The arithmetic signal is a coherent amplitude vector
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Normalization makes the signal a unit vector without changing its relative phases. Define \(u_k=1/2\) for \(k\in U_{12}\) and zero otherwise. Then \(\|u\|=1\), and
\[
\psi=Fu,\qquad\psi_n=\frac{c_{12}(n)}{\sqrt{48}},\qquad F^\dagger\psi=u.
\]
The normalization is exact because \(\sum_n c_{12}(n)^2=48\). The matrix \(F\) is unitary, so no probability mass is lost during this coherent change of basis.