Here's an answer, any corrections welcome:

in sats, the original subsidy is 5 billion = 5*10^9 = 5^10*2^9

That means that after 9 halvings we have a pure power of 5 number of sats. So after the 10th halving that's no longer true and rounding has to happen.

After 9th halving: 9765625 sats (=5^10).

After 10th, 4882812.5 (no longer whole number so pattern breaks).

I believe it always rounds down, but the internal calculation, for the next halving, includes the full number, meaning the pattern 'internally' continues all the way to the end (think of a decimal number as an infinite sequence of powers of 10)