As I understand it, the whole reason a majority that is not before us does not receive the status of a majority that is before us is that I do not know that a statistical sample really corresponds to reality. But once we have learned, by a scriptural decree, that a majority determines the matter, majority can also determine the question whether a statistical sample faithfully represents reality—and from there we can apply a majority that is not before us.
It reminds me of Hume’s principle of induction—that induction itself proves the reliability of induction.

Even if you say that the distinction between “before us” and “not before us” is whether the case at hand is part of the majority known to us, the question whether a sample represents the whole world includes that very distinction within it (because if a sample really represents reality, then the case at hand is also part of reality, and the majority is a majority that is before us).

A continuation that came up in another thread:
I’ll phrase it a bit differently. As I understand it, a majority that is not before us would not differ from a majority that is before us if we rely on a representative sample. If so, after the Torah taught us that we follow the majority, one can learn regarding a representative sample that it is indeed reliable on the basis of majority—that is, most representative samples were measured as faithfully representing reality. From here we can learn the force of a majority that is not before us.
Something like what Hume says regarding the question of induction.

That’s Chinese to me.

Why? The Torah taught that majority is effective, but I cannot derive from that anything about a majority that is not before us, because how do I know that the sample I know really represents the whole population? But this can be learned by means of majority, since most representative samples showed correct results.

It is not true that most representative samples showed correct results. How do you know that? You assume it, but you haven’t substantiated it.
And even if it were true (and it isn’t), how would you derive a majority that is not before us from a majority that is before us?
As stated, this is Chinese to me, or it just doesn’t get off the ground.

It’s not Chinese, because there is a claim here that can be discussed.
As for the first claim—are you really saying that most representative samples did not show correct results? That isn’t right.
True, statistical samples that the Rabbi knows by that name did not show correctly, and that is because they are not talking about nature. A majority that is not before us does speak about nature. We all understand that a statistical sample that deals with nature generally does show a correct result, since research is a statistical sample—induction. And from where does induction get its authority within the walls of the study hall, within the authority of Jewish law? From a majority that is before us, because most inductions have shown us claims that turned out to be correct about reality—that is Hume’s point. And there is a possibility of checking this. For example, in the topic in Yevamot regarding the possibility of a minor and his minor yevamah being together, we rely on the fact that most of the world are not aylonit women. If I actually had the knowledge that most of the world are not aylonit women, this would not be a majority that is not before us, because the woman would emerge from within the totality familiar to us—the population of the world. But how do I know that indeed most women are not aylonit women? The judge knows only a small number from among the world’s population, so the judge knows it by virtue of the reliability he grants to a statistical sample.
After the judge rules that this woman is not an aylonit, and when she grows up that indeed turns out to be the case, it has been proven that one may rely on a statistical sample.

When I write that it’s Chinese, what I mean is that maybe there is a claim here, but I didn’t understand it.
I can only repeat what I said, and with this I’ll finish.
The Talmud brings a source for a majority that is before us. Regarding a majority that is not before us, there is no source there (in Rashi at the end of the passage there are two possibilities, and both are of course strained). You are trying to build a majority that is not before us on a majority that is before us (though you wrote that you built it on itself, on a majority that is not before us, which is circular to begin with).
Regarding building it on a majority that is before us, I am saying that you have no evidence whatsoever that most samples of a majority that is before us are correct. We have no way to check that, and I elaborated on this in my columns about a majority that is before us and not before us, and majority in a religious court. Therefore there is no way to build a majority that is not before us on a majority that is before us. Moreover, even if there were a test showing that a majority that is before us works in most cases, that itself would be a majority that is not before us. And of course it makes no sense to build a majority that is not before us on a majority that is not before us itself. Moreover, even with a majority that is not before us, it is not true that we checked and found that in most cases it works. Usually there is no way to check that at all. Try checking whether, in most cases where we decided that a woman gave birth at nine months, she really did give birth at nine months. That is not the same thing as saying that in most cases a woman gives birth at nine months. But as I said, the discussion is unnecessary, because even if it were possible to check, it would not help.