Nanocircuits in loop structures: Continuous waves preclude gauge invariant wavelengths

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.