Context
The current RTG simulations use a cubic \(L^3\) lattice, introducing artificial Cartesian symmetries. We propose a spherical lattice structure to reflect radial and isotropic interactions, aligned with \(\Delta\omega^* = (1.45 \pm 0.08) \times 10^{23} \, \text{s}^{-1}\).
Effective Potential Foundation
The effective potential, derived from node resonance, is: \[ V(r) = \sigma r – \frac{\kappa’}{r} \]
Where \(\sigma = \Delta\omega^* \cdot a \approx 1.45 \times 10^{23} \, \text{s}^{-1} \cdot 2.066 \times 10^{-16} \, \text{m} \approx 0.03 \, \text{J/m}\), and \(\kappa’ = \hbar c / (2\pi a) \approx 2.43 \times 10^{-8} \, \text{J·m}\) (with \(a = c / \Delta\omega^* \approx 2.066 \times 10^{-16} \, \text{m}\)).
Note: All energy units in \(V(r)\) are expressed in J unless explicitly converted to GeV.
Key Goals
- Replace Cubic Lattice with Spherical Shell Lattice
Define nodes on \(N_r = 10\) concentric shells with \(\Delta r = 2.066 \times 10^{-16} \, \text{m}\) (Ten shells provide ~1.5 fm resolution while keeping matrix sizes tractable for 3D tensor contractions), using a Fibonacci grid (\(\theta_k = \arccos(1 – 2k/N)\), \(\phi_l = 2\pi l / N_\phi\)) for \(\sim 100\) nodes per shell. - Reformulate Lattice Hamiltonian in Spherical Coordinates
Incorporate volume elements: \[ H = \sum_{n,m} \mathcal{R}_{nm} \cdot \frac{1}{(\Delta r)^2} \cdot r_n^2 \sin\theta_{nm} \, \Delta r \] where \(\mathcal{R}_{nm} = \frac{3}{4} [1 + \cos(\phi_n – \phi_m)] (1 + s_n s_m) e^{-(\omega_n – \omega_m)^2 / (\Delta\omega^*)^2}\). - Redefine Path Integral and Continuum Action
Use spherical integration: \[ S = \int_0^\infty \int_0^\pi \int_0^{2\pi} \mathcal{L}(r, \theta, \phi) \, r^2 \sin\theta \, d\phi \, d\theta \, dr \] with \(\mathcal{L} = \frac{1}{2} (\partial_r \phi)^2 + \frac{1}{r^2} (\partial_\theta \phi)^2 + \frac{1}{r^2 \sin^2\theta} (\partial_\phi \phi)^2 – V(r) \cdot \langle \mathcal{R} \rangle\), and \(\langle \mathcal{R} \rangle \approx 1 \pm 0.1\). - Revise CHSH and Decoherence Simulations
Test noise (\(\pm 1–7 \times 10^{22} \, \text{s}^{-1}\)) on \(r\) and \(\theta, \phi\), targeting CHSH drop to \(0.725 \pm 0.04\) at \(\delta\omega = 0.5 \Delta\omega^*\). Use spherical harmonics \(Y_{lm}\)-based projections for tangent-plane measurements, defining Alice/Bob angles via \(Y_{1m}\) projections, to probe radial vs. angular decoherence asymmetries. - Compute Binding Energies and Mass Spectra
Solve \(-\frac{\hbar^2}{2m} \frac{d^2\psi}{dr^2} + V(r)\psi = E\psi\) with finite differences, estimating binding energy \(\sim 0.89 \, \text{GeV}\), compared to 8 MeV/nucleon (proton/neutron) and 28 MeV (alpha particle), with ±0.05 GeV from Monte Carlo variance. - Compare with QCD and Empirical Data
Match \(\sigma \approx 0.89 \, \text{GeV/fm} \approx 0.197 \, \text{J/m}\) and \(\kappa’ \approx 0.2 \, \text{GeV·fm} \approx 3.2 \times 10^{-8} \, \text{J·m}\), adjusting \(a\) and \(\kappa’\).
Future option: Explore embedding the spherical lattice in a curved (e.g., hyperbolic or AdS-like) 3D volume with \(k = -1\) curvature to test cosmological predictions.
Timeline
- Week 1: Develop spherical lattice code
- Week 2–3: Reformulate Hamiltonian and action
- Month 2: Run CHSH simulations
- Month 3: Compute binding energies
- Month 4: Compare with QCD, finalize document
Deliverables
- Spherical lattice generator code (Python, outputs 1000 nodes using Mayavi visualizer)
- Hamiltonian/action derivation (LaTeX, with \(\mathcal{R}_{ij}\))
- CHSH-vs-noise simulations (plot CHSH vs. \(\sigma\), \(0.725 \pm 0.04\) target; see also: CHSH drop simulation)
- Effective potential plots (show \(V(r)\) fit to 0.89 GeV/fm)
- Binding energy calculator (output \(0.89 \pm 0.05 \, \text{GeV}\))
- HTML document with embedded LaTeX, including JSON config files for reproducibility
- Spherical lattice schematic (showing shell radius, node distribution, radial vs. angular links, and sample CHSH pair angles, pending)
Conclusion
This rethinking of RTG’s lattice from cubic to spherical structures aligns its mathematical underpinnings with physical intuition—favoring isotropy, radial decay, and angular momentum quantization. It refines potential modeling, improves CHSH simulation fidelity, and provides a platform for high-precision nuclear and cosmological predictions.
| Parameter | Symbol | Typical Value | Units | Notes |
|---|---|---|---|---|
| Lattice spacing | \(\Delta r\) | \(2.066 \times 10^{-16}\) | m | Matches \(\Delta\omega^* / c\) |
| String tension | \(\sigma\) | 0.197 | J/m or GeV/fm | QCD-matched |
| Short-range term | \(\kappa’\) | \(3.2 \times 10^{-8}\) | J·m or GeV·fm | Fit from potential |
| Shell count | \(N_r\) | 10 | – | ~1.5 fm resolution |
| Nodes per shell | – | 100 | – | Fibonacci or HEALPix |
| Decoherence noise | \(\delta\omega\) | up to \(0.5 \Delta\omega^*\) | s⁻¹ | For CHSH loss |
Created: June 26, 2025 · Authors: Mustafa Aksu, Grok 3, ChatGPT-4.5