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\noindent The least fixed point of 
\begin{align*}
    h & : (\mathbb{Z}_\bot \to \mathbb{Z}_\bot) \to (\mathbb{Z}_\bot \to \mathbb{Z}_\bot) \\
    h(F)(n) & = 
    \begin{cases}
        1 & n = 1 \\
        (\phi^{n/2} + (-\phi)^{-n/2})\, F_{n/2} & n \textrm{ is even} \\
        \sqrt[3]{\frac{\phi^{3n+1}-\phi F_{3n+1}}{10} + \frac{\sqrt{25 \phi^{3n+1} - 25\phi F_{3n+1} - 20}}{50}} + \sqrt[3]{\frac{\phi^{3n+1}-\phi F_{3n+1}}{10} - \frac{\sqrt{25 \phi^{3n+1} - 25\phi F_{3n+1} - 20}}{50}} & \textrm{otherwise} \\
    \end{cases}
\end{align*}
is the Fibonacci sequence if and only if the Collatz conjecture is true.

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