Quantum Phases and Structural Transitions of Bosons in Bubble-Shaped Confinements

We investigate the structural properties and superfluid behavior of soft-core bosons confined in a spherical bubble trap at large particle numbers ($N\le 1800$). Using Path Integral Monte Carlo simulations with worm algorithm sampling in continuous space, we show that increasing $N$ drives a sequence of layering transitions: beyond the established bilayer supersolid, a third concentric layer of clusters emerges at $N\approx 1800$. By analyzing the distribution of permutation cycles distribution, we demonstrate that a finite superfluid fraction ($f_s\approx 10\mathrm{\%}$) persists even in this multi-layer regime. These findings clarify the role of particle number in driving layer formation on curved manifolds, offering a concrete guide for future experiments with bubble-trapped bosons.