Dress Conjecture for the rank of the 3D rigidity matroid via rigid clusters

Establish that for every graph G, the rank r_3(G) of the 3-dimensional generic rigidity matroid equals |F| + Σ_{X∈C}(3|X| − 6) − Σ_{h∈H(C)}(d_C(h) − 1), where C is the family of rigid clusters (maximal cliques in the R_3-closure) of size at least 3, F is the set of edges belonging to rigid clusters of size 2, H(C) is the set of hinges (vertex pairs lying in ≥2 clusters), and d_C(h) is the number of clusters containing hinge h.

Background

Maxwell’s sparsity counts are insufficient in 3D (e.g., the double banana graph). Dress et al. proposed that the correct rank formula incorporates overlaps among rigid clusters via hinge corrections.

A proof would yield a combinatorial rank oracle for 3D rigidity and significantly advance the structural theory of r_3.