Binary Black Hole Phase Space Discovers the Signature of Pair Instability Supernovae Mass Gap

The rapidly expanding catalog of gravitational-wave detections provides a powerful probe of the formation history of compact binaries across cosmic time. In this work, we extend the Binary Compact Object (BCO) phase-space framework to the full set of events in the GWTC-4 catalog to map the observed binary formation scenarios in a data-driven way. Applying this framework, we identify distinct regions of phase-space associated with different channels and discover for the first time a unique mass-cutoff scale in a data-driven way. The mapping of these on different formation channels reveals a population of first-generation (1G) black holes sharply truncated at approximately 45.5 $M_\odot$, consistent with the theoretically predicted pair-instability supernova (PISN) mass gap. These findings demonstrate the capability of the BCO phase-space to disentangle overlapping formation pathways, establish robust connections between gravitational-wave observations and binary evolution, and highlight the potential of upcoming observing runs to reveal rare populations and exotic origins.