This project implements a frequency-based spectral model (HPO2) that reconstructs the prime density function π(x)/x using the first N Riemann zeta zeros as modal frequencies. Based on the theory of Non-Hermitian Spectral Networks (NHSN), this approach interprets Riemann zeros as frequency-locking resonance points in a dynamic spectral field.
We model the prime density structure as a modal frequency projection:
[ \rho(x) = \frac{1}{\log x} + \sum_{n=1}^{N} A_n \cos(t_n \log x + \theta_n) ]
Where:
- ( t_n ) = imaginary parts of the first N Riemann zeta zeros (ζ zeros)
- ( A_n ), ( \theta_n ) = optimized modal amplitudes and phases
- ρ(x) ≈ π(x)/x when parameters are optimized
| Feature | Traditional HPO | HPO2 / NHSN |
|---|---|---|
| ζ zeros | Static eigenvalues | Emergent modal resonances |
| Operator type | Hermitian (self-adjoint) | Non-Hermitian (asymmetric kernel) |
| Spacing | Linear / Regular | GUE-like, irregular |
| Information | Not transmittable | ψ-paths can encode structured info |
| Output | Real λₙ ~ tₙ² | √λₙ ~ tₙ + δ(x) |
- ✅ Loads prime density data π(x)/x via
sympy.primepi - ✅ Uses the first 15 Riemann ζ zeros as modal frequencies ( t_n )
- ✅ Optimizes ( A_n ), ( \theta_n ) using L-BFGS-B
- ✅ Computes modal projection ρ(x)
- ✅ Measures reconstruction accuracy via MSE (δ²)
- ✅ Outputs:
- Matplotlib plot of ρ(x) vs π(x)/x
- CSV table of modal parameters
- LaTeX table for reports
- StructureLang ψ-paragraph
| File | Description |
|---|---|
hpo2.py |
Main script to perform modal fitting |
modal_projection_parameters.csv |
Output CSV of ( t_n, A_n, \theta_n ) |
projection_result.png |
Visual comparison (if saved) |
README.md |
This file |
Zeta Zeros as Emergent Modes in a Noncommutative Spectral Network https://doi.org/10.5281/zenodo.15353621
pip install numpy matplotlib scipy sympy
python3 hpo2.py