This thread appears to be returning to dormancy, so I'll summarize the interesting issues that have emerged. It is a point of some frustration to me that I have made the following mathematically bulletproof arguments yet objections have been raised that don't seem to understand the fundamental points I'm making, so I'll try one last time to condense them into the best and purest form I can, and the reader can take it or leave it as they like. I personally think there is a lot to be learned about what probability can be used for, and even more interestingly, what it can't.

Bottom line: there are two very different ways to state the "Carter catastrophe" conjecture, and they are both examples of bad probability, even in their purest forms with no consideration of whether or not our asking the question itself changes anything (which is a severe problem for the conjecture but I don't need it). Here are the two ways, and what is wrong with them:

1) the conjecture stated in terms of birth number: This goes something like this. 90% of humanity will have a birth number that will ultimately prove to be in the last 90% of humans born. Thus any human may expect with 90% certainty that they fall in the last 90%. That is true as stated, but it is no longer true if that human uses their own birth number (let's say ours is about 15 billion) to infer a coming "catastrophe" in the next 150 billion (to achieve the incorrect 90% likelihood). I have referred to this as "using the M value" to make predictions about N. Others have claimed that is not part of the "catastrophe" scenario, but they have not justified that claim with any suggestion of how there is a "catastrophe" without it. You see, the only way to not invoke M is for a human who has no idea what their birth number is, even vaguely, to then say "I'm 90% likely to be in the last 90% of humans". Note that is perfectly true, but see the important difference? Where's the catastrophe! If the person's birth number might be 100 trillion, for all they know, then humanity could easily live to a quadrillion. Or a million times that-- this is what it means to have no idea what your birth number is, and is clearly violated by the Carter logic. If any use is made of our actual birth number around 15 billion, all bets are off-- the probability argument is simply false at that point, as is easily verified by choosing any arbitrary distribution over N that you like.

2)the conjecture stated in terms of exponential growth rate: Here it goes something like, since humanity is growing exponentially with an e-folding time of (let's say) 50 years, then a version of the birth number argument states we are 90% likely to be within about 2.4 efolds of the end of humanity, say less than 120 years. This version does not require an M value, so takes advantage of the magnitude-free form of an exponential distribution, but it fails for far less subtle reasons. After all, who in their right mind would think it is a valid expectation that if humanity has been e-folding every 50 years for the last handful of centuries, that it should continue to do so for the next handful? In five minutes I could list 100 contradictions to that assumption from everyday experience, it's no better than the pseudoscientific arguments used by creationists (indeed, they do use invalid extrapolations all the time). All you can say is that any exponential distribution cannot be generically expected to extrapolate more than a few efolds into the future, to which I say: duh. But that's not a "catastrophe", merely an expectation that the magnitude-free form of human population growth should be expected to change fairly soon. Again, duh. (Note that the argument goes through perfectly well if the growth slows to, say, an efold of 100 years-- no catastrophe there. We can only be said to be generically chosen from the subclass that shares our growth rate-- unless we make specific reference to M to obtain a catastrophe prediction.)