Macroscopic Feynman Cycles and Poisson--Kingman Universality in Bose Condensation

We prove a canonical limit theorem for the macroscopic Feynman cycles of finite-volume ideal Bose gases. Cycles carry marks in a general Polish space $\mathsf{M}$, encoding spatial, geometric, spectral, or internal data. After removing a deterministic background density $\rho_{\mathrm{bg}}$, the marked macroscopic cycle process converges in the canonical ensemble to a marked Poisson--Kingman bridge of total mass $\rho - \rho_{\mathrm{bg}}$. The bridge is constructed from a marked Poisson point process with intensity $x^{-1}\eta_x(dm)\,dx$, conditioned on total mass $\rho - \rho_{\mathrm{bg}}$, where the kernel $x \mapsto \eta_x$ and its total-mass profile $\phi(x) = \eta_x(\mathsf{M})$ are determined by the low-energy spectral data visible on the scale $j \sim V_L$. When $\phi$ is constant, the bridge reduces to a Gamma bridge and the ranked cycle lengths follow the Poisson--Dirichlet law. We verify this for the ideal Bose gas in dimension $d > 2$ under periodic, Dirichlet, and Neumann boundary conditions: in all three cases $\phi \equiv 1$ and the ranked lengths converge to $\mathrm{PD}(0,1)$, while the mark kernels distinguish the three models through their winding, killed-bridge, and reflected-bridge geometry. When $\phi$ is not constant, the bridge is no longer Gamma and the ranked lengths are not Poisson--Dirichlet. As a concrete example, a critical double-well potential whose tunnelling splitting satisfies $V_L \Delta_L \to \gamma$ gives $\phi_\gamma(x) = 1 + e^{-\beta\gamma x}$; more generally, a finite-type visible spectrum with $R$ components yields $\phi(x) = \sum_{r=1}^{R} \theta_r e^{-\beta\lambda_r x}$. These results identify Poisson--Kingman bridges as the canonical universality class for marked macroscopic Bose cycles, with the visible low-energy spectrum selecting the particular bridge.