Ring Spacing from a Fourth-Order Radial Feedback Green Function in a Keplerian Accretion Disk

ALMA observations of protoplanetary disks reveal ubiquitous concentric ring structures whose origin remains debated. We present an exactly solvable local model for ring spacing in a Keplerian disk. The passive advection--diffusion problem with a general vertical diffusivity profile separates into a vertical Sturm--Liouville spectrum and a radial modified-Bessel Green function. This passive kernel is smooth and does not by itself generate a periodic ring train. We therefore introduce a minimal fourth-order radial feedback closure for the ring-averaged surface density. For a localized steady ring source, the resulting Green function has a damped oscillatory exterior branch whenever the decaying spatial roots are complex. Under an observationally motivated AU scaling, the same dimensionless solution gives an illustrative spacing of order 12 AU, distinct from the fastest-growing temporal wavelength. The model separates the passive transport kernel from the feedback mechanism that selects the observable radial scale. Because this mechanism is internal to the disk, it does not require pre-existing planets. The vertical diffusivity profile affects the background kernel but is not the source of the Green-function oscillation.