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Examples of calcium-aluminum-rich inclusions (CAIs) surrounded by thick chondrule mantles have been found in chondritic meteorites and cast doubt on the conventional belief that CAIs and chondrules possessed different spacetime origins in the primitive solar nebula. We study specific processes by which such objects, and the more common ordinary CAIs and chondrules, might have formed by flare heating of primitive rocks interior to the inner edge of a gaseous accretion disk that has been truncated by magnetized funnel flow onto the central proto-Sun. Motivated by the appearance of the chains of Herbig-Haro knots that define collimated optical jets from many young stellar objects (YSOs), we adopt the model of a fluctuating X-wind, where the inner edge of the solar nebula undergoes periodic radial excursions on a timescale of ~30 yr, perhaps in response to protosolar magnetic cycles. Flares induced by the stressing of magnetic fields threading both the star and the inner edge of the fluctuating disk melt or partially melt solids in the transition zone between the base of the funnel flow and the reconnection ring, and in the reconnection ring itself. The rock melts stick when they collide at low velocities. Surface tension pulls the melt aggregate into a quasi-spherical core/mantle structure, where the core consists mainly of refractories and the mantle mainly of moderate volatiles. Orbital drift of rocks past the inner edge of the disk or infall of large objects from the funnel flow replaces the steady loss of material by the plasma drag of the coronal gas that corotates with the stellar magnetosphere. In quasi-steady state, agglomeration of molten or heat-softened rocks leads to a differential size-distribution in radius R proportional to R-3e, where tL ~ 20 yr is the drift time of an object of fiducial radius L ≡ 1 cm and t is the time since the last inward excursion of the base of the funnel flow and X-wind. Thus, during the ~30 yr interval between successive flushing of the reconnection ring, flash-heated and irradiated rocks have a chance to grow to millimeter and centimeter sizes. The evaporation of the moderately volatile mantles above large refractory cores, or the dissolving of small refractory cores inside thick ferromagnesian mantles before launch, plus extended heating in the X-wind produce the CAIs or chondrules that end up at planetary distances in the parent bodies of chondritic meteorites.