Certified First 1,000 Zeros of the Riemann Zeta Function

This repository contains the final certified dataset of the first 1,000 nontrivial zeros of the Riemann zeta function
[ \zeta!\left(\tfrac12 + it\right), ] produced using a dual-evaluator method, a hexagonal argument-principle contour, strict Krawczyk uniqueness isolation, and an automatic refinement pipeline that corrects any multi-zero contours or weak contraction regions.

The goal is to provide a clean, reproducible, and verifiable reference dataset for research, analysis, numerical experiments, or independent verification.

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Contents

data/ zeros_1_to_1000_final.json # Final certified dataset

scripts/ unified_zeta_framework_v2.5.py # Full certification engine zero_analysis_and_scaling.py # Spacing analysis + stability metrics merge_zero_certs.py # Utility to merge per-range JSONs These are the only files needed to reproduce the dataset from scratch.

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Method Summary

1. Dual ζ Evaluators (Consistency Check)

Each contour evaluation uses two independent ζ functions:

• mpmath.zeta(s)
• Dirichlet η-series partial summation

The maximum disagreement (max_evaluator_diff_on_contour) confirms numerical stability.

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2. Hexagonal Contour + Argument Principle

Each zero is enclosed inside a hexagonal contour.
Winding numbers are computed for both evaluators:

• wA_int = 1
• wB_int = 1

Any contour that encloses more than one zero (w = 2) automatically triggers refinement.

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3. Wavelength-Limited Sampling

Contour sampling is governed by a Nyquist-style bound using the local phase speed of
[ \frac{\zeta'}{\zeta}, ] ensuring correct resolution of phase jumps and preventing aliasing.

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4. Krawczyk Uniqueness Test

Each zero is validated with a 2D Krawczyk operator, verifying:

• β < 1 (contraction)
• ρ ≤ r_box (isolation)
• exactly one zero exists in the box

If any condition fails, refinement is automatically applied.

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5. Automatic Refinement

For any zero where the contour or Krawczyk test fails:

• contour radius is reduced
• Krawczyk box is reduced
• evaluator agreement rechecked
• the full certification cycle repeats

This continues until the zero satisfies:

• wA_int = wB_int = 1
• β safely below 1
• ρ ≤ r_box
• evaluator agreement is stable

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Dataset

The file: data/zeros_1_to_1000_final.json

contains, for each zero:

• zero_index
• approx_zero.t (the height)
• modulus bounds (min_abs_zeta_on_contour)
• evaluator agreement (max_evaluator_diff_on_contour)
• winding numbers (wA, wB, wA_int, wB_int)
• full Krawczyk isolation fields (beta, rho, r_box, success)
• contour geometry parameters

This dataset is ready for:

• visualization
• GUE spacing experiments
• analytic number theory research
• replication or extension

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Usage

Certify a new range:

python scripts/unified_zeta_framework_v2.5.py --range 101 150

Analyze spacing statistics:
python scripts/zero_analysis_and_scaling.py --analyze data/zeros_1_to_1000_final.json
Merge multiple certificate files:
python scripts/merge_zero_certs.py --output merged.json zeros_*.json


License

MIT License — free for academic, commercial, and independent research use.