The Constant Inside the Logarithm: Bethe's Mean Excitation Energy and the Leverage of an Input

Bethe's 1930 stopping-power formula is a derivation: from the target's oscillator-strength distribution it returns the energy loss of a fast charged particle with a shape — the 1/\beta^2 rise, the logarithmic relativistic increase — that remains the working law of dosimetry and detector physics. The material itself enters the derivation through one number, the mean excitation energy I, a logarithmic average over the target's whole spectrum, which the paper does not compute for any material: it is taken from experiment, and later from semi-empirical tables. This paper evaluates the input's leverage. The bench returns the stopping power for a declared proton in four declared materials at the record's tabulated I, then moves I by ten percent: the stopping power moves by about one percent — 0.959144\% to 2.923813\% across the grid — and across the record's own tabulated range for a single material, water at 69.000000 to 80.000000 eV, the quoted stopping power moves by up to 2.577616\% under the formula's unchanged name. The logarithm shelters the constant; the shelter is the value, and percent-accuracy fields use it silently. Census headline, unchanged: NOT SOLVED / PARTIAL · producing-count 0.