Geometrically Significant Surfaces of Black Holes from a Single Scalar

Black hole spacetimes contain several geometrically distinguished hypersurfaces, including event and Cauchy horizons, stationary-limit surfaces, and curvature singularities. These structures are usually identified by different geometric or causal criteria. We show that, for the Kerr--Newman black hole, the membrane-paradigm pressure of a stretched horizon admits an analytically continued scalar representative whose fully factorized form has zeros, poles, and singular factors aligned with these standard Kerr--Newman loci. Its zeros occur at the outer and inner horizons, its poles occur at the outer and inner stationary-limit surfaces, its higher-order divergence contains the ring-singularity factor, and its large- r behavior records the asymptotic decay of the scalar. The construction is not proposed as a new invariant characterization of the spacetime, nor as a physical extension of the membrane fluid into the black-hole interior. Rather, it gives a compact membrane-pressure-based diagnostic whose analytic structure reorganizes several familiar surfaces of the Kerr--Newman geometry. We also note a secondary, purely algebraic analogy with generalized multi-component van der Waals-type equations of state.