The popular explanation of Hawking radiation — in which virtual particle pairs are born near the event horizon, the horizon tears them apart, one particle falls in with negative energy and the other escapes — has become the canonical image of black hole physics since Hawking's "A Brief History of Time" (1988). This paper subjects that picture to systematic quantitative and conceptual scrutiny using Sagittarius A* as a concrete test case. We present four quantitative checks and a thought experiment with three observers that together expose fundamental failures of the popular picture: (1) the predicted radiation composition contradicts the thermal spectrum — electron-positron pairs cannot be produced at temperatures 24 orders of magnitude below the pair-creation threshold; (2) the naive count of virtual particles in a Planck-thickness shell near the horizon (~10⁸⁹) implies radiation power exceeding the luminosity of the observable Universe by 46 orders of magnitude, while the actual Hawking luminosity is ~10⁻⁴² W; (3) the characteristic wavelength of emitted photons (~79 Schwarzschild radii ≈ 6.3 AU for Sgr A*) precludes localization of any "pair" at the horizon; (4) no physical mechanism determines which particle escapes, and classical mechanics in such a gravitational field predicts capture of both particles with zero net mass change. A Gedankenexperiment with three observers — distant, freely falling, and hovering — yields three independent contradictions: a 137-order-of-mag