Compactness for true cofinalities of products

Determine whether, when HOD is cofinality-correct, κ is a strong limit singular cardinal of uncountable cofinality, a club sequence ⟨κ_α∣α<ω_1⟩ belongs to HOD, and the true cofinalities of the corresponding products agree stationarily below κ, the true cofinalities of the full products in HOD and V must agree at κ.

Background

The paper discusses coding failures of GCH and the long scales witnessing them into HOD. It asks whether stationary agreement of the true cofinalities of initial products necessarily propagates to the product indexed by cf(κ). This is a possible PCF-theoretic analogue of the compactness phenomena established elsewhere in the paper.

References

Suppose that $\HOD$ is cofinality-correct and $\kappa$ is a strong limit singular cardinal with $(\kappa)\geq \omega_1$. Let $\langle \kappa_\alpha\mid \alpha < \omega_1\rangle\in \HOD$ be a club on $\kappa$. Assuming that ${\alpha <\kappa\mid \mathrm{tcf}(\prod{\HOD}_{\beta < \alpha}\kappa_\beta)=\mathrm{tcf}(\prod_{\beta < \alpha}\kappa_\beta)}$ is stationary, must $\mathrm{tcf}(\prod{\HOD}_{\alpha < (\kappa)}\kappa_\alpha)=\mathrm{tcf}(\prod_{\alpha < (\kappa)}\kappa_\alpha)$.

— Compactness phenomena in HOD and the Optimality of Magidor's Covering theorem  (2608.24190 - Benhamou et al., 25 Aug 2026) in Section "Open questions", third displayed Question