Semi-supervised 3D medical image segmentation reduces the need for dense voxel-level annotations by exploiting unlabeled volumes. Although existing methods such as consistency regularization, pseudo-labeling, and co-training improve prediction-level robustness, they often provide insufficient feature-space organization for anatomically complex structures, especially small organs and ambiguous boundary regions with large intra-class variations. To address this issue, we propose Variation-Conditioned Distributional Proxy Learning (VCDP), a plug-and-play training-only regularization module for semi-supervised 3D medical image segmentation. VCDP represents each class with a learnable Gaussian distribution for shared class semantics and multiple variation prototypes for fine-grained intra-class patterns. A unified variation-conditioned compatibility score is further formulated to fuse distributional similarity and soft variation aggregation, guiding voxel embeddings to align with both global organ identity and local anatomical variations. VCDP is attached to decoder features during training and removed during inference, introducing no additional inference cost. Experiments on multi-organ segmentation benchmarks show that VCDP improves most evaluated baselines, particularly for small, ambiguous, and highly variable organs. Our anonymous code is released at https://anonymous.4open.science/r/VCDP_code-41ED.
A small Wasserstein distance does not certify that a transformation is admissible. In evidence-constrained, semantic, causal, physical, monotone, or risk-sensitive learning, one must ask not only how far two probability laws are, but whether mass has moved in a direction allowed by available information. We introduce conditional random ordered transport spaces (CROTS), a class of \(L^0\)-valued spaces of random probability measures equipped with a Wasserstein ambient metric, a closed stochastic order, hard and soft ordered transport discrepancies, and a conditional risk functional for evaluating order violation under an evidence sigma-field. The central object is an order-admissible transport geometry for random measure-valued dynamics, distinct from cone-valued metrics, ordered Kantorovich constructions, random Wasserstein spaces alone, and model-specific residuals for generative paths. We develop the foundations of CROTS as a space theory for reliable distributional learning. The results include well-posedness and duality for hard and soft ordered transport, soft-to-hard variational convergence, measurability and completeness of the random lifted space, reductions to classical Wasserstein and ordered geometries, ordered geodesics, constrained barycenters and projections, conditional risk-transport duality, and separation of order-violating distributions. The main stability theorem shows that random learning dynamics may converge in the ambient Wasserstein metric while its local admissibility leakage follows a separate conditional order-risk recursion. The resulting asymptotic order-risk floor provides a mathematical language for evidence overreach, ordered distribution shift, robustness failure, and admissible distributional dynamics.
Idioms can be analysed in terms of their decomposability, the extent to which constituent meanings contribute to the figurative whole. Decomposability is thought to predict syntactic flexibility. Usage-based accounts instead attribute idiom behaviour to distributional experience, such as speaker familiarity and predictability. We examine these views using contextualised language models as controlled distributional learners. We propose a model-internal measure of decomposability and relate it to human ratings, syntactic flexibility, and predictability while tracking idiom learning during pretraining. Model-derived decomposability correlates weakly with human judgments and shows a small but consistent negative relationship with syntactic flexibility. Pretraining analyses show that stabilisation of idiom representations in models is not explained by frequency alone. Instead, surprisal, decomposability, and frequency all contribute, with decomposability showing the strongest training-dependent effect.