My Bike Friday does not flex. There is no hinge in the frame itself. The fold is not quite as neat as the Brompton as a result, but the ride is markedly better, more rigid. A bike which somehow folds more square than rectangular but still does not have a hinge in the frame would be ideal. Not quite sure how one would do this in terms of engineering, but perhaps look at the Helix for ideas. Hinges are the primary reason for squeaks as well.
https://www.kickstarter.com/projects...t-folding-bike