  Two methodologies have been presented in the literature which connect
relativistic three-particle scattering amplitudes with lattice QCD spectra --
the ``relativistic effective field theory'' approach and the ``finite-volume
unitarity'' method. While both methods have been shown to be equivalent in
various works, it has not been shown how to arrive at the relativistic
effective field theory results directly from $S$ matrix unitarity. In this
work, we provide a simple proof of the relativistic effective field theory form
of the scattering equations directly from unitarity. Motivated by the
finite-volume unitarity approach, we then postulate a set of quantization
conditions which relate the finite-volume energy spectra to the $K$ matrices
which drive the short-distance physics in the scattering equations, obtaining
all previously known results for three identical particles. This work also
presents new relations which provide a new pathway to generalize the results to
arbitrary systems.
