Emergent Time in a TAGC‑LQG‑RG Toy Model: Numerical Ensembles, Bifurcation Maps, and Sensitivity Analysis

We present a concise numerical study of emergent time in a TAGC‑LQG‑RG toy model, combining ensemble sampling, mesh convergence tests, bifurcation mapping and local sensitivity analysis. Using large ensembles and systematic mesh refinement, we obtain a robust adimensional estimate for the emergent time, t∗≈5.77×10−10t^* \approx 5.77\times10^{-10}, with a narrow ensemble spread and clear convergence as nstepsn_{\text{steps}} is increased. A coarse parameter sweep in (Γ0,ρ0)(\Gamma_0,\rho_0) reveals little variation of t∗t^* at the explored resolution, while targeted sensitivity experiments show a reproducible response of t∗t^* to multiplicative perturbations in ρ\rho. Independently, the data support a scaling relation for time dilation consistent with Δt/t∝ρ1/4\Delta t/t \propto \rho^{1/4}, which is used to produce order‑of‑magnitude physical estimates reported in the figures. All numerical procedures, diagnostics and provenance metadata are provided to ensure reproducibility; note that a subset of exploratory runs employed a documented fallback implementation of the time‑operator and are explicitly flagged in the archive. These results are intended as a transparent, reproducible numerical characterisation of the toy model and as a basis for further analytical and numerical refinement.