Abstract
Black-hole thermodynamic topology assigns integer charges to equilibrium branches as zeros of an off-shell reservoir-mismatch one-form. We formulate this construction for a specified thermodynamic potential, active variables, fixed reservoir intensities and admissible compactified state space. The total charge is its Brouwer degree, equivalently the Morse degree of the potential and the oriented intersection number of the equation-of-state and reservoir Lagrangians. Regular Legendre changes of polarization preserve this degree after orientation transport; this covariance does not imply ensemble independence. We prove homotopy rigidity and boundary-jump formulae and relate auxiliary-angle windings, residues and gradient-flow representatives to the same degree. Standard Schwarzschild, Schwarzschild–AdS, fixed-charge RN–AdS, BTZ and logarithmically corrected examples follow from endpoint data. For grand-canonical RN–AdS and dyonic sectors, a positive-determinant desingularization of the coupled charge coordinates extends smoothly to zero size and identifies |Φ0|=1 and Φe2+Φm2=1 as boundary walls of the reservoir intersection problem, not local-stability walls or coordinate artifacts. A mixed dyonic window realizes the Poincaré–Hopf count on the physical fluctuation plane. In Kerr–Newman–AdS, one reservoir supports a coupled size–spin saddle and a companion minimum. At the saddle both diagonal curvatures in the extensive (S, J) variables are positive, whereas the Hessian determinant is negative. Thus the restricted radius index is +1 while the full fixed-potential Hessian index is −1, exhibiting information accessible only on the multi-variable fluctuation space.