A Multi-Region Folding Identity for Mask-Robust Parity Tests in Sky Surveys

Parity-odd statistics on the sphere vanish under full antipodal integration but survive under hemispheric restriction, as captured by the Hemispheric Folding Identity. Current anisotropy analyses often rely on ad hoc mask corrections or patchwise comparisons, which obscure this cancellation-to-additivity mechanism. Here we propose a Multi-Region Folding (MRF) identity that generalises hemispheric folding to arbitrary survey footprints. By representing survey masks as unions of base regions and enforcing linear constraints on coefficients, the MRF preserves odd-parity additivity while cancelling mask-edge and intersection artefacts. This framework provides an algebraic, mask-robust analogue of the East–West method and suggests a systematic route to parity-odd statistics across multi-survey mosaics.