Numerical Evidence for Linear Scaling Laws in Goldbach-induced Oscillator Networks

The Nedelchev Structural Law and Spectral Goldbach Prediction Method Hristo Nedelchev August 24, 2026 We present a unified spectral framework for analyzing Goldbach partitions, built upon the Nedelchev Structural Law: κc=σ(P)λmax⁡(P)∼Nα,α≈0.9985κc=λmax(P)σ(P)∼Nα,α≈0.9985 This law establishes that the prime distribution possesses a self-similar spectral structure, which is directly inherited by Goldbach partitions. Building on this foundation, we introduce a Spectral Goldbach Prediction Method that: Constructs a binary vector v(n)v(n) encoding all prime pairs (p,n−p)(p,n−p) Applies FFT to extract spectral entropy and amplitude Generates a predictive score S(n)S(n) that correlates with the true partition count g(n)g(n) Key results (validated at LIMIT = 2,000,000): DENSE champions: 100.00% enrichment (vs HL: 93.33%) SPARSE champions: 97.78% enrichment (vs HL: 6.67%) Correlation: 0.999978 (vs HL: 0.983356) Statistical significance: t=4.24t=4.24 (DENSE), t=50.21t=50.21 (SPARSE) The method significantly outperforms the classical Hardy–Littlewood approach, confirming that the Nedelchev Structural Law provides a complete spectral framework for analyzing additive prime structures. All results are fully reproducible using the open-source implementation provided. **Overview:** This dataset and associated source code provide a comprehensive numerical investigation into a novel class of complex networks where the coupling topology is governed by the additive properties of prime numbers (the Goldbach Conjecture). We demonstrate that such arithmetic structures lead to predictable and robust dynamical behaviors in non-linear systems. **Key Scientific Contributions:** * **The Nedelchev Scaling Law:** We report the discovery of a universal linear scaling relationship, $\kappa_c \propto N$, between the critical coupling strength ($\kappa_c$) required for global synchronization and the system size ($N$). * **Deterministic Topology:** Unlike stochastic or scale-free networks, the Goldbach-induced network is purely deterministic, constructed by linking nodes whose prime indices sum up to a specific even integer. * **Spectral Invariance:** Numerical verifications confirm that the spectral radius of the coupling operator remains constant ($\rho(G) = 1$), indicating that the synchronization threshold is driven by frequency dispersion rather than topological growth. * **The Global Bridge Effect:** Observation of a unique synchronization transition ("The Nedelchev Effect"), where the system reaches coherence through specific arithmetic pathways. **Contents of this Record:** 1. **Source Code:** Optimized Python/Jupyter scripts (Google Colab compatible) for simulating Kuramoto, Winfree, and Stuart-Landau models on Goldbach topologies. 2. **High-Resolution Figures:** Visualizations of the scaling law ($R^2 = 1.00000$) and phase transition dynamics. 3. **Technical Preprint:** A comprehensive document (PDF) detailing the theoretical framework and numerical results. **Significance:** This work bridges the gap between analytic number theory and statistical physics, offering a new perspective on how fundamental mathematical conjectures can dictate the collective behavior of physical systems. * **High-Density Scaling ($N=400$):** Extension of the model to large-scale systems, achieving near-perfect global synchronization ($R \approx 0.995$) by utilizing the full density of the Goldbach partition ensemble.* **Spectral Order Proof:** Numerical evidence through Level Spacing Distribution analysis, confirming that Goldbach-induced coupling follows an ordered Poissonian regime, fundamentally distinct from the Wigner-Dyson statistics of random matrices.* **The Nedelchev Bridge:** A topological architecture that successfully suppresses high frequency entropy in prime-valued oscillator systems.