Assuming that $p$ denotes a particular OTM program, let $x$ denote a real which is accidentally writable on the initial segment of length $\beta$ of the tape of $p$ at a particular time $\alpha$, where $\alpha$ is an arbitrary countable ordinal.
Question: does there exist an OTM program $P$ which (given an input real with a suitable encoding of the combination of $\alpha, \beta, x$) can simulate the computational process of $p$ and provide the correct answers to any question of the following type:
Is it true that $x$ is accidentally writable on the initial segment of length $\beta$ of the tape of $p$ at time $\alpha$?
I am interested in two situations: (i) $\beta = \omega$ (the smallest limit ordinal); (ii) $\beta$ is an arbitrarily large countable ordinal.
That is, if $x$ does, in fact, occur on the initial segment of length $\beta$ of the tape of $p$ at time $\alpha$, the $P$ program should output "Yes". Otherwise, the program should always output "No."