Regularity in the P-forcing extension when o(α) = α+

Determine whether, in the forcing extension by the projected Prikry forcing P_{U,t} defined in Section 6, every cardinal α with o(α) = α+ remains regular; that is, establish whether cf^{V^{P_{U,t}}}(α) = cf(α) holds whenever the length parameter o(α) of the coherent sequence of supercompact measures at α equals α+.

Background

Section 6 introduces the projected forcing P_{U,t}, obtained from the supercompact Radin forcing R_{U,t} via a projection. The paper develops a detailed analysis of cardinals in the corresponding generic extension V[H] by P_{U,t}, proving preservation and cofinality results (Theorem 6.17).

Earlier, Lemma 5.8 establishes that in the R_{U,t}-generic extension V[G], if o(α) ∈ {α, α+}, then cf(α) = ω. For the projected P_{U,t} extension, the authors can show various preservation results, but the precise behavior at cardinals α with o(α) = α+ remains unsettled. They explicitly conjecture regularity in this case, i.e., that α is not singularized by P_{U,t}.

Here o(α) denotes the ordinal assigned to α by the function o from Lemma 3.4, which equals the length of the coherent sequence of measures U_{α,i} at α.