Robin boundary conditions in global AdS$_4$: exact double-trace thermodynamics and a soft-mode instability
We consider a conformally coupled scalar field in four-dimensional global anti-de Sitter space with Robin boundary conditions, parametrized by an angle $\alpha$. On the boundary cylinder $\mathbb{R}\times S^{2}$ these conditions realize the double-trace deformation $\tfrac12\lambda\!\int O^{2}$ of the dimension-one operator $O$ in the alternate quantization with $\lambda=\cot\alpha/L$. Because the conformal map to one half of the Einstein static universe is exact, the boundary integral equation can be diagonalized, and the deformed two-point function follows in closed form, $\widehat{\mathcal G}_{\alpha}=\widehat{\mathcal G}_{N}/(1+\lambda\widehat{\mathcal G}_{N})$. Its poles give the normal-mode spectrum, and its determinant gives the free energy exactly within this Gaussian sector. After three local boundary counterterms, the Casimir energy reaches the stability endpoint with a finite square-root cusp. At any finite coupling the bulk $T^{4}$ and $T^{3}$ terms are independent of $\alpha$ and cancel in the difference from Neumann, leaving $\tfrac{\pi}{3}\cot\alpha\,LT^{2}$ as the leading $\alpha$-dependent term. All nonanalyticity comes from one static homogeneous mode, which becomes soft at $\alpha_{\rm crit}$, in agreement with the known classical stability threshold. The susceptibility diverges with exponent $\gamma=1$ and the gap closes with exponent $1/2$. Beyond this angle the mode is tachyonic, and a stable phase would require a stabilizing interaction. In the flat-space limit the physical coupling scales to zero at fixed energy, so the Robin dependence survives only in the soft-frequency sector, which we characterize by a meromorphic Mellin transform in the boost weight. The Robin angle thus gives a control parameter for a Gaussian stability endpoint that can be followed exactly, and raises the analogous question for relaxed boundary conditions in AdS gravity.