Distribution shift is the defining challenge of real-world machine learning. The dominant paradigm--Unsupervised Domain Adaptation (UDA)--enforces feature invariance, aligning source and target representations via symmetric divergence minimization [Ganin et al., 2016]. We demonstrate that this approach is fundamentally flawed: when domains are unequally informative (e.g., high-quality vs degraded sensors), strict invariance necessitates information destruction, causing "negative transfer" that can be catastrophic in safety-critical applications [Wang et al., 2019]. We propose a decision-theoretic framework grounded in Le Cam's theory of statistical experiments [Le Cam, 1986], using constructive approximations to replace symmetric invariance with directional simulability. We introduce Le Cam Distortion, quantified by the Deficiency Distance $δ(E_1, E_2)$, as a rigorous upper bound for transfer risk conditional on simulability. Our framework enables transfer without source degradation by learning a kernel that simulates the target from the source. Across five experiments (genomics, vision, reinforcement learning), Le Cam Distortion achieves: (1) near-perfect frequency estimation in HLA genomics (correlation $r=0.999$, matching classical methods), (2) zero source utility loss in CIFAR-10 image classification (81.2% accuracy preserved vs 34.7% drop for CycleGAN), and (3) safe policy transfer in RL control where invariance-based methods suffer catastrophic collapse. Le Cam Distortion provides the first principled framework for risk-controlled transfer learning in domains where negative transfer is unacceptable: medical imaging, autonomous systems, and precision medicine.