Confusion with the proof of constraints for a particular adiabatic quantum evolution
The norms are the usual norms of a ket vector, i.e. $|||a\rangle||^2 = \langle a|a\rangle$.

What did we do on the bottom of page 3? It is really running the argument from the An Analog Analogue paper backwards.

Imagine we start from the state $|g_x\rangle$ (a particular ground state of the final Hamiltonian $H_{P}$, related to $|x\rangle$), and evolve backwards with an inverse algorithm. Similarly to AAA, we are looking at whether evolving backwards with no purpose (without knowledge of $x$) makes the evolution any different from evolving with a Hamiltonian that involves $x$. We show that for a successful algorithm, it is necessary that the quantity $S(0)$ must be large enough. However, running back in time from $T$ to $0$, this $S(t)$ doesn't change much, which allows us to show that $T$ must be large.