  Using Patchkoria--Pstr\k{a}gowski's version of Franke's algebraicity theorem,
we prove that the category of $K_p(n)$-local spectra is exotically equivalent
to the category of derived $I_n$-complete periodic comodules over the Adams
Hopf algebroid $(E_*, E_*E)$ for large primes. This gives a finite prime result
analogous to the asymptotic algebraicity for $\mathrm{Sp}_{K_p(n)}$ of
Barthel--Schlank--Stapleton.
