Resonant Spacetime Hypothesis 2.0

   \[\Box \delta g_{\mu\nu} + \kappa \partial_\mu \partial_\nu \mathcal{R}_{\text{boundary}} = 0,\]  

showing exponential damping of high-frequency instabilities (unlike inflation's tachyonic modes).  

Fractal Spacetime Manifold 

We model the early universe as a Sierpiski-like 3-manifold with Hausdorff dimension \(D \approx 2.7\), matching the observed cosmic web.  

Key Equations

1. Fractal Laplacian

 \[\Delta_F \psi_n \equiv \lim_{r \to 0} \frac{1}{r^D} \int_{B_r} \psi_n(x') \, d\mu(x') = \lambda_n \psi_n,\]  

where \(d\mu\) is the Hausdorff measure. Eigenvalues \(\lambda_n\) scale as \(n^{D/3}\) (not \(n^2\)), explaining **log-periodic galaxy clustering**.  

2. Spiral Harmonics

   Solutions to \(\Delta_F \psi_n = \lambda_n \psi_n\) in spherical coordinates take the form:  

   \[ \psi_{nlm}(r, \theta, \phi) \propto r^{-\beta} e^{im\phi + \beta \ln r} Y_{lm}(\theta), \]