Let $J$ denote [Jensen's modification](https://en.wikipedia.org/wiki/Jensen_hierarchy) 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](https://mathoverflow.net/questions/67933), and a relevant paper is "[Patterns of Projecta](https://www.jstor.org/stable/2273621)" by A. Krawczyk (1981). In the MO question's [accepted answer](https://mathoverflow.net/a/81168), 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.



Ordering patterns of projecta by least witness
$<\_\rho$ is a wellorder essentially by definition. The ordertype is $\omega^{\omega}$ (ordinal exponentiation of course).


In fact $s<\_\rho t$ iff either $\mathrm{lh}(s)<\mathrm{lh}(t)$,
or $\mathrm{lh}(s)=\mathrm{lh}(t)$ and $s<\_{\mathrm{lex}}t$,
i.e. letting $i$ be least such that $s(i)\neq t(i)$, we have $s(i)<t(i)$. This easily yields the ordertype $\omega^{\omega}$.


This is just by a slight variant of Philip Welch's construction.
Suppose first $\mathrm{lh}(s)=\mathrm{lh}(t)$.
Let $\beta$ be an ordinal which instantiates the pattern of $t$. Let $i$ be least such that $s(i)<t(i)$. So $\beta$ also instantiates the pattern of $s\upharpoonright i$. But $$\rho\_{t(k)+1}^{L\_\beta}<\rho\_{t(k-1)+1}^{L\_\beta}<\ldots<\rho\_{t(i)+1}^{L\_\beta}<\rho\_{t(i)}^{L\_\beta}=\rho\_{s(i)+1}^{L\_\beta}=\rho\_{s(i)}^{L(\beta)},$$
and the $\rho\_{t(k)+1}^{L\_\beta},\ldots,\rho\_{t(i)+1}^{L\_\beta}$ are each cardinals in $L\_\beta$. Let $\kappa\_j=\rho\_{t(j)+1}^{L\_\beta}$ for $j\in[i,k]$. Now form the hull $$\mathrm{Hull}\_{\Sigma\_{s(i)+1}}^{L\_\beta}(\kappa\_i\cup\{x\}),$$
where $x$ is an appropriate finite set that this is $\Sigma\_{s(i)+1}$-elementary and reflects the relevant information
(including the relevant standard parameters etc). Let $\beta\_i$ be such that $L\_{\beta\_i}$ is the transitive collapse of the hull. Note that $\beta\_i<\beta$, and since $\kappa\_i$ is a cardinal in $L\_\beta$, therefore so
$$\rho\_\omega^{L\_{\beta\_i}}=\kappa\_i=\rho\_{s(i)+1}^{L\_{\beta\_i}}<\rho\_{s(i)}^{L\_{\beta\_i}}$$
and $L\_{\beta\_i}$ instantiates $s\upharpoonright(i+1)$. Since $\kappa\_k<\kappa\_{k-1}<\ldots<\kappa\_i$ and these are cardinals of this model, we can continue forming hulls in this manner at elementarities corresponding to the entries in $s$, and since $\mathrm{lh}(s)=\mathrm{lh}(t)$, there are enough cardinals to do that. This results in an ordinal $\beta\_k$ instantiating $s$, and $\beta\_k<\beta$, as desired.


Now suppose $\mathrm{lh}(s)<\mathrm{lh}(t)$ and let $\beta$ instantiate $t$. Then we get $\mathrm{lh}(t)$-many $L\_\beta$-cardinals $\kappa\_k<\ldots<\kappa\_0<\beta$ as the resulting projecta. Now $\rho\_\omega^{L\_{\kappa\_0}}=\kappa\_0$, so we can proceed like before, taking a $\Sigma\_{s(0)+1}$-elementary hull of $L\_{\kappa\_0}$ in parameters in $\kappa\_1\cup\{x\}$ for an appropriate $x$, etc. This shows that $s<\_\rho t$.


Note also (by calculation as above) that $\sup\_{s}\alpha\_s$,
where $\alpha\_s$ is the least instantiation of $s$, is just the stack of the minimal models of $n$th order arithmetic, for $n<\omega$.


