The Deep Acceleration Scale in a Single-Medium Emergent Gravity: ξ = 1.38 ± 0.11 from SPARC, Planck, and Cross-Habitat Constraints

The empirical proximity between the galaxy acceleration scale a₀ and the cosmological scale cH₀/2π is well known, but its status remains interpretive. In the patched single-medium branch developed in Papers I–III, the deep acceleration scale is not fitted independently of cosmology: it is tied at core level to Λₑff through a₀ᵈeep = ξ c (2π) ⁻¹ √ (Λₑff/3), while the quantity measured in galaxy phenomenology is a habitat-matched scale a₀ᵉff. We use Planck 2018 cosmological parameters and public SPARC-based determinations of the radial-acceleration scale to infer ξ. Using the hierarchical SPARC inference of Desmond as the headline observational input, we obtain ξ = 1. 38 ± 0. 11 and R_Λ0 = 1. 14 ± 0. 09, with all comparison determinations remaining of order unity. We also test the public binned SPARC radial-acceleration relation against a minimal family of interpolation functions. The mean relation is compatible with the deep-branch asymptotics, while the transition region remains non-discriminating at the level of the frozen public binned data. The paper does not claim that the framework is validated in full. It quantifies a structural coefficient, shows compatibility with the public mean SPARC relation, and serializes the unresolved cross-habitat and per-galaxy residual tests as explicit next-step protocols rather than hidden caveats.