The Nedelchev Structural Law: Spectral Invariance and Dynamical Scaling in Goldbach-Partitioned Oscillator Networks

Overview 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 scales linearly with the system size ($\kappa_c \approx 2N$), with a statistical precision of $R^2 = 1.00000$. 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. Global Phase Transition: By applying adaptive scale normalization, the system overcomes the "Arithmetic Echo" interference, leading to a stable global order parameter ($R > 0.45$) 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 Contents: nedelchev_structural_law.py: Proof of spectral invariance. goldbach_vs_random_benchmark.py: Structural uniqueness validation. dynamical_scaling_v4.py: Kuramoto-based dynamical simulations. results_data.csv: Raw experimental data (Scales $N=200$ to $N=1000$). Nedelchev_Law_v5_Technical_Paper.pdf: Full technical derivation and formal proof. 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