I don't see how you can formulate an entirely different problem and begin to draw an analogy to my twin prime proof. In fact, come to think of it, I can dispense with all my talk about if the twin prime conjecture is false, it will take one and only one special prime to eliminate all the 6n-1 and 6n+1 pairs at some point in its elimination process. This has been the main statement that a lot of you have been hanging on to. I still believe in this statement but it doesn't even need to be in the proof because the mere fact that every prime, 5 or greater, will perpetually keep hopping over un-eliminated 6n-1 and 6n+1 pairs during its turn on the number line with all multiples of 2 and 3 removed is enough to prove the twin prime conjecture is true. Every prime having this property means that these pairs can never run out. No one, as yet, has come up with a way that these pairs can run out apart from one prime doing the job if this was possible. All you keep doing is formulating new problems and trying to use analogical arguments. That doesn't work for me. Stick to this twin prime problem at hand and explain if there is any other way for all these pairs to be eliminated at some point if the twin prime conjecture is false. None of you can because there is no other way i.e. if the twin prime conjecture is false. A fraction of infinity is a lesser infinity but still infinity. With my way that fraction of infinity is 1. This means that this prime takes out all the infinity number of pairs and not say a third or half or a quarter of infinity. Of course this latter part of my discussion is based on if the twin prime conjecture is false. I don't need to read some difficult philosophical books about infinity by the 'Greats' to come up with my own common sense concept of infinity.