Arrival-Direction Isotropy: Dipole Sensitivity as a Function of Event Count

How isotropic is the ultra-high-energy sky, and how strongly do the event counts let anyone say so? For N events the dipole amplitude carries a statistical uncertainty of about two over the square root of N, so samples of one thousand, ten thousand, and one hundred thousand events resolve dipoles of 6.325e-02, and correspondingly finer scales. Auger's reported dipole of 6.5 percent above 8e18 eV, on a sample of roughly thirty-two thousand events with amplitude uncertainty 1.115e-02, has an isotropic chance probability of 1.718e-15: the sky at those energies is not isotropic, but its departure is a broad dipole, not the sharp pointing that light primaries from a few sources would produce. Detecting a 6.5 percent dipole at five standard deviations requires of order 2.367e+04 events, which explains why the result awaited a large observatory's full exposure. Magnetic smearing of a heavier composition reconciles a real dipole with the absence of point sources, and no frame quantity enters the statistics. Simulated and computed; not a measurement.