First-law-consistent thermodynamic topology and exact outer-horizon separation in ultraspinning Kerr–AdS black holes

We develop a first-law-consistent thermodynamic defect construction on constrained one-parameter state spaces and apply it to ultraspinning Kerr–AdS black holes. Pulling the thermodynamic work form back to a prescribed curve gives a work-completed off-shell functional whose stationary points select the Hawking temperature wherever the entropy varies along the curve. This field is distinct from the fixed-rotation-parameter field, which instead selects a path temperature containing rotational work contributions. In the five-dimensional maximal rotation sector, the additional small-radius pair of the fixed-parameter field lies below the unique extremal mass minimum and therefore belongs to the inner horizon sector. For every odd spacetime dimension d ≥ 5 and arbitrary nonzero residual rotations, without an equal-rotation assumption, we prove that the entropy stationary radius is unique and lies strictly below the unique extremal radius. This establishes, over the full parameter range, that the complete positive small-radius defect branch is separated from the event-horizon domain. On the connected outer-horizon component, the first-law-completed field has topological degree W o u t = + 1 for every positive control temperature; any additional outer zeros can only occur in index-cancelling pairs. The resulting construction separates the formal winding of a chosen parameterized field from the equilibrium topology of the outer horizon and supplies an exact classification criterion for odd d ≥ 5.