Why is it impossible to do without sidebands in amplitude modulation?

Question

Radio communication

Good time of the day!
In learning how to amplitude modulate an audio signal, I found out that sidebands (or one sideband) are always used in addition to (or without) the carrier signal. In this case, the audio signal is, as it were, decomposed into frequencies, and each individual frequency in a modulated form is transmitted over its corresponding radio frequency. Thus, audible sound in a band up to 3 kHz occupies about 6 kHz of the radio bandwidth with amplitude modulation (or 3 kHz with single-sideband amplitude modulation).
The question arises why not modulate the signal so that it is transmitted at one frequency. An analogy is a gramophone that reproduces a signal recorded on a single track.
I could not find a clear explanation on this topic, and I really hope that there will be someone who can explain in detail why this method will or will not work.

Answer 1

Because the sidebands are generated by the modulating signal - when it is superimposed on the carrier. You already modulate the signal at one frequency, but in space (when you left the antenna) it decomposes into this spectrum - this is physics - you can’t get rid of this in any way, it’s like rivers that flow down.
Roughly speaking, 2 spectra are added - carrier and modulating signals
There are still a bunch of formulas that show how it works.
In general, amplitude modulation is practically history - it is used only in Chinese toys because the circuits are cheap, and now they are switching to modern types of modulation with narrower spectra, because the chips cost a penny.