... GRAPHICAL OFF THE AXIS35In this part, we prove Theorem I.0.8 when the surface is in a slab, illus-trating the key points (the full theorem, using the results of this part, will beproved ... completedin the second paper. The third and fourth papers of this series will deal withhow the multi-valued graphs fit together and, in particular, prove regularity of the set of points of large ... curvature – the axis of the double spiral staircase. The reader may find it useful to also look at the survey [CM8] and the expository article [CM9] for an outline of our results, and their proofs, andhow...