In this work, we address the challenge of generalizable audio deepfake detection (ADD) across diverse speech synthesis paradigms—including conventional text-to-speech (TTS) systems and modern diffusion or flow-matching (FM) based generators. Prior work has mostly targeted individual synthesis families and often fails to generalize across paradigms due to overfitting to generation-specific artifacts. We hypothesize that synthetic speech, irrespective of its generative origin, leaves behind shared structural distortions in the embedding space that can be aligned through geometry-aware modeling. To this end, we propose RHYME, a unified detection framework that fuses utterance-level embeddings from diverse pretrained speech encoders using non-Euclidean projections. RHYME maps representations into hyperbolic and spherical manifolds—where hyperbolic geometry excels at modeling hierarchical generator families, and spherical projections capture angular, energy-invariant cues such as periodic vocoder artifacts. The fused representation is obtained via Riemannian barycentric averaging, enabling synthesis invariant alignment. RHYME outperforms individual PTMs and homogeneous fusion baselines, achieving top performance and setting new state-of-the-art in cross-paradigm ADD.