Localization in Stationary Black-Hole Thermodynamics: Observers, Algebras, Cuts, and Systems

Statements that black-hole temperature or entropy is "located at the horizon" often combine inequivalent notions of localization. We formulate a four-component localization profile that separates geometric carrier, formal/state-space data, operational readout, and thermodynamic system assignment, together with a domain-and-interface test for claims connecting those components. For a stationary nonextremal Killing horizon, this test yields a precise domain separation. The normalized stationary Killing congruence exists only at nonzero lapse; its proper acceleration and, under a KMS/detailed-balance calibration, the proper-time parameter T_db = T_H/N diverge in the horizon limit, while no massive stationary detector occupies the exact null generator. By contrast, the Killing flow, conserved Killing-energy flux, canonical horizon charge, and--where an appropriate equilibrium state exists--the exterior Kubo-Martin-Schwinger (KMS) parameter remain well defined in their respective domains. Freely falling detector trajectories also remain timelike at a regular horizon crossing. We establish the separation invariantly, define two inequivalent detector families, and derive exact asymptotics of their leading-order response functionals in a four-dimensional Rindler-Minkowski calibration. We compare the geometric endpoint in Schwarzschild, de Sitter, and Kerr, then apply the same framework to the stationary and physical-process first laws, Wald and generalized entropy, stretched horizons, Type III local algebras, and gravitational crossed products. The resulting classification identifies the additional interface data--detector coupling, normalization, boundary identity, regulator limit, or relational constraint--needed to pass between local, geometric, canonical, and algebraic descriptions. Exact-horizon rest-frame thermodynamics is thereby excluded without relocating or eliminating black-hole thermodynamics. No claim about microscopic state counting, evaporation unitarity, or a resolution of the information problem is required.