The Drift Dichotomy: Growing versus Saturating Drift as the Operator Origin of the Jamming–Equilibrium Genealogy Divide in Full Replica Symmetry Breaking

The companion classification placed the Parisi-Zamponi matching front in the Kingman class (discrete, pushed) and the equilibrium full replica-symmetry-breaking (fullRSB) cascade in the Bolthausen-Sznitman class (continuum, pulled), with the spectral type of the forward operator as the discriminant. It left open the operator origin of that spectral type. This note supplies it. We attach to every level x in [0,1] of the Parisi hierarchy the reciprocal pair of the constellation -- the linearization of the backward Parisi equation and its forward adjoint -- and identify the single feature that fixes the spectral type: whether the level-x drift b_x(h) = x qdot(x) m(x,h) grows or saturates as h -> +-infinity. Saturation of the local magnetization, m -> +-1, makes the conjugated Schroedinger potential asymptotically flat; by Weyl's theorem the essential spectrum is then non-empty, a continuum exists, and the front can be pulled (Bolthausen-Sznitman). A scaling-pinned drift V ~ omega_R t, set by the marginal fixed point rather than by a saturating field, makes the potential confining; the essential spectrum is then empty, there is no continuum to be pulled by, and the front is robustly pushed (Kingman). The dichotomy is exactly the presence or absence of essential spectrum. Three consequences follow. (i) The Sherrington-Kirkpatrick model is uniformly pulled: because its local magnetization satisfies |m|