  Context. Star-forming regions are excellent benchmarks for testing and
validating theories of star formation and stellar evolution. The Perseus
star-forming region being one of the youngest (<10 Myr), closest (280-320 pc),
and most studied in the literature, is a fundamental benchmark. Aims. We aim to
study the membership, phase-space structure, mass, and energy (kinetic plus
potential) distribution of the Perseus star-forming region using public
catalogues (Gaia, APOGEE, 2MASS, PanSTARRS). Methods. We use Bayesian
methodologies accounting for extinction to identify the Perseus physical groups
in the phase-space, retrieve their candidate members, derive their properties
(age, mass, 3D positions, 3D velocities, and energy), and attempt to
reconstruct their origin. Results. We identify 1052 candidate members in seven
physical groups (one of them new) with ages between 3 and 10 Myr, dynamical
super-virial states, and large fractions of energetically unbound stars. Their
mass distributions are broadly compatible with that of Chabrier for masses >0.1
$M_\odot$ and do not show hints of over-abundance of low-mass stars in NGC1333
with respect to IC348. These groups' ages, spatial structure, and kinematics
are compatible with at least three generations of stars. Future work is still
needed to clarify if the formation of the youngest was triggered by the oldest.
Conclusions. The exquisite Gaia data complemented with public archives and
mined with comprehensive Bayesian methodologies allow us to identify 31% more
members than in previous studies, discover a new physical group (Gorgophone: 7
Myr, 191 members, and 145 $M_\odot$), and confirm that the spatial, kinematic,
and energy distributions of these groups support the hierarchical
star-formation scenario.
