Sharp defect bounds for regular-matroid Jacobians

Let M be a finite connected regular matroid of rank r ≥ 2 on m elements. Let e be the exponent of its Jacobian and put δ = er − 2m. Then either δ = 0 and M is the cycle matroid of a complete graph with constant positive edge multiplicity, or r ≤ δ, e + r − 1 ≤ 3δ, m ≤ δ². In the second case, equality in the rank or size bound holds precisely for the cone over a doubled complete graph. The restriction to elements with ePᵢᵢ = 2, where P is the orthogonal cut projection, is a direct sum of complete-graph matroids with constant multiplicities. If this restriction is proper, its rank is less than r and its complement has rank r. The dual cut lattice, with its inner product multiplied by e, is integral. Every connected regular matroid of Jacobian exponent at most five is graphic or cographic; the least exponent of one which is neither is six. The integral cycle and cut lattices contain sublattices generated by signed circuits and cocircuits of size at most e, respectively, and both quotients are annihilated by e. Mathematics Subject Classification (2020): 05B35, 05C25, 11H56.