Five Terms and a Remainder: The Semi-Empirical Mass Formula, Priced Term by Term

The Bethe–Weizsäcker semi-empirical mass formula evaluates the nucleus as a charged liquid drop: five terms — volume, surface, Coulomb, asymmetry, pairing — with five coefficients fitted to the mass record. It is the most successful fitted formula in nuclear physics, and this paper does not dispute its success; it evaluates its structure. Against eight declared measured binding energies spanning A = 4 to A = 238, the formula's mean absolute residual is 4.252 MeV, but the residuals are not noise: they are −1.5271 MeV per nucleon at He-4 and −0.0054 at Sn-120 — the formula fails exactly where the nucleus stops being a drop and starts being a shell structure, and the record's shell closures themselves depart from the harmonic-oscillator closures by shifts (−12, −20, −30, −42) supplied entirely by one further assumed term, the spin–orbit coupling. Five fitted terms yield the bulk; the structure they miss is the physics.