Hemispheric Folding and the East–West Method: A General Identity for Odd Functions on the Sphere

Draft version of a paper to prove a general identity for odd functions on the sphere with respect to antipodally symmetric measures, which we call the Hemispheric Folding Identity. The identity shows that folding by a hemisphere kernel does not cancel odd contributions but instead doubles them, converting global cancellation into hemispheric additivity. This simple observation has broad implications for parity-odd statistics on the sphere. A key result is that the classical East–West method of cosmic ray anisotropy analysis is recovered as a special case of the folding identity, with the folding axis chosen along the local east direction.