Spectral Type as the Genealogical Discriminant: Classification of the Parisi Reciprocal-Pair Coalescents and the Confinement Origin of the Jamming Exponents

The genealogy of the Parisi-Zamponi matching front was placed in the Kingman class -- binary mergers, finite reproduction variance, non-cascade -- while the equilibrium full replica-symmetry-breaking (fullRSB) state is ultrametric, the Bolthausen-Sznitman cascade. This note resolves the apparent tension and turns it into a classification. Building on the reciprocally weighted self-adjoint pair (A_c in L2(rho^-1), L in L2(rho)) whose duality is generic to all fullRSB (the companion 'Beyond the Shadow'), we identify the single quantity that fixes the genealogy: the spectral type of the forward operator L -- a discrete bound state (pushed) versus a state at the edge of the essential spectrum (pulled) -- together with the conditioning (stationary versus killed). We establish three numerically-verified facts and one classification. (I) The operator parameters are genealogically inert. Sweeping the asymmetry c = omega_R - omega_L over its physical range, the top eigenvalue of L stays strictly negative (the fully-pushed signature) with a strictly positive gap for every c; the empty-side (leading) edge is Gaussian-confined at rate omega_R/c = (1+c)/(2c) >= 1 throughout, so the front is fully pushed at every asymmetry. The closed-form reproductive-value tail exponent obeys the exact identity nu_v(c) = -(1 + c + 2 lambda)/(1 - c), whose only divergence is the kinematic factor 1/(1-c), tied to the vanishing left-confinement curvature kappa_L(c) = (1-c)^2/(8c). (II) Spectral type is diagnosable by box scaling: the gap of L is independent of the truncation length L (a discrete eigenvalue) whereas a free operator's gap scales as 1/L^2 (a continuum). (III) The pushed-pulled transition is a spectral delocalization: scaling the confining part of the L-potential to zero dissolves the arithmetic ladder of the companion continuously and flips the gap from L-stable to 1/L^2 at the singular endpoint of vanishing confinement. These give the classification: discrete x stationary -> Kingman (jamming); discrete x killed -> the Brownian coalescent point process (Schertzer-Tourniaire); continuum -> Bolthausen-Sznitman (Derrida-Spohn / the random energy model / the Parisi cascade). The genealogical bridges are classical and are cited, not reproved. The synthesis is structural and, to our knowledge, new; its sharpest consequence inverts a common intuition: the model-specific jamming exponents are the discrete spectral data of confinement -- precisely what the generic, universal cascade lacks. Model-specific spectral richness and replica-symmetry-breaking complexity are opposite faces of the same spectral type. Every numerical statement is reproduced by a consolidated driver (jam_ops.py + paper6_repro.py), included in full in the appendix.