Define the Fibonacci terms $t_{n}$ for all $n\geq 1$ by letting $t_{1}(x,y)=y,t_{2}(x,y)=x$, and $t_{n+2}(x,y)=t_{n+1}(x,y)*t_{n}(x,y)$ for all $n$. The Fibonacci terms are used in order to approximate the composition operation on an algebra of elementary embeddings with application.

Let $\mathcal{E}_{\lambda}$ denote the set of all elementary embeddings $j:V_{\lambda}\rightarrow V_{\lambda}$, and let $\mathcal{E}_{\lambda}^{+}$ denote the set of all non-identity elementary embeddings from $V_{\lambda}$ to $V_{\lambda}$. Recall that $\mathcal{E}_{\lambda}$ is endowed with a self-distributive operation $*$ called application defined by $j*k=\bigcup_{\alpha<\lambda}j(k|_{V_{\alpha}})$. Let $\equiv^{\gamma}$ denote the congruence on $\mathcal{E}_{\lambda}$ where $j\equiv^{\gamma}k$ precisely when $j(x)\cap V_{\gamma}=k(x)\cap V_{\gamma}$ for all $x\in V_{\gamma}$.

Suppose that $j,k\in\mathcal{E}^{+}_{\lambda}$ and $\mathrm{crit}(k)\geq\mathrm{crit}(j)$. Then what are some lower bounds of the necessarily finite cardinality $$|\langle j,k\rangle\equiv^{\mathrm{crit}(t_{2n+1}(j,k))}|?$$ What are some lower bounds of the necessarily finite cardinality $$|\{\mathrm{crit}(l)|l\in\langle j,k\rangle,\mathrm{crit}(l)\leq\mathrm{crit}(t_{2n+1}(j,k))\}|?$$

didanswer both Aaron and Gerry. For instance, Aaron's second comment was in response to Joseph's reply, and Joseph replied to it as well. (And, after a few days, he deleted his comments.) $\endgroup$ – Andrés E. Caicedo Oct 26 '17 at 14:10