Phase-Space Contractions of Carrollian Black-Hole Thermodynamics

We study double-scaling contractions of extended AdS black-hole thermodynamics in Carrollian limits. The construction combines the Hamiltonian-generator normalization ${\xi }_{\lambda }=c^{-\alpha }{\mathrm{\partial }}_t$ with the Newton-coupling scaling $G=c^{\gamma }{G}_C$, while geometric black-hole parameters are held fixed. The extended Iyer–Wald identity with variable cosmological constant identifies the renormalized bulk contribution proportional to $\delta \mathrm{\Lambda }$ as the generator-normalized pressure-volume term $V_{\xi }\delta P$. We also formulate the magnetic Carroll-AdS Hamiltonian action, constraint surface, canonical presymplectic form, coframe symplectic current, and boundary variation in the static sector. The Schwarzschild-AdS spatial geometry satisfies the magnetic Carroll Hamiltonian constraint, and its Carroll boundary term gives the same finite first-law products obtained from the Lorentzian contraction. For stationary AdS black-hole families, the generator-normalized extended first law scales as $c^{1-\alpha -\gamma }$, so finite contractions lie on $\alpha +\gamma =1$. This presents a phase-space complement to intrinsic Carroll black-hole constructions and places the pressure-volume sector on the same footing as energy, entropy, charge, and angular-momentum work terms. The Lorentzian finite-clock endpoint is separated from Carrollian finite branches, where $T_{\lambda }\to 0$, $S\to \mathrm{\infty }$, $P\to \mathrm{\infty }$, and $V_{\lambda }\to 0$, while $T_{\lambda }\delta S$ and $V_{\lambda }\delta P$ remain finite. The construction is applied to Schwarzschild-AdS, Reissner–Nordström-AdS, Kerr-AdS as a Lorentzian rotating phase-space test, and $D$-dimensional Schwarzschild-AdS, with a three-form appendix for variable $\mathrm{\Lambda }$.