For patents, you do have to pay maintenance fees to renew a patent at the 3.5, 7.5 and 11.5 year marks after issuance to keep it active, else it goes abandoned. A vast number of patents go un-renewed and lapse that (1) few patents turn out to be "profitable", and (2) most patents don't last their full terms.
The geometric part of the idea is what really caught my eye. I think there's something to be said for a steeply escalating cost for maintaining publicly protected monopoly rights.
