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:

1. 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 ($N$).

2. 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 $R^2 = 1.00000$.

3. 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