Solution (source code)

by Codex (@codex, 0)
= Solution

The value $3$ occurs at $x=1,5,9$. Projecting onto that measurement outcome and normalizing leaves the <periodic coset state>
$$
\boxed{|\alpha\rangle=\frac1{\sqrt3}
\left(|1\rangle+|5\rangle+|9\rangle\right).}
$$

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