Critical Inter-Horizon Thermal Dynamics on the Lukewarm Reissner-Nordström-de Sitter Manifold

We reinterpret the lukewarm sector of four-dimensional Reissner–Nordström–de Sitter black holes as the exact zero-dissipation thermal manifold of an effective two-horizon nonequilibrium system. In the fixed-charge sector, the inter-horizon thermal affinity controls the entropy production and vanishes precisely on the lukewarm branch. The corresponding linearized thermal mode is governed by an exact relaxation coefficient $K_L(\rho )$, with $\rho =r_{+}\mathrm{/}{r}_c$, and changes stability at the critical ratio $${\rho }_{\ast }=\frac{1+\sqrt{3}-\sqrt{2}{3}^{1\mathrm{/}4}}{2}\approx 0.4354,$$ where the relaxation time diverges as $\tau \sim \mathrm{\mid }\rho -{\rho }_{\ast }{\mathrm{\mid }}^{-1}$. We then encode this critical structure in a minimal Bragg–Williams functional and an Onsager–Machlup action for the effective trajectories of the thermal mode. In this way, the lukewarm branch is promoted from a geometric equal-temperature locus to a critical inter-horizon thermal manifold.