Problems in additive number theory, VII: The structure of additive $h$-bases for $n$

In additive number theory, a finite set $A$ of integers is an $h$-basis for $n$ if every integer in $\{0,1,2,\dots ,n\}$ can be represented as the sum of exactly $h$ not necessarily distinct elements of $A$. This paper introduces a new class of problems for these and related additive bases. The problems are designed, in part, to be susceptible to solution by AI.