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 (<10Myr), closest (280-320pc), and most studied in the literature, is a fundamental benchmark. 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, and Pan-STARRS). We used Bayesian methodologies that account 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 10Myr, dynamical super-virial states, and large fractions of energetically unbounded stars. Their mass distributions are broadly compatible with that of Chabrier for masses >~0.1M_{sun}_ 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. The exquisite Gaia data complemented with public archives and mined with comprehensive Bayesian methodologies allow us to identify 31% more members than previous studies, discover a new physical group (Gorgophone: 7Myr, 191 members, and 145M_{sun}_), and confirm that the spatial, kinematic, and energy distributions of these groups support the hierarchical star formation scenario.