Coprimality selects primitive roots
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A selected residue is a label for a wave whose phase has exact order twelve. Set \(\zeta=e^{2\pi i/12}\). The phase \(\zeta^a\) first returns to \(1\) after \(12/\gcd(a,12)\) powers. Thus the coprime labels \(1,5,7,11\) select exactly the primitive twelfth roots.
\[
\gcd(a,12)=1\quad\Longleftrightarrow\quad\operatorname{ord}(\zeta^a)=12.
\]
Arithmetic has supplied a precise selector. Nothing probabilistic has happened at this stage.