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As explained in the book, how far the induction can reach (that is, how strong can we obtain a core model) is intimately tied up with how much determinacy we can prove (and so, measuring the strength of the core models becomes a problem of the strength of determinacy assumptions). This is because to obtain closure under the appropriate operators requires that we prove different instances of "mice capturing", for which determinacy appears essential. Nowadays, we understand that the relevant descriptive set theory and inner model theory are so intimately related, that we talk of Descriptive inner model theory.

As explained in the book, how far the induction can reach (that is, how strong can we obtain a core model) is intimately tied up with how much determinacy we can prove (and so, measuring the strength of the core models becomes a problem of the strength of determinacy assumptions). This is because to obtain closure under the appropriate operators requires that we prove different instances of "mice capturing", for which determinacy appears essential. Nowadays, we understand that the relevant descriptive set theory and inner model theory are so intimately related that we talk of Descriptive inner model theory.

The answer probably depends on how we define core model. At the level of "there are no Woodin cardinals in any inner model", we can finally show that core models exist, provably in $\mathsf{ZFC}$. The result was known before, of course, but we needed extra assumptions (such as: There is a measurable cardinal $\kappa$) that allowed us to build the model, but only locally (say, in $V_\kappa$). That $\mathsf{ZFC}$ suffices has recently been established by Jensen and Steel, in $K$ without the measurable.

Making this precise requires that we be able to "patch" together local structures into global ones, which is typically why we need additional assumptions (such as, $V$ is closed under sharps). The precise nature of these assumptions obviously ends up depending on the setting we work with, but we can axiomatize the whole process. This leads to the core model induction which, finally, we can truly describe as an induction, The specific requirement on $V$ then becomes closure under appropriate "mice operators". An excellent description of this approach, and how far it can get, can be seen in the Schindler-Steel monograph The core model induction.

The expectation Sargsyan has is that the Solovay hierarchy should at least reach as far as the large cardinal hierarchy. The core models we build this way would have strength measured in terms of the Solovay sequence, and to identify their strength in traditional large cardinal terms would then require a further translation. This is now described in some detail in various places, see for instance his paper on The core model induction beyond $L(\Bbb R)$ in the arXiv. As you see there, the final translation (in this case, from "there is an inner model $M$ containing $\mathbb R$ and satisfying $\mathsf{AD}_{\mathbb R}+\Theta$ is regular" to something past "there is an inner model of $\mathsf{ZFC}$ where there is a proper class of Woodin cardinals and a proper class of strong cardinals") requires a non-trivial amount of work. It usually builds on results of Neeman (on models with a Woodin limit of Woodin cardinals) and Jensen-Schimmerling-Schindler-Steel on Stacking mice.

The answer probably depends on how we define core model. At the level of "there are no Woodin cardinals in any inner model", we can finally show that core models exist, provably in $\mathsf{ZFC}$. The result was known before, of course, but we needed extra assumptions (such as: There is a measurable cardinal $\kappa$) that allowed us to build the model, but only locally (say, in $V_\kappa$). That $\mathsf{ZFC}$ suffices has recently been established by Jensen and Steel, in $K$ without the measurable.

Making this precise requires that we be able to "patch" together local structures into global ones, which is typically why we need additional assumptions (such as, $V$ is closed under sharps). The precise nature of these assumptions obviously ends up depending on the setting we work with, but we can axiomatize the whole process. This leads to the core model induction which, finally, we can truly describe as an induction, The specific requirement on $V$ then becomes closure under appropriate "mice operators". An excellent description of this approach, and how far it can get, can be seen in the Schindler-Steel monograph The core model induction.

The expectation Sargsyan has is that the Solovay hierarchy should at least reach as far as the large cardinal hierarchy. The core models we build this way would have strength measured in terms of the Solovay sequence, and to identify their strength in traditional large cardinal terms would then require a further translation. This is now described in some detail in various places, see for instance his paper on The core model induction beyond $L(\Bbb R)$ in the arXiv. As you see there, the final translation (in this case, from "there is an inner model $M$ containing $\mathbb R$ and satisfying $\mathsf{AD}_{\mathbb R}+\Theta$ is regular" to something past "there is an inner model of $\mathsf{ZFC}$ where there is a proper class of Woodin cardinals and a proper class of strong cardinals") requires a non-trivial amount of work. It usually builds on results of Neeman (on models with a Woodin limit of Woodin cardinals) and Jensen-Schimmerling-Schindler-Steel on Stacking mice.

Part of what Sargsyan has done is to identify what seems to be the right hierarchy of mice within models of determinacy. From earlier work of Woodin we knew that the relevant $\mathsf{ZFC}$ models to study where the relativizations of the class $\mathsf{HOD}$ to the models of determinacy. The appropriate hierarchy of hybrid mice we now call hod mice. Their "hybrid" nature means that they are not pure mice, but add fragments of the relevant iteration strategies to them. Their strength is then measured via the Solovay sequence of the determinacy models we consider. In turn, we use these mice to prove determinacy of stronger models, so the process is truly inductive. Here, the Solovay sequence is the sequence of "local" $\theta$ ordinals: $\theta_0$ is the smallest non-zero ordinal not the surjective image of $\mathbb R$ via an ordinal definable map. We can then define $\theta_1$ as the smallest non-zero ordinal not the surjective image of $\mathbb R$ via maps that are ordinal definable in $A$, where $A$ is any set of reals of Wadge degree $\theta_0$, etc. The hierarchy stops once we reach (true) $\Theta$, the first non-zero ordinal not the surjective image of $\mathbb R$.

Part of what Sargsyan has done is to identify what seems to be the right hierarchy of mice within models of determinacy. From earlier work of Woodin we knew that the relevant $\mathsf{ZFC}$ models to study were the relativizations of the class $\mathsf{HOD}$ to the models of determinacy. The appropriate hierarchy of hybrid mice we now call hod mice. Their "hybrid" nature means that they are not pure mice, but add fragments of the relevant iteration strategies to them. Their strength is then measured via the Solovay sequence of the determinacy models we consider. In turn, we use these mice to prove determinacy of stronger models, so the process is truly inductive. Here, the Solovay sequence is the sequence of "local" $\theta$ ordinals: $\theta_0$ is the smallest non-zero ordinal not the surjective image of $\mathbb R$ via an ordinal definable map. We can then define $\theta_1$ as the smallest non-zero ordinal not the surjective image of $\mathbb R$ via maps that are ordinal definable in $A$, where $A$ is any set of reals of Wadge degree $\theta_0$, etc. The hierarchy stops once we reach (true) $\Theta$, the first non-zero ordinal not the surjective image of $\mathbb R$.