A Spectral Foundation for the Genealogy of the Parisi–Zamponi Matching Front: Gap, Transport, and the Kingman Coalescent at the Frozen Scale

This is the genealogy companion to the Parisi–Zamponi jamming deposit (Zenodo 10.5281/zenodo.20572800; companion to arXiv:2606.03300). At the frozen scale of the matching front it puts the genealogical reading of the deposit's closing section on a rigorous spectral footing, and establishes three results. (I) The confining replicon operator A_c = (c/2)∂² − V∂ + cf is self-adjoint in the weighted space L²(ρ⁻¹), with ρ = exp((2/c)∫V), and unitarily equivalent to a Schrödinger operator H = −(c/2)∂² + 𝒱 with the closed-form potential 𝒱 = V²/2c − V'/2 − cf, whose wings are asymptotically harmonic and asymmetric, of frequencies ω_R = (1+c)/2 and ω_L = (1−c)/2. At the selected c = c* the spectrum is an arithmetic ladder with top a = ω_L = (1−c)/2 (the velocity-selection marginality of the deposit, the condition fixing c*) and spectral gap equal to the harmonic mean Δ = (1−c²)/2 > 0; the spine is exponentially ergodic, of Ornstein–Uhlenbeck type. (II) On this foundation the many-to-few (k-spine) transport closes. The forward, mean-density (genealogical) operator L = (c/2)∂² + V∂ + c(1−2f) is itself self-adjoint, but in the reciprocal weighted space L²(ρ), hence unitarily equivalent to its own Schrödinger form H_L; it has a strictly negative top eigenvalue λ = −0.5277 (the fully-pushed signature) and a strictly positive forward gap Δ_L = 0.4679, so the reproduction variance is finite and the pairwise merger rate is time-homogeneous. Its reproductive value v = ρ^{1/2}Φ₀ is polynomial and its density φ = ρ^{−1/2}Φ₀ Gaussian, so the spine measure vφ = Φ₀² is bulk-localized. (III) Together — a positive gap (time-homogeneous merger), closed transport (finite reproduction variance), and the front's stationary, constant-size profile — these place the frozen-scale genealogy in the Kingman class, with precise object Kingman's coalescent: the constant-size member of the class (Lambert–Schertzer), as opposed to the Brownian coalescent point process that Schertzer–Tourniaire obtain for the critical, killed realization. The front genealogy is therefore not the Brunet–Derrida / Bolthausen–Sznitman cascade. All spectral and numerical values are computed at c* = 0.4157355 from the symmetric Schrödinger forms (scipy.linalg.eigh) and a Langevin simulation of the spine; the consolidated driver and its helper module are reproduced in full in the appendix.