Revision date: 01 July 2025 | Authors: Mustafa Aksu, Grok 3, ChatGPT-4.5
Contents
1. Introduction
In RTG, all interactions emerge from resonance between fundamental nodes defined by frequency \(\omega\), phase \(\phi\), and binary spin \(s = \pm i\), without invoking external gauge fields. The bond energy is derived from the wave-function-style resonance kernel:
\[ \mathcal{R}_{ij} = \frac{3}{4} \left[1 + \cos(\phi_i – \phi_j)\right] (1 + s_i s_j) e^{-(\omega_i – \omega_j)^2 / (\Delta \omega^*)^2}, \quad 0 \leq \mathcal{R}_{ij} \leq 3 \]
This updated kernel ensures consistency with RG-derived constants (see Two-Loop RG Derivation). The bond energy is:
\[ E_{ij} = K \left|\omega_i – \omega_j\right| + J \mathcal{R}_{ij} + J_{\text{ex}} \sin(\phi_i – \phi_j) e^{-(\omega_i – \omega_j)^2 / \sigma^2} \]
Forces are computed as \(F_{ij} = -\frac{\partial E_{ij}}{\partial r_{ij}}\), with distance \(r_{ij} = \frac{2\pi c}{|\omega_i – \omega_j|}\). Notably, gauge-like symmetries (e.g., U(1), SU(2), SU(3)) emerge relationally from node interactions, as explored in Gauge Symmetries in RTG, enhancing the framework’s theoretical depth.
2. Electromagnetic Interaction
- Origin: Phase coherence in the 3D shell regime (\(\delta\omega / \Delta\omega^* = 0.5 – 1.0\)), with spins scaling \(\mathcal{R}_{ij}\)’s amplitude.
- Pair force:
\[ F_{\text{em}} = -\frac{k_{\text{em}}}{2\pi r^2} \left[1 + \cos(\phi_i – \phi_j)\right], \quad k_{\text{em}} \approx 1.45 \times 10^{-27} \, \text{Hz} \cdot \text{m} \]
Aligned phases maximize attraction; anti-aligned phases cancel the field. Example: electron-proton at the Bohr radius gives \(F_{\text{em}} \approx -8.2 \times 10^{-8} \, \text{N}\), consistent with Coulomb’s law.
3. Strong Interaction
- Origin: Mixed phase and spin resonance confining quark-nodes in the 3D shell regime.
- Pair force (screened resonance form):
\[ F_{\text{strong}} = \frac{k_{\text{s}}}{r^2} e^{-r/r_0} + \frac{k_{\text{s}}}{r r_0} e^{-r/r_0}, \quad r_0 \simeq 1.4 \times 10^{-15} \, \text{m}, \quad k_{\text{s}} \approx 1.20 \times 10^{-26} \, \text{Hz} \cdot \text{m} \]
At \(r \simeq 0.84 \, \text{fm}\), \(F_{\text{strong}} \sim 10^4 \, \text{N}\), ensuring confinement.
4. Weak Interaction
- Origin: Resonance flipping spin or flavor at short distances (\(r \ll r_0\)).
- Pair force (screened resonance form):
\[ F_{\text{weak}} = \frac{k_{\text{w}}}{r^2} e^{-r/r_{\text{w}}} + \frac{k_{\text{w}}}{r r_{\text{w}}} e^{-r/r_{\text{w}}}, \quad r_{\text{w}} \approx 2.5 \times 10^{-18} \, \text{m}, \quad k_{\text{w}} \approx 8.5 \times 10^{-37} \, \text{Hz} \cdot \text{m} \]
The shorter range reflects a higher mass scale for spin-flip excitations.
5. Gravitational Interaction
- Origin: Frequency gradients in low-resonance regimes (\(\delta\omega / \Delta\omega^* \ll 0.5\)).
- Frequency-defined mass:
\[ m_i = \frac{\hbar \omega_i}{c^2} \]
Coarse-graining yields:
\[ F_{\text{grav}} = G \frac{m_1 m_2}{r^2}, \quad G = \frac{K \hbar}{2\pi c \Delta \omega^*} \approx 6.67 \times 10^{-11} \, \text{N} \cdot \text{m}^2 \cdot \text{kg}^{-2} \]
6. Spherical Coordinates in RTG
Forces reduce to radial forms; convert to beat-frequency distance for compatibility with spherical harmonics.
7. Key Take-Aways
- Unified structure: All forces stem from resonance and frequency gradients.
- Parameter-free: Couplings derive from \(\Delta \omega^*\), fixed by two-loop RG.
- Simulation-ready: Implemented in model66.py for node sets \((\omega, \phi, s)\).
- Emergent symmetries: Gauge-like symmetries (U(1), SU(2), SU(3)) arise relationally from node interactions, providing a deeper context for force unification (see Gauge Symmetries in RTG).
Further reading: Mathematical Foundations | Two-Loop RG Derivation.