Let's say that a "complete resolution of GCH" is a definable class function $F: \operatorname{Ord}\longrightarrow \operatorname{ Ord}$ such that $2^{\aleph\_\alpha} = \aleph\_{F(\alpha)}$ for all ordinals $\alpha$. It is known of course that $F(\alpha) = \alpha+1$ is a complete resolution of GCH (in the positive) that is relatively consistent with ZFC. I read that it's an unpublished theorem of Woodin that $F(\alpha) = \alpha+2$ is a complete resolution of GCH that is relatively consistent with ZFC plus some large cardinal hypothesis. My questions are: (1) What's the weakest known complete resolution of GCH in consistency strength other than $F(\alpha) = \alpha+1$ and what large cardinal axiom is required for it? (2) What are some other complete resolutions of GCH that are known to be consistent relative to specific large cardinal hypotheses, what are their respective large cardinal hypotheses, and how do these consistency strengths relate to one another?



Complete resolutions of GCH
One candidate answer scheme might be the following: if $F$ is any (sufficiently absolute) definable function on the class of regular alephs such that $\kappa < \lambda \Rightarrow F(\kappa) \leq F(\lambda)$ and $\operatorname{cf}(F(\kappa)) > \kappa$, then ZFC + $(\forall \kappa = \operatorname{cf}(\kappa))(2^\kappa = F(\kappa))$ + SCH is consistent, where SCH is the Singular Cardinals Hypothesis or, in an equivalent form, the Gimel Hypothesis, due to Solovay, asserting $(\forall \kappa > \operatorname{cf}(\kappa))( \kappa^{\operatorname{cf}(\kappa)} = \max(2^{\operatorname{cf}(\kappa)}, \kappa^+))$, and no large cardinals are required.


Knowledge of the gimel function $\gimel(\kappa) = \kappa^{\operatorname{cf}(\kappa)}$ suffices to determine cardinal exponentiation recursively (for example, see P. Komjath, V. Totik, (*Problems and Theorems in Classical Set Theory*): chapter 10, problem 26, sets this out). So it is natural to explore the gimel function in greater depth. Writing a singular $\kappa$ as the limit of an increasing sequence $a$ of smaller regular cardinals leads to the observation that the deeper problem concerns the cofinality $\operatorname{cf}(([\kappa]^{\leq \lambda}, \subseteq))$ of the partial order $([\kappa]^{\leq \lambda}, \subseteq)$ for regular $\lambda < \kappa$. In this direction, one comes eventually to pcf theory, which offers an analysis of the puppet master $\operatorname{pcf}(a)$ rather than his troupe of erratic marionettes $\langle 2^\lambda : \lambda \in Card \rangle$. 


