# The Nedelchev Structural Law: Prime Synchronization & Spectral Stability This research dataset and software framework formalize the **Nedelchev Structural Law**, a fundamental discovery linking Additive Number Theory with the synchronization dynamics of non-linear systems. The project provides a mathematical and physical bridge between Goldbach partitions and the Kuramoto model, proving that arithmetic structures dictate the stability of complex networks. ### Core Scientific Discoveries: * **The Nedelchev Invariant (Spectral Stability):** We demonstrate that the Goldbach adjacency matrix possesses a scale-invariant spectral radius ($\lambda_{max} = 1.000$). This structural property ensures that the network's internal stability remains constant regardless of the arithmetic scale ($N$). * **Dynamical Scaling Law:** Through high-resolution simulations, we established that the critical coupling threshold ($\kappa_c$) required for global resonance is governed by the spectral properties of the Goldbach network, providing a universal constant for synchronization. * **The Stability Gap:** Comparative benchmarks against randomized topologies prove that this resonance is not a result of node density, but a unique product of the Goldbach arithmetic symmetry. ### From Local to Global Synchronization: * **Localized Resonance (The Nedelchev Effect):** The emergence of order begins at the "arithmetic bridge" level, where Goldbach pairs ($p_i + p_j = N$) form the first stable resonant clusters. This holds true both with and without auxiliary constraints (the "crutches" logic). * **Global Phase Transition:** By applying the Scaling Law, the system overcomes interference, leading to a stable global order parameter across the entire prime spectrum. ### Target Applications & Interdisciplinary Impact: The Nedelchev Law provides a new engine for optimization in several cutting-edge fields: * **Telecommunications (6G/7G):** Interference filtering and phase-locking in Massive MIMO systems using prime-based distribution. * **Neuromorphic Engineering:** Modeling synchronization states and phase-locking in artificial neural networks. * **Cybersecurity:** Development of structural encryption keys based on Goldbach weights. * **Swarm Robotics:** Decentralized coordination of autonomous agents through localized arithmetic resonance. ### Dataset & Software Contents: 1. **`goldbach_spectral_core.py`**: The clean, minimal algorithm for calculating the Goldbach matrix and its spectral properties (The "Heart" of the Law). 2. **`prime_sync_full_simulation.py`**: High-resolution dynamical simulations using Kuramoto and Stuart-Landau oscillators. 3. **`Nedelchev_Universal_Prime_Sync_Law.pdf`**: The final technical manuscript (Version 3) as submitted to the Journal of Mathematical Physics (JMP). 4. **`results_data.csv`**: Raw experimental data covering scales from $N=200$ to $N=1000$. ### Conclusion: The Nedelchev Law identifies prime numbers not as isolated entities, but as the **"structural skeleton"** of resonant systems. This framework validates the hypothesis that arithmetic order is a precursor to physical stability in high-entropy environments. Source Code and Simulations: https://github.com/icobug/prime-synchronization-theorem