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While radiobiological data are conveniently fitted by a linear quadratic formula to data of limited dynamic range at doses typically exceeding 1 Gy, they are extrapolated linearly to doses below a milligray for the evaluation of low dose RBEs. However a single relativistic electron passing through a cell nucleus deposits a “dose” there in the neighborhood of a milligray. The validity of the linear extrapolation then rests on the demonstration that a single electron transit through a cell nucleus can cause inactivation or mutation or can lead to cancer induction. The extrapolation made is a huge one, of some three orders of magnitude, from thousands of electrons traversing a cell to a single cell traversal, and with only qualitative and rhetorical foundation (one track-two tracks), and ignoring the statistical fluctuations of track intersections with targets. Existing data obtained with beams of electrons, protons, X ray photons, incorporated tritium, and 125I demonstrate that hundreds of electrons may traverse a cell for inactivation and millions may be required for cancer induction. If linear extrapolation were valid these numbers would be reduced to one. These contradictions suggest serious reconsideration of accepted radiation protection standards.