"Verifiable information": is this a known concept?
$K\subseteq\{0,1\}^\omega$ is $\mathsf{R}$-verifiable if and only if $K$ is a $\Pi^0_1$ class (in Cantor space), a concept that has been studied extensively in computability-theory. They are also called effectively closed sets.

A set $K$ is a $\Pi^0_1$ class iff it is the set of infinite paths through a recursive (computable) tree, and this is the version of the concept that you defined.

A monograph devoted to them:

Effectively Closed Sets (Douglas Cenzer and Jeffrey B. Remmel), Perspectives in Logic, Cambridge U. Press, 350 pages, to appear.