  Tunnel junctions for quantum computing require discrete spectra from
continuous waves on a doubly connected coordinate or loop. For an electron on a
metal ring discrete spectra follow from discontinuous Bloch waves. Can both
propositions be true? We find using a gauge function originating in the
Lagrangian that continuity on a ring or loop violates gauge invariance of the
de Broglie wavelength. This same gauge function shows that Lagrangians for the
electron on a ring and the charge on a junction are mutual transforms. Thus
persistent current on a metal ring and the Coulomb blockade on a tunnel
junction seem to be the same dynamical theory based on discontinuous Bloch
waves on the compact perimeter of a circle
