Although the cap may have once stood proud and tall, the undulating brass was not the problem. This shades upper aperture ring was pulling loose and had it been placed upon a lamp base it would have pulled completely through. It is a great example of the need for proper reinforcement.
After carefully removing the ring and cleaning the glass, new foil was applied to the edges of the top pieces and the ring was replaced. Finally, reinforcement wires were added to the inside to give strength and the cap was soldered good and solid.