Fractal-Layered Holographic Cosmology: A Testable Framework for Hubble Tension, CMB Anomalies, and Early Structure Formation

P(k)k(3DH)P(k) \sim k^{-(3 - D_H)}

Interpretation:

If DH=2.0D_H = 2.0, a common observationally inferred value for large-scale structure:
P(k)k1P(k) \sim k^{-1}

For a truly homogeneous distribution (DH=3D_H = 3):
P(k)k0P(k) \sim k^0 (white noise, uncorrelated)

This scaling emerges without inflation, arising instead from renormalization group (RG) flows and topological coarse-graining across layered spacetime structures. It naturally incorporates:

Self-similarity across scales,

Void-dominated volume bias,

Deviations from isotropy and statistical homogeneity at scales above the homogeneity scale 150 Mpc\sim 150 \text{ Mpc}.

3. Key Differences and Observational Discriminants

4. Observational Testing

Potential discriminants include: