Earlier quoted context omitted.
Why are they identical? A program may use more steps than memory.
A Turing machine can either go left or right on its tape (some versions have a stay step as well, doesn’t matter). If all your program does is step right forever, than it will use the maximum amount of memory, but only ever a finite amount, that is equal to the number of steps taken. So if you don’t have infinite time (you don’t have), and you have big enough memory for the particular use case so that you don’t get O…
Although the naive way of deciding halting requires exponential time in program's memory bound, AGIs will speed that up for many programs by using clever math.