Let stratified $\sf Z$ have all axioms of $\sf Z$-$\sf Reg.$ with Infinity stipulated by existence of a Dedekind infinite set, and with Separation restricted to use only stratified formulas, where those are defined after Quine's New Foundations. AC is not assumed.

We add a new primitive unary function symbol $U$ to the language of set theory.

The idea is to have a hierarchy $$\cdots \subset U_{-2} \subset U_{-1} \subset U_0 \subset U_1 \subset U_2 \subset \cdots$$, where each stage (rank) $U_{i+1}$ is the powerset of its predecessor $U_{i}$

I call this "inverted hierarchy" because ranks can have an infinite descent.

Axiom I: $\forall i \in \mathbb Z: U_i \subset U_{i+1} \land U_{i+1} = \mathcal P (U_i) $

We call each $U_i$ an ultra-rank, denoted "$u$.rank"

Axiom II: $\forall x \exists i \in \mathbb Z: x \in U_i$

Axiom III: $\forall i \in \mathbb Z \exists x: x=U_i$

Axiom VI: $ \forall i \in\mathbb Z: \forall \alpha \, ( V_\alpha \in U_i)$

Where $V_\alpha$ is defined in the usual manner, i.e. as the $\alpha^{th}$ iterative power of $\varnothing$.

[NOTE]: $\mathbb Z$ is internal. The $+$ operator on $\mathbb Z$ is defined so that $i+1$ is one type higher than $i$ in stratification. For example we take $\mathbb Z$ to be the set of all acyclic infinitely iterative singletons $\{\{\{\cdots \space \cdots\}\}\}$, fix one of its elements to be $0$ and define $i+1 = \{i\}$. Needed axioms to support such internal grasp of $\mathbb Z$ to be added. $U$ can be used in Separation. The arguments of $U$ receive equal type indices during stratification, so $U_i$ and $i$ receive the same type.

If the above is consistent can we have an external automorphism that shifts $u$.ranks one-step downwardly? That is, there is an external automorphism $j$ such that for some $i \in \mathbb Z$, we have: $U_{j(\alpha) + 1} = U_\alpha$, for all $\alpha \in \mathbb Z, \alpha < i $.