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Let $J$ denote Jensen's modification of the constructible hierarchy. For an ordinal $\alpha$ and an $n\in\mathbb N^+$, let $\rho_n^{J_\alpha}$ denote the $\Sigma_n$-projectum of $J_\alpha$, the least $\delta\leq\alpha$ such that there is a $\Sigma_n(J_\alpha)$-definable subset of $\omega\rho$ not in $J_\alpha$. (Note that $x<y$ implies $\rho_x^{J_\alpha}\geq\rho_y^{J_\alpha}$.) For a finite increasing sequence of positive naturals $s=(s_1,\ldots,s_m)$, call $s$ a pattern of projecta if there is some $\alpha$ where the points of decrease in the sequence $(\rho_i^{J_\alpha})_{1\leq i\leq m}$ are just the integers in $s$, i.e. for any $i\in\mathbb N^+$, $\rho_i^{J_\alpha}>\rho_{i+1}^{J_\alpha}$ iff $i$ is an entry of $s$. In this way a pattern of projecta encodes when the sequence of $\Sigma_i$-projecta will "drop" (if $i$ is in the sequence), or will "stay" (if $i$ is not).

What ordering results if we order the patterns of projecta by the size of the least $\alpha$ where $J_\alpha$ has that pattern? We will call this ordering $<_\rho$: set $s<_\rho t$ if the least $\alpha$ where $J_\alpha$ has pattern of projecta $s$ is less than the least $\beta$ where $J_\beta$ has pattern of projecta $t$.

What are some of the basic properties of $<_\rho$, e.g. its order type? If $<_\rho$ is a well-ordering, what ordinal is it isomorphic to?

What I do know is that all finite sequences of naturals are patterns of projecta, a relevant question is MO #67933, and a relevant paper is "Patterns of Projecta" by A. Krawczyk (1981). In the MO question's accepted answer, Philip Welch gives an explicit construction, for arbitrary sequence $s$, of a $J_\alpha$ whose pattern of projecta is $s$. However, at each step of the Skolem hulling for $1\leq j\leq m-1$, I don't know any guarantee that $\pi(H_m)$ is the minimal $J$-rank with the necessary segment of the pattern so far, most importantly at the $j=m-1$ step. So this construction may not help us determine $<_\rho$ due to it not giving information about the minimal $\alpha$.

Let $J$ denote Jensen's modification of the constructible hierarchy. For an ordinal $\alpha$ and an $n\in\mathbb N^+$, let $\rho_n^{J_\alpha}$ denote the $\Sigma_n$-projectum of $J_\alpha$, the least $\delta\leq\alpha$ such that there is a $\Sigma_n(J_\alpha)$-definable subset of $\omega\rho$ not in $J_\alpha$. (Note that $x<y$ implies $\rho_x^{J_\alpha}\geq\rho_y^{J_\alpha}$.) For a finite increasing sequence of positive naturals $s=(s_1,\ldots,s_m)$, call $s$ a pattern of projecta if there is some $\alpha$ where the points of decrease in the sequence $(\rho_i^{J_\alpha})_{1\leq i\leq m}$ are just the integers in $s$, i.e. for any $i\in\mathbb N^+$, $\rho_i^{J_\alpha}>\rho_{i+1}^{J_\alpha}$ iff $i$ is an entry of $s$. In this way a pattern of projecta encodes when the sequence of $\Sigma_i$-projecta will "drop" (if $i$ is in the sequence), or will "stay" (if $i$ is not).

What ordering results if we order the patterns of projecta by the size of the least $\alpha$ where $J_\alpha$ has that pattern? We will call this ordering $<_\rho$: set $s<_\rho t$ if the least $\alpha$ where $J_\alpha$ has pattern of projecta $s$ is less than the least $\beta$ where $J_\beta$ has pattern of projecta $t$.

What are some of the basic properties of $<_\rho$, e.g. its order type? If $<_\rho$ is a well-ordering, what ordinal is it isomorphic to?

What I do know is that all finite sequences of naturals are patterns of projecta, a relevant question is MO #67933, and a relevant paper is "Patterns of Projecta" by A. Krawczyk (1981). In the MO question's accepted answer, Philip Welch gives an explicit construction, for arbitrary sequence $s$, of a $J_\alpha$ whose pattern of projecta is $s$. However, for this construction to be used in computing $<_\rho$, we need information about the minimal $\alpha$ in particular. Sufficient for this would be if at each step of the Skolem hulling for $1\leq j\leq m-1$, $\pi(H_m)$ were the minimal $J$-rank with the necessary segment of the pattern so far.