A Mathematical Theory of Phase-Consistent Information Bottleneck for Cross-Domain Generalization.

We propose a mathematical framework for domain generalization in medical image segmentation built on dual-tree complex wavelet transform (DTCWT) and variational information theory. The core premise is that, under adequate spatial normalization and acquisition-style shifts, DTCWT phase components are more closely associated with anatomical structure, whereas amplitude components are more sensitive to domain-specific intensity and style variations. We formulate this as a local phase-magnitude complementarity premise and construct an information bottleneck that operates on structured subband representations. The framework provides several key theoretical results under explicit structural assumptions: an information bound showing when DTCWT amplitude subbands better isolate domain-related information than global Fourier representations; a variational information bottleneck encoder that compresses domain-specific amplitude information into low-dimensional latent codes; a triple constraint mechanism (domain supervision, KL compression, and orthogonality) that controls domain-task information leakage; and a predictive feature modulation scheme with O(1) spatial complexity. We further analyze test-time adaptation via calibrated uncertainty, deriving a sufficient condition under which a two-pass inference strategy reduces the expected generalization gap. Finally, we include illustrative public-dataset checks on FeTS 2022 and BraTS 2023 to test the central phase-amplitude premise and the feasibility of DTCWT-front-end segmentation. All theorems are stated with their assumptions and verifiable conditions, offering a physically motivated approach to domain generalization in medical imaging.