Elements of $\bigcap_n \iota(E)U^{p^n}$: closure of $\iota(E)$ times prime-to-$p$ torsion

Let $p$ be a prime, $K$ a number field, $U$ the semilocal units at $p$, $\iota:E\to U$ the diagonal embedding of the global units, and $\bar E=\bigcap_{n}\iota(E)\,U^{p^n}$ (the platform's unitClosure). Then $u\in \bar E$ if and only if $u=c\,t$ where $c$ lies in the topological closure $\overline{\iota(E)}$ of $\iota(E)$ in $U$ and $t\in U$ is a torsion element of order prime to $p$. In other words

where $\mu'(U)$ denotes the prime-to-$p$ torsion of $U$. This corrects the identification $\bar E=\overline{\iota(E)}$ in the source: the intersection may be strictly larger (e.g. $K=\mathbb Q$, $p=7$ gives $\mu_6$ versus $\{\pm1\}$), though only by a finite group, so the $\mathbb Z_p$-ranks agree.