  The two-fluid Maxwell system couples frictionless electron and ion fluids via
Maxwell's equations. When the frequencies of light waves, Langmuir waves, and
single-particle cyclotron motion are scaled to be asymptotically large, the
two-fluid Maxwell system becomes a fast-slow dynamical system. This fast-slow
system admits a formally-exact single-fluid closure that may be computed
systematically with any desired order of accuracy through the use of a
functional partial differential equation. In the leading order approximation,
the closure reproduces magnetohydrodynamics (MHD). Higher order truncations of
the closure give an infinite hierarchy of extended MHD models that allow for
arbitrary mass ratio, as well as perturbative deviations from charge
neutrality. The closure is interpreted geometrically as an invariant slow
manifold in the infinite-dimensional two-fluid phase space, on which two-fluid
motions are free of high-frequency oscillations. This perspective shows that
the full closure inherits a Hamiltonian structure from two-fluid theory. By
employing infinite-dimensional Lie transforms, the Poisson bracket for the
all-orders closure may be obtained in closed form. Thus, conservative
truncations of the single-fluid closure may be obtained by simply truncating
the single-fluid Hamiltonian. Moreover, the closed-form expression for the
all-orders bracket gives explicit expressions for a number of the full
closure's conservation laws. Notably, the full closure, as well as any of its
Hamiltonian truncations, admits a pair of independent circulation invariants.
