Resonant Spacetime Hypothesis 2.0

- Spiral harmonics: \\( \psi_{nlm} \propto e^{im\phi + \beta \ln r} \\).  

3. Mathematical Derivations

- Eigenmode Solutions:  

- Quantized spectra for 3-torus (\\( k_n = 2\pi n/L \\)) and dodecahedral space (icosahedral harmonics).  

- Mode coupling via perturbation theory: \\( \delta \rho \sim \sum_{n,m} A_n A_m \psi_n \psi_m \\).  

- Fractal Helmholtz Equation:  

- Proof of self-similar solutions using renormalization group methods.  

4. Numerical Simulations  

- Methods:  

-Code: Custom C++ solver with MPI parallelization for 3D eigenmode evolution.  

-Parameters: