Answer

There is a dispute here among the medieval authorities (Rishonim) and the halakhic decisors, so what would my opinion add for you on this matter? Most of the public does not follow Rabbeinu Tam, and if you don’t have your own position, you can certainly rely on that.
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Questioner (another one):
If he remains in doubt, he’ll be forced to be stringent because in a Torah-level doubt we rule stringently, at least regarding Torah-level prohibited labors at the end of the Sabbath. I think the question is whether one must take Rabbeinu Tam’s view into account because this is seemingly a Torah-level doubt, or whether one may from the outset be lenient like the Geonim—and if so, why?
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The Rabbi:
When you have your own position, you should act accordingly. When you have no position and there is a dispute among halakhic decisors / medieval authorities (Rishonim), you are in a state of doubt. There are several rules in the laws of doubt, only one of which is that a Torah-level doubt is treated stringently and a rabbinic-level doubt leniently. For example, in monetary law: the burden of proof is on the one seeking to extract money from another; or they divide it; or “whoever is stronger prevails.” And in prohibitions, passive omission is preferable; one follows custom; and so on. Here there is a clear custom of the majority of the public, and those who are concerned for Rabbeinu Tam are a very small minority. Therefore one may follow the custom as though Jewish law has been ruled that way, and then you are no longer in doubt, so the rule of Torah-level doubt requiring stringency does not apply. It seems to me that the rule “Torah-level doubt is treated stringently” is the last rule to be used—only when all the other rules are not applicable.
This is the reason we do not always act stringently in accordance with every opinion that appears among the halakhic decisors / medieval authorities (Rishonim) (as the Brisker approach would have it), and when there is a clear custom we follow it. The custom is considered like a ruling that selected one of the opinions, and therefore we are not in doubt. This is the concept of “error in legal judgment,” which the Talmud in tractate Sanhedrin explains as referring to someone who goes against “the accepted practice of the world” (= what is commonly accepted in practice, and that is considered as though Jewish law was ruled that way). The Talmud there does not distinguish between doing so leniently or stringently (although in monetary law, of course, there is no such thing as leniency and stringency).
Of course, one is allowed to be stringent in accordance with another opinion (and regarding being lenient in accordance with another opinion, I hesitate whether that is permitted, and perhaps he would be considered one who combines the leniencies of this view and that view and is called wicked), and perhaps that is even proper. But there is no obligation.
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Questioner:
Seemingly, according to what you said, regarding the sky-blue thread we should have followed the accepted practice of the world, and yet there you said that one should apply “Torah-level doubt is treated stringently,” and not “follow the custom.” What is different there from here?
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The Rabbi:
There the custom is a foolish custom. There is no halakhic position that exempts one from the sky-blue thread; rather, there are those who decided that this is a Zionist craziness, and therefore exempt themselves from the commandment. The positions that are taken into account are reasoned positions, not nonsense. The doubt there is factual—whether this really is the sky-blue thread—and I am not familiar with good arguments claiming that it is not.
Besides, as I wrote, if I have my own position I must follow it. The whole discussion applies only when I have no position, and then perhaps one should apply the laws of doubt.

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Ofek:

It is written that there is no halakhic position exempting one from the sky-blue thread, but it says explicitly that white does not invalidate sky-blue and sky-blue does not invalidate white—how much more so when there is doubt whether this is really the sky-blue thread.

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The Rabbi:
That is a common mistake. The fact that the sky-blue thread does not invalidate the white does not mean that the sky-blue thread is voluntary. Tefillin for the arm also do not invalidate those for the head (in that same Mishnah in Menachot). Does that mean that tefillin for the head are also not obligatory? When two things do not invalidate one another, that means that if you did not do one, you still fulfilled your obligation of the other. But the first one, in itself, can still be a full obligation. Therefore
when you put on fringes of white alone, you have neglected the positive commandment of the sky-blue thread, but that did not invalidate the white, which you did fulfill. And if there is doubt whether this is the sky-blue thread, the regular rules of doubt apply.
See details in my article here.
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Ofek:
But that is exactly the point. Since the sky-blue thread does not invalidate the white, and there is doubt regarding the correctness of this sky-blue thread, there is no obligation here—and that is precisely why this halakhah exists. After all, you cannot obligate a person in a commandment he cannot fulfill, and according to the laws of doubt he is not obligated at all, because the moment there is doubt about a commandment there is no obligation to fulfill it. The tefillin analogy/example distorts the issue, because we have no doubt about what tefillin are or whether the head exempts the arm, etc. The correct analogy/example is to ask whether there is an obligation to put on Rabbeinu Tam tefillin. And the Vilna Gaon’s answer is valid here as well. All the more so when there is more than one view regarding the sky-blue thread, and unlike tefillin, it is impossible to fulfill all the possibilities.
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The Rabbi:
Hello Ofek.
I explained the matter, and I will repeat it again.
First we have to examine the case where the sky-blue thread is certainly genuine. In that case, the fact that the sky-blue thread does not invalidate the white certainly does not mean that the sky-blue thread is voluntary. Agreed? That is why I brought the “distorted” example from tefillin. I would be happy if you explained to a poor mind like mine what exactly is distorted here (aside from the interpretation you gave my words).
So when there is doubt about the sky-blue thread, one should fulfill it out of doubt like any other commandment. There is not the slightest connection to the question of whether it invalidates the white. This is an ordinary case of doubt regarding a positive commandment. Whatever you say about a doubtful positive commandment—and I mentioned that the later authorities (Acharonim) disagreed about this, but according to the great majority of them one should be stringent even with doubt about a positive commandment—you would say here as well.
The existence of several views regarding the sky-blue thread is also irrelevant. The windings and wrappings, even if they are not correct, do not invalidate it according to almost any position. So what does that have to do with our discussion?
The example of Rabbeinu Tam tefillin is not relevant here, because by the same logic you could also ask about Rashi tefillin. So what then? Clearly, the accepted custom does not see this as a doubt at all; rather there is a decision in accordance with Rashi, and the question is only whether to be stringent like Rabbeinu Tam. This is not a question in the ordinary laws of doubt.
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Ofek:
Thank you for the further explanation, but you didn’t bring anything new that I hadn’t already understood from the earlier remarks. First of all, as is well known, a Torah-level doubt is treated stringently. There is doubt that this is the sky-blue thread, and the moment a person wears fringes with doubtful sky-blue thread, he enters the category of a doubtful fulfillment displacing a certain one—or in other words, wearing sky-blue thread that is not the real sky-blue thread causes a certain nullification of the commandment of fringes. Therefore, in such a case and similar ones, one should not fulfill the doubtful element at the expense of the certain one. And as stated, since there is no proof that this is sky-blue thread, one should not wear it, and the reason is clear. As for what you wrote regarding tefillin, that “the accepted custom does not see this as a doubt but as a decision in accordance with Rashi, and the question is only whether to be stringent like Rabbeinu Tam…” that is completely incorrect. There are many halakhic decisors, from the Arizal to Rabbi Ovadia of blessed memory, who determine that this is not a doubt but an obligation. So as you wrote, this is not an ordinary doubt—but not in the way you meant, and again this is not a valid metaphor.
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The Rabbi:
I did not understand your remarks. Even if the sky-blue thread is not genuine, it does not nullify the commandment of fringes. And regarding tefillin, I disagree. But as I already remarked on your words, I could just as well have asked about Rashi tefillin too.
Be that as it may, this whole discussion does not have the slightest connection to the question of invalidation, and that is what I was talking about.