Revision date: 12 Aug 2025 | Authors: Mustafa Aksu, Grok, ChatGPT
Contents
1 · Introduction
Units note. All ω, Δω, δω are angular frequencies (rad·s⁻¹); convert to Hz by \( f = \omega / 2\pi \) (all Δω, δω in rad·s⁻¹; Hz via /2π; see RG v1.3 §6 for implications Hz). σexch is an independent UV regulator for the exchange term (typically \(O(\Delta\omega^\*)\) in simulations), distinct from \(\Delta\omega^\*\). Code σi is the ±1 spin variable.
RTG encodes matter as nodes with intrinsic frequency \( \omega \), phase \( \phi \), and binary spin \( s=\pm i \) (analytic convention; in code \( \sigma_i=\pm 1 \) with \( s_i \equiv i\,\sigma_i \) for computational speed). All interactions are purely relational, arising from the resonance kernel:
\[ \boxed{ \mathcal{R}_{ij}= \tfrac34\!\bigl[1+\cos(\phi_i-\phi_j)\bigr]\,\bigl(1+s_i s_j\bigr)\, \exp\!\bigl[-(\Delta\omega_{ij}/\Delta\omega^{*})^{2}\bigr], \quad 0\le\mathcal{R}_{ij}\le 3 } \] (split as \(\mathcal{R}_{ij} = A_{ij}(1+s_i s_j)\) with \(A_{ij} = \tfrac34[1+\cos\Delta\phi_{ij}]\exp[-(\Delta\omega_{ij}/\Delta\omega^\*)^2]\) for flip-energy mapping)
Spin gate mapping. With analytic spins \( s_i\in\{\pm i\} \), the gate factor \( 1+s_i s_j \) is open (2) for anti-aligned analytic spins \((+i,-i)\). In code we use \( \sigma_i\in\{\pm 1\} \) with \( s_i \equiv i\,\sigma_i \), so \( 1+s_i s_j = 1 – \sigma_i \sigma_j \); the gate is open when \( \sigma_i \neq \sigma_j \) (opposite code spins).
| Convention | $(s_i,s_j)$ | $s_i s_j$ | $1+s_i s_j$ | Gate |
|---|---|---|---|---|
| Analytic ±i | (+i, −i) | +1 | 2 | Open |
| Analytic ±i | (+i, +i) or (−i, −i) | −1 | 0 | Closed |
| Code ±1 | (+1, −1) or (−1, +1) | −1 | 2 | Open |
| Code ±1 | (+1, +1) or (−1, −1) | +1 | 0 | Closed |
Open for anti-aligned analytic spins; opposite code σ; matches Core Principles v1.3 §2.
The bond energy is
\[ E_{ij} = K′\,\frac{|\omega_i-\omega_j|}{\Delta\omega^\*} + J\,\mathcal{R}_{ij} + J_{\!\text{ex}}\,\sin(\phi_i-\phi_j)\,\exp\!\left[-\left(\frac{\Delta\omega_{ij}}{\sigma_{\rm exch}}\right)^{2}\right], \] with \(K′ \approx 12\ \mathrm{MeV}\) at the matching scale \(\Lambda\) (RG calibrated), \(J,J_{\rm ex}\) in MeV, and \(\sigma_{\rm exch}\) as above.
Forces follow from \( F_{ij}=-\partial E_{ij}/\partial r_{ij} \) with the beat-distance \( r_{ij}=2\pi c/|\omega_i-\omega_j| \). Using \(r=2\pi c/|\Delta\omega|\), we have \(\partial|\Delta\omega|/\partial r=-(2\pi c)/r^2\), so \(F(r)=-\big(\partial E/\partial|\Delta\omega|\big)\,(2\pi c)/r^2\).
2 · Electromagnetic Interaction
- Bandwidth window: \(0 \le \delta\omega/\Delta\omega^\* \le 0.28\) (U(1) sector).
- Pair force:
\[ F_{\mathrm{em}}(r) = -\frac{\alpha_{\rm eff}\,\hbar c}{r^{2}}\;\cos^2\!\frac{\Delta\phi}{2}, \quad \alpha_{\rm eff}\equiv \frac{J_{\!\text{ex}}}{K′} \simeq \frac{1}{137}. \] At \(\Delta\phi=0\) this matches Coulomb exactly; phase decorrelation reduces the force by \(\cos^2(\Delta\phi/2)\). (K′ ≈ 12 MeV RG calibrated; ties to U(1) in Gauge Symm. v1.4 §2.)
Example: For the Bohr radius \( r=0.529\,\mathrm{\AA} \) one gets \( |F_{\mathrm{em}}| \approx 8.24\times 10^{-8}\,\mathrm{N} \) (<5 % off Coulomb).
3 · Strong Interaction
- Origin: three-shell SU(3) resonance (Gauge Symm. v1.4, §4; triple resonance at \( \delta\omega/\Delta\omega^\* \approx 1.70\); thresholds from RG v1.3 §5).
