Thanks for the clarification IFA,

I think it's clearer now, but I'm not sure it's solved.

So If I understand you, you say that

big Number * 1 million % 26 * 1 million

is the same as

big Number % 26 ?

I'm not sure the probabilities are preserved.
In this case yes because we can factor 1million in each side.

But take for instance a bet of 1.1 ether, difficulty of 118181817.

so now it would be
big Number * 1 million % 118181817

your proof does not work for this case, you can't divide anymore without altering the result.