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Using Patchkoria--Pstragowski's version of Franke's algebraicity theorem, we prove that the category of Kₚ (n) -local spectra is exotically equivalent to the category of derived Iₙ-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 Sp₊䂹 (₍) of Barthel--Schlank--Stapleton.
Torgeir Aambø (Tue,) studied this question.