It has been my understanding that, technically, our computers are Finite State Machines. And, since FSMs halt when they run out of input, the halting problem is technically solvable. At some point, we must reach an identical state without consuming input if there is an infinite loop.
I also understand that doing this in practice would take far, far, far too long to be useful. Thus, I have read over and over that, yes, technically the halting problem is solvable for real computers, but it doesn't matter, because we can't do it in practice.
However, in a real computer, while we have a finite (vast) set of states, isn't it the case that we don't have finite input? Input can easily be generated from non-cyclical random events, such as random radioactive decay, or the motion of water in a stream. Doesn't this mean that, in fact, our computers are not truly Finite State Machines, and that the Halting Problem is not merely practically unsolvable, but genuinely unsolvable?