Electromagnetism from two matter spaces: mutual helicity and the nondegenerate completion

Abstract We show that, locally, every Maxwell field can be represented within the matter-space framework by at most two independent matter-space flows. In the one-flow formulation the electromagnetic field strength is the pull-back of a two-form on a three-dimensional matter space and therefore satisfies $F\wedge F=0$, so that a single flow captures only the degenerate sector that may carry conserved self-helicity. For a nondegenerate field, Darboux's theorem provides a decomposition $F=G^{(1)}+G^{(2)}$ into two closed simple sectors, and every closed simple two-form is shown to be locally the pull-back of a closed two-form on a three-dimensional matter space, so that each sector is realized by an independent flow. Each sector is individually degenerate, $G^{(I)}\wedge G^{(I)}=0$, while the full field satisfies $F\wedge F=2G^{(1)}\wedge G^{(2)}$; the invariant excluded by the one-flow theory is thus recovered as an inter-flow quantity. The Chern--Simons helicity three-form decomposes into two self-helicity contributions and a mutual term whose exterior derivative is $F\wedge F$, so that a nondegenerate Maxwell field necessarily has a nonclosed mutual-helicity three-form with source \(F\wedge F\). We characterize the freedom in the decomposition, distinguish it from gauge freedom, and illustrate the construction with a constant field with $\mathbf E\parallel\mathbf B$ and with counter-propagating polarized plane waves. A variational principle for the total field recovers the sourced Maxwell equation on every region where $F\wedge F\neq0$. The one-flow sector contains the frozen-in fields of ideal magnetohydrodynamics and force-free electrodynamics, whereas the pseudoscalar invariant, and every coupling to it, is carried entirely by the mutual sector.