The Estimated Constant: Mott's Screening Criterion, Priced

In 1949 N. F. Mott argued that when free carriers screen a Coulomb centre, the bound state vanishes at a critical density, and the criterion built on that argument, n_c^(1/3) a_H* = kappa, became the most-used first estimate of a metal-insulator critical density in physics. I credit the argument at full strength: the transition's existence and suddenness are the paper's permanent return, and the canonical form organises six named systems on my bench within a factor 0.813378 to 2.116886 of measurement. What I evaluate is the derivation status of the criterion's constant. The vanishing is argued from the screening picture, not derived from the many-body Hamiltonian, and the constant is an estimate: Mott's own form carries kappa approximately 0.20, the canonical form 0.250000, and the record's inverted form carries estimates of C from 2.0 to 4.2 (kappa 0.500000 down to 0.238095). Because the density goes as the constant's cube, that spread is a factor of 15.625000 in the density the criterion predicts, at every Bohr radius. My bench returns the criterion at every constant of the record's set for six named systems (Si:P, Si:Bi, CdS:In, n-GaAs, n-InSb, VO2) and inverts each measured pair into the constant that system itself implies: 0.194704, 0.212093, 0.220486, 0.221388, 0.267820, 0.253703 — a scatter of 2.602585 in density at fixed radius, and not one of them the canonical 0.250000. Counted plainly: 1 screening criterion carrying 1 estimated constant, 0 derivations in 1949 of the criterion or of its constant. Grade: REASON FOUND / STILL TURNING — PREMISE-ONLY, explicitly not a kill and not a falsification. Program headline: vacuum catastrophe NOT SOLVED / PARTIAL, producing-count 0.