Let M be an excluded minor for the class of P-representable matroids for some partial field P, and let N be a 3-connected strong Pstabilizer that is non-binary. We prove that either M is bounded relative to N , or, up to replacing M by a ∆-Y -equivalent excluded minor, we can choose a pair of elements {a, b} such that either M \{a, b} is N -fragile, or M * \{a, b} is N * -fragile.Date: September 21, 2018. The first, fourth, and fifth authors were supported by the New Zealand Marsden Fund. 1 Lemma 2.1. Let e be an element of a matroid M , and let (X, {e}, Y ) be a partition of E(M ). Then e ∈ cl(X) if and only if e / ∈ cl * (Y ). Connectivity. The following results are well known. Lemma 2.2. Let M be a 3-connected matroid. If X is a rank-2 subset of E(M ) and |X| ≥ 4, then M \x is 3-connected for all x ∈ X. Lemma 2.3 (Bixby's Lemma [2]). Let M be a 3-connected matroid, and let e ∈ E(M ). Then si(M/e) or co(M \e) is 3-connected.The next three results state elementary properties of 3-separations that we shall use frequently. We use the notation e ∈ cl ( * ) (X) to mean e ∈ cl(X) or e ∈ cl * (X).Lemma 2.4. Let X be an exactly 3-separating set in a 3-connected matroid, and suppose that e ∈ E(M ) − X. Then X ∪ e is 3-separating if and only if e ∈ cl ( * ) (X). 4 N. BRETTELL, B. CLARK, J. OXLEY, C. SEMPLE, AND G. WHITTLE Lemma 2.5. Let (X, Y ) be an exactly 3-separating partition of a 3connected matroid M . Suppose |X| ≥ 3 and x ∈ X. ThenWe say that a partition (X, {z}, Y ) is a vertical 3-separation of M when both (X ∪ {z}, Y ) and (X, Y ∪ {z}) are vertical 3-separations and z ∈ cl(X) ∩ cl(Y ). We will write (X, z, Y ) for (X, {z}, Y ). If (X, z, Y ) is a vertical 3-separation of M , then we say that (X, z, Y ) is a cyclic 3-separation of M * .A path of 3-separations of M is a partition (P 1 , . . . , P n ) of E(M ) such that (P 1 ∪· · ·∪P i , P i+1 ∪· · ·∪P n ) is a 3-separation of M for each i ∈ {1, . . . , n−1}. In particular, a vertical 3-separation (X, z, Y ) is a path of 3-separations. Lemma 2.7 ([23, Lemma 3.5]). Let M be a 3-connected matroid, and e ∈ E(M ). The matroid M has a vertical 3-separation (X, e, Y ) if and only if si(M/e) is not 3-connected.Let k be a positive integer, and let (P, Q) be a k-separation. We call the set cl(P ) ∩ cl(Q) the guts of (P, Q), and cl * (P ) ∩ cl * (Q) the coguts of (P, Q). We also say that an element z ∈ cl(P ) ∩ cl(Q) is a guts element, and z ∈ cl * (P ) ∩ cl * (Q) is a coguts element.We write "by uncrossing" to refer to an application of the next result.Lemma 2.8. Let M be a 3-connected matroid, and let X and Y be 3separating subsets of E(M ). Then the following hold.Series classes. We will use the following two results on series classes. We omit the easy proof of the first lemma.Lemma 2.9. Let M be a matroid such that co(M ) is 3-connected. If S and S ′ are distinct series classes of M , then either S ∪ S ′ is independent, or co(M ) ∼ = U 1,3 .When S is a series class of size two, we say S is a series pair.Lemma 2.10. Let M be a 3-connected matroid, and let u ∈ E(M ) be a...