  An oriented hypergraph is an oriented incidence structure that extends the
concepts of signed graphs, balanced hypergraphs, and balanced matrices. We
introduce hypergraphic structures and techniques that generalize the circuit
classification of the signed graphic frame matroid to any oriented hypergraphic
incidence matrix via its locally-signed-graphic substructure. To achieve this,
Camion's algorithm is applied to oriented hypergraphs to provide a
generalization of reorientation sets and frustration that is only well-defined
on balanceable oriented hypergraphs. A simple partial characterization of
unbalanceable circuits extends the applications to representable matroids
demonstrating that the difference between the Fano and non-Fano matroids is one
of balance.
