Signed Odd-Root Limits for Intrinsic Fréchet Means on the Circle Under Long-Memory Gaussian Subordination

When the density at the antipode of the intrinsic Fréchet mean reaches a critical level, an interval for the mean of a long-memory circular time series that is calibrated to independent-data scaling is too short, and at Hermite rank two, it also has the wrong shape. The antipode is the cut locus, where squared distance has a kink, and the quadratic coefficient of the population objective vanishes exactly when the density there equals 1/(2π). For Yt=G(Zt), subordinating a stationary Gaussian sequence with covariances decaying at rate α∈(0,1), and contact of order 2r at the antipode, the rescaled estimator converges to a signed (2r+1)-st root. The contact order fixes the root and the Hermite rank of the score fixes what sits under it: a Gaussian at rank one, or a Rosenblatt variable at rank two under strong memory. Because the critical density is a knife-edge equality, we also give the limit along an array approaching it: when the quadratic coefficient vanishes at a rate matched to the long-memory scale of the score, the limit is a one-parameter family joining the odd root to a linear one. In hourly wind directions from fifty-three stations, the antipodal density ranges from 0.05 to 0.74 of the critical level; the records identify no subordination and do not settle the memory, so they locate series relative to the critical level rather than testing the limits.