Riemann‑ID

Validation of the Real‑Imaginary Duality Principle: from the Riemann Hypothesis to prime density generation

This repository contains the complete source code and generated figures for the paper “An Information‑Dynamics Proof of the Non‑Trivial Zeros of the Riemann Zeta Function: From Dynamic Real‑Imaginary Coupling to Prime Density Generation” (Kai Huang, 2026).

What is inside

File	Description
prime_density.py	Numerical integration of the generalized Ginzburg–Landau (GL) equation driven by the coupling matrix $K(x)=\sqrt{2x\ln x}$. Evolves the complex information field from random noise and outputs fig1_prime_density.pdf.
prime_density_binned_validation.py	Binned $\pi(x)$ comparison. Divides [2, 100] into 10 sub‑intervals and compares the model‑predicted prime count with the exact count (using sympy.primepi). Outputs output_prime_density_binned_validation.txt.
GUE_spacings.py	Generates a 1500×1500 Gaussian Unitary Ensemble (GUE) random matrix, computes normalized nearest‑neighbour spacings, and compares them with the first 100 Riemann zeros (Odlyzko data). Reports the Kolmogorov–Smirnov statistic. Outputs fig3_gue_spacings.pdf.
coupling_matrix.py	Plots the dynamic coupling matrix $K(x)$ together with the prime density $1/\ln x$. Outputs fig2_coupling_matrix.pdf.
zero_mapping.py	Illustrates the schematic mapping from eigenvalues $\lambda_n^2$ of the squared operator $\Delta_K$ to critical‑line zeros via $s(1-s)=\lambda_n^2$. Outputs fig4_zero_mapping.pdf.
fig1_prime_density.pdf	Self‑organised steady‑state density vs. target $1/\ln x$ and relative error (log scale).
fig2_coupling_matrix.pdf	Coupling matrix $K(x)$ and prime density $1/\ln x$.
fig3_gue_spacings.pdf	Nearest‑neighbour spacing distribution: GUE model vs. first 100 Riemann zeros.
fig4_zero_mapping.pdf	Schematic of the eigenvalue‑to‑zero mapping on the critical line.

Reproducing the results

Requirements

• Python ≥ 3.8
• NumPy, SciPy, Matplotlib, SymPy

Install dependencies:

pip install numpy scipy matplotlib sympy

Generate the figures

python prime_density.py
python prime_density_binned_validation.py
python GUE_spacings.py
python coupling_matrix.py
python zero_mapping.py

All PDF figures will be written to the current working directory.

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9. License

This project is licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License (CC BY-NC-SA 4.0).

This license allows you to:

• Share — copy and redistribute the material in any medium or format.
• Adapt — remix, transform, and build upon the material.

Under the following terms:

1. Attribution (BY) — You must give appropriate credit, provide a link to the license, and indicate if changes were made.
2. NonCommercial (NC) — You may not use the material for commercial purposes without prior written permission.
3. ShareAlike (SA) — If you remix, transform, or build upon the material, you must distribute your contributions under the same license.

To view a copy of this license, visit https://creativecommons.org/licenses/by-nc-sa/4.0/.

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Citation

If you use this code or data in your research, please cite:

Kai Huang. An Information‑Dynamics Proof of the Non‑Trivial Zeros of the Riemann Zeta Function: From Dynamic Real‑Imaginary Coupling to Prime Density Generation. Zenodo, 2026.
DOI: 10.5281/zenodo.20054082

@misc{huang2026riemann,
  author       = {Kai Huang},
  title        = {An Information‑Dynamics Proof of the Non‑Trivial Zeros of the 
                  Riemann Zeta Function: From Dynamic Real‑Imaginary Coupling 
                  to Prime Density Generation},
  year         = 2026,
  publisher    = {Zenodo},
  doi          = {10.5281/zenodo.20054082},
}

Contact

Kai Huang – hkaiopen@foxmail.com
Project repository: https://github.com/hkaiopen/Riemann-ID