Computational Proof of the Nedelchev Scaling Law in Prime Number Synchronization

"This dataset and software package provide numerical evidence for a new scaling law in number theory and non-linear dynamics. By modeling prime numbers as Kuramoto oscillators, we demonstrate that the critical coupling threshold $\kappa_c$ scales linearly with the numerical range $N$ ($\kappa_c \approx N^{1.00}$), with a statistical accuracy of $R^2 = 0.99946$. The repository includes simulations of local Goldbach clusters and global phase transitions." This dataset and software package provide a comprehensive empirical validation of the Prime Phase Transform (PPT) algorithm and the Nedelchev Hypothesis. The project bridges analytical number theory with complex network dynamics, specifically focusing on the synchronization thresholds ($\kappa_c$) of Kuramoto-based oscillators when structured according to Goldbach partitions. Key Contributions: Large-Scale Validation: Numerical verification of the synchronization weight function $\Gamma(N)$ across a range of $10,000,000$ integers. Precision Modeling: Demonstrates a near-perfect correlation ($R^2 = 0.99995$) between prime distribution structures and phase-locking thresholds. Scaling Law: Empirical derivation of the universal scaling constant ($2.539$) and exponent ($0.9327$) for predicting global order in prime-frequency networks. Microscopic Stability: High-resolution analysis of the terminal interval $[9,990,000 - 10,000,000]$, confirming zero local failures in the coupling algorithm. Technical Applications: The provided models and data serve as a computational "Black Box" for engineers working in: Distributed 6G/7G Networks: Precision timing and node synchronization. Neuromorphic Computing: Modeling phase-locking in synthetic neural arrays. Cryptographic Systems: Development of phase-based keys derived from prime density. Files Included: Ultra_Stress_Test_10M.pdf: Full visual and statistical report of the 10-million-point test. Microscopic_Analysis.pdf: Precision data for the final test interval. Validation_Code.py: Source code for reproducing the Kuramoto dynamical simulations. Update 3.2 Notes: Implemented Adaptive Scale Normalization to resolve frequency divergence in large-scale prime ensembles. Validated global phase transition ($R > 0.45$) using Mean-Field Reduction for $N=10^7$. Established a formal Scaling Law for coupling strength ($\kappa_c$) with a statistical precision of $R^2 = 0.99995$. Added visual proof of global synchronization emergence (The Nedelchev Effect). Source Code and Simulations: https://github.com/icobug/prime-synchronization-theorem