What are the chances of finding the digits of φ within π ?
I realize we can find irrational numbers within π. Even by simply shifting the decimal, we can end up with new irrational numbers, which we can say are "within π".
But what are the chances of finding the digits of φ in it? Say, could at some point the digits of π be identical to those of φ and continue in the exact same pattern from there on?
On
If $\varphi$ and $\pi$ were identical from some point on they would differ by a rational number. That would make $\pi$ algebraic, so it can't happen.
It's likely that any finite sequence of digits occurs in $\pi,$ because we suspect that $\pi$ is a normal number.
For your last question, the answer is that the digits cannot eventually match exactly. If they did, then there would be rational numbers $a$ and $b$ such that $\pi=a+b\varphi$.
The rest of the question is that we don't know for example whether at some point there is never again a $9$ in the remainder of the decimal expansion of $\pi$. We are not in a position to determine whether all of the digits of $\varphi$ as a subsequence can be found.