Real Bers embedding on the line: Fisher-Rao linearization, Schwarzian curvature, and scattering coordinates

We develop a real-analytic counterpart of the Bers embedding for the Fr\'echet Lie group $\Diff^{-\infty}(\R)$ of decay-controlled diffeomorphisms of the line, and establish its connection to $L^p$ Fisher-Rao geometry on densities. For $p\in[1,\infty)$, the $p$-root map $\varphi\mapsto p(\varphi'^{1/p}-1)$ isometrically linearizes the homogeneous $\dot W^{1,p}$ Finsler metric on $\Diff^{-\infty}(\R)$, yielding explicit geodesics and a canonical flat connection whose Eulerian geodesic equation is the generalized Hunter-Saxton equation; for $p=\infty$, logarithmic coordinates $\varphi\mapsto\log\varphi'$ provide a global isometry and the Schwarzian derivative emerges as the projective curvature. We construct a real Bers map $\beta^{-\infty}\colon\Diff^{-\infty}(\R)/\Aff(\R)\to W^{\infty,1}(\R)$ via this Schwarzian, prove it is a Fr\'echet-smooth injective immersion whose linearization admits a tame right inverse given by an explicit Volterra operator, and characterize its image through Sturm-Liouville spectral theory. We introduce an $L^p$-Schwarzian family $S_p$ that interpolates between affine and projective cocycles, establish full asymptotic expansions as $p\to\infty$ in Fr\'echet and Orlicz-Sobolev scales, and extend the Bers embedding to Orlicz diffeomorphism groups. Through the Jacobian correspondence, these structures transfer to a manifold of densities asymptotic to Lebesgue measure, where the nonlinear Eulerian transport reduces to a pointwise Riccati law and the Schwarzian becomes the score curvature governing Fisher information. The compact-manifold $L^p$ Fisher-Rao linearization of Bauer, Bruveris, Harms, and Michor is recalled as a guiding framework.