Direct triangulation-independence proof for canonical forms

Establish a direct proof that the formula assigning the canonical form of a pair consisting of a loopless oriented matroid and a chirotope, namely the signed sum over a triangulation of the boundaries of basis elements in the reduced Orlik–Solomon algebra, is independent of the chosen triangulation without using the recursive characterization of the canonical form.

Background

The paper defines the canonical form of a loopless oriented matroid and chirotope by the formula $(M,\chi)=\sum_{B\in\mathcal T}\chi(B)\partial e_B$, where $\mathcal T$ is a triangulation of the oriented matroid. The main theorem proves that this element is independent of the triangulation by establishing a recursive characterization through residue maps and then using that characterization in the proof.

The authors explicitly state that they do not know how to prove triangulation independence directly, without passing through the recursive characterization. They also note that the flip-graph of triangulations of an oriented matroid need not be connected, even for realizable oriented matroids, which helps explain why a direct argument is nontrivial.