Ring or Spot? The Azimuthal Smearing Factor That Separates a Latitude Ring from a Localised Cold Spot, and the Signal Each Leaves in Annulus Means

Two rival accounts of the Cold Spot predict different things in the azimuthal average about the reported preferred axis: a profile in which the spot is one point on a cold ring at polar angle 123.40 degrees predicts an annulus mean of -17.7 microK at that latitude at the registered amplitude cap, while a localised spot predicts only the small residue left when a compact feature is smeared around a full ring. This paper computes the smearing exactly by numerical integration: a Gaussian spot of width five degrees and central depth -150 microK contributes an annulus mean of -5.43 microK, a smearing fraction of 0.0362. The measured annulus mean on Planck PR3 SMICA, -5.08 microK, lies 0.35 microK from the localised prediction and 12.62 microK from the ring prediction: it sits closer to the localised reading. The two predictions are separated by 1.44 annulus standard deviations, so the azimuthal average can discriminate between the accounts, and on the published numbers it favours the localised spot. The study is computed from stated assumptions, not a new measurement.