Bookkeeping, Not a Derivation: The Pronic Identity Charged Against One Spin-Orbit Strength

AF-7 found that the shell closures' departures from the oscillator closures — 12, 20, 30, 42 — are exactly the pronic numbers k(k+1), each the capacity of one oscillator shell, one shell down, and charged the successor: treat the identity as a derivation charge on the spin–orbit strength. This paper separates the charge into its two halves and grades both. The count half is bookkeeping, exactly: shell k holds (k+1)(k+2) states, its top j = l + ½ orbit holds 2k+2, and if exactly that orbit drops one shell the departure is (k+1)(k+2) − (2k+2) = k(k+1) with no remainder and no strength anywhere in the arithmetic. The strength half fails: in the declared bench — harmonic oscillator plus the Nilsson-form spin–orbit term with ONE dimensionless strength κ for all shells — dropping shell 3's top orbit requires κ > 1/3, while not over-dropping shell 6's second orbit requires κ ≤ 1/4. The window is empty by 1/12. Every κ regime is enumerated in §2: too weak drops nothing or only the heavy closures; strong enough for all four drops over-drops the heavy shells and returns departures 22 and 32 where the record says 30 and 42. The identity tells any future derivation exactly what it must return; one strength in this form does not return it. AF-2's deficit — the spin–orbit term fitted, not derived — stands untouched, now with its target and its simplest would-be supplier both evaluated.