Periodicity in the cumulative hierarchy

We investigate the structure of rank-to-rank elementary embeddings at successor rank, working in $\mathrm{ZF}$ set theory without Axiom Choice. Recall that set-theoretic universe is naturally stratified by cumulative hierarchy, whose levels $V\_\alpha$ are defined via iterated application power operation, starting from $V\_0=\emptyset$, setting $V\_{\alpha+1}=\mathcal{P}(V\_\alpha)$, and taking unions limit stages. Assuming $$ j:V\_{\alpha+1}\to V\_{\alpha+1} a (non-trivial) embedding, we show fundamentally different $V\_{\alpha+1}$: $j$ definable parameters over $V\_{\alpha+1}$ iff $\alpha+1$ an odd ordinal. The definability uniform $j$. also give characterization $j:V\_{\alpha+2}\to V\_{\alpha+2}$ terms ultrapower maps certain ultrafilters. For ordinals $\lambda$, prove if $j:V\_\lambda\to V\_\lambda$ $\Sigma\_1$-elementary, then not $V\_\lambda$ parameters, $\beta<\lambda$ $j:V\_\beta\to fully $\in$-cofinal, likewise definable. If there Reinhardt cardinal, for all sufficiently large $\alpha$, indeed $j:V\_\alpha\to V\_\alpha$, therefore hierarchy eventually periodic (with period 2).