Thank you for the reply.
So let us assume that the Sages did have a reason to say that fixed is like half-and-half. Presumably this is not a fully sufficient reason, since they feel a need to derive this law from a verse.
So then I have another question: how do we learn from capital law to the entire Torah? One could challenge it: "What is unique about capital cases? They involve examination and cross-examination, and we do not execute on the basis of majority in a fixed case. Will you then say the same for the whole Torah, where there is no examination and cross-examination?"
If so, then for the rest of the Torah we are left only with an incomplete logical argument, without a derivation from a verse.

That is harder for me to answer. It seems that at least after deriving it from the verse, they understood that the reasoning is itself sound, and if so, this is not a leniency specific to capital law but a sweeping halakhic rule. In other words, if this were a formal halakhic principle not grounded in probability, then there would be room to say that it is just a leniency in capital cases. But they apparently understood that there is a probabilistic claim here, even though a verse was needed in order to infer it, and therefore there is no reason to distinguish between capital cases and other halakhic contexts.

If I understand the Rabbi's suggestion correctly, there are several stages in this exposition:
1. Logical reasoning
2. Applying that reasoning to capital law by means of exposition of the verse "and rises against him and lies in wait for him"
3. Understanding (mistakenly?) that the original reasoning really was probabilistic
4. Extending the law to the whole Torah

Is that what the Rabbi means?

(Too bad—the Rabbi's article actually convinced me that the law of fixed cannot be explained in probabilistic terms, and that it would be better to look for other kinds of explanations—legal ones, for example…)

First of all, the truth is never "too bad."
Second, I did not understand your point 3. The reasoning is not mistaken.
And third, I no longer remember what was in that column, but I also suggested statistical explanations for the law of fixed (if only because the law of fixed is said only regarding a majority that is present before us, and that itself is not probabilistic). But even if the explanation is legal rather than statistical, if it is not merely formal but has some logic to it—legal logic—it is still correct to extend it to other areas.

The law of fixed is indeed learned from a verse that speaks about capital cases, but the exemption there belongs to the category of "majority"—just as, for example, I might learn the laws of kindling from adding oil to a Hanukkah menorah [whether that counts as having lit the Hanukkah light or not] to the category of kindling on the Sabbath, which is a labor [according to one opinion] punishable by stoning—and then ask how one can learn from rabbinic Hanukkah law to the labors of the Sabbath. [If I remember correctly, they do compare such things, but the principle is clear and very common in Talmudic discussions: to learn rules in this way when what is learned does not stem from the same subject as the source that teaches it. [And perhaps this is also what the Rabbi means.]

That is exactly what he asked: why is this a general principle? It is stated in capital cases, and the extension can be challenged: what is unique about capital cases? We are far more lenient there because of the rule "and the congregation shall save."

It seems to me that the answer is simple. The leniencies in capital law apply when there is doubt whether the defendant is in fact the murderer, and that is where "and the congregation shall save" is said—to save the one who may be innocent. But there is no logic at all in exempting despicable murderers because of a line of reasoning based on "and the congregation shall save," because we really do not want to save murderers. Therefore, if according to the ordinary rules the person who threw a stone into a group is considered a murderer (because the majority are Israelites), it cannot be that he is exempted because of "and the congregation shall save." Unless this is a general Torah rule that anything fixed is like half-and-half, and this murderer's exemption is not because of "and the congregation shall save" (except indirectly: according to the ordinary rules he is considered half-and-half, and then our friend "and the congregation shall save" comes along and saves him just as it saves others).

Very unlikely. "And the congregation shall save" speaks about situations where it is clear that he is the murderer.

Could you give an example of such a situation? (Then my suggestion would be completely refuted and not just very unlikely.) What I wrote—that "and the congregation shall save" deals with cases where there is still doubt, and with an extreme insistence on clarifying every possible argument for acquittal—I did not write from knowledge, only from the details I saw in the Mishnah at the beginning of tractate Sanhedrin, that the differences are that capital cases require twenty-three, and one may not open with arguments for conviction, and one may not incline based on one judge for conviction, and one may not reverse to conviction, etc.

What is the logic of saving someone when it is clear that he is the murderer? If the Torah does not want us to kill murderers, then it should not tell us to kill murderers.

You can see a survey here: https://asif.co.il/wpfb-file/avn-15-9-pdf/
For example, in disjoint testimony there is no real doubt about the facts, and there are others as well.
And if in capital cases we follow the majority, then there is no real concern about factual doubt. Yet in a fixed case we still do not follow the majority.

I see that my suggestion really is refuted. But I did not understand what you meant at the end of your remarks—"Yet in a fixed case we still do not follow the majority"—because from that I inferred (according to the suggestion that was refuted) that the law of fixed cannot be because of "and the congregation shall save," as a way of avoiding reliance on the majority in such a case and saving the murderer, but rather that it is a general rule in the Torah.

My point was that even in a case of majority there is doubt, and nevertheless we follow it. Jewish law does not hesitate to rely on a majority.

The Rabbi's words: "But even if the explanation is legal rather than statistical, if it is not merely formal but has some logic to it—legal logic—it is still correct to extend it to other areas."

That is exactly what I am asking about, and I will spell it out:

Either way:

If the reasoning is not strong enough on its own, that is, without a verse, then it can be learned only:
– in the case described in the verse
– and in cases that are no more novel than the case of the verse (if there are any such cases)
So we should learn it only in capital cases and in other cases that have "and the congregation shall save" (if there were any such cases).

If the reasoning is strong enough and does not require anchoring in a verse, then the fact that the Torah chose to mention it דווקא in a verse speaking about capital law, where there is "and the congregation shall save," proves that this reasoning should not be applied in other laws where there is no "and the congregation shall save."

And to that I answer again: the reasoning is not enough to introduce the law without a verse. But after the verse, we understand that this indeed is the reasoning. And since there is reasoning here, and I understand its essence to be a fundamental innovation in the laws of majority and not a leniency in capital law, I apply it to all other contexts in Jewish law.
We are repeating ourselves. We've exhausted it.