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Code for "Quantum Statistical-Gauge Geometry"

Pham Van Hai

Published Jun 2, 2026• 1 citation
Publisher

Abstract

This repository contains the Python notebooks used to generate and verify the numerical results presented in the article "Quantum Statistical-Gauge Geometry". The notebooks implement exact-diagonalization calculations for finite-dimensional lowest-Landau-level torus models, including tests of bulk Lie closure, finite-temperature hyperforce Ward identities, and flux-threading/holonomy diagnostics on the twist torus.

DOI
10.5281/zenodo.20503381

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This repository contains the Python notebooks used to generate and verify the numerical results presented in the article "Quantum Statistical-Gauge Geometry". The notebooks implement exact-diagonalization calculations for finite-dimensional lowest-Landau-level torus models, including tests of bulk Lie closure, finite-temperature hyperforce Ward identities, and flux-threading/holonomy diagnostics on the twist torus.
Exact equilibrium constraints in quantum many-body systems, such as sum rules and Ward identities, can be hard to expose because local currents are distributional and equal-time commutators may contain anomaly-like contact terms. We present a simple geometric organization of the recently identified quantum shift symmetry. The local shifting superoperator is viewed as a statistical-gauge connection acting on observables, so hyperforces are covariant derivatives and force balance becomes a thermal Ward identity expressed with the Kubo-Mori inner product. For compact smearings we prove bulk flatness: smeared shift generators close on the Lie bracket of vector fields with no bulk Schwinger term. This closure yields an iterative, all-orders Ward-Bianchi hierarchy of equal-time constraints that closes on a finite set of force insertions and a bilinear force-gradient kernel, leading to general antisymmetry relations and positivity bounds. We fix the microscopic generator by quantizing the diffeomorphism moment map, where Weyl and half-density quantization agree and give a self-adjoint operator without bulk ordering anomalies. Although the bulk connection is flat, global topology remains through holonomy, which links naturally to twisted boundary conditions and flux threading. Background electromagnetic fields deform the algebra through explicit density-weighted curvature insertions rather than a bulk central extension. As an application we treat fractional quantum Hall fluids on a torus, connect the deformation to the guiding-center GMP-W_{∞} algebra, and verify the key identities with exact diagonalization.
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
This repository contains the complete Python notebook and numerically generated data supporting Sections 7.1–7.5 and Figs. 1–3 of the accompanying article, “Eigenbasis Conditioning and Fisher-Information Singularity Classes Near a Second-Order Exceptional Point.” The archive includes the exact canonical EP2 benchmark, a reproducible ensemble of 300 noncanonical continuous complex operator paths, the Bernoulli Fisher-information readout models, the relative Fisher–conditioning analysis, and the numerical precision, unitary-invariance, and nonunitary-similarity tests. The repository contains the complete ensemble-generation parameters, recorded pseudorandom seeds, processed numerical data, software-environment specifications, and files underlying the reported figures. All data in the archive are numerically generated. The notebook code is released under the MIT License. The generated numerical data, figures, and documentation are released under the Creative Commons Attribution 4.0 International License.