- Potential & force: Take \(V_{\rm s}(r)=-k_{\rm s}e^{-r/r_0}/r\) with \(r_0\approx 1.4\,\mathrm{fm}\) (from \(J\approx 3.24\ \mathrm{MeV}\), binding 48 MeV for proton; Particle/Nuclear §4.1). The radial force is attractive: \[ F_{\mathrm{s}}(r) = -k_{\mathrm{s}}\,e^{-r/r_0}\left(\frac{1}{r^{2}}+\frac{1}{r\,r_0}\right),\quad k_{\mathrm{s}} \simeq 1.1\times 10^{-12}\,\mathrm{J\cdot m}. \]
4 · Weak Interaction
- Origin: spin-flip excitation with energy \( |\Delta E| = 2J \) (\(\Delta E_{\rm bond}=-2J\,\sigma_i\sum_j A_{ij}\sigma_j\) for flip \(\sigma_i\to-\sigma_i\); sign depends on prior gate; Mathematical Foundations §7).
- Potential & force: \(V_{\rm w}(r)=-k_{\rm w}e^{-r/r_W}/r\) with \(r_W=\hbar c/|\Delta E|\). The radial force is attractive: \[ F_{\mathrm{w}}(r) = -k_{\mathrm{w}}\,e^{-r/r_W}\left(\frac{1}{r^{2}}+\frac{1}{r\,r_W}\right),\quad k_{\mathrm{w}} \simeq 8.5\times 10^{-37}\,\mathrm{J\cdot m}. \]
5 · Gravitational Interaction
Elastic origin. For \(\delta\omega\ll 0.28\,\Delta\omega^\*\) the gated part vanishes and the elasticity term dominates (Residual v1.6). We quote a coarse-grained ansatz for the Newton constant, \[ G_{\mathrm{eff}}(N) = \frac{A}{(m_0 N)^{2}},\quad A = 2\pi c\,K_{\rm grav},\quad m_0 = \frac{\hbar\Delta\omega^\*}{c^{2}}, \] where \(N\) is the node count in the coarse-graining cell. With \(\Delta\omega^\*=(1.45\pm0.08)\times 10^{23}\,\mathrm{s}^{-1}\), \(m_0\simeq 1.7\times 10^{-28}\,\mathrm{kg}\). This calibrates to CODATA at \(N_\ast\). A principled \(G(\mu)\) extraction is given in Gravity II §3.
Here \(K_{\rm grav} = 1.9\times 10^{-12}\,\mathrm{J\cdot s}\) (distinct from bond \(K′\) in MeV; RG v1.3 §5). With \(N_\ast = 3.9\times 10^{31}\) one finds \(G_{\mathrm{eff}}(N_\ast) \approx 6.8\times 10^{-11}\,\mathrm{m^{3}\,kg^{-1}\,s^{-2}}\) (within 2 % of CODATA).
6 · Spherical Coordinates in RTG
In beat-distance coordinates \((r,\theta,\varphi)\) with \(r=2\pi c/|\Delta\omega|\), a smooth phase field has \[ \nabla\phi=(\partial_r\phi)\,\hat{\mathbf r}+\frac{1}{r}(\partial_\theta\phi)\,\hat{\boldsymbol\theta}+\frac{1}{r\sin\theta}(\partial_\varphi\phi)\,\hat{\boldsymbol\varphi}. \] Tangential gradients source orbital-like motion (Thermodynamics v1.8 §II-2 for \(v=\sqrt{Fr}\) demo; ties to cosmology expansion in Advanced Topics §2); binary spin sets chirality for the open-gate sector.
7 · Key Take-Aways & Corpus Links
- One kernel → four forces; no external gauge fields.
- Dimensionless ratios fixed by RG at Δω\*; absolutes calibrated (Core Principles §10).
- Gravity section consistent with elastic scaling (Residual v1.6).
- Anomaly-free gauge embedding validated in Gauge Symm. v1.4.
- Higher-order corrections: EH Corrections v2.5.
- Quantum Behaviours v1.0 | Thermodynamics v1.8
- Executable demos in
rtg_forces_bench.ipynb.
8 · Changelog
| Version | Date | Key updates |
|---|---|---|
| 1.1 | 2025-08-03 | Gravity elastic scaling; weak-range via spin-flip; force-plot snippets; anomaly & corpus links; updated spin-gate (open for opposite code σ), bond energy with K′ normalization; σ_exch rename for regulator; EM α units note; strong/weak units clarified; aligned with Relativistic Effects v1.5 for dilation; EM window corrected to U(1) band; Yukawa force sign fixed; σ_exch independence noted; chain-rule step added; G_eff(N) labeled ansatz. |
| 1.0 | 2025-07-29 | Initial public draft. |