Beyond the Average: Azimuthal Fourier Modes About a Preferred Axis as a Test of Axisymmetry, and the Leakage Budget of a Localised Spot

The azimuthal average throws away everything about a ring except its mean, including the information that distinguishes a true ring from an arc or a spot. This paper develops the complementary statistic, the azimuthal Fourier decomposition at fixed polar angle, and computes its discriminating power at the Cold Spot's latitude. A genuine ring places all its power in the zeroth azimuthal mode; a localised Gaussian spot of the Cold Spot's dimensions places only 0.059 of its azimuthal power there, spreading the rest over modes up to about m equals 17, with 0.941 of the power in modes one and above. An isotropic sky's rings, by contrast, return a zeroth-mode fraction averaging 0.103. The azimuthal spectrum thus separates the three cases, ring, spot, and ordinary sky, on a single ring's data, and it identifies the measurement that a future claim of ring structure must publish: not the average, but the mode fractions. The study is computed, not measured.