Horizon, Photon Sphere and Shadow on the Sky: The Lensing Magnification Ladder

Three concentric scales structure every black hole image: the horizon at four gravitational radii of angular diameter, the photon sphere at six, and the shadow at 2 sqrt(27), a lensing magnification of 2.598 applied to the horizon's apparent size. For M87* the three angular diameters are 15.3, 22.9 and 39.7 microarcseconds; for Sagittarius A* they are 20.5, 30.7 and 53.3 microarcseconds. The measured rings, at 2.75 and 2.53 times the horizon's angular diameter for the two sources, therefore do not resolve the horizon itself, which remains a factor of about two and a half below the shadow in apparent size and below current resolution. Statements that the telescopes have photographed a black hole's horizon are quantitatively wrong, and this paper supplies the ladder by which the correct statement, that they have resolved its capture cross-section, can be checked. Computed; not measured.