The effort to generalize iteration theorems and forcing axioms to higher cardinals faces significant barriers. For example, the axiom for meeting \(\aleph_2\) many dense subsets of any countably closed, \(aleph_2\)-cc forcing either fails (under CH) or is trivial (under not-CH). Moreover, naive generalizations of classical iteration theorems provably fail at higher cardinals, with problems arising at low-confinality limit stages.

In this talk, we present a class of forcings from work in progress with Curial Gallart that overcomes these barriers and is itself closed under iterations with appropriate support. While not as general as one might desire, the class still has many interesting examples. We plan to prove the iteration theorem for the class of \(< \kappa\)-strategically closed forcings that are exactly proper for kappa-models and derive the consistency of a corresponding forcing axiom. After mentioning a few consequences, we will focus more in depth on two issues: (1) the value of \(b_\kappa\) under this forcing axiom, (2) the influence of the axiom on coherent families of injections, relating to work of Scheepers and Clontz-Dow.