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

# 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.

 

"This document provides a technical clarification regarding the Goldbach Interaction Operator $W_N$. It formally defines the operator as a partial permutation matrix, distinguishing it from standard graph-theoretic adjacency matrices¹. The note explains the structural origin of the constant spectral radius $\rho(W_N) = 1$ and clarifies how the observed linear scaling of the synchronization threshold $\kappa_c(N) \propto N$ relates to intrinsic frequency dispersion rather than operator growth²²²²². This ensures the mathematical consistency of the results reported in the main research³."