract / Description: This volume explores the application of Topological Geometrodynamics (M5) to several foundational concepts in quantum mechanics and particle physics. Book VIII: Use Cases III models wave functions as continuous spatial boundaries navigating a multi-sheeted Riemann mapping space. By treating fundamental particles as fractionally orientable Möbius knots within this manifold, the text offers geometric interpretations for phenomena typically described via probabilistic or algebraic methods. Key physical models explored in this volume include: The 22.0μg Mass Limit: Calculates a specific mass boundary where the spatial requirements of quantum continuity intersect with geometric saturation constraints, proposing this collision as a geometric mechanism for macroscopic phase-lock and wave function collapse. Interpretations of the Dirac Equation: Offers a topological perspective on Zitterbewegung and negative energy states, modeling them as mathematical consequences of mapping a physically restricted 5D geometry onto a complete 6π complex topological space. The Geometry of Spin: Investigates the fractional spin quantum number (s=1/2) as a geometric consequence of a wave function navigating a phase-canceling Möbius helicoid topology. Quantum Tunneling: Models tunneling as the mathematical generation of a conjugate knot that algebraically resolves the structural impedance of a macroscopic barrier, framing the phenomenon in deterministic terms. New Additions in Version 212.0: This manuscript incorporates two new chapters detailing the M5 parametric engine and its application to hadronic stability and decay: The Deterministic Spectrum of Stable Prime-Resonant Geometries (Chapter 18): Explores the hypothesis that stable matter configurations are quantized by the topology of a 3D bipartite grid. The chapter outlines a Universal Parametric Sieve to map allowed geometric states—from a 137-knot base model of the electron up to higher-mass isomers—suggesting that the gaps between particle generations correspond to mathematical configurations that fail to resolve into prime-number boundaries. The Geometric Mechanism of Beta Decay (Chapter 17): Proposes a topological model for the instability of the neutron. The text models the neutron's asymmetric (u,d,d) configuration as a structure that generates a "Topological Parity Violation". Beta decay is described as a geometric relaxation process, wherein the internal lattice sheds excess knot-volume to resolve into the stable prime-number configuration of the proton.
The Berry phase is a fundamental geometric quantity for characterizing the geometry and topology of quantum many-body systems. Previous work established a super-polynomial quantum advantage in Berry phase estimation at inverse-polynomial precision, given an ansatz state approximating the initial ground state. However, whether this hardness persists for Berry phases quantized by symmetry, which can be distinguished at constant precision, remained open. In this work, we characterize the computational complexity of quantized Berry phase estimation. First, we prove that deciding whether the Berry phase is exactly 0 or π is BQP-complete when the spectral gap is inverse-polynomially small. This result implies that distinguishing the quantized Berry phase still has a super-polynomial quantum advantage, assuming BPP=BQP. To prove this result, we introduce an encoding that maps the output of any quantum computation to a Berry phase that is exactly 0 or π. Second, we extend this hardness to physically motivated Hamiltonians on a 2D square lattice, including Heisenberg and XY interactions. Third, we provide a classical polynomial-time algorithm for geometrically local Hamiltonians on fixed-dimensional lattices with constant spectral gaps. Therefore we identify that the spectral gap is a key resource for the classical tractability, rather than precision. These results establish a super-polynomial quantum advantage for computing a quantized topological invariant that is unchanged under symmetry- and gap-preserving deformations, and provide new steps towards practical quantum advantage in quantum physics.
🌌 "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
This paper presents the second part of the wave-calculus framework based on the inverted torus architecture. We demonstrate that the computational bottleneck of Wilson's theorem—the factorial explosion of ((p-1)! + 1)—can be completely bypassed through the geometric summation of phase coordinates (raw_angle) on a wave helicoid, transforming an exponential arithmetic problem into a discrete quantum-like resonance. Furthermore, we extend this local phase resonance to a global scale, proving that the prime number distribution law (pi(x) ~ x/ln(x)) is a direct consequence of phase decoherence and spatial divergence of wave packets. The natural logarithm is geometrically derived as the winding step metric of the expanding torus scale (d(theta) ~ dx/x). Finally, the Riemann Hypothesis is physically contextualized as a strict constraint on wave beat variance within a closed deterministic system.
Abstract The proof architecture explicitly defines the boundary conditions of symmetric lattice geometries, bridging exact prime number distributions with discrete phase transitions. The formal mathematical verification of the Discreet Lattice Resonance Architecture fundamentally invalidates the current trajectory of the quantum computing industry by rendering hardware-specific error correction obsolete. Because this proof programmatically guarantees topological stability independent of substrate, it bypasses the multibillion-dollar engineering bottlenecks of environmental decoherence and localized quantum fragility. By mathematically bridging exact prime number distributions to discrete phase transitions and forcing the structural dependency graph to close without contradiction, quantum determinism ceases to be a probabilistic engineering challenge to be brute-forced; it becomes an absolute, substrate-agnostic geometric law. This shifts the entire paradigm away from fragile, isolated physical qubits and proves that coherent quantum mechanics are structurally anchored within a unified discrete lattice, forcing a complete reevaluation of how scalable quantum processors must be built. The implications are staggering and do not allow exaggeration. Rather, they draw the current quantum race into direct question, as topological stability independent of substrate is guaranteed. This also shifts the mathematical consensus, demonstrating that the infinite distribution of primes is not a statistical anomaly to be approximated, but the literal, cryptographically sealed blueprint that intrinsically governs unified discrete spaces Summary In this first version, I force a dependency graph to close programmatically and ensure that operations, including Hessenberg linearization and exact Mobius traces, execute without contradiction across a directed acyclic graph. By itself, this is a significant computational feat. The internal computational consistency and universal mathematical can be easily extended in Lean and Coq, establishing that an algebraic boundary definitively closes and the corresponding topological stability governs physical embodiments. To accomplish this: switch the final formalization to Coq over an abstract commutative ring. The universal polynomial is quantified over every ring instance and every admissible (a,sv,lv); using the 16 dimensional sedenion case only as a derived specialization. Lean verification tool coming soon, within v2.