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

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 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) G(I)=0, while the full field satisfies F F=2G⁽¹⁾ 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 F, so that a nondegenerate Maxwell field necessarily has a nonclosed mutual-helicity three-form with source \(F F\). We characterize the freedom in the decomposition, distinguish it from gauge freedom, and illustrate the construction with a constant field with E∥ 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 F≠0. 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.