Consistency of \(V= \text{HOD}\) with the wholeness axiom (Q1568713)

The Wholeness Axiom (WA) is an axiom schema that can be added to the axioms of ZFC in an extended language \(\{\in, j\}\), and that asserts the existence of a nontrivial elementary embedding \(j: V\to V\). The well-known inconsistency proofs are avoided by omitting from the schema all instances of Replacement for \(j\)-formulas. We show that the theory \(\text{ZFC}+ V=\text{HOD}+ \text{WA}\) is consistent relative to the existence of an \(I_1\) embedding. This answers a question about the existence of Laver sequences for regular classes of set embeddings: Assuming there is an \(I_1\)-embedding, there is a transitive model of \(\text{ZFC}+ \text{WA}+\) ``there is a regular class of embeddings that admits no Laver sequence''.