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Recall question "Can we have this sequence where choice fails and returns?"

Can that theory be extended with requiring the $\mathcal V_n$'s to fulfill a choiceless large cardinal extension of $\sf ZF$? For example, we add a primitive unary function $\mathcal J$, then axiomatize: $$\forall n \in \omega \, ( \mathcal J_n: \mathcal V_n \prec \mathcal V_n)$$, where $\prec$ refers to non-trivial elementary embedding over signature $\{=,\in, \mathcal J_n\}^\dagger$. Now, can we add to it the axiom: $$\forall n \in \omega: (\mathcal V_n, \in^{\mathcal V_n}, \mathcal J_n) \models \sf ZF_j + Reinhardt \ cardinal$$

If we can do that, then we can take the least super-transitive model that satisfy that theory, and its ordinal index would be strictly larger than Reinhardt's cardinal, and yet compatible with choice!

The bigger question is if we can use this method to render any choiceless large cardinal be interpretable in a theory in which choice holds and proves the existence of a larger cardinal?

$^\dagger$ by that I mean the symbols $\mathcal J$ and $n$ must be used together in the formula and only appearing as $\mathcal J_n$

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