The Crossover Phenomenon in Hardy-Littlewood Goldbach Formula - Computational Evidence Across Five Orders of Magnitude

This dataset accompanies the research paper "The Crossover Phenomenon in Hardy-Littlewood Goldbach Formula: Computational Evidence of Scale-Dependent Performance and Asymptotic Dominance" by Ruqing Chen (2026). DATASET OVERVIEW:The dataset contains 21,511 strategically sampled verification points spanning five orders of magnitude (N = 10³ to N = 10⁸), documenting a fundamental scale-dependent performance crossover between the classical Hardy-Littlewood series expansion and logarithmic integral formulation for predicting Goldbach representations. KEY FINDINGS DOCUMENTED:1. Critical crossover phenomenon at N ≈ 10⁵ where optimal method transitions from series to integral2. Three distinct computational regimes with different error characteristics3. Peak accuracy advantage of 843-fold at favorable large-scale points (N ≈ 5.5×10⁷)4. Non-monotonic variation in advantage ratios (2-843×) revealing complex arithmetic resonances in asymptotic error structure5. Factor-of-2 correction in counting methodology (ordered pairs vs unordered) FILES INCLUDED:- COMPLETE_DATASET_1k_to_100M_FINAL.csv (2.1 MB): Full dataset with 21,511 points- THE_FINAL_MASTERPIECE.png (6.1 MB): Main 7-panel visualization (300 DPI)- THE_FINAL_MASTERPIECE_highres.png (17 MB): High-resolution version (600 DPI)- README_FOR_ZENODO.txt: Complete documentation METHODOLOGY:- Counting method: Ordered-pair Goldbach representations (consistent with Hardy-Littlewood circle method)- Prime generation: Sieve of Eratosthenes to 100M (5.76M primes)- Series: 4th-order expansion in 1/log(N)- Integral: Adaptive Gaussian quadrature (limit=200)- Sampling: Stratified strategy across logarithmic scales- Validation: Cross-checked against OEIS A006307 DATA FORMAT (CSV columns):1. N: Even integer tested2. Series_Bias: (Series_prediction - Actual) / Actual3. Integral_Bias: (Integral_prediction - Actual) / Actual4. Abs_Bias_Series: |Series_Bias|5. Abs_Bias_Integral: |Integral_Bias| USAGE:This dataset enables reproduction of all results in the paper and supports further research on Hardy-Littlewood asymptotic formulas, scale-dependent numerical methods, and arithmetic properties of Goldbach representations. CITATION:Chen, R. (2026). The Crossover Phenomenon in Hardy-Littlewood Goldbach Formula: Computational Evidence of Scale-Dependent Performance and Asymptotic Dominance. [Journal to be added]. Dataset: https://doi.org/10.5281/zenodo.18123132 CONTACT:Ruqing ChenGUT Geoservice Inc.Montreal, Quebec, CanadaEmail: ruqing@hotmail.com CODE AVAILABILITY:Analysis code available at: https://github.com/Ruqing1963/goldbach-crossover-phenomenon