  Two recent papers by Kawarabayashi, Thomas and Wollan, "A New Proof of the
Flat Wall Theorem" (arXiv:1207.6927) and "Quickly Excluding a Non-Planar Graph"
(arXiv:2010.12397) provide major improvements over Robertson and Seymour's
original proof of the structure theorem for finite graphs that exclude a given
graph. The first paper redefines the notion of a flat wall. Unfortunately, this
new notion is too strong. As a result, the new Flat Wall Theorem in that paper
is incorrect. A counterexample is given in Appendix A. A follow-on lemma in the
first paper, about the transitivity of flatness, is also incorrect, a fact that
was noticed by Dimitrios Thilikos et al in arXiv:2102.06463. However, that
error is derivative and not the main issue. This paper provides a weaker
definition of the notion of a flat wall, provides a correction to the proof of
the new Flat Wall Theorem and a new proof of flatness transitivity. The notion
of a tight rendition as presented here differs from Thilikos' definition but is
defined much more simply, and the notion of a proper cycle is introduced. The
notions of certificates and tilted walls used by Thilikos turn out of be
unnecessary and transitivity is preserved in its original simplicity and
generality. Most importantly, it looks like the new weaker definition of
flatness is all that is really necessary to carry through the proof of the
structure theorem in the second paper of Kawarabayashi, Thomas and Wollan.
