Varying Gravity from a Modified Fractional Model: Observational Constraints and Slow–Fast Dynamics

We investigate a fractional gravity model in which both the Hubble parameter and the gravitational constant evolve dynamically due to fractional renormalization-group effects. The model incorporates a scalar field coupled to a time-varying G, generating nonlocal corrections characteristic of fractional-action cosmology. Analytical and numerical solutions reveal oscillatory regimes, cyclic phases, and rapid variations with implications for BBN and early-universe evolution. A robust numerical framework is developed to integrate the regularized system and compare the resulting H(z) evolution with observational data from the Hubble parameter, baryon acoustic oscillations, Type Ia supernovae, gravitational lensing, and black-hole shadows, thereby enabling a consistent reconstruction of cosmographic quantities. A Bayesian analysis shows that the fractional model with μ=0 is the only statistically viable variant. The inferred Hubble parameter is stable across models (h≃0.72), while the fractional parameters are significantly better constrained in the μ=0 case (α=1.20−0.14+0.25, ζ=0.43−0.29+0.39). The dynamical sector yields m=30.8−20.9+28.0 and Γ=108.3±1.1, leading to a positive discriminant and a relaxation timescale τ+=219−162+1980 Gyr, confirming an overdamped regime. Although the μ=0 model attains a slightly lower χmin2 than ΛCDM, the BIC strongly favors ΛCDM due to its smaller parameter space. Overall, the model reproduces late-time acceleration and mimics ΛCDM while introducing distinctive cosmographic signatures. The dynamical systems analysis clarifies the stability structure and parameter dependence, indicating that fractional nonlocal corrections may offer new pathways toward addressing the H0 and S8 tensions.