The Formula Carried Over Whole: Kramers and Heisenberg's Translation Rule and the Half of Dispersion Nobody Derived

In 1925 Kramers and Heisenberg produced the dispersion formula for radiation scattered by an atom — the expression from which the scattering of light by matter has been computed ever since — by a method with no precedent: they took the classical oscillator formula over term by term and replaced each classical Fourier amplitude by a transition amplitude between stationary states, the "virtual oscillators." The replacement was asserted from the correspondence principle, a demand of asymptotic agreement, not derived from any mechanics of the atom then in existence. This paper evaluates the rule in the formula's own currency. On a declared two-level atom the bench returns the translated cross-section whole and then with its anti-resonant term dropped — the one-term reading a later age performs silently: at zero frequency the translated amplitude is 2.000000 against the truncation's 1.000000, a factor of 4.000000 in the static cross-section, and the anti-resonant term carries 69.750000\% of the full cross-section at a tenth of the resonance frequency, falling to 0.997475\% at resonance itself. The two halves of the formula are one classical term and one asserted term; every dispersion number since 1925 uses both. Census headline, unchanged: NOT SOLVED / PARTIAL · producing-count 0.