TITLE: Only admissibles start gaps in clockable ordinals
QUESTION [7 upvotes]: This is a question about ITTM model introduced by Hamkins et al. In this paper it is proven that no admissible ordinal is clockable, so it either starts or lies within a gap in clockable ordinals. I seek for reference concerning sort of opposite result - that if an ordinal starts such gap, then it's necessarily admissible.
P.D.Welch claims here to have proof of this fact, however he doesn't give one.
I'm asking for either a reference to full proof or at least a sketch of the methods used in obtaining this result.
Thanks for help in advance!

REPLY [3 votes]: I'd like to add to Joel's generous and thorough answer. He is right to recall that if $\beta$ is clockable, then $\beta$ is writable in order type at most $\beta$-steps (thus improving the Lemma).  One can argue this directly and it sounds like you have found the paper, but for completeness it is in:
P.D. Welch ``Characteristics of discrete transfinite Turing machine models:
halting times, stabilization times, and Normal Form Theorems'' in Theoretical Computer Science, vol. 410, Jan. 2009, 426-442 doi: 10.1016/j.tcs.2008.09.050,
But actually there is a nice argument, which involves seeing how an ITTM can compute directly codes for levels of the $L$-hierarchy up to $\Sigma$, at the same time as computing the theories of each such level.
Theorem. There is an ITTM program so that for limit $\alpha$ (i) it has at time $\omega^2\cdot\alpha$ on a section of its tape: a set $T_\alpha \subseteq \mathbb{N}$ so that (via some usual goedel coding) the complete $\Sigma_2$-theory of $(L_\alpha, \in)$ is uniformly r.e. in $T_\alpha$.
(ii) at time $\omega^2\cdot\alpha +\omega\cdot2$ on another section of the tape there is a code $E_\alpha$ for $(L_\alpha, \in)$  (meaning that $(\omega, E_\alpha) \cong   (L_\alpha, \in)$).
Because we are below $\Sigma$, (uniformly) recursive in the $\Sigma_2$-theory of $(L_\alpha, \in)$ is a wellorder of order-type $\alpha$. This leads to:
Corollary If a limit $\alpha$ is clockable, then a code for $\alpha$ is writable by time $\alpha + \omega$ 
(just add on the extra steps to write out (a code for) the wellorder).
With some more fiddling around and consideration of cases that ``$+\,\omega$'' can be removed, and also for successor cases... 
The theorem appears in:
S-D Friedman & P.D. Welch Two Observations concerning Infinite Time Turing Machines, in Bonn International Workshop on Ordinal Computability, Ed: I. Dmitriou, J. Hamkins & P. Koepke, 2007, Bonn, 44-48.
http://www.math.uni-bonn.de/ag/logik/events/biwoc/report.pdf
There the theorem is proven for the Jensen $J_\alpha$ hierarchy, but with work it can be done for the $L_\alpha$'s too.