= Solution
Fix $\delta<\kappa^+$ with $\operatorname{cf}(\delta)>\omega$. The limit points of $C_\delta$ form a <club set> in $\delta$. For each such point $\gamma$, coherence gives
$$
\operatorname{otp}(C_\gamma)=\operatorname{otp}(C_\delta\cap\gamma).
$$
As $\gamma$ increases through those limit points, these order types strictly increase. Hence the <club set> contains at most one member of a fixed fiber $E_\xi$. Removing an initial segment past that member leaves a <club set> disjoint from $E_\xi$. Therefore $E_\xi\cap\delta$ is not stationary. This proves \b[each $E_\xi$ is non-reflecting], by the <nonreflection of fixed order-type fibers>.
For uncountable regular $\kappa$, part (a) gives type $\kappa$ at <cofinality> $\kappa$, and the size clause gives type less than $\kappa$ at smaller cofinalities. All nonzero <limit ordinals> below $\kappa^+$ fall into one of these cases. Thus
$$
\lim(\kappa^+)=\bigsqcup_{\xi\le\kappa}E_\xi.
$$
The index set $\kappa+1$ has cardinality $\kappa$, so this is a disjoint union of $\kappa$ non-reflecting <subsets>. Empty fibers can be omitted; if nonempty pieces are required, a fiber of size $\kappa^+$ can be split further, since nonreflection is inherited by <subsets>. Such a fiber exists because $\kappa$ sets of size at most $\kappa$ cannot cover $\kappa^+$ points.
At $\kappa=\omega$, the nonreflection definition is vacuous for every <subset> of $\omega_1$, since no smaller <ordinal> has uncountable <cofinality>. The partition conclusion is therefore still true by an arbitrary countable partition, although the particular fibers from the weak printed square assumptions need not cover all <limit ordinals>. Under the standard order-type-bounded square definition, the displayed fiber partition works in this case too.
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