The Level Density — the cavity's ladder against the measured a-parameter

How many energy levels a nucleus has near a given excitation energy is a measured quantity across the table — the level density — and it matters for every reactor calculation and every stellar reaction rate. Its measured shape is an exponential of a square root, , governed by a single parameter with famous systematics: a A/8 MeV, dipping at magic numbers. The framework's core is a spherical cavity whose mode ladder — , 4.4934, 5.7635, the zeros of the spherical Bessel functions — is the same ladder the video series draws and the same sphere ( fm) the ledger's un-fitted Coulomb coefficient is built from. Fill that ladder with A nucleons and count: the number of ways to distribute excitation energy among the modes is integer-partition counting, partitions grow as , and that is the measured exponential shape. Almost everything in that paragraph is prior art, and this paper's first job is to say so. Bethe derived exactly this in 1936; the partition asymptotics are Hardy–Ramanujan, 1918. What is the framework's to test is narrower: the same cavity that prices the masses fixes the single-particle spectrum with zero new parameters — so it fixes , and can be scored. It was scored. The kind of shape survives; the scale and the shell positions fail exactly where every bare well has failed since 1949, and the failures are printed at full size. [computed for the shape in kind; null-fail on scale and shell positions; all mechanisms credited] This record holds one paper on its own and is refreshed every couple of months. The complete deposit, 10.5281/zenodo.21788037, holds every paper with its code and data and is updated first; for the newest version of this paper, go there.