>If the ask is about the probability of rolling a sequence and you've already rolled some, then you can't ignore what's been rolled so far.

This is exactly what you do. If you want to know the probability of rolling 6 6s in a row, you calculate it for that. If you've rolled 3 out of 6 and want to know the probability of making it to 6 straight, you do the very basic 6 - 3 = 3, and then crunch the probability for only 3 rolls, because you are only actually calculating the probability of 3 rolls.

Your "preservation of information" is a simple subtraction. It's not some mystical connection influencing future rolls. All those future rolls are still entirely independent of the ones you've already made.

The probability of rolling the same on 6 dice, and the probability of rolling 3 of the same having already rolled 3 are identical, because you're still rolling 6 dice.

The probability of rolling 3 more of the same after any arbitrary amount is not the same question. This is where your confusion is coming from. Here, you only need the probability of 3 rolls, no matter how many you've rolled previously.