A chain theorem for internally 4-connected binary matroids

Let M be a matroid. When M is 3-connected, Tutteʹs Wheels-and-Whirls Theorem proves that M has a 3-connected proper minor N with | E ( M ) − E ( N ) | = 1 unless M is a wheel or a whirl. This paper establishes a corresponding result for internally 4-connected binary matroids. In particular, we prove that if M is such a matroid, then M has an internally 4-connected proper minor N with | E ( M ) − E ( N ) | ⩽ 3 unless M or its dual is the cycle matroid of a planar or Möbius quartic ladder, or a 16-element variant of such a planar ladder.