The Bland–Jensen Conjecture Through Quotient Geometry

The family \(MI_n\) becomes larger in every obvious sense as \(n\) increases: its ground set, rank, corank, and Bland–Jensen equation system all grow. Our approach is to separate this growth from the part that controls representability. We place the \(n\)-dependent directions in a common \(n\)-dimensional subspace and project the remaining incidence data to a quotient of dimension two or three.

In one representative case, all of the relevant incidence information is encoded by seven vectors in \(\mathbb Q^3\). The proof strategy we explore has two parts: a fixed collection of parity equations shows that every \(MI_n\) is not weakly orientable, while six quotient constructions represent its single-element minors over \(\mathbb Q\). Together these steps yield infinitely many pairwise nonisomorphic excluded minors for the class of weakly orientable matroids, and hence rule out a finite forbidden-minor characterization.

Why a finite forbidden list matters

Throughout this post, all matroids are finite.

A matroid packages dependence without coordinates. Graph cycles, linear dependence among vectors, and many incidence structures all fit into the same language. An oriented matroid remembers additional sign information. Weak orientability asks for less: signed circuits and signed cocircuits need to satisfy the Bland–Jensen orthogonality condition only when their supports meet in exactly two elements.

The property is minor-closed. Delete or contract elements of a weakly orientable matroid and the result remains weakly orientable. For a minor-closed class, an excluded minor is a minimal obstruction: it lies outside the class, but every proper minor lies inside. If there were only finitely many excluded minors, membership in the class could be characterized by avoiding a finite list.

Bland and Jensen conjectured in 1987 that weak orientability has no such finite characterization. In 2015, Jesús De Loera, Jon Lee, Susan Margulies, and Jacob Miller proposed a concrete route to the conjecture. They constructed matroids \(MI_n\), proved that none is weakly orientable, checked minimality for \(n\leq 2\), and conjectured that every \(MI_n\) is an excluded minor. The remaining task is therefore to prove minimality uniformly in \(n\).

A uniform nine-equation obstruction

The obstruction has a uniform form. For a circuit \(C\) and a cocircuit \(D\) with \(C\cap D=\{e,f\}\), the Bland–Jensen characterization produces an equation over \(\mathbb F_2\):

In \(MI_n\), take the four circuits \(T,C_1,C_2,C_3\) and the three cocircuits \(D_i=E_n\setminus H_i\). There are nine relevant circuit–cocircuit pairs: \((C_j,D_i)\) for \(i\neq j\), together with \((T,D_i)\) for \(i=1,2,3\).

Add their nine equations. Every variable on the left occurs exactly twice, so the left side vanishes in \(\mathbb F_2\). The right side is nine copies of \(1\), hence equals \(1\). The whole obstruction is

The same nine equations give the contradiction for every \(n\). Thus the non-weak-orientability argument has a complexity independent of the size of the matroid.

Why minimality is the difficult part

To prove that \(MI_n\) is an excluded minor, it is enough to show that every single-element deletion and contraction is weakly orientable. Symmetry reduces these single-element minors to six representatives:

It is enough to represent these six matroids over \(\mathbb Q\). A rationally representable matroid is weakly orientable because \(\mathbb Q\) is an ordered field. Once every single-element minor is weakly orientable, minor-closedness handles every longer deletion–contraction sequence.

Absorbing the growing part into a common core

A direct approach would solve a larger Bland–Jensen system for every \(n\). Instead, for each representative minor we choose an \(n\)-dimensional space \(W_n\) and a quotient \(Q_d\) with \(d\in\{2,3\}\), and write

Every element in one of the large blocks \(X,Y,Z\) receives a subspace of the form

where \(p\) is a label in the fixed quotient. The special elements \(1,2,3,4\) receive fixed one-dimensional subspaces. For mixed subsets, the entire \(n\)-dimensional contribution to the relevant dimension count is absorbed into \(W_n\). What remains is fixed-dimensional incidence geometry among the quotient labels; the finitely many purely special subsets will be checked separately.

Consider the deletion \(MI_n\setminus x_n\). After writing \(X'=X\setminus\{x_n\}\), take \(Q_3=\mathbb Qa\oplus\mathbb Qb\oplus\mathbb Qc\). Label the three blocks by \(a,b,c\), and use

for the four special elements. The resulting quotient calculation depends only on the seven labels \(a,b,c,q_1,q_2,q_3,q_4\) in the three-dimensional space \(Q_3\), independently of \(n\).

Rado’s theorem and independent representatives

We still need actual representing vectors. For each ground-set element \(e\), choose a vector \(v_e\in L_e\). Rado’s independent-transversal theorem says that a set \(B\) admits linearly independent representatives precisely when every subset \(J\subseteq B\) has enough ambient dimension:

Call \(J\) mixed if it contains at least one block element. For mixed \(J\), the common core \(W_n\) is already present in the sum. Projection to the quotient gives the key identity

where \(P(J)\) is the span of the quotient labels appearing in \(J\). The dependence on \(n\) is now confined to the explicit additive term on the right. For an intended basis of \(MI_n\setminus x_n\), which has size \(n+3\), the only potentially deficient mixed subsets are those whose quotient labels span a line or a plane. These reduce to incidence questions in \(Q_3\).

Classifying the possible rank-two obstructions

The coordinates are designed so that the prescribed nonbases are dependent. For example, \(q_4=b+q_2\) places the labels of \(C_2\) in a plane, while \(q_1=q_2-q_3\) makes the special circuit \(T=\{1,2,3\}\) dependent. The converse is the substantive step: we must show that no intended basis lies in another low-rank configuration.

In the representative deletion, the maximal rank-two label configurations can be classified by computing the \(3\times3\) determinants of the seven quotient labels. Five large label planes are exactly the five hyperplane or circuit conditions that should define nonbases. The purely special plane accounts for \(T\). The few remaining rank-two sets cannot contain an \((n+3)\)-element mixed set.

This finite classification verifies the Rado inequalities. Rank three is automatically large enough; rank two can fail only for the whole candidate basis, in which case the list above places it in a prescribed nonbasis; and rank one has capacity at most \(n+1\). The purely special subsets are handled separately. The verification is therefore reduced to a fixed finite configuration in projective geometry.

Choosing representatives simultaneously

Rado’s theorem initially gives an independent choice of representatives for each intended basis separately. A matroid representation requires one global choice that works for every basis simultaneously, so an additional argument is needed.

For each intended basis \(B\), its determinant is a polynomial \(\Delta_B\) in the free coordinates of the block vectors. Rado tells us that \(\Delta_B\) is not the zero polynomial. There are only finitely many bases for a fixed \(n\), so

is still nonzero. Because \(\mathbb Q\) is infinite, there is a rational point where \(F\neq0\). At that point every intended basis is independent at once, while the prescribed nonbases remain dependent for structural reasons. This produces a single rational representation, not a separate representation for each basis.

From one quotient picture to the full theorem

The other five representative minors use the same architecture. The coordinates change, and contractions use a two-dimensional quotient rather than a three-dimensional one, but the growing part is always absorbed into \(W_n\). All possible excess-capacity obstructions for mixed subsets live in \(Q_2\) or \(Q_3\), where they can be classified once and for all; the purely special subsets are checked directly in each configuration.

Thus the six representative single-element minors are representable over \(\mathbb Q\), hence weakly orientable. Symmetry covers every single-element deletion and contraction. Minor-closedness then covers every proper minor. Combined with the nine-equation contradiction, this makes each \(MI_n\) an excluded minor.

Since this cardinality is strictly increasing with \(n\), the family contains infinitely many pairwise nonisomorphic excluded minors. Any forbidden minor contained in a minimal obstruction must be the obstruction itself, so a finite forbidden-minor list would have to contain a representative of every excluded-minor isomorphism class. This proves the conclusion proposed by Bland and Jensen in 1987.

What the Lean formalization checks

The companion Lean 4 development follows the same mathematical chain from the signed circuit–cocircuit definition of weak orientability. It constructs the actual matroids \(MI_n\), proves the nine-equation contradiction, represents the six actual deletions and contractions, passes from single-element minors to all proper minors, and formalizes the failure of every finite forbidden-minor characterization.

The development has no unfinished proof holes or added mathematical assumptions, and every finite verification is checked by Lean’s kernel. It records the logical chain from the signed definition of weak orientability to the statement that no finite forbidden-minor characterization exists.

The structure of the argument

The two parts of the argument have the same structural feature. Non-weak-orientability is witnessed by nine equations independent of \(n\). Minimality is proved using fixed-dimensional quotient geometry and finitely many purely special checks, again independent of \(n\); symmetry and minor-closedness then extend the six rational representations to all proper minors.

The common subspace \(W_n\) absorbs the growing part of the construction, while the possible obstructions to representability remain in \(Q_2\) or \(Q_3\). This separation turns the uniform problem into a finite collection of incidence and determinant calculations.