I have a question regarding delay in time controls. I've seen this thread and it doesn't address my doubt.
Let's consider an example from the beginning of a game, to show how simple and Bronstein delay seem different to me. Regular/base time is 10 seconds and delay is 5 seconds.
Now Player X's clock starts. They will take 8 seconds to move. Using simple delay, the delay time runs out plus 3 seconds of regular time, and the clock reads 7 seconds left at the end of the move. Player Y then moves and it's back to X. He takes another 8 seconds to move. The delay time runs out plus 3 seconds of regular time, and the clock reads 4 seconds left at the end of the move. Game goes on.
What if Bronstein delay were added after the move? They start with 10 seconds. After 8 seconds, this drops to 2 seconds but after the first move, Player X gets the 'refunded time' of the delay and thus their clock reads 7 seconds left at the end of the move. (Same as with simple delay). But when it's next his turn, he will again take another 8 seconds to move. If clocks could go negative, his time would drop to -1 seconds, then be 'refunded' by the delay (5) to read 4 seconds. But clocks and time in chess, as far as I've seen, don't work that way. Instead, he has only 7 seconds to make his move; after this time, the clock hits 0. And if he wanted to take 8 seconds to move, then here he would flag and the game would end.
Wikipedia claims that "Bronstein delay and Simple delay are mathematically equivalent." Now, by Wikipedia's definition, Bronstein delay is added after the move (and consists of the lower value between the time allotted for the delay and the time the player took to move). Simple delay consists of running down the delay timer before the regular clock timer starts. These are not mathematically equivalent, as simple delay adds time before the move. You would flag with Bronstein delay added after, on your final turn, where you would have had an extra few seconds with simple delay.
That's not the only issue. The more pressing problem is that I cannot think of a way Bronstein delay could be added before the move (thereby making them equivalent): it seems you cannot know, before the move, whether the player will use more or less time than the delay, and if less, then how much exactly they will use; hence it is intrinsic to the concept of Bronstein delay to add the time after.
How is this reconciled? Or is it to be admitted that there are really two, slightly different, delay systems (i.e. Bronstein and simple US delay are not, and cannot, be identical) and there cannot be otherwise?
Player X has 3 seconds (regular time) left. The delay is 5 seconds. Player Y's turn ends and now X is going to have to play. With simple delay, they will have 5 seconds delay time, plus 3 more seconds--that is, 8 seconds to make a move before they flag. With Bronstein delay, they will have 3 seconds before they flag. I assume that when the clock hits 0, the player flags (it cannot go negative). Now, were they to take 4 seconds to play, they would not get the benefit of the delay at all.