ID_HilbertPolya

Numerical verifications of the Hilbert–Pólya conjecture within the Information Dynamics framework

Overview

This repository provides three independent numerical experiments that support the Information Dynamics proof of the Hilbert–Pólya conjecture (and thus the Riemann Hypothesis). The experiments demonstrate that:

1. Level spacing statistics – The spacings of Riemann zeta function zeros follow the GUE distribution, matching eigenvalues of random Hermitian matrices.
2. Transition matrix elements – Hydrogen atom $1s \to np$ oscillator strengths are strongly correlated with the prime density $1/\ln n$, as predicted by the projection of the coupling matrix $K(x)=\sqrt{2x\ln x}$.
3. Information purity transition – As information purity $p_{\mathrm{ID}}$ decreases from 1 to 0, the level spacing distribution continuously transitions from Wigner (GUE) to Poisson, demonstrating the decoherence effect.

These experiments quantitatively confirm key predictions of the Information Dynamics framework.

Repository structure

ID_HilbertPolya/
├── README.md
├── requirements.txt
├── level_spacing/                      # Experiment 1: GUE vs Riemann zeros
│   ├── gue_vs_riemann.py
│   └── level_spacing_comparison.png
├── transition_matrix/                  # Experiment 2: Transition vs prime density
│   ├── transition_vs_prime.py
│   ├── transition_matrix_elements&residual_plot.png
└── purity_transition/                  # Experiment 3: p_ID effect on level statistics
    ├── purity_transition.py
    ├── purity_transition.png
    └── ks_vs_purity.png

Getting started

1. Clone the repository

git clone https://github.com/hkaiopen/ID_HilbertPolya.git
cd ID_HilbertPolya

2. Install dependencies

pip install -r requirements.txt

Dependencies: numpy, scipy, matplotlib.

3. Run experiments

Each experiment is self-contained. Run the scripts from the respective subdirectories:

python level_spacing/gue_vs_riemann.py
python transition_matrix/transition_vs_prime.py
python purity_transition/purity_transition.py

Each script will:

• Print statistical results (KS statistic, Pearson correlation, fitted parameters).
• Generate and save the corresponding figures.

Experiment details

1. Level spacing statistics (level_spacing/)

Compares the nearest-neighbor spacing distribution of GUE random matrices with that of Riemann zeta function zeros (simulated). The KS test confirms they are statistically indistinguishable.

Key output: KS statistic ≈ 0.045, p-value ≈ 0.816 → consistent with GUE.

2. Transition matrix elements vs prime density (transition_matrix/)

Computes hydrogen $1s\to np$ oscillator strengths (Bethe–Salpeter formula) and correlates them with prime density $1/\ln n$. Pearson correlation ≈ 0.8366 (p ≈ 1.57e-8), confirming strong positive correlation.

Key output: Scatter plot with regression line and 95% confidence ellipse, plus residual plot.

3. Information purity transition (purity_transition/)

Generates mixed spectra as a function of information purity $p_{\mathrm{ID}}$. At $p_{\mathrm{ID}}=1$ (pure GUE), spacing follows Wigner; at $p_{\mathrm{ID}}=0$ (classical), follows Poisson; intermediate values show continuous transition. KS distance from GUE increases monotonically as $p_{\mathrm{ID}}$ decreases.

Key output: Six-panel histogram transition + KS vs $p_{\mathrm{ID}}$ plot.

Reproducibility

All scripts use fixed random seeds where applicable. The results are deterministic and can be reproduced exactly.

Citation

If you use this code in your research, please cite the accompanying paper:

K. Huang, “A Proof of the Hilbert–Pólya Conjecture within the Information Dynamics Framework,” Zenodo, 2026. DOI: 10.5281/zenodo.20560128

and the foundational Information Dynamics papers:

K. Huang et al., “Information Dynamics: From Quantum Mechanics to Information Superconductor,” Zenodo, 2026. DOI: 10.5281/zenodo.19413342

K. Huang, “Mathematical Principles of Information Dynamics,” Zenodo, 2026. DOI: 10.5281/zenodo.20432225

Author

Kai Huang – hkaiopen@foxmail.com – GitHub