The vaccine itself or booster doesn't remain in the system ready to fight off any potential infection, it doesn't work like that.
It works by introducing the immune system to the virus which kick starts it into making antibodies.
The body's immune system then remembers and is primed ready to do the same again should the need arise.
How long it takes for the body to forget in the instance of Covid is unknown and wont be known for years yet.
Some vaccines are for life such as TB and some for many many years like tetanus and some are not that long lived like flu.
I personally can't see how a double dose would be ineffective to the point where it would need boosting after only 10 weeks which is what the current system is suggesting but what do i know.