The Strong Spin–Orbit Term, Priced: Mayer's Inserted Coupling and the Magic Numbers It Returns

Census pricing (row CC-98) of the strong spin-orbit term of Mayer's 'On Closed Shells in Nuclei. II,' Phys. Rev. 75, 1969-1970 (1949), and of Haxel, Jensen & Suess, Phys. Rev. 75, 1766 (1949), against a fixed bench (what is postulated, what is derived, what the postulates return numerically, what they cannot reach). The term -- a spin-orbit coupling of structural strength, with the sign lowering the j = l + 1/2 partner -- is postulated (its origin in the paper is Fermi's question, acknowledged there), riding on the independent-particle mean field. Computed by exact counting in the notebook (14 checks, exit 0): without the term the oscillator shells close at 2, 8, 20, 40, 70, 112, 168 and the measured magic numbers 28, 50, 82, 126 do not occur; the splitting spreads (2l+1)/2 select the highest-l intruder of each shell, whose capacities 8, 10, 12, 14 return 28, 50, 82, 126 and the full measured list 2, 8, 20, 28, 50, 82, 126 at all seven entries. The strength is priced as crossing thresholds kappa > 2/N = 0.667, 0.500, 0.400, 0.333 of the shell spacing for N = 3..6 -- a coupling of order the spacing itself. Remainders at full size: the mean field is a second, undischarged assumption; the coupling's microscopic origin in the two-nucleon force is later work at another address, credited there; the counting returns closures only, not the spectrum. EXTENSION. Vacuum catastrophe NOT SOLVED; quantum gravity NOT SOLVED.