Let $\mathcal{E}_{\lambda}$ be the set of all elementary embeddings $j:V_{\lambda}\rightarrow V_{\lambda}$. Suppose that $(\alpha_{n})_{n}$ is an increasing cofinal sequence in $\lambda$. Give $\mathcal{E}_{\lambda}$ the metric $d$ where $d(j,j)=0$ and where if $j\neq k$, then $d(j,k)=\frac{1}{n+1}$ where $n$ is the least natural number such that $j|_{V_{\alpha_{n}}}\neq k|_{V_{\alpha_{n}}}$. Then $(\mathcal{E}_{\lambda},d)$ is a complete metric space. The uniform structure of $(\mathcal{E}_{\lambda},d)$ does not depend on the choice of ordinals $\alpha_{n}$.
Let $\mathcal{E}_{n}(\lambda)$ be the set of all $\Sigma_{2n}^{1}$ elementary embeddings $j:V_{\lambda}\rightarrow V_{\lambda}$, and let $\mathcal{E}_{\omega}(\lambda)$ be the set of all elementary embeddings $j:V_{\lambda}\rightarrow V_{\lambda}$ that extend to elementary embeddings $j^{+}:V_{\lambda+1}\rightarrow V_{\lambda+1}$. See 1 for more details on the algebras $\mathcal{E}_{n}(\lambda)$ for $n\in\omega+1$.
Suppose now that $X$ is closed separable subalgebra of $(\mathcal{E}_{\lambda},*,\circ)$. Then $X$ is a Polish space. If $r\in\omega+1$, then what is the descriptive set theoretic complexity of the subset $\mathcal{E}_{r}(\lambda)\cap X$ of $X$? For example, is $\mathcal{E}_{r}(\lambda)\cap X$ a Borel or analytic set?
Recall that the classical Laver table $A_{n}$ is the unique algebra $(\{1,...,2^{n}\},*_{n})$ that satisfies
$x*_{n}(y*_{n}z)=(x*_{n}y)*_{n}(x*_{n}z)$, and
$x*_{n}1=x+1\mod 2^{n}$.
Let $E\subseteq\varprojlim_{n}A_{n}$ be the set of all threads $(x_{n})_{n}\in\varprojlim_{n}A_{n}$ where for all $N$ there exists an $M\geq N$ such that $x_{n}*_{n}2^{N}<2^{n}$ whenever $n\geq M$. Then $E$ is a $G_{\delta}$ subset of the inverse limit $\varprojlim_{n}A_{n}$ (which is compact), so $E$ is a Polish space.
Suppose now that $j\in\mathcal{E}^{+}_{\lambda}$ and $k\in\overline{\langle j\rangle}$. Then define $$\phi_{j}(k)=(x_{n})_{n\in\omega}\in\varprojlim_{n}A_{n}$$ where $k\equiv^{\mathrm{crit}_{n}(j)}j_{[x_{n}]}$. For each elementary embedding $j\in\mathcal{E}^{+}_{\lambda}$, the mapping $\phi_{j}:\overline{\langle j\rangle}\rightarrow E$ is a homeomorphism.
Suppose that $j\in\mathcal{E}_{n}(\lambda)$ and $n\in(\omega+1)\setminus\{0\}$. Does the set $\phi_{j}[\overline{\langle j\rangle}\cap\mathcal{E}_{n}(\lambda)]$ depend on $j$ and $n$? If the set $\phi_{j}[\overline{\langle j\rangle}\cap\mathcal{E}_{n}(\lambda)]$ does not depend on $j$, then is there a purely algebraic description of $\phi_{j}[\overline{\langle j\rangle}\cap\mathcal{E}_{n}(\lambda)]$? If $\phi_{j}[\overline{\langle j\rangle}\cap\mathcal{E}_{n}(\lambda)]$ does depend on $j$, then what purely algebraic and topological axioms (i.e. axioms that do not refer to large cardinals) must the set $\phi_{j}[\overline{\langle j\rangle}\cap\mathcal{E}_{n}(\lambda)]$ satisfy?
