Compactness phenomena in HOD

We prove two compactness theorems for HOD. First, if $\kappa $ is a strong limit singular cardinal with uncountable cofinality and for stationarily many $\delta <\kappa $ , $(\delta ^+)^{\mathrm {HOD}}=\delta ^+$ , then $(\kappa ^+)^{\mathrm {HOD}}=\kappa ^+$ . Second, if $\kappa $ is a singular cardinal with uncountable cofinality and stationarily many $\delta <\kappa $ are singular in $\operatorname {\mathrm {HOD}}$ , then $\kappa $ is singular in $\operatorname {\mathrm {HOD}}$ . We also discuss the optimality of these results and show that the first theorem does not extend from $\operatorname {\mathrm {HOD}}$ to other $\omega $ -club amenable inner models.