D-exp-Zr: Structural Origin of the Quantum Phase Transition in Zirconium Isotopes — A PDL Framework Analysis of the Lifetime Measurements of Pasqualato et al. (2024)

This exploratory document (D-exp series, Projective Dynamic Logo framework) analyses the quantum phase transition (QPT) at N=60 in the zirconium isotopic chain in the light of two established theorems of the PDL corpus. The experimental motivation is the precision lifetime measurement of Pasqualato, Ansari, Heines, Görgen, Korten, Ljungvall, Clément et al. (arXiv:2410.17004, 2024), which provides the most reliable B(E2) values to date for ⁹⁸⁻¹⁰⁴Zr using the Recoil Distance Doppler Shift technique in γγ-coincidence mode at GANIL with AGATA and VAMOS++. Two theorems of the PDL corpus are applied: (1) The nuclear saturation threshold Z_sat = R_sea(n)/R_surf(p) = 9960/(310φ) = 19.86 ≈ 20 (D40), which yields Z(Zr)/Z_sat = 2.014, a deviation of 0.72% from the exact integer 2. Zirconium (Z=40) is the element whose proton number equals twice the PDL saturation threshold, derivable from the PDL integers alone. (2) The harmonic-oscillator PDL level table with spin-orbit splitting s = 1/12 (exact rational, D47), which reproduces all seven nuclear magic numbers {2, 8, 20, 28, 50, 82, 126} exactly (7/7, zero free parameter). Reading this table: the QPT neutron number N=60 falls within the 5d₃/₂ neutron level (cumulative occupancy 59–62), 2 neutrons above the 5s₁/₂ closure at N=58 and 3 neutrons below the opening of the 5g₇/₂ level at N=63. Furthermore, π-g₉/₂ (filling Z=41–50) and ν-g₇/₂ (filling N=63–70) are spin-orbit partners in the same oscillator shell (N_osc=4, ℓ=4), related by the Mirror Lemma of D47. Zirconium (Z=40) is the last proton number before π-g₉/₂ opens. A structural conjecture is proposed: the QPT in Zr is the consequence of the simultaneous proximity of Z=40 to the π-g₉/₂ opening and of N=60 to the ν-g₇/₂ opening, mediated by the Mirror Lemma. This conjecture is explicitly labelled as requiring formal derivation from axioms C1–C4 and constitutes a well-defined open problem. All results are verified by a self-contained Python script deposited alongside this document. No free parameter is introduced. The epistemic status of every claim (theorem, arithmetic corollary, conjecture, open problem) is stated explicitly throughout. This document is part of the D-exp exploratory series of the PDL framework and does not belong to the core corpus D01–D55.