Hyperbolic vision-language models are well suited to modeling the hierarchical structure of multimodal data, because hyperbolic volume grows exponentially with radius, enabling low-distortion representations of tree-like taxonomies from general to specific concepts. However, existing pretraining objectives for hyperbolic vision-language models entangle semantic alignment with entailment modeling, forcing radial coordinates to simultaneously encode semantic matching and hierarchical structure, leading to conflicting geometric constraints. In this paper, we introduce AURORA, a hyperbolic vision-language framework that decouples the objectives through angular alignment and a novel Radial-Angular Order Cone (RAOC) for entailment modeling. Angular alignment captures semantic similarity while preserving the radial structure for representing hierarchies. Unlike conventional entailment cones, RAOC determines angular allowance from relative parent-child radial gaps. It is incorporated into hierarchical entailment contrastive training to capture partial-order relations. We further introduce a norm regularization term that prevents undesirable radial degeneration for stable training. We train models on millions of image-text pairs and evaluate them across zero-shot classification, image-text retrieval, hierarchical reasoning, and compositional understanding. Our model consistently outperforms strong Euclidean and hyperbolic vision-language baselines, demonstrating the effectiveness of decoupling semantic alignment from hierarchical entailment modeling.

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