In short, how does a 5-watt CB produce 12W PEP in the SSB mode?

Once the carrier and one sideband are filtered out of the signal to
produce a SSB signal, and this is fed to the CB's RF amplifier (which
would generate 4-5W of AM), why wouldn't the result just be 4-5W of
SSB?

It is often described as the power being focused into one sideband,
and the increased bandwidth efficiency is clear, but: if the power
input to the final amplifier is the same as with AM, where do the
"extra watts" come from? Or is the amplification somehow applied in a
different way to the SSB signal? Not sure how to conceptualize this.

Thanks!