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Infinite time Turing machine models with tape length $\alpha$, denoted$T_\alpha$, strengthen the machines of Hamkins and Kidder [HL00] with tapelength $\omega$. A new phenomenon is that for some countable ordinals $\alpha$,some cells cannot be halting positions of $T_\alpha$ given trivial input. Themain open question in [Rin14] asks about the size of the least such ordinal$\delta$.  We answer this by providing various characterizations. For instance, $\delta$is the least ordinal with any of the following properties: (a) For some$\xi<\alpha$, there is a $T_\xi$-writable but not $T_\alpha$-writable subset of$\omega$. (b) There is a gap in the $T_\alpha$-writable ordinals. (c) $\alpha$is uncountable in $L_{\lambda_\alpha}$. Here $\lambda_\alpha$ denotes thesupremum of $T_\alpha$-writable ordinals, i.e. those with a $T_\alpha$-writablecode of length $\alpha$.  We further use the above characterizations, and an analogue to Welch'ssubmodel characterization of the ordinals $\lambda$, $\zeta$ and $\Sigma$, toshow that $\delta$ is large in the sense that it is a closure point of thefunction $\alpha \mapsto \Sigma_\alpha$, where $\Sigma_\alpha$ denotes thesupremum of the $T_\alpha$-accidentally writable ordinals.

Reachability for infinite time Turing machines with long tapes