AM signal =
sin 2fct + ½ m cos (2fc - 2fm)t - ½ m cos (2fc + 2fm)t
where: fc is the carrier frequency, fm is the modulating frequency
You recognize the first term as the carrier, the second term as the lower sideband and the third term as the upper sideband.
A simple view of single sideband would discard the carrier term and one of the sidebands. This could be implemented, for example, using a sharp filter. Examining the term that is left shows a constant sinusoid (cosine function) at a frequency above or below the original carrier frequency by an amount equal to the constant modulating frequency.
During our discussion I postulated that if I transmit a pure (single frequency) audio tone on my SSB transmitter and did not tell you where I was tuned (e.g. 3740 kHz), you could not tell, by tuning your receiver, what audio tone frequency I was transmitting. This is supported by an illustration in the Modulating Sources chapter of my copy of the Handbook. It shows a spectrum analyzer display with a single peak and an oscilloscope view of a constant amplitude RF envelope. The caption labels it as “an unmodulated carrier or single-tone SSB signal”.
Another way of saying this is: Suppose another ham tunes up with a carrier at 3738 kHz. What do you hear at 3740 kHz on your SSB receiver on LSB? You hear a 2 kHz audio tone. Now suppose I transmit a 2 kHz audio tone on my SSB transmitter on LSB on 3740 kHz. If you are listening on LSB on 3740 kHz you hear a 2 kHz tone. The effect is the same.
In the absence of more rigorous analysis, I maintain that a constant pure audio tone transmitted on SSB is equivalent to an unmodulated carrier. Of course, the real world equipment generating such a signal will add some distortion, making it not precisely identical to an unmodulated carrier. Also, a voice waveform is highly complex, with multiple varying frequencies and amplitudes.
The more rigorous treatment of SSB (example) uses math that is equivalent to the phasing method of generating SSB. It is a notch up in level of complexity compared to what is presented in the Handbook. It is also the basis of many communications systems that we take for granted today: broadband Internet access, digital TV, cell phones, etc.