Existence of V_{κ_{n+1}} and inaccessibility of κ_{n+1} under cofinal Reinhardt embeddings in ZF−_j

Ascertain, in the setting of ZF−_j with a cofinal Reinhardt embedding j: V → V and letting κ_n = j^n(crit(j)), whether the rank-initial segment V_{κ_{n+1}} exists and whether κ_{n+1} is inaccessible; clarify the large-cardinal-like properties of these critical points without assuming Replacement or Powerset.

Background

In the proof of Hayut’s result that a cofinal Reinhardt embedding cannot exist over ZF−j, the argument references properties of critical points κ_n and notes that certain steps would be straightforward if V{κ0} (and by extension V{κ_{n+1}}) existed and had inaccessibility properties. The author explicitly remarks that these existence and inaccessibility facts are unclear in the choiceless, replacement-weakened context, highlighting a gap in the structural understanding of ranks around the critical sequence.