Resolving the Hubble Tension through Scale-Dependent Expansion in the Fractal Cascade Framework

We propose that 𝐻0 is not a constant but a scale-dependent parameter reflecting the ongoing creation of new cascade levels as the universe expands. In the Fractal Mechanics framework, new quantum nodes (qubits) emerge at rate ̇𝑛 = 𝐻(𝑡)/ ln 𝜑 as the universe grows, contributing to energy density as non-relativistic matter. This leads to the prediction: 𝐻0(𝑧) = 𝐻0_base × √[1 + 𝛼/(1 + 𝑧)³] with 𝛼 = 0.197 ± 0.045. Fitting this single-parameter model to six independent 𝐻0 measurements spanning 𝑧 ≈ 0–1100, we obtain 𝜒²/dof = 0.73, 𝑝 = 0.569, and significance 4.36𝜎 for 𝛼 ≠ 0, reproducing both CMB and local 𝐻0 values simultaneously. The model predicts 𝐻0(𝑧 = 0.5) ≈ 69.6 km/s/Mpc and 𝐻0(𝑧 = 1.0) ≈ 68.5 km/s/Mpc, testable with DESI and Euclid within 2029. Keywords: Hubble constant, Hubble tension, dark energy, cosmic expansion, cosmological parameters