= Solution
For uncountable regular $\kappa$, let $\theta=\operatorname{otp}(C_\zeta)$. Since $C_\zeta$ is a <club set> in $\zeta$, $\operatorname{cf}(\theta)=\operatorname{cf}(\zeta)=\kappa$, so $\theta\ge\kappa$.
If $\theta>\kappa$, its <cofinality> ensures $\theta>\kappa+\omega$: <ordinals> strictly between $\kappa$ and $\kappa+\omega$ are successors, and $\kappa+\omega$ has <cofinality> $\omega$. In the continuous increasing enumeration $c$ of $C_\zeta$, take $\gamma=c(\kappa+\omega)$. It is a limit point of $C_\zeta$ and has <cofinality> $\omega<\kappa$. Coherence gives $C_\gamma=C_\zeta\cap\gamma$, with <order type> $\kappa+\omega$ and size $\kappa$. This contradicts the size clause for <ordinals> of <cofinality> below $\kappa$. Thus
$$
\boxed{\operatorname{otp}(C_\zeta)=\kappa\quad\text{for uncountable regular }\kappa.}
$$
\b[The printed hypotheses need this qualification.] At $\kappa=\omega$, choose $C_\zeta=\zeta$ for every nonzero countable <limit ordinal>. These <club sets> are coherent, and the small-cofinality clause is vacuous. But at $\zeta=\omega^2$ their <order type> is $\omega^2$, not $\omega$. The standard <square principle> includes an order-type bound $\operatorname{otp}(C_\zeta)\le\kappa$, which also repairs this countable case.
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