Towards a splitter theorem for internally 4-connected binary matroids V

Let M be an internally 4-connected binary matroid and N be an internally 4-connected proper minor of M. In our search for a splitter theorem for internally 4-connected binary matroids, we proved in the third paper in this series that, except when M or its dual is a cubic Möbius or planar ladder or a certain coextension thereof, either M   has a proper internally 4-connected minor M′M′ with an N  -minor such that |E(M)−E(M′)|⩽3|E(M)−E(M′)|⩽3, or, up to duality, M has a triangle T and an element e of T   such that M\eM\e has an N  -minor and has the property that one side of every 3-separation is a fan with at most four elements. The fourth paper in the series proved that, when we cannot find such a proper internally 4-connected minor M′M′ of M, we can incorporate the triangle T into one of two substructures of M, a good bowtie or a good augmented 4-wheel. The goal of this paper is essentially to eliminate the need to consider good augmented 4-wheels by showing that, when M contains such a substructure, either it also contains a good bowtie, or, in an easily described way, we can obtain an internally 4-connected minor of M with an N-minor.