Large-$N$ Carrollian Thermodynamics from AdS Black-Hole Phase-Space Contractions

We develop a boundary large-$N$ interpretation of finite Carrollian black-hole thermodynamics and formulate its thermal Carrollian correlators in a celestial conformal-primary basis. The bulk input is a phase-space contraction, developed in earlier work, in which the time generator and Newton constant are scaled so that the extended AdS first law has a finite Carrollian limit. We show that this finite sector is naturally realized holographically as a double-scaled low-temperature, large-$N$ ensemble: the Carrollian temperature decreases while the effective number of boundary degrees of freedom grows, leaving the thermodynamic products in the first law finite. The large-$N$ dictionary is anchored by the standard ${\mathrm{A}\mathrm{d}\mathrm{S}}_5\mathrm{/}{\mathrm{C}\mathrm{F}\mathrm{T}}_4$ normalization and by the Brown–Henneaux central charge in ${\mathrm{A}\mathrm{d}\mathrm{S}}_3\mathrm{/}{\mathrm{C}\mathrm{F}\mathrm{T}}_2$. We construct the finite Carrollian Brown–York stress tensor on the contracted AdS boundary and show that its global energy charge reproduces the contracted bulk Hamiltonian. The boundary first law rewrites the bulk pressure term as a combined variation of spatial volume and holographic normalization, and identifies the Hawking–Page locus with the zero of the corresponding chemical potential conjugate to the holographic normalization. The same energy is the constant mode of a Carrollian time-shift charge, so the first law is the thermal zero-mode sector of a Carrollian time-shift Ward identity. Finally, we formulate the thermal celestial-basis correlators as positive/negative-frequency Mellin transforms of Carrollian spectral kernels.