Proposition: A scalar source family cannot certify all coherences

The identity channel and computational dephasing agree on

for every $\theta$. Yet their half-diamond distance is $1/2$.

Proof. Both maps fix every displayed diagonal state. On $|+⟩$, their outputs are $|+⟩⟨+|$ and $I/2$, with trace distance $1/2$. Conversely, dephasing is $(\mathrm{id}+{\mathrm{Ad}}_Z)/2$, so its half-diamond distance from identity is at most $1/2$.