When does $\Pi_2$-reflection on $X$ fail to imply iterated $\Pi_1$-reflection on $X$?

Question

Let lowercase Greek letters denote ordinals. Recall from Richter and Aczel's "Inductive definitions and reflecting properties of admissible ordinals", for a set of formulae $\Gamma$ and a class of ordinals $X$, the definition of $\Gamma$-reflection on $X$: an ordinal $\alpha$ is $\Gamma$-reflecting on $X$ if for every formula $\phi(\vec x)$ from $\Gamma$ with parameters $\vec x$ from $L_\alpha$, we have $(L_\alpha\vDash\phi(\vec x))\rightarrow\exists(\beta\in\alpha\cap X)(L_\beta\vDash\phi(\vec x))$. In particular I'm interested in the cases where $\Gamma$ is the Levy hierarchy's $\Pi_1$ or $\Pi_2$.

We may also iterate this convention: (this is not in the paper) for $n$ a natural number say an ordinal $\alpha$ is $0$-fold $\Gamma$-reflecting on a class $X$ iff $\alpha\in X$, and $\alpha$ is $n+1$-fold $\Gamma$-reflecting on $X$ iff $\alpha$ is $\Gamma$-reflecting on the class $\{\sigma\mid\sigma\textrm{ is }n\textrm{-fold }\Gamma\textrm{-reflecting on }X\}$. This "thins $X$ down" by repeatedly applying $\Gamma$-reflection to it, obtaining a thinner and thinner class after each iteration.

The same paper argues that $\Pi_2$-reflection is a recursive analogue of stationarity, but recently I learned a point at which their behavior diverges, which is that $\Pi_2$-reflection on a class $X$ need not imply iterated $\Pi_1$-reflection on $X$, e.g. $2$-fold $\Pi_1$-reflection on $X$. This is in contrast to being stationary on a class $X$, which necessarily implies being an $n$-fold limit point of $X$ for all natural $n$.

How much is known about cases where in ths way reflection fails to imitate stationarity? For instance, are there cases where we may iterate $\Pi_1$-reflection further before its failure, which mimic stationarity more closely? Is a concrete "topographical" analysis of the points of failure possible, like the section in Marek and Srebrny's 1973 paper "Gaps in the Constructible Universe" that gives an elucidating bird's-eye view of the structure of the ordinals under study?

Currently I'm only aware of one such example, where we have an ordinal $\alpha$ that is $\Pi_2$-reflecting on a class $X$, and also $\Pi_1$-reflecting on $X$, but not $\Pi_1$-reflecting on the class of ordinals that are $\Pi_1$-reflecting on $X$ (i.e. we fail to have $2$-fold $\Pi_1$-reflection on $X$):

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Let $\alpha$ be the least nonprojectible ordinal, and $X=\{\beta\in\alpha\mid L_\beta\prec_{\Sigma_1}L_\alpha\}$. A proof which I take no credit for of the fact that $\alpha$ is $\Pi_2$-reflecting on $X$ goes as follows: let $\phi(x,\vec p)$ be an arbitrary $\Sigma_1$ formula such that $L_\alpha\vDash\forall x\phi(x,\vec p)$, where $\vec p$ is a finite sequence of parameters. In the context of reflecting down from $L_\alpha$, we restrict all parameters of $\phi$ (namely $x$ and all entries of $\vec p$) to $L_\alpha$. By choice of $\phi$ we have $\forall(x\in L_\alpha)\phi(x,\vec p)$. Until the end of this paragraph fix an arbitrary ordinal $\beta\in X\cap\alpha$. We also have $\forall(x\in L_\beta)\phi(x,\vec p)$ (this follows from the standard definition of $\forall(x\in y)\psi$ as $\forall x(x\in y\rightarrow\psi)$.) For any $x\in L_\beta$, we have $\phi(x,\vec p)$ (i.e. $L_\alpha\vDash\phi(x,\vec p)$ since we're working in $L_\alpha$). Since $L_\beta\prec_{\Sigma_1}L_\alpha$, we have $L_\beta\vDash\phi(x,\vec p)$. As $x\in L_\beta$ is arbitrary, we showed $\forall(x\in L_\beta)(L_\beta\vDash\phi(x,\vec p))$, which I believe is enough.

According to some literature such as Rathjen's "The Art of Ordinal Analysis", $X$ has order type $\omega$, therefore it has precisely one limit point, namely $\alpha$. Since $\Pi_1$-reflection on a class is equivalent to being a limit point of that class (according to RichterAczel74), we have that $\alpha$ is $\Pi_1$-reflecting on $X$.

However, for $\Pi_1$-reflection on the set of limit points of $X$, there are no limit points of $X$ below $\alpha$, so a formula such as $\phi(x)\equiv\ulcorner x=x\urcorner$ fails to reflect down to some $L_\xi$ with $\xi\in\alpha$ that's a limit point of $X$.

Answer 1

Re the question "For instance, are there cases where we may iterate $\Pi_1$-reflection further before its failure, which mimic stationarity more closely?": Yes: For ordinals $\alpha$ let $X_\alpha=\{\beta<\alpha\bigm|L_\beta\preccurlyeq_1 L_\alpha\}$. Then for each $n<\omega$, there are ordinals $\alpha<\omega_1^L$ such that $\alpha$ is $\Pi_2$-reflecting for $X_\alpha$ and $n$-fold non-$(n+1)$-fold $\Pi_1$-reflecting for $X_\alpha$. For this just note that if the ordertype of $X_\alpha$ is of the form $\gamma+\omega^{n+1}$ for some $\gamma$ then this holds for $\alpha$, much as in the $n=0$ case you mentioned, where the ordertype of $X_\alpha$ was just $\omega$. But it is easy to find many such $\alpha$ - just note that $X_\alpha$ is always closed below $\alpha$, and the ordertype of $X_{\omega_1^L}$ is $\omega_1^L$, and if $\alpha\in X_{\omega_1^L}$, then $X_\alpha=X_{\omega_1^L}\cap\alpha$. Note that these considerations also lead to iterated $\Pi_1$-reflection of higher orders than just finite.

Remark: There are related things going on in the construction of $\square$-sequences in $L$.