The Prime Synchronization Theorem: An Empirical Scaling Law for Goldbach Partitions and Kuramoto Dynamics

This research presents an empirical framework linking additive number theory with non-linear dynamics. Specifically, it explores the relationship between the distribution of Goldbach prime partitions and the synchronization thresholds ($\kappa_c$) in networks of coupled oscillators governed by the Kuramoto model.

Methodology:

The study utilizes numerical simulations where the natural frequencies of oscillators are derived from the logarithms of primes within Goldbach pairs for a given even number $N$. Through extensive computational runs, we identify a stable, universal scaling law that governs the transition from chaos to phase-locked synchronization.

Key Findings:

Numerical verification across the tested range indicates a near-perfect correlation ($R^2 \approx 0.9999$) for the proposed scaling relationship:

where $\Gamma(N)$ represents the Goldbach rank (the number of prime partitions). This result suggests that prime number distributions possess inherent physical properties that can be mapped to spectral gaps in complex networks.

Applications:

The "Nedelchev Hypothesis" offers potential new methodologies for:

• Information Security: Analysis of prime-based synchronization for cryptographic protocols.

• Biomedical Engineering: Modeling pathological synchronization in neural networks (e.g., epilepsy).

• Power Grid Stability: Using arithmetical-physical models to predict phase transitions in energy networks.

Note on Status:

This work is submitted as an Empirical Hypothesis. It provides numerical evidence and a computational framework for further theoretical investigation. The author invites collaboration from the mathematical and physics communities for formal analytical proof and experimental laboratory validation.

Keywords:

Goldbach Conjecture, Kuramoto Model, Prime Numbers, Synchronization, Non-linear Dynamics, Computational Physics.

Source Code and Simulations: https://github.com/icobug/prime-synchronization-theorem