  Consider a spectrally positive Stable($1+\alpha$) process whose jumps we
interpret as lifetimes of individuals. We mark the jumps by continuous
excursions assigning "sizes" varying during the lifetime. As for
Crump-Mode-Jagers processes (with "characteristics"), we consider for each
level the collection of individuals alive. We arrange their "sizes" at the
crossing height from left to right to form an interval partition. We study the
continuity and Markov properties of the interval-partition-valued process
indexed by level. From the perspective of the Stable($1+\alpha$) process, this
yields new theorems of Ray-Knight-type. From the perspective of branching
processes, this yields new, self-similar models with dense sets of birth and
death times of (mostly short-lived) individuals. This paper feeds into projects
resolving conjectures by Feng and Sun (2010) on the existence of certain
measure-valued diffusions with Poisson--Dirichlet stationary laws, and by
Aldous (1999) on the existence of a continuum-tree-valued diffusion.
