Chromatic Cardinalities via Redshift

Using higher descent for chromatically localized algebraic $K$-theory, we show that the higher semiadditive cardinality of a $\pi$-finite $p$-space $A$ at the Lubin-Tate spectrum $E_n$ is equal to the higher semiadditive cardinality of the free loop space $LA$ at $E_{n-1}$. By induction, it is thus equal to the homotopy cardinality of the $n$-fold free loop space $L^{n}A$. We explain how this allows one to bypass the Ravenel-Wilson computation in the proof of the $\mathrm{\infty }$-semiadditivity of the $T(n)$-local categories.