Possible answer to the 123 black hole

Here is a explanation of the mathemagical black hole (123) in my opinion. If we consider the transformation on x as a function f(x), such that f(x) = E(x) O(x) T(x)

where E(x) = Number of even digits in x, O(x) = Number of odd digits in x and T(x) = Number of digits in x

Then for all x,

f(x) < x

So eventually as we recurse this process, at some point f(x) will fall below 1000. In that case the only possibilities for E(x) O(x) and T(x) are

1 2 3

2 1 3

3 0 3

Each of the above in the next iteration will give 123. Hence the proof.