Definitions of nice recursive analogues of cofinality? The cofinality of an ordinal is the minimal order type of a cofinal subset of that ordinal. In the study of admissible ordinals, some concepts that mimic the idea of cofinality appear. One common example is the $\Sigma_1$-cofinality of an ordinal $\alpha$, whose definition is the same as the cofinality of $\alpha$ except instead of considering all $\alpha$-unbounded subsets we only consider those that are also ranges of $\alpha$-recursive functions. (We will call these $\alpha$-recursive sets.)
If we use this definition for "recursive" or "effective" cofinality, we get some results that scale nicely: like how regular ordinals are those where $\alpha=\textrm{cof}(\alpha)$, the "recursively regular" (i.e. admissible) ordinals are those where $\alpha=\Sigma_1\textrm{-cof}(\alpha)$. However, because according to Sacks's Higher Recursion Theory there exists a $\omega_1^{CK}$-recursive injection from $\omega_1^{CK}$ into $\omega$, from it we can construct a $\omega_1^{CK}\cdot 2$-recursive subset of $\omega_1^{CK}\cdot 2$ with order type $\omega$. So $\Sigma_1\textrm{-cof}(\omega_1^{CK}\cdot 2)=\omega$. This isn't what we want, since $\textrm{cof}(\omega_1\cdot 2)=\omega_1$. So $\Sigma_1$-cofinality, while okay in some cases, isn't what we want for a good concept of effective cofinality.
Are there any known concepts of "effective cofinality" that the behavior of $\textrm{cof}$ scales nicely to? I've asked two other people and neither was aware of one that didn't have this unwanted behavior in particular.
 A: I don't actually think that the situation you're describing really is a problem.
Remember that $\Sigma_1$ definability (along with any notion of effectivity, really) is relative to a domain. Modifying Sacks' notation a bit, it's not really that $$\sigma1\mathrm{cf}(\omega_1^{CK})\not=\sigma1\mathrm{cf}(\omega_1^{CK}\cdot 2),$$ but rather that $$\sigma1\mathrm{cf}^{\omega_1^{CK}}(\omega_1^{CK})\not=\sigma1\mathrm{cf}^{\omega_1^{CK}\cdot 2}(\omega_1^{CK}\cdot 2).$$ Here "$\sigma1\mathrm{cf}^\alpha(\beta)$" is the smallest ordinal with a $\Sigma_1$-over-$L_\alpha$ cofinal map into $\beta$, while "$\sigma1\mathrm{cf}(\beta)$" is $\sigma1\mathrm{cf}^\beta(\beta)$. For all limit $\alpha$ and all $\beta\cdot 2\le\alpha$ we clearly have $\sigma1\mathrm{cf}^\alpha(\beta)=\sigma1\mathrm{cf}(\beta\cdot2)$ as desired. Meanwhile, the discrepancy between $\sigma1\mathrm{cf}(\beta)$ and $\sigma1\mathrm{cf}(\beta\cdot 2)$ in general is due to the fact that $L_{\beta\cdot 2}$ is generally somewhat richer - if only in a pathological way! - than $L_{\beta}$.
The key insight that higher recursion theory has here is that we're usually not interested in some overarching notion of effective cofinality full stop, but rather in how deeply a single ordinal can "see" into itself. Change the ordinal, even in seemingly mild ways, and you change that capacity - so you should see (at least sometimes) corresponding changes in combinatorial invariants.
