In this question, I asked about how slowly the Fibonacci terms in a algebra of elementary embeddings could grow, but I did not inquire if there was a limit to how quickly the Fibonacci terms could grow.
Define the Fibonacci terms $t_{1}(x,y)=y,t_{2}(x,y)=x$ and where $t_{n+2}(x,y)=t_{n+1}(x,y)*t_{n}(x,y)$ for $n\geq 1$. Let $\mathcal{E}_{\lambda}$ be the set of all elementary embeddings $j:V_{\lambda}\rightarrow V_{\lambda}$. Then $\mathcal{E}_{\lambda}$ can be endowed with an operation $*$ defined by $j*k=\bigcup_{\alpha<\lambda}j(k|_{V_{\alpha}})$. Recall that $\mathrm{crit}(j)$ is the least ordinal where $j(\alpha)>\alpha$.
Suppose that $n$ is a natural number. Then does there exist a natural number $m$ such that if $\lambda$ is a cardinal and $X$ is an $n$-generated subalgebra of $\mathcal{E}_{\lambda}$ and $$\gamma=\min(\{\mathrm{crit}(t_{2n+1}(j,k))|j,k\in X,\mathrm{crit}(j)\leq\mathrm{crit}(k)\}),$$ then $$|\{\mathrm{crit}(j)|j\in X\}\cap\gamma|\leq m?$$ Are there any good lower or upper bounds on the size of the least such $m$?
If $\gamma<\lambda$ and $\gamma$ is a limit ordinal, then recall that $j\equiv^{\gamma}k$ if and only if $j(x)\cap V_{\gamma}=k(x)\cap V_{\gamma}$ for each $x\in V_{\gamma}$. The motivation for this question comes from the fact that a negative answer shows that the algebras $\mathcal{E}_{\lambda}/\equiv^{\gamma}$ generalize to a purely algebraic setting in two very different ways and I still do not know if these two definitions are equivalent.
