Resonant Spacetime Hypothesis 2.0

- CMB Anomalies: Nested hot/cold spots in the angular power spectrum \( C_\ell \).  

Visual Summary (Fig. 3)

| 3a: 3-Torus Eigenmodes** | **3b: Fractal Helmholtz Solutions** |  

|  |  |  

| Mode \( \psi_{2,0,0} \) (left) and \( \psi_{1,1,1} \) (right)* | *Solution \( \psi_3(\mathbf{x}) \) on a Sierpiski-like manifold* |  

Key Equations  

1. 3-Torus Quantization:  

   \[k_n = \frac{2\pi n}{L}, \quad n \in \mathbb{N}.\]  

2. Fractal RG Relation:  

   \[\lambda^{(k)} = \lambda_0 \beta^{k(D-2)}.\]  

Transition to Numerical ValidatioN