A basis change makes the hidden distinction observable
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The signed state recovers all four coprime labels. The unsigned state recovers none.
The exact unnormalized DFT of the unsigned integer signal is
\[
(16,0,4,0,4,0,16,0,4,0,4,0).
\]
After state normalization and unitary transformation, its probabilities are \(4/9\) at \(0,6\), \(1/36\) at \(2,4,8,10\), and zero elsewhere. Since \(|c_{12}(n)|\) has period six, its odd Fourier labels vanish.