This question implies that we have fixed: (i) a particular enumeration of Ordinal Turing machines; (ii) a particular way to encode an ordinal by an infinite binary sequence.

The class of $(1)$-machines is defined as the $1$st iteration of the strong jump operator for Ordinal Turing Machines. That is, a machine is equipped with an oracle able to answer any question of the following form (note that $(0)$-machines are Ordinal Turing Machines with no oracles):

Does an $i$-th $(0)$-machine halt given an infinite binary sequence $x$ as the input?

The ordinal $alpha_1$ is defined as the supremum of ordinals *eventually writable* by $(1)$-machines with *empty input*.

Let $m_i(x)$ denote a computation performed by an $i$-th $(0)$-machine, assuming that the input is $x$. If $m_i(x)$ eventually writes a countable ordinal $alpha$, then $M_i(x) = alpha$. Otherwise, $M_i(x) = 0$.

Then the function $F(i)$ is defined as follows: if the value of $sup {M_i(x) | x in mathbb{R}}$ is a countable ordinal, then $$F(i) = sup {M_i(x) | x in mathbb{R}};$$ otherwise, $F(i) = 0$. Here “$x in mathbb{R}$” implies that we take into account *all* infinite binary sequences.

The ordinal $alpha_2$ is defined as follows: $$alpha_2 = sup {F(i) | i in mathbb{N}}.$$.

The ordinal $eta$ is defined as the least ordinal $gamma$ such that $L_gamma$ and $L$ have the same $Sigma_2$-theory (see part 3 of Lemma 3.11 in the paper “Recognizable sets and Woodin cardinals: Computation beyond the constructible universe”). As far as I understand, $eta$ is equal to the supremum of ordinals *eventually writable* by Ordinal Turing Machines ($(0)$-machines) with empty input.

Question: which ordinal is larger, $alpha_1$ or $alpha_2$? Is any of these two ordinals larger than $eta$?