Prolongable satisfaction classes and iterations of uniform reflection over $\mathrm{PA}$

We study nonstandard models of iterations of uniform reflection over $\mathrm{PA}$ that contain a partial inductive satisfaction class. Our original motivation was to prove an analogue of the Enayat and Visser theorem that each partial inductive satisfaction class can be prolonged in an end-extension to a full satisfaction class (proved independently and unpublished) in the case of the $\Delta_0$ inductive satisfaction class.
To achieve our goal we introduced the notion of a prolongable satisfaction class: in words a class $S$ on $M$ is prolongable if there exists an elementary end extension $N$ of $M$ and a partial inductive satisfaction class $S'$ on $N$ which “covers” $M$ and prolongs $S$. $S$ is $n$-prolongable if this can be repeated $n$-times starting from $S$.
It turned out that the existence of an $n$-prolongable partial inductive satisfaction class characterizes the models of $n$-iterated uniform reflection over $\mathrm{PA}$ ($\mathrm{UR}^n (\mathrm{PA})$):
Theorem 1. For a nonstandard model $M \vDash \mathrm{PA}$ having a partial inductive satisfaction class $S$ the following are equivalent

• $M \vDash \mathrm{UR}^n (\mathrm{PA})$
• $S$ can be restricted to an $n$-prolongable satisfaction class.

As a limit we obtain our desired theorem:
Theorem 2. For a nonstandard model $M \vDash \mathrm{PA}$ having a partial inductive satisfaction class $S$ the following are equivalent

• $M \vDash \mathrm{UR}^{\omega} (\mathrm{PA})$
• There exist a restriction $S'$ of $S$, an elementary end extension $N$ of $M$ and a full $\Delta_0$ inductive satisfaction class $S''$ on $N$ such that $S' \subseteq S''$.

Our methods consist in internalizing the standard existence arguments for partial inductive satisfaction classes. Moreover, Theorem 2, provides a new proof of conservativity of the theory $\mathrm{CT}_0$ over $\mathrm{UR}^{\omega} (\mathrm{PA})$ (the first was presented in [1]).