This was posted as a side question in Formal definition of this ordinal? and was split as a separate question based upon suggestion in comments there.

Assume an ordinary ORM model (call it $C_1$). Suppose we add an extra instruction of the form to $u:=u+\omega_1$ (where $u$ is a variable) to the definition of an ordinal register style program (and one or two more basic commands in case they are necessary). Or just consider an OTM with a single parameter $\omega_1$ given to it. Call this model $C_2$. Then define $p$ as the supremum of halting times of these programs (no input). Now as the answer linked below uses $\theta$ to denote $\omega_{\omega_1+1}^{CK}$, I will use the same symbol for uniformity.

Which of the following possibilities can actually occur: (a) $p < \theta$ (b) $p = \theta$ (c) $p > \theta$.

Now given the discussion in comments, assuming the answer given in the linked topic to be correct, it is clear that the possibility (b) above is ruled out. But what about the other two possibilities. Another thing is confinality difference. Consider the possibility $p>\theta$. For that to be true, not only would it have to be true that there exists a program (in model $C_2$) which halts beyond $\theta$ on empty input. But it would also have to be true that there exists some value $\beta<\theta$ such that for all $t$ (where $\beta<t<\theta$) there exists no program which halts at time $t$ on empty input (if this weren't true $cf(\theta)$ would be countable).

Is there a nice/easy way to visualize $\beta$?

The ordinal $q$ was also defined in the previous version of question as follows: One could also consider well-orders of $\omega_1$ that these programs can calculate. For example, say $q$ is the least upper bound of all the ordinals $x$ such that: "there is some program (from model $C_2$) that computes a well-order relation ($\omega_1^2$ to $\{0,1\}$) describing well-order of/on $\omega_1$ with order-type $x$".

One can also ask about the relation of $q$ with $p$,$\theta$ (certainly it seems to me that $p \geq q$ should be true).

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