Topological changes and response singularities in black hole thermodynamic branch structure

Different diagnostics of black hole thermodynamic branch structure do not always identify the same local event. We examine this issue in charged AdS black hole examples where a thermodynamic topological phase transition is known to occur. In these examples, previously reported in [JHEP 06 (2024) 213}], the inverse temperature $\tau_a$ at which the total topological number changes is separated from the inverse temperature $\tau_b$ at which the fixed charge parameter branch response considered here becomes singular. Reanalyzing the same zero point branch geometry, we find that the two diagnostics track different local structures. The change at $\tau_a$ is a boundary membership and index flow event of the positive radius physical branch set: a zero point with nonzero winding enters or leaves the counted domain. The singularity at $\tau_b$ is instead caused by a turning point at finite radius, where a pair of branches annihilates. Since the two branches carry opposite windings, this event changes the branch count but preserves the total topological number. Thus, in the analyzed examples, the thermodynamic topological phase transition and the branch response singularity are distinct diagnostics of the same zero point branch geometry, with different local origins.