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 $\mu=0$ is the only statistically viable variant. The inferred Hubble parameter is stable across models ($h\simeq 0.72$), while the fractional parameters are significantly better constrained in the $\mu=0$ case ($\alpha=1.20^{+0.25}_{-0.14}$, $\zeta=0.43^{+0.39}_{-0.29}$). The dynamical sector yields $m=30.8^{+28.0}_{-20.9}$ and $\Gamma=108.3\pm1.1$, leading to a positive discriminant and a well-determined relaxation timescale $\tau_{\rm rel}\simeq 9$ Gyr, confirming an overdamped regime. Although the $\mu=0$ model attains a slightly lower $\chi^2_{\min}$ than $\Lambda$CDM, the BIC strongly favors $\Lambda$CDM due to its smaller parameter space. Overall, the model reproduces late-time acceleration and mimics $\Lambda$CDM 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 $H_0$ and $S_8$ tensions.