?????
I’m just asking whether, in your view, the thirteen principles of interpretation (by which the Torah is expounded, of course, not the thirteen attributes of mercy written in the Torah) were given at Sinai, or whether they were invented or developed later on.

This appears only very rarely in the words of the Sages. All the medieval authorities agree about it. I have no criterion for determining whether yes or no, and in my opinion nobody else does either. In the second book on Talmudic logic I explained why, when people say about the principles that they are a law given to Moses at Sinai, they do not mean that they came down from there as is and reached us that way. There is fairly clear evidence that they underwent development, expansion, and conceptual formulation over the generations. What I show in the book is that this development is based at every stage on what came before it. That is, this is not an invention but a conceptualization and refinement of what was transmitted in the tradition.

Let me sharpen what I’m actually trying to ask — this isn’t necessarily about the thirteen principles specifically, but about midrashic interpretation in general. Is derash “correct” — meaning, does it reflect the intent of the Giver of the Torah, which would seemingly be true only if He said it — or is it an invention?
If we have no good reason to believe it was given at Sinai, then everything learned through it is seemingly mistaken from the outset, and then as I understand your approach, we are not bound by it (because we are bound by the Talmud only if it is not based on an error from the outset).

There is derash in the Hebrew Bible (Tanakh) as well — in Ezekiel, in the book of Chronicles, and in Psalms.
Rabbi Jonathan Sacks expands on this point in his book Crisis and Covenant.

I already answered that in my view the principles are a law given to Moses at Sinai in the flexible sense. The fact that the principles came down from Sinai does not mean that derash reflects the intent of the Giver of the Torah. The Sages who expound use their own reasoning, and they can make mistakes. But this is the authoritative interpretation of the word of God, and therefore it is binding.

Honorable Rabbi, if the Sages interpreted and used their own reasoning, and indeed can make mistakes, what do you do when you discover that?

I have an example where the Sages derive something using the thirteen principles, but do not derive it in another similar case. How can that be?

After all, the thirteen principles have a logic to them. What is stronger — the thirteen principles or some other rules besides the rules of the thirteen principles? How do you infer that?

When that is discovered, you change it if there is authority to do so (when there is a Sanhedrin). If not, then you follow the ruling of the Sages despite the mistake.
But it is not clear to me how one would discover that there is a mistake there. Do you know enough about the methods of derash to reach such a conclusion? I don’t.
Applying the thirteen principles is not mechanical. It takes into account quite a few additional factors: reasoning, the textual context, tradition, various difficulties in the plain meaning, and more.