Conformal Dark Sector and LQG Spin Coarse-Graining: Two Roads to the Same Running Gravitational Constant (Paper IV)

We demonstrate that Loop Quantum Gravity (LQG) spin network coarse-graining (Papers I–III) and classical conformal invariance with Coleman–Weinberg (CW) radiative symmetry breaking independently converge to the same functional form for the running effective gravitational constant: Gₑff (z) /GN = 1 − ε₀/1+ (z/zc) ⁿ. We identify the precise theoretical dictionary between the two frameworks and introduce a dynamical dark matter mass mₑff (z) ∝ (1+z) ^β from the conformal sector, providing a new handle on the Hubble tension absent from Papers I–III. A formally converged 6-parameter MCMC (64 walkers × 8000 steps, τₘax = 106, Nₑff = 4818, acceptance rate 0. 434) targeting Planck 2018 CMB lensing, DES Y6, KiDS Legacy, BOSS DR12 BAO, SH0ES H0, and FIRAS μ-distortion yields: ε₀ = 0. 241 ± 0. 12, log₁₀ (zc) = 0. 124 ± 0. 38, n = 2. 041 ± 0. 48, β = 0. 090 ± 0. 038, γ₀ = 5. 06 ± 3. 41, log₁₀ (g/λ) = 0. 283 ± 0. 38. The critical exponent n = 2. 041 ± 0. 48 is consistent with the Z2 universality class predicted analytically — confirmed across four independent MCMC runs. At best fit (ε₀=0. 356, β=0. 088), the model reduces the S8 tension from 4. 38σ to 3. 00σ (DES Y6) and from 2. 60σ to 1. 31σ (KiDS Legacy), and the Hubble tension from 5. 60σ to 4. 10σ (SH0ES). The Bayes factor ln (B₁₂) ≈ +9. 80 constitutes decisive evidence in favor of the Bio-Cubic model over ΛCDM on the joint S8+H0 dataset — the first such result in this series. We formalize the 'cubic universe' geometric intuition as a discrete conformal automaton and present the theoretical dictionary connecting the LQG and conformal descriptions.