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To be more precise, suppose $f\\in{_2\\pi_n^s}$ pulls back to $g\\in{_2\\pi_n^s}P$ through the Kahn-Priddy map $\\lambda:QP\\to Q_0S^0$ such that $g$ projects nontrivially to an element $g'\\in{_2\\pi_n^s}P_{t(n)}$ with $h(g')=0$ where $h:{_2\\pi_*}QP_k\\to H_*QP_k$ is the unstable Hurewicz map, and $t(n)=\\lceil n/2\\rceil$. 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