Non-Local Gravitational Leakage: A Spectral Propagator Ansatz from the Dark Dimension

We present a first-principles derivation of non-local gravity embedded in a 5D Einstein-Dilaton "Dark Dimension" framework. We explicitly solve the bulk field equations, identifying tensor perturbations as Bessel modes with a spectral density ρ(m) ∼ m2ν-1. We resolve the hierarchy problem by linking the leakage scale to the Species Scale (N ∼ 10³⁰), yielding a geometric suppression ϵ ∼ 10⁻⁶¹. Analyzing 118 SPARC galaxies (Q=1) with a robust Bayesian likelihood, we find decisive evidence against MOND (n=1) with a significance of Z ≈ 3.81σ. The model achieves a remarkable goodness-of-fit with a reduced chi-squared χ²_ν ≈ 0.86 and negligible bias (-0.01 dex). The Bullet Cluster offset is explained via a WKB tunneling delay (Δτ ≈ 40 ± 8 Myr), and we derive the modified growth equation predicting S8 = 0.78 ± 0.03.