A planet recently discovered was explored and proved unworthy of terraforming: It is mostly made of carbonaceous minerals, classifying it as a carbon planet. However, one of the explorers does not see that as a hurdle. He seizes the opportunity to mine the abundant carbon and build space elevators using carbon nanotubes fibers. The space elevators will serve to hold-on a sheet made-out of carbon fibers as well. This sheet is the foundation for the "flat earth". The setup looks like this:
The image is not to scale, and the number of "ropes" holding the basket to the surface of the planet is not final. The idea is that the basket spins along with the planet. The content comprising the atmosphere, water, soil and biomass is held-down to the surface using the centrifugal force. The flat Earth has rims which serve as the walls holding the volatiles on the surface. edit: This is a side-view. The basket is a disk with rims holding the atmosphere. The anchorage points to the basket may serve as anchorage as well as mountain tops, therefore simplifying engineering issues. Each anchorage point is connected to several distant points on the planet's surface. This setup should prevent the basket from "folding" and spilling volatiles into outer space.
At its final state, the planet will serve as a "moon" to the dwellers of the flat Earth. It will, however, remain at the Zenith and will only change phases along the day/night cycle.
So, is my setup stable enough? Will the surface remain flat, or will it bend and fold? Assume there are enough "ropes", and planetary "anchors" are sufficiently massive to avoid being lifted away.