No Single Pseudopotential: The Isotope Table's Defect, Priced Honestly

Change nothing about a metal except the weights of its atoms, and its superconducting transition temperature moves. That is the isotope effect, and the first theory of superconductivity made its size a forced number: T_c proportional to M^(-alpha) with alpha = 1/2. The measured table does not read one half down the column: it reads one half for a few elements, less for most, and zero for ruthenium and zirconium. The standard repair is the Coulomb pseudopotential mu*: heavier atoms vibrate slower, the electron repulsion gets more time to be dodged, and the exponent falls below the half by an amount the McMillan formula computes. This paper executes that repair as a closure test and prints the standard theory's answer beside the result in full. Executed honestly, no single mu* closes the eleven-element isotope table: the best global value, mu* = 0.139, still leaves a worst-row defect of 0.249, about six times the table's demonstrated noise, and swapping in the second standard compilation raises the floor to 0.279. Per-element alpha-implied mu* values span 0.00 to 0.169 and sit in tension with the same elements' T_c-implied values on the independently witnessed rows, zinc above all. Printed fairly: mu* was never promised to be one number; fitted element by element it closes every row, and an earlier overclaim to the contrary is withdrawn in the paper, in the open. Grade: NO SINGLE PSEUDOPOTENTIAL — DEFECT 0.249, PRICED WITH ITS NOISE. This paper stands outside my work on the vacuum catastrophe; it draws nothing from that work and repairs nothing in it.