[petercerno]

> Plus, you can actually empirically compute a finite set of initial digits (a specific Turing machine can be analyzed to see if it terminates or not).

Well, the fact that the number is not computable means that there will exist an index i, for which you will not be able to compute a_i (no matter how hard you try). In other words, you will not be able to analyze the Turing Machine i, i.e. it will not be possible to prove the termination or non-termination of the Turing Machine i.

So this specific digit will be a mystery forever, and you would not be able to compare it to anything.