  It is generally accepted that H. Friedman's gap condition is closely related
to iterated collapsing functions from ordinal analysis. But what precisely is
the connection? We offer the following answer: In a previous paper we have
shown that the gap condition arises from an iterative construction on
transformations of partial orders. Here we show that the parallel construction
for linear orders yields familiar collapsing functions. The iteration step in
the linear case is an instance of a general construction that we call
`Bachmann-Howard derivative'. In the present paper, we focus on the unary case,
i.e., on the gap condition for sequences rather than trees and,
correspondingly, on addition-free ordinal notation systems. This is partly for
convenience, but it also allows us to clarify a phenomenon that is specific to
the unary setting: As shown by van der Meeren, Rathjen and Weiermann, the gap
condition on sequences admits two linearizations with rather different
properties. We will see that these correspond to different recursive
constructions of sequences.
