How Far Should a Preferred Axis Wander? Fisher Geometry of Axis Estimation and the Consistency of Axes Reported at Different Multipole Ranges

Preferred axes are reported from different multipole ranges, different statistics, and different experiments, and their directions never quite agree; the disagreements are then argued over as though an axis were a point rather than an estimate. This paper computes how much wander an axis estimate should show, from Fisher geometry on the sphere. An axis defined by a pattern standing at signal-to-noise ratio two has a mean estimation offset of 37.2 degrees and a ninetieth-percentile offset of 64.7 degrees; two independent estimates of the same true axis at that quality typically disagree by about 53.1 degrees. At signal-to-noise three the mean offset falls to 24.3 degrees, and at five, to 14.4 degrees. Most of the directional disagreement among the large-angle anomaly axes is therefore within the wander their own significances predict, and claims that two reported axes coincide, or fail to coincide, should be made in units of this wander rather than in raw degrees. The study is simulated, not measured.