Hyperbolic Gravitational Cone: A Physically-Inspired Approach for Partial Order Embeddings

Partial-order modeling is essential for capturing asymmetric entailment semantics in knowledge graph reasoning and multimodal hierarchical alignment, which necessitates an inductive bias capable of strictly preserving transitivity. While hyperbolic entailment cones provide rigorous geometric guarantees for this property, their efficacy is limited by two constraints: geometric rigidity and optimization stagnation. Specifically, isotropic apertures mask semantic heterogeneity, and binary objectives sever gradient flow for entailed pairs. To address these issues, we propose Hyperbolic Gravitational Cone (HGC), inspired by the physical principle that mass governs semantic influence. The HGC formulates the entailment scope as a dynamic gravitational field, yielding a closed-form cone aperture modulated jointly by a learnable anisotropic mass and radial depth. To ensure hierarchical consistency, this mass function is strictly governed by positivity, radial monotonicity, convexity, and directional anisotropy. Furthermore, a continuous gravitational energy is derived to replace rigid binary constraints. By employing a soft-gated mechanism to evaluate angular deviation and coupling it with parental mass and axial distance, this energy function provides informative gradients even within the cone interior, driving precise structural optimization. Crucially, radial transitivity is mathematically guaranteed by imposing strict monotonicity on mass decay, preserving logical nesting along any given radial direction. Empirical evaluations demonstrate that HGC achieves superior performance on standard partial-order benchmarks, where the learned mass distributions reveal the underlying structure of semantic hierarchies and quantify the intrinsic breadth of concepts.