Resonant Spacetime Hypothesis 2.0

- Sierpiski (1915) -- Fractal geometry foundations.  

- Calcagni (2012, *PRD*) -- Fractal field theory.  

- LIGO Collaboration (2023) -- Current GW sensitivity.  

CHAPTER 3. Mathematical Derivations  

Eigenmode Solutions

1. Quantized Spectra on a 3-Torus

For a spatially flat universe modeled as a 3-torus with side length \( L \), periodic boundary conditions enforce quantized wavevectors:  \[\mathbf{k}_n = \frac{2\pi}{L} \mathbf{n}, \quad \mathbf{n} \in \mathbb{Z}^3,\]  

where \( \mathbf{n} = (n_x, n_y, n_z) \) are integers. The eigenmodes of the Laplacian \( \Delta ) are plane waves:  

\[\psi_n(\mathbf{x}) = \frac{1}{\sqrt{L^3}} e^{i \mathbf{k}_n \cdot \mathbf{x}},\]  

with eigenvalues \( k_n^2 = \|\mathbf{k}_n\|^2 \). This discretization naturally explains hierarchical structure formation, as modes with \( n = |\mathbf{n}| \) correspond to comoving scales \( \lambda_n = L/n \).  

2. Dodecahedral Space and Icosahedral Harmonics