Beyond the Shadow: Reciprocal Self-Adjointness, the Exact Spectral Offset, and the Limits of Integrability of the Parisi–Zamponi Matching Front

The spectral foundation of the genealogy of the Parisi–Zamponi matching front rests on two reciprocally weighted self-adjoint operators — the confining replicon operator A_c in L²(ρ⁻¹) and the forward genealogical operator L in L²(ρ), with ρ = exp((2/c)∫V) — whose spectra were observed to be an arithmetic ladder with spacing the harmonic mean (1−c²)/2 of the asymmetric confinement frequencies ω_R = (1+c)/2, ω_L = (1−c)/2. That observation left a single open line in the ledger of the spectral foundation: whether the ladder is exactly arithmetic, and what algebraic structure, if any, it reflects. This note resolves that line and determines the complete operator structure of the reciprocal pair. The operator-mechanism deposit (Zenodo 10.5281/zenodo.20572800) showed the jamming identity to be the shadow of a self-adjoint structure; this note goes literally beyond that shadow, to the precise content of its spectrum. We prove three things. (I) The ladder is not exactly arithmetic: the spacings drift monotonically — A_c from below, L from above — and converge to the harmonic mean, which is therefore the WKB (Bohr–Sommerfeld) asymptote and not an exact algebraic gap. The exact spectrum of the intrinsic asymmetric oscillator is a transcendental parabolic-cylinder (Weber) matching condition, an explicit Γ-function determinant, of which the arithmetic ladder is the semiclassical shadow. (II) The two operators are co-spectral up to an exact rigid shift: their Schrödinger potentials differ by exactly ω_L + 2cf (a consequence of the Boolean cancellation V'+cf = b), and the two ladders are offset by the closed-form constant ω_L + 2c·ω_R = (1+c+2c²)/2, established as a k → ∞ limit and confirmed by Richardson extrapolation to 10⁻⁵. They are not isospectral, not similar, and not Darboux/supersymmetric partners of any finite order: the rigid offset is a non-integer — generically irrational — multiple of the spacing, so the two ladders share no eigenvalue, which excludes intertwining of every order. The only exact structure is the reciprocal self-adjointness together with this closed-form offset. (III) The reciprocal duality itself is generic to all full replica-symmetry-breaking: the forward operator is the adjoint of the linearized backward Parisi operator, independently of the model. What is specific to jamming is the front specialization — the cancellation V'+cf = b, the exponents, and the exact offset — which is universal within the jamming class (the spherical perceptron inherits it by the SAT–UNSAT universality of Franz–Parisi–Sevelev–Urbani–Zamponi). The two ways in which the pair fails to be an exact oscillator are governed by the asymmetry c = ω_R − ω_L, itself the nontriviality of the exponent (b − a = c): the spectral mismatch is linear, δ ∼ c (odd, set by the Boolean orientation of the front), and the ladder anharmonicity is quadratic, ∼ c² (even, the orientation-blind oscillator shape). Both vanish as c → 0. Every numerical statement is reproduced from the verified profile by the consolidated driver, reproduced in full in the appendix.