The operator mechanism behind the Parisi-Zamponi jamming identity - a conditional closure of the matching-profile selection

Parisi and Zamponi recently proved the identity a+b=1 relating the infinite-dimensional jamming critical exponents, from the scaling fullRSB equations and via a bilinear test-function identity. Taking that result as our starting point, we set down the operator mechanism behind the identity: we complete what the matching-profile formulation closes at once, and chart a research program for what remains. This paper explains why the identity holds, and isolates exactly what in its derivation is unconditional and what is not.

Our central result is structural and unconditional at the level of the profile/eigenvalue system: given sufficiently regular f, p satisfying the stated profile and eigenvalue equations and the required decay, the adjoint pairing and the cancellation V'+cf=b imply the identity exactly. The replicon eigenvalue problem and the matching-profile equation are linked by an adjoint bilinear pairing, and a single algebraic cancellation V'+U=b — forced by the Boolean contact constraint of hard spheres — collapses that pairing to a one-parameter family of exponent identities c(D−2N)=(λ−a−c)K, indexed by the constant λ:=V'+U, of which the Parisi–Zamponi relation is the jamming member λ=b. In this operator form the test function ξ=f(1−f) is the logistic reaction nonlinearity, the choice that collapses the pairing onto exactly the three marginality integrals. Parisi and Zamponi reach the same identity by integration by parts; we show it is the shadow of a self-adjoint structure — the replicon and matching-front operators are an adjoint pair of Schrödinger operators, self-adjoint in the reciprocal weighted spaces L²(ρ^{±1}), not identified in their analysis. Recognizing this turns a one-step cancellation into a framework, and it is the framework that makes the remaining exponent a computable spectral crossing, the two-sided enclosure of c* effective, and the front genealogy accessible; the structure and these consequences are, to our knowledge, new.

Building on this structure we establish three further results, with their logical status made explicit. (i) We recast the determination of the remaining exponent as a scalar spectral crossing, a(c)=(1−c)/2, where a(c) is the top eigenvalue of the operator A_c (bounded above; a confining Schrödinger operator after a sign flip), and — on the physical branch 0≤f≤1 — the analytic bound c*>1/3. (ii) We reformulate the matching profile as a stationary Fisher–KPP front, give an exact renormalization-group reduction and a weighted gradient structure, prove monotonicity for the autonomous flow, and isolate a precise, explicitly conditional criterion under which the scaling-limit selection closes and yields 0≤f≤1, hence a+b=1; we do not construct the scaling limit, which we take — exactly as Parisi–Zamponi and Charbonneau–Kurchan–Parisi–Urbani–Zamponi do — as external input. (iii) We give an effective, refinable two-sided bracket c*∈[0.41359, 0.4162] and spectral diagnostics against standard closed-form mechanisms (no supersymmetric or shape-invariant factorization detected).

A closing section records the branching-front reading of the matching profile. The front carries a pulled-front linear edge (unweighted growth rate +c), but its selected profile decays Gaussian-fast — steeper than any exponential — which places it in the fully pushed class; we give computational and spectral evidence that its genealogy therefore lies in the Kingman class — Kingman's coalescent for the confined, constant-size front, the Brownian coalescent point process being the critical (killed) member — not the Brunet–Derrida / Bolthausen–Sznitman cascade, and the front genealogy does not reproduce the Parisi cascade. Every symbolic identity and every numerical value is reproducible from the protocols and accompanying scripts of the Supplement; the numerical statements are effective and reproducible rather than certified.

Companion structural analysis to Parisi & Zamponi (arXiv:2606.03300). The deposit comprises the main paper, a Supplement (symbolic and numerical protocols), and a Code Appendix reproducing the verification scripts (SymPy / NumPy / SciPy), with a one-shot driver verify_all.py (full mode ~3 min; --fast < 30 s). The numerical statements are effective and reproducible, not interval-certified.

In the closing branching-genealogy section the matching front decays Gaussian and is therefore fully pushed; computational and spectral evidence places its genealogy in the Kingman class — Kingman's coalescent for the constant-size confined front, the Brownian coalescent point process being the critical (killed) member — and the conjectured identification of the front genealogy with the Parisi cascade is refuted.