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We prove that for each $m\\ge1$, exactly $F(m+1)$ odd integers in $\\{1,\\ldots,2^m\\}$ have the property that their orbit under $T$ avoids the residue class $4\\pmod6$ during steps $2,\\ldots,m$, where $F(m+1)$ is the $(m+1)$-th Fibonacci number; the proportion decays at rate $(\\varphi/2)^m$, $\\varphi=(1+\\sqrt{5})/2$. The proof uses the directed graph $G$ of Collatz transitions modulo $6$ and its unique absorbing strongly connected component $G'=G[\\{1,2,4,5\\}]$. 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We prove that for each $m\\ge1$, exactly $F(m+1)$ odd integers in $\\{1,\\ldots,2^m\\}$ have the property that their orbit under $T$ avoids the residue class $4\\pmod6$ during steps $2,\\ldots,m$, where $F(m+1)$ is the $(m+1)$-th Fibonacci number; the proportion decays at rate $(\\varphi/2)^m$, $\\varphi=(1+\\sqrt{5})/2$. The proof uses the directed graph $G$ of Collatz transitions modulo $6$ and its unique absorbing strongly connected component $G'=G[\\{1,2,4,5\\}]$. 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