They say it does not activate an electrical circuit but only changes the current. Though it is not really clear what is special, in the questioner’s eyes, about SMS more than any other change in current.

Indeed, this is just an example of any other change in current.
There is no building through changing current, so why is it prohibited?

Logically, changing a current in a way that creates a new phenomenon (typing letters) is not merely a change but a creation.

On what grounds is it prohibited?

Writing, rabbinically prohibited, or creating something new. It is creating something new, since the prohibition of creating something new is general and was not stated specifically only about current. Therefore, just as they prohibit creating current, there is no reason they would permit creating writing by means of changing current.

Rabbi,
So if we are not talking about writing a text message, but about another action on the phone, for example opening an app, or playing on the phone, and the like — then there is no basis here for creation, right?

B. I did not understand at all how people want to compare electricity to creating something new. How can one claim that there is such a concept as creating current?
1. It is not perceptible at all — unlike smell, and you cannot see it. And even then, this is at most just changing current, not generating current entirely.
2. Nor can it be compared to creating smell (where, as far as I know, it is permitted to continue adding smell).
3. It is an entirely temporary creation; the moment you open the circuit, the current stops completely.

A. Why is creating an app through changing current different from creating writing? Even in the laws of writing, making a mark is a derivative of writing.
B. 1. It is very perceptible through its results. Precisely for that reason I said that the change in current in itself is not the problem, but rather the perceptible results it brings about. The same is true in generating current: what is perceptible is the result. By the way, that is why in my opinion going down in a Sabbath elevator, where your extra weight increases the current in the elevator, is not a problem on the Sabbath. The reason is that the elevator travels in exactly the same way, and the added current does not create anything distinctly perceptible.
2. Same here. The smell is the result. Also in generating current, what is perceptible is the result of the current and not the current itself.
3. Smell too is temporary. In any case, it is significant enough to be prohibited rabbinically.

So in practice, what follows from your words is that if there is a device with a button such that when you press it, a program is activated that performs a very complicated calculation and uses millions of transistors, etc., as long as I do not see or hear any result from it, then there is no prohibition at all, correct?

You are not looking at the atomic level but at the dimension of the result. As the verse says, “Far from the eye, far from the heart.”

2. How do you define the prohibition of creating something new? Very briefly. I did not find any lectures by you on the drive about this topic, only about the prohibition of electricity as a prohibition of building.

1. Indeed. If I remember correctly, Rabbi Rabinovitch in Siach Nachum raises such a line of reasoning in his well-known permission to use a magnetic card to open hotel doors (although there it seems more that there is a result — a door that has opened).
2. Bringing something new into the world.

1. Thanks! But there is something new here, after all — and we hold that “new is prohibited by the Torah…”
2. And is opening an app something new? What is new about that?

1. ?
2. You create a situation in which it becomes possible to do all sorts of things. How is that different from writing a text message (you said that if not writing a text message, then let us discuss opening an app)? If that is creating something new, then that is too.

Before you ask about writing a text message on the Sabbath, ask about writing a text message on a weekday.
After all, in order to write a text message you need a non-kosher phone, and anyone who has such a device is suspected of sexual immorality and disqualified from testimony, according to the ruling of our master, the prince of Torah, Rabbi Chaim Kanievsky.
Besides that, he is also breaching a fence and treating the words of the Sages with contempt, and his punishment is death, and he causes harm to himself and to his household, and causes his descendants not to be accepted into kosher educational institutions.
In short, someone who is not afraid to permit himself texting on a weekday can also permit himself to text on the Sabbath.

And if someone needs to send messages because of danger to life, should he try to write as briefly as possible? (Of course, if that will not interfere with saving lives.)

Yes. The claim that this is entirely permitted, and therefore there are no limitations, sounds problematic to me.