oMEGACat. X. Shedding light on the disrupted dwarf galaxy of Omega Centauri

Omega Centauri ($\omega\,$Cen) is the most massive and chemically complex star cluster in the Milky Way and is widely regarded as the surviving nuclear star cluster of an accreted dwarf galaxy. However, its parent host remains uncertain. Here, we investigate a scenario in which Sequoia, Thamnos, and Gaia--Enceladus (GE) are debris from a single disrupted progenitor, the $\omega\,$Dwarf, whose nucleus survives today as $\omega\,$Cen. Using APOGEE and GALAH abundances together with Gaia astrometry, we reconstruct the chemical structure across this progenitor adopting orbital energy as a proxy for pre-merger radius. We find that the chemically evolved (younger Al-N-He-rich) population is strongly concentrated toward the inner regions, representing a population formed after/during the merger, while the primordial population represents a dwarf-galaxy-like population, supporting a common dwarf-galaxy origin for its components. The metallicity profile shows an inverted U-shaped gradient similar to those observed in present-day nucleated dwarf galaxies. At the same time, the inner regions ($\omega\,$Cen+Thamnos) are more $\alpha$-enhanced than the outskirts, pointing to shorter and more efficient star formation and indicating that the nucleus may have assembled through the merger of inspiraling globular clusters. Neutron-capture abundances reveal a Eu-rich, r-process-dominated outskirts and inner regions enhanced in [Ba/Eu] and [La/Eu], requiring delayed enrichment and more complex chemical evolution. Finally, our analysis shows that Sequoia and Thamnos naturally fit an outside-in stripping sequence around $\omega\,$Cen, whereas the connection with GE remains unsure.