arXiv:2610.31806v1 Announce Type: new Abstract: Let $p$ be a prime and $n\geq 1$. We prove that the Tate-Frobenius of the $\operatorname{K}(n)$-local sphere induces an equivalence between $L_{n-1}L_{\operatorname{K}(n)}Y$ and $(L_{\operatorname{K}(n)}Y)^{t\Sigma_p} \cong (L_nY)^{t\Sigma_p}$ for any finite $p$-local type $n-1$ spectrum $Y$ with trivial $\Sigma_p$-action.
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