Assuming that $s$ denotes a particular symbol that can appear on the tape, $\alpha$ denotes any limit ordinal such that $\alpha \ge \omega$ and $C_i[\tau]$ denotes the symbol on the $i$-th cell of the tape at stage $\tau$, I need an answer to the following question:
given an arbitrary limit ordinal $\alpha$ and an arbitrary natural number $n$, what is $C_n[\alpha]$?
As far as I understand, the answer can be formulated as follows:
$C_n[\alpha] = s$ if there exists an ordinal $\beta_1 < \alpha$ such that for any ordinal $\beta_2 > \beta_1$, the condition $C_n[\beta_2] = C_n[\beta_1] = s$ is true (assuming that $\beta_2 < \alpha$);
$C_n[\alpha] = 1$ if an ordinal $\beta_1$ does not exist.
Is the above answer mathematically correct? Are there any missing details?