Kernel-smoothed QUMOND: Solar-System quadrupole and wide binaries as probes of a finite-range MOND transition

The Cassini bound on the anomalous Solar-System quadrupole, Q_2 = (3 +/- 3) x 10^-27 s^-2, is in 8.7 sigma tension with the radial acceleration relation (RAR) in classical AQUAL and QUMOND, and several Gaia analyses find wide binaries to be Newtonian. A length scale that screens MOND effects in small systems has been proposed as a way out. We develop the nonlocal branch of this idea: a normalized, symmetric convolution kernel K of range l smooths only the MOND part of the QUMOND Lagrangian. The theory follows from an action, conserves momentum, recovers Newtonian gravity exactly and reduces to QUMOND for r >> l; rotation curves change only at order (l/L)^2. We show that Q_2 is set by the ultraviolet behaviour of the kernel. If K(k) falls faster than k^-3/2, Q_2 = -GMA/(15 l^3) for a Yukawa kernel, with A the logarithmic slope of the interpolating function at the Galactic field: about 10^-29 s^-2 for l = 0.2 pc, for the simple, standard and RAR functions alike. A kernel with K(k) ~ 1/k, as produced by free streaming with exponential attenuation over a length lambda, gives Q_2 ~ (0.8-1.4) x 10^-27 s^-2 (1 pc / lambda), at the present ephemeris sensitivity, but also a phantom density of 10^2-10^3 M_sun pc^-3 in the planetary region, so the ephemeris density bound requires lambda >= 1.65 pc for the RAR function. The wide-binary velocity boost grows as (s/l)^3 for the Yukawa kernel and as (s/lambda)^2 for the free-streaming kernel; Gaia DR3 requires l >= 0.14 pc or lambda >= 0.40 pc, nearly independently of the interpolating function and the external field. Dwarf spheroidals bound the effective range to <= 5-10 pc. Future ephemerides can separate the Yukawa class from both the free-streaming class and local screening, while the wide-binary profile and open star clusters probe the kernel shape further.