No, it's not known that Pi is normal. This is a very, very open question. We don't even have a way of tackling that question at the moment.
You're correct that if Pi is normal, it's normal in all bases. But that's precisely the point I'm getting at - if Pi is normal, you need to use base 2 for this to work because your codomain is just {0,1}.
Mapping a number to another number such that each digit becomes its own parity is materially different from converting that number to base 2. They aren't the same thing whatsoever, and you can't generally take a normal number and create another normal number this way.
So even if we accept the reasonable conjecture that Pi is normal, you still need to map it to the same base as your codomain in order for its entropy to be preserved. The proposed function is injective and destroys entropy.
> But that's precisely the point I'm getting at - if Pi is normal, you need to use base 2 for this to work because your codomain is just {0,1}.
You can use any base that is a multiple of 2 (like 10) and then apply a parity function to the digits. If pi is normal, then the digit parity sequence is also normal.
Sure you lose some entropy, but in some sense the digits of pi have 0 entropy anyway since they can be calculated. And in the other sense of treating the digits of pi as an unknown random sequence, there is infinite entropy, so throwing away 70% of it doesn't matter.
Yeah someone else pointed out a similar point about the bijection of evens and odds, good point. My initial claim is incorrect, so I'll concede that. I was operating from the idea that there is no bijection between the naturals and a parity check function, but you're right that if one sequence is normal the second set should also be normal regardless of invertibility.
You're correct that if Pi is normal, it's normal in all bases. But that's precisely the point I'm getting at - if Pi is normal, you need to use base 2 for this to work because your codomain is just {0,1}.
Mapping a number to another number such that each digit becomes its own parity is materially different from converting that number to base 2. They aren't the same thing whatsoever, and you can't generally take a normal number and create another normal number this way.
So even if we accept the reasonable conjecture that Pi is normal, you still need to map it to the same base as your codomain in order for its entropy to be preserved. The proposed function is injective and destroys entropy.