arXiv:2610.32760v1 Announce Type: new Abstract: We establish polarization for memoryless sources taking values on the unit-perimeter circle, or on more general compact groups, under Ar{\i}kan's polar transform. For sources with finite differential entropy, all but an $O((\log N)^{-1/12})$ fraction of the conditional differential entropies in successive cancellation order lie within any fixed positive distance of zero, where $N$ is the block length. Examples indicate that the true polarization rate is indeed very slow: an inverse-logarithmic rate does not underestimate the polarization speed at moderate block lengths.