TNR has been practiced in England since the 1960s and in the U.S. since the late 1980s.  There has never been any proof offered to support the claim "TNR eliminates cat colonies."  All we have are testimonials, like the ads that used to be in the back of magazines, such as "Shelley G. of Tampa, Florida says, 'I lost 20 pounds in 20 days with Miracle-Go, and I was never hungry!!!!'"  Let's have a controlled experiment comparing TNR with trap-and-remove.  Find 2 cat colonies of comparable size and setting, in the same city.  Do TNR in the usual way on one colony (no adoptions, because then you are confounding TNR with trap-and-remove) and trap and remove all the cats in the other colony, remove all cat food sources, and promptly trap and remove any new cats that show up.   Count the cats every year, and after ten years, see which colony has more cats.  Repeat (preferably simultaneously) in several other cities with different research teams.  Then we will have some evidence.