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Thought:
The key aspect that will make the space elevator possible will be, I think:
The cable/tether will be made from a material which can be continuously repaired while in use by enzymatic processes.
Maybe it's #graphene or #cnt but it might also be some bio polymer like lignin or chitin.
I also think the structure of the cable will be made up of tubes and it'll be porous, so that it can be locally infused with a solution containing enzymes and other molecules for repair.
