Resonant Spacetime Hypothesis 2.0

\[\Delta_F \psi_n + \lambda_n \psi_n = 0,\]  

where \( \Delta_F \) is the fractal Laplacian, defined via the Hausdorff measure \( \mu \):  

\[\Delta_F \psi(\mathbf{x}) = \lim_{r \to 0} \frac{1}{r^D} \int_{B_r(\mathbf{x})} \left[ \psi(\mathbf{x}') - \psi(\mathbf{x}) \right] d\mu(\mathbf{x}').\]  

Self-Similar Solutions via Renormalization Group (RG) 

1. Scale Invariance: The fractal's recursive structure allows solutions to satisfy:  

   \[\psi_n(\mathbf{x}) = \alpha \psi_n(\mathbf{x}/\beta),\]  

where \( \alpha \) and \( \beta \) are scaling factors.  

2. RG Flow: Iteratively coarse-grain the manifold by a factor \( \beta \), deriving a recursion relation for \( \lambda_n \):  \[\lambda^{(k+1)} = \beta^{D-2} \lambda^{(k)},\]  

leading to a geometric spectrum \( \lambda_n \propto \beta^{n(D-2)} \).  

Observable Implications  

- Log-Periodic Clustering: Galaxy separations \( r_k \propto \beta^{k} \).