General Relativity as the Padé Approximant of Density Field Dynamics: Refractive gravity on flat space and its relation to Schwarzschild geometry

We show that general relativity arises as the simplest rational truncation of Density Field Dynamics (DFD) in the gravitational clock-rate sector. Defining the lapse-squared scalar L (u) = c²/|gₜt| with u = GM/ (ϱc²), the isotropic-coordinate Schwarzschild form of GR gives LGR (u) = (1+u/2) / (1−u/2) ², while DFD's exterior solution gives LDFD (u) = exp (2u). The exact identity LGR (u) = P (1, 1) (exp (u) ) ² extends to a Padé hierarchy in which P (m, m) (exp (u) ) ² = exp (2u) + O (u^ (2m+1) ) for every m ≥ 1, with each finite-m truncation carrying a Padé pole that recedes to infinity only as m → ∞. GR is the m = 1 slot; DFD is the entire-function limit. The Schwarzschild horizon at r = 2GM/c² is the Padé pole of the m = 1 truncation; DFD's exponential has no finite pole, and r = 2GM/c² appears instead as a photon sphere. The two theories agree through O (u²) by construction — consistent with all gravitational-redshift and clock observations to date and reproducing the post-Newtonian parameter beta = 1 — and first differ at O (u³), generating a ~4. 6% larger black-hole shadow. The identity is in the lapse-squared scalar, which controls clock rates, redshift, the Newtonian limit, beta, and horizon structure; the spatial metric, ray-optics, and the PPN parameter gamma are not its consequences and are established separately for DFD via the physical metric. Several prior constructions also produce exponential lapses (Papapetrou 1954, Yilmaz 1958, Dicke 1957, Puthoff polarizable vacuum 1999/2002, Broekaert 2008), so the Padé identity itself holds for any of them; what makes the present statement an inter-theory reduction rather than a tautology is that DFD is not embedded inside GR. The Yilmaz exponential is itself a GR solution interpretable as a wormhole with exotic matter, whereas DFD's flat-R³ elliptic field equation admits no throat and requires no exotic matter. Current Event Horizon Telescope data on M87* and Sgr A* are consistent with both GR and DFD at the present precision; the predicted ~4. 6% shadow excess is the proximal observational discriminator